Matthias Gerdts

Approximation of reachable sets by direct solution methods for optimal control problems

A numerical method for the approximation of reachable sets of linear control systems is discussed. The method is based on the formulation of suitable optimal control problems with varying objective function, whose discretization by Runge–Kutta methods leads to finite-dimensional convex optimization problems. It turns out that the order of approximation for the reachable set depends on the particular choice of the Runge–Kutta method in combination with the selection strategy used for control approximation. For an inappropriate combination, the expected order of convergence cannot be achieved in general. The method is illustrated by two test examples using different Runge–Kutta methods and selection strategies, in which the run times are analysed, the order of convergence is estimated numerically and compared with theoretical results in similar areas.

Collision-Free Path Planning of Welding Robots

In a competitive industry, production lines must be efficient. In practice, this means an optimal task assignment between the robots and an optimal motion of the robots between their tasks. To be optimal, this motion must be collision-free and as fast as possible. It is obtained by solving an optimal control problem where the objective function is the time to reach the final position and the ordinary differential equations are the dynamics of the robot. The collision avoidance criterion is a consequence of Farkas’s lemma. The criterion is included in the optimal control problem as state constraints and allows us to initialize most of the control variables efficiently. The resulting model is solved by a sequential quadratic programming method where an active set strategy based on backface culling is added.

Numerical Approaches Towards Bilevel Optimal Control Problems with Scheduling Tasks

In this paper, we consider the problem of scheduling N robots interacting with a moving target. Both, the sequence of the robots and their trajectories are unknown and subject to optimization. Such type of problems appear in highly automated production plants and in the simulation of virtual factories. The purpose of the paper is to provide a mathematical model and to suggest a numerical solution approach. Our approach is based on the formulation of the problem as a bilevel optimization problem, where the lower level problem is an optimal control problem, while the upper level problem is a finite dimensional mixed-integer optimization problem. We approach the problem by exploitation of necessary optimality conditions for the lower level problem and by application of a Branch & Bound method for the resulting single level optimization problem. Two settings are taken into account. Firstly, no state constraints are assumed on the lower level problem, thus the local minimum principle applies directly. Secondly, the problem setting is augmented by pure state constraints, which are being handled by virtual controls in order to regularize the problem.