The ``Michael Schumacher'' problem

D.E. Stewart

Introduction

The essence of this problem is to solve an optimal control problem where the dynamics involve Coulomb friction in an essential way. Discretized in time, it can be treated as a Mathematical Program with Equilibrium Constraints (MPEC).

In outline, the problem is to drive a car around a circuit in minimum time; limits on accelerations arise from the Coulomb friction between the tires and the road. More realistic physical models than the one given below can be used, and a greater range of physical phenomena could be included, but the essence of the problem should remain the same.

As given, this is a state-constrained optimal control problem; penalty methods could be used to approximate the constraint of not leaving the track without violating the spirit of the problem.

Model

A simplified model of the behavior of a car on a road is used where Coulomb friction is assumed to hold only perpendicular to the orientation of the car. The controls are the level of the accelerator pedal (a) and the angle of the steering wheel ( $\psi$ ). The state variables are the coordinates of the center of mass of the car (x,y) and its angle with respect to the x-axis ( $\theta$ ), and the translation velocity $(u,v)=(\dot{x},\dot{y})$ .The track is a set A in ${\bf R}^{2}$ with smooth boundary as illustrated in Figure 2. The model of the racing car is illustrated in Figure 1.

**Figure 1:** Illustration of physical model
$\begin{figure} {\par\centering \includegraphics {formulaVa.eps} \par} \end{figure}$

**Figure 2:** A simple track layout (using ellipses)
$\begin{figure} {\par\centering \includegraphics {track.eps} \par} \end{figure}$

Level 1 model

This uses just one-dimensional coulomb friction, and does not incorporate friction forces tending to rotate the car. The differential equations are

$\begin{displaymath} \begin{array} {rcl} m\ddot{x} & = & (a-C(\dot{x}^{2}+\dot{y}... ...a +F\cos \theta ,\ \dot{\theta } & = & \beta \psi \end{array} \end{displaymath}$

where $F\in \mu mg\cdot \textrm{Sgn}(\sin \theta \cdot \dot{x}-\cos \theta \cdot \dot{y})$ ,

$\begin{displaymath} \textrm{Sgn}(w)=\left\{ \begin{array} {cl} \{-1\}, & \textrm... ...extrm{if }w=0,\ \{+1\}, & \textrm{if }w\gt.\end{array}\right. \end{displaymath}$

Control constraints have the form $\vert\psi \vert\leq \psi _{max}$ and $\vert a\vert\leq a_{max}$ .

Research issues

1.

Discretization in time . For the initial analysis, use the implicit Euler method, which leads to inclusions (generalized equations) of the form

$\begin{displaymath} \textrm{solve for }z:\quad 0\in g(z)+\textrm{Sgn}(h(z))\end{displaymath}$

which can be represented by the box variational inequality:

$\begin{displaymath} \begin{array} {rcl} 0 & = & g(z)+\alpha ,\ h(z)\gt & \Right... ...alpha \leq +1,\ h(z)<0 & \Rightarrow & \alpha =-1.\end{array} \end{displaymath}$

This leads to large-scale but sparse MPECs -- there is a staircase structure associated with the dynamics.

2.

Infinite-dimensional algorithms . Since the basic problem is infinite-dimensional, it is probably appropriate to find algorithms that work in infinite dimensions. Although such algorithms are not implementable, they can guide researchers toward algorithms whose complexity (in terms of number of iterations) is independent of the problem size, and hopefully that the method is asymptotically optimal.

3.

Pontryagin-type conditions . What are appropriate Pontryagin-type conditions for this problem? It seems appropriate that the adjoint equations must have solutions with jumps, which leads to measure differential equations and related ideas. How should these equations/conditions be formulated? How should they be solved numerically?

Acknowledgements

I would like to thank Stefan Scholtes for suggesting the name of the problem.

David Stewart
6/14/1999