Neighboring extremal nonlinear model predictive control of a rigid body on SO(3)

Abstract

AbstractThe issue of implementing nonlinear model predictive control (NMPC) on mechanical systems evolving on special orthogonal group (SO(3)) is taken into consideration in the first place. Necessary conditions of optimality are extracted based on Lie group variational integrators, leading to a two-point boundary value problem (TPBVP) which is solved using sensitivity derivatives and indirect shooting methods. Fast Newton-like methods referred to as fast solvers which are commonly used to solve the TPBVP are established based on the repetition of a nonlinear process. The numerical schemes employed to alleviate the computation burden consist of eliminating some constraint-related but non-essential terms in the trend of sensitivity derivatives calculation and for solving the TPBVP equations. As another claim, assuming that a first attempt to resolve the NMPC problem is accessible, the problem subjected to some changes in its initial conditions (due to some re-planning schemes) can be resolved cost-effectively based on it. Instead of solving the whole optimization process from the scratch, the optimal control inputs and states of the system are updated based on the neighboring extremal (NE) method. For this purpose, two approaches are considered: applying NE method on the first solution that leads to a neighboring optimal solution, or assisting this latter by updating the NMPC-related optimization using exact TPBVP equations at some predefined intermediate steps. It is shown through an example that the first method is not accurate enough due to error accumulations. In contrast, the second method preserves the accuracy while reducing the computation time significantly.