Abstract
The problem of motion planning for nonholonomic dynamic systems is studied. A model for nonholonomic dynamic systems is first presented in terms of differential- algebraic equations defined on a phase space. A nonlinear control system in a normal form is introduced to completely describe the dynamics. The assumptions guarantee that the resulting normal form equations necessarily contain a nontrival drift vector field. We show that the linearized control system always has uncontrollable eigenvalues at the origin. However, any equilibrium is shown to be strongly accessible and small time locally controllable. A motion planning approach using holonomy is developed for nonholonomic Caplygin dynamical systems, i.e. nonholonomic systems with certain symmetry properties which can be expressed by the fact that the constraints are cyclic in certain of the variables. The theoretical development is applied to physical examples of systems that we have studied in detail elsewhere: the control of motion of a knife edge moving on a plane surface and the control of motion of a wheel rolling without slipping on a plane surface. The results of the paper are also applied to the control of motion of a planar multibody system using angular momentum preserving joint torques. Results of simulations are included to illustrate the effectiveness of the proposed motion planning approach.
Supported by in part by NSF Grant DMS-9002136 and NSF PYI Grant DMS-9157556
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References
V.I. Arnold, Dynamical Systems III, Vol.3, Springer-Verlag, 1988.
A.M. Bloch and N.H. McClamroch,“Control of Mechanical Systems with Classical Nonholonomic Constraints,” Proceedings of IEEE Conference on Decision and Control, 1989, Tampa, Fla., 201–205.
A.M. Bloch, N.H. McClamroch and M. Reyhanoglu,“Controllability and Stabilizability Properties of a Nonholonomic Control System,” Proceedings of IEEE Conference on Decision and Control, 1990, Honolulu, Hawaii, 1312–1314.
A.M. Bloch, M. Reyhanoglu and N.H.McClamroch,“Control and Stabilization of Nonholonomic Dynamic Systems,” Submitted to IEEE Transactions on Automatic Control, 1991.
A.M. Bloch, M. Reyhanoglu and N.H.McClamroch,“Control and Stabilization of Nonholonomic Caplygin Dynamic Systems,” Proceedings of IEEE Conference on Decision and Control, 1991, Brighton, UK.
R.W. Brockett, “Control Theory and Singular Riemannian Geometry”, in New Directions in Applied Mathematics, ed. P.J. Hilton and G.S. Young, Springer-Verlag, 1982.
P.S. Krishnaprasad,“Eulerian Many-Body Problems,” in Dynamics and Control of Multibody Systems, eds. J.E. Marsden, P.S. Krishnaprasad and J. Simo, Contemporary Math. Vol.97, AMS, 1989.
P.S. Krishnaprasad,“Geometric Phases and Optimal Reconfiguration for Multibody Systems,” Proceedings of American Control Conference, 1990, San Diego, 2440–2444
G. Laffarriere and H.J. Sussmann,“Motion Planning for Completely Controllable Systems Without Drift,” Proceedings of Conference on Robotics and Automation, 1991, Sacramento, California, 1148–1153.
Z. Li and J. Canny,“Motion of Two Rigid Bodies with Rolling Constraint,” IEEE Transactions on Robotics and Automation, Vol. 6, No. 1, 1990, 62–71.
Z. Li and R. Montgomery,“Dynamics and Optimal Control of a Legged Robot in Flight Phase,” Proceedings of Conference on Robotics and Automation, 1990, Cincinnati, Ohio, 1816–1821.
J. Marsden, R. Montgomery and T. Ratiu, Reduction, Symmetry and Berry’s Phase in Mechanics, Memoirs of the A.M.S., 1990.
R. Montgomery,“Isoholonomic Problems and Some Applications,” Communications in Mathematical Physics, Vol. 128, 1990, 565–592.
R. Murray and S. Sastry,“Steering Nonholonomic Systems Using Sinusoids,” Proceedings of IEEE Conference on Desicion and Control, 1990, Honolulu, Hawai, 2097–2101.
Ju. I. Neimark and F.A. Fufaev, Dynamics of Nonholonomic Systems, Vol. 33, A.M.S. Translations of Mathematical Monographs, A.M.S., 1972.
M. Reyhanoglu and N. H. McClamroch,“Controllability and Stabilizability of Planar Multibody Systems with Angular Momentum Preserving Control Torques,” 1991 American Control Conference, Boston, 1102–1107.
M. Reyhanoglu and N. H. McClamroch, “Reorientation of Space Multibody Systems Maintaining Zero Angular Momentum,” 1991 AIAA Guidance, Navigation and Control Conference, New Orleans, 1330–1340.
A. Shapere and F. Wilczek, Geometric Phases in Physics, Wold Scientific, 1988.
N. Sreenath,“Nonlinear Control of Multibody Systems in Shape Space,” Proceedings of IEEE Conference on Robotics and Automation, 1990, Cincinnati, Ohio, 1776–1781.
H.J. Sussmann,“A General Theorem on Local Controllability,” SIAM Journal on Control and Optimization, Vol. 25, No. 1, 1987, 158–194.
H.J. Sussmann and V. Jurdjevic, “Controllability of Nonlinear Systems,” J. Differential Equations, 12, 1972, 95–116.
A. M. Vershik and V. Ya. Gershkovich,“ Nonholonomic Problems and the Theory of Distributions,” Acta Applicandae Mathematicae, 12, 1988, 181–209.
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Reyhanoglu, M., McClamroch, N.H., Bloch, A.M. (1993). Motion Planning for Nonholonomic Dynamic Systems. 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_6
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DOI: https://doi.org/10.1007/978-1-4615-3176-0_6
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