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.
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