In this paper, the mobility and geometrical analysis of a novel mobile robot that utilizes two actuated spoke wheels is presented. Intelligent Mobility Platform with Active Spoke System (IMPASS) is a wheel-leg hybrid robot that can walk in unstructured environments by stretching in or out three independently actuated spokes of each wheel. First, the unique locomotion scheme of IMPASS is introduced and the definitions of the coordinate systems are developed to describe the kinematic configurations. Since this robot is capable of utilizing its metamorphic configurations to implement different types of motion, its topology structures are classified into different groups based on the cases of ground contact points. For each contact point case, the mobility analysis is performed using the conventional Grubler and Kutzbach criterion. However, as for the cases in which the structure is overconstrained, the Modified Grubler and Kutzbach criterion based on reciprocal screws are implemented to obtain the correct number of degrees of freedom. Line geometry is adopted to assist in the process. Additionally, the geometrical constraint equations of the robot are derived. The results in this work lay the foundation of the future research on inverse and forward kinematics, instantaneous kinematics, dynamics analysis and motion planning of this unique locomotion robot.