Page - 217 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

Energy Optimal Control of anIndustrial Robot by using theAdjoint Method ThomasLauss,Peter Leitner, Stefan Oberpeilsteiner, andWolfgang Steiner University of Applied Science UpperAustria SchoolofEngineeringand EnvironmentalSciences Stelzhamerstraße 23, 4600Wels, Austria {thomas.lauss,wolfgang.steiner}@fh-wels.at Abstract The main goal of this contribution is to determine the excitation of an industrial robot, such that the energyconsumptionbecomesaminimumduringthemanipulationof the toolcenterpoint (TCP) from astartposition toa givenendpointwithina predefined time. Such taskscan berestatedasoptimiza- tion problems where the functional to be minimized consists of the endpoint error and a measure for theenergy. Thegradientof this functionalcanbecalculatedbysolvinga lineardifferentialequation, called the adjoint system. On the one hand the minimum of the cost functional can be achieved by the method of steepest descent where a proper step size has to be found or on the other hand by a Quasi-Newton algorithm where the Hessian can be appreciated. The theory is applied to a six-axis robotand the identification leads to areductionof 47%of thesignalenergy. Keywords: optimalcontrol,multibodydynamics,adjoint system,optimization,calculusof variation. 1. Introduction In this contribution an approach to such inverse dynamical problems is presented. It starts from an optimal control formulation of the problem by introducing a cost functional which has to be min- imized subject to a system of differential equations (c.f. [1, 2]). The gradient computation of the cost functional is based on the so called adjoint method. Due to better convergence a Quasi-Newton methodisused insteadof thesimplegradientmethod. Therefore, theHessianmatrix isapproximated byusing theBFGS-algorithm. Theadjointmethodisalreadyusedinawiderangeofoptimizationproblemsinengineeringsciences. Especially, in the field of multibody systems, the computation of the gradient of the cost function is often the bottleneck for computational efficiency and the adjoint method serves as the most efficient strategy in this case. The basic idea of the adjoint method is the introduction of additional adjoint variables determined by a set of adjoint differential equations from which the gradient can be com- puted straightforward. This main idea directly corresponds to the gradient technique for trajectory optimizationpioneeredby Brysonand Ho [3]. Various authors have utilized the adjoint method in the sensitivity analysis of multibody system, as e.g., [4, 5]. Bottasso et al. [6] presented a combined indirect approach of the adjoint method in multibodydynamics for solving inversedynamics and trajectory optimizationproblems, also similar to the ideas presented in [7]. 217