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Abstract:

Carrying out a task optimally with respect to time or energy is of significant importance for automated systems in production plants. Determining such an optimal solution, however, is mathematically very challenging. To simplify the problem, it is often divided into two steps: a path planning stage, where a geometric path is determined only accounting for geometric constraints such as collision avoidance, and a path following stage, where an optimal velocity profile along the path is determined taking into account the system dynamics and limitations. This research deals with (time) optimal path following of mechatronic systems. Via a transformation of variables, the optimization problem can be solved efficiently for a variety of systems and situations: For a simplified robot model the path following problem can be cast as a convex optimization problem. For such problems the global solution can be calculated very efficiently and reliably, allowing for an online implementation of this strategy. In many practical situations convexity of the problem is lost, but it can still be cast as a so-called convex-concave optimization problem which can be solved reliably by a series of convex optimization problems using Sequential Convex Programming. Most applications only require solving 4 to 5 convex problems. When multiple robots are working together in shared workspaces, collision avoidance constraints become important. The robots are modeled as a union of convex bodies and collision avoidance constraints can be derived using the concept of Lagrangian duality in optimization. In this case, we can no longer decompose the problem as a convex-concave one and therefore must rely on general nonlinear optimization solvers. In many applications the geometric path need not be followed exactly but within a certain (machining) tolerance, so-called optimal tube following. By exploiting this additional freedom, solutions may be found that are significantly more optimal