Attitude Feedback Control: Unconstrained and Nonholonomic Constrained Cases

Abstract

The problem is considered of asymptotically driving, the attitude of a spacecraft via feedback control. First, a simple feedback control structure based on the measure of the equivalent angle axis is determined for a fully actuated system. Then, the same reasoning is used to e nd the solution for a spacecraft operating in failure mode, that is, whenever the angular velocity vector is constrained to lie in one of the spacecraft' s coordinate planes. The basic aim is to show that with a simple choice for the attitude parameterization, a simple and effective structure for the feedback control law can be obtained. HE problem of asymptotically driving the attitude of a space- craft via feedback regulation is a standard one and can be ad- dressed via a number of attitude representations that give rise to different and effective control solutions. It is of greater interest to understand if any of the feasible solutions under nominal opera- tion conditions could be adopted (or extended) in case of failure. In particular, we are interested in the case of failures that limit the spacecraft' s attitude maneuvering capabilities and which constrain (in a purely cinematic framework ) the controllable vehicle angular velocities that lie in a given plane e xed with respect to any space- craft' s reference frame. The general aspects characterizing the constrained problem have received a certain amount of attention in the past few years because of their relevance in the area of nonlinear control of nonholonomic systems (see Ref. 1). A complete survey on this subject may be found in Ref. 2. Within this framework, the problem of stabilizing the attitude of a spacecraft subject to nonholonomic constraints has been investi- gated in depth. In particular, in Ref. 3, it has been proven that such a classofsystems,evenifcontrollable,cannotbestabilized toanarbi- trary equilibrium point if 1 )a nonzero initial angular velocity exists directed along the nonactuated axis and 2 ) smooth state feedback controllawsare searchedfor.Moreprecisely,ifsuch an initialangu- lar velocityexists,the systemcan bestabilizedto anyarbitrary equi- librium point only if discontinuous state feedback laws are allowed. In the case where smooth state-feedback laws are adopted and in presence initial velocities as those just indicated, it has been shown that the system is anyway stabilizable with respect to an attractor, representedbyacircularmotionaboutthedesiredequilibriumpoint. From the results of Ref. 3, only initial angular velocities com- patible with the nonholonomic constraints allow attainment of any desired equilibrium attitude using smooth state-feedback control laws. For this particular condition, a nonlinear stabilizing smooth feedback law based on the use of a three-dimensional Euler-angles parameterizationforthespacecraftattitudehasbeenproposed.More recently, a new and very specie c kind of parameterization for rep- resenting the spacecraft attitude was introduced. This approach al- lowed simplifying the structureofpreviouslyproposed smooth con- trol laws, including those presented in Ref. 3 and in many of the references listed in Refs. 4 and 5, while also enhancing the overall control performances. In the present work we shall consider the well-known equiva- lent angle-axis (Euler axis plus angle of rotation ) parameterization.