Model Predictive Control for a 3D Pendulum on SO(3) Manifold Using Convex Optimization

Abstract. Conventional model predictive control (MPC) methods are usually implemented to systems with discrete-time dynamics laying on smooth vector space R n . In contrast, the configuration space of the majority of mechanical systems is not expressed as Euclidean space. Therefore, the MPC method in this paper has developed on a smooth manifold as the configuration space of the attitude control of a 3D pendulum. The Lie Group Variational Integrator (LGVI) equations of motion of the 3D pendulum have been considered as the discrete-time update equations since the LGVI equations preserve the group structure and conserve quantities of motion. The MPC algorithm is applied to the linearized dynamics of the 3D pendulum according to its LGVI equations around the equilibrium using diffeomorphism. Also, as in standard MPC algorithms, convex optimization is solved at each iteration to compute the control law. In this paper, the linear matrix inequality (LMI) is used to solve the convex optimization problem under constraints. A numerical example illustrates the design procedure. Keywords.