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).

An additional restriction severely limits the motions of the car: it is constrained to move forward-only and left-only. It can turn left in varying degrees, but is unable to turn right or go straight!

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).

Another solution is shown, taken from a second run.

A third solution.

An animation of the previous solution.

A more challenging example.

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