Figure 1: An example of robot trajectory optimization.
The application of numerical optimization methods to trajectory optimization for industrial robots has shown that large improvements are possible compared with traditional path planning methods (e.g., , ). But the use of sophisticated numerical optimization methods requires several necessities as, e.g., the proper description of the dynamic behavior of the robot as an n-body system (in minimal coordinates, e.g., the relative angles between successive joints of the robot)
where is the torque control, M is the positive definite and symmetric ()-matrix of moments of inertia and are the moments resulting from centrifugal, coriolis, gravitational and frictional forces. Thus the first basic problem is the modelling of the dynamic system (1) of the robot . The identification of unknown dynamic parameters of the specific robot (as, e.g., moments of inertia or friction parameters) is a second must (see problem P2).
Once the dynamic equations have been modelled and determined in a proper way, the optimization of trajectories can be investigated. Different tasks for robots require different objectives for optimal trajectories, e.g., in welding optimal tracking of the prescribed path might be required. Otherwise if only the initial and the final position of the robot are prescribed a fast point-to-point movement might be requested. It has been demonstrated that the fastest point-to-point motion () exhibits quite often a surprisingly structure, impacts enormous stress on the links and can even often not be realized in practice. Here, fast minimum energy motions () offer a compromise between time and stress . The robot trajectories also have to satisfy several constraints as, e.,g., constraints on the controls, the angles and the angular velocities. Further constraints result from the geometrical design of the robot's working cell and require an efficient modelling of collision avoidance.