Facing the challenge of rapidly increasing energy consumption in robotics, this study proposes a physical model to optimize motor motion and efficiency. By implementing an innovative spherical coordinate to Cartesian coordinate transformation, we achieve precise calculation of the robotic arm's end-effector position while completing comprehensive drive motor safety verification. By implementing spherical-to-Cartesian transformation, we achieve precise end-effector positioning while reducing computational overhead. Furthermore, the research incorporates quintic polynomial trajectory planning combined with a trigonometric phase-shift model to achieve coordinated multi-joint motion analysis. Furthermore, combining quintic polynomials with a trigonometric phase-shift model ensures smooth trajectories and coordinated joint motion. The proposed methodology addresses a critical gap in robotic systems by harmonizing motion accuracy with energy conservation. By focusing on physical modeling and trajectory optimization, this work provides a viable technical pathway for tackling energy consumption challenges in industrial robots. This framework demonstrates that strategic motion planning significantly reduces energy usage without compromising performance, supporting the development of sustainable industrial automation. This research contributes to sustainable robotics by offering a practical solution that balances operational demands with environmental considerations, paving the way for wider adoption of energy-conscious practices in industrial automation.

spherical coordinate-cartesian coordinate conversion, polynomial interpolation, mathematical modeling