Rolling bodies with regular surface: controllability theory and applications

Pairs of bodies with regular rigid surfaces rolling onto each other in space form a nonholonomic system of a rather general type, posing several interesting control problems of which not much is known. The nonholonomy of such systems can be exploited in practical devices, which is very useful in robotic applications. In order to achieve all potential benefits, a deeper understanding of these types of systems and more practical algorithms for planning and controlling their motions are necessary. In this paper, we study the controllability aspect of this problem, giving a complete description of the reachable manifold for general pairs of bodies, and a constructive controllability algorithm for planning rolling motions for dexterous robot hands.

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

[2]  Ole Jakob Sørdalen,et al.  Conversion of the kinematics of a car with n trailers into a chained form , 1993, [1993] Proceedings IEEE International Conference on Robotics and Automation.

[3]  Alessandro De Luca,et al.  Control of nonholonomic systems via dynamic compensation , 1993, Kybernetika.

[4]  Yoshihiko Nakamura,et al.  Exploiting nonholonomic redundancy of free-flying space robots , 1993, IEEE Trans. Robotics Autom..

[5]  A. Agrachev,et al.  An intrinsic approach to the control of rolling bodies , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[6]  Eduardo Sontag.  Control of systems without drift via generic loops , 1995, IEEE Trans. Autom. Control..

[7]  Ilya Kolmanovsky,et al.  Developments in nonholonomic control problems , 1995 .

[8]  A. Chelouah,et al.  Extensions of differential flat fields and Liouvillian systems , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[9]  Richard M. Murray,et al.  Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems , 1994, Math. Control. Signals Syst..

[10]  Susumu Tachi,et al.  Dynamic control of a manipulator with passive joints in an operational coordinate space , 1991, Proceedings. 1991 IEEE International Conference on Robotics and Automation.

[11]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[12]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[13]  M. Fliess,et al.  Flatness, motion planning and trailer systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[14]  S. Sastry,et al.  Trajectory generation for the N-trailer problem using Goursat normal form , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[15]  D. Normand-Cyrot,et al.  An introduction to motion planning under multirate digital control , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[16]  O. J. Sørdalen,et al.  Exponential stabilization of nonholonomic chained systems , 1995, IEEE Trans. Autom. Control..

[17]  David J. Montana,et al.  The Kinematics of Contact and Grasp , 1988, Int. J. Robotics Res..

[18]  Mitsuji Sampei,et al.  Arbitrary path tracking control of articulated vehicles using nonlinear control theory , 1995, IEEE Trans. Control. Syst. Technol..

[19]  Antonio Bicchi,et al.  Dexterous manipulation through rolling , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[20]  S. Shankar Sastry,et al.  Kinematics and control of multifingered hands with rolling contact , 1989 .

[21]  G. Jacob.  MOTION PLANNING BY PIECEWISE CONSTANT OR POLYNOMIAL INPUTS , 1992 .

[22]  Antonio Bicchi,et al.  Planning motions of rolling surfaces , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[23]  Gerardo Lafferriere,et al.  Motion planning for controllable systems without drift , 1991, Proceedings. 1991 IEEE International Conference on Robotics and Automation.

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

[25]  Antonio Bicchi,et al.  Planning Motions of Polyhedral Parts by Rolling , 2000, Algorithmica.

[26]  M. Levi.  Geometric phases in the motion of rigid bodies , 1993 .

[27]  A. Marigo,et al.  Dexterity through rolling: manipulation of unknown objects , 1999, Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C).

[28]  H. Sussmann.  Orbits of families of vector fields and integrability of distributions , 1973 .

[29]  Joel W. Burdick,et al.  Geometric Perspectives on the Mechanics and Control of Robotic Locomotion , 1996 .

[30]  C. Samson.  Control of chained systems application to path following and time-varying point-stabilization of mobile robots , 1995, IEEE Trans. Autom. Control..

[31]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[32]  S. Sastry,et al.  Nonholonomic motion planning: steering using sinusoids , 1993, IEEE Trans. Autom. Control..

[33]  H. Sussmann,et al.  A continuation method for nonholonomic path-finding problems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[34]  Zexiang Li,et al.  Motion of two rigid bodies with rolling constraint , 1990, IEEE Trans. Robotics Autom..

[35]  R. Brockett,et al.  On the rectification of vibratory motion , 1989, IEEE Micro Electro Mechanical Systems, , Proceedings, 'An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots'.

[36]  Woojin Chung,et al.  Design of a nonholonomic manipulator , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.