Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003 zyxwvutsrq TuM02-4 zy Asymptotic Stability zyxw of Rigid Body Attitude Systems Jinglai Shen, Ainit K. Sanyal, N. Harris McClainroch Department of Aerospace Engineering University of Michigan, Ann Arbor, MI 48109 {j inglais asanyal nhm}@engin. umich. edu Abstract A rigid body, supported by a fixed pivot point, is free to rotate in three dimensions. Two cases are studied: the balanced case, whose dynamics are described by the Euler equations for a free rigid body, and the un- balanced case, whose dynamics are described by the heavy top equations. Both cases include linear passive dissipation effects. For each case, conditions are pre- sented that guarantee asymptotic stability for relevant equilibrium solutions. The developments are based on a careful treatment of nonlinear coupling in applying LaSalle’s invariance principle. Emphases are given to the partial damping cases; an approach based on the polynomial structure of the dynamics is used to obtain asymptotic stability conditions for these cases. 1 Introduction The triaxial attitude control testbed (TACT) is an experimental facility built in the Attitude Dynamics and Control Laboratory at the University of Michigan. Its physical properties are described in detail in [2], and a detailed derivation of mathematical models for the TACT is given in [5, zyxwvutsrqp 61. The TACT is based on a spherical air bearing that provides a near-frictionless pivot for the rigid base body. The stability problems treated in this paper are motivated by possible set-ups of the TACT. This paper presents results on the asymptotic stability of an abstraction of the TACT. This abstraction consists of a rigid body, supported at a fixed pivot, that is free to rotate in three dimensions; the pivot contact provides linear dissipation. Two cases are studied. The first case, referred to as the balanced case, occurs if the pivot is located at the center of mass of the rigid body. The second case, referred to as the unbalanced case, occurs if the pivot is not located at the center of mass of the rigid body. These two cases have different equilibrium structures. IThis research has been supported in part by NSF under grant ECS-0140053. This paper is closely related to [13], where we treated a similar problem on the asymptotic stability of multi- body attitude systems. In some sense, the balanced case studied in the present paper can be considered as a special case of the problems studied in [13]. However, the results in the present paper, restricted to a rigid body, are more complete and simpler to obtain. The unbalanced case, although restricted to a single rigid body, involves more complicated dynamics. The results for both cases expose clearly how asymptotic stability can arise from nonlinear mechanisms. Both the present paper and [13] have been motivated by [lo], in which Miller and Michel studied a multi-body system consisting of an interconnection of ideal mass elements and elastic springs. Lyapunov function argu- ments, based on the system Hamiltonian, were used to develop damping assumptions that guarantee asymp- totic stability of the equilibrium. One of their key contributions was the use of observability properties to guarantee asymptotic stability, based on LaSalle’s invariance principle. Their arguments were based on the observability rank condition for linear Hamiltonian systems. In contrast, our results on the asymptotic sta- bility of a rigid body require a fundamentally nonlinear analysis. 2 Asymptotic Stability of Rigid Body Attitude Systems: the Balanced Case Consider the balanced case; this is equivalent to a free rotating rigid body in three-dimensions. The attitude dynamics with linear damping are described by zy JW = JW x zyxw w - CW, (1) where w zyxw E R3 denotes the angular velocity expressed in the base body frame, J is the reduced inertia matrix assumed to be symmetric and positive definite, and zy C E denotes a constant damping matrix that satisfies C = CT >_ 0. The’equilibrium of (1) is w = 0. 0-7803-7924-1/03/$17.00 02003 IEEE 544