IFAC PapersOnLine 50-1 (2017) 10456–10461 ScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2017.08.1975 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: optimal control, discrete mechanics, variational mechanics, Denavit-Hartenberg, Legendre-Transformation, Furuta double pendulum, swing-up, trajectory planning 1. INTRODUCTION For many years inverted pendulum has been associated as the classical problem in many disciplines such as math- ematics, physics, dynamics and control due to its chal- lenging system properties like nonlinearity and underac- tuation. Its practical correspondence to stabilization of humanoid robot, vehicle during the rollover, rocket launch- ing and a lot of others make the stabilization of inverted pendulum widely used. However, despite the vast studies on the swing-up maneuver from the downward position to the upright position it is still a challenging problem. The swing-up problem is approached either on a nonde- terministic way or on a deterministic way. Energy based techniques as in ( ˚ Astr¨om and Furuta, 2000) and (Spong, 1995), fuzzy based techniques as in (Wang et al., 1996) and (Tao et al., 2008), bang-bang based techniques as in (Awrejcewicz et al., 2012) are nondeterministic, whereas a proper trajectory planning as in (Gl¨ uck et al., 2013) are deterministic. In (Graichen et al., 2007) a solution of a two- point boundary problem with free parameters is proposed. In (Jung and Wen, 2004) a nonlinear model predictive control is deployed on a Furuta single pendulum. Up until now, most of the existing techniques are applied on other versions of pendulums like the inverted pendulum on a cart or on a wheeled pendulum. However, due to its rotation na- ture an inverted Furuta double pendulum introduces more centrifugal and coriolis forces and thus raises the challenge of the swing-up. In this study, the swing-up of the inverted Furuta double pendulum has been deployed using the efficiency feature of the Discrete Mechanics and Optimal Control (DMOC) (Junge et al., 2006). In a merge of the op- timal control, variational integrators (Hairer et al., 2006) and discrete mechanics (Marsden and West, 2001) the concept of DMOC offers a novel alternative for formulating an optimization problem. Direct methods ((Hargraves and Paris, 1987), (Von Stryk, 1993)) and indirect methods (Rao, 2009) in both the objective function and the dif- ferential equations are given in a continues form. Derived from the variation of the Lagrange-d’Alembert principle the Euler-Lagrange equations results in the differential equations. Most of certain optimization applications are based on either the direct method as in (Petropoulos et al., 2002), (Somavarapu et al., 2016) and (Geiger et al., 2006) or the indirect method as in (Yoshimura and Yamanaka, 1982) and (Ranieri and Ocampo, 2006). The direct method is mostly used due to its convenient implementation and the widely used optimization techniques. However, for efficient optimization the indirect method seems to have an advantage, since it deals with a less computational cost for solving the canonical boundary value problem resulting from the Pontryagin’s maximum principle. In (Graichen, 2012) and (B¨ achle et al., 2013) the indirect method is employed for efficient optimization in the con- text of model predictive control. In contrast, DMOC pre- sumes a discrete cost function and discrete Lagrange- d’Alembert principle. Due to its variation, a discrete Euler- Lagrange with discrete forces is obtained, which depends only on the position q instead of both position q and speed ˙ q like in the continues Euler-Lagrange. In fact, this discretization occurs at an earlier stage than in the direct and indirect methods. For the boundary conditions the explicit dependency of the speed is however solved using the Legendre-Transformation (Marsden and West, 2001). As a result, the forced discrete Euler-Lagrange equations together with the discrete boundary conditions are ob- tained, which serve as equality constraints for a finite dimensional nonlinear optimization problem (NLP). To * Institute of Control Systems, University of Kaiserslautern P.O. Box 3049, 67653 Kaiserslautern, Germany (e-mail: ismail@eit.uni-kl.de, sliu@eit.uni-kl.de). Abstract: This paper presents a more accurate model of a Furuta double pendulum for efficient planning of optimal trajectories using the new approach of discrete mechanics and optimal control (DMOC). Based on the variation of discrete mechanics a direct discretization of the Lagrange-d’Alembert principle enables a novel formulation of the optimization problem. This leads to less optimization parameters and thus more computational efficiency. To shed the light on the advantages of the proposed method the swing-up maneuver of the Furuta double pendulum is used to demonstrate the point-to-point trajectory planning. For this purpose, a model of Furuta double pendulum based on Denavit-Hartenberg convention with higher accuracy is presented additionally. Jawad Ismail * Steven Liu * Efficient Planning of Optimal Trajectory for a Furuta Double Pendulum Using Discrete Mechanics and Optimal Control