Abstract
We consider mobile robots made of a single body (car-like robots) or several bodies (tractors towing several trailers sequentially hooked). These robots are known to be nonholonomic, i.e., they are subject to nonintegrable equality kinematic constraints involving the velocity. In other words, the number of controls (dimension of the admissible velocity space), is smaller than the dimension of the configuration space. In addition, the range of possible controls is usually further constrained by inequality constraints due to mechanical stops in the steering mechanism of the tractor. We first analyze the controllability of such nonholonomic multibody robots. We show that the well-known Controllability Rank Condition Theorem is applicable to these robots even when there are inequality constraints on the velocity, in addition to the equality constraints. This allows us to subsume and generalize several controllability results recently published in the Robotics literature concerning nonholonomic mobile robots, and to infer several new important results. We then describe an implemented planner inspired by these results. We give experimental results obtained with this planner that illustrate the theoretical results previously developed.
Similar content being viewed by others
References
Aho, A. V., Hopcroft, J. E., and Ullman, J. D.Data Structures and Algorithms. Addison-Wesley, Reading, MA, 1983.
Alexander, H. L. Experiments in Control of Satellite Manipulators. Ph.D. Dissertation, Department of Electrical Engineering, Stanford University, 1987.
Arnold, V. I.Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 1978.
Barraquand, J. and Latombe, J. C. Robot Motion Planning: A Distributed Representation Approach. Report STAN-CS-89-1257, Computer Science Department, Stanford University, Stanford, CA, May 1989.
Barraquand, J. and Latombe, J. C. Robot Motion Planning: A Distributed Representation Approach.The International Journal of Robotics Research,10(6) (1991).
Barraquand, J. and Latombe, J. C. On Nonholonomic Mobile Robots and Optimal Maneuvering.Proc. Fourth IEEE International Symposium on Intelligent Control, Albany, New York (1989), pp. 340–347.
Barraquand, J. and Latombe, J. C.: On Nonholonomic Mobile Robots and Optimal Maneuvering.Revue d'Intelligence Artificielle,3(2) (1989), 77–103.
Barraquand, J. and Latombe, J. C. Controllability of Mobile Robots with Kinematic Constraints. Report No. STAN-CS-90-1317, Department of Computer Science, Stanford University, June 1990.
Brooks, R. A. and Lozano-Pérez, T. A Subdivision Algorithm in Configuration Space for Find-Path with Rotation.Proc. Eighth International Joint Conference on Artificial Intelligence, Karlsruhe, FRG (1983), pp. 799–806.
Cai, C. Instantaneous Robot Motion with Contact Between Surfaces, Report No. STAN-CS-88-1191, Department of Computer Science, Stanford University, 1988.
Chow, W. L.: Uber Systeme von Linearen Partiellen Differentialgleichungen ester Ordnung.Mathematische Annalen,117 (1939), 98–105.
Cutkosky, M.Grasping and Fine Manipulation, Kluwer Academic, Boston, 1985.
Faverjon, B. Object Level Programming Using an Octree in the Configuration Space of a Manipulator.Proc. IEEE International Conference on Robotics and Automation, San Francisco (1986), pp. 1406–1412.
Faverjon, B. and Tournassoud, P. A Local Based Approach for Path Planning of Manipulators with a High Number of Degrees of Freedom.Proc. IEEE International Conference on Automation and Robotics, Raleigh, NC (1987), pp. 1152–1159.
Fortune, S. and Wilfong, G. Planning Constrained Motion.Proc. STOCS, ACM, Chicago (1988), pp. 445–459.
Gouzénes, L.: Strategies for Solving Collision-Free Trajectories Problems for Mobile and Manipulator Robots.The International Journal of Robotics Research,3(4) (1984), 51–65.
Greenwood, D. T.Principles of Dynamics, Prentice-Hall, Englewood Cliffs, NJ, 1965.
Haynes, G. W. and Hermes, H.: Non-linear Controllability via Lie Theory.SIAM Journal of Control,8 (1970), 450–460.
Hermann, R. On the Accessibility Problem in Control Theory.International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics (1963), pp. 325–332. Academic Press, New York.
Hermann, R. and Krener, A. J.: Nonlinear Controllability and Observability.IEEE Transactions on Automatic Control,22(5) (1977), 728–740.
Isidori, A.Nonlinear Control Systems: An Introduction, Springer-Verlag, New York, 1985.
Jacobs, P. and Canny, J. Planning Smooth Paths for Mobile Robots.Proc. IEEE International Conference on Robotics and Automation, Scottsdale, AZ (1989), pp. 2–7.
Jacobs, P., Laumond, J. P., and Taix, M. Efficient Motion Planners for Nonholonomic Mobile Robots.Proc. IEEE/RSJ International Workshop on Intelligent Robots and Systems, Osaka, Japan (1991), pp. 1229–1235.
Krener, A. J.: A Generalization of Chow's Theorem and the Bang-Bang Theorem to Nonlinear Control Problems.SIAM Journal of Control,12 (1974), 43–52.
Lafferriere, G. and Sussman, H. J. Motion Planning for Controllable Systems without Drift: A Preliminary Report. Report SYCON-90-04, Rutgers Center for Systems and Control, Rutgers University, New Brunswick, NJ, June 1990.
Latombe, J. C.Robot Motion Planning, Kluwer Academic, Boston, 1991.
Latombe, J. C. A Fast Path Planner for a Car-Like Indoor Mobile Robot.Ninth National Conference on Artificial Intelligence, Anaheim, CA (1991), pp. 659–665.
Laugier, C. and Germain, F. An Adaptative Collision-Free Trajectory Planner.International Conference on Advanced Robotics, Tokyo, Japan (1985).
Laumond, J. P. Feasible Trajectories for Mobile Robots with Kinematic and Environment Constraints.Proc. International Conference on Intelligent Autonomous Systems, Amsterdam (1986), pp. 346–354.
Laumond, J. P. Finding Collision-Free Smooth Trajectories for a Nonholonomic Mobile Robot.Proc. Tenth International Joint Conference on Artificial Intelligence, Milano, Italy (1987), pp. 1120–1123.
Laumond, J. P., Siméon, T., Chatila, R., and Giralt, G. Trajectory Planning and Motion Control of Mobile Robots.Proc. IUTAM/IFAC Symposium, Zurich, Switzerland (1988).
Laumond, J. P. and Siméon, T. Motion Planning for a Two Degrees of Freedom Mobile Robot with Towing. Technical Report No. 89-148, LAAS/CNRS, Toulouse, 1989.
Laumond, J. P. Nonholonomic Motion Planning Versus Controllability Via the Multibody Car System Example. Report STAN-CS-90-1345, Department of Computer Science, Stanford University, December 1990.
Li, Z. and Canny, J. F. Robot Motion Planning with Nonholonomic Constraints. Memo UCB/ERL M89/13, Electronics Research Laboratory, University of California, Berkeley, February 1989.
Li, Z., Canny, J. F., and Sastry, S. S. On Motion Planning for Dexterous Manipulation, Part I: The Problem Formulation.IEEE International Conference of Robotics and Automation, Scottsdale, AZ (1989), pp. 775–780.
Lobry, C.: Contôlabilité des systémes non lineaires.SIAM Journal of Control,8 (1970), 573–605.
Lozano-Pérez, T.: Spatial Planning: A Configuration Space Approach.IEEE Transactions on Computers,32(2) (1983), 108–120.
Lozano-Pérez, T.: A Simple Motion-Planning Algorithm for General Robot Manipulators.IEEE Journal of Robotics and Automation,3(3) (1987), 224–238.
Nakamura, Y. and Mukherjee, R. Nonholonomic Path Planning and Automation, Scottsdale, AZ (1989), pp. 1050–1055.
Nilsson, N. J.Principles of Artificial Intelligence, Morgan Kaufmann, Los Altos, CA, 1980.
Poston, T. and Stewart, I.Catastrophe Theory and its Applications, Pitman, London, 1978.
Reeds, J. A. and Sheep, R. A.: Optimal Paths for a Car that Goes both Forward and Backward.Pacific Journal of Mathematics,145(2) (1991), 367–393.
Sussmann, H. J. and Jurdjevic, V. J.: Controllability of Nonlinear Systems.Journal of Differential Equations,12 (1972), 95–116.
Taix, M. Planification de Mouvement pour Robot Mobile Non-Holonome. Doctoral Dissertation, LAAS, Toulouse, France, January 1991.
Tournassoud, P. and Jehl, O. Motion Planning for a Mobile Robot with a Kinematic Constraint.Proc. IEEE International Conference on Robotics and Automation, Philadelphia (1988), pp. 1785–1790.
Warner, F. W.Foundations of Differentiable Manifolds and Lie Groups. Springer-Verlag, New York, 1983.
Wilfong, G. Motion Planning for an Autonomous Vehicle.Proc. IEEE International Conference on Robotics and Automation, Philadelphia (1988), pp. 529–533.
Author information
Authors and Affiliations
Additional information
Communicated by Bruce Randall Donald.
This research was partially funded by DARPA contract DAAA21-89-C0002 (Army), CIFE (Center for Integrated Facility Engineering), and Digital Equipment Corporation.
Rights and permissions
About this article
Cite this article
Barraquand, J., Latombe, J.C. Nonholonomic multibody mobile robots: Controllability and motion planning in the presence of obstacles. Algorithmica 10, 121–155 (1993). https://doi.org/10.1007/BF01891837
Issue Date:
DOI: https://doi.org/10.1007/BF01891837