Linear Parameterization of Nonlinear Spacecraft Control by Pseudoinversion of the Control Coefficient

Abstract

A novel approach for rigid spacecraft control is presented, based on pointwise-linear parametrization of nonlinear redundant solutions using the concept of pseudoinversion. The procedure is to form a difierential error equation involving the difierence between the spacecraft attitude variables and their desired values. The evaluation of this equation along the trajectories deflned by the equations of motion of the spacecraft yields a pointwise-linear relation in the control variables. These can be solved for by utilizing the MoorePenrose pseudoinverse of the involved control coe‐cient row vector. The resulting control law consists of auxiliary and particular parts, residing in the null space of the control coe‐cient and the range space of its pseudoinverse, respectively. The pseudo-control vector in the auxiliary part parameterizes all control variables that are required for spacecraft control, so that the desired attitude error dynamics is realized. It is chosen in this paper to be proportional to the spacecraft angular velocity vector. Based on the choice of the quasi-linear proportionality gain matrix, two tracking control laws are derived. One control law is produced by a control Lyapanov function, and provides exact following of the desired attitude error dynamics. Nevertheless, employing the Moore-Penrose pseudoinverse of the control coe‐cient may result in a numerical instability as the coe‐cient becomes singular. Robustness against the instability due to control coe‐cient singularity is obtained by thresholding the control coe‐cient. The other control law is feedback linearizing, and provides approximate following of the desired error dynamics with a guaranteed avoidance of the instability due to coe‐cient singularity. The spacecraft attitude trajectories can be brought arbitrarily close to the desired trajectories by increasing the natural frequency of the desired error dynamics, and thresholding the control coe‐cient is useful in smoothing the control and angular velocity variables. Both controllers are based on the concept of perturbed null space projection matrix, and both yield bounded spacecraft state and control variables and a global attraction to the desired attitude trajectories. The attitude state variables of the spacecraft are chosen to be the modifled Rodrigues parameters because of their validity in describing large angular conflgurations.