In order to improve the obstacle-crossing ability, motion stability and load-bearing capacity of mobile robots for different terrains, the Rubik’s Cube mechanism (RCM) with strong coupling and variable topology is introduced into the field of mobile robots, and a wheel-legged mobile robot (WLMR) based on RCM is proposed. Aiming at the problems of the classical three-order RCM, such as small internal space, difficult processing and demanding precision, a new type of chute third-order RCM is designed, and its mechanical characteristics analysis and feasibility analysis are carried out. What’s more, a driving configuration analysis method is established according to different driving configuration relationships, and the configuration of WLMR is determined by the configuration stability analysis. Then, a WLMR with polymorphism is designed, and gait planning and gait stability analysis are conducted. Eventually, the co-simulation and prototype experiments are performed to verify the efficiency of the WLMR’s straight motion, in-situ rotation, obstacle-crossing and morphology transformation in complex environments. This research not only provides a reference for the design of polymorphous mobile robots, but also opens up ideas for the application of the RCM in daily production and life.