Movies

Optimal Control of Flows

The animation shows the solution of a tracking-type optimal control problem subject to the instationary Stokes equation with a distributed control. The Stokes equation was considered as an index-2 partial differential-algebraic equation (PDAE). The velocities are considered as differential variables and the pressure is an algebraic variable.

The animations show the time-minimal motion of a robot moving a load from one point to another. The optimal control problem was solved by OC-ODE.

The animations show the motion of a bouncing ball without and with friction. A complementarity problem has to be solved in each step of the underlying discretization scheme for the mechanical multibody system in order to satisfy certain contact conditions. The complementarity problem is solved by the nonsmooth Newton's method.

The animations show the "Elk-test" and the slalom manoeuvre with the VW Iltis. The VW Iltis is modeled using SIMPACK as a mechanical multibody system in descriptor form (index 3 DAE system). The driver is modeled as an optimal test driver by formulating an optimal control problem with nonlinear state constraints, e.g. the observation of the boundaries of the lane. The optimal control problem was solved using SODAS . The animations are created with POVRAY . The mpeg-files (0.6 MB and 1.6 MB) can be viewed with, e.g., xanim or mpeg_play with additional option -dither color. more elk-test, more slalom

Again, the "Elk-test" is simulated with an alternative car-model. The model equations are stated as an index-1 differential-algebraic equation system of dimension 45 (41 differential equations and 4 algebraic equations). The initial velocity of the car is 34 m/s and the manoeuvre lasts 5.15 seconds. The underlying optimal control problem was solved using SODAS . In addition, a real-time approximation of the optimal control was computed w.r.t. a perturbation in the height of the car's centre of gravity as well as a perturbation in the offset of the test-course. The motion with the real-time optimal control approximation applied for a perturbation of 10 percent in the course's offset and 30 percent in the height of the car's centre of gravity is colored blue, whereas the nominal (unperturbed) motion is colored red. The animation was created by Karsten Nikkel and POVRAY . The mpeg-files (approximately 1.6 MB) can be viewed with, e.g., xanim or mpeg_play with additional option -dither color. more

A slalom manoeuvre is simulated. The model equations are stated as an index-1 differential-algebraic equation system of dimension 45 (41 differential equations and 4 algebraic equations). The initial velocity of the car is 20 m/s and the manoeuvre lasts 13.6 seconds. The underlying optimal control problem was solved using SODAS . The animation was created with the help of Karsten Nikkel and POVRAY . The mpeg-file (approximately 1.1 MB) can be viewed with, e.g., xanim or mpeg_play with additional option -dither color. more

A test-drive along a test-course of length 1200 m is simulated using a moving horizon technique. The model equations of the car are stated as an index-1 differential-algebraic equation system of dimension 45 (41 differential equations and 4 algebraic equations). The initial velocity of the car is 10 m/s, the highest achieved velocity is approximately 40 m/s. The simulation time over all is 52.5 seconds. In addition a skidding car is simulated. The underlying optimal control problem was solved using SODAS . The animations were created with the help of Karsten Nikkel and POVRAY . The mpeg-files (approximately 7 MB) can be viewed with, e.g., xanim or mpeg_play with additional option -dither color. more

The motion of a pendulum chain linked to a vehicle is simulated (uncontrolled). The equations of motion of the pendulum chain are given in descriptor form (index 3 DAE system). The 33 link pendulum chain consists out of 375 differential-algebraic equations with 104 algebraic equations. The motion of the vehicle is controlled by an additional force acting in horizontal direction. An optimal control problem is given by minimization of a linear combination of the steering effort and the horizontal excitation of the vehicle. The optimal control problem was solved using SODAS . In addition, a real-time approximation of the optimal control was computed w.r.t. a perturbation in the link masses as well as a perturbation in the final horizontal position of the vehicle. The motion with the real-time optimal control approximation applied for a perturbation of 20 percent in both parameters is colored red, whereas the nominal (unperturbed) motion is colored green. The animations were created by Tina Silberhorn, Andrea Kölz and POVRAY . The mpeg-files can be viewed with, e.g., xanim or mpeg_play with additional option -dither color. more

Numerical solution of the heat equation in 2 and 3 space dimensions by the method of lines. In addition the sensitivity w.r.t. the constant of diffusivity is depicted.