Research in robotics is one of the promising areas in mobile robot development, which is planned to be implemented in extreme dangerous conditions of areas explored by humans. This article aims at developing and improving a prototype of zoomorphic mobile robots that are designed to repeat the existing biological objects in nature. The authors performed a detailed analysis on the structure and dynamics of the geometrical family caterpillar movement, which is passed on a practical design implemented to perform the dynamic movement on uneven vertical surfaces. Based on the obtained analysis, the design and kinematic scheme of the movement is developed. Also, the structural control scheme via the Internet technologies that allow carrying out remote control is presented in this paper, considering the dangerous mobile robot work zones. To test the recommended solutions, the authors developed detailed 3D printed models of the mobile robot constructions for the implemented hardware. The model of the mobile robot is constructed, and the control system with examples of the user program code implementations is performed. Several experiments were performed, which showed the efficiency of the achieved mobile robot for solving problems of vertical movement on uneven metal surfaces. Moreover, the obtained slow motion of the designed robot proves that the simulated robot behaves similarly to the natural behavior of caterpillar movement.