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. تفاصيل المقالة

In this paper, a method based on the indirect solution of optimal control problem is presented to specify the optimal trajectory of spatially suspended cable robot in point to point motion with considering the counterweights. In fact, an optimal trajectory planning problem is outlined in which states, controls and the values of counterweights must be calculated simultaneously in order to minimize the given performance index. The value of the pulley torques is considered for the performance index (objective function). Using the fundamental theorem of a calculus of variations, the necessary conditions for optimality of cable robot are achieved. For the three-cable spatial robot, a two-point boundary value problem is achieved which can be solved with bvp4c command in MATLAB. The obtained results show that optimal balancing in comparison with the unbalancing method can reduce the performance index significantly. تفاصيل المقالة