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

Conventional model predictive control (MPC) methods are usually implemented to systems with discrete-time dynamics laying on smooth vector space $ mathbf{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.