Synergistic Hybrid Feedback for Global Rigid-Body Attitude Tracking on $\hbox{ SO }(3)^{\ast}$ ${\ssr {SO}}(3)^{\ast}$

In this paper, we propose two hybrid feedbacks based on “synergistic” potential functions to achieve robust global asymptotic tracking of rigid-body attitude, a task that is impossible by classical state feedback-be it smooth, nonsmooth, or periodic-due to the topological structure of the special orthogonal group. Both hybrid feedbacks are based upon a proportional-derivative structure where the proportional term is generated from a finite family of control-induced artificial potential energies and the derivative term is due to a damping injection. In a patient-yet-greedy fashion, the hybrid feedback hysteretically switches to the minimum control-induced artificial potential energy in a finite family. The synergy property-which requires that for each undesirable critical point of each potential energy, there exists a lower potential energy in the family-guarantees the globality of robust asymptotic tracking. The first hybrid feedback is the natural extension of existing PD controllers and results in discontinuous jumps in the control signal due to a direct switch the potential energy. Using a backstepping procedure, we design a second hybrid feedback that smooths the jumps in the control torque by dynamically interpolating between the switching potential-energy terms. We show that while the class of “modified trace functions” is not wide enough to generate a “centrally synergistic” potential function, relaxing the centrality assumption allows one to construct a synergistic family and we provide explicit guidelines for doing so.

[1]  Thor I. Fossen,et al.  Singularity-free tracking of unmanned underwater vehicles in 6 DOF , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[2]  J. Wen,et al.  The attitude control problem , 1991 .

[3]  J.S. Yuan,et al.  Closed-loop manipulator control using quaternion feedback , 1988, IEEE J. Robotics Autom..

[4]  Frank L. Lewis,et al.  Hybrid control for a class of underactuated mechanical systems , 1999, IEEE Trans. Syst. Man Cybern. Part A.

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

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

[7]  J. Milnor.  Lectures on the h-cobordism theorem , 1965 .

[8]  A. Veselov,et al.  Integrable Gradient Flows and Morse Theory , 1995, dg-ga/9506004.

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

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

[11]  Carlos Silvestre,et al.  Combination of Lyapunov and Density Functions for Stability of Rotational Motion , 2011, IEEE Transactions on Automatic Control.

[12]  Emilio Frazzoli,et al.  Trajectory tracking control design for autonomous helicopters using a backstepping algorithm , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[13]  Richard M. Murray,et al.  Tracking for fully actuated mechanical systems: a geometric framework , 1999, Autom..

[14]  Theodore Frankel,et al.  Critical Submanifolds of the Classical Groups and Stiefel Manifolds , 1965 .

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

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

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

[18]  A. Morse,et al.  Applications of hysteresis switching in parameter adaptive control , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[19]  Bong Wie,et al.  Quaternion feedback for spacecraft large angle maneuvers , 1985 .

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

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

[22]  Ricardo G. Sanfelice,et al.  On Path-Lifting Mechanisms and Unwinding in Quaternion-Based Attitude Control , 2013, IEEE Transactions on Automatic Control.

[23]  V. Kapila,et al.  A quaternion-based adaptive attitude tracking controller without velocity measurements , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[24]  Eduardo D. Sontag,et al.  Mathematical control theory: deterministic finite dimensional systems (2nd ed.) , 1998 .

[25]  Dennis S. Bernstein,et al.  Asymptotic Smooth Stabilization of the Inverted 3-D Pendulum , 2009, IEEE Transactions on Automatic Control.

[26]  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..

[27]  Perry Y. Li,et al.  Passive velocity field control (PVFC). Part II. Application to contour following , 2001, IEEE Trans. Autom. Control..

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

[29]  Mamoru Mimura,et al.  Lusternik-Schnirelmann category of non-simply connected compact simple Lie groups , 2005 .

[30]  Karl Johan Åström,et al.  A Stabilizing Switching Scheme for Multi Controller Systems , 1996 .

[31]  Rita Cunha,et al.  Output-feedback control for stabilization on SE(3) , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

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

[33]  N.H. McClamroch,et al.  Global stabilization of a fully actuated mechanical system on a Riemannian manifold: controller structure , 2006, 2006 American Control Conference.

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

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

[36]  Robert E. Mahony,et al.  Attitude control of rigid body dynamics from biased IMU measurements , 2007, 2007 46th IEEE Conference on Decision and Control.

[37]  Ricardo G. Sanfelice,et al.  Further results on synergistic Lyapunov functions and hybrid feedback design through backstepping , 2011, IEEE Conference on Decision and Control and European Control Conference.

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

[39]  Rita Cunha,et al.  Almost global stabilization of fully-actuated rigid bodies , 2009, Syst. Control. Lett..

[40]  J. Aubin,et al.  Differential inclusions set-valued maps and viability theory , 1984 .

[41]  N. McClamroch,et al.  Almost global attitude stabilization of an orbiting satellite including gravity gradient and control saturation effects , 2006, 2006 American Control Conference.

[42]  João Pedro Hespanha,et al.  Scale-Independent Hysteresis Switching , 1999, HSCC.

[43]  D. H. S. Maithripala,et al.  Almost-global tracking of simple mechanical systems on a general class of Lie Groups , 2005, IEEE Transactions on Automatic Control.

[44]  J. Magnus,et al.  Matrix differential calculus with applications to simple, Hadamard, and Kronecker products. , 1985 .

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

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

[47]  P. Olver.  Nonlinear Systems , 2013 .

[48]  F. Wilson.  The structure of the level surfaces of a Lyapunov function , 1967 .

[49]  Alexander Leonessa,et al.  Nonlinear system stabilization via hierarchical switching control , 2001, IEEE Trans. Autom. Control..

[50]  Jan F. Broenink,et al.  A spatial impedance controller for robotic manipulation , 1997, IEEE Trans. Robotics Autom..

[51]  Pierluigi Pisu,et al.  Attitude Tracking With Adaptive Rejection of Rate Gyro Disturbances , 2007, IEEE Transactions on Automatic Control.

[52]  Ricardo G. Sanfelice,et al.  Synergistic Lyapunov functions and backstepping hybrid feedbacks , 2011, Proceedings of the 2011 American Control Conference.

[53]  Taeyoung Lee,et al.  Geometric tracking control of a quadrotor UAV on SE(3) , 2010, 49th IEEE Conference on Decision and Control (CDC).

[54]  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).

[55]  R. Mortensen.  A globally stable linear attitude regulator , 1968 .

[56]  J. Stuelpnagel.  On the Parametrization of the Three-Dimensional Rotation Group , 1964 .

[57]  Robert E. Mahony,et al.  Attitude tracking of rigid bodies on the special orthogonal group with bounded partial state feedback , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[58]  Johan Eker,et al.  Hybrid control of a double tank system , 1997, Proceedings of the 1997 IEEE International Conference on Control Applications.

[59]  D. Koditschek.  The Application of Total Energy as a Lyapunov Function for Mechanical Control Systems , 1989 .

[60]  Carlos Silvestre,et al.  Combination of Lyapunov functions and density functions for stability of rotational motion , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.