sketches
LimblessLocomotionNotes
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As it is this project may profit from my work. I wonder what the updated attempt has achieved.
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RI | Publications | Limbless Locomotion: Learning to Crawl with a Snake Robot
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Limbless Locomotion: Learning to Crawl with a Snake Robot
K. Dowling
doctoral dissertation, tech. report CMU-RI-TR-97-48, Robotics Institute, Carnegie
Mellon University, December, 1997.
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Abstract
Snake robots that learn to locomote
Robots can locomote using body motions; not wheels or legs. Natural analogues, such as snakes, although
capable of such locomotion, are understood only in a qualitative sense and the detailed mechanics, sensing
and control of snake motions are not well understood.
Historically, mobile vehicles for terrestrial use have either been wheeled, tracked or legged. Prior art reveals
several serpentine locomotor efforts, but there is little in the way of practical mechanisms and flexible control
for limbless locomoting devices. Those mechanisms that exist in the laboratory exhibit only the rough features
of natural limbless locomotors such as snakes.
The motivation for this work stems from environments where traditional machines are precluded due to size or
shape and where appendages such as wheels or legs cause entrapment or failure. Example environments include
tight spaces, long narrow interior traverses, and movement over loose materials and terrains. Several applications,
including industrial inspection and exploration of hazardous environments, compel serpentine robots.
This research develops a general framework for teaching a complex electromechanical robot to become mobile
where sequences of body motions alone provide progression. The framework incorporates a learning technique,
physical modeling, metrics for evaluation, and the transfer of results to a snake-like mobile robot. The mechanism
and control of a 20 degree of freedom snake robot is described and multiple gaits are demonstrated including novel
non-biological gaits. This research furthers the design and control of limbless robots.
Notes
Sponsor: NASA
Grant ID: N00014-95-1-0591, IRI-9224521
Number of pages: 150
Text Reference
K. Dowling, Limbless Locomotion: Learning to Crawl with a Snake Robot,
doctoral dissertation, tech. report CMU-RI-TR-97-48, Robotics Institute, Carnegie Mellon University, December, 1997.
BibTeX Reference
@phdthesis{Dowling_1997_460,
author = "Kevin Dowling",
title = "Limbless Locomotion: Learning to Crawl with a Snake Robot",
school = "Robotics Institute, Carnegie Mellon University",
month = "December",
year = "1997",
address = "Pittsburgh, PA"
}
The Robotics Institute is part of the School of Computer Science, Carnegie Mellon University.
For updates and comments, please see these instructions.
This page maintained by robotwebmaster@ri.cmu.edu
Limbless Locomotion: Learning to
Crawl with a Snake Robot
Submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy in Robotics
Kevin J. Dowling
Advised by William L. Whittaker
The Robotics Institute
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213
December 1997
This research was supported in part by NASA Graduate Fellowships 1994, 1995 and
1996. The views and conclusions contained in this document are those of the author and
should not be interpreted as representing the official policies, either expressed or
implied, of NASA or the U.S. Government..? 1997 by Kevin Dowling.
.Limbless Locomotion:
Learning to
Crawl
Snake robots that learn to locomote
Submitted in partial fulfillment of the requirements for the degree of Doctor of
Philosophy in Robotics by Kevin Dowling
The Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213
Robots can locomote using body motions; not wheels or legs. Natural analogues, such
as snakes, although capable of such locomotion, are understood only in a qualitative
sense and the detailed mechanics, sensing and control of snake motions are not well
understood.
Historically, mobile vehicles for terrestrial use have either been wheeled, tracked or
legged. Prior art reveals several serpentine locomotor efforts, but there is little in the
way of practical mechanisms and flexible control for limbless locomoting devices.
Those mechanisms that exist in the laboratory exhibit only the rough features of natural
limbless locomotors such as snakes.
The motivation for this work stems from environments where traditional machines are
precluded due to size or shape and where appendages such as wheels or legs cause
entrapment or failure. Example environments include tight spaces, long narrow interior
traverses, and movement over loose materials and terrains. Several applications,
including industrial inspection and exploration of hazardous environments, compel
serpentine robots.
This research develops a general framework for teaching a complex electromechanical
robot to become mobile where sequences of body motions alone provide progression.
The framework incorporates a learning technique, physical modeling, metrics for
evaluation, and the transfer of results to a snake-like mobile robot. The mechanism and
control of a 20 degree of freedom snake robot is described and multiple gaits are
demonstrated including novel non-biological gaits. This research furthers the design
and control of limbless robots.
? 1997 by Kevin Dowling.We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time
T.S. Eliot.Acknowledgments
Research is hard but involves great joy as well. The greatest of joys has been working
with the people here at CMU. An observer might have thought I was working alone -
but the critical mass of people here in the Robotics Institute meant that I could always
find the help and encouragement on all issues.
Red - Friend, mentor, and force of nature. Thank you Red.
Hans Moravec - Once upon a time, Hans hired yours truly, an eager but inexperienced
undergrad, to help build his robots. Hans always has a fresh perspective, new insight,
and a wonderful way of looking at things. I will dearly miss the discussions.
Mike Blackwell - Friend and officemate of fifteen years, Mike understands the
combination of hardware and software better than anyone I know and has been an
invaluable help in answering a zillion questions over the years. Thanks Mike.
John Bares - John, Your review, help and advice and friendship have been invaluable.
Dave Wettergreen - The clearest of voices, the best of reviewers and good friend.
Dave Simon - Always weighing possibilities and thinking out issues. Thanks for the
advice and support over the years.
Hagen Schempf - A potent combination of enthusiasm, talent and experience. Thanks
Hagen, for advising, helping and making things happen.
Ben Brown - The best designer I know, and a sounding board on many technical issues
herein including metrics and design.
Chris Leger - A remarkable programmer and Quake enthusiast. Chris developed a
software toolkit that I used in this work.
Anton Staaf - A remarkable undergrad who is destined to do great things. Thanks for
the caterpillar and the discussions, Anton.
Sundar Vedula, David Baraff, Martial Hebert, Andrew Moore, Howie Choset and Joel
Burdick all provided advice and insights into several areas of this research. I greatly
appreciate their time, perspectives and assistance..Acknowledgments
Thanks to Tony Nolla, Jesse Eusades and Dave Vehec for their assistance on some
wiring and drawings.
Thanks also to Takeo and Raj who have also advised, mentored, and supported me
through the years. It?s been an enormous and beneficial influence.
The members of the Field Robotics Center, the Robotics Institute, and friends
throughout Carnegie Mellon. This is the best place in the world for robot research.
Most of all, Mary Jo and Ashlinn, and our most recent research project: Aidan. Your
love, support, advice and understanding are monumental. I love you..i
Contents
Overview and Rationale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
Introduction 3
Why Serpentine Locomotion? 5
Applications 9
Challenges of Limbless Locomotion 12
Summary 12
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
Biological Systems 13
What?s Missing? 20
Robotic Systems 20
Learning 32
Summary 34
Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
Overview 37
Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
Metrics 43
Efficiency 44
Dimensionless Metrics 49
Scale 53
Summary and Selection 55
Learning and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
Optimization Techniques 57
Representation 61
Summary 67.Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Actuation 69
Design 77
Electronics 87
Sensing 89
Other Subsystems 90
Experimental Setup 92
Physical Modeling 93
Summary 94
Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Power Calculations 97
Units 100
Velocity Calculations 101
Gaits 101
Summary 119
Summary and Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Contributions 121
Future Work 122
Wisdom 123
Servo Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Link Weight Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Derivation of Actuator Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3
Chapter 1
Overview and Rationale
Overview and Rationale profiles the content of this dissertation and examines the rationale
for serpentine robots and their application. This chapter offers strong motivation for
serpentine mechanisms; this includes the advantages and disadvantages of serpentine
locomotion as well as application areas where such mechanisms can make a powerful
impact.
Introduction 1.1
Biological snakes are pervasive across the planet; their diverse locomotion modes and
physiology make them supremely adapted for the wide variety of terrains, environments
and climates that they inhabit. It would be wonderful to capture these broad features of
movement and capability in man-made equivalents. While wheeled and walking
machines have undergone decades, even centuries, of development, they are still limited
in the types of terrain they can traverse. A snake-like device that could slide, glide and
slither could open up many applications in exploration, hazardous environments,
inspection and medical interventions.
Serpentine robots have a number of useful features and applications, are fascinating to
observe, and may answer questions about biological equivalents. One of the
fundamental issues is understanding their locomotion. In a qualitative sense, propulsion
with wheels or legs is more readily apparent and understandable versus the movement
of limbless locomotors. A wheel turns; the vehicle moves. A leg pushes; the vehicle
moves. How a snake moves is not so evident. This dissertation addresses robots that
crawl and slither without the use of wheels or legs; where body motions alone enable
progression.
A worthwhile snake robot has the ability to wriggle into confined areas and traverse
terrain that would pose problems for traditional wheeled or legged robots. The useful.
Overview and Rationale
4
features of snake robots include stability, terrainability, good traction, high redundancy
and completely sealed mechanisms. Robots with these properties open up several
critical applications in exploration, reconnaissance, medicine and inspection.
This research creates a robot snake that can locomote in novel ways, develops a
framework for teaching the snake to locomote and also develops and integrates the
many technologies required to make this happen. The research culminates in the
demonstration of an effective mechanism and the enabling gait generation techniques.
The research is distinguished from prior work by the development of more general
locomotion, a better mechanism capable of traversing more complex terrain and the
demonstration of multiple locomotion modes.
Prior work in understanding real snakes has been limited to providing qualitative
descriptions of snake movement and the classifications of gaits. While a few snake
robot devices have been built, they have shown only limited locomotion in the form of
single types of gaits.
Another hurdle is the evaluation of gaits and the determination of what is a good gait.
After examining and evaluating many criteria and measures, a non-dimensional metric,
specific resistance, offers the most useful measure of gait efficacy. However, gait
evaluation requires that gaits be generated so they can be tested. Most prior work in gait
generation provides an explicit description of a gait. However, this assumes a good
model or good imagination to provide a variety of gaits. Another way, developed here,
is to let a learning process generate gaits and test them using the performance metric as
an evaluation function.
Rather than simply generating gaits in a random fashion however, a guided learning
process is utilized. Stochastic learning processes, genetic algorithms (GA) and
probabilistic-based incremental learning (PBIL) methods provide the most general
technique for generating and testing of gaits. The results of this work are not only
several snake-like gaits but several novel gaits, not found in real snakes.
Because real mechanisms are susceptible to breakdown and wear, especially when
confronted with rapid changes and extreme ranges of inputs, it would not be possible to
test all gaits on a real mechanism. Thus, physical simulation is required; using the
computer to accurately simulated the physics of the body, commands and resulting
motions from those commands. Physical simulation offers the potential of quickly
evaluating many gaits and motion sequences.
The design and implementation of a snake robot is the confluence of several
technologies: actuation, form and structure, electronics, control, sensing, etcetera.
Mature technologies for emulating characteristics of biological muscle lie in the future.
However, electromagnetic motors offer many options, and the robot snake is designed
around these technologies.
The framework that ties all of this together is, in itself, a separate development. By
incorporating learning, evaluation, simulation and the robot, a complex system ia made
to locomote in surprising and new ways. Another surprising development is the
resulting simplicity of the gait formulation; it took a long march to come back to a
simple formulation..Overview and Rationale
5
Outline 1.1.1
Overview and Rationale presents the advantages, disadvantages and applications of
snake-type robots. Background looks at prior efforts in understanding limbless
locomotion in animals such as snakes and discusses what is still not understood in these
animals. Structure and locomotion modes, in particular, are examined in detail.
Additionally, several prior snake-like robots, their mechanism and control contributions
as well as their limitations are examined.
Framework presents a structure or architecture for learning locomotion and control of
a serpentine robot. The structure or framework itself represents the outline for the
remaining chapters and each component is treated at length in the following chapters.
What is a good way to evaluate locomotion? What makes one gait or sequence better
than another? Learning requires a quantitative measure of performance and
Performance Metrics examines ways to evaluate performance in locomotion and
selects specific resistance as a metric for learning limbless locomotion. This selection
is preceded by an examination of a variety of units, dimensions and dimensionless
metrics.
Learning and Optimization explores machine learning techniques to maximize
movement. A stochastic method, probabilistic-based incremental learning, is selected
from several learning methods.
Implementation describes the robot developed in the course of this research. The
configuration and design includes actuation selection, mechanical design and
electronics and hardware control. Additionally, the experimental set-up and other
details of implementation are shown.
Locomotion codifies the outcome of the experiments in simulation and with the
physical robot and reports on the successes and failures of the many experiments. A
number of gaits, both snake-like and non snake-like, are revealed.
Finally, Summary & Conclusions codifies the results and contributions of this work,
and looks towards the future where new contributions can be made. A little wisdom is
also dispensed.
The appendices include a detailed look at evaluation of a class of actuators, parameter
derivation and weight tables. Finally, a detailed bibliography completes the dissertation.
Why Serpentine Locomotion? 1.2
For centuries, people have created a menagerie of machines whose appearance and
movement have mirrored animals to an astonishing degree. There are anthropomorphic
figures that resemble man and mobile machines that resemble animals. However, the
strongest reactions are not simply to outward appearance; after all, costumes, statues
and puppets are extant throughout history. What attracts and holds attention are the
animated behaviors and motions without apparent human connection.
However, these historical devices are mere simulcra, controlled by unseen hands; only
in the last few decades have researchers and designers begun to replicate the general
movements of animals in mechanisms..Overview and Rationale
6
The general motivation for serpentine locomotors are environments where traditional
machines are precluded due to size or shape and where appendages such as wheels or
legs cause entrapment or failure. Example environments include tight spaces, long
narrow interior traverses, and travel over loose materials and terrains.
Serpentine mechanisms hold particular fascination due to the singular motions usually
associated with animals such as snakes and tentacles. Few terrestrial mobile devices
move without the use of wheels or legs; those that exist in the laboratory have exhibited
only the rough features of natural limbless locomotors such as snakes. Serpentine
features include serial chains of actuators capable of subtending small curvatures.
However many of these prior efforts incorporated non-biological features: the use of
casters for support and propulsion or the use of fixed pins for support and traction.
Other broad features of these prior robots include the use of models that explicitly
describe the shape of the robot, the use of tensor mechanisms that limit curvatures and
forms and mechanism designs that are impractical for application. There are significant
challenges in designing, building and controlling practical limbless mechanisms that
are capable of locomoting without traditional forms of propulsion and actuation. These
challenges include configuration, design and geometry of the form, determining the
number and arrangement of actuators, routing power and signal distribution, and robot
control.
Wheels offer smooth and efficient locomotion but often require modifications to
terrains for best use; even all-wheel-drive mechanisms are limited in the type and scale
of terrain that can be traversed [Bekker 61]. Walking mechanisms offer extreme terrain
negotiation for a given scale and provide discrete, rather than continuous, contact. This,
in principle, benefits efficiency and traversibility. The current state of the practice,
outside of the laboratory, are walkers such as the Komatsu RECUS, CMU?s Dante, the
Finnish Plustech forestry machine, and Honda?s anthropomorphic biped [Ishino
83][Apostolopoulos 95] [Plustech 96][Honda 96]. The number of efficient and practical
walkers is small and there is much development and incentive necessary for walkers to
become viable.
Advantages of Serpentine Robot Locomotion 1.2.1
Serpentine locomotors possess a number of potential advantages beyond the
capabilities of most wheeled and legged vehicles.
Stability 1.2.1.1
Unless a serpentine robot purposefully slithers off a cliff, it can?t fall over. In contrast,
stability is of great concern to wheeled and legged machines in rough terrain; they can
fall over. Terrain contacts in vehicles form a constellation of points on the terrain; if the
center of gravity moves beyond the bounds of the convex polygon formed by these
points, it tips over. In a serpentine robot, the potential energy remains low in most
situations; therefore there are few concerns for stability and no need for the support
polygons formed by wheel or leg contact points. Even in the case where a free-fall
occurs, the serpentine device may survive better than most mobile devices because
potential failure points such as the connections between the body and wheels or legs do
not exist..Overview and Rationale
7
Terrainability 1.2.1.2
Terrainability is the ability of a vehicle to traverse rough terrain. Terrain roughness is
often measured by scale of features, power spectral density, distribution of obstacles
such as rocks and geographic forms [Bekker 69], or even its fractal dimension
[Arakawa 93]. A serpentine mechanism holds the promise of climbing heights many
times its own girth; this feature can enable passage through terrain that would encumber
or defeat similarly scaled wheeled and legged machines.
Additionally, a serpentine robot can climb steps whose heights approach its longest
linear dimension. This is an attribute that few, if any, wheeled or legged mechanisms
possess. This assertion assumes quasi-static systems; mobile leaping systems, like the
Russian Phobos vehicle, might jump many times their height or length [Klaes 90].
While there have been numerous wheeled trackless ?trains,? or coupled-mobility
devices, that use powered wheels, they still suffer from the limitations of wheel traction
and terrain shear [Hirose 93][Odetics 88][Gowenlock 96][Haddock 94]. Many coupled-mobility
devices make use of active or passive wheels to move and body joints to
accommodate obstacles. However, the wheel still limits locomotion over soft and
viscous materials. Additionally the serpentine mechanism has no appendages that can
become stuck unlike the shank-rocking of a leg or the descent of a wheel into a hole.
Traction 1.2.1.3
Traction is the force that can be applied to propel a vehicle. Traction is usually limited
to the product of the vehicle weight and the coefficient of friction. Tractive forces can
be quite high for natural snakes; a moving snake can exert a force up to a third of its
own weight. The distribution of the snake mass over such a large area, in comparison to
mass equivalent legged or wheeled vehicles, results in forces that can be below the
thresholds of the plastic deformation of the soil. In comparison, load concentration
resulting from most wheels or leg designs results in soil work. Because of the large
contact area, serpentine vehicles may result in little or no soil work. If restrained or
locomoting in certain modes, natural snakes can sustain a pull of up to four or five times
its weight [Parker 63]. As an unscaled comparison, large man-made vehicles under
good conditions and slow speed may exert drawbar pull of 90% of their weight
[Caterpillar 94]. Limbless locomotion may prove superior in marginal or soft terrains
where plowing and shearing actions restrict wheel mobility.
Efficiency 1.2.1.4
Snakes achieve efficiencies and performance equivalent to biomechanisms of similar
scale and mass [Walton 90]. Reasons include reduced costs associated with less lifting
of the center of gravity as compared to legged animals, elimination of the acceleration
or deacceleration of limbs, and low cost for body support. This begs the question: why
wouldn?t natural snakes be more efficient than similarly sized animals? The answer is
that energy losses in snakes include greater frictional losses into the ground, lateral
accelerations of the body that do not contribute to forward motion, and the cost of body
support for partial body elevations during movement. These advantages and
disadvantages appear to balance total energy use to a comparable level as animals of
similar mass [Secor 92]..Overview and Rationale
8
Size 1.2.1.5
Depending on the mechanism design, the small frontal area of snake mechanisms
allows penetration of smaller cross-sectional areas than mass-equivalent legged or
wheeled vehicles. If the volume of a snake, a cylindrical form, is kept the same and the
diameter is reduced by half, the length becomes four times greater. Cross-sectional area
for mechanisms of similar density and mass may result in very long vehicles.
Redundancy 1.2.1.6
Candidate configurations for serpentine robots may employ many simple motion
actuators in sequence. During operation, the loss of short segments would still permit
mobility and maneuverability. That is, the mechanism is still able to move and
maneuver even if a number of actuators failed. The penalty, of course, is reduced
efficacy in mobility.
Sealing 1.2.1.7
With its continuous unperforated surface a serpentine device has no appendages to
impede progress or be exposed to surroundings. This allows better sealing between the
mechanism internals and the environment. This provides advantage to applications in
hostile environments.
Disadvantages of Serpentine Robot Locomotion 1.2.2
If snake mechanisms are so good, why aren?t there lots of them? One answer is they are
difficult to design, build and control. Another is that there are some disadvantages to
these configurations.
Payload 1.2.2.1
Much locomotion has to do with work; the transport of materials from one place to
another. There is no integral platform for attaching payloads. It?s difficult to envision
transport of materials using snake-like robots unless an integral conduit can be used to
deliver materials.
Degrees of Freedom 1.2.2.2
To subtend the various curves needed for locomotion requires a larger number of
actuators than most wheeled or legged vehicles. The number of DOFs in vehicles can
range from two up to eighteen and even more for some walkers. However, a relatively
flexible snake mechanism may require even more. A large number of DOFs may
introduce reliability problems; if one actuator has a given failure rate then robots with
large numbers of units have a higher chance of having any unit fail. Fortunately, for the
serpentine mechanism, sufficient redundancy can allow the robot to continue to
function in a limited manner. While the control for a serpentine mechanism involves
more motions to control, an advantage is that complex planning for footholds and wheel
contacts is obviated; the system can simply follow its head.
Related to this issue is designing actuators and structures that are strong, efficient, and
elegant; the need is for high forces in small packages. However, this need is not unique
to serpentine mechanisms, and many applications await the emergence of actuators
with these properties!.Overview and Rationale
9
Thermal Control 1.2.2.3
Obviously, snakes are not even close to being spherical; the volume to surface area ratio
is worse than for animals of similar mass. Even though limbed animals have protruding
limbs and appendages, the surface area to volume ratio is significantly less than for
snakes. The Meeh coefficient, k, in the equation S = kM 0.67 , where S is surface area and
M is body mass, is higher for snakes than for many other mammals and fish. [Schmidt-Nielsen
84]. The effect of this may be that thermal control is more difficult in a
serpentine mechanism. On the other hand, if the application allows the use of the
environment as heat-sink or heat-source, then this works in the snake?s favor and is of
great benefit to actuator systems.
Speed 1.2.2.4
The fastest natural snakes, under ideal conditions, can move at 3.0 m/s and appear to
have a length to circumference ratio of about 10-12 [Bauchot 94]. It seems unlikely that
a robot system, in the near future, will develop speeds anywhere near this. Most snake
locomotion is fairly slow, but the motion is deceptively fast however; the lateral motions
of the body often give the impression of higher speeds. However, the bottom line is that
robot snakes are likely to be slower than their natural counterparts.
Applications 1.3
What good are snake robots? Where would they be used? Consultation with potential
users, and examination of many application areas suggests a number of areas where
serpentine robots can make an impact.
In the past, a recurring litany of robotics application areas included nuclear plants,
medical applications and the inspection of hard-to-reach areas. The difficulty in
accepting these speculations resulted from immaturity of the technology and
techniques. Robots prefer strongly structured applications, and many applications do
not offer structured environments.
However, maturing and evolving technology in sensing, control and machine learning
has enabled the successful deployment of operational field robots in unstructured
environments and this will be true of serpentine robots as well. Each of the following
applications offers a compelling scenario for self-propelled serpentine devices. Each
application offers pratfalls and failure for wheeled or legged robots; problems and
issues where serpentine robots could succeed.
There is a separate issue of fixed-base serpentine devices and several of these
applications would benefit from serpentine manipulators as well as serpentine
locomotors. However, this work is concerned with locomotion and not simply
manipulation.
Exploration 1.3.1
In unpredictable environments, there are zones of uncertainty and footing is insecure or
unknown. A snake-like device can distribute its mass over a large area for support so
that even if footing gives way, self-support between secure points enables continued
operation. Such environments include planetary surfaces and extreme terrains with
loose rubble and inclines near the slope of repose..Overview and Rationale
10
Inspection 1.3.2
Many inspection techniques in industry and medicine rely on fixed-base mechanisms
such as borescopes, videoscopes and fiberscopes. These devices are primarily used to
inspect cavities that cannot be seen directly by the eye. Inspection applications include
airline engine maintenance, quality control in manufacturing, and process monitoring
and inspection in utilities and chemical plants. Simple direct-view borescopes have
proven useful, but articulated self-advancing devices forming and following complex
paths could open many more applications.
To eliminate some of the difficulties with current borescope use, plant equipment is
modified with portals, but this requires additional design and manufacturing resources
but doesn?t address needs of older or legacy plants. Such equipment would not require
such alterations if a device capable of reaching those points were available. Another real
need is the inspection of power station cooling tubes which can be up to 18m long and
only 10mm in diameter. [Olympus 94][VIT 95][Westinghouse 97].
Utilities, chemical processors and manufacturers have large and complex pipe networks
that often require inspections or determination of blockage. Guesswork, followed by
trenching or cutting operations, is a very expensive technique even discounting the
associated downtime costs. Mobile pipeline devices are used by pipeline service
companies but these pipe pigs, as they are termed, are of limited use. A snake-like
device would prove very useful. With a serpentine tool, in-situ inspections and accurate
localization could lower costs and downtime significantly.
Aware of their limitations, developers of inspection equipment are keenly interested in
self-propelled inspection devices. However, the industry is highly competitive, and it is
difficult to get information on their efforts. Yet, their interest is evidenced by a growing
number of patents on ?walking? and self-propelled videoscopes [Welch-Allyn
94][Olympus 94]. None of these devices are yet mature.
Medical 1.3.3
Snake-like devices have received attention as a potential medical technology.
Minimally-invasive surgery reduces or eliminates the need to cut open large sections of
skin and tissue. It is currently estimated that 35% of the 21,000,000 surgeries performed
each year could be done with minimally invasive techniques [Grundfest 94]. There
could be dramatic reductions in hospital stays, patient suffering, and costs.
Laparoscopic devices, which are rigid tools inserted into the abdominal wall, and
endoscopic devices are used in these types of surgical procedures. A snake-like robot
could subtend the curvatures of interior tissues and enable further diagnosis and
treatment.
In recent years, non-invasive surgery has met with wide acceptance and produced
phenomenal results. The surgical tools of the trade, however, are often difficult to
manage and have their limitations. There are many needs for dextrous and articulated
tooling and surgical devices that can advance through organs and tissue. As one
example, about 60% of the gastro-intestinal tract is inaccessible to conventional
endoscopic tooling.
However, substantial impact to existing procedures could be made in other areas, and
gastro-intestinal endoscopy is one such example. Two leading companies have several.
Overview and Rationale
11
patents in this area, but neither have self-propelled endoscopic devices on the market
yet. According to one company, a market does not yet exist, but the primary reason is
that the costs are high [Welch-Allyn 94][Olympus 94].
Hazardous Environments 1.3.4
Human activity is precluded in many areas where there are extremes of radiation,
temperature, chemical toxicity, pressure or structural weakness. However, it is often
necessary to explore and survey these areas to insure safety and ascertain status. A
variety of small tracked or wheeled machines have been constructed for such
applications, but these have limitations in their ability to traverse and maneuver through
hazardous terrain [VIT 95] [Gothard 90][RSI 94][Sasaki 85][Eguchi 84]. A serpentine
mechanism could fare better due to the advantages cited earlier.
Other dangerous areas include those following disasters such as earthquakes,
explosions, cave-ins, hurricanes, fires etcetera. The search for survivors and removal of
material is often thwarted by loose rubble that might be penetrable by a snake. Outfitted
with sensors such as ammonia or pyroelectric IR detectors, a snake-like mechanism
would enable sensing of humans in rubble. These are applications that would eliminate
life-endangering alternatives such as using heavy construction equipment to move loose
material from accident sites.
Another application is a mine accident probe. Following a cave-in or roof collapse, a
small articulated device would penetrate and burrow through the loose material to effect
a survey and establish communication to survivors.
Reconnaissance 1.3.5
Subterfuge and reconnaissance offer some novel applications of serpentine
mechanisms. The ability to command small roving eyes and ears offers attractive
possibilities to law enforcement agencies; the penetration of dense vegetation by
serpentine robots could provide information not otherwise possible to obtain. My own
work has resulted in inquires from law enforcement agencies including the FBI and
Special Forces.
Routing 1.3.6
Much effort in the wiring of existing structures requires routing of cables and lines
through narrow passages behind existing walls and through pipes. A variety of manual
tools for feeding the lines, such as fishtapes (metal bands), are useful for short runs but
become more difficult to use in longer runs. While some specialized devices have been
designed for wire and cable routing they are not used in practice [Hill 65].
These tasks involve long reaches, wrestling with tools, and stretching and working in
awkward positions. In practice, snake-like devices would maneuver through crowded
plenums and pull the initial lightweight tapes that are then used to pull the actual cables.
I have shown a variety of applications where serpentine robots could provide significant
advances in productivity over existing methods..Overview and Rationale
12
Challenges of Limbless Locomotion 1.4
While the features and advantages and the applications for serpentine robots are
attractive, there remain many challenges in realizing such robots. To create a truly
successful snake robot requires that all areas be addressed and solved. These must be
pondered and evaluated concurrently; design affects function. Integration is
complicated, even intractable, if individual areas are not thought of in the whole.
Configuration and Design 1.4.0.1
The challenge of configuration is determining the form of a robot. The challenge of
actuation is determining the technology that drives the mechanism. The questions are
sometimes mundane but essential to answer: How long should segments be? What
angle should they subtend? Are there actuation techniques that can provide smoother
curves? Determining both the result and implications of each decision is a challenge.
These are addressed in later chapters.
Infrastructure and Electronics 1.4.0.2
Supplying and routing power and signals in complex robots is often underestimated as
a design task. Serpentine robots must be compact and small to accrue the advantages
shown in the previous section. Small size burdens the tasks of wire routing and
actuation support.
Control and sensing 1.4.0.3
Finally, the greatest challenge: how to learn to control such a device? A larger issue is
determining the process, method and framework to achieve this.
Summary 1.5
The advantages of snake locomotion suggest a number of applications for their use. The
application areas detailed here are useful and compelling for serpentine mechanisms,
but exploration and inspection are probably the initial venues for a serpentine robot.
Other applications, such as medical, introduce other issues such as miniature scale and
government regulation. Although the scale for devices across these applications varies
from tiny medical devices to large inspection devices, the principles of the
configuration remain the same. Without a doubt, the development of small self-propelled
limbless devices would open up areas currently intractable to the tools and
technologies available today.
There are a number of serpentine applications that could provide opportunities that are
both technically tractable and economically attractive. The application areas need not
be exotic to make sense, and many of these areas compel further serpentine
developments.
The challenges in development of a robot that can fulfill these promises are many, and
I address them in this dissertation. It is possible that a serpentine robot can be built and
can learn to locomote..13
Chapter 2
Background
There is prior work in snake robots and snake locomotion. However, efforts and results
in these areas are relatively limited in terms of scope, understanding and results.
Background examines prior work in three areas: biological snakes, robot snakes and
machine learning for physical simulation and real devices. The biological history
provides few insights into design but great deal of information on the varied forms of
limbless locomotion. The next section, prior work in serpentine locomotors, is
surprising in its breadth, but little of the research builds on prior work. As a result the
work does not have a history of continued or incremental development. For further
background on serpentine robots, including manipulators, see [Dowling 97b].
Snakes are the ultimate example of limbless animals; the modes and quality of their
locomotion exceeds all other biological limbless locomotors. I have avoided review of
invertebrate limbless locomotors such as worms, however, because of their limited
mode of locomotion and because their hydrostatic structures do not parallel or
correspond to the mechanical robots I propose. I also briefly cover skeletal structure,
musculature, surfaces, and forms of serpentine locomotion.
Biological Systems 2.1
Biological snakes, as existing limbless locomotors, offer lessons in design and function.
The difficulty, as shown, is the codification and extrapolation from biological animals
to man-made mechanisms.
There is a temptation to use the animal as a model or blueprint for robotic mechanisms.
It is inherent in the nomenclature; the use of the terms such as ?serpentine? implies that
the study of snakes can lead to ideas and forms for such mechanisms. There is some
danger in this assumption. The canonical example is that of bird flight; manned flight
bears little resemblance to bird flight with the exception of curved wing surfaces. In.Background
14
addtion, many biological forms are scale dependent, and biological selection
commonly reflects compromises among multiple events or influences in biological
evolution.
Commonly, biological evolution also leaves vestiges and forms that do not directly
relate to a particular function such as running or slithering, so emulating these
structures may be a misdirected effort [Bertram 94].
It?s worth keeping this in mind as lessons and ideas are drawn from snake morphology.
Skeletal Structure 2.1.1
The snake is a vertebrate, an animal with a backbone, and has the largest number of
vertebrae of any animal: between 100-400 vertebrae, depending on the species. Snake
vertebral articulation is one of the most complex of all vertebrates. Although only a few
limited motions and amplitudes are possible between adjacent vertebrae, concatenation
of these articulations can produce large angular excursions. The vertebrae of the snake
form ball-and-socket joints with additional projections that eliminate torsional motion
to protect the spinal cord. This remarkable design uses a series of surfaces to allow the
limited lateral and ventral excursions (respectively 10-20 degrees and 2-3 degrees for
most snakes) but eliminate torsion which would otherwise twist the spine. The
projections can be seen below in Figure 2-1.
The backbone stretches very little; snakes are not elastic like many invertebrates and
they remain a constant length. Snake skeletal form and structure is quite simplified in
number and type in comparison to other vertebrates. Unlike limbed vertebrates, whose
skeleton has many different parts, snake skeletons have only three types of bones: skull,
vertebrae and ribs.
The interesting lessons from snake skeletons are the simplicity of a repeated structure
and the relatively limited motions between adjacent pieces. These aspects are worth
examining in a mechanism design.
Figure 2-1: Snake vertebrae provide lateral and ventral flexing without permitting torsion..Background
15
Forms of Limbless Locomotion 2.1.2
Snakes and other limbless animals have been objects of study for centuries. However,
until recently, little research has focused on the detailed mechanics of serpentine
locomotion. Yet, there is a fair amount of information on the qualitative aspects of snake
locomotion. There are several broad classes of limbless locomotion; these include
concertina, lateral undulation, sidewinding, rectilinear, slide-pushing and other less
common forms. These classes are, in fact, gaits, a term normally associated with legged
animals. Gaits are repetitive patterns of movement used to change speed, adapt to
terrain, and improve stability. Gaits are often chosen because they are more economical
for a particular situation [Alexander 92].
Lateral Undulation 2.1.2.1
Lateral undulation is the most frequently used form of snake locomotion for most
snakes. All parts of the body move simultaneously, experiencing continuous sliding
contact with the ground. It is a sliding motion with all parts moving at the same speed
that occurs through the propagation of waves from the front to rear of the snake. The
snake remains in contact with surface and the motion is similar to a swimming motion.
As shown in [Walton 90], energy consumption is comparable to that of legged animals
of similar scale. During lateral undulation, the snake pushes against features in the
environment to facilitate forward movement.
Lateral undulation is the only form of biological snake locomotion that doesn?t use
static contacts between snake and substrate. The ideal path is a single track along which
the snake slides. Lateral undulation requires a minimum of three contact points for
continuous forward progress: two to generate force and the third to balance forces to
move in a particular direction [Gray 68].
Lateral undulation is unsuited for smooth, low-friction surfaces and narrow corridors.
Nor is it well suited for short stouter animals or for large heavy-bodied snakes because
they are unable to either subtend the curves required or the body mass and environment
tend to significantly reduce its efficacy [Gans 74].
Both wheels and legs use static contacts for propulsion but lateral undulation in snakes
offers an interesting variant using sliding or dynamic friction. This is not as inefficient
as it might first appear. However, the complexity of snake anatomy may make it difficult
to realize these advantages in mechanisms.
Concertina 2.1.2.2
The concertina gait derives its name from a small accordion-like instrument because of
the shape and motion of the snake body. Concertina progression provides a base in
which parts of the body stop for purchase and other parts move forward. The sequence
repeats, and the snake moves forward. It is usually used in confined areas, such as
tunnels, where the snake cannot utilize the full amplitude of other gaits. As shown in
Figure 2-3, the trunk straightens forward of each contact site and is simultaneously set
down in a curved pattern at the rearward end of each site. As a result the musculature
needs to be activated at or near moving portions of the trunk. The key element of
concertina locomotion is the utilization of the difference between high forces with the
static coefficient of friction and low forces with the dynamic coefficient of friction
along different parts of the body..Background
16
Due to momentum changes, static friction, and slower speeds, concertina is a relatively
inefficient mode of locomotion [Walton 90], but forms of concertina allow traverses not
otherwise possible, such as moving along wires and cables as well as through tree
branches.
Figure 2-2: Lateral undulation uses continuous sliding contacts to propel the body.
Figure 2-3: Concertina locomotion is usually used in enclosed areas..Background
17
Concertina movement resembles, in some ways, the motion of worms; parts of the body
remain in place and other parts move forward. It would also appear to be simpler,
perhaps, to implement in a mechanism, than other forms of snake locomotion.
Sidewinding 2.1.2.3
Sidewinding is probably the most enchanting gait to observe; among all serpentine gaits
it evokes the most curiosity. Sidewinding is the use of continuous and alternating waves
of lateral bending. A downward force is exerted for purchase on low shear surfaces like
sand or loose soil; this mode establishes rolling static contacts to cross relatively
smooth substrates. There are only two contact patches while the snake is in motion. The
technique minimizes slippage and is even more efficient than lateral undulation [Secor
92]. Some sidewinding snakes have been observed to travel kilometer-length distances
continuously [Mosauer 30]. Sidewinding is used primarily by snakes in desert regions
where loose soils and sands are prevalent. The development of sidewinding may be
related both to the need for traction on low shear surfaces such as sand and the need to
avoid the high temperatures of desert terrain. As shown in Figure 2-4, sidewinding can
be thought of as the ?peeling? of the body from one track to the next. The tracks, or lines,
show the rolling of the body contacts during locomotion.
Rectilinear 2.1.2.4
Rectilinear progression uses movements of skin with respect to the skeleton to ?rachet?
the body along the ground. Rectilinear motion is a slower, creeping motion using the
belly to provide traction through anchoring and is typically used by larger snakes.
Rectilinear motion was once conjectured to result from ?Rib-walking,? an active
movement of the ribs. However, this was conclusively disproved in [Lissman 50]
through x-ray observations of a snake in motion. Muscles connected from the ribs to the
elastic skin provide the propulsive motions through reciprocating or racheting
movements.
Figure 2-4: Sidewinding locomotion results from rolling or static contacts.
Direction of Motion.Background
18
In rectilinear locomotion, several portions of the body are in contact with the ground at
any moment, and the gait uses symmetrical rather than staggered waves of contraction.
A section of the skin of the belly is drawn forward so belly scales are bunched. This part
of the body is then pressed down, and ventral edges engage the surface. Then the body
slides forwards within the skin until it is in normal alignment with skin, and the motion
repeats. Only small vertical motions are needed for rectilinear locomotion.
Other Snake Locomotion Modes 2.1.2.5
Other forms of limbless locomotion include slidepushing, saltation, burrowing and
climbing. Slidepushing is a gait used in times of stress where anteriorly propagating
waves move more quickly backwards than the snake moves forwards. A great deal of
sliding and motion occur without a corresponding forward progression. Saltation is the
jumping the near-vertical walls and trunks of trees. Some saltating snakes can leap gaps
of a meter or more, sometimes vertically. This requires storage and release of a lot of
energy and, additionally, involves a ballistic phase of motion during which control is
difficult. Other extraordinary modes are used by certain asian tree snakes that glide
through the air by opening the rib cage to form a gliding surface. The amazing thing
about this mode of snake travel it is not how well it flies, but that it flies at all! These
modes are exotic and I do not explore these forms of locomotion in robots.
Skin and Musculature 2.1.3
Snakes are covered with scales whose distal sections are loose and overlap other scales.
They are dry and highly polished with a coefficient of friction of between 0.3 and 0.4
[Gans 84] [Jayne 88]. The belly scales are much wider than the scales along the sides
and back of the snake. The skin, to which the scales are attached, is highly elastic. This
Figure 2-5: Rectilinear motion couples muscle action between ground and vertebrae..Background
19
is most evident in feeding since the snake eats everything in a single gulp; enough food
for over a year in some cases!
Snake posture is established by muscle groups as shown in Figure 2-6. Many such
bundles interconnect vertebra to each other, to the ribs on each side and in several bands
to the skin. In contrast to early observations of snake locomotion the ribs do not ?walk?
or move while the snake moves forward [Gray 46]. Specialized musculature in some
snakes allows 50% of the body length to be extended above the ground without support
and constriction.
Both the skin and musculature of the snake are highly refined and specialized. It is not
possible, within the scope of this research, to replicate their capability in sensing and
control. There are few, if any, commercial actuators like muscles, no sensing like snake
skin and no controller equivalent to the nervous system of such a complex animal.
However, I show that it is possible to replicate general characteristics of serpentine
locomotion.
Analysis of Limbless Locomotion 2.1.4
[Fokker 27] and [Jones 33] show that body curvature is a key element of the lateral
undulatory form of locomotion. The snake body tends to propel best at portions of the
body that are undergoing the greatest amount of curvature change [Gray 68].
To gain a better understanding of force application by limbless animals, it would be
useful to externally monitor forces of snake motion. Gray did this with pendulums and
force plates and Gans instrumented pegs on a low friction surface [Gray 68]. But the
small number of discrete points gives only a little information about a few discrete
points. No force plates or other techniques seem capable of providing this information.
Full and others at Berkeley?s Polypedal lab have a number of means of monitoring
forces even for small insects, but their clever photo-elastic techniques are limited to
discrete contacts [Zimmer 94][Harris 78]. Force plate techniques, used in biomechanics
Figure 2-6: Snake ribs, vertebrae and skin are linked by complex woven muscle bundles..Background
20
studies, are too coarse to provide good information but some sensing pads used in these
studies may be useful tools [Novel 97]. Hirose also showed measurement techniques
for measuring forces in snake locomotion using strain gauges and a support mechanism
[Hirose 93].
From observation of snake motions it becomes obvious that the control mechanism
utilizes local information about the terrain to quickly and effectively adapt to changing
conditions to propel. Since the position of the contact sites is not known by the snake
and these contact sites can move or deform, the snake must make continual selections
by monitoring external forces and contact sites. In addition, a feedback mechanism
exists that responds to this information so that following portions of the body adapts its
curvatures and provides appropriate forces to the terrain [Gans 85]. While comparisons
are often made between fish swimming and the lateral undulation of a snake, swimming
is significantly different for a fish where buoyancy, fluid dynamics, and hydrodynamic
mass play the critical role in propulsion.
A key difference between gait selection in snakes and most other animals is that gait
selection in legged animals is a function of speed in a given environment. For snakes,
gait selection appears to be more a function of environment, not of speed. This
difference may strongly affect learning results in a fixed environment.
What?s Missing? 2.2
The impetus for examining snakes was to discover what could be transferred to a
serpentine mechanism. What has been described so far is the consensus regarding
qualitative forms of limbless locomotion and when they are used. However, there is
much that is unknown. For example, snakes crawl on their bellies but propulsive forces
are generated along vertebrae axis. At this time it is not known clearly how this is done.
It is conjectured that the complexity of muscle structure may be a reflection of the
control system, not just and not simply the structural basis of force generation. This is
not known although it likely that the anatomy and control co-evolved.
There are no detailed and complete analyses of the force propagation that results in
motion in snakes although there are some analyses for invertebrates [Keller 83][Niebur
91]. The full sequence of muscle control is also unknown. Additionally, the feedback
mechanism for contact force and slip is not well understood. More studies on efficiency
are needed. Additional work is needed to make definitive comparisons of energy use in
locomotion. Many other questions include: How are gait selection strategies decided?
What is the distribution of control?
I do not propose to answer these questions; they are tightly related and coupled to the
anatomy and nervous system of the snake. However, it may be that serpentine robots
will eventually offer insights to those who study the natural organisms, although this
work does not make this claim.
Robotic Systems 2.3
Nearly all mobile vehicles built by man for terrestrial use have either been wheeled or
legged. Wheeled vehicles date back several thousand years; walking devices can be.
Background
21
traced back to the 19th century. Locomotion without the attributes of legs and wheels
is represented by only a few examples, mostly within the past twenty years and almost
entirely within the laboratory rather than in the field.
In my search for undulating and locomoting serpentine mechanisms, the earliest
mechanisms I came across were developed by a Russian constructivist artist of the
1920?s, Petr Miturich. He designed a series of utilitarian designs for undulators, termed
volnoviki, that moved by wriggling. He made many designs for volnoviki that were to
operate on land, in air, or water. He applied for several patents on the ideas and built
models, but none included power and control [Lodder 83].
There are also obscure references to clockwork snakes and caterpillars built by
craftsmen such as Faberg?, but most had small hidden wheel drives and did not
locomote through body motions alone.
Hirose 2.3.1
Work by Hirose and Umetami, in the early 1970?s, was among the first to explore and
develop limbless locomotors. Hirose has a sustaining interest in limbless locomotion
and designed and built several robots over decades. He termed the devices Active Cord
Mechanisms or ACMs. Hirose focused on developing robots that could perform lateral
Figure 2-7: Miturich developed a wide variety of undulating designs as art..
Background
22
undulation and later developed a series of wheeled coupled-mobility devices that
followed from this work.
Hirose?s development of modeling and control first derived expressions of force and
power as functions of distance and torque along the curve described by the snake. The
curve was then derived and compared with results from natural snake locomotion. The
curve, termed serpenoid, has curvatures that vary sinusoidally along the length of the
body axis. These equations are shown below:
This curve is different from sinusoidal or even clothoid curves. Comparisons with
natural snakes across constant friction surfaces showed close agreement between the
serpenoid curve and the empirical data.
Hirose then went on to develop models for the distribution of muscular (actuator) forces
along the body. This was done for normal and tangential forces as well as power
distribution. Again, the developed models closely correlated to muscle exertion data
and force measurements from natural snake movements.
Figure 2-8: Hirose?s Adaptive Cord Mechanism utilized a series of
articulated links with passive wheels.
x ( s) sJ o ( a) 4l
p ---- - ( ?1)
m
2m -------------- J 2m ( s mp-- ? l?
? ? a)sin
m 1 = ? + =
y ( s) 4l
p ---- - ( ?1)
m 1 =
?
?
m 1 ? J 2m 1 ? ( a)
2m 1 ? ------------------------ 2m 1 ?
2 ---------------- p s -- ? l?
? ? sin =.Background
23
The experiments to this point were primarily of a uniform nature, but Hirose recognized
that snakes quickly adapt locally to variations in terrain and environment. The next
issue was to characterize this adaptation. From observation it was noticed that snake
locomotion is not necessarily a two-dimensional problem; in fact during higher speed
motions, snakes use ventral motions to actively distribute their weight to those areas
where propulsion is maintained.
Further study developed relationships between amplitudes and wavelengths of the
motion and local friction conditions, as well as morphological features of the snake
such as vertebrae motion and muscles (actuators). Models for locomotion in rough
terrain where obstacle contact is made were also developed and correlated with snake
motions.
Hirose examined the construction of mechanisms that were able to perform lateral
undulation. Several views of these machines are shown in Figure 2-8, Figure 2-9, and
Figure 2-10. By calculating torques, velocities and power required, Hirose was able to
provide design guidelines for the actuators and drivetrains. The next development was
a distributed control scheme wherein each link could respond independently. In
Hirose?s work the control took the form of angle commands at each joint. The variables
were simply related closely to the amplitude, wavelength and velocity of the body axis.
Figure 2-9: A close-up of the first ACM link
showing the body and drive..Background
24
Steering of the robot was accomplished by biasing the control to adjust curvature in a
section of the body.
A 20 link mechanism weighing 28kg was constructed. Link actuation was
accomplished with DC motors coupled to a caster board and potentiometers were used
for feedback. Later, after a motor change, the weight was reduced to 13kg.
To accommodate unknown environments required tactile sensing; this was the next step
in Hirose?s work. Small contact switches provided this information to the controller. As
shown in Figure 2-10, this robot could negotiate and propel itself through winding
tracks. The developments included a control technique called lateral inhibition tactile
signal processing, which provided for contact and reflex motions. The shape of the body
was varied according to the second derivative of the sensed contact pressure and
responded appropriately to provide forward progress.
All of Hirose?s locomotors used either powered wheels or passive casters and the only
locomotion mode studied was lateral undulation. Hirose and his colleagues have gone
on to develop an elastic elephant-like trunk, a large serpentine mechanism for interior
inspection of turbines and small manipulators for surgical applications. The best overall
paper on Hirose?s work is [Hirose 90]. It succinctly covers many years of development
in serpentine mechanisms. More recently, Hirose?s book Biologically Inspired Robots
is an excellent overview of his work [Hirose 93].
Hirose?s work in serpentine robots is probably the most complete of all work in this
area. He dealt with issues of mechanism, control, sensing and modeling of natural
animals. However, the mechanisms used wheels, the terrains for the ACM?s were 2D
only, and the mechanism used only lateral undulation as the locomotion mode. The
configuration, while not practical for application use, was a great advance in serpentine
robots.
Figure 2-10: Hirose?s locomoting Adaptive Cord Mechanism..Background
25
Burdick and Chirikjian 2.3.2
Joel Burdick and his students at Caltech, especially Greg Chirikjian 1 , have pursued
work in serpentine manipulation and locomotion for several years. Chirikjian?s thesis
presented a framework for kinematics and motion planning of serpentine mechanisms.
Curves in three dimensions, R 3 , are defined to provide a general means of
parameterizing curves and sets of reference frames. In addition to describing the curve
shapes, they extended features to allow roll distribution along the curve and extensions
and contractions along curve segments. These were then used to specify serpentine
configurations [Chirikjian 92].
Since most manipulators do not describe continuous curves, there remained the
problem of fitting rigid link devices to the desired curve. A general parallel algorithm
was found for fitting manipulator segments to the desired curve. The modal approach
was then used to resolve the ?excess? degrees of freedom (DOFs) in hyper-redundant
robots to carry out specified tasks. The modal approach provides a means of
characterizing the shape and motions without developing full inverse kinematics, which
have an infinite number of solutions. A series of specified functions could be specified
in modal form, and the problem became finding modal participation factors to satisfy,
as best can be done, the task constraints. Optimal techniques for minimizing measures
of bending, extension, etcetera, were then developed via the calculus of variations. One
issue with these optimal techniques is the selection of cost functions to evaluate
configurations. That is, how to determine the ?goodness? of a particular solution.
Chirikjian described obstacle avoidance using this set of tools and it was assumed that
paths were provided through traditional motion planning techniques. An additional
issue addressed is that of time, that is, velocity, for the solutions. A series of arcs and
lines were used to create a path along which the manipulator sections can move. But,
independent of the path formulations, the previous solutions to kinematics could fit
manipulator configurations and trajectories.
Locomotion through sequences and patterns of geometries was developed next. The
extensible locomotion modes were traveling wave, similar to rectilinear motion in
1. Now at Johns Hopkins University
Figure 2-11: Burdick?s Snakey, a VGT-style hyper-redundant manipulator and locomotor..Background
26
snakes or caterpillars, and stationary wave, similar to inchworm motion where the
advancing wave remains in the same position with respect to body coordinates.
The extensible modes are similar to earthworm locomotion where segments provide
extension and contraction to propel the robot. To avoid the need for differential friction,
portions of the body can be raised to facilitate this motion. Descriptions of techniques
for non-flat floors are also developed. Intriguing ideas were also introduced using
serpentine robots to provide grasping and manipulation capabilities. The mechanism
could contact and wrap about an object; the propagation of a wave or extension of the
links caused the object to move in a desired direction. These techniques could be used
to simultaneously grasp, move and manipulate objects.
A mechanism, a variable geometry truss configuration, was designed and built and is
shown in Figure 2-11. The mechanism was comprised of commercial linear actuators;
a simple modular and maintainable approach to design was used. A variety of tests
using the methods described above were conducted and a number of successful
experiments in control and locomotion were carried out.
Key to Chirikjian and Burdick?s work was the modeling of the robot as a 3D shape and
sequence of shapes. This enabled a variety of techniques in trajectory planning and path
generation. The mechanism also allowed exploration of non-snake-like extensible gaits.
The mechanism however was primarily a fixed base device and a couple of limited gaits
were demonstrated on the robot. Additionally, rachet wheels were used in locomotion
[Chirikjian 95]. Sidewinding was also formulated in piecewise continuous curves in
[Burdick 95] and, although the exact shape of the body was not necessarily snake-like.
the general form of the motion was identical to that of snakes.
Later work by Burdick with J. Ostrowski explored the use of geometric mechanics to
formulate general notions of locomotion. Two systems were evaluated in this context:
a ?snake-board? which is an actively articulated skate board, and Hirose?s ACM
[Ostrowski 96]. Other related work at Caltech included the work on geometric phases
to describe robot locomotion [Kelly 95].
Choset also developed path planning methods for highly articulated robots such as
snake robots. He developed the Generalized Voronoi Graph (GVG) and Hierarchical
GVG for use in sensor-based planning motion schemes. The techniques utilize concise
descriptions of the topological spaces to build paths. A key feature of the work is that
is does not require a priori knowledge of the world [Choset 96a][Choset 96b].
Shan 2.3.3
Shan?s work was primarily in obstacle accommodation: motion planning that makes use
of obstacles rather than strictly avoiding them. The mechanism in this work, shown
below, used a form of concertina locomotion. The device Shan built, shown in Figure
2-12, uses solenoids at the joints to drive vertical pins into the surface. This establishes.Background
27
fixed contact points from which the rest of the mechanism can move [Shan 92][Shan
93a][Shan 93b].
The configuration and locomotion of Shan?s robot were limited to flat floors and a
concertina mode that required a great deal of space; far more than the cross-section of
the mechanism suggests. The length of the links or, more importantly, the ratio of length
to diameter, play a large role in the robot?s inability to traverse tight spaces.
Ikeda and Takanashi 2.3.4
In 1995, the giant Japanese electronics company, NEC, announced the development of
a snake robot which was dubbed ?The Quake Snake? and designed to enter the rubble-strewn
aftermath of earthquakes and explosions to search for survivors. The device,
called Orochi, utilized an active universal joint, a novel form of a Hooke?s joint
designed by Ikeda and Takanashi. The seven segment device is shown in Figure 2-13
Figure 2-12: Shan?s snake mechanism uses a concertina-like motion..Background
28
and the joint design in Figure 2-14 [Ikeda 87] [NEC 96]. A small video camera was also
deployed at the head of the mechanism and used to by the operator to assist in guiding
the snake.
The mechanism is one of the best mechanical designs in serpentine robots, and some
automatically generated gaits have been used on this mechanism [Burdick 97]. Control
as shown in the videos is done manually and the single gait used is akin to a rectilinear
or inchworm gait. Additionally, in some footage of the device, but not shown in Figure
2-13, are small brackets used to stabilize the snake as it moves; effectively they are wide
?feet.? This class of mechanism has great promise for serpentine robots in real
applications. Another version of this joint was used in a snake built for JPL.The key
lesson in this robot was effective packaging of the mechanism, the slim design and the
modularity of the links.
Figure 2-13: The NEC ?Quake Snake? utilized a novel universal-type joint
between links for a total of 12 DOF.
Figure 2-14: The rotating joint developed by Ikeda and Takanashi provides for a
smooth and wide range of motion..Background
29
Nilsson 2.3.5
Martin Nilsson of the Swedish Institute for Computer Science in Sweden, as part of the
PIRAIA project, has developed a novel universal serpentine link that is a roll-pitch-roll
joint. Multiple links give it the ability to subtend some very non-snakelike modes of
locomotion that incorporate a rolling motion. In one instance, the snake might ?hug? a
tree and, using the side rolling capability, roll directly up the tree. The joint has another
nice feature in that cables passing through the joint cannot twist if the joint is controlled
properly. This joint is equivalent to universal joint, but unlike a normal universal joint,
the input angular velocity equals the output angular velocity for all angles. The
mechanism, while relatively complex, can be realized with standard components.
Additional work by Nilsson showed learning techniques for locomotion using
differences between static and dynamic coefficients of friction and sliding contact
surfaces. A simple pair of paddles joined by a single degree of freedom was capable of
learning to move itself through a simple physical model [Nilsson 95] [Nilsson 97a]
[Nilsson 97b].
Nilsson?s mechanism is very different from other work and from natural animals. The
two roll motions at each joint enable wheel-like effectiveness in locomotion, but also
complicate internal mechanics. Since the intent of this dissertation is to replicate more
standard snake-like locomotion modes, this design does not appear to have many
parallels to this work.
Figure 2-15: The PIRAIA link provides a roll-pitch-roll capability..Background
30
2.3.6
Paap 2.3.7
Karl Paap and his group at GMD in Germany developed a snake-like device to
demonstrate concepts and developments for real-time control. The device is a tensor
device that uses short sections with cable winding mechanisms to effect curvatures
along several segments. The device is shown in Figure 2-17 [Paap 96]. The curvatures
are continuous along those sections but the joining segments, where the drive
mechanisms are located, do not bend or move. Some very limited locomotion has been
shown in the mechanism and the cable drives have been a design challenge.
Figure 2-16: The PIRAIA links incorporate power as well as drive mechanisms in
a unique roll-pitch-roll joint.
Figure 2-17: The GMD Snake mechanism.Background
31
IS Robotics 2.3.8
IS robotics built a small snake-like machine, Kaa, for prehensile grasping of pipes and
locomoting. Not an effective locomotor, the robot was initially designed for moving in
and through networks of pipes and support structures. It is probably the first completely
self-contained snake locomoting robot. Using RC-servos as actuators, the robot
propagated a ripple down the body to effect a straight-line motion on a flat surface
[Desai 95]. The robot did not locomote in the position shown in Figure 2-18 but instead
lay flat upon the ground so that actuator motion was in vertical plane only. The
movement was limited and the large box in the middle of the robot, housing power and
computing, made locomotion problematic on the ground.
Coupled-Mobility Devices 2.3.9
Coupled mobility devices, sometimes called overland trains, are similar to trains of
vehicles linked together. Although Hirose?s ACM robots resemble a coupled-mobility
device, all wheels were passive and the robot ?skated? because of body movements.
However, a number of machines have been built that are similar to small trains for off-road
navigation and used active wheel drives. The largest of these were LeTourneau?s
huge Sno-trains [Gowenlock 96] and a much earlier overland steam train [Anon 95].
Figure 2-18: The Kaa snake is a self-contained locomoting device..Background
32
Others, such as the Odetics ATMS, All-Terrain Mobility System, as shown in Figure 2-
19, were coupled-mobility devices with active couplings designed to traverse a variety
of terrains. In this device, both link and wheel motions were explicitly described for
movement [Odetics 88].
Coupled mobility devices, while bearing resemblance to snake-like robots, are not
limbless. They use wheels to drive or support, and because of this, they do not offer
some of the advantages of snake locomotion as described earlier. However, it is possible
that such mechanisms could offer advantages of the wheel and some advantages of
snake-like robots in future developments.
Learning 2.4
Physical modeling is a key element of a serpentine mechanism. In all prior work
described for serpentine robots, control was generated in an explicit fashion. Results
included limited modes of locomotion and little adaptation. There is a great deal of
research in machine learning, but much of it resides within computer models and
databases. Little work in learning has been applied to physical mechanisms that are
more complex than, for example, a robot learning to throw a ball. Two pieces of
research that are relevant to my work are the work of Karl Sims and David Barrett.
Sims 2.4.1
Karl Sims developed a learning tool that simultaneously evolved both morphology and
control in simulation using genetic algorithms. The various creatures evolved both their
geometric forms and their control structure using simple physical models to determine
fitness. Metrics for fitness were mostly based on speed but also incorporated the notion
of winning margins where fitness evaluation depended on the margin of victory
between pairs of creatures [Sims94a][Sims94b]. However, Sims did not extend his
work beyond simulation, but did illustrate the possibilities of learning using physical
modeling. The physical models were not detailed and were fairly simple simulations
Figure 2-19: The ATMS could cross and climb obstacles..Background
33
but the end result was both inspiring and enchanting. As can be seen from Figure 2-20,
the bodies of the evolved creatures were simple constructs of intersecting rectangular
volumes. No realism or accuracy was required; the work was not intended to be a true
or realistic predictive simulation. The creatures themselves were simple constructs of
intersecting polygonal objects without real joints or pivot points.
Barrett 2.4.2
David Barrett?s work at MIT used genetic algorithms to teach a swimming robot, a
tunafish, to swim efficiently. A large externally powered, cable-driven mechanism was
used to control the motion of the tuna. The metric for the robot was an efficiency
measure formed from the ratio of the mechanical power output to the power input to
undulate the tuna?s body.
Locomotion was evolved in situ, not in simulation. The hydrodynamics were too
complicated to model and compute so, in essence, the water acted as a large analog
computer with infinite resolution. Seven parameters related to fluid flow were chosen
for the optimization criteria. Small populations of 10 were used and convergence was
accomplished within only a few generations [Barrett 97] [Triantafyllou 95].
Figure 2-20: An example of evolution from Sim?s creatures..Background
34
Summary 2.5
Biological Understanding 2.5.1
Biological analogues to serpentine robots offer remarkable performance and many
issues might be understood through the study of these natural animals. However, much
is not understood in snake control and locomotion and perhaps serpentine robots will
offer explanations for biologists! In the meantime, snake locomotion modes offer
striking examples of the promise of limbless locomotion. Additionally, useful lessons
from structure and morphology of biological snakes can be applied to the mechanisms.
Skin, in particular, appears to have a strong influence on locomotion but prior robot
mechanism work has not addressed this issue.
Robotics Developments 2.5.2
Although there have been several projects related to serpentine manipulation and
locomotion there have been far fewer robots built, and little success towards practical
mechanisms. In fact, only a few serpentine manipulator mechanisms ever made it as far
as a commercial venue; the Toshiba Multijoint Inspection Robot [Asano 83][Asano
84][Nakayama 88][Toshiba 89], and the Spine robot [Drozda 84][Grunewald 84].
Neither were successful in the marketplace. I am aware of only one serpentine
manipulator manufacturer, Kinetic Sciences, who is attempting to market their
?Tentacle Arm? [Immega 95].
The developments of Sims, Barrett and other in the machine learning community show
great promise, but also show the immense amount of work and computational
requirements required in modeling and evolution of locomoting vehicles. The control
and planning work of Burdick, Ostrowski, Choset and others is furthering the
Figure 2-21: The RoboTuna internals and foam latex covering..Background
35
techniques necessary for the control of these and other highly articulated mechanisms.
The most promising emerging developments are the non-linear control and geometric
mechanics literature Burdick, Murray, Kelly, Ostrowski and others. They are forming
general frameworks for all locomotion and have used serpentine locmotion as one of
several case studies.
Technical Needs 2.5.3
There are many technical developments required for useful and successful serpentine
mechanisms. Mechanical structures have limits to rigidity resulting from both technical
and economic considerations. Additionally, long and narrow structures raise issues of
dynamics and oscillations that must be addressed. Other technical areas include
sensing; accurate sensing needs arise because of the indeterminacy of individual joint
positions. Another need is the development of sensor ?skins? that allow joints to
independently react to local obstacles and control. Research is underway in each of
these areas but only at formative stages. Additionally, most of the technology efforts are
not directly related to serpentine mechanisms but focus instead on the needs of industry
and mechanism control.
The work to date shows great promise in understanding and configuring robots, but
there remain many hurdles on the path to develop, control and deploy practical
serpentine robots. The quest is to demonstrate that robots can learn to locomote even
when they have no wheels or legs. The following sections show a new approach to the
problem of teaching limbless robots to locomote.
Bridging the Gap 2.5.4
As shown in Figure 2-22, while several works have bridged physical simulation and
learning and others have bridged learning for robots, little has been done in bridging all
three areas. As systems grow in complexity and as classical modeling techniques
become more intractable, then connecting these areas will be critical. This work takes
a step towards bridging the areas of mechanism, learning and simulation.
Each of the areas shown in Figure 2-22 are open research areas with many people
working hard in the areas of physical simulation tools, learning methods and mobile
robots. Rarer are the works that bridge these areas including the gap between physical
simulation and learning and the gap between learning and mobile robots. A few people,
Physical
Simulation Learning Robots
[Barrett 97]
[Sims 94] [Maes 90]
[Ngo 93]
Figure 2-22: Bridging physical simulation, learning and robot mechanism is key to the control of
complex mobile robots .
[Schneider-j 95]
Mobile.Background
36
as listed in the figure, have begun the work required in these areas. What has not been
bridged is the long path between simulation, learning and mobile robots. This requires
a great deal of ground work in all areas. It is this bridge that I construct in this thesis.
The results of this background study provide lessons and results from prior approaches
and in several cases provide inspiration that directly influenced this research. For
example, simplicity in design and the avoidance. Questions arose as to why the snake
robots were designed in particular ways and compelled the formulation of a methodical
approach to answering questions regarding morphology and form. Other influences
included the learning approaches used by Sim and Barrett for simulation and robots
respectively. Snakes themselves inspire by example and the wonder they incurr but also
the short, repeated structure that forms their backbone. The prior work influences and
inspires through example and approach..37
Chapter 3
Framework
How can we teach a limbless robot to move? What is the structure, the form, or the
architecture for making this happen? For a robots to learn to locomote, a structure or
framework is necessary to support learning. Evaluation is also critical to making this
occur; that is, how is performance evaluated? Framework presents such an architecture
for simulating, designing and testing serpentine locomotion with a transfer of those
results to a physical device. Sections of this chapter correspond to later chapters in this
dissertation.
Overview 3.1
A snake robot mechanism is relatively complex; the design is a repeated structure of
many identical links, all of which need to be coordinated and each of which have to be
controlled. The fundamental issue is determining the sequences of motion of the
elements that move the mechanism in a particular direction. While it may be possible
to construct sequences by hand that move the system, this is fairly tedious, inefficient,
and likely to miss a variety of interesting gaits. Schemes tried in previous works, see
Background, appear inadequate for determining and expressing a variety of gaits for this
mechanism; you need to know the gait before attempting it. Ideally, the system would
move forward after learning how to move. The problem then becomes: how to teach the
robot to move.
The fundamental idea then, is to develop a structure or framework to allow both
representation and development of gaits. The need is then a modular framework with
the following elements: learning, simulation, and evaluation. Other elements include
techniques or protocal for communicating and passing information from one element
to another and selection of parameters and the form of the parameters that are passed
back and forth within the framwork..Framework
38
Physical simulation is a useful tool for the configuration and control of complex
mechanisms and allows observation of these robots in a simulated environment. The
rationale for physical modeling and simulation is to represent a physical device such
that inputs and resultant outputs are reflected accurately and, in turn, provide an
understanding of mechanism behavior. A useful simulation tool for robots provides the
capabilities to model physical entities, physical laws, interactions, and incorporates
monitoring tools to see input effects.
Given such a tool, it can be incorporated into a larger scheme where control and
evaluation take place. This scheme would also provide a means for testing the results
of various control methods. Expanding on this idea further, a machine learning
technique, using an appropriately chosen metric, could interface to the modeling and
control techniques for its own use.
In particular, control parameters are selected and are operated on by a learning function.
These parameters, in turn, become the inputs to the physical simulation. As the
simulation learns to behave in a desired fashion, using a metric to be defined later, then
these results can be ported to a physical device for testing and further refinement. This
testing can still be done in the context of this original framework however by simply
substituting the robot for the modeled device. This assumes that the framework
provides an approximation to the robot sufficient to demonstrate viability.
As shown in Figure 3-1, the framework is comprised of several areas including
optimization or learning, evaluation, control and physical simulation. A loop of test and
iteration is created by the results of performance in physical testing. The results in each
set of tests provide input into the next series of tests. In the following subsections we
will take a brief look at each component of the framework.
Evaluation and Metrics 3.1.1
As the physically simulated model executes, a performance measure is used to
determine how well the model is doing. For example, one such measure might be the
Evaluation
Learning & Optimization
Coriolis
Parameters Metric
Control
Figure 3-1: Framework for learning control of a physical device.
Physical Modeling.Framework
39
maximum distance the device moves in a given period of time. This measurement, or
metric, is a quantitative assessment of the relative ?goodness? of that set of parameters.
This metric can take time, energy, distance and other measures into account in
determining the efficacy of that set of parameters. In a sense, this is a measure of the
efficiency of the particular sequence of body motions that effect forward movement.
Learning is best when the results are easily and quickly evaluated, so the metric should
also be easy to evaluate. An overview and discussion of metrics as well as metric
determination and evaluation is detailed in Performance Metrics.
Learning and Optimization 3.1.2
A crucial part of the framework provides the generation and selection process for new
sets of parameters to determine the ?best? set of these values. In Learning and Optimization,
the types and methods of machine learning techniques are detailed. These are used in
the course of determining and tuning efficient gaits for the mechanism.
Control 3.1.3
From the optimization, a set of parameter values is chosen and these can be used to run
a program or controller to implement and execute a run. Although Control is shown as
a distinct box within the framework, it is really two separate components; one to interact
with Physical Modeling and the other to interact with the physical robot. This work is
detailed in Implementation.
Physical Modeling 3.1.4
The key element for the framework is the ability to simulate the geometry of the
physical robot and its interactions with external surfaces and contacts. Physical
simulation allows the modeling of complex shapes and their interaction. Physical
quantities that are modeled include torques, forces, velocities, acceleration, both kinetic
and potential energies. Physical simulation, until recently, was intractable for
mechanisms because of algorithm complexity and the computing power required.
While many of the physical principles were understood, the tools were lacking. The
convergence of increased computing power and usable programs for creating physical
models has enabled the use of physical simulation as a tool in research, design and
testing.
Modeled quantities in physical simulation also include gravity, dynamics, masses, and
material properties such as friction and density. The user is required to configure
geometries, define physical relationships between geometries, set forces and any time
dependent relationships. The physical modeler then creates the simulated world and
allows the user to observe the subsequent behaviors and interactions. The whole process
of physical modeling promises to be a wonderful extension to existing design tools; it
facilities the creation and evolution of designs.
While simulation is used for resolving inputs and outputs into the physical modeler, it
is also used to learn sequences of body configurations that enable the model to move
..Framework
40
That is, the modeler is used in a larger framework that allows observation and reaction
to gain a desired output; a closed-loop system.
The framework is also designed to be used on the robot mechanism. The physical
modeler is replaced by the robot mechanism, as shown in Figure 3-3, and the same
criteria and optimization techniques can be used on the real mechanism after the
physical simulation produces useful results. Even with strong effort, the simulation
does not model the system perfectly; the vagaries of the real world prohibit accurate and
high fidelity predictions of behavior.
However, the initial physical model is used to provide general classes of gaits that can
be refined on the real system.
Figure 3-2: An accurate 3D model of the physical snake.
Evaluation
Learning & Optimization
Coriolis
Parameters Metric
Control
Figure 3-3: The same framework is used for the robot mechanism
Robot.Framework
41
Summary 3.1.5
The confluence of a new generation design tool, physical modelers, recent advances in
learning and a novel mechanism promise to bring advances in robot design. Physical
modeling, a relatively new tool primarily developed for use in the graphics community,
can provide the designer feedback and provide a tool in a larger context; evolutionary
design. The framework shown here, comprises modeling, simulation, evaluation and
control of the robot under consideration. The following chapters detail each of these
aspects and their use in the design process..Framework
42.43
Chapter 4
Performance Metrics
Teaching a serpentine robot to crawl requires evaluation of locomotion strategies or the
use of metrics. Metrics are measures of performance. The selection of good metrics is
critical to learning methods because they provide a means to compare the results,
techniques and methods of locomotion. In addition to providing assessment and
comparison, they should be easily calculated and measured. Metrics are used to
evaluate and drive the learning. Thus, the simpler and more straightforward the metric,
the easier it is to track and guide learning.
The general question is: how to evaluate the performance of mobile robots? Specific
questions for evaluating performance include: how fast does it move? how much energy
does it use? how far can it go? how well does it carry out a task?
Performance Metrics examines criteria for evaluating the performance of locomoting
vehicles including limbless robots. Performance metrics can be used for objective
analyses of different locomotion techniques and be used to optimize these criteria
during learning. I show several formulations of metrics used in the analysis of vehicle
and animals, and I also derive useful measures and provide insight into scaling issues.
Metrics 4.1
Metrics are used to compare the performance of vehicles, tools, computers and human
endeavor. As an example, a simple metric of performance might be speed. However,
speed, by itself, is both insufficient and misleading because, without compensating for
size, scale or energy use, the comparison between different systems is comparing apples
and oranges; they aren?t the same. In an absolute sense, if speed is the only metric of
concern, then the development of vehicles favors large size and high power.
In most forms of vehicle racing, for example, the only metric that counts is speed, but
a wide variety of constraints and narrow classifications result in a relatively narrow
.Performance Metrics
44
range of parameters that can be adjusted to maximize speed. This discussion, while
interesting to racers, is moot; speed or velocity is an insufficient measure of
performance because scale matters, power matters and configuration matters.
Additionally, the very simplest metrics, such as speed, are often insufficient because
energy sources are typically power- or energy-limited for small robots [Dowling 97a].
Thus, minimizing energy use over time and distance is important for mobile vehicles.
I present a number of examples of metrics and, importantly, their dimensional units and
physical meaning. Units reveal metrics in a way that is intuitive and allows comparison
with other metrics.
Efficiency 4.2
Efficiency is the ratio of power output to the power input of a system. It can be the
theoretical mechanical output power versus measured input electrical power. It
represents how effectively a system utilizes power to effect motion or how much energy
it takes to move a given distance. However, to say that one system is more ?efficient?
than another is often invalid because of environment or other constraints. Indeed while
size, weight and speed are probably related to efficiency they are usually not directly
taken into account in the measure.
Also, it?s not obvious how to determine efficiency. For example, the mechanical
efficiency of constant speed locomotion over level ground is zero because there is no
work done. Work, the product of force and distance, is zero at constant speed and with
horizontal motion. Energy is expended, of course, but no physical work is done.
Does efficiency always matter? A recurring litany in walking machine literature is the
potential efficiency of walking. With few exceptions, such as planetary explorers,
efficiency is not likely to be important for walkers. The metric that compels walking
machines is likely to be task performance and efficacy, not how little energy is used.
However, planetary robots are often energy limited and efficiency does matter for that
application. However, the use of snake robots may be driven by the ability to complete
tasks that are intractable to other vehicles, not energy efficiency.
The major use of energy in animals is for locomotion and thus there are distinct
advantages in keeping these energy costs low. Advantages include the ability to run for
periods of time from predators or surviving without food for indeterminate periods.
Interestingly, for most terrestrial animals, energy use for a given distance traveled is
independent of the speed [Schmidt-Nielsen72]. However, the amount of energy used is
related to the size of the animal; where larger animals use less energy per weight to
travel a given distance. This effect is related to the amount of time that the animal?s legs
are in contact with the ground during movement and is due to the storage and release of
energy between tendons and muscles during that time. Strangely enough, the muscles
do not directly drive most locomotion but in fact are a mechanism to store energy in
tendons which transmit the forces to ground. This technique results in high efficiencies
for animal locomotion [Roberts97].
There are several ways of defining efficiency including:
? Gross efficiency Ratio of work accomplished to energy expended..
Performance Metrics
45
? Net efficiency Ratio of work accomplished to energy expended during rest.
? Work efficiency Ratio of physical work accomplished to energy expended with
no load.
? Delta efficiency Ratio of change in power output to the change in energy
expended at each power output.
These forms of efficiency, when used as design principles or performance evaluation,
can affect decisions of configuration or control. Efficiency is clearly important and is
also used to assess energy use in animal locomotion. A clear difference between
animals and man-made vehicles is that the latter do not typically exhibit the
regenerative properties of the muscle and tendons of animals.
Regeneration is the recapture of expended energy through energy storage mechanisms
such as the tendons in animals or springs in mechanisms. Regeneration has been
explored in robots but, for the most part, regeneration has not been successful. The
additional complexity, volume and mass of regenerative systems has been a break-even
proposition at best [Waldron 97]. Efficiency then, is the objective of minimizing energy
use, rather than regeneration. Regeneration is likely to be an important issue in
electrically driven machines; results in this area will offer tremendous benefits.
Energy 4.2.1
There are many parameters that affect energy usage for vehicles. These include: speed,
forces or torques, terrain, and infra-structure such as computing, cooling or sensing.
Energy over time gives a power measure; this is useful for instantaneous power or
simply integrating energy use over time. Yet it provides no relation to size, weight or
mass of the vehicle nor the distance over which it exerts this power. Simply looking at
power is an ineffectual exercise.
One simple measure then, is the net amount of energy used per distance traveled as
shown in Equation[4-1]. At this point, steady-state power draw is not considered, but
rather the net power draw for locomotion.
[4-1]
This metric, or its reciprocal, is commonly used in assessing energy or fuel
consumption. Common examples include the liters-per-100km or miles-per-gallon
measures used in evaluating fuel consumption for automobiles. However, this metric
doesn?t indicate size, mass, weight, or even payload of the vehicle.
For animals this measure, when normalized for mass, is smaller for larger animals.
Surprisingly, the cost of traveling a unit distance in animals is not proportional to body
mass but to (body mass)2/3 . This appears counter-intuitive since mass increases as the
cube of length and you would expect the energy-per-distance measure to be
proportional to mass. The reason, as pointed out earlier, is that tendons in legged
animals store and return energy to the animal. In general, muscles are more efficient
when they develop force more slowly. This is the case with larger animals, where the
strides are longer and the time in contact with the ground is greater. [Roberts97] [Kram
90].
power
velocity --------------------- energy
distance ------------------- - joules
meter -------------- newtons = ==.Performance Metrics
46
For animals, locomotion energetics are determined by placing the animal on a treadmill
and simultaneously measuring oxygen intake over a range of speeds. Of course, with
animals there is a steady state rate of oxygen consumption for basic metabolic processes
but any incremental intake is due to muscle exertion. In biomechanics, this energy use
is measured in terms of oxygen intake. Such measures include ml O 2 , m 3 O 2 , moles O 2 ,
or calorie equivalents. One such measure of energy use is the Cost of Transportation.
Cost ofTransportation 4.2.1.1
The Cost of Transportation is defined as the mass-specific aerobic power input over the
speed of locomotion [Alexander 92]. In the biological literature, this is expressed in cal
g -1 km -1 . In physical units this is equivalent to energy per distance per mass.A Net Cost
of Transportation is determined by subtracting the steady state energy consumption
from the total metabolic expenditure.
[4-2]
This is also dimensionally equivalent to a measure of force per mass. This measure is
germane to biomechanics studies because, for animal locomotion, exerting force is
more important than doing work [Kram 90]. This is because size differences in energy
cost are proportional to stride frequency at equivalent speeds. That is, smaller animals
have a higher stride frequency at a given speed than larger animals. The contact time
with the ground, when the support force is generated, is the determining factor in the
cost of transportation. Big animals take big steps and the contact time with the ground
is longer. This gives additional time for the muscles to develop force and this requires
less energy. So bigger animals expend less energy per unit mass to move than smaller
animals [Roberts97].
Power to Mass Ratio 4.2.1.2
Another measure, shown in Equation [4-3], is energy per mass per time; this is
equivalent to a power/mass ratio.
[4-3]
This use of energy per mass per time is a measure of power density and is often
examined with respect to speed to give trends and indications of energy use at different
speeds. These types of measures are usually more interesting and useful when using
weight instead of mass. This is because locomotion is typically in a gravity field and the
force exerted on terrain and energy expended is primarily due to gravity. Two such
ratios incorporating weight have interesting physical dimensions: energy-to-weight and
power-to-weight ratios.
Energy toWeight Ratio 4.2.1.3
The dimension of energy to weight ratio is length:
energy
distance ------------------- -?
?
? ?
? ?
mass ------------------------
joules
meter --------------?
?
? ?
? ?
kg ------------------ Cost of Transportation ==
energy
mass ----------------?
?
? ?
? ?
time --------------------
joules
kg -------------- ? ?
second -------------------- watts
kg ------------ = =.Performance Metrics
47
[4-4]
This results in a dimension of distance that is equivalent to how high an energy storage
system can lift its own weight in a 1g field. This gives an intuitive feel for energy
capacity and the value is independent of the size of the system. For example, using
specific energy values of 50Wh/kg for lead acid batteries means they can lift their own
weight 18,000 meters.
Power to Weight Ratio 4.2.1.4
Similarly, the dimension of power to weight ratio is velocity:
[4-5]
This has an interesting physical meaning; it represents how quickly a system could
climb vertically, given a power-to-weight ratio in a 1g field. It is the terminal velocity
the system could achieve if it could devote all of its power to vertical motion against
gravity. In the battery example above, if the battery could provide 500W continuously,
the upward velocity would be 51 m/s for about 6 minutes. For most applications, this
has more pragmatic implications for climbing hills than it does for making vertical
ascents, but it can be used to directly relate power limitations to velocity limitations in
traversing terrain.
Computing Metrics 4.2.2
Another, more general, measure relates locomotion to computing resources; the cost
and complexity of planning and executing movement. A proposed NASA program of
the mid-1980?s involved sending a mobile robot to Mars for exploration. Space-qualified
computers of the time offered little computing power compared to what was
needed for sensing, control and other compute intensive areas. As a result, a computing
metric was proposed to provide a measure and comparison for computing power. The
metric is shown in Equation [4-6] and is in units of millions of instructions per second
required per meter of travel divided by the number of seconds.
[4-6]
This is equivalent to instructions per meter and is a measure of computation required
per distance traveled. It can be used when locomotion is compute bound and not limited
by mechanical or energy issues. It may be a useful measure for comparisons of
navigation techniques; if computing power is at a premium, this metric could justify
certain methods and algorithms over others. However, metrics for these types of
missions continue to be about efficiency of vehicles and not directly with computation
required. Power is typically an even scarcer resource than computing!
For snake robots, at least initially, this type of metric should not be central to evaluation
but if locomotion is limited by sensing and evaluation, it may be useful to revisit this
metric.
energy
weight ---------------- joules
newtons ------------------- meters = =
power
weight --------------- - joules/second
newtons -------------------------------- meters
second ---------------- = =
mips
meter ------------- ? ?
? ?
second ------------------- instructions
meter ---------------------------- =.Performance Metrics
48
Work Metrics 4.2.3
Metrics are also needed for evaluating working machines that move payloads from one
place to another. One such metric is called normalized work, coined by [Binnard 95].
Normalized work, as shown in Equation [4-7] is the product of the payload to mass ratio
and the bodylengths per time. Unfortunately, this has a strange dimension of time -1 and
implicitly favors shorter machines. The metric is really the frequency at which the
vehicle can move a load equal to its own mass, one body length. This metric unfairly
penalizes long machines so that machines such as trains, which are actually quite
efficient at moving enormous payloads, have low normalized work.
[4-7]
4.2.4
Because the dimensions of this metric are peculiar, normalized work is not equivalent
to the physical notion of work. Additionally, the incorporation of length in the metric
favors small machines. Ants can lift more than elephants because strength to weight is
better at smaller scales. But elephants can move a lot more material at higher speed
[Eltringham 91]. Yes, an equivalent mass of ants can move more material, but overall
payload velocity must be evaluated as well. If the intention was to compare working
machines to determine payload ratios and velocities then a better metric would not
incorporate length at all.
I suggest a better metric: the product of velocity and payload to mass ratio as shown in
Equation [4-8]. This assumes horizontal movement of payloads. In this way, a small
robot that weighs half as much but carries half the payload of a large robot, but moves
twice as fast, can move the same amount of material over time. The dimensions of
payload velocity are distance per time but these are normalized for payload.
[4-8]
4.2.5
In this way, if an elephant-mass-equivalent number of ants can carry ten times the
payload per mass but only move at one-tenth the speed of the elephant then the payload
velocity is equivalent. In general, although load carrying increases energy use in all
animals, the cost is relatively higher for smaller animals [Schmidt-Nielsen 84].
In many cases though, payloads are irrelevant, especially in research machines or where
the sensor payload is a small fraction of overall mass. Additionally, in earthmoving
vehicles, such as used in construction, the economics (another pervasive metric!) favor
larger machines especially where large forces are required, where maintenance is
expensive, and where scheduling and traffic control for large numbers of machines is
too complex.
However, the scenario of larger numbers of smaller machine has some advantages. One
advantage is overall uptime; if a few machines fail, the task can be continued, albeit at
reduced productivity. Additionally, the larger numbers of small machines are decoupled
and perform multiple tasks and explore more opportunities. However, these features are
not readily quantified as a metric and are strongly task dependent.
normalized work speed
length -------------- - = payload mass
mass -------------------------------- ?
payload velocity speed = payload mass
mass -------------------------------- ?.Performance Metrics
49
Another example of a comparison might be body lengths per time but this introduces a
scale invariant measure that, again, favors very short vehicles. Clearly, a snake is at a
significant disadvantage with this measure! For further overview of performance
measures for ground vehicles see [Bekker 69].
What does all this mean for snake robots? For serpentine robots, it is unlikely that
delivery cycles and work are defining characteristic. Serpentine robot applications will
mostly involve communications; a transfer of information that is only loosely coupled
to payloads.
Dimensionless Metrics 4.3
As shown, many of these metrics can be reduced to simpler dimensions of time, mass
and distance. These units constitute the dimensions of the metric. However, if a metric
formulation results in no units, it is a dimensionless number. For example, the ratio of
height to width has no dimension.
Dimensionless relationships are important because they can reduce the number of
physical variables in a problem, thus reducing the dimensionality of the design space.
This is because a dimensionless construct can minimize the number of substantial
variables (force, power, velocity, etcetera) necessary to determine or measure. Since the
dimensionless numbers are products or ratios of the physical variables, the construction
of dimensionless metrics always reduces the number of variables in a system. They also
do not require reference to some external standard; e.g. the length of someone?s thumb
or the wavelength of Krypton. Another reason they are important is that they can
provide similitude at different scales. This allows fidelity of comparisons and
predictions that extrapolate from models of these systems.
Dimensionless numbers can also provide insight into a problem and reduce
experimental effort. The elimination of all but the essential variables brings
simplification and the result of this is the clarification or elimination of
interdependencies. These features of dimensionless numbers led to a wide variety of
dimensionless variables in engineering and science. Commonly used dimensionless
numbers include Mach numbers, strain, angular measurement in radians, Reynolds
numbers in fluid flow, and friction coefficients [Ipsen 60].
It is possible to construct a variety of dimensionless metrics from the units of interest
in any problem. One such dimensionless metric that does not utilize time in its
formulation is energy per distance per weight:
[4-9]
This metric is a non-dimensional unit akin to a coefficient of friction. The reciprocal of
Equation [4-9] has been termed the Net Propulsive Efficiency, NPE, for transportation
vehicles whose dimensional formula is ton-miles/gal. This is equivalent to the product
of weight and distance per energy [Rice72].
A similar non-dimensional metric has been used for electric vehicles; the driving
distance per unit of electric energy delivered by the power source. This is equivalent to
energy
distance ------------------- -?
?
? ?
? ?
weight ------------------------ energy
weight distance ?
---------------------------------------- joules
newton meter ?
----------------------------------- ==.
Performance Metrics
50
energy used per unit of vehicle weight and distance and is sometimes shown as Wh per
ton-mile. The lower the number the higher the efficiency [Kalhmmer 95]. This is simply
the reciprocal of the NPE metric cited above.
Many of these dimensionless metrics involve power or energy, distance or velocity and
weight. All of these appear to be useful measures of performance; combining these may
provide a useful and relevant metric.
Specific Resistance 4.3.1
An equivalent dimensionless metric can be formed from the ratio of power to velocity,
P/V, a tractive force represented in Newtons, and the weight of the vehicle. This is
shown in Equation [4-10]. This metric, first proposed by Gabrielli and von K?rm?n in
[Gabrielli 50], is similar to a global friction coefficient and is called the specific tractive
force or specific resistance. The tractive force and weight form a thrust-to-weight ratio.
Specific resistance can be thought of as the inverse of the lift-to-drag ratio; a term used
in aeronautics where ?drag?, rather than simply referring to aerodynamic losses, is a
general term that refers to all energy dissipation mechanisms. Specific resistance then
becomes a measure of the energetic cost of locomotion. The results shown indicate, in
general, that any mode of locomotion has a relatively narrow band of efficient
locomotion and vehicles that are large and move slowly, such as ships and trains, are
most efficient.
[4-10]
Note that Equation [4-10] is the same as Equation [4-9] but with the addition of the time
dimension. For vertical motion, the specific resistance = 1 and for horizontal motion,
assuming frictionless motion with no air resistance, specific resistance = 0. Think of this
in terms of the previously discussed power-to-weight ratio; the power-to-weight ratio is
a velocity corresponding to vertical motion. If this is divided by the system velocity and
the result is 1, then all power is efficiently devoted to vertical motion. For horizontal
motion with no external forces acting against the motion then no power is used and the
specific resistance is also zero. There is also a physical limit to power and speed shown
by the line in the right of the chart. This represents a limit due to aerodynamic or
hydrodynamic drag. That is, there is a minimum value of specific resistance that is
related to the speed. It?s likely that this limit is related to the physical limits of materials.
That is, to reduce weights or increase speeds requires a more efficient use of materials
or stronger materials. It is beyond the scope of this work, but it would be an interesting
exercise to evaluate the specific resistance of vehicles over the past fifty years to see
specific resistance power
weight velocity ? ---------------------------------------- - watts
newton meters
second ---------------- ?
---------------------------------------- ==.Performance Metrics
51
how they improved, or not. Some additional figures of specific resistance for robots
were made in [Gregorio 94].
Although Gabrielli and von K?rm?n used gross weight for their analysis they also
proposed using the useful load or payload in determining the specific resistance. For
both animals and vehicles, the net energy and net weight can be used as an appropriate
metric for specific resistance. Thus, a classification of four types of specific resistance
are possible: net or total weight combined with net or total energy [Hirose 84].
Table 4-1 shows parameters from a variety of walking machines and robots and the
corresponding measure of specific resistance. Two of the best performances, the
Mecant and ASV walking robots, are also plotted in Figure 4-1. Even the best
performing walking machines reveal that a lot of improvement is needed in the design
and control of these mechanisms to approach the performance of other vehicles and
animals.
The claim is often made that walking machines exhibit their mettle best in extreme
terrains and areas in which other vehicles, especially wheeled vehicles, would prove
inadequate at best. However, most walking machines, aside from animals, have yet to
demonstrate all of the supposed benefits of walking efficiently, even in benign terrain.
For smooth terrain and open areas, it is doubtful a robot snake will be as efficient as a
Mecant
ASV
Figure 4-1: Chart of specific resistance for a wide variety of vehicles including
two walking robots. From [Gabrielli 50]
PostScript error (undefined, inary).
Performance Metrics
52
wheeled machine, but for the purposes of self comparison, specific resistance offers a
good measure of relative efficacy of locomotion.
Specific resistance is an attractive measure to use for several reasons. The weight,
obviously, is unchanging and becomes a constant in the calculations. Power is readily
measured in the vehicle and can be determined in a straightforward manner from
simulation as well. Velocity, determined by tracking the center of mass of the vehicle,
can also be calculated. It is a metric that takes both energy and distance into account,
thus providing a simple and effective measure of progress for comparison purposes.
In examining the graph in Figure 4-1, one clear trend across most vehicles is that the
specific resistance goes up with velocity. This is not too surprising since energy
consumption is related to the increased aerodynamic and hydrodynamic forces required
to move faster. This also means that, for a given vehicle, if a low specific resistance is
desirable, then this corresponds to low velocities. Thus, if the metric for a given vehicle
is to minimize the specific resistance, then a low velocity is the result. For the serpentine
robot, however, the effects of aerodynamics or hydrodynamics do not play a role in
power consumption, so that velocity effects will be minimized.
Table 4-1: Specific resistance for a variety of walking robots. Some data from [Wettergreen 96].
It?s tempting, but misleading, to compare specific resistances across very different
systems. The applications, tasks, missions and environments are too different for any
meaningful comparisons between all the vehicles shown in Figure 4-1 or Table 4-1.
Some of the robots were designed to study a particular configuration or control method
and not to minimize energy use or maximize payload. In addition, some vehicles are
configured for specific environments. However, for a particular vehicle in a given
environment, specific resistance is a useful measure of how well the vehicle is
performing.
Robot Speed
[m/s]
Mass
[kg]
Power
[W]
P/WV
[1]
ARL Monopod 1.0 18 60 0.34
Mecant 0.5 1050 3500 0.68
ASV 1.0 3200 26000 0.83
Ambler 0.016 2700 1900 4.49
PV II 0.02 10 10 5.10
Aquabot 0.03 23 500 7.39
TUM 0.3 23 500 7.39
Dante I 0.02 725 1500 10.56
Dante II 0.01 770 1000 13.25
Melwalk 0.01 35 80 23.32.Performance Metrics
53
For a serpentine robot, specific resistance also offers a measure that is straightforward
to calculate. Power consumption can be determined both in simulation and in the
vehicle, weight is invariant and velocity can be tracked easily as well. The
dimensionless quality is nice in that it provides a measure against other vehicles, but
this is not the focus of the metric. Comparing the robot?s performance against itself is
the only key criteria.
Scale 4.4
Is it better to be big or small? The motions, shape and structure of animals and
mechanisms depends upon size or scale. The answer for robots might depend on the
task but there are advantages and disadvantages to different scales. In general, large
systems use less energy per distance per mass than small systems. In animals for
instance, this expression is proportional to body mass -1/3 and this relationship holds for
insects, reptiles and mammals. Thus, as shown earlier, the energy cost of traveling a unit
distance is proportional to body mass 2/3 . Energy use does increase linearly with speed,
but for smaller animals the rate is steeper than for large animals [Alexander 92]
[Schmidt-Nielsen 84].
There are also issues of scale related to geometric and dynamic similarity. Shapes are
geometrically similar if scaled by uniform factors of length. Two motions are
dynamically similar if they can be made identical by uniform changes of the scales of
0.1 1
Velocity (m/s)
1
10
100
Specific Resistance
Melwalk
Dante II
Dante I
Aquabot
PV II
Ambler
TUM
Mecant ASV
ARL Monopod
Figure 4-2: Plot of data from Table 4-1 of specific resistance versus velocity..Performance Metrics
54
length, time and force. The metric of comparison often used for scale comparison is the
Froude number, a scale-invariant measure that is often used to compare the dynamics
of vehicles [Alexander 92].
The Froude number, shown in Equation [4-11], is a dimensionless number that is the
ratio of inertial to gravitational forces.
[4-11]
Alexander showed that leg length in animals could be used in the Froude number
formulation to compare scale. In walking and running, different animals run in
dynamically similar fashions at speeds only where their Froude numbers are equivalent.
For robots, the story is similar, but Froude numbers of robots tend to be very low
compared to animals.
Wettergreen and Full provide an evaluation of walking robots using Froude numbers but
concluded that the Froude number could not be used to compare robots because the
robots were too dissimilar. The performance of the robots versus animals was quite
poor. The exception, Raibert?s running robots, are the only ones to approach the Froude
numbers of animals. Primarily this is because most of the robots compared were
statically stable and incapable of running whereas Raibert?s hoppers were quite
dynamic [Wettergreen 96]. Perhaps another reason is that all power and computing for
the running machines was offboard the robot. The animals do better primarily due to
energy recovery in muscles and tendons which allows higher speeds.
One issue with Froude numbers is that they don?t reveal the suitable optimization
criteria that result in different gaits. In [Alexander 84], it?s postulated that two primary
criteria are maximizing stability and minimizing energy consumption for different
animals. Clearly, in biped and quadruped animals, gaits are a function of speed and
minimize energy expenditures for that given speed, whereas serpentine gaits appear to
be a function of environment.
An obvious problem with the formulation of the Froude number is that it depends on a
measure or dimension of length. For walking robots, this could be the length of legs,
strides, or steps. The problem for limbless robots is determining what the analogous
height or length dimensions should be. Simply growing the length of a snake robot
might drastically affect the dimensionless value without changing the velocity. Hence
the Froude number for a snake is contrived.
Since most serpentine gaits exhibit characteristic patterns or waves, wavelength might
make physical sense for limbless robots. In the example below, lambda is the
characteristic wavelength of the body of the robot [McMahon 96].
[4-12]
The problem with this, as shown in Background, is that snake gait selection and
progression do not appear to relate directly to undulating frequency. As demonstrated
Froude number speed 2
g length ?
---------------------------- =
Froude number speed 2
gl
---------------- =.Performance Metrics
55
in [Secor 92], if the mean frequency and forward speeds are the same for different gaits,
then the mean distance travelled per cycle must also be equal. The problem was that the
energy use difference between gaits was significantly different. Thus, the Froude metric
does not show the difference in its evaluation.
Another measure which may be appropriate for undulating vehicles is amplitude per
wavelength. It is not clear, however, if this is to be maximized, minimized or made to
approach unity!
Velocity effects from scaling can produce other effects. At higher velocities the effects
of air resistance are significant. However, air resistance is much worse at smaller scales
as a percentage of energy output if velocity is constant with size. This is because surface
and cross-sectional area increase more slowly than volume (mass).
Surface area increases as the square of the size whereas volume and mass increase as
the cube. Thus, wind-drag, which is dependent on surface area, is proportionally
smaller for a heavier and larger object than a smaller one of similar shape and
composition. An everyday example is that of falling dust and rocks. They are the same
shape and composition and differ only in size, yet the dust settles much more slowly
than the rocks.
For similar reasons, small things tend to do more work against friction because surface
friction effects tend to be proportional to area; the ratio of viscous forces to inertial
forces increases as size decreases. This has a severe damping effect on very small
vehicles. In some ways, however, small scale can assist the designer. Small things have
better strength-to-weight ratios and may require proportionately less structural mass.
Air resistance does not affect the serpentine robots considered here, and issues with
very small size, other than strength to weight ratio, are not relevant for this system.
Small animals consume more energy to carry a unit mass a given distance than larger
animals. Although energy use is proportional to speed, energy use is also lower per unit
mass for larger animals; smaller animals use relatively more energy to move a load over
a given distance. An excellent discussion of this and other issues of scale can be found
in [McMahon 83].
Summary and Selection 4.5
Metrics have been only cursorily examined in many papers, especially for robot
performance evaluation. This discussion was necessary to set the stage for a final
selection of a metric and provide a comprehensive background on the subject.
I selected specific resistance as the measure of performance for learning locomotion. It
provides a notion both of energy, time and weight of the robot. It utilizes two
measurements that can be found from both the physical model and from simulation. It
is easily and quickly calculated and provides a clean and understandable metric for
evaluation during the learning process.
Whatever the particular metric value, it is not a good idea to draw too many conclusions
or provide close comparisons to other robots. It?s too easy to contrive a metric that
favors a particular robot. It?s also too easy to draw conclusions about vehicles that don?t
take environment and task into account..Performance Metrics
56
However, it is important to realize that metrics reveal only how well a vehicle did on a
particular performance measure. It does not reveal why, although it can provide clues,
and, finally, it does not directly reveal how to make the performance better. It can be
used as a tool to ascribe trends through the use of small changes in the control
techniques and hence, develop a better understanding of what makes a better gait. The
metric developed is used as part of the learning process and placed into the overall
framework for teach the snake robot to locomote..57
Chapter 5
Learning and Optimization
Machine learning techniques evaluate past data to form insights on future performance;
learning provides improved performance through experience. Learning and
Optimization examines learning locomotion for simulated mechanisms and actual
robots as well as criteria and structures for learning. This section also looks at the
critical area of parameter representation. While the technique selection is important,
knowing what to optimize, and knowing how to evaluate a solution are even more
critical. What to Optimize is a metric, specific resistance. How to Evaluate is
determining this from simulation or measuring from the robot. With the metric and the
learning method combined with physical modeling a complete framework can be
constructed.
Optimization Techniques 5.1
A wide variety of search methods have been developed to find solutions to problems in
mathematics and computer science. A few techniques are widely used and work well in
small domains but bog down to intractable levels for large high-dimensional search
spaces. I examined and tested a number of these techniques in the course of this
research.
Random Search 5.1.1
Random search is unsuited for all but the smallest problems. The technique simply
chooses parameters at random and evaluates the metric by simply keeping the highest
value around. If the search space is not large, then this is a simple, albeit not efficient,
strategy.
Hillclimbing 5.1.2
Hillclimbing, a simple and sometimes very effective method, simply looks at the local
gradient and moves ?uphill? in the direction of better solutions. Hillclimbing techniques.Learning and Optimization
58
may take a random parameter, but use adjacent points in the space and evaluate the
gradient to find the direction of higher scoring evaluations. Hill climbing is at the root
of many successful techniques which use more advanced strategies including steepest
ascent or conjugant gradient techniques which better optimize the direction of
subsequent search. While a great deal of effort in research is focused on pther search
techniques, often these more straightforward techniques still prove very successful.
A general issue with hillclimbing techniques is that the search tends to find the nearest
and highest hill; the local maximum. Peaks that are higher still are not found because
they lie across valley floors and other lower elevation areas. Finding a global optimum
then requires new techniques or a combination of techniques.
Simulated Annealing 5.1.3
Based on an analogy to the slow cooling of alloys, simulated annealing is similar to hill
climbing with an added ?earthquake? component. By adding energy to the values, the
current search areas can be jumped or bounced to other search areas. It?s a stochastic
technique for finding near globally minimum cost solutions to large optimization
problems. Simulated annealing is sometimes much better at avoiding local minima that
may trap pure hill-climbing. However, there are limits to the cost function and
simulated annealing can be very computationally intensive for finding satisfactory
optimal results for many problems.
Neural Nets 5.1.4
Neural nets utilize a network of units, each of which has inputs and a simple weighting
function to provide an output to other layers within the network. Neural networks work
well at providing classification and regression functions but typically require a ?master?
or training set that is used to initially provide the weights within the network.
Optimization using a training set establishes the weights of a number of interacting
?units? within the neural net. In this case, the lack of training sets is a formidable
obstacle. It is possible that the neural network can provide an architecture, however, for
a local distributed control scheme for the snake wherein local joints listen and respond
to adjacent joints.
Response Surface Methods 5.1.5
A response surface method (RSM) is a graphical representation of a relationship
between some simple-to-evaluate metric such as yield and a large number of variables.
Typically, you wish to find the values of the variables to maximize the metric. It differs
from ordinary optimization techniques because gradients often cannot be found, it is an
unknown function and the measurements of the function are typically noisy.
Response surface methods can provide good experiment design plans and good
statistical analysis but the underlying structure relies on linear multivariate regression
analysis. This type of analysis can behave badly when non-linearities are present. It is
also typically used where there are slight variations in parameters rather than large
changes. This makes it well suited for industrial process control but difficult to apply to
applications where widely varying parameters are likely to be searched. Typically, RSM
is used in conjunction with human judgement as well as statistical analysis [Box 87]..
Learning and Optimization
59
Genetic Algorithms 5.1.6
Genetic Algorithms (GAs) use a stochastic process to enable the testing and evaluation
of many individuals over time. Genetic algorithms are a guided random selection
process that utilize the following ideas:
? Selection and coding of parameters
Parameters are selected and represented with binary strings at a sufficient resolution so
that rapid changes of the output do not occur with small changes in parameter values.
? Search with population of points or ?individuals.? This is accomplished by generating
multiple sets of parameters.
? Objective functions to evaluate each individual. The evaluation of each individual is
carried out and a fitness value is returned.
? Probabilistic transition rules. These are used to cull or glean individuals that result in
high evaluations and uses them to generate the next generation of individuals. This
process is repeated and eventually converges to a solution, but not necessarily a global
one.
Table 5-1: Sample set of parameters for GA used for caterpillar robot.
Table 5-1 shows a sample genome and parameter encoding for a three parameter
system. The genome values show the size of the bitstring and not the actual value used.
I implemented a series of tests using GA?s to control a 2D caterpillar robot. For the GA
tests I instantiated a number of parameters including population size, number of
generations, mutation, and crossover. For the program execution, I used tests of 100
generations and populations of 30.
The following example, in pseudo-code, shows a sample form of the objective function:
float Objective(GAGenome& c)
{
write parameter file (?parameters?);
system call (?snakes -t 5 -nodisp?);
// read and return the metric result
}
The parameters represent a specific set of values generated by the learning technique.
These values are passed through a file to the program, snakes, which creates and runs
the simulation. The simulation runs for a fixed amount of time and the evaluation metric
is then written out to a file. This occurs for each and every set of parameters that are
tested in the learning algorithm. This metric value is read and then returned to the
learning function.
parameter magnitude frequency phase
genome 1111111111 1111111111 111111111
values 0-10 0-100 0-10
resolution ~0.01 ~0.1 ~0.01.Learning and Optimization
60
Although writing files is not as efficient as passing information through other means,
such as sockets, it makes for a method that is easy to implement, maintain and
understand. The system call runs another program which, in turn, calls the physical
simulation program and runs it without a display and for a specified period of time.
When snakes exits, it writes a file that contains the value of the metric. This value is
then read by the program and used in the regeneration of the parameter statistics. The
overhead in writing these tiny files, less than 1K, is quite small compared to the
overhead of running the simulation itself.
PBIL 5.1.7
Baluja, in [Baluja 95a], introduced Probabilistic-Based Incremental Learning (PBIL) as
a means to provide the same functionality as other stochastic methods such as GAs but
in a more efficient manner. Rather than implicitly maintaining statistics within a
population, as GA?s do, PBIL explicitly maintains the statistics for the genome.
In this pseudo-code example derived from [Baluja 95a], the program shows the sequence
of generation, update, and evaluation.
// initialize probability vector
for i = 1 to LENGTH do P[i] = 0.5;
while (NOT termination_condition) {
// generate samples
for i = 1 to SAMPLES do
sample_vectors[i] = generate_sample_vector(P);
evaluations[i] = evaluate_solution(sample_vectors[i]);
Generations
Metric
Evaluation
Figure 5-1: Example of GA convergence to maximum value of metric.
X Graph
output1.dat
Y
X 0.00
0.50
1.00
1.50
2.00
2.50
0.00 100.00 200.00.Learning and Optimization
61
find_best_vector(sample_vectors, evaluations);
// update probability vector towards best solution
for i = 1 to LENGTH do
P[i] = P[i] * (1-LR) + (best_vector[i] * LR);
// mutate probability vector
if (random(0,1) <MUT_PROB then
if (random(0,1) > 0.5 then mutate_dir = 1;
else mutate_dir = 0;
P[i] = P[i] * (1-MUT_SHIFT) + (mutate_dir * MUT_SHIFT);
}
User defined constants (example values)
SAMPLE: number of vectors generated before new update (100)
LR: Learning rate, how fast to exploit search (0.1)
LENGTH: number of bits used in generated vector (30)
MUT_PROBABILTY: Probability of a mutation occurring in each
position. (0.02)
MUT_SHIFT: amount a mutation can alter a value in the bit
position.(0.05)
This encoding of the solutions as statistics can, in many cases, be far more efficient than
traditional GA methods. The executions are more efficient, use fewer cycles, and
converge more quickly. The complete coding, as shown in this example, is about all that
is required for implementation. find_best_vector, for example, in my
implementation is not written as a separate function but simply integrated into the
evaluation loop. Thus, PBIL provides similar, if not better, performance than GA?s with
lower overhead.
Representation 5.2
Gaits or sequences of motion can be represented in a variety of ways and by several
criteria, described below, can be used to evaluate different representation techniques.
The desired result is a single representation that, with carefully selected or generated
values for the parameters, represent all the forms of snake locomotion that are periodic
in nature. This includes sidewinding, lateral undulation, rectilinear and concertina.
Although the problem is of greater dimension than a landscape, imagine a varying
landscape that represents the solution space for gait generation. The peaks and valleys,
by analogy, represent the good and bad metrics that result from parameter values. The
task is to find peaks and ridges where high values correspond to good gaits. The task is
to peer through the fog that covers the landscape and discern, not only the high peaks
and ranges, but patterns and structure. Trends may make it possible to classify regions
and areas into styles of gait.
However, even given a means of learning, there still remains the problem of selecting a
way of representing the information presented to the technique. The general problem is
to represent a wide variety of progressing waveforms that provide a systematic
displacement of the robot. The waveform parameters are adjusted during the learning
process to provide efficient motions of the snake joints and thus provide locomotion.
There are several competing issues for any representation and these include:.
Learning and Optimization
62
? Compactness - the ability to represent the most information possible in the most
concise manner.
? Calculation - how much overhead does the representation require?
? Complexity - how involved is the creation, debugging and evaluation of the
representation? This is really an implementation issue.
? Comprehension - How easy is it to interpret and understand the results? This is
different from complexity although it can be related.
? Correspondence - How easily is the information mapped from representation in
learning to parameter values for simulation and control?
In this section we?ll investigate several means of representing the parameters for
learning locomotion gaits.
Trigonometric forms 5.2.1
In the first set of tests on the caterpillar, a simple trigonometric sine function was used
for representing the traveling wave on the robot. This form has only three parameters:
magnitude, frequency and phase. Magnitude and frequency are obvious and phase
represents the shift of the waveform along the body. An additional parameter, offset,
can be used to provide a nominal starting configuration. For example, the body may
form a helix which is deformed to provide locomotion. Sinusoidal patterns are easy to
represent and parameterize. The problem is they appear too restrictive in representing
arbitrary time-varying waveforms.
Fourier 5.2.2
Ideally, a relatively simple function like a trigonometric function or a combination of
such functions is both simple to represent and easy to parameterize. However, a counter
example that can not easily be represented in this manner is shown in [Hirose 93] for
lateral undulation; the serpenoid function that he derived represents only lateral
undulation. The serpenoid curve is derived by assuming that curvatures vary in a
uniform fashion along the length of the curve defined by the snake. The curvature of a
sine function, however, does not smoothly vary along the curve. This derived curve, the
serpenoid curve, was compared to that of natural snakes performing lateral undulation
and shown to closely approximate their actual motions.
Hirose?s derived formulation uses Bessel functions which are the results of an integral
that cannot be expressed in a simple closed form. However, it is possible to use a Fourier.
Learning and Optimization
63
series with parameters that represent coefficients of the individual terms up to some
cutoff frequency (spatial frequency).
The Fourier series which can represent any periodic function as a sum of exponential
terms, usually sine curves. Figure 5-2 shows an example end result where the angular
values of the angle between links are a function of time and link. Since all gaits, by
definition, are periodic sequences, the Fourier series can represent them. In Figure 5-3
the mapping of the time and joints to the coefficients of the Fourier series which are
magnitude and frequency values. This table of values, in turn, can be represented in a
binary string which facilities operation within the learning framework. The values in the
string can be the coefficients of the values in the array.
Parametric curves 5.2.3
Alternative representations are the parametric forms shown in [Chirikjian 92] or the 3D
splines used in graphics. Linear polynomials in R 3 are another, but the problem is the
number of coefficients, at least 20, that need to be represented as well as the sequence.
Figure 5-2: A representation of time and joint link versus angular value.
time
joints
frequency
magnitude
binary representation
Figure 5-3: Mapping from time varying representation to fourier representation to genome..
Learning and Optimization
64
Higher order polynomials also require a relatively large number of parameters and
representing all time varying sequences can be an issue for learning and evaluating.
Wavelets 5.2.4
Another means to represent forms in a concise manner are wavelets which, unlike
Fourier series, can be used to represent non-periodic time-varying functions.
Complicating the use of wavelets are the selection of the mother wavelet and a wide
variety of choices for the representation use. Since gaits are periodic in nature, there
doesn?t seem to be particular advantages of the wavelet approach, although
representation of gait transistions and terrain accomodation may be more easily
managed in a wavelet framework.
Tables and Masks 5.2.5
Rather than trying to explicitly represent all manner of waveforms or trying to identify
parameters to refine for different modes of locomotion, a conceptually simpler route is
to directly represent the joint angles over time. By explicitly representing the angular
position of the links within a one-dimensional tape, the joints can be essentially
?masked? off and the snake shifted through the tape, adjusting the positions to reflect
the tape values. By adjusting the set of tape values in physical simulation and looking
for effective modes, a variety of locomotion modes can be represented, including those
that are not exhibited in natural snakes. This representation then forces the time-histories
of the individual joints to be identical but only shifted in time.
A more general extension of this method is to represent the joint angles in a column of
a 2-dimensional array and, with each timestep, march across the columns where the
column entries represent time steps for a particular waveform. This should work well
for periodic forms, but also for forms such as concertina where the time history of
different joints is not the same. The array becomes the representation of each joint angle
or, even more concisely, the difference between the angles in adjacent time slices.
The values in the table can also be constrained to represent the likely sequences. This
constrains movement between slices and even joints. This has a two-fold benefit: first
a 1 a 2 a 7 a 8 a 5 a 4 a 3 a 6 a 9 a 12 a n a 11 a 10 ...
Figure 5-4: A snake ?tape? defining joint angles at each time step for a periodic waveform..
Learning and Optimization
65
it prevents abrupt jumps and it culls unlikely gait patterns and body contortions, and
finally, it reduces the gait space significantly.
Another issue is the size of the array. Using some rough numbers, if a given gait
sequence takes 2 seconds before repeating and the robot has 20 DOF, then using a 10Hz
update rate gives 400 parameters. Each of the parameters might use a full 8 bits and if
differences are used, 2-3 bits. This gives a a total of a few thousand total bits, which
creates a very large set of parameters to adjust. However, while the space of
configurations described in this model is large, the representation is simple, easily
understood and easily mapped to the robot. Another possibility is to operate on number
values themselves rather than their binary representation.
The time to cross the table is directly related to the frequency of a gait. For legged
animals the stride frequency is inversely related to the square root of the leg length, so
that even for long legged animals the stride frequency is typically less than a second or
so. Leg length is also proportional to the cube root of the mass, so that the stride
frequency can be shown to be proportional to the sixth root of mass [Alexander 92]. If this
proportion holds for snakes, then the stride (undulating) frequency appears to be less
than two to three seconds even for the largest, presumably slower, snakes. Thus, the
time across the table can be up to two seconds and the partitioning should be on the
order of 8 or 16 numbers to capture the variations in the joints during that time.
Filling in the table 5.2.5.1
Given the tabular representation, the next issue is generating values. The most straight-
forward way might appear to simply fill in the table with random values and insure that
the changes across the rows and columns are less than or equal to some threshold. For
example, although this is straightforward to implement, the problem with this method
a 11 a 12 ... a 1M a 15 a 14 a 13 a 16
a 21 a 22 ... a 2M a 25 a 24 a 23 a 26
a 31 a 32 ... a 3M a 35 a 34 a 33 a 36
a 41 a 42 ... a 4M a 45 a 44 a 43 a 46
... ... ... ... ... ... ... ...
a N1 a N2 ... a NM a N5 a N4 a N3 a N6
Figure 5-5: A 2D array can be used to represent all joint motions over time..Learning and Optimization
66
is that, since the table wraps around, the random walk distance is effectively cut in half.
This well-known result is that the distance of a random walk is approximately the
square root of the number of time steps. Thus, for 32 total time steps, the random walk
distance is sqrt(32/2) or four times the change that is allowed from step to step. So
unless very large steps are allowed or the array is made quite large this method does not
do well for describing large motions of the joints.
The irony is that the constraint should keep the changes manageable but to also allow
significant overall change in the positions of the joints. As just shown, the random walk
doesn?t accomplish this requirement.
To reduce the universe of search spaces it is possible to ?seed? the table with mediocre
hand-tuned locomotion modes and by iterating between the simulation and physical
simulation to arrive at efficient modes of locomoting. The result is likely to converge to
a local maximum.
Another way is to use a fourier series, in a different manner, to represent the angle in
the following manner: since a fourier series uses magnitude and phase and relates them
to frequency, it is possible to define both to create a waveform with the right properties.
In this case, of course, it the spatial frequency of the joint angles. The magnitude
function can be defined as simply as 1/f, the power spectrum for noise. This creates a
rolling cutoff to prevent high spatial frequencies where the joint angles change abruptly.
The phase is then seeded with random values and the coefficients are calculated and
then the series is created.
The magnitude function can be tuned to effect a good sweep of values across the
positions. Several tests resulted in only a small modification of the magnitude function
to provide an earlier cutoff.
It isn?t necessary or even desirable to create an array that defines all timesteps in the
process. Too many and the search space explodes and convergence, even if possible,
takes an inordinate amount of time. Too few and the coarseness results in ineffective
gaits.
Most natural serpentine gaits appear to take on the order of 2-3 seconds. By partitioning
this time into about 10 slices/second gives 20-30 numbers to describe the gait over time.
For these reasons and pragmatic coding reasons I chose 32 slices for the array. One
result of the fourier seeding is shown in Figure 5-7. Again, this is one possible gait
sequence generated by the fourier technique.
Phase
Magnitude
Frequency
a n = M n cosP n
b n = M n sin P n
F ( x) a n cos(nx) b n sin(nx) +
n 1 =
length
? =
M
P
Figure 5-6: A fourier series is used to generate joint positions for tabular method..Learning and Optimization
67
The landscape of gaits in this representation is quite extensive and the process of
determining good gaits in this landscape is to identify the peaks and ranges that
correspond to demonstrated efficacy in simulation and in the robot.
Summary 5.3
Learning provides improved performance with frequent testing and evaluation. After
examining and testing a number of the learning methods described here, I selected a
stochastic technique, genetic algorithms or GAs, and, later, probabilistic-based
incremental learning or PBIL. PBIL?s compactness, ease-of-implementation,
effectiveness and simplicity made it a good choice for evaluating metrics in physical
simulation.
The representation chosen for the framework is the tabular array with entries that
directly represent desired positions of body segments. This representation is the most
general and does not depend on combinations of mathematical functions to attempt all
possible configurations; it represents those configurations directly. The danger in the
tabular approach is that it opens the search space further, but the generality appears
worth the risk. The trigonometric approach to reprensentation turns out to be
surprisingly powerful one and this was used for a number of gait applications as well.
The next step in the process integrates the learning technique and learning
representation into the framework.
Figure 5-7: Table generated from the fourier technique; columns represent
joints angles and rows represent time history..Learning and Optimization
68.69
Chapter 6
Implementation
Implementation details the physical configuration of the serpentine robot. Configuration
includes actuation selection, morphology, and design of the mechanics and electronics
as well as the experimental setup and physical modeling. For each area alternatives are
presented that were explored and evaluated, as well as the final selection and form of
the serpentine robot.
There is a cyclic design process at the core of the framework described here; the use of
simulation can assist the design process. Designs that do not work in simulation are
unlikely to work in the real world. Hence, an analysis of form and its effect on
locomotion is productive and useful. Similarly, the choice of actuation technology can
set constraints on the robot design.
As a result, actuation technologies were closely examined and a short summary of that
evaluation is provided here. The final selection, small off-the shelf-servo actuators, are
analyzed to provide a good model for simulation. An important geometric analysis
reveals that the angular excursions of joints can be small and that robot link aspect
ratios, the relationship of length to width should also be small.
The final mechanism, is a lightweight assembly of hardware, bracketry and servos
connected by a very small wiring harness to provide control signals from a set of small
controller boards. Skins, an aspect of snake robots that has been ignored is also
examined and candidate materials were identified. The final assembly of the snake
robot provides a highly articulated twenty degree of freedom machine with integral
controllers and control bus.
Actuation 6.1
To create motions and sequences of body shapes requires devices that move or actuate.
There are many technologies that are capable of creating motion but there are also many.
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other issues involved in the selection process. These include fidelity, response, power,
speed, torque, required infrastructure, etcetera. The selection of actuation technology
also directly affects the configuration and control and, as a result, the actuator selection
process is critical and integral to the design process. A wide variety of actuation
technologies were closely examined and evaluated in the course of this work.
Some of these technologies were initially examined with the intent of using scaled
snake vertebrae in a robotic mechanism. I performed a high density scan of a vertebrae
in a Magnetic Resonance Imaging (MRI) machine at a hospital and then constructed a
3D model of this data. The model is shown, along with the actual vertebrae, in Figure
6-1 [Carnegie 95]. The intention was to carry this into a rapidly prototyped model using
stereolithography and then use muscle-like actuators for providing motions of linked
vertebrae bodies. The problem, as we will see, was the immaturity of the technologies
that provide the muscle-like action.
Several areas of actuation were examined in detail and the following section on
actuation technologies briefly summarizes each area.
Actuation Technologies 6.1.1
Polymer Gels 6.1.1.1
Polymers are groups of small molecules that form long chains that are essentially
repeated units of the smaller groups. They exhibit a wide range of properties and most
of the synthetic materials used in our daily lives are polymer-based. Some polymers are
capable of converting chemical energy to mechanical work in isothermal conditions.
These polymers significantly change their length in response to chemical changes
involving altered temperature, pH, or applied electric fields. Volume changes can be as
high as a factor of 1000. Gel polymer networks are a balance of such properties as
rubber elasticity, polymer-polymer affinity and hydrogen ion pressures and changing
the balance of these determines the volume change.
For polymer gels to be useful there are many technical issues to resolve. There are
issues of strength, response, stress-strain relations, fatigue life, thermal and electrical
conductivity. Other issues include efficiency, power and force densities and power
Figure 6-1: Northern Anaconda vertebrae and 3D model constructed from MRI data..
Implementation
71
limits. Finally there remain engineering considerations of supply and delivery of power,
construction, manufacturing, and modeling of these actuators [Brock 91][Caldwell 89].
Shape Memory Alloys 6.1.1.2
Nickel-titanium alloys and their useful properties were discovered by the Naval
Ordinance Laboratory decades ago and the material was termed NiTiNOL. These
materials have the intriguing property that they provide actuation by means of current
cycling through the materials. The alloy undergoes a reversible phase change exhibited
as force and motion in the wire. At room temperature, nitinol wires can be easily
stretched by a small force. However, when conducting an electric current, the wire heats
and changes to a harder form that returns to the unstretched shape; the wire shortens in
length with a usable amount of force.
Nitinol can be stretched by up to eight percent of their length and will recover fully, but
only for a few cycles. However, when used at smaller strains, such as three to five
percent, nitinol wires can run for millions of cycles with very consistent and reliable
performance. Strain is much higher than piezos; as high as 8% with corresponding
forces as high as 600N/mm 2 .
However, the response time is contingent on heat removal and is relatively slow. As a
result the efficiency is very low, as is the stiffness. Typical efficiencies of Nitinol
materials are on the order of 5-6%. Hysteresis is a problem and fatigue life is relatively
low. Additionally, the generated heat can be an issue in many applications.
A wide variety of grippers, manipulators, dextrous hands and even small swimming
devices utilize SMA materials. For practical robot applications however, there is little
beyond a few demonstration devices such as small walkers and heat engines [Dario
89][Golestaneh 84].
Piezoelectric Devices 6.1.1.3
Certain classes of crystal materials change length with applied electric potential.
Conversely, they can produce electric fields when put under pressure. These materials
are termed piezoelectric or PE and can produce high forces at good efficiencies.
The difficulty for some applications is that the motions produced are extremely small;
The strains are on the order of 1% or less. Advantages of piezoelectrics include the
ability to control small (sub-micron) displacements with applied voltages, very high
stiffness and very fast response. Loads into the hundreds or thousands of Newtons are
easy to achieve. They are very stiff as well; the modulus of elasticity, E, can be up to
100 GPa. As a comparison, Steel is about 200GPa and Aluminum is about 70GPa.
Disadvantages of piezoelectrics include very small displacements; 30ppm is typical. An
additional concern for piezos is that high electric fields can cause breakdown and failure
Another general problem is non-linear response and high hysteresis and creep [Petrucci
94].
Electrostriction Devices 6.1.1.4
Unlike PE?s electrostrictive crystals are symmetric. The electrostrictive strain is
proportional to square of electric field. This property is independent of piezoelectric
effect and is due to rotation of polar domains in ceramic through the field..
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72
In general, linearity and hysteresis are better than PEs and lower voltages are used.
Movement ranges to 105 microns are available in commercial products but the ratio of
length to change in length is still low and is on the order of 0.1%.
The motion has low hysteresis and very small thermal expansion coefficient and the
non-linearity can be overcome by operating at a bias voltage. Most electrostriction
material properties are similar to piezoelectrics. A number of commercial
electrostrictive devices are now offered. The differences from PE?s offer advantages in
some applications when there are issues with the high voltage and hysteresis associated
with PEs.
The combination of high-voltages, small strain, similar to PEs make these difficult for
a large actuator implementation. PEs may be the technology of choice for very small
scales however.
Magnetostriction 6.1.1.5
Magnetostriction is the mechanical deformation of a ferromagnetic material when
subjected to a uniform magnetic field. Unlike piezoelectrics, the displacement per unit
field actually increases with length. Internal stresses in the material due to anisotropy
energy are required to magnetize it in certain directions relative to the crystal axes and
vice versa. The strains and displacements can be significantly more than piezoelectrics
but piezoelectrics can be stacked to give nearly the same stroke per length. Terfenol-D,
used in several magnetostrictive commercial products, offers high forces and good
strain [Dyberg 86]
MEMS 6.1.1.6
Micro-Electrical Mechanical Systems are a relatively recent development by which
fabrication techniques, normally associated with integrated circuit design, are used to
build mechanical structures that can be moved and controlled. The field is rapidly
developing and already micro robots on the mm scale demonstrate motion control,
mobility and sensing capabilities. As of this writing however, no techniques for
coupling the microscopic motions directly with macroscopic motions has been
achieved.
The physical principle used in most MEMS actuation is electrostatics. This is based on
the force on electrons in an electric field. Any two electrodes separated by an insulating
barrier will be attracted to each other and the force is related to the square of field.
Whereas electromagnet forces depend on volume of magnet present, electrostatic
forces become significant at small gaps. Traditional-sized motors using this principle
result in very low forces and torques. But at very small scales, electromagnetic motors
become very inefficient. Most of the power is consumed as heat and the torque is very
low.
Below 1mm though, electrostatics looks very promising. The charge on a particle is
large compared to particle volume if particle is small. However, there are issues of
temperature and humidity suggesting use of dielectrics other than air. Electrostatic
techniques can work well in vacuum too. Even though the force change is nonlinear it
is a very exact relationship extending over several orders of magnitude..
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Recent work by Flynn, has identified a gap in actuator technology between the MEMS
technologies and traditional motor technologies. This gap is roughly between 100um
and 1mm. Flynn identified PE wave motors as one potential technology to fill this gap
in actuation technology [Flynn 92][Stix 92][Congress 91].
Thermal Actuators 6.1.1.7
For decades, thermostats in automotive cooling systems utilize the expansion of
materials as a means to actuate valves. A typical automotive thermostat uses a paraffin
or wax actuator to open coolant flow to the radiator. After reaching a set temperature
the wax undergoes a phase transition, i.e. it melts. The wax expands as it melts in a
small confined chamber and squeezes a rubber boot that pushes out a small piston. This
form of actuation has high force, long lifetime and it is very reliable. Response time
depends on power input but can take many seconds. The transition temperature can be
set to any value simply by changing the wax mixture.
Spacecraft use thermal actuators and many reliable designs have been built [Starsys 95].
A new development in this form of actuation is the use of thermopolymers as the phase-transition
material. The key attributes of the new material are the rapid response and
cycle times.
As with all thermal actuators the removal of heat is a critical issue. Based on
calculations from published specs, such actuators are about 5% or less efficient. This is
low, but not lower than other types of actuation such as shape memory alloys [Tcam
95][Schneider 91][Schneider 93].
Electro-magnetic Motors 6.1.1.8
The interaction of magnetic fields and current-carrying conductors produces a force
which is harnessed in the form of a motor. The design of motors is relatively mature
compared to the other technologies discussed here. The principles have not changed in
over a century but better magnetic materials, better tolerances and improved control
have resulted in the continuing evolution of small, high performance motors coupled to
efficient drivetrains. Thermal considerations and magnetic field densities appear to limit
motor technology at this point in time, but continued improvements in power density
and the advent of superconductive motors will further performance and design.
Actuator Selection 6.1.2
After examining each of these technologies and the commercially available versions, I
concluded that the technologies, in most cases, are either immature or unsuitable or
require substantial development beyond the scope of this work. As a result, I re-examined
the use of small electromagnetic DC motors and configurations of gear and
lever drives. After closely examining a variety of small gear motors and drivetrains I
selected a packaged actuator used in a variety of applications.
Servos 6.1.3
Scale models of planes, cars, boats and helicopters and other vehicles use modular
actuators for steering, moving surfaces and for controlling larger actuators such as high
power and high speed DC motors. Radio control, or R/C, servos are small geared DC
motors that provide closed-loop position control. New generation devices are rapid,.
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74
precise, lightweight and cost effective compared to small DC motor and gearhead
alternatives or packaging separate components.
R/C servos provide closed-loop position control of angular position and newer servos
also provide control of linear position. As shown in Figure 6-2, the control signal is a
pulse that is repeated every 10-30 milliseconds. The width of the pulse determines the
position of the servo. A pulse-width change from 1 to 2 milliseconds will sweep the
position of the actuator from one extreme to the other. Typical angular excursions of
servos are about 60 degrees, though many servos can mechanically provide 180 degrees
of motion.
There have been significant improvements in R/C servo design and construction over
the past several years. Recent designs use coreless motors, integrated electronics using
surface mount technologies, custom integrated circuits, strong roller bearing support
and O-ring-type seals for use in adverse environments. I undertook an evaluation of a
variety of manufacturers products to compare the products. Appendix A details the
servo specifications and resultant figures for about forty servos from six different
manufacturers. Conversations with users and marketers also indicated that some
manufacturers also slightly inflate specifications.
The final servo selected, the JR4721, showed significantly better performance over
other servos in metrics that included power to weight and torque to volume ratios.
Specifications for the servo are shown in Table 6-1. The power is calculated as one-quarter
the product of speed and maximum torque. This is a typical rule-of-thumb for
brushed motors; the product of the maximum no-load speed and maximum stall-torque.
On a torque speed curve this is generally a linear relationship where the maximum
power is halfway between the two points and, thus, is given by one fourth of the
product.
10-30ms
1-2ms Control Signal
Mechanism
Figure 6-2: Servos use small and efficient geartrains integrated with a positioning
control loop.
motor
feedback
electronics
geartrain.
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Table 6-1: Specifications for the selected servo.
Servo Modeling 6.1.4
A model of the servo is needed for the simulated physical model of the serpentine robot.
The servo is treated, appropriately enough, as a black box and the output of the actuator
is examined for a specified input. The actuator is loaded and then the response observed
in reaction to reference commands that move the servo to a given position. The
relationship of time and angular position gives a response which is analyzed to provide
parameters of the servo. Since, in the simulation, the various gains can be modeled, it
remains to provide a model of the servo by testing the physical device. An experimental
setup using a position tracking device fixed to a bracket and then attached to the servo
is used to find the angular position versus time.
From observation, R/C servos appear to use a simple proportional control, but because
of the high gear ratios, the servos do quite well in tracking the reference signal. The
gearing ratio of servos is on the order of 300:1. Since the reflected inertia of the drive
train is proportional to the square of the transmission ratio, this provides a fair amount
of inertia but this has benefits to controlling varying loads as well.
The optical target, an active LED, traces out a section of a circle as it moves and the
position is returned at about 100Hz to the tracking device. Thus, a commanded position
is sent to the servo and the resultant motions are tracked with high fidelity.
The X,Y,Z positions are fitted to a plane and then these positions are fit to a circle. The
angle is now be computed for each position and the relationship between time and angle
is then be plotted as shown in Figure 6-4. The response of the servo, shown in the time
Model Max Torque
[Nm]
Mass
[kg]
Speed
[s/60???]
Power
(W)
Power/Weight
[W/N]
Torq/Weight
(Nm/N)
JR4721 0.84 0.049 0.22 1.21 2.52 1.77.Implementation
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response curve, is used to determine the natural frequency, w, of the servo. This is then
fed into the simulation model for each servo.
Figure 6-3: Test setup for determining servo model.
Servo
Support bracket
Tracking LED
Response plot.Implementation
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The results of one of the tests is shown in Figure 6-4, provided the step response to an
input command to move to a desired reference position. The swept angle shown is about
100 degrees. From this information, several parameters can be found, including
stiffness and damping coefficients, and the inertia.
For a second order system (mass, spring, damping) the coefficients were determined by
deriving these values from experimental data. See Appendix C: Derivation of Actuator
Parameters for details. Once the parameters were found from the experimental data, they
were plugged into the physical simulation for the actuator values. The derived values
are for a spring damper and inertia system and the simulation values determined are the
proportional and derivative gait. The behavior of the simulation had the general
characteristics of the actuator. This was determined by plotting the response of the
simulated snake robot to a given input and observing the angular velocity of the
simulated actuator.
Design 6.2
Actuation is closely tied to the structural design that supports the robot and I examined
and discarded many ideas and iterated a number of configurations to resolve this issue.
Mechanisms examined included push-rods, linkages, bellcranks and clevis joints to
increase leverage and provide higher torques. The additional complexity of these
0.0 1.0 2.0 3.0 4.0 5.0
0.5
1.0
1.5
2.0
2.5
3.0
Figure 6-4: The servo, as measured, exhibits a classic underdamped response.
Time (seconds)
Angle
(radians).Implementation
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mechanisms did not warrant the additional design, fabrication and maintenance that
they required. By directly tying actuation to output, the mechanism was simplified and
made very compact even though the torque requirements increased. A beneficial
cascade effect occurred that shortened and lightened joints, thus reducing structural
stresses and loads.
Geometry 6.2.1
In natural snakes, as we saw in Background, links are formed from the concatenation of
many similar vertebrae. The angular motion between vertebrae is fairly small but,
because the vertebrae are relatively short, they can subtend small body curvatures.
Thus, the relationship between link length and the angular motion that a joint can
subtend is important.
In the right hand side of Figure 6-5, a right angle corridor of equal passage width, W,
provides a geometry for determining the relationship between the aspect ratio of the
links and the joint angle they subtend. One relationship between the joints and links is
that the angle between links in a circular arc must be equal to the arc angle divided by
the number of segments in that arc. However, this doesn?t reveal the relationship
between the joint excursions and the aspect ratio of the links.
In Figure 6-5, R o is the radius of the outside arc envelope of the link configuration, R i
is the corresponding inside radius and d and L are the respective diameter (or width)
and length of the individual links. Finally, q is the angle between links. In Equation [6-
1], the outside radius is shown as a function of L and d and the inside radius. The inside
radius, R i , shown in Equation [6-2], is similarly derived from the figure and finally, in
Equation [6-3], the corridor width, W, is shown as a function of the two radii.
L
R i
R o
q
q
W
Figure 6-5: The relationship between link apsect ratio and joint angle
motion for a right angle corridor.
W
L
q
d.Implementation
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[6-1]
[6-2]
[6-3]
Now substituting for Ro and Ri gives W as a function of link length and the angular
excursion. If d, the link width or diameter, is set to 1, then the length of the link, L,
represents the aspect ratio of the joint length to width. Substitution gives the equation
below:
[6-4]
In Figure 6-6, this function is plotted as a function of both q and L.
The important aspects of Figure 6-6 are that, as might be expected, the arc through
which the links can move is linearly related to the length of the link. That is, the longer
the link, the broader the arc. However, the angular excursion between links has an
interesting property: there are rapidly diminishing returns from increasing the angular
excursion of the joints. Beyond 0.3 radians, about 20 degrees, of motion for almost any
link length, the corridor through which the connected links can pass does not
appreciably decrease in size.
This argues for link designs to be as short as possible and that designing links whose
rotation is beyond +/- 20 degrees is a misguided effort. However, there may be other
reasons such as deployment and setup that require greater ranges of motion for the
links. This is a general argument that shorter links are better than longer links and that
large ranges of motion are probably unnecessary to subtend tight geometries.
This result is important, and although the analysis is specific to this geometry, the
general lessons are: shorter links with small aspect ratios are better and the angular
motion need not be large to gain benefit. More strongly, large range of motions do not
appear to benefit movement through tight geometries. This argument does not reflect a
particular gait, only a passage of entry and exit.
A related question is whether locomotion through open environments is served better
by large angular excursions. The primary benefit of smaller angular motions is that the
body of the robot can better describe arcs due to the smaller variation in distances
between the links and the nominal arc. For large angular excursions the fit is worse; the
maximum distance between link and arc becomes greater.
R o
L
2 --- ? ?
? ?
2
R i d + ( ) 2 + =
R i
L
2 q
2 --- tan
-------------- - d
2 -- - ? =
W R o
2
2 -------R i ? =
W 1
2 -- - L 2 4 L
2 q
2 --- ? ?
? ? tan
-------------------- 1
2 -- - ?
? ?
? ?
? ?
? ?
? ? 2
4L
q
2 --- ? ?
? ? tan
----------------- ++2
2 ------- L
2 q
2 --- ? ?
? ? tan
-------------------- 1
2 -- - ?
? ?
? ?
? ?
? ?
? ?
+ =.
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Mechanism 6.2.2
The initial link design, constructed of aluminum, used material over 3mm thick. A
single link was built and constructed to test assembly, clearances and strength. This
link, shown in Figure 6-9, proved the concept and provides two orthogonal motions of
up to 180 degrees each. While the preceding analysis showed that the large range of
motion is probably unnecessary, the motion came at little cost to the design.
The distance between parallel axes on adjacent links is 100 mm. Between adjacent
links, the joint motions are 40mm apart but are 60mm apart between the two motions
in a given link. This slight asymmetry doesn?t appear to have any adverse effects on
performance. A second generation link was designed and built using thinner material;
aluminum about 1.6 mm thick with a number of modifications for reducing weight and
increasing clearances in the structure. This thinner and lighter structure is the final
design. Specific features included a mounting plate that is integral to the servo structure
and housing. The bearing capture for the pivoting arm was also tied into this plate.
Servo fasteners were used to hold the mounting plate to the servo. This eliminated
modifications to the servos and makes for a strong and modular mechanism.
Following the selection of servo actuators, I completed a design of the links and
mechanism for a 3D snake. An undergraduate working with me, Anton Staaf, also
began the development of a 2D caterpillar system shown in Figure 6-7. The design
Figure 6-6: Plot of passage width as a function of link aspect ratio and
angular excursion between the joints.
W
L.
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utilizes eight links and eight parallel degrees of freedom. The caterpillar is capable of
traveling wave gaits and enabled testing of the experimental setup while the 3D robot
was developed.
Figure 6-7: The Caterpillar crawling robot utilizes eight linearly-linked servos.
Figure 6-8: Exploded view of link mechanism..
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The 3D snake link design utilizes two orthogonal DOF?s each with approximately 170
degrees of motion limited by the mechanics of the servos. Typical servo excursions are
about 90 degrees, but can be commanded to nearly 170 degrees. Figure 6-9 shows an
earlier version of the link. The aluminum pieces are over 3mm thick and no weight
reduction was performed on the design. The plate upon which the servos are mounted
is attached to the servos using the servos own case mounting screws. This provided
great simplification of attachment and a solid and direct mounting. The mounting plates
on the opposing side of the servo horn has a threaded hole for mounting a shoulder
screw. This attaches the rotating section to the servo very securely and takes up moment
loads from the adjacent links of the mechanism.
Figure 6-9: The first generation 3D link is comprised of two orthogonal servos.
Figure 6-10: Link mechanism design.
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The complete 3D snake, shown in Figure 6-11, has ten links for a total of twenty DOFs.
Improvements in the design ranged from significant reduction in the wall thickness, a
corresponding reduction in the weight of the material and improved fabrication using
bending instead of machining. Bearing capture and support was changed and simplified
from the earlier versions and test designs.
3D Robot Specifications 6.2.2.1
Here are several general specifications for the serpentine robot:
? Mass: Mechanism is 1.32 kg. About 2/3 of the mass is the servos. The rest is metal
and hardware. See Link Weight Distribution in Appendix B for more detail. Total mass,
including wiring and controllers is 1.48kg.
? Length: 102 cm (10 links, each is 1.02cm long)
? Diameter: 6.5 cm
? Power: 24.2W max total mechanical output. ~75W max total electrical input.
Quiescent: 1.15W (9.5mA@6VDC)
The figures result in an overall robot density of about 0.39g/cm 3 , less than water, due
to the spaces in the rotating sections between joints and thin sections of material. This
does not include final wiring or the skin which bring this figure to about 0.5g/cm 3 .
Skin 6.2.3
The vehicle-terrain interface for wheeled and legged vehicles has been the topic of
many research works, but is neglected for serpentine robots. Wheels, feet and ankle
designs and tire tread and forms have been extensively examined in the literature.
However, previous serpentine robot research has not evaluated skins and surfaces and
surprisingly little attention has been paid to this area. The integument, or skin, provides
terrain and environment contact and is the key component of interaction between the
mechanism and the terrain.
Recall from Biological Systems that the skin of the biological snake is comprised of
smooth, dry, highly polished overlapping scales. The underlying skin is elastic, like
ours, and accommodates the varied shapes and motions of the body. To provide similar
Figure 6-11: 3D snake utilizes 10 links with 20 servos.
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properties for a robot, I examined, evaluated and tested a wide variety of materials for
the serpentine skin.
Another effect which can be used is that of preferential friction, where the coefficient
of friction varies depending on the direction. This makes it easier for a variety of gaits
to demonstrate progress. The extreme case, of course, is a one-way bearing where the
friction is negligible in one direction and free-rolling in the other. Materials with a nap,
like velvet, show this quality and some specialty materials for material transfer exhibit
this property as well.
Bellows 6.2.3.1
Bellows material is often used to protect exposed surfaces of machinery. These are
usually made of heavy cloth materials or segmented plastic or metal frames. The
pleating of the material provides a corrugation that gives form to the bellows as well as
allowing the compression of the bellows itself. This form then, allows discrete line or
point contact with the ground and a way must also be provided for the bellows to attach
to the underlying mechanism.
Cable Chains 6.2.3.2
Cable chains are used for the protection of wire harnesses during the movements of
machine tools. They are usually made of tough plastic materials, but are also available
in metal links. Only a few cable chains allow motion out of the plane, the igus Triflex ?
series being one of them. [igus95]. This is an attractive possibility for several reasons, it
provides the rotation axes, it is of the right scale for the mechanism and, through the use
of sliding surfaces, provides some degree of protection from the environment. The
downside is the weight of the links and a simple means to couple the actuation to the
plastic frame. In addition, the angular excursion at each link is limited to about 10
degrees. Weight is also an issue; for a 1 meter length the weight of the chain alone is
nearly 1.5kg for the square 50mm interior model.
Flexible ducts 6.2.3.3
Ducts are often comprised of steel springs embedded within latex or vinyl cover to
provide a conduit for air or wires. Dryer duct hose, commonly found in hardware stores
is one example. The compound material is flexible and provides a high ratio of
extension to compression. However, the rigidly rounded profile requires a means of
tying the hose to the structure and does not conform well to underlying mechanism and
Figure 6-12: igus Triflex 3D cable chain..
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structure. Plastic corrugated materials such as vacuum hose were also examined but
found to be too stiff.
Rubber 6.2.3.4
Natural and man-made latex materials exhibit substantial stretch and are available in
very thin films. They are highly elastic and are very thin in a wide variety of forms.
However, even very thin materials take a fair amount of energy to distort and stretch and
prevention of wrinkling and folds is a difficult challenge.
Fabrics 6.2.3.5
A number of fabric materials were tested including spandex materials, which provide
high elasticity and durability. Spandex is unusual in that it is a fiber that acts like an
elastomer. Spandex materials can be stretched over 500% without breaking and
completely recover. The material is a polyurea-urethane elastomer chain that has soft
segments that provide elasticity and rigid segments in the chain that act like the
crosslinks in natural rubber.
Spandex is stronger and more durable than rubber and also resists pilling, the build-up
and consolidation of short fiber segments on the surface of the material. Compared to
other fibers though, it has poor strength and so Spandex is usually blended with other
fibers including polyester, nylon and cotton. Typically percentages of spandex are less
than 20% in blended materials. Commercial spandex fibers include Lycra ? by Dupont,
Cleerspan ? and Glospan ? by Globe and Dorlastan ? by Bayer.
Other, more exotic fabrics, tested included four-way stretch velvet materials, (92%
Polyester, 8% Lycra) which have the interesting property of differential friction,
commonly termed the ?nap? of the material. Another set of surfaces tested included
sequined materials with 100% overlapping coverage on a Lycra-base. These are
particularly gaudy materials but provided dry and polished lapped surfaces similar to
scales.
Textile qualities are not only dependent on the material that they are made of but also
on the fiber treatment, processing and the fabrication or weaving technique. An ideal
material might also be impervious to water and, in fact, some new woven materials
using microfiber (<1.0 dpf) are water resistant.
Braided materials 6.2.3.6
Another material tested, and eventually used, are polyethylene-based braids. These are
typically used to bundle and protect wires, cables and hoses. They offer good abrasion
protection and are very lightweight and durable. A wide variety of materials and types
are available including teflon, halar, copper and steel. Additionally the weave and mesh
can vary. I selected a sleeve made from a 0.25 millimeter polyethylene monofiliment
yarn of polyethylene terephthlate or PET. A variety of diameters were tried to evaluate
fit and motion. A 3.8cm nominal diameter braid was finally selected and stretched over
the 6.5 cm diameter mechanism
Final Skin 6.2.3.7
A Lycra spandex sleeve with a single dorsal seam is slid over a PET braid to form a two
layer skin. The seam utilizes an elastic thread and cross-stitch to minimize loss of
stretch across the seam. Figure 6-13 shows the 3D snake atop several of the skins made.
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for testing. All the underlying fabric material is identical with the exception of color but
the surfaces are different. The surfaces range from no treatments, to small studded hard
materials to sequins.
Tread 6.2.3.8
While the fiber materials provide excellent stretch characteristics and form
accommodation, they also result in a smooth, low friction surface. This, in turn, can
result in slipping and is an obvious problem for traction.
Surface treatments such as polymeric paints were applied in a wide variety of patterns
to fabric surfaces to experiment with tread configurations. These paints or coatings are
especially designed for fabric applications. In some cases, after the material was
partially cured, steam is applied to the surface. The high heat and humidity combine to
raise the surface of the applied coatings above the base material.
Many of these coating materials were applied by squeezing drops of the materials
across the material in different patterns. Coefficients of friction were compared between
the surfaces and patterns. This was done by wrapping the material around a wooden
block and then laying it atop a surface and determining at what angle the material would
slide at a constant velocity. The base material used was aluminum. The tangent of this
angle is the coefficient of friction. For the materials shown in Figure 6-13, the
coefficients of friction are shown in Table 6-2.
Figure 6-13: A variety of skins made using spandex fibers as the base fabric with a
variety of surface treatments..
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Table 6-2: Skin materials and corresponding coefficients of friction.
Electronics 6.3
Electronics provide and distribute information and power to the robot. Due to the large
number of actuators in this type of robot there can be a correspondingly large number
of conductors carrying signals and power. However, the actuation chosen for the robot
reduced some of this infrastructure because the servos provide local closed loop control
of position. For the many controlled degrees of freedom in robots there are numerous
feedback signals, motor winding and commutation signals etcetera. Servos, however,
require only power, ground and the reference signal. Even so, this can be a difficult
number; every additional conductor is multiplied by twenty.
Ideally, both power and data would be supplied over just two wires; one for ground and
the other providing power with a superimposed electronic signal. The concept of a bus
that multiplexes both power and data has been in use in homes and industry for a
number of years: it superimposes a digital signal atop the power lines in the home and
allows the control of lights, appliances, etc. from a single or multiple locations [X10
97]. The more recent IEEE 1394, Firewire, standard has a similar capability. This bus
concept would be ideal for a snake-like robots. Every link and degree of freedom
requires both power for motion and signal for control and the structure of serpentine
robots facilities such mundane issues as routing and termination.
Digital Command Control 6.3.0.0.1
One such bus meant for small scale devices is Digital Command and Control, or DCC.
DCC was initially designed for scale railroad control and utilizes the track for
transmitting power to all devices and onboard electronics to listen to the superimposed
signal. Each device is individually addressed and when a particular address is signaled
the following data is directed to that particular device. The small boards provide outputs
that can drive small DC brushed motors at up to an amp or so. When used with a small
and efficient geared motor DCC can provide simple control of a large number of motors
[DCC 94]. I built a design on a PC board with a serial interface and used a dual-H-bridge
driver to provide power with a superimposed signal. Combined with control software,
this provides motion control that is easy to implement and use.
Spandex Material mk
Patterned w/ rhinestones 0.48
Plain Spandex 0.32
Red Sequined 0.30
Larger Rhinestone patterns 0.36
Diffractive material 0.42
Polyethylene braid 0.26.Implementation
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The advantages include the ability to provide both power and signal information over
the same two wires to each electronic device. Thus, this ?snake bus? is enormously
simplified over running power and information to each actuator. The main disadvantage
to DCC is that it is open loop and the commands control acceleration and velocity but
not position. There are some proposed future enhancements to DCC to provide these
features but no commercial versions exist at this time. I designed another bus system
using RS-485, a multi-drop serial bus, in conjunction with ICs that can provide simple
interfaces for A/D converters but the size, complexity and cost became prohibitive. The
eventual selection of R/C servos as actuators eliminated many of these issues with
feedback and control.
Servo Controller 6.3.0.0.2
While the input signals to the servos are logic level signals that can be provided by any
digital I/O board and software routines, I chose a convenient controller board that
provides serial control of up to eight servos per board [Scott 96]. The boards can be daisy-chained
to provide control of up to 256 servos. The communication format is a three
byte sequence for talking to any particular servo and is as follows: <255> <servo>
<position> where the position is the total excursion divided into 256 possible
settings. This provides servo motion and control from one extreme to the other. The
servo is responsible for maintaining that position through its own feedback and control
electronics. For twenty servos, the initial serpentine configuration, three boards are
required. The board size is approximately 35mm by 41mm.
In wiring the robot, the power is easily parallelized and a two wire power bus was
integrated into the mechanism. The signal wires also needed to be connected and there
are two possibilities for integrating them. One is to put the servo controllers within the
snake at intervals that minimizes the overall length of the wiring. For initial testing
however, I created a ?tail? that appends the boards to the end of the robot. The tail
provides a stacked arrangement of the controller boards and a tightly integrated harness
to connect power and signals to all three boards.
Wiring 6.3.0.0.3
Since the signal wires are not multiplexed along the length of the snake it?s necessary
to route them along the mechanism to both minimize bulk and maximize flexibility.
These constraints suggested very small and finely stranded wires. After investigating a
variety of small cables including ribbon cables, small gauge wire and specially
manufactured cabling, I selected a fine gauge hearing aid wire [Siemans 97]. This fine
insulated wire uses seven strands, each about 0.05mm and the total cable diameter,
including insulation, is less than 0.4 mm. A combined bundle of the required 20 wires
is only about 2mm making the combined bundle very manageable.
A rule of thumb for small wires is that the bend radius should be no smaller than ten
times the wire diameter. Thus, the bend radius in this case should be greater than or
equal to about 4mm. The solution is to route the wires directly across the points of
rotation at each axis minimized bending of the wire harness.
In early testing, there was significant jitter throughout the robot and a network of
decoupling capacitors was devised between power and ground. However, the addition.
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of other return lines was sufficient to remove most of the noisy signals and crosstalk in
the wires.
Sensing 6.4
A key element of biological snakes is sensing. It enables rapid adaptation to varying
terrains during locomotion. Since the terrain is unknown, such sensing is necessary for
traversal. If, magically, the terrain were known and both the position and configuration
of the robot were also known it would be simple to provide appropriate information to
guide the robot during locomotion. A minimum form of sensing is to simply provide
contact sensing to determine whether or not contact has occurred. The best form, of
course, is full contact sensing with knowledge of terrain and forces.
Implementation for contact and force sensing is difficult. Proximity and contact devices
for industrial use are bulky and difficult to integrate into small mechanisms. The
technologies exist and include optical, capacitive, force sensing resistors, and even
piezo-electric pads. The difficulty is not just identifying the technology, but integrating
it into the robot.
However, I identified and evaluated several candidates:
? Small tactile switches - binary, with trigger threshold
? Force sensing resistors - analog, wide range
? Capacitive arrays - flexible, good resolution, good sensitivity.
The switches are the simplest, in terms of operation, interface and mechanism. The
drawback is that they only provide contact information above a force threshold required
to trip the switch. A variety of force sensing resistors were tested but for most of these
devices there are significant issues with curved surfaces. The stress of forming a shape
other than on a flat surface makes the response of these devices unusable for this
application. One device, however, a thick film polymer with force resistive properties
works well for measuring forces on a surface that is curved along one axis. This is the
middle device shown in Figure 6-14. Another device uses piezoelectric-generated
Figure 6-14: Two resistive force sensors and a small tactile switch that were
evaluated..
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acoustic signals to find deformation in a soft pad. While the resolution is high the
cabling, packaging and cost make this type of sensor untenable for this application.
The final device appears to have the desired sensing properties for tactile sensing, high
resolution, fast response, accurate measurements, good spatial resolution, curved
surface use. The drawbacks are cost and the electronics packaging and infrastructure
[Novel 97].
Additional sensing such as local range information could provide useful data for
locomotion, obstacle avoidance and grasping. Small IR sensors could be used to detect
local surfaces without contact. This offers other opportunities for gait selection and
modification but was not explored further in this work. The sensing devices were
evaluated but not emplaced on the snake robot due to cost and time considerations.
Other Subsystems 6.5
A robot is a substantial integration of several technologies. Not only mechanism,
actuation and sensing but communications, power and computing. Each of these
subsystems cannot be fully independent of the others and considerations for each must
be taken into account during the design process.
Communications 6.5.1
Communications is handled by a serial-based RS-232 device. At 9600 baud, using the
three byte command stream, meant that each servo could be updated about 16 times per
second. This rate results from 10 bits/byte, 3 bytes/command and 20 servos. The
diagram for the electronics is shown in Figure 6-15. Each link, shown as the small boxes
with numbers representing the 20 servos, are connected via lines to one of the three
servo controllers, C 0 -C 2 . These, in turn, are daisy-chained to a serial line connected to
the computer.
Computing 6.5.2
Since the learning experiments required high-performance platforms and the
communications could be either to the simulation or the robot it made sense to use the
Computer
C 0 C 1 C 2
0 12 34 567
RS232
8 910 11 12 13 14 15 16 17 18 19
Links/Servos
8 8 4
Servo Controllers
Figure 6-15: The electrical system utilizes three servo controllers with a serial connection.
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same platform for simulation and device control. Most of the development and testing
is done on Silicon Graphics (SGI) workstations.
For these implementations, programs are written in C++ and compiled for execution on
SGI platforms. However, even the high speed workstations, such as the 195MHz
R10000 computers, cannot run the simulation in anything close to real-time. The
framework is executed on a single processor machine for most of this research although
it can be partitioned across multiple machines. In fact, optimization and learning can
parcel out the tasks across multiple machines so that evaluation can be parallelized
significantly. For most testing however, single machine execution is sufficient and runs
take several hours or so on R5000-based computers. All display code for the 2D
systems is written in C++ using OpenGL calls and an Xforms interface. For the 3D
simulations, Inventor is used for display.
Power 6.5.3
The servos are typically powered by 4.8V batteries. However, many servos can be
powered at 6V or even 7.2V with corresponding increases in power but perhaps reduced
operating lifetimes due to brush arcing on the commutators. For most testing, a DC
switching power supply is used but a variety of battery technologies were also
investigated during the course of this work [Dowling 97a].
Key attributes of any power system are power and energy density. Energy for long term
operation and power density for high demand periods. Even with the advent of many
new technologies, Nickel-Cadmium (Ni-Cd) batteries offer close to the best power
density of any battery technology. They are also widely available, and are available in
a large number of configurations and at reasonably low cost.
Although battery testing and selection occurred, most operation was done with an
offboard power supply during testing and evaluation.
Video 6.5.4
A small CCD camera was also added to the snake to provide video feedback for
operators of the device. The device, shown in Figure 6-16, provides, a snake?s eye view
of the area directly in front of the robot. Because the robot is highly articulated, the.
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forward joints can act as positioners for the camera for pan and tilting the image. The
robot is tethered for power and data, as is the video. For an eventual self-contained
robot, aself-contained robot could use one of several micro-miniature transmitters for
wireless video transmission.
Experimental Setup 6.6
As shown in Figure 6-17, a means of measuring and calculating the metric was
accomplished through power monitoring and a tracking device. The tracking device, an
Optotrak device, provides data rates of up to 1000Hz and tracking accuracies to 0.1mm.
It utilizes 3 very accurate PID chips to view multiple IR LED?s each of whose position
can be tracked. Ideally, the position data would be for the center of mass of the robot,
but this would require tracking LEDs at each link and wires to connect them to the
strobers. A single tracking LED was used near the middle of the robot to provide travel
distance over the experimental time. Although a single LED does not provide the true
center of mass, it gives a reasonable approximation to movement of the robot.
Figure 6-16: A camera, shown by the arrow, provides a ?snake?s eye? view.
Optotrak
[Position]
Computer
[Control]
[Power] Snake
Figure 6-17: Experimental setup of the robot and tracking and power monitoring.
Active leds used for tracking.Implementation
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Physical Modeling 6.7
Physical modeling programs are recent developments and only one commercial
package is available at the time of this writing [Knowledge 97]. Coriolis is a toolkit
developed at CMU to support interactive simulation. The toolkit is implemented as a
set of C++ classes that are instantiated to represent relationships and forces in the
simulated world. Class types within Coriolis include Bodies, Constraints and
Influences. Bodies are used to represent physical objects and their geometries.
Constraints describe relationships between different bodies and, finally,
Influence classes are used to describe the forces that act upon Bodies [Baraff 97].
Coriolis does not provide graphics, I/O formats or an interface and these must be
provided by the user.
2D 6.7.1
My implementation for the 2D simulation uses OpenGL for the graphics and a variety
of input and output methods including command-line arguments, files, and direct user
interaction with keyboard and mouse. Finally, I wrote an Xforms user interface on top
of the simulation which provided an OpenGL window, interaction and more
importantly, a variety of monitoring tools for observation of the simulation in action.
The Coriolis window in Figure 3-1 utilizes a number of these tools including power
usage, distance traveled by the simulated robot, elapsed time, and real-time geometric
information such as joint angles. I built in a number of options, including the ability to
turn off graphics for improved performance. Other interaction includes window
panning and scrolling capabilities and direct interaction with the simulation time step
functions. I also enabled user interaction that allows the user to click and drag
simulation pieces around to facilitate experimental setup or quick retesting without
having to quit and restart the simulation.
An example of the toolkit and how it is used to create the 2D snake within the
simulation is shown below:
void create_snake(Material *mat, UniformField *grav)
{
for (all_links);
AtomicShape *link_poly = makelink(geometry)
links.set_geometry(*link_poly);
links.set_material(*mat);
links.compute_inertials();
grav->add(links);
}
void SnakeMotor::update()
{
double now = bs->current_time;
torque = magnitude * sin(now*freq + link*phase);
vector j(3, 0.0, 0.0, torque);
exert_force(*body1, j);
exert_force(*body2, -j);
}.Implementation
94
create_snake takes as one of its arguments the material type which includes density,
coefficient of friction, coefficient of restitution (bounce), color etc. Additionally the
function describes the forces acting on the snake. This is set to a uniform force field in
one direction, namely gravity. Then for all the links, the geometry is created, the
geometry and materials are instantiated, inertial properties are calculated and the
gravity property is added to all bodies.
SnakeMotor is a torque applied to each body segment that, in this example, describes a
simple time-varying torque function for each of the joints. The torque is applied to each
body and the negative to the adjacent body segment. In this example, the form of the
torque function is a sinusoidal wave whose amplitude, frequency and phase are
adjusted. These values can be altered for each execution of the program and the metric
can be tested for efficacy. A different representation is required for more realistic and
expeditious locomotion.
3D 6.7.2
Coriolis, augmented with supplementary classes, is used to define 3D body geometries
and their connections [Leger 97]. Parts are created, connectors defined and pieces are
assembled into more complex geometries. For position control of the bodies however,
control is determined by setting a number of parameters, including gains, for position
control. This is because, in Coriolis, all motion is effected by Influences and not
simply by position commands. Open Inventor is used for all display functions. Figure
3-2 shows the 3D model of the physical snake structure. The model is created by
individually creating the vertex and face list for each part and by specifying the relative
geometries and assemblies.
The physical model of the serpentine robot is shown in Figure 3-2. The model is
generated from the mechanism design and used in the physical simulation.
Summary 6.8
A robot is a complex electro-mechanical integration of many technologies and
decisions on configuration affect software, planning and control. Issues as mundane as
packaging and wiring can slow and arrest development unless carefully addressed. The
adverse and cascading effects of improperly chosen subsystems can stop research in its
tracks.
Because of the central nature of actuation (it affects mechanism, control and a host of
other issues) this technology was carefully investigated. The technology chosen, small,
well-packaged electro-magnetic motors and spur gear drive trains, was the most mature
of the alternatives. Since the focus of this research is not the development of actuator
technologies, this was a good decision for implementation. Eventually, other nascent
technologies such as electrostrictive polymers, will be viable for small mechanisms but
at this time the immaturity of these alternative technologies makes them unsuited for
this research.
The efforts in design include a geometric analysis relating link length, diameter and
angular range of motion. The result is that short links are better and that, interestingly,
the range of angular motion need not be large to be effective..Implementation
95
The mechanism design proceeded through several iterations to simplify connection and
support and reduce weight and complexity of the design. The final result, using
lightweight formed plates and a simple bearing support provides good capability and
reduces overhead in fabrication and assembly.
Modeling of the robot is accomplished through the use of a powerful tool kit that
models a large range of physical phenomena and provides for both control and
monitoring of these forces.
Electronics design proceeded from several tested concepts to a separate power bus and
signal control system that is implemented in a straightforward manner. Additional work
in power and sensing have provided insight into future developments. Finally I
examined skins, a first for snake robots, for the purposes of providing a compliant and
tractive surface.
Although design is an appreciable effort, very often testing takes the greatest amount of
time and mundane issues such as connectors and cabling conspire to thwart even the
best intentions in design and fabrication..Implementation
96.97
Chapter 7
Locomotion
Locomotion presents details of experimental work; calculation, evaluation and
demonstration of the snake in both simulation and on the physical system. This
culminates in the integration of all the elements of the framework: optimization,
evaluation, simulation, and the robot. This is prefaced by a detailed look at how values
for the specific resistance metric are calculated or measured.
A number of interesting gaits were developed and, surprisingly, many of these were
non-snake-like. The gaits in simulation were able to provide a good measure of
performance relative to specific resistance. It was also found that the ideal path from
simulation coupled to learning and then learning coupled with the robot did not provide
the ideal path for gait development; the ideal serial process became an iterative one.
Power Calculations 7.1
Since the metric for evaluation, specific resistance, uses a value of power in its
formulation, that measure is required for both physical simulation and the robot.
For an electrically driven mechanism, the measurement of power can be determined by
simply monitoring current, voltage and power factor if there are large inductive loads.
The product of current and voltage is the input power to the robot, but it can be difficult
to determine where the dissipative losses occur and what fraction of that power
measurement goes into motion.
In simulation, the calculation of total power in the robot is the sum of the products of
torques and angular velocities for all the links of the serpentine robot. In Equation
Equation [7-1] m is the number of links or motions..
Locomotion
98
[7-1]
Assuming a power limited source, which a robot has, this value provides both an
indication of power levels and when cutoff thresholds are reached. Average power is
determined by dividing the total time into the total power used. Peak power, obviously,
is the maximum power reading.
However, an issue with Equation [7-1] is determining negative work where the torque
applied is in the opposite direction of the motion. For a walking machine this is a critical
issue and one that has adverse consequences on vehicle efficiency. An example of
negative work is when an actuator is used as a brake. Power is expended, but not in a
manner that enables forward motion. This issue is germane in legged robots where
improperly designed configurations result in large expenditures of energy without
corresponding forward progress.
Interestingly, in animals, negative work occurs when muscles are stretched while they
are still developing tension [Roberts97]. This extra tension has little extra energy cost
and this, in turn, has strong implications to locomotion: people can descend stairs with
far less effort than it takes to ascend even though the muscles still exert considerable
force. Thus, negative work requires less total energy than positive work in animals.
Geometric work in mechanisms is a form of negative work where actuator backdriving
occurs opposite to the direction of motion of the vehicle. It?s inefficient because other
actuators have to make up backdriven energy losses in addition to the energy required
to move forward [Waldron 81].
Gravity loads, while probably not significant for a snake, do contribute additional losses
every time part of the robot is lifted and placed down on the ground. Gravity loads do
affect contact forces and sliding friction however. For forms of sliding locomotion the
additional frictional losses can be significant. It is amazing that snakes are at all
efficient.
One solution is to take positive and negative work separately to assess the contributions
from both types of work. The net energy input, shown in Equation [7-2], should be
equal to the friction losses in the system.
Geometric work can be minimized by decoupling propulsion from support or lifting
motions. In the case of serpentine robots, however, this may prove difficult to do. The
opposing issues are generating sufficient traction to maximize forward progress and
enabling low friction to minimize losses into the ground. One thought is to provide a
skin that provides differential friction and this topic is explored in a later section on skin
design.
Other significant losses can occur from isometric work where opposing forces are
generated through the ground. This coupling typically results in force generation and
energy losses without motion. Without good coordination this can prove to be a
significant factor in energy losses. Legged machine designers must consider issues in
isometric work, but designers must also address this issue for wheeled vehicles that are
highly articulated, or that require multiple degree-of-freedom of control.
Total Power T n q ? n
n 1 =
m
? =.Locomotion
99
The most straightforward analysis of power usage is the sum of the products of torque
and angular velocity for all links as shown below.
[7-2]
This gives an instantaneous measure of power use by the robot. An average power
estimate over a fixed period of time can then be calculated to give a measure of total
energy used. The power for a particular joint is found over a specified time interval by
assuming a fixed level during the interval. If the time intervals are short then this will
generally hold true. Otherwise, the calculation is more complex and, in any case, does
not reveal great difference in the calculations which are used to only determine
differences between evaluations, not highly accurate values. Thus, the power
calculation for a joint becomes the sum of the power during the all the intervals.
Average power is determined by dividing by the number of intervals and the power for
the whole robot in simulation becomes the sum of these products over all the joints.
As shown in Equation [7-3], the total power is then summed over, n, the number of time
intervals and m, the numbers of joints. The angular velocity is actually the difference
between the angular velocity of the two links. As shown in Figure 7-1, this is w1 - w2 .
Thus, the equation for total power over the time interval becomes the formulation
shown in Equation [7-3].
[7-3]
In Figure 7-2 are shown two plots of power for a robot run. The snake was a small two
segment, four DOF system. The data was captured at 100Hz for thirty seconds during
locomotion. Notice the power spikes, mostly negative, in the top figure. These spikes
result from the surface impact when a joint section makes contact with the surface. A
short rebound results in motion that is opposite that of actuation and this happens very
quickly. Thus, the product of the current actuator force and the rapid change in the
velocity give rise to a spike or discontinuity in the data. In the bottom graph in Figure
7-2 is shown the result of a running average (20 data points wide). The power average
remains about the same in both cases, about 85mW. The power values from simulation
are much smaller than that of the actual robot. This is primarily due to the friction and
Power input T n q ? n
n 1 =
links
? =
t 1 t 2 t n
|Power|
Figure 7-1: Power is measured at discrete intervals by the product of torque applied at
the joint and the angular velocity.
T
w 1 w 2
Power 1
n -- - T ij q ? ij
i 1 =
n
? ? ?
? ?
? ?
j 1 =
m
? =.Locomotion
100
inefficiencies of the robot drivetrain. Modeling all of these physical phenomena within
the drivetrain is impractical, but the general results should be valid. That is, a good gait
in simulation should correspond to a good gait on the physical robot.
Units 7.2
In physical simulation, the consistent use of units is critical for relevant modeling of the
snake robot. Consistent use of physical definitions is one such requirement; units of
mass, length and time can be used to define a complete physical system. For practical
reasons within the simulation however, the values of these units must allow good
calculations of other physical concepts such as inertias and densities. Thus, the physical
units used in simulation are kg for mass, meters for length and seconds for time; an
MKS system. For defining snake geometries, and for simulation purposes, I used a CGS
system. This may appear to be a simple matter of scaling but at issue are the final values
within the simulation that are used to determine forces, frictions, motions, moments
etcetera. These values should not be small because calculation errors will creep and
eventually result in erratic performance. Thus, MKS units result in very small numbers
for the relatively small mechanism where links are at cm scales and masses are at gram
scales.
0.0 10.0 20.0 30.0 -100.0
0.0
100.0
200.0
300.0
0.0 10.0 20.0 30.0 -100.0
0.0
100.0
200.0
300.0
Figure 7-2: Measured power data from run of physical simulator. Top graph is all data with
impact spikes and bottom uses running average for smoothing.
Time(seconds).Locomotion
101
Velocity Calculations 7.3
Another piece of the specific resistance evaluation is the velocity metric. In simulation,
this is given by tracking the distance traveled by the system center of mass. This is
calculated by the square root of the XY distance traveled by the center of mass. The
center of mass of the whole system is determined from the average of the X and Y
positions of the centers of the individual links.
For the robot, the center of mass is a little trickier. Because tracking all of the individual
links is difficult, one way is an approximation using the distance traveled by one of the
central links. This can be tracked using a camera-based tracking system that provides
rapid and accurate feedback. The central link tracking gives only an indication of
motion. However, it appears that the maximum distance that the center of mass can be
from the center link is one-quarter of the length of the snake robot; this is a pathological
configuration. No proof is presented but the result should be obvious. Most
configurations, especially symmetric ones, should result in small differences between
center of mass and center link position.
Gaits 7.4
Several types of gaits were generated including some non-biological modes of
locomotion. In many cases, the gaits could also be reduced to a simpler, more general
waveform which provided good insight into how the gaits work. The gaits are detailed
here into snake-like and non-snake modes of locomotion..Locomotion
102
Snake-like Locomotion 7.4.1
The next several pages reveal figures of various stages of locomotion modes. The first
few are those snake-like modes demonstrated in simulation. This are interesting
because they replicate existing modes in snakes.
Sidewinding 7.4.1.1
Sidewinding is a relatively efficient mode of locomotion with little sliding ground
contact but with an odd means of moving laterally. Sidewinding is really two waves, one
ventral and one lateral that are out of phase. Together they produce a motion that moves
a body section to the side and the rest moves along and settles the body down to form
successive parallel tracks as it moves..Locomotion
103
Figure 7-3: Sidewinding locomotion..Locomotion
104
Rectilinear 7.4.1.2
Rectinlinear locomotion propagates a wave along the length of the body. This reverse
moving wave provides for locomotion by lifting a portion of the body and using length
in the wave to move the head forward and down. An effective gait that does not slip or
slide much along the ground..Locomotion
105
Figure 7-4: Rectilinear locomotion..Locomotion
106
Lateral Undulation 7.4.1.3
True lateral undulation provides for a continuous sliding motion along the ground. The
issue in the simulation is that the surface the robot moves along is flat; the ground plane
could be populated with an array of small objects that the system could push against.
This locomotion configuration also slightly lifts the outward lateral wave. This is the
the ?sinus lifting? mentioned by Hirose..Locomotion
107
Figure 7-5: Lateral undulation..Locomotion
108
Non-Snake Gaits 7.4.2
Lateral rolling 7.4.2.1
An intriguing gait is formed by a U-shaped bbody and providing oscillating motions
about each joint. It appears, at first glance, that there is a continuous rotation of the
joints but this, of course, is not possible and is unecessary. The gait is similar to
sidewinding in that it uses two waves out of phase: a lateral sine wave and a ventral
cosine wave. However, in this case the phase of the waves is zero; all joints on similar
axes move together..Locomotion
109
Figure 7-6: Lateral rolling..Locomotion
110
Traveling wave rotor 7.4.2.2
This gait is similar to a spinning coin as it come slowly to rest. The ventral wave, in this
case, is not a rigid body but a wave that propogates around the body of the circle formed
by the snake robot body. This is also similar to the principle of ultrasonic motors that
uses a traveling wave to move objects. Typically the ultrasonic motors use a low
amplitude, high frequency wave to achieve motion of a plate..Locomotion
111
Figure 7-7: Traveling wave rotor..Locomotion
112
Wheel 7.4.2.3
One of the most intriguing results was the reinvention of the wheel. Essentially a
wrapping of the ventral joints and a coordinated motion provide a rotating section very
much like a wheel or track.
[Brackenbury 97] details the odd locomotion of caterpillars that perform this feat; they
can roll up and move in a protective coil. By doing so, the animal can recoil and retreat
very quickly; much faster than it could locomote in a normal ?inching? manner.
This mode of locomotion was attempted on the caterpillar robot and was partially
successful. The wiring complicated the attempt and the robot could quickly become
unstable and fall sideways. However, in at least one test the robot rolled nearly a full
revolution.
This mode could be reduced to a simple offset of the joints to form the circle and then
a low amplitude wave is propagated along the periphery to roll the robot.
Several experments revealed that interpenetration of the snake with itself occurs. In
image five of the sequence, in this example, you can see the inter-penetration of the ends
of the snake. To reduce computational requirements, the body segments are allowed to
inter-penetrate. This reduces the amount of collision detection needed at each iteration
and with most gaits this is not an issue because inter-penetration does not occur. With
the wheel gait however, occasionally there is a self contact - but this does not appear to
be critical to the gait..Locomotion
113
Figure 7-8: The wheel..Locomotion
114
Flapping Locomotion 7.4.2.4
Another sideways mode of locomotion, this mode uses in-phase motions of the ends to
swing forward, come down in contact and the lift or drag the center of the body forward.
This is similar to the motions of a swimmer performing the butterfly stroke..Locomotion
115
Figure 7-9: Flapping locomotion.Locomotion
116
Rolling Collar 7.4.2.5
If the configuration of lateral roller motion, shown in Figure 7-6, closes upon itself and
forms a circle, it forms a rolling collar which looks similar to a smoke ring when in
motion. By itself, on flat ground, this produces no locomotion. However, if this form is
used to surround a pipe or other convex object or is used internally, then this motion
produces locomotion. It produces a motion akin to that described by Nilsson in [Nilsson
97a], but without the use of the complex roll-pitch-roll joint. As in lateral rolling the
body of the snake acts as a rolling wheel to move..Locomotion
117
Figure 7-10: Rolling Collar locomotion.Locomotion
118
Helical rolling
Another feature is to use a constant offset for joints. For example, forming a hoop and
then rippling a wave along the hoop. The formulation then becomes:
Amplitude * Sin(time/period + phase * i) + offset[i]
A slightly different class of gaits can be formed by providing fixed offsets of the joints
so that a general configuration can be initiated and then operated upon during motion.
Concertina 7.4.2.6
An issue for varied gaits is that the time history of all the joints will be exactly the same.
This works for many gaits, but concertina is one example where it does not. All of these
gaits assume the same fixed frequency of motions along the snake, but the body
frequency of an intermittent gait, such as concertina, is different. The wavelength of the
body varies along the body and over time.
Discussion 7.4.3
The development of the gaits did not proceed in the manner originally envisoned; In
many cases these gaits can be described by elementary formulations. However, it is also
clear that natural snakes do not use these formulations. A number of snake gaits use a
simple family of forms to describe joint and subsequent body motions.
Amplitude * Sine(Time/Period + Phase(link number)) + Offset
The phase is usually the product of a phase value and the joint number. This produces
a traveling or propagating wave down the body of the robot. Zero phase, obviously,
produces the same motion at all joints simultaneously, whereas a phase of Pi produces
alternating and opposite motions in adjacent links. The offset can also be particular to
a specific joint. For example, the body could describe a nominal 3D shape, such as a
helix and use that as a base from which to propogate other waveforms.
Note that this formulation describes the motion of the links and not the waveform of the
body. In fact, it is the phase that most nearly describes the bodyform. This formulation
can also be scaled to be proportional to the link number of the inverse. This gives some
additional interesting motions.
180 degrees, +/-1.5 radians, of motion is the limit for individual joint excursions. In
simulation, some interesting patterns emerge beyond that limit, including some
intriguing lissajous-like patterns, but it makes no sense for mechanisms since hard
physical constraints prevent these motions.
Additional features of these formulations include an amplitude factor that is
proportional to the link number. This allows heading changes in the gait so that the
system can be ?steered? in a desired direction.
Time 7.4.4
The time for simulation for a very small two-link snake on a powerful workstation could
run twice as quickly as real time. However, longer snakes took substantially longer; run-times
for longer snakes went up quadratically and, in some cases, even worse. The
runtime depends mostly on the amount of contacts which, for a snake, are numerous. In
addition, there is a lot of coupling between links that is closed through the ground or.
Locomotion
119
surface. These closed kinematic chains plus the large number of contacts contribute to
substantially longer run-times.
Figure 7-11 shows the dramatic performance drop when additional segments are used
on the snake. This data was found from running a long series of dynamic simulations
on snakes of varying length using a slowly propagating wave gait but not using high
forces or large excursions. Each segment corresponds to two degrees of freedom. In
general, the additional computation time is a function of the number of contact points;
this appears to be a quadratic relationship, although it can be even worse [Baraff 97].
The numbers for actual runs became much worse, as much as 500 times longer than
real-time for some configurations, taking several hours for an individual run.
Real-time playback 7.4.5
As a result of the runtime issue, simulation is quite slow for the larger interconnected
physical systems. For this reason, a method of storing the state of the system at each
timestep was developed. This stored the state vector of the entire system at each
iteration of the simulation and appended it to a file. The file could later be played back,
as rapidly as a purely kinematic model, to show the performance in real time.
Summary 7.5
Lessons 7.5.1
At this point in time the computational requirements for generalized learning of
complex physical systems seems prohibitive. The 100?s:1 ratio of real-time to simulated
time is sufficient to explore a single model, but when thousands must be explored it
becomes intractable. The obvious approach is the use of massive parallelization such as
2.0 4.0 6.0 8.0 10.0
number of segments
0.0
50.0
100.0
150.0
200.0
real- time/ simulated time
Figure 7-11: Performance drops significantly with increased number of segments.Locomotion
120
that by Sims or the further simplification of the physical model. Simplification brings
its own set of dangers for model fidelity; the results may be useless.
Another issue with the simulation system is the sensitivity to a variety of parameters.
This includes the dimensioned quantities such as length, mass, density etcetera. If other
values that are calculated from these values are too small then the resulting
discrepancies will eventually result in poor physical modeling. The other major
parameter that physical simulation is sensitive to is the size of the timestep between
iterations. Although the time step is adaptive, depending on the state of the simulation,
there is still a large sensitivity to timestep values. If too large, errors will almost
certainly occur and if too small, the amount of time required to run will increase. The
problem is determining what value of the timestep will provide the best trade-off of
accuracy and speed. There is no hard and fast rule. In the graphics community, this is
nearly a moot point; if it looks good, it is good.
There is no doubt that within a few years that the combination of computational power
and parallelizing techniques over networks will allow rapid high fidelity simulation and
testing for learning in physical robots..121
Chapter 8
Summary and Conclusions
Summary and Conclusions summarizes the results and contributions of this research and
looks into the future to see what research lies ahead in this area.
The old aphorism, ?you have to crawl before you can learn to walk?, has not applied to
mobile robotics. There have been many walking machines over the decades but very
few crawling machines. In fact, crawling appears to be a harder problem. In this
dissertation, research demonstrated that a snake robot can learn to crawl and it can crawl
in several different ways.
The conclusion of this research is that robots can learn to locomote even when they have
no wheels or legs. In this dissertation I provide a general framework to teach a complex
electromechanical robot to become mobile that includes a learning method, metrics for
evaluation, physical simulation and the transfer of results to a robot.
Is this thesis extendable to other mechanisms and forms? The framework and loop of
learning, testing and evaluation is certainly applicable to a wide variety of domains for
physical control. For locomotion, all patterns of motion, gaits, can be described in terms
of cyclical or periodic forms and this architecture lends itself well to learning those
modes of locomotion.
Contributions 8.1
There were a number of new and interesting developments in this work.
Contributions included:
? A novel and practical design for snake-like locomotors. With the exception of the
NEC snake, prior serpentine robots have paid scant attention to practical packaging of
devices such as actuators and wiring..Summary and Conclusions
122
? Learning to locomote with a limbless locomotor. Prior works have utilized explicit
models and relied on gaits contrived by humans. However, even the process described
here wasn?t as clean as simulate, then learn, then try on robot; it really became an
iterative process.
? Varying Multiple gaits using a single mechanism. Prior locomotors have shown only
one or two varients of a gait. Burdick?s Snakey did show three gaits: traveling wave,
stationary wave and an extensible wave; a type not possible in a snake or this robot. In
other works, ground contacts used wheels or pointed metal pins to provide ground
purchase. This has limited the types of locomotion that they can achieve. Snake-like
gaits shown in simulation and on the robot included rectinlinear, sidewinding and
lateral rolling. A wide variety of novel gaits and motions were also shown including
lateral undulation, varient of lateral rolling called the smoke ring and other novel gaits
including the ventral wave and the butterfly gait. These extraordinary gaits can achieve
locomotion without parallels in the animal kingdom.
? General learning technique and framework for representing periodic gaits. The
architecture proved valuable in development. It set the stage for creating and developing
gaits.
? The first development of skins for limbless locomotors. This included an evaluation
of many materials for use as ground contact interface and protective sheath. This is very
different from the point contacts and exposed mechanisms done by others in this area.
The resulting skins made from stretchable fabrics provide protection, a low-pass
filtering of the mechanics akin to a spline, and a smooth ground contact interface. This
is accomplished while being faithful to the shape and movement of the underlying
mechanism.
? A comprehensive examination of performance metrics and metric evaluation
including the development of a new metric - payload velocity to describe capbabilities
of working machines that transport material.
? Novel gaits including the flapping locomotion and the lateral rolling gaits. The wheel
has previously been shown in simulation by Yim.
Future Work 8.2
As with any sequence of work and discovery, you always discover how you can do it
better. There are many future advances in this work including mechanism and control.
I believe this mechanism works well but that there are numerous changes and
refinements to the current robot to improve it. Here are a few areas for further work:
? Gaits - the testing and evolution of gaits has only begun. While a number of intriguing
gaits have been shown, there is much more that can be done. Varied terrains,
experimenting further with direction, vertical climbing and more. Additional work in
steering and gait transistions is necessary for more general locomotion.With each test,
with each gait, new ideas suggest themselves and I forsee fruitful work in this area.
? Mechanism - There are a number of potential improvements to the mechanism. These
include: an easier-to-disassemble joint structure with a rapid mechanical and electric
connection; perhaps similar to bayonet style connectors. The use of lighter materials,.
Summary and Conclusions
123
such as polymers and composites, fabricated from molding processes will not only
lighten the structure but result in a beneficial cascade effect of requiring even smaller,
lighter actuators.
? Power - As with so many other applications, power is the critical technology for
deploying small robots. Long term energy and short term power needs dictate
limitations and capability. Recent advances in battery technologies and evolution of
technologies such as fuel cells will further this and many other applications.
? Sensing - The addition of sensing takes two forms: the sensor itself and the means to
use the sensor information. Sensing can utilize the small high-resolution force sensors
described herein but utilizing sensing in an appropriate manner will require further
research.
? Electronics- Wiring is a real issue, constant use involves wear and tear; wire exposure
results in abrasion, wear and failure. One suggestion is to develop a simple bus using
small PIC or ASIC devices to run motion control and feedback for each joint. This will
minimize wiring and increase the control and flexibility at end joint.
? Learning - Faster computing is inevitable. This will enable exploration of even more
of the search space. Stochastic learning techniques will benefit strongly from this and
the use of transparent parallelization across computing platforms. Planning is also
required; the ability to traverse 3D terrains will require substantial planning issues;
although a case can be made for a reactive strategy for overcoming obstacles and
marginal terrains.
? Physical Simulation - The first pass at real simulators is barely adequate. Coriolis and
a recent commercial package, Working Model, are steps in the right direction, but
computation needs and high fidelity modeling capability are sorely needed for complex
systems. The selection of simulation parameters, such as the time step and iteration
method can radically change the output of the simulation for the same starting
conditions.
Wisdom 8.3
It is possible to go full term from design and simulation to the locomotion control of
complex mobile robots. This can be done in the context of learning which can be used
both in simulation and on the real machine.
From here, many possibilities suggest themselves for the design and control of complex
mechanisms. Snake robots, in particular, can offer a variety of useful applications and
uses ranging from exploration to inspection.
Metrics are germane to evaluation. However, if taken too seriously for all locomotion,
metrics will tend to favor particular classes of vehicles. This is important to keep in
mind during the design or learning process..Summary and Conclusions
124.125
Appendix A
Servo Evaluation
Table 1: R/C Servo Comparison a
Model Torque
[N-m]
Dimensions
[mm]
Mass
[g]
Speed
[sec/60
deg]b
Power c
[Watts]
Torque
/Mass
[Nm/kg]
Torque/
Volume
[Nm/m3
/1000]
Power/
Weight
[W/N]
JR 341 0.23 12.70 28.45 29.72 17.86 0.24 1.18 12.613 20.98 6.73
JR 321 0.21 14.73 33.02 25.91 21.83 0.23 1.13 9.446 16.36 5.26
JR 3021 0.26 14.73 33.02 25.91 23.81 0.22 1.51 11.121 21.01 6.48
JR 3025 0.21 14.73 33.02 25.91 45.64 0.15 1.73 4.518 16.36 3.86
JR 901 0.30 18.03 34.80 33.53 37.71 0.27 1.42 8.072 14.47 3.83
JR 9021 0.41 18.03 34.80 33.53 42.53 0.22 2.32 9.549 19.30 5.56
JR 507 0.28 18.54 38.61 33.53 41.67 0.25 1.43 6.829 11.86 3.50
JR 517 0.28 18.54 38.61 33.53 44.79 0.25 1.43 6.354 11.86 3.26
JR 4000 0.52 18.54 38.61 33.53 49.90 0.19 3.44 10.417 21.66 7.03
JR 4131 0.64 18.54 38.61 33.53 42.53 0.23 3.49 15.012 26.60 8.36
JR 4721 d 0.84 18.54 38.61 33.53 48.76 0.22 4.82 17.321 35.19 10.09
JR 4735 0.64 18.54 38.61 33.53 48.76 0.15 5.32 13.034 26.48 11.13
JR 703 0.66 22.35 43.94 23.62 32.89 0.51 1.62 20.014 28.37 5.03
JR 7000 0.44 22.35 43.94 23.62 41.11 0.19 2.92 10.754 19.05 7.25
JR 7005 0.44 22.35 43.94 23.62 37.14 0.19 2.92 11.904 19.05 8.03
JR 3321 0.42 14.73 33.02 33.02 26.93 0.36 1.47 15.680 26.29 5.58
JR 605 0.98 32.00 63.50 58.42 134.66 0.28 4.41 7.295 8.27 3.34
Fut S125 0.91 22.35 39.62 42.93 65.21 0.62 1.85 14.004 24.02 2.89
Fut S132H 0.18 17.27 36.32 29.97 31.19 0.13 1.71 5.661 9.39 5.58.Servo Evaluation
126
a. All units converted to SI units. All figures using manufacturers specifications.
b. Servo speed is time to rotate 60 degrees. This is converted to rpm for power calculations.
c. Power is calculated using 25% of product of max stall torque and max rpm.
d. Selected for serpentine device.
Fut S3302 0.78 28.96 58.93 50.04 102.06 0.19 5.13 7.611 9.10 5.13
Fut S3303 1.41 28.96 58.93 50.04 107.73 0.26 6.82 13.111 16.54 6.46
FUT S9303 0.70 20.07 40.39 35.56 65.21 0.19 4.62 10.722 24.26 7.23
FUT S9304 0.49 20.07 40.39 35.56 48.20 0.22 2.80 10.184 17.03 5.93
FUT S9403 0.31 20.07 40.39 35.56 48.20 0.16 2.47 6.521 10.91 5.22
FUT S9601 0.25 15.75 30.73 29.97 31.19 0.17 1.88 8.175 17.57 6.16
Tower ts-72 0.94 58.42 27.94 50.80 99.23 0.22 5.36 9.466 11.33 5.51
Tower ts-55 0.30 40.64 20.32 38.10 45.36 0.20 1.91 6.695 9.65 4.29
Tower ts-11 0.21 27.94 13.72 27.94 17.29 0.15 1.77 12.251 19.79 10.47
Air 94831 0.27 37.08 18.03 29.97 31.19 0.17 1.98 8.605 13.39 6.49
Air 94510 0.78 47.50 22.86 39.12 65.21 0.33 2.96 11.914 18.29 4.63
Air 94501 0.14 26.92 12.45 26.92 18.43 0.33 0.54 7.665 15.65 2.98
Air 94401 0.23 30.99 14.99 30.99 26.93 0.26 1.13 8.653 16.19 4.26
Air 94403 0.18 30.99 14.99 30.99 26.93 0.20 1.11 6.555 12.27 4.20
Air 94737 0.39 39.37 20.32 35.56 53.87 0.15 3.25 7.211 13.65 6.16
Condor MS-747WB 1.18 54.61 52.07 26.67 113.40 0.26 5.70 10.400 15.55 5.13
Condor SSPS-105 35.31 130.05 55.12 111.00 779.63 0.60 73.90 45.291 44.38 9.67
Hitec 605-BB 0.54 40.89 19.81 39.88 49.05 0.16 4.27 11.087 16.83 8.88
Hitec rcd-apollo15 1.20 30.48 48.26 58.42 85.05 0.23 6.55 14.116 13.97 7.86
Hitec HS-615 0.76 38.10 20.32 40.64 60.10 0.21 4.52 12.573 24.02 7.67
Hitec HS-805BB 1.58 55.88 30.48 60.96 119.07 0.20 9.93 13.285 15.24 8.51
Hitec HS-205MG 0.30 33.02 17.78 33.02 31.19 0.20 1.91 9.738 15.66 6.24
Table 1: R/C Servo Comparison a
Model Torque
[N-m]
Dimensions
[mm]
Mass
[g]
Speed
[sec/60
deg]b
Power c
[Watts]
Torque
/Mass
[Nm/kg]
Torque/
Volume
[Nm/m3
/1000]
Power/
Weight
[W/N].127
Appendix B
Link Weight Distribution
The weight distribution of the link components is servos 68%, hardware 9%, and
brackets 23%. Since the actuators are 2/3 of the weight, there are diminishing returns
in attempting to reduce the weight of the hardware or brackets.
Table 2-1: Weight of snake link components.
Weight (g) Quantity Total weight (g)
Hardware 4mm bearing 0.60 1 0.60
4-40 FHCS, 1/4? 0.37 1 0.37
4-40 BHCS, 1/4? 0.35 8 2.80
4-40 nut 0.48 8 3.84
2-56 BHCS, 1/4? 0.20 8 1.60
2-56 nut 0.18 8 1.44
horn screw 0.50 2 1.00
bearing spacer 0.50 2 1.00
Hardware Total 12.65
Brackets servo plate 9.40 1 9.40
bearing plate 3.80 2 7.60
horn plate 4.00 2 8.00
octo plate 3.30 2 6.60
Bracket Total 31.60
Servo JR4721 45.00 2 90.00
horn 1.40 2 2.80
Servos Total 92.80
Link Total 137.05.
Link Weight Distribution
128.129
Appendix C
Derivation of Actuator Parameters
Developing an externally-based model of a control system requires observation of input
signals and a measured response of output. The actuator is treated as a second order
system incorporating stiffness and damping coefficients and a mass or inertia. This
model neglects non-linearities because the experimental results appeared to fit a second
order system very well. To solve for these requires observations combined with a
derivation based on the observable response. This is accomplished by determining the
undamped natural frequency, the damping ratio, and the period from experimental data
and the solving for the second-order parameters as follows:
[ C-1]
Solving for k:
[ C-2]
The damping ratio, shown below, is the ratio of the actual damping value, b, over the
critical damping value.
[ C-3]
Solving for b:
[ C-4]
Now, solving for another natural frequency by adding a small mass to the system:
w n 1
k
I 1
------ undamped natural frequency ==
k w n 1
2 I 1 =
z 1
b
2 kI 1
--------------- damping ratio ==
b 2 z 1 kI 1 =.Derivation of Actuator Parameters
130
[ C-5]
Solving for k gives:
[ C-6]
Now setting the two systems equal to each other:
[ C-7]
Then solve for I 1 :
[ C-8]
Now, substituting for k in Equation [ C-4]:
[ C-9]
Substituting for I 1 gives:
[ C-10]
Substituting for I1 into [ C-2] then gives:
[ C-11]
The three parameters, k, b, and I 1 , can now be used to find the parameters for the system
model. z is determined directly from the actuator response from Equation [ C-12] where
x n is the first amplitude peak, x n is the final value of the output and the period is the time
between zero crossings. See Figure C-1.
w n 2
k
I 1 I 2 + ------------------ - =
k w n 2
2 I 1 I 2 + ( ) =
w n 1
2 I 1 w n 2
2 I 1 I 2 + ( ) =
I 1
I 2
w n 1
2
w n 2
2 -------- 1 ?
? ?
? ?
? ?
? ?
----------------------- =
b 2 z 1 I 1 w n 1
2 2 z 1 I 1 w n 1
2 = =
b
2 z 1 I 2 w n 1
2
w n 1
2
w n 2
2 -------- 1 ?
? ?
? ?
? ?
? ?
------------------------ =
k
w n 1
2 I 1
w n 1
2
w n 2
2 -------- 1 ?
? ?
? ?
? ?
? ?
----------------------- =.Derivation of Actuator Parameters
131
[ C-12]
To do this, however, several parameters have to be determined experimentally, The
natural frequency, wn, and the damped natural frequency, wd, were determined
experimentally by optically tracking the output of the actuator. A tracking LED was
attached to an arm connected to the servo. Data from the position of the arc was
recorded at 1ms intervals during motion. Since the motion describes an arc, a circle was
fit to the data and used to determine the angle in radians as a function of time. This
decaying oscillation was measured directly from the data to determine the period of
oscillation (time between zero crossings). See Figure C-1. The waveform was also used
to determine the damping ratio based upon the amplitude decay [Ogata 78].
[ C-13]
and finally,
[ C-14]
Thus, the response parameters are found from experiments and the stiffness, damping
and inertial coefficients are found from those values.
Now, the problem is to map these model parameters onto the gains used in a typical
control system. Typical control system gains include proportional, derivative and
integral terms or k s , k d and k i respectively. The integral term can reduce or eliminate
steady state error in a system at the expense of settling time and longer term oscillations.
Even worse, it can introduce uncontrollable limit cycling. From observation of the
actuator output it does not appear that integral gain is used in it?s internal control
system; even if it is, it appears to have a negligible influence on the control system. The
z
1
n 1 ? ----------- - ? ?
? ? x 1
x n
----- ln()
4p 2 1
n ? ----------- - ? 1?
? ? x 1
x n
----- ln()
? ?
? ?
? ? +
2
------------------------------------------------------------ - =
w n
w d
1 z 2 ?
------------------
1
n 1 ? ----------- - ? ?
? ? x 1
x n
----- ln()
zT -------------------------------- ==
w d 2p T ? =
Figure C-1: Modeling the decaying oscillation of the servo actuator
x 1
t 1
T
t n
x n.Derivation of Actuator Parameters
132
system is quite stiff and the force does not appear to increase over time, only with
angular error. For this reason, our model is:
[ C-15]
Where b n is the natural damping of the system as separated from the derivative gain.
Rearranging terms gives
[ C-16]
The Laplace transform equivalent is
[ C-17]
Dividing through by I and equating equivalent coefficients gives the following
relations:
[ C-18]
and
[ C-19]
These terms define the total system gains. The problem is to now distinguish the two
contributions. The b, k, and I terms solved for previously are actually the numerator
sums in the left hand side of [ C-18] and [ C-19]. For the spring stiffness, k n , and the
proportional gain, k s , however, we will assume the contribution is entirely from the
system and that the actuator stiffness is infinite. This does not neglect stiffness, it
merely transfers all the effects into the overall system. Now the problem becomes the
separation of system damping from actuator damping; the b n + k d term. To eliminate
the closed loop term from the natural system term requires another experiment. The
actuator will be allowed to move under a gravity load, as a pendulum, and its response
observed. The settling time and oscillation provides a response of the natural system
and not closed loop behavior.
[ C-20]
For small angles, sinq = q and shifting from angular velocity terms, w, to angular terms,
q, results in:
[ C-21]
Again, similar to above, diving through by I, the Laplace version is as follows:
[ C-22]
Then, equating like terms and also, for a pendulum, the inertia is equal to the product
of the mass and the square of the length.
b n x? kx Ix??++ k s x d ? ( x)k d x? d ? ( x?) + =
b n k d + ( )x? kk s + ( )x Ix??++k s x d k d x? d + =
s 2 2 zw n s w n
2 ++
b n k d +
I -----------------2zw n =
k n k s +
I ---------------- w n
2 =
Iw? b n w +mg ?Lq sin =
Iq ?? b n q ? mgLq ++ 0 =
q( s)s
2 b n
I -----s mgL
I ----------- ++ 0 =.Derivation of Actuator Parameters
133
[ C-23]
Also
[ C-24]
Solving for b gives
[ C-25]
Thus, given the response of the system we can solve for the damping coefficient of the
actuator independent of the closed loop value shown earlier. The difference of the two
gives the derivative gain.
What is the end result of these derivations and underlying meaning? Can we establish
that the values are sufficient for simulation and modeling purposes? The purpose, when
we began, was to develop a model of sufficient fidelity that physical simulation results
are valid for transfer to the real system. Vagaries and idiosyncrasies of the physical
simulation tool make it difficult to ascribe figures of almost any accuracy, but the proof
is in the results - they appear to approximate the motions and dynamics of the real
system. With nearly all control systems, the control parameter gains are adjusted in an
iterative process because the pervasive effects of friction and contact are intractable to
model perfectly. It is no different in this process; values were determined and later
adjusted to reflect the actual actuator performance. However, the magnitude of these
adjustments was relatively small. The end result is a better understanding of actuator
performance and a model sufficient for simulation purposes.
2 zw n
b n
I ----- b n
mL 2 ----------- ==
w n
2 mgL
I ----------- mgL
mL 2 ----------- g
L --- ===
b n mL 2 2 z g
L --- =.Derivation of Actuator Parameters
134.135
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