Optimal Swing up of Double Inverted Pendulum using Indirect Method

Document Type : Original Article


1 Faculty of Mathematics, Statistics and computer Science, Semnan University, Iran

2 Robotics and Control Lab, Faculty of Mechanical Engineering, Semnan University, Iran


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.


Main Subjects

[1]     Graichen, K., Treuer, M., and Zeitz, M., Swing-up of the Double Pendulum on a Cart by Feedforward and Feedback Control with Experimental validation, Automatica, Vol. 43, No. 1, 2007, pp. 63-71.
[2]     Bettayeb, M., Boussalem, C., Mansouri, R., and Al-Saggaf, U., Stabilization of an Inverted Pendulum-Cart System by Fractional PI-State Feedback, ISA Transactions, Vol. 53, No. 2, 2014, pp. 508-516.
[3]     Astrom, K. J., Furuta, K., Swinging up a Pendulum by Energy Control, Automatica, Vol. 36, No. 2, 2000, pp. 287-295.
[4]     Aracil, J., Gordillo, F., A Family of Smooth Controllers for Swinging up a Pendulum, Automatica, Vol. 44, No. 7, 2008, pp. 1841-1848.
[5]     Wang, J. J., Stabilization and Tracking Control of X–Z Inverted Pendulum with Sliding-Mode Control, ISA Transactions, Vol. 51, No. 6, 2012, pp. 763-770.
[6]     Gluck, T., Eder, A., and Kugi, A., Swing-up Control of a Triple Pendulum on a Cart with Experimental Validation, Automatica, Vol. 49, No. 3, 2013, pp. 801-808.
[7]     Adhikary, N., Mahanta, C., Integral Backstepping Sliding Mode Control for Underactuated Systems: Swing-up and Stabilization of the Cart–Pendulum System, ISA Transactions, Vol. 52, No. 6, 2013, pp. 870-880.
[8]     Cruz, J. D., Leonardi, F., Minimum‐Time Anti‐Swing Motion Planning of Cranes Using Linear Programming, Optimal Control Applications and Methods, Vol. 34, No. 2, 2013, pp. 191-201.
[9]     Kahvecioglu, S., Karamancioglu, A., and Yazici, A., Nonlinear Model Predictive Swing-up and Stabilizing Sliding Mode Controllers, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, Vol. 3, No. 9, 2009, pp. 1041-1046.
[10]  Ast, J. M. V., Babuska, R., and Schutter, B. D., Novel ant Colony Optimization Approach to Optimal Control, International Journal of Intelligent Computing and Cybernetics, Vol. 2, No. 3, 2009, pp. 414-434.
[11]  Al-Janan, D. H., Chang, H. C., Chen, Y. P., and Liu, T. K., Optimizing the Double Inverted Pendulum’s Performance via the Uniform Neuro Multiobjective Genetic Algorithm, International Journal of Automation and Computing, Vol. 14, No. 6, 2017, pp. 686-695.
[12]  Oliver, J. P. O., Sanchez, O. J. S, and Morales, V. L., Toward a Generalized Sub‐Optimal Control Method of Underactuated Systems, Optimal Control Applications and Methods, Vol. 33, No. 3, 2012, pp. 338-351.
[13]  Chernousko, F., Reshmin, S., Time-Optimal Swing-up Feedback Control of a Pendulum, Nonlinear Dynamics, Vol. 47, No. 1-3, 2007, pp. 65-73.
[14]  Mason, P., Broucke, M., and Piccoli, B., Time Optimal Swing-up of the Planar Pendulum, IEEE Transactions on Automatic Control, Vol. 53, No. 8, 2008, pp. 1876-1886.
[15]  Paoletti, P., Genesio, R., Rate Limited Time Optimal Control of a Planar Pendulum, Systems & Control Letters, Vol. 60, No. 4, 2011, pp. 264-270.
[16]  Merakeb, A., Achemine, F., and Messine, F., Optimal Time Control to Swing-up the Inverted Pendulum-Cart in Open-Loop Form, 11th International Workshop In Electronics, Control, Measurement, Signals and their Application to Mechatronics, 2013,  pp. 1-4.
[17]  Gregory, J., Olivares, A., and Staffetti, E., Energy‐Optimal Trajectory Planning for the Pendubot and the Acrobot, Optimal Control Applications and Methods, Vol. 34, No. 3, 2013, pp. 275-295.
[18]  Horibe, T., Sakamoto, N., Optimal Swing up and Stabilization Control for Inverted Pendulum via Stable Manifold Method, IEEE Transactions on Control Systems Technology, Vol. 26, No. 2, 2018, pp. 708-715.
[19]  Nikoobin, A., Moradi, M., and Esmaili, A., Optimal Spring Balancing of Robot Manipulators in Point-to-Point Motion, Robotica, Vol. 31, No. 4, 2013, pp. 611-621.
[20]  Korayem, M. H., Vatanjou, H., and Azimirad, V., New Hierarchical Method for Path Planning of Large-Scale Robots, Latin American Applied Research, Vol. 41, No. 3, 2011, pp. 225-232.
[21]  Hull, D. G., Sufficiency for Optimal Control Problems Involving Parameters, Journal of Optimization Theory and Applications, Vol. 97, No. 3, 1998, pp. 579-590.