Immersion and Invariance Based Adaptive Backstepping Control for Body-Fixed Hovering Over an Asteroid

An adaptive control law is proposed in this paper for body-fixed hovering over an asteroid with an unknown gravity field and an unknown rotation rate. The unknown parameters and irregular shape of an asteroid make the hovering problem challenging. To facilitate the controller design, first, it is assumed that the required parameters are attainable, and then a body-fixed hovering controller is designed using the backstepping approach. Based on the immersion and invariance theory, an adaptive law is devised to estimate the unknown parameters in conjunction with the backstepping controller. To overcome the integrability obstacle arising in the adaptive law design, the regressor matrix is decomposed to two matrices: a Jacobian matrix and a non-Jacobian matrix, and an auxiliary matrix is introduced into the latter to form a Jacobian matrix. Then, a bounded function is employed to dynamically scale the estimate errors and its dynamics is elaborately constructed to stabilize the system. A Lyaponuv function is constructed for the entire closed-loop system to achieve the stabilization conditions by LaSalle's theorem, and the bounded scaling function much simplifies the analysis. Finally, numerical simulations demonstrate the effectiveness of the proposed control scheme.