It has been demonstrated that even a simple manoeuvre can exhibit quite a difficult switching structure in the time optimal motion of an industrial robot. The state constraints on the angular velocities play an important role in the time optimal motion as they often become active. The knowledge of the fastest possible motion provides reliable bounds for fast minimum energy motions. Hereby, the stress on the links of the robot can be significantly decreased if an increase in time of about ten percent is accepted. Thus lifetime and reliability of the robot will increase.
The second link of the Manutec r3 robot is the weakest.
This is indicated also by several other time optimal movements investigated
by the authors where the constraints on 
become
active during most parts of the motions.
Thus a better design of robots might be possible
if the investigation of optimal trajectories is
included in the development phase.
The combination of direct and indirect methods,
namely direct collocation and multiple shooting,
is an efficient hybrid approach for solving
highly complex, nonlinear, real life optimal control problems
that amalgamates the benefits of both methods.
Acknowledgement. The authors acknowledge
the helpful discussions with
Priv.-Doz. Dr. H.J. Pesch
and the colleagues from the Numerical Analysis and Optimal
Control Group of Prof. R. Bulirsch at the
Munich University of Technology
and the valuable
support by the colleagues
from the Robotics Group of Prof. G. Grübel at the
Laboratory of Robotics and System Dynamics of the DLR,
esp. J. Franke, S. Lewald, and M. Otter.