Optimal Control of Multibody Systems using the Adjoint Variable Approach

In the last years, the complexity of models in multibody dynamics has grown tremendously. In particular, industrial simulations of large systems including several flexible bodies such as complete vehicles or robots result in models with a vast number of degrees of freedom. Moreover, there is also an increased demand of both, the research, as well as the industry for developing efficient and reliable algorithms for solving optimal control problems in multibody dynamics. A general approach to solve an optimal control problem is the formulation as an optimization task. In this case, one looks for actuating forces or torques of a multibody system, which minimizes a cost functional, e.g. the mean deviation of a system output from a measured signal. The focus of the presented thesis lies on the efficient gradient computation of the cost functional by using the adjoint variable approach. Therefore, the cost functional is extended by the equations of motion, which are multiplied by the adjoint variables. Then, the variation of the extended cost functional has to be determined and the adjoint variables have to be chosen, such that this variation is directly related to the variation of the inputs. This leads to a system of additional differential-algebraic equations, from which the gradient of the cost functional with respect to the inputs can be computed. However, the numerical solution of the adjoint differential equations raises several questions with respect to stability and accuracy. Hence, an alternative and maybe more natural approach is the discrete adjoint method. A finite difference scheme is constructed for the adjoint system directly from the numerical solution procedure, which is used for the solution of the equations of motion. The method delivers the gradient of the discretized cost functional subjected to the discretized equations of motion. Moreover, the efficient computation of the gradient allows the application of gradient-based optimization methods, which speed-up the whole solution procedure significantly. Finally, the described methods are applied to academic and industrial applications. First of all, an optimal control of a swing-up maneuver of a cart double pendulum is computed. As a second academic example the force on a cart and the torque on the winch are determined for an underactuated planar overhead crane, such that the load follows a prescribed path. The energy optimal trajectory planning of a robot is presented as a first industrial application. Finally, a wheel suspension of a racing car is investigated as an inverse dynamic problem.