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# 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 animation shows the solution of a tracking-type optimal control problem subject to the instationary Navier-Stokes equation with a distributed control. The Navier-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. Computations and animation were done by Martin Kunkel from the University of Hamburg.

## Optimal Control of Robots

front view             top view                  top view

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. In addition, the optimal control was used to control a LEGO robot.

## Optimal Control of cooperative Robots

front view                      top view

The animations show the time-minimal motion of two robots moving. Additional state constraints ensure that the robots do not collide. The optimal control problem was solved by OC-ODE.

## Contact Problems

The animations show the motion of a bouncing ball without and with friction, two colliding balls, and a falling rod (thanks to Fabio Carfagno for computing the colliding balls and the falling rod). 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.

## Testdrives

Car-to-car communication and automatic driving strategies

The movie illustrates the outcome of a model-predictive control strategy for a crossing scenario. The cars use car-to-car communication to adapt their driving strategies in order to avoid collisions and to reach a given individual target position.

Pro-Active Optimal Control for Vehicle Suspension

The movie shows an video capture of the software sensUp. The optimal controls for singular events like potholes and thresholds are calculated in real-time using sensitivity analysis of the undelying parametric optimal control problem.

Test-drives created by Hannes Gruschinski, University of Magdeburg:
test-course
race track of Hockenheim

This movie was created by Hannes Gruschinski (University of Magdeburg) using OC-DAE1 and the software DYNAanimation by Tesis. The above youtube video can be found here.

Collision avoidance and active steering

overtaking maneuver

The animation shows the results of a collision avoidance algorithm for an overtaking maneuver. The red car, which is driving at 100 km/h, intends to overtake the blue car next to it, which is driving at 75 km/h. Using the underlying algorithm it is possible to detect whether a collision with the car approaching from the right at 100 km/h can be avoided. The resulting trajectory could be used in an active steering driver assistance system. The width of the road is 7 m and the width of the cars was assumed to be 2.6 m. The single track car model was used to model the red car and includes realistic tyre characteristics that obey Kamm's circle. Moreover, additional constraints ensure that the red car does not leave the road and that it remains in a secure state.

The video shows an optimal collision avoidance maneuver. The robot approaches the obstacle and measures the distance to the obstacle using a sonar sensor until it reaches the minimum distance that allows to perform a collision avoidance maneuver. Then, an optimal avoidance maneuver given by the solution of an optimal control problem is applied.

Test-drives by Matthias Gerdts using full car models:

 Version 1 Version 2 Slalom

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 animations are created with POVRAY.

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. 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

Slalom

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 animation was created with the help of Karsten Nikkel and POVRAY

 Version 1 Version 2 Skidding car 1 Skidding car 2

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 animations were created with the help of Karsten Nikkel and POVRAY .

## Optimal Control of a Pendulum chain

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.  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.

## Heat conduction

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.