Discrete mechanics modelling and real-time optimal control method based on Lie group variational integrator

Under discrete mechanics, a real-time optimal control method is developed based on Lie group variational integrator. First, the Hamilton-Pontryagin principle is promoted as a variational principle called the d´ Alembert-Pontryagin principle by introducing virtual displacement and virtual work, that would be applied to the mechanical system with nonconservative force and control. The left-trivialized d´ Alembert-Pontryagin equation of a controlled mechanical system is derived in continuous time and discrete time, respectively. Second, the first-order necessary optimality condition of a discrete-time optimal control problem is derived by using discrete variation method on Lie group, and thus a two-point boundary value problem is created. Then, referring to C/FD-GMRES method, an algorithm is developed based on receding horizon control for solving the two-point boundary value problem. Finally, the validity of the presented algorithm is verified by simulation example of optimal control problem on SE(2).