Time-varying Feedback Stabilization of the Attitude of a Rigid Spacecraft with two controls

Rigid body models with two controls cannot be locally asymptotically stabilized by continuous state feedbacks. Existence of a locally stabilizing smooth time-varying feedback has however been proved. Here, such a feedback is ezplicitely derived. 1 Introduction The attitude control of a rigid spacecraft operating in degraded mode, i.e with only one or two controls, has already been much studied in the literature. With respect to other contributions on the subject, the present paper focuses on feedback stabilization . The related but simpler problem consisting of stabilizing the angular velocity of the spacecraft with one or two actuators has been investigated by several authors (see e.g [l]). Smooth stabilization of the attitude seems to have been previously ruled out due t o that the resulting system, although controllable, cannot be stabilized via continuous state feedback (as easily proved by application of Brockett's necessary condition [2]). An article by Samson [8] has recently triggered the discovery that many systems of this type can in fact be stabilized by smooth "time-varying" feedback. Research on time-varying control has then expanded quickly (see e.g [4], [7]). In particular, results by Kerai [5] and Coron [4] ensure that the attitude of a controllable spacecraft with two actuators can be locally stabilized by continuous time-varying feed-backs. The purpose of this note is to describe a smooth (C") stabilizing time-varying feedback. 2 Main results The angular velocity vector of the inertial frame Fo with respect to a fixed frame F1, expressed in the basis of Fo, is denoted as w. The matrix representation of the cross product (x-x A w) is denoted as S(W). It is common to use Euler angles in order to work with a minimal parametriza-tion of SO(3). We prefer here the paranietriza-tion X = sin! U , with (cos:, sin: U) being the uni-tary quaternion associated with the rotation matrix representing the attitude of Fl with respect to Fo. = C3WlW2 where the U; (i = 1,2) are the torques applied to the rigid body, and the parameters cj (j = 1,2,3) are deduced from the coefficients of the body's inertia matrix. We assume that c 3 # 0, since otherwise the system would not be controllable. Moreover, by an adequate change of variables, we may also assume that c3 > 0. We first consider the following reduced order system obtained by taking w 1 = 7J1 and w2 = v 2 as control …