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 can move in both forward or reverse. The steering angle appears as a fourth state variable, and it required to be continuous over time.

The equations of motion are:

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

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

dx[2] = Speed*u[0]*tan(x[3])/CarLength;

dx[3] = u[1];

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 4D state space by using only xy coordinates of the vertices).

An alternative solution, obtained from a second run.

An animation of the previous solution.

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