Optimal Swing up of Double Inverted Pendulum using Indirect Method

In this paper, optimal swing up of a double inverted pendulum (DIP) with two underactuated degrees of freedom (DOFs) is solved using the indirect solution of optimal control problem. Unlike the direct method that leads to an approximate solution, the proposed indirect method results in an exact solution of the optimal control problem, but suffers from its limited convergence domain which makes it difficult to solve. In order to overcome this problem, an inversion-based method is used to obtain the required initial solution for the indirect method. In the proposed methodology, dynamic equations are derived for a general inverted pendulum using Euler-Lagrange formulation. Then the necessary optimality conditions are derived for a DIP on the cart using the Pontryagin’s maximum principle (PMP). The obtained equations establish a two-point boundary value problem (TPBVP) which solution results in optimal trajectories of the cart and pendulums. In order to demonstrate the applicability of the presented method, a simulation study is performed for a DIP. The simulation results confirm the superiority of the proposed method in terms of reduced effort.