In This Issue
Centennial of Aviation
March 1, 2004 Volume 34 Issue 1

Mechanics, Control, and Applications of Biomimetic Robotic Locomotion

Wednesday, December 3, 2008

Author: Joel W. Burdick

Remarks delivered by Joel W. Burdick at the National Academy of Engineering 1999 National Meeting, Arnold and Mabel Beckman Center, Irving CA.

Since mobility can be an essential requirement for the operation of many autonomous systems, robotic locomotion has been actively studied for over three decades. Most mobile robots use wheels, since they provide the simplest means for mobility. However, some terrains are inaccessible to wheeled vehicles, and wheels are undesirable for a number of applications. Biomimetic locomotion refers to the movement of robotic mechanisms in ways that are analogous to the patterns of movement found in nature. Biomimetic robotic locomotors do not rely on wheels, tracks, jets, thrusters, or propellers for their propulsion.

Despite more than three decades of research effort, biomimetic robotic locomotors have largely remained laboratory curiosities. However, promising applications of biomimetic mobility still motivate research in this area. The potential for robots to assist the elderly in their homes has recently motivated the Honda Motor Co. (1998) to invest heavily in bipedal walking research, since anthropomorphic robots can adapt better to preexisting environments that were designed for humans. In the 1980s, the Defense Advanced Research Projects Agency provided significant support for Ohio State University’s Adaptive Suspension Vehicle, a six-legged, 7,000-pound. locomotor that was intended to support combat operations in complex terrain (Song and Waldron, 1989). Nonlegged locomotory machines might find significant applications as well. For example, efforts are under way at Caltech to develop a "snakelike" robotic endoscope that would enable minimally invasive access to the human small bowel system, which is currently inaccessible by conventional endoscopes. Snakelike robots have also been investigated for use in urban search-and-rescue operations following earthquakes or other natural disasters. Biomimetic fluid propulsion based on changes in the mechanism’s shape offers an alternative to traditional underwater vehicle propulsion based on propellers and control surfaces. Such fluid propulsion may be very maneuverable at many size scales and is free from the motor noise, vibrations, and propeller cavitation associated with propellers.

Current Status
Quasi-static legged locomotion (where the locomotor’s center of mass is always supported by at least three legs in ground contact) has been the most extensively studied type of biomimetic locomotion, and many four- and six-legged robots have been successfully demonstrated (Song and Waldron, 1989). Beginning with Raibert (1986), legged hopping robots have received considerable experimental and analytical attention (M’Closkey and Burdick, 1993). Bipedal walking and running also has been an active area of study (McGeer, 1990; Honda Motor Co., 1998). "Snakelike" robots can potentially enter environments that are inaccessible to legged or wheeled vehicles. Significant work in snakelike locomotion was initiated in the 1970s by Hirose and Umetani (1976) and more recently by Chirikjian and Burdick (1995). Realistic efforts to develop fishlike robots have emerged only recently (Barrett, 1996; Kelly et al., 1998).

Research Needs and Objectives
There are many issues that limit widespread deployment of biomimetic locomotors, including limitations in actuation technology, onboard power-carrying capability, and sensing. While all of these issues merit serious attention, this paper briefly discusses the associated limitations in theory. In an attempt to derive useful results for specific examples, prior biomimetic locomotion studies have generally focused on a particular robot morphology (such as a biped or quadruped). Unfortunately, results derived for one morphology typically do not extend to other morphologies. To enable future widespread deployment of cheap and robust robotic locomotion platforms, we must ultimately seek a more unifying and comprehensive framework for biomimetic robotic locomotion engineering. This framework should have the following properties:

  • the analysis, design, and control methodologies can be uniformly applied to a broad class of locomotory problems;
  • significant aspects of the framework can be encoded in automated software tools; and
  • the underlying methods are sufficiently rigorous to predict and enable robust system performance.

Realization of such a framework would enable more widespread deployment of effective biomimetic locomotors. One strategy being pursued at Caltech to realize this framework is to (1) establish general forms for the equations of motion of locomotion systems, (2) develop a control theory for this class of nonlinear equations, (3) abstract motion planning and feedback control algorithms from the control theory, and (4) develop paradigms (perhaps rules of thumb) for designing systems for specific applications. Caltech’s research program, whose underlying concepts are briefly sketched in the next section, has reached a good understanding of the underlying mechanics principles. The associated control theory and algorithms are currently in development.

Principles of Biomimetic Propulsion
Biomimetic propulsion is typically generated by a coupling of periodic mechanism deformations to external constraints (i.e., mechanical interactions with the environment). The forces generated by these constraint interactions (e.g., pushing, rolling, sliding) induce net robot movement. The creeping, sidewinding, and undulatory gaits of snakes rely on no-slip, or nonholonomic, constraints. Slug and snail movement depends on the viscous fluid constraint of slime trails, while amoebae and paramecia move via a constraint between their surfaces and the surrounding fluid. Fish use a variety of fluid mechanical constraint principles. Surprisingly, common principles underlie the mechanics and control of these seemingly different systems (Ostrowski and Burdick, 1998).

The language of geometric mechanics has proven to be a useful way to precisely phrase these intuitive notions. Two simple concepts motivated by geometric thinking have proven useful in developing a comprehensive basis for the mechanics and control of biomimetic locomotion. The first key observation is that it is always possible to divide a locomoting robot’s configuration variables into two classes. The first class of variables describes the position of the robot - that is, the displacement of a robot fixed coordinate frame with respect to a fixed reference frame. The set of frame displacements is SE(m), m 3, or one of its subgroups - that is, a Lie group. The second set of variables defines the mechanism’s internal configuration or shape. The set of all possible shapes (the "shape space") is a manifold, M. The Lie group, G, together with the shape space, M, form the total configuration space of the system, denoted by Q = G M. The configuration space of both terrestrial and aquatic biomimetic locomotors is a trivial principal fiber bundle.

The importance of the principal fiber bundle structure of the configuration space of locomoting systems is related to the following facts. The shape and position variables are coupled by the constraints acting on the robot. By making changes in the shape variables, it is possible to effect changes in the position variables through the constraints. A central goal of locomotion analysis is the systematic derivation of an expression that answers the question: If I wiggle the body, how far does the mechanism locomote? Formally, this all-important relationship between shape changes and position changes can be described via a connection, which is an intrinsic geometric feature of principal fiber bundles. Recent Caltech efforts have shown that there is a systematic way to derive the connection for a very large class of locomotion problems. The connection provides not only a unified way of thinking about mechanics but also one about motion planning and control (Radford and Burdick, 1998).

The second key idea is the use of symmetries in locomotion analysis. Because a locomotor is a mechanical system, we can assume there exists a Lagrangian and a set of constraint equations that describe the interaction principle underlying the given locomotion scheme. A symmetry corresponds to a group of transformations that leave the Lagrangian (and possibly the constraints) invariant. In the absence of constraints, these symmetries correspond to the well-known principles of momentum conservation. Unfortunately, conservation laws are not necessarily preserved in the presence of most constraints, which are essential to locomotion. However, it is possible to extend classical theory (see Bloch et al., 1996, for the case of nonholonomic constraints) to develop a generalized momentum equation that describes the evolution of the momenta due to the interaction constraints. The mixture of symmetry and interaction constraints can give rise to the ability to increase or control momentum via the action of internal forces. This is an extremely important effect in generating biomimetic locomotion. The aforementioned connection and the use of invariance principles (and their associated momenta equations) yield a comprehensive framework for the fundamental analysis of biomimetic locomotion mechanics and control.

Barrett, D. S. 1996. Propulsive Efficiency of a Flexible Hull Undersea Vehicle. Ph.D. dissertation. Massachusetts Institute of Technology, Cambridge.

Bloch, M., P. S. Krishnaprasad, J. E. Marsden, and R. M. Murray. 1996. Nonholonomic mechanical systems with symmetry. Archive for Rational Mechanics and Analysis 136(l):21-99.

Chirikjian, G. S., and J. W. Burdick. 1995. The kinematics of hyper-redundant robot locomotion. IEEE Transactions on Robotics and Automation 11(6):781-793.

Hirose, S., and Y. Umetani. 1976. Kinematic control of active cord mechanism with tactile sensors. Pp. 241-252 in Proceedings of the Second International CISM-IFT Symposium on Theory and Practice of Robots and Manipulators, A. Morecki and K. K?edzior, eds. New York: Elsevier Scientific Publishers.

Honda Motor Co., Ltd. 1998. Humanoid robot. [Online]. Available: html

Kelly, S., R. Mason, C. Anhalt, R. Murray, and J. Burdick. 1998. Modeling and experimental investigation of carangiform locomotion for control. Paper presented at the American Control Conference, Philadelphia, Pa., June 24-26, 1998.

McGeer, T. 1990. Passive dynamic walking. International Journal of Robotics Research 9(2):62-82.

M’Closkey, R. T., and J. W. Burdick. 1993. On the periodic motions of a hopping robot with vertical and forward motion. International Journal of Robotics Research 12(3):197-218.

Ostrowski, J. P., and J. W. Burdick. 1998. The mechanics of undulatory robotic locomotion. International Journal of Robotics Research 17(7):683-701.

Radford, J., and J. Burdick. 1998. Local motion planning for nonholonomic control systems evolving on principal bundles. Paper presented at the International Symposium on Mathematical Theory of Networks and Systems (MTNS ’98), Padova, Italy, July 6-11, 1998.

Raibert, M. H. 1986. Legged Robots that Balance. Cambridge, Mass.: MIT Press.

Song, S. M., and K. J. Waldron. 1989. Machines that Walk: The Adaptive Suspension Vehicle. Cambridge, Mass.: MIT Press.

About the Author:Joel W. Burdick is Associate Professor, Department of Mechanical Engineering, California Institute of Technology, Pasadena, California. The author was a participant in the 1998 Frontiers of Engineering Symposium. The Symposium publication Reports on the Leading Edge: Engineering from the 1998 NAE Symposium on Frontiers of Engineering is available online