Abstract
We propose a general strategy for solving the motion planning problem for real analytic, controllable systems without drift. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added to the right-hand side. The main point then is to use formal calculations based on the product expansion relative to a P. Hall basis, to produce another control that achieves the desired result on the formal level. It then turns out that this control provides an exact solution of the original problem if the given system is nilpotent. When the system is not nilpotent, one can still produce an iterative algorithm that converges very fast to a solution. Using the theory of feedback nilpotentization, one can find classes of non-nilpotent systems for which the algorithm, in cascade with a precompensator, produces an exact solution in a finite number of steps. We also include results of simulations which illustrate the effectiveness of the procedure.
Work supported in part by the National Science Foundation under NSF Grant DMS-8902994.
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References
A.A. Agrachev and R.V. Gamkrelidze. The exponential representation of flows and chronological calculus. Math. Sbornik, 107(149):467–532, 1978.
J.A. Bathurin. Lectures on Lie algebras. Akademie Verlag, Berlin, 1978.
R.W. Brockett. Control theory and singular Riemannian geometry, pages 11–27. New Directions in Applied Mathematics. Springer-Verlag, New York, 1981.
M. Fliess. Réalisation locale des systèmes non linéaires, algèbres de lie filtrées transitives et series generatrices non commutatives. Invent. Math., 71:521–537, 1983.
H. Hermes. Distributions and the lie algebras their bases can generate. In Proc. Amer. Math. Soc. 106, No. 2, pages 555–565, 1989.
H. Hermes, A. Lundell, and D. Sullivan. Nilpotent bases for distributions and control systems. J. Diff. Eqs., 55(3):385–400, 1984.
J. Hauser, S. Sastry, and P. Kokotovic. Nonlinear control via approximate input-output linearization: the ball and beam example. In Proc. 28th IEEE CDC, pages 1987–1993, Tampa, Florida, Dec. 1989.
G. Lafferriere and H.J. Sussmann. Motion planning for controllable systems without drift: a preliminary report. SYCON report 90–04, Rutgers Center for Systems and Control, Rutgers University, New Brunswick, New Jersey, July 1990.
R.M. Murray and Sastry S.S. Grasping and manipulation using multifingered robot hands. Memorandum UCB/ERL M90/24, Electronics Research Laboratory, Univ. of California, Berkeley, Berkeley, California, March 1990.
R.W. Murray and S.S. Sastry. Steering nonholonomic systems using sinusoids. In Proc. 29th IEEE CDC, pages 2097–2101, Honolulu, Hawaii, Dec. 1990.
R. Palais. Global formulation of the Lie theory of transformation groups, volume 22 of Mem. Amer. Math. Soc. AMS, 1957.
H.J. Sussmann and G. Lafferriere. Motion planning for controllable systems without drift. In preparation.
S. Sastry and Zexiang Li. Robot motion planning with nonholonomic constraints. In Proc. 28th IEEE CDC, pages 211–216, Tampa, Florida, Dec. 1989.
H.J. Sussmann. Lie brackets and local controllability: a sufficient condition for scalar input systems. S.I.A.M. J. Control and Optimization, 21(5):686–713, 1983.
H.J. Sussmann. A product expansion for the Chen series, pages 323–335. Theory and Applications of Nonlinear Control Systems, C. Byrnes and A. Lindquist Eds. North-Holland, 1986.
H.J. Sussmann. A general theorem on local controllability. S.I.A.M. J. Control and Optimization, 25(1):158–194, 1987.
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Lafferriere, G., Sussmann, H.J. (1993). A Differential Geometric Approach to Motion Planning. In: Li, Z., Canny, J.F. (eds) Nonholonomic Motion Planning. The Springer International Series in Engineering and Computer Science, vol 192. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3176-0_7
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DOI: https://doi.org/10.1007/978-1-4615-3176-0_7
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