This is the dynamics that has been used in the following paper:
https://arxiv.org/pdf/1908.02243

Abstract:
This paper discusses the design, modeling, and control of Norma, a novel 2 DOF mobile spherical robot (SR). The propelling mechanism of this robot consists of two actuators: a slider, and a rotational pendulum located on the SR's diagonal shaft. The slider can translate along the shaft and shift the robot's center of gravity towards the robot's sides. The pendulum rotates around the shaft to propel the SR to roll forward and backward. These two actuators enable the SR to perform both rolling and turning maneuvers as a nonholonomic robot. The advantage of the proposed mechanical design lies in its convenience of physical implementation, agility, and accurate mathematical model. The Euler Lagrange approach is utilized to derive the dynamics of the proposed mechanical structure using minimum simplifications possible. Further, a path tracking control scheme is introduced for a smooth trajectory. Finally, simulations are carried out in MATLAB to verify the accuracy of the mathematical model and the effectiveness of the controller against experimental results.