Car.html

Maple worksheet. Study of a flat model of a car with

Intrinsic and Apparent Singularities.

Joint work with Yirmeyahu J. Kaminski, Jean Lévine, François Ollivier

This function draws the car : x and y are the coordinates of the rear axle center, theta the angle of the car body and phi the angle of the front wheel. A red dot provides the orientation of this wheel.

If the global variable car_display is 0, the body of the car is a magenta polygon (used for animations). If not is just a black curve (used for superpositions).

This functions computes the position of a car, described by the trajectory (f,g) of the middle of the rear axle. The functions f and g depend on the variable t and t0 that must be of type numeric, stands for the current time. The value of theta is stored using global variable theta_prev and theta is computed modulo Pi in order to minimize |theta_prev - theta|. The same thing is done for phi_prev.

The trajectory is given by theta and psi= sin(theta)x-cos(theta)y. This may be seen as the limit of flat outputs given by a point of the rear axle, when the point goes to infinity. There is no more ambiguity about the value of theta, but theta_prev is stored together with phi_prev to allow combined use with rear_axle function.

Parametrization of a trajectory. Functions x and y are used if

Views is the list of times to be computed. N_size the size of the output picture. The global variable param_display controls the style of the output: an animation if param_output=0 a superposition of views if not.

We denote the position of the car by the quintuple (x,y,theta,phi,u). Here the car goes from (0,10,0,0,10) at time t=0 to (0,-10,Pi,0,10) at time t=1.

Plot_2d

Plot_2d

Plot_2d

Here we want to go to from point (0,1,0,0,40) at t=0 to point (0,1,0,0,-40) at t=1, meaning that we start forward and arrive backward. For this, we need to cross a cusp where x~c_1-c_2*t^2, y~c3*t^3.

The presence of the cusp implies that diff(x,t) and diff(y,t) have a common factor that we need to remove to avoid trouble like: division by 0, or an extra loop in the curve, due to floating point inaccuracies, and the value of theta increase by Pi!

Plot_2d

Plot_2d

Plot_2d

Like in the previous example, we want to go to from point (0,1,0,0,-40) at t=0 to point (0,1,0,0,-40) at t=1, meaning that we start forward and arrive backward. But this time, we want to have a better control on the value of phi and avoid it to reach Pi/2. For this, we need to cross a rhamphoid cusp with x~c_1-c_2*t^2, y~c3*t^5.

The presence of the rhamphoid cusp implies that diff(x,t) and diff(y,t) have a common factor that we need first to remove. But this is not enough, as we will have at this point x'=y'=theta'=0.