This article presents a kinematic analysis of a four-wheeled mobile robot in three-dimensions, introducing computational mechanics. The novelty lies in (1) the type of robot that is analyzed, which has been scarcely dealt with in the literature, and (2) the methodology used which enables the systematic implementation of kinematic algorithms using the computer. The mobile robot has four wheels, four rockers (like an All-Terrain Mobile Robot), and a main body. It also has two actuators and uses a drive mechanism known as differential drive (like those of a slip/skid mobile robot). We characterize the mobile robot as a set of kinematic closed chains with rotational pairs between links and a higher contact pair between the wheels and the terrain. Then, a set of generalized coordinates are chosen and the constraint equations are established. A new concept named ''driving modes'' has been introduced because some of the constraint equations are derived from these. The kinematics is the first step in solving the dynamics of this robot in order to set a control algorithm for an autonomous car-like robot. This methodology has been successfully applied to a real mobile robot, ''Robotnik,'' and the results are analyzed.