Stability of obstacle–crossing and structural optimization are important issues in the research of tracked mobile robots. In this paper, in order to fully understand the obstacle–surmounting ability of the robot, the relationship between the position of the center of gravity and the posture of the front and rear swing arms is analyzed. Based on the motion mechanism of the robot crossing obstacles, the geometric model and the dynamic model are established for the key states in the obstacle crossing process. Based on these models, a multi-objective optimization problem for the maximum obstacle–crossing height and minimum driving torque is established during the obstacle crossing process of the robot, which must meet geometric, slip, and stability constraints. To effectively handle the optimization problem of tracked mobile robots, an improved non–dominated sorting genetic algorithm with elite strategy version II based on adaptive genetic strategy (NSGA-II-AGS) is proposed in this paper. Some meaningful relationships between the objective function and the design variables are obtained through sensitivity analysis. Finally, the robot's obstacle-crossing ability was verified through virtual simulation and prototype experiments. These excellent performances enable the proposed NSGA-II-AGS to be qualified for dealing with the multi-objective optimization problem.