Evaluation of Steering Algorithm Optimality for Single-Gimbal Control Moment Gyroscopes

Analytic optimization methods typically used to derive optimal steering algorithms for single-gimbal control moment gyros do not consider the structure of the Jacobian matrix mapping the gimbal rates onto the desired torque within their cost function. Many of the steering algorithms resulting from these optimization methods systematically take first and second derivatives, forming the Jacobian and Hessian matrices to obtain a solution. However, the optimality is usually a local result and cannot be mapped back to its resulting performance. It is shown that the majority of steering algorithms are optimal with respect to one specific cost function previously published and that the design of the weighting matrices within the cost is what distinguishes steering algorithms. The author analytically shows how the blended inverse, Moore-Penrose pseudoinverse, generalized inverse steering law, singularity robust inverse, generalized singularity robust inverse, singular direction avoidance, local-gradient methods, and the hybrid steering logic are derived from the same optimizations but their sense of optimality is lost because the structure of singularities is not considered in the optimization process. In addition, the author also points out that the design of the quadratic costs weighting matrix used for optimization and desired gimbal rate is of the highest importance in differentiation between steering law performance.