## On optimal control simulations for mechanical systems

• The primary objective of this work is the development of robust, accurate and efficient simulation methods for the optimal control of mechanical systems, in particular of constrained mechanical systems as they appear in the context of multibody dynamics. The focus is on the development of new numerical methods that meet the demand of structure preservation, i.e. the approximate numerical solution inherits certain characteristic properties from the real dynamical process. This task includes three main challenges. First of all, a kinematic description of multibody systems is required that treats rigid bodies and spatially discretised elastic structures in a uniform way and takes their interconnection by joints into account. This kinematic description must not be subject to singularities when the system performs large nonlinear dynamics. Here, a holonomically constrained formulation that completely circumvents the use of rotational parameters has proved to perform very well. The arising constrained equations of motion are suitable for an easy temporal discretisation in a structure preserving way. In the temporal discrete setting, the equations can be reduced to minimal dimension by elimination of the constraint forces. Structure preserving integration is the second important ingredient. Computational methods that are designed to inherit system specific characteristics – like consistency in energy, momentum maps or symplecticity – often show superior numerical performance regarding stability and accuracy compared to standard methods. In addition to that, they provide a more meaningful picture of the behaviour of the systems they approximate. The third step is to take the previ- ously addressed points into the context of optimal control, where differential equation and inequality constrained optimisation problems with boundary values arise. To obtain meaningful results from optimal control simulations, wherein energy expenditure or the control effort of a motion are often part of the optimisation goal, it is crucial to approxi- mate the underlying dynamics in a structure preserving way, i.e. in a way that does not numerically, thus artificially, dissipate energy and in which momentum maps change only and exactly according to the applied loads. The excellent numerical performance of the newly developed simulation method for optimal control problems is demonstrated by various examples dealing with robotic systems and a biomotion problem. Furthermore, the method is extended to uncertain systems where the goal is to minimise a probability of failure upper bound and to problems with contacts arising for example in bipedal walking.

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Verfasserangaben: Sigrid Leyendecker urn:nbn:de:hbz:386-kluedo-27615 978-3-942695-04-6 Lehrstuhl für Technische Mechanik Kaiserslautern Ralf Müller, Peter Betsch, Michael Ortiz Habilitation Englisch 12.01.2011 12.01.2011 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 12.01.2011 06.10.2011 constrained systems; discrete variational mechanics; flexible multibody dynamics; optimal control; variational integrators 196 Fachbereich Maschinenbau und Verfahrenstechnik 6 Technik, Medizin, angewandte Wissenschaften / 620 Ingenieurwissenschaften und Maschinenbau 34-XX ORDINARY DIFFERENTIAL EQUATIONS 37-XX DYNAMICAL SYSTEMS AND ERGODIC THEORY [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] 49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] 53-XX DIFFERENTIAL GEOMETRY (For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx) 74-XX MECHANICS OF DEFORMABLE SOLIDS Standard gemäß KLUEDO-Leitlinien vom 27.05.2011

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