Animated simulation results of trajectory tracking control for the kinematic model of a car. Comparison with flatness-based control.

The controllers are designed in such a way that the magenta point of the car has to follow the magenta point moving on the circle.

  • To start animation, click on one of the initial positions of the car.
  • To stop animation, and return to initial picture, click return button.
  • To see the kinematic scheme and the equations of car, click here.

S.V.Gusev and I.A.Makarov, arXive:math/0507567.

 
M.L.Fliess et al., Int.J.Cont. (1995) 61, 1327-1361

Demo animation Start animation #1 Start animation #2 Start animation #3 Start animation #4 Start animation #5 Start animation #5
The trajectory tracking is performed by the static control law. It is singular only when the front and rear axles of the car are orthogonal. (Such a position is impossible in practice.) Starting from a nonsingular position the system never reaches the singular one. The control law can stabilize any sufficiently smooth trajectory of the car.   The trajectory tracking is performed by the dynamic control law. It is singular when the speed of the rear axle midpoint is zero. In particular this happens when the front and rear axles of the car are orthogonal. Starting from a nonsingular initial state the system can reach the singular one. This is illustrated by the simulation. For any initial state of the system there exist a trajectory that cannot be stabilized due to singularity in the control law.

Animated simulation of a road train control