Swing-up of the moving double pendulum on a cart with simulation based LQR-Trees

This paper presents the control design problem for the real-time swing-up and stabilization of a double pendulum on a cart from a large set of non-stationary initial conditions. The proposed algorithm generates a tree of feedback stabilized trajectories leading to the upper equilibrium of the pendulum. For each trajectory in the tree, the algorithm determines the set of states near the nominal trajectory from which the system can be transfered to the upper equilibrium without violating the state and input constraints using the feedback controller of the trajectory. This set of states is defined as the funnel of the stabilized trajectory. The design of a control strategy for a large set of initial states is equivalent to the coverage of the desired region in state space using funnels. Several modifications of the original LQR-trees algorithm are introduced in this contribution, which includes a discretization based approximation of the funnel, the exploration of symmetric property of the double pendulum and the coverage of a design set which has a lower dimension than the state space. We show both simulation results and experimental validation of the proposed method.