Robotics design and applications become significant worldwide. In this work, an improved and upgraded Legged based robot is designed and modeled with the mathematical framework to enable rough terrain exploration. This work aims to analyze existing robots design and use it to design a better and more efficient robot that could be used in surveillance and exploration. In the proposed design, the robot’s stability is the main target. The new proposed design of the robot considers such a critical parameter. In designing the optimized and improved robot, weight, cost, and the closed-loop control algorithm for this robot are closely examined, described, and analyzed with promising results. The resulting design with solar panels as a partial power supply is simulated with mathematical modeling and analysis. The special case of the robot’s whole body covered with solar panels is described, characterizing curves relating drag force to solar power. The effect of safety factors and velocity is also characterized in the work. The resulting mathematical model describing curves showed a linear dependency between solar power (driving power) and drag force, with similar findings for the safety factor. However, a less linear, close-to-exponential relationship is demonstrated for velocity about the drag force. Such dynamic-legged design with supporting springs is numerically modeled using the Jacobian element, which proved to be the most suitable.