Dynamic analysis of a spherical mobile robot in rough terrains

In this article, the dynamic equations of a spherical mobile robot, named BYQ-III, are derived by utilizing the Lagrange method. There is no simplification throughout the whole dynamic analysis and the derived dynamic equations can be used for more precise studies of spherical mobile robots' behavior. Considering any possible differentiable function for the terrain's curve, only assuming that the spherical shell will remain in contact with the ground and the elastic effect of the spherical shell is ignored, the effect of the terrain's unevenness is completely described in the dynamic equation evaluation. Although there are complicated and nonlinear relations between the spherical shell and rough terrain, proper choice of generalized coordinates leads to the general closed form dynamic equations of motion, and finally results in the effective reduction of simulation time. But there is no need for the numerical method to solve the complex dynamic equation due to the closed form derivation. In the dynamic equation all variables are highly coupled together and their individual effect cannot be decoupled exactly. From this proposed complete model a simplified model for controller design can be extracted and the proposed model description can give an insight about the performance of different controllers of the spherical robots' motion. Simulations with the same initial conditions on a flat surface and rough terrain show that a rough terrain has a considerable effect on the dynamic behavior of the spherical robots. And as the unevenness of the terrain increases, its effect in the dynamic analysis becomes greater and cannot be neglected.