Construction of Synergistic Potential Functions on SO(3) With Application to Velocity-Free Hybrid Attitude Stabilization

We propose a systematic and comprehensive procedure for the construction of synergistic potential functions, which are instrumental in hybrid control design on SO(3). A new map, via angular warping on SO(3), is introduced to generate a central family of potential functions allowing an explicit determination of the critical points and the synergistic gap. Some optimization results on the synergistic gap are also provided. The proposed synergistic potential functions are used for the design of a global velocity-free hybrid attitude stabilization scheme relying solely on inertial vector measurements.

[1]  R. Sanfelice,et al.  Hybrid dynamical systems , 2009, IEEE Control Systems.

[2]  Andrew R. Teel,et al.  On the topological structure of attraction basins for differential inclusions , 2011, Syst. Control. Lett..

[3]  Andrew R. Teel,et al.  Synergistic Hybrid Feedback for Global Rigid-Body Attitude Tracking on $\hbox{ SO }(3)^{\ast}$ ${\ssr {SO}}(3)^{\ast}$ , 2013, IEEE Transactions on Automatic Control.

[4]  Christopher G. Mayhew.  Hybrid Control for Topologically Constrained Systems , 2010 .

[5]  J. Hespanha,et al.  Hybrid systems: Generalized solutions and robust stability , 2004 .

[6]  M. Morse.  The Calculus of Variations in the Large , 1934 .

[7]  J. Dieudonne,et al.  Encyclopedic Dictionary of Mathematics , 1979 .

[8]  Darryl D. Holm,et al.  Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions , 2009 .

[9]  Robert J. Vanderbei,et al.  Linear Programming: Foundations and Extensions , 1998, Kluwer international series in operations research and management service.

[10]  P. Hughes.  Spacecraft Attitude Dynamics , 1986 .

[11]  Christopher G. Mayhew,et al.  Hybrid control of planar rotations , 2010, Proceedings of the 2010 American Control Conference.

[12]  Taeyoung Lee.  Global Exponential Attitude Tracking Controls on ${\mathsf {SO}}({\mathsf 3})$ , 2015 .

[13]  Andrew R. Teel,et al.  Global asymptotic stabilization of the inverted equilibrium manifold of the 3-D pendulum by hybrid feedback , 2010, 49th IEEE Conference on Decision and Control (CDC).

[14]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[15]  Robert M. Sanner,et al.  A coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise , 2003, IEEE Trans. Autom. Control..

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

[17]  Andrew R. Teel,et al.  Hybrid control of spherical orientation , 2010, 49th IEEE Conference on Decision and Control (CDC).

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

[19]  Pedro Casau,et al.  A globally asymptotically stabilizing trajectory tracking controller for fully actuated rigid bodies using landmark‐based information , 2015 .

[20]  Abdelhamid Tayebi,et al.  Unit Quaternion-Based Output Feedback for the Attitude Tracking Problem , 2008, IEEE Transactions on Automatic Control.

[21]  Andrew R. Teel,et al.  Synergistic potential functions for hybrid control of rigid-body attitude , 2011, Proceedings of the 2011 American Control Conference.

[22]  Abdelhamid Tayebi,et al.  Attitude stabilization of a VTOL quadrotor aircraft , 2006, IEEE Transactions on Control Systems Technology.

[23]  Ricardo G. Sanfelice,et al.  Quaternion-Based Hybrid Control for Robust Global Attitude Tracking , 2011, IEEE Transactions on Automatic Control.

[24]  Andrew R. Teel,et al.  Hybrid control of rigid-body attitude with synergistic potential functions , 2011, Proceedings of the 2011 American Control Conference.

[25]  Andrew Roberts,et al.  Inertial Vector Measurements Based Velocity-Free Attitude Stabilization , 2013, IEEE Transactions on Automatic Control.

[26]  N. McClamroch,et al.  Rigid-Body Attitude Control , 2011, IEEE Control Systems.

[27]  Ricardo G. Sanfelice,et al.  Invariance Principles for Hybrid Systems With Connections to Detectability and Asymptotic Stability , 2007, IEEE Transactions on Automatic Control.

[28]  Daniel E. Koditschek,et al.  Application of a new Lyapunov function to global adaptive attitude tracking , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[29]  M. Shuster.  A survey of attitude representation , 1993 .

[30]  D. Bernstein,et al.  Inertia-Free Spacecraft Attitude Tracking with Disturbance Rejection and Almost Global Stabilization , 2009 .

[31]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[32]  Taeyoung Lee.  Robust global exponential attitude tracking controls on SO(3) , 2013, 2013 American Control Conference.

[33]  Robert E. Mahony,et al.  Nonlinear Complementary Filters on the Special Orthogonal Group , 2008, IEEE Transactions on Automatic Control.

[34]  Jerrold E. Marsden,et al.  Symmetry and bifurcation in three-dimensional elasticity, part I , 1982 .

[35]  Rafal Goebel,et al.  Solutions to hybrid inclusions via set and graphical convergence with stability theory applications , 2006, Autom..

[36]  Malcolm D. Shuster.  Survey of attitude representations , 1993 .

[37]  G. Meyer,et al.  DESIGN AND GLOBAL ANALYSIS OF SPACECRAFT ATTITUDE CONTROL SYSTEMS , 1971 .