Compared with traditional wheeled and footed robots, spherical tensegrity robots offer advantages such as a high strength-to-mass ratio, excellent cushioning performance, and superior terrain adaptability, making them highly promising for deep space exploration. While cable-driven modes are commonly used for tensegrity robots, the excessive number of actuators requires for walking complicates manufacturing and control. This study proposes a novel driving mode based on the post-buckling deformation of flexible rods. Numerical simulations of the walking process of a spherical tensegrity robot are conducted, and the efficiencies of cable-driven and rod-post-buckling-driven modes are compared. The exact solution for the post-buckling deformation of a single rod is obtained using the elliptic integral method. Based on this, a rigid-flexible coupling dynamics simulation model of the spherical tensegrity robot is established in ADAMS, with considering the post-buckling deformation of the rods, as well as contact and friction. The walking gait of the spherical tensegrity robot is determined through joint simulation using ADAMS and Simulink software, employing a greedy search algorithm. A control system model is established in Simulink to facilitate the robot’s walking control to any target points under the rod-post-buckling-driven mode. Compared to the conventional cable-driven mode, the post-buckling-driven mode reduces the number of actuators required for continuous robot walking from 18 to 6 and increases the walking speed by 43.78%. The results provide theoretical guidance for the design and manufacture of new tensegrity robots.