The dynamics of a 2D rigid body that moves in the plane can be handled as a special case of a 3D body. Let be a 2D body, expressed in its body frame. The total external forces acting on can be expressed in terms of a two-dimensional total force through the center of mass and a moment through the center of mass. The phase space for this model has six dimensions. Three come from the degrees of freedom of , two come from linear velocity, and one comes from angular velocity.
The translational part is once again expressed as
All rotations must occur with respect to the -axis in the 2D formulation. This means that the angular velocity is a scalar value. Let denote the orientation of . The relationship between and is given by , which yields one more component of the state transition equation.
At this point, only one component remains. Recall (13.92). By inspecting it can be seen that the inertia-based terms vanish. In that formulation, is equivalent to the scalar for the 2D case. The final terms of all three equations vanish because . The first terms of the first two equations also vanish because . This leaves . In the 2D case, this can be notationally simplified to
The state transition equation for a 2D rigid body in the plane is therefore
Steven M LaValle 2020-08-14