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Limbless Locomotion: Learning to Crawl with a Snake Robot the requirements for the degree of Kevin J. Dowling Advised by William L. Whittaker Carnegie Mellon University 5000 Forbes Avenue Pittsburgh, PA 15213 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 1997 by Kevin Dowling. Limbless Locomotion: Learning to Crawl Snake robots that learn to locomote Submitted in partial fulÞllment 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 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 Þrst 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 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 ofÞcemate of Þfteen years, Mike understands the 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. ItÕs been an enormous and beneÞcial inßuence. 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 Why Serpentine Locomotion?5 Challenges of Limbless Locomotion12 . 69 Design 77 Electronics 87 Sensing 89 Other Subsystems 90 3 Overview and Rationale Overview and Rationale proÞles 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 Introduction Overview and Rationale 4 features of snake robots include stability, terrainability, good traction, high redundancy Overview and Rationale 5 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 Why Serpentine Locomotion? For centuries, people have created a menagerie of machines whose appearance and movement have mirrored animals to an astonishing degree. There are anthropomorphic Þgures that resemble man and mobile machines that resemble animals. However, the strongest reactions are not simply to outward appearance; after all, costumes, statues Overview and Rationale 6 The general motivation for serpentine locomotors are environments where traditional 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 Þxed 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 signiÞcant challenges in designing, building and controlling practical limbless mechanisms that are capable of locomoting without traditional forms of propulsion and actuation. These Overview and Rationale 7 Terrainability 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 Òtrains,Ó or coupled-mobility devices, that use powered wheels, they still suffer from the limitations of wheel traction Overview and Rationale 8 Size Depending on the mechanism design, the small frontal area of snake mechanisms 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 ßexible 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, sufÞcient 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, efÞcient, 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 Overview and Rationale 9 Thermal Control 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 signiÞcantly less than for snakes. The Meeh coefÞcient, k, in the equation S = kM , where S is surface area and M is body mass, is higher for snakes than for many other mammals and Þsh. [Schmidt- Nielsen 84]. The effect of this may be that thermal control is more difÞcult 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 snakeÕs favor and is of 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. 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 difÞculty in 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 Þeld 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 There is a separate issue of Þxed-base serpentine devices and several of these applications would beneÞt from serpentine manipulators as well as serpentine locomotors. However, this work is concerned with locomotion and not simply 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 Overview and Rationale 10 Many inspection techniques in industry and medicine rely on Þxed-base mechanisms such as borescopes, videoscopes and Þberscopes. 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 difÞculties with current borescope use, plant equipment is modiÞed with portals, but this requires additional design and manufacturing resources but doesnÕt 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 Overview and Rationale 11 mechanisms. The ability to command small roving eyes and ears offers attractive work has resulted in inquires from law enforcement agencies including the FBI and Special Forces. Much effort in the wiring of existing structures requires routing of cables and lines Overview and Rationale 12 Challenges of Limbless Locomotion 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. ConÞguration and Design Summary The advantages of snake locomotion suggest a number of applications for their use. The 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 fulÞll these promises are many, and I address them in this dissertation. It is possible that a serpentine robot can be built and 13 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 Biological snakes, as existing limbless locomotors, offer lessons in design and function. The difÞculty, as shown, is the codiÞcation and extrapolation from biological animals It is inherent in the nomenclature; the use of the terms such as ÔserpentineÕ 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 ßight; manned ßight bears little resemblance to bird ßight with the exception of curved wing surfaces. In Background 14 addtion, many biological forms are scale dependent, and biological selection commonly reßects compromises among multiple events or inßuences in biological evolution. Commonly, biological evolution also leaves vestiges and forms that do not directly structures may be a misdirected effort [Bertram 94]. ItÕs worth keeping this in mind as lessons and ideas are drawn from snake morphology. Figure 2-1: Snake vertebrae provide lateral and ventral ßexing without permitting torsion. Background 15 Forms of Limbless Locomotion Snakes and other limbless animals have been objects of study for centuries. However, tend to signiÞcantly reduce its efÞcacy [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 inefÞcient as it might Þrst appear. However, the complexity of snake anatomy may make it difÞcult to realize these advantages in mechanisms. Concertina 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 conÞned areas, such as tunnels, where the snake cannot utilize the full amplitude of other gaits. As shown in Background Due to momentum changes, static friction, and slower speeds, concertina is a relatively inefÞcient 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 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 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 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 efÞcient than lateral undulation [Secor Figure 2-4: Sidewinding locomotion results from rolling or static contacts. Background In rectilinear locomotion, several portions of the body are in contact with the ground at Figure 2-5: Background 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 ÔwalkÕ 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 Both the skin and musculature of the snake are highly reÞned and specialized. It is not 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 Analysis of Limbless Locomotion [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]. Figure 2-6: Snake ribs, vertebrae and skin are linked by complex woven muscle bundles. Background 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 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 WhatÕs Missing? 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 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 mechanisms I came across were developed by a Russian constructivist artist of the Figure 2-7: Background undulation and later developed a series of wheeled coupled-mobility devices that followed from this work. HiroseÕs development of modeling and control Þrst 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 Figure 2-8: HiroseÕs Adaptive Cord Mechanism utilized a series of articulated links with passive wheels. -------------- ------------------------ ---------------- Background 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 Figure 2-9: A close-up of the Þrst ACM link showing the body and drive. Background 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 Figure 2-10: HiroseÕs locomoting Adaptive Cord Mechanism. Background Burdick and Chirikjian Joel Burdick and his students at Caltech, especially Greg Chirikjian , have pursued work in serpentine manipulation and locomotion for several years. ChirikjianÕs thesis presented a framework for kinematics and motion planning of serpentine mechanisms. Curves in three dimensions, R , are deÞned to provide a general means of 1.Now at Johns Hopkins University Figure 2-11: BurdickÕs Snakey, a VGT-style hyper-redundant manipulator and locomotor. Background 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-ßat ßoors 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. Background Þxed contact points from which the rest of the mechanism can move [Shan 92][Shan The conÞguration and locomotion of ShanÕs robot were limited to ßat ßoors 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 Figure 2-12: ShanÕs snake mechanism uses a concertina-like motion. Background 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. 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 Figure 2-13: The NEC ÔQuake SnakeÕ utilized a novel universal-type joint Figure 2-14: The rotating joint developed by Ikeda and Takanashi provides for a smooth and wide range of motion. Background 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 ÔhugÕ a tree and, using the side rolling capability, roll directly up the tree. The joint has another 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 Figure 2-15: The PIRAIA link provides a roll-pitch-roll capability. Background Paap 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 Figure 2-17: The GMD Snake mechanism Background 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 Figure 2-18: The Kaa snake is a self-contained locomoting device. Background 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 Figure 2-19: The ATMS could cross and climb obstacles. Background 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. Figure 2-20: An example of evolution from SimÕs creatures. Background Summary Biological Understanding 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 inßuence on locomotion but prior robot mechanism work has not addressed this issue. Robotics Developments 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 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]. Figure 2-21: The RoboTuna internals and foam latex covering. Background Learning Robots Figure 2-22: Bridging physical simulation, learning and robot mechanism is key to the control of complex mobile robots . [Schneider-j 95] Background as listed in the Þgure, have begun the work required in these areas. What has not been 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 results to a physical device. Sections of this chapter correspond to later chapters in this Overview 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 Framework Physical simulation is a useful tool for the conÞguration 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 reßected 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 Evaluation Figure 3-1: Framework for learning control of a physical device. Physical Modeling Framework maximum distance the device moves in a given period of time. This measurement, or Framework That is, the modeler is used in a larger framework that allows observation and reaction 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 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 Þdelity predictions of behavior. However, the initial physical model is used to provide general classes of gaits that can Figure 3-2: An accurate 3D model of the physical snake. Evaluation Figure 3-3: The same framework is used for the robot mechanism Framework Summary The conßuence 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 Framework EfÞciency EfÞciency is the ratio of power output to the power input of a system. It can be the 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 Þeld. This gives an intuitive feel for energy capacity and the value is independent of the size of the system. For example, using speciÞc energy values of 50Wh/kg for lead acid batteries means they can lift their own Another example of a comparison might be but this scale invariant measure that, again, favors very short vehicles. Clearly, a snake is at a signiÞcant 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 deÞning characteristic. Serpentine robot applications will mostly involve communications; a transfer of information that is only loosely coupled how they improved, or not. Some additional Þgures of speciÞc resistance for robots were made in [Gregorio 94]. Although Gabrielli and von K‡rm‡n used gross weight for their analysis they also Mecant ASV Figure 4-1: wheeled machine, but for the purposes of self comparison, speciÞc resistance offers a good measure of relative efÞcacy of locomotion. SpeciÞc 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 For a serpentine robot, speciÞc resistance also offers a measure that is straightforward 4.4 0.11 10 100 Melwalk Ambler Figure 4-2: Plot of data from Table 4-1 of speciÞc resistance versus velocity. speed l = 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 Summary and Selection Machine learning techniques evaluate past data to form insights on future performance; learning provides improved performance through experience. Learning and examines learning locomotion for simulated mechanisms and actual robots as well as criteria and structures for learning. This section also looks at the Optimization Techniques Learning and Optimization Learning and Optimization Learning and Optimization Although writing Þles is not as efÞcient as passing information through other means, 0.00100.00200.00 Learning and Optimization ()()// mutate probability vectorif (random(0,1) MUT_PROB then&#xMUT_;&#xPROB;&#x the;&#xnTj ;
.8 ;&#x-1.2;&#x TD ;&#xif r; ndo;&#xm0,1;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 deÞned constants (example values) This encoding of the solutions as statistics can, in many cases, be far more efÞcient than 5.2 Learning and Optimization ¥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. Learning and Optimization frequency Figure 5-3: Mapping from time varying representation to fourier representation to genome. Learning and Optimization ... Figure 5-4: A snake ÔtapeÕ deÞning joint angles at each time step for a periodic waveform. Learning and Optimization it prevents abrupt jumps and it culls unlikely gait patterns and body contortions, and Þnally, it reduces the gait space signiÞcantly. 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 a a a a ......... ...... a Figure 5-5: A 2D array can be used to represent all joint motions over time. Learning and Optimization 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 Frequency Figure 5-6: Learning and Optimization The landscape of gaits in this representation is quite extensive and the process of Summary Learning provides improved performance with frequent testing and evaluation. After Figure 5-7: Table generated from the fourier technique; columns represent joints angles and rows represent time history. Learning and Optimization Implementation 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 other issues involved in the selection process. These include Þdelity, response, power, Figure 6-1: Northern Anaconda vertebrae and 3D model constructed from MRI data. 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 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 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 Control Signal Mechanism Figure 6-2: Servos use small and efÞcient geartrains integrated with a positioning control loop. feedback electronics geartrain Table 6-1: SpeciÞcations for the selected servo. Servo Modeling 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 speciÞed 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 Figure 6-3: Servo Support bracket Tracking LED 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 Actuation is closely tied to the structural design that supports the robot and I examined and discarded many ideas and iterated a number of conÞgurations 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 0.01.02.03.04.05.0 1.0 1.5 2.0 2.5 3.0 Figure 6-4: The servo, as measured, exhibits a classic underdamped response. Time (seconds) mechanisms did not warrant the additional design, fabrication and maintenance that they required. By directly tying actuation to output, the mechanism was simpliÞed and made very compact even though the torque requirements increased. A beneÞcial cascade effect occurred that shortened and lightened joints, thus reducing structural In the right hand side of Figure 6-5, a right angle corridor of equal passage width, W, R q W Figure 6-5: q [6-2][6-3]Now substituting for Ro and Ri gives W as a function of link length and the angular 2 q 2 tan d 2 Ð = 2 R = -------------------- -------------------- Mechanism 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. utilizes eight links and eight parallel degrees of freedom. The caterpillar is capable of Figure 6-8: Exploded view of link mechanism. The 3D snake link design utilizes two orthogonal DOFÕs 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 simpliÞcation 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 Figure 6-9: The Þrst generation 3D link is comprised of two orthogonal servos. Figure 6-10: Figure 6-12: igus Trißex 3D cable chain. structure. Plastic corrugated materials such as vacuum hose were also examined but found to be too stiff. Rubber 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 Tread Figure 6-13: Table 6-2: Skin materials and corresponding coefÞcients of friction. Electronics 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 The advantages include the ability to provide both power and signal information over the same two wires to each electronic device. Thus, this Ôsnake busÕ is enormously simpliÞed 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 Servo Controller 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 . 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: where the position is the total excursion divided into 256 possible 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 conÞguration 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 Figure 6-14: Two resistive force sensors and a small tactile switch that were evaluated. acoustic signals to Þnd 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 Þnal 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 A robot is a substantial integration of several technologies. Not only mechanism, actuation and sensing but communications, power and computing. Each of these be taken into account during the design process. Communications 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 . These, in turn, are daisy-chained to a serial line connected to the computer. Since the learning experiments required high-performance platforms and the 4567 1011 1213 1415 1617 1819 Links/Servos 884 Servo Controllers Figure 6-15: The electrical system utilizes three servo controllers with a serial connection 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 signiÞcantly. For most testing however, single machine execution is sufÞcient 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 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 forward joints can act as positioners for the camera for pan and tilting the image. The Figure 6-16: A camera, shown by the arrow, provides a ÔsnakeÕs eyeÕ view. [Position] [Control] [Power] Figure 6-17: Active leds used for tracking Physical Modeling 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 takes as one of its arguments the material type which includes density, Summary A robot is a complex electro-mechanical integration of many technologies and decisions on conÞguration 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 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, The mechanism design proceeded through several iterations to simplify connection and support and reduce weight and complexity of the design. The Þnal 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 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 Þrst 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. Locomotion Power Calculations 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 The most straightforward analysis of power usage is the sum of the products of torque and angular velocity for all links as shown below. This gives an instantaneous measure of power use by the robot. An average power estimate over a Þxed 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 speciÞed time interval by assuming a Þxed 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 |Power| Figure 7-1: w 1 n T ú 1 = n ç÷ 1 = m In physical simulation, the consistent use of units is critical for relevant modeling of the snake robot. Consistent use of physical deÞnitions is one such requirement; units of 0.010.020.030.0 0.0 100.0 0.010.020.030.0 0.0 100.0 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. Velocity Calculations 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 The next several pages reveal Þgures of various stages of locomotion modes. The Þrst few are those snake-like modes demonstrated in simulation. This are interesting because they replicate existing modes in snakes. Sidewinding Sidewinding is a relatively efÞcient mode of locomotion with little sliding ground contact but with an odd means of moving laterally. Sidewinding is really two waves, one Figure 7-3: Sidewinding locomotion. 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 Figure 7-4: Lateral Undulation 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 ßat; the ground plane This locomotion conÞguration also slightly lifts the outward lateral wave. This is the the Ôsinus liftingÕ mentioned by Hirose. Figure 7-5: Lateral undulation. Lateral rolling An intriguing gait is formed by a U-shaped bbody and providing oscillating motions 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 Figure 7-6: Lateral rolling. Traveling wave rotor 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. Figure 7-7: Traveling wave rotor. 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. Figure 7-8: The wheel. 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 butterßy stroke. Figure 7-9: If the conÞguration 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 ßat 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 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. Figure 7-10: surface. These closed kinematic chains plus the large number of contacts contribute to 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 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 conÞgurations, taking several hours for an individual run. Real-time playback As a result of the runtime issue, simulation is quite slow for the larger interconnected Summary complex physical systems seems prohibitive. The 100Õs:1 ratio of real-time to simulated time is sufÞcient 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 2.04.06.08.010.0 50.0 100.0 200.0 Figure 7-11: Performance drops signiÞcantly with increased number of segments Summary and Conclusions Summary and Conclusions summarizes the results and contributions of this research and The old aphorism, Òyou have to crawl before you can learn to walkÓ, 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 Contributions 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 ¥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 wasnÕt 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. BurdickÕs Snakey did show three gaits: traveling wave, stationary wave and an extensible wave; a type not possible in a snake or this robot. In Future Work As with any sequence of work and discovery, you always discover how you can do it Summary and Conclusions such as polymers and composites, fabricated from molding processes will not only lighten the structure but result in a beneÞcial cascade effect of requiring even smaller, ¥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 described herein but utilizing sensing in an appropriate manner will require further ¥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 ßexibility at end joint. ¥Learning - Faster computing is inevitable. This will enable exploration of even more 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 Þrst 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 Þdelity modeling capability are sorely needed for complex complex mobile robots. This can be done in the context of learning which can be used From here, many possibilities suggest themselves for the design and control of complex Summary and Conclusions Servo Evaluation Table 1: R/C Servo Comparison Torque Dimensions[mm]Mass[g]Speed[sec/60 deg] Power [Watts] Torque Torque/ Volume /1000]Power/ Weight JR 3410.2312.7028.4529.7217.860.241.1812.61320.986.73 JR 3210.2114.7333.0225.9121.830.231.139.44616.365.26 JR 30210.2614.7333.0225.9123.810.221.5111.12121.016.48 JR 30250.2114.7333.0225.9145.640.151.734.51816.363.86 JR 9010.3018.0334.8033.5337.710.271.428.07214.473.83 JR 90210.4118.0334.8033.5342.530.222.329.54919.305.56 JR 5070.2818.5438.6133.5341.670.251.436.82911.863.50 JR 5170.2818.5438.6133.5344.790.251.436.35411.863.26 JR 40000.5218.5438.6133.5349.900.193.4410.41721.667.03 JR 41310.6418.5438.6133.5342.530.233.4915.01226.608.36 0.8418.5438.6133.5348.760.224.8217.32135.1910.09 JR 47350.6418.5438.6133.5348.760.155.3213.03426.4811.13 JR 7030.6622.3543.9423.6232.890.511.6220.01428.375.03 JR 70000.4422.3543.9423.6241.110.192.9210.75419.057.25 JR 70050.4422.3543.9423.6237.140.192.9211.90419.058.03 JR 33210.4214.7333.0233.0226.930.361.4715.68026.295.58 JR 6050.9832.0063.5058.42134.660.284.417.2958.273.34 Fut S1250.9122.3539.6242.9365.210.621.8514.00424.022.89 Fut S132H0.1817.2736.3229.9731.190.131.715.6619.395.58 Servo Evaluation Fut S33020.7828.9658.9350.04102.060.195.137.6119.105.13 Fut S33031.4128.9658.9350.04107.730.266.8213.11116.546.46 FUT S93030.7020.0740.3935.5665.210.194.6210.72224.267.23 FUT S93040.4920.0740.3935.5648.200.222.8010.18417.035.93 FUT S94030.3120.0740.3935.5648.200.162.476.52110.915.22 FUT S96010.2515.7530.7329.9731.190.171.888.17517.576.16 Tower ts-720.9458.4227.9450.8099.230.225.369.46611.335.51 Tower ts-550.3040.6420.3238.1045.360.201.916.6959.654.29 Tower ts-110.2127.9413.7227.9417.290.151.7712.25119.7910.47 Air 948310.2737.0818.0329.9731.190.171.988.60513.396.49 Air 945100.7847.5022.8639.1265.210.332.9611.91418.294.63 Air 945010.1426.9212.4526.9218.430.330.547.66515.652.98 Air 944010.2330.9914.9930.9926.930.261.138.65316.194.26 Air 944030.1830.9914.9930.9926.930.201.116.55512.274.20 Air 947370.3939.3720.3235.5653.870.153.257.21113.656.16 Condor MS-747WB1.1854.6152.0726.67113.400.265.7010.40015.555.13 Condor SSPS-10535.31130.0555.12111.00779.630.6073.9045.29144.389.67 Hitec 605-BB0.5440.8919.8139.8849.050.164.2711.08716.838.88 Hitec rcd-apollo151.2030.4848.2658.4285.050.236.5514.11613.977.86 Hitec HS-6150.7638.1020.3240.6460.100.214.5212.57324.027.67 Hitec HS-805BB1.5855.8830.4860.96119.070.209.9313.28515.248.51 Hitec HS-205MG0.3033.0217.7833.0231.190.201.919.73815.666.24 Table 1: R/C Servo Comparison Torque Dimensions[mm]Mass[g]Speed[sec/60 deg] Power [Watts] Torque Torque/ Volume /1000]Power/ Weight Link Weight Distribution The weight distribution of the link components is servos 68%, hardware 9%, and Link Weight Distribution -damping ratio 2 I Solving for k gives: --------1 2 --------1 ------------------------ --------1 ----- -------------------------------- p T ¤ = Figure C-1: Modeling the decaying oscillation of the servo actuator system is quite stiff and the force does not appear to increase over time, only with angular error. For this reason, our model is: Where bn is the natural damping of the system as separated from the derivative gain. Rearranging terms gives The Laplace transform equivalent is Dividing through by I and equating equivalent coefÞcients gives the following and[ C-19]These terms deÞne 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 and [ C-19]. For the spring stiffness, k , however, we will assume the contribution is entirely from the system and that the actuator stiffness is inÞnite. This does not neglect stiffness, it merely transfers all the effects into the overall system. Now the problem becomes the 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 Also[ C-24]Solving for b gives Thus, given the response of the system we can solve for the damping coefÞcient 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 sufÞcient for simulation and modeling purposes? The purpose, when we began, was to develop a model of sufÞcient Þdelity that physical simulation results are valid for transfer to the real system. Vagaries and idiosyncrasies of the physical simulation tool make it difÞcult to ascribe Þgures of almost any accuracy, but the proof is in the results - they appear to approximate the motions and dynamics of the real References Anonymous, ÒSteam Power 1899,Ó Automotive Industries, July 1995, p36. [Alexander 84] Alexander, R., ÒThe Gaits of Bipedal and Quadrupedal Animals,Ó International Journal of Robotics Research, Vol. 3, No. 2, Summer 1984, pages 49-59. [Alexander 92] Alexander, R., , ScientiÞc American Library, W.H. Freeman, New York, NY, 1992. Apostolopoulos, D., Bares, J., ÒConÞguration of a Robust Rappelling RobotÓ, IROS Õ95. Pittsburgh, PA. [Arakawa 93] Arakawa, K., Krotkov, E., ÒFractal Surface Reconstruction for Modeling Natu- References [Baraff 97] Baraff, David, ÒCoriolis Documentation,Ó Carnegie Mellon University and Physical Effects, Inc., 1997. Bares, J., ConÞguration of Autonomous Walkers for Extreme Terrain Dissertation in Civil Engineering, Carnegie Mellon University, May 1991. References Chirikjian, G.S., Theory and Applications of Hyper-Redundant Manipulators PhD Thesis, California Institute of Technology, Pasadena, CA 1992. Chirikjian, G.S., Burdick, J.W., ÒThe Kinematics of Hyper-Redundant Locomo- tion.Ó IEEE Transactions on Robotics and Automation, Vol. 11, No. 6, Decem- References Gabrielli, G., von K‡rm‡n, T.H., ÒWhat Price Speed?,Ó Mechanical Engineer- Gans, C., Biomechanics: An Approach to Vertebrate Biology References Hill, Pneumatic Self-Propelled Apparatus , US Patent 3224734, Assignee: Her References References Ogata, K., System Dynamics, Prentice-Hall, Englewood Cliffs, NJ. ISBN 0-13- Olympus, Brochures and conversations with sales representatives. 1994. [Ostrowski 96] Ostrowski, J.P., The Mechanics and Control of Undulatory Locomotion Thesis, California Institute of Technology, 1996. [Paap 96] Paap, K.L., Dehlwisch, M., Klaassen, B., ÒGMD-Snake: A Semi-Autonomous Snke-like Robot,Ó 3rd International Symposium on Distributed Autonomous Robot Systems (DARS 96), October, 29-31, 1996, RIKEN, Saitama, Japan. [Parisi 97] Parisi, P, ÒSnake Charmers,Ó Cinefex, No. 70, June 1997, pp. 61ff. [Anaconda movie special effects] [Parker 63] Parker, H.W., References References [Waldron 84] Waldron, K.J., Pery, A., McGhee, R.B., ÒConÞguration Design of the Adaptive Suspension Vehicle,Ó International Journal of Robotics Research, V 3, No. 2, [Waldron 97] Waldron, K.J., Personal conversation, 1997. [Walton 90] References