Nonlinear Controllability Assessment of Aerial Manipulator Systems using Lagrangian Reduction

This paper analyzes the nonlinear Small-Time Local Controllability (STLC) of a class of underatuated aerial manipulator robots. We apply methods of Lagrangian reduction to obtain their lowest dimensional equations of motion (EOM). The symmetry-breaking potential energy terms are resolved using advected parameters, allowing full SE(3) reduction at the cost of additional advection equations. The reduced EOM highlights the shifting center of gravity due to manipulation and is readily in control-affine form, simplifying the nonlinear controllability analysis. Using Sussmann’s sufficient condition, we conclude that the aerial manipulator robots are STLC near equilibrium condition, requiring Lie bracket motions up to degree three.

[1]  J. Ostrowski.  The mechanics and control of undulatory robotic locomotion , 1995 .

[2]  Joel W. Burdick,et al.  Reduced dynamical equations for barycentric spherical robots , 2016, 2016 IEEE International Conference on Robotics and Automation (ICRA).

[3]  Simon Parsons,et al.  Principles of Robot Motion: Theory, Algorithms and Implementations by Howie Choset, Kevin M. Lynch, Seth Hutchinson, George Kantor, Wolfram Burgard, Lydia E. Kavraki and Sebastian Thrun, 603 pp., $60.00, ISBN 0-262-033275 , 2007, The Knowledge Engineering Review.

[4]  Seungwon Choi,et al.  Aerial manipulation using a quadrotor with a two DOF robotic arm , 2013, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems.

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

[6]  A. D. Lewis,et al.  Geometric control of mechanical systems : modeling, analysis, and design for simple mechanical control systems , 2005 .

[7]  Darryl D. Holm,et al.  The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories , 1998, chao-dyn/9801015.

[8]  Gerald W. Schwarz.  SMOOTH FUNCTIONS INVARIANT UNDER THE ACTION OF A COMPACT LIE GROUP , 1975 .

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

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

[11]  Matthew Ryan Burkhardt.  Dynamic Modeling and Control of Spherical Robots , 2018 .

[12]  J. Ostrowski.  Reduced equations for nonholonomic mechanical systems with dissipative forces , 1998 .

[13]  P. Krishnaprasad,et al.  Nonholonomic mechanical systems with symmetry , 1996 .

[14]  A. D. Lewis,et al.  Geometric Control of Mechanical Systems , 2004, IEEE Transactions on Automatic Control.

[15]  R. Hermann.  On the Accessibility Problem in Control Theory , 1963 .