This problem involves differential (nonholonomic) constraints on a car-like robot. The robot has a limited steering angle, and is required to roll along the ground (i.e., it cannot move sideways). The robot moves forward only.

The equations of motion are:

dx[0] = Speed*u[0]*cos(x[2]);

dx[1] = Speed*u[0]*sin(x[2]);

dx[2] = tan(u[1])/CarLength;

in which dx represents dx/dt, u represents the input vector, and
x represents the state vector.

An RRT that takes into account these differential constraints
is shown below.

An example solution path for a simple problem is shown below. The RRTs are shown (projected from a 3D configuration space by using only xy coordinates of the vertices).

An alternative solution, obtained after a second run, is shown below.

An animation of the solution.

A solution for a more challenging problem.

This problem is similar to the previous one, except that the maximum steering angle is smaller. Note the small curvature along the path.

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