Controllability and motion planning of a multibody Chaplygin's sphere and Chaplygin's top

SUMMARY This paper studies local configuration controllability of multibody systems with nonholonomic constraints. As a nontrivial example of the theory, we consider the dynamics and control of a multibody spherical robot. Internal rotors and sliders are used as the mechanisms for control. Our model is based on equations developed by the second author for certain mechanical systems with nonholonomic constraints, e.g. Chaplygin’s sphere and Chaplygin’s top in particular, and the multibody framework for unconstrained mechanical systems developed by the first and third authors. Recent methods for determining controllability and path planning for multibody systems with symmetry are extended to treat a class of mechanical systems with nonholonomic constraints. Specific results on the controllability and path planning of the spherical robot model are presented. Copyright q 2007 John Wiley & Sons, Ltd. Received 16 February 2007; Revised 23 May 2007; Accepted 23 June 2007

[1]  Sangbum Cho.  Dynamics and control of underactuated multibody spacecraft , 2002 .

[2]  A. D. Lewis,et al.  Configuration Controllability of Simple Mechanical Control Systems , 1997 .

[3]  Francesco Bullo,et al.  A series describing the evolution of mechanical control systems , 1999 .

[4]  Alexey V. Borisov,et al.  The Rolling Body Motion Of a Rigid Body on a Plane and a Sphere. Hierarchy of Dynamics , 2003, nlin/0306002.

[5]  V. JURDJEVlC.  The Geometry of the Plate-Ball Problem , .

[6]  Jinglai Shen,et al.  NONLINEAR CONTROL OF MULTIBODY SYSTEMS WITH SYMMETRIES VIA SHAPE CHANGE , 2002 .

[7]  Anthony M. Bloch,et al.  Local equilibrium controllability of multibody systems controlled via shape change , 2004, IEEE Transactions on Automatic Control.

[8]  David Schneider.  Non-holonomic Euler-Poincare´ equations and stability in Chaplygin's sphere , 2002 .

[9]  Ranjan Mukherjee,et al.  Design considerations in the development of a spherical mobile robot , 2001, SPIE Defense + Commercial Sensing.

[10]  Edward John Routh,et al.  The elementary part of A treatise on the dynamics of a system of rigid bodies , 2009 .

[11]  Velimir Jurdjevic.  The geometry of the plate-ball problem , 1993 .

[12]  H. Sussmann.  A general theorem on local controllability , 1987 .

[13]  Anthony M. Bloch,et al.  Controllability and motion planning of multibody systems with nonholonomic constraints , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[14]  H. Sussmann.  Lie Brackets and Local Controllability: A Sufficient Condition for Scalar-Input Systems , 1983 .

[15]  K. Lynch.  Nonholonomic Mechanics and Control , 2004, IEEE Transactions on Automatic Control.

[16]  H. Zhang,et al.  Simple Mechanical Control Systems with Constraints and Symmetry , 2002, SIAM J. Control. Optim..

[17]  Richard Cushman.  Routh's sphere , 1997 .

[18]  Francesco Bullo,et al.  On mechanical control systems with nonholonomic constraints and symmetries , 2002, Syst. Control. Lett..

[19]  N. Harris McClamroch,et al.  Translational and Rotational Maneuvers of an Underactuated Space Robot using Prismatic Actuators , 2002, Int. J. Robotics Res..

[20]  Francesco Bullo,et al.  Series Expansions for the Evolution of Mechanical Control Systems , 2001, SIAM J. Control. Optim..

[21]  Richard M. Murray,et al.  Controllability of simple mechanical control systems , 1997 .

[22]  Richard M. Murray,et al.  Configuration Controllability of Simple Mechanical Control Systems , 1997, SIAM Rev..

[23]  R. Mukherjee,et al.  Motion Planning for a Spherical Mobile Robot: Revisiting the Classical Ball-Plate Problem , 2002 .

[24]  A. D. Lewis.  Local configuration controllability for a class of mechanical systems with a single input , 1997, 1997 European Control Conference (ECC).

[25]  Andrew D. Lewis,et al.  Simple mechanical control systems with constraints , 2000, IEEE Trans. Autom. Control..

[26]  L. Dai,et al.  Non-holonomic Kinematics and the Role of Elliptic Functions in Constructive Controllability , 1993 .