Consider a differential drive robot on a horizontal plane with two rear driving wheels and a passive front spherical ball bearing, as shown in Fig. 1. We assign three reference frames to describe the kinematics of the robot: An inertial reference frame, *X*–*Y*, a body-fixed reference frame $eu\u2212ev$ attached to the center of mass (CM) located at *G*, and another body-fixed reference frame $eud\u2212evd$ attached to the spherical ball. A comprehensive rigid body model of the robot has 6DOFs. There are the two coordinates of the CM with respect to the *X*–*Y* frame, $x\xaf,y\xaf$, the robot yaw angle measured from the *X*-axis to $eu$, *θ*, and three angles of the spherical ball. These three angles consist of a rotation about the vertical axis, *θ*_{d}, which measures a deviation of the ball from the robot axis $eu$, a rotation about a rotated horizontal axis, *ψ*_{r}, which measures the rolling of the ball, and a rotation about the rotated vertical axis, *ψ*_{s}, which measures the spin of the ball. These *ψ*_{r} and *ψ*_{s} angles are not shown in the figure. However, if the mass of the ball is negligible in comparison with the masses of the wheels and the frame, $m\xafball\u226am\xafwheel<m\xafframe$, the ball's degrees-of-freedom are inconsequential for the robot dynamics. Note that the bar symbol “ ¯ ” is used to denote dimensional quantities.