Geometric Controllability and Stabilization of Spherical Robot Dynamics

Geometric control of a spherical robot rolling on a horizontal plane with three independent inertia disc actuators is considered in this note. The dynamic model of the spherical robot in the geometric framework is used to establish the strong accessibility and small-time local controllability properties. Smooth stabilizability to an equilibrium fails for the nonholonomic spherical robot. A novel contribution of this note is a smooth, asymptotically stabilizing geometric control law for position and reduced attitude, which corresponds to an equilibrium submanifold of dimension one. From Brockett's condition, this is the best possible dimension of a smoothly stabilized equilibrium submanifold. We also present a novel smooth global tracking controller for tracking position trajectories.

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

[2]  Puyan Mojabi,et al.  Introducing August: a novel strategy for an omnidirectional spherical rolling robot , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[3]  Qiang Zhan,et al.  Design, analysis and experiments of an omni-directional spherical robot , 2011, 2011 IEEE International Conference on Robotics and Automation.

[4]  Atsushi Koshiyama,et al.  Design and Control of an All-Direction Steering Type Mobile Robot , 1993, Int. J. Robotics Res..

[5]  Yan Wang,et al.  Motion control of a spherical mobile robot , 1996, Proceedings of 4th IEEE International Workshop on Advanced Motion Control - AMC '96 - MIE.

[6]  Sanjay P. Bhat,et al.  Controllability of nonlinear time-varying systems: applications to spacecraft attitude control using magnetic actuation , 2005, IEEE Transactions on Automatic Control.

[7]  Gianna Stefani,et al.  Controllability along a trajectory: a variational approach , 1993 .

[8]  Alexey V. Borisov,et al.  How to control Chaplygin’s sphere using rotors , 2012 .

[9]  Wouter Saeys,et al.  Modeling and control of a spherical rolling robot: a decoupled dynamics approach , 2011, Robotica.

[10]  Ranjan Mukherjee,et al.  Reconfiguration of a Rolling Sphere: A Problem in Evolute-Involute Geometry , 2006 .

[11]  Ravi N. Banavar,et al.  The Euler-Poincaré equations for a spherical robot actuated by a pendulum , 2012 .

[12]  Jean-Baptiste Pomet.  Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift , 1992 .

[13]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[14]  Claude Samson,et al.  Time-varying Feedback Stabilization of Car-like Wheeled Mobile Robots , 1993, Int. J. Robotics Res..

[15]  Ravi N. Banavar,et al.  Motion analysis of a spherical mobile robot , 2009, Robotica.

[16]  Mehdi Keshmiri,et al.  Stabilization of an autonomous rolling sphere navigating in a labyrinth arena: A geometric mechanics perspective , 2012, Syst. Control. Lett..

[17]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[18]  Mark A. Minor,et al.  Simple motion planning strategies for spherobot: a spherical mobile robot , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[19]  Sunil K. Agrawal,et al.  Spherical rolling robot: a design and motion planning studies , 2000, IEEE Trans. Robotics Autom..

[20]  Anthony M. Bloch,et al.  Controllability and motion planning of a multibody Chaplygin's sphere and Chaplygin's top , 2008 .

[21]  Sun Hanxu,et al.  The Design and Analysis of A Spherical Mobile Robot , 2004 .

[22]  S. Bhat,et al.  A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon , 2000 .

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

[24]  Antonio Bicchi,et al.  Introducing the "SPHERICLE": an experimental testbed for research and teaching in nonholonomy , 1997, Proceedings of International Conference on Robotics and Automation.

[25]  Shigeyuki Hosoe,et al.  Motion Planning Algorithms for a Rolling Sphere With Limited Contact Area , 2008, IEEE Transactions on Robotics.