A Study on Robot Path Jacobian Using a Nonlinear Differential Equation Framework for Motion Analysis, Trajectory Planning, and Control

Abstract

Robot motion analysis and trajectory planning are fundamental aspects of modern robotics, requiring precise mathematical models to achieve accurate positioning, smooth motion, and effective control. This study presents a nonlinear differential equation framework for the analysis of the robot path Jacobian, providing a comprehensive mathematical approach for modeling the kinematic and dynamic behavior of robotic manipulators. The proposed framework integrates nonlinear differential equations with Jacobian matrix analysis to describe the relationship between joint variables and end-effector motion while accounting for nonlinear effects arising from actuator dynamics, joint coupling, and external disturbances. The robot's motion is formulated as a system of nonlinear differential equations, and the Jacobian matrix is employed to establish the mapping between joint-space velocities and Cartesian-space trajectories. The mathematical model is analyzed to investigate the existence and uniqueness of solutions, stability characteristics, singularity conditions, and trajectory-tracking performance. Furthermore, nonlinear control strategies are incorporated to improve path-following accuracy and enhance the robustness of the robotic system under varying operating conditions. Numerical simulations are performed to validate the proposed framework using representative robotic trajectories. The simulation results demonstrate that the nonlinear differential equation model accurately captures the dynamic behavior of the robotic manipulator while providing reliable Jacobian-based motion analysis and efficient trajectory planning. Compared with conventional linear approaches, the proposed framework exhibits improved stability, reduced tracking error, and enhanced adaptability to nonlinear system dynamics. The findings of this study highlight the significance of integrating nonlinear differential equations with robot path Jacobian analysis for advanced robotic motion planning and control. The proposed mathematical framework provides a valuable foundation for the development of intelligent robotic systems in industrial automation, autonomous vehicles, medical robotics, and precision manufacturing, where accurate trajectory generation and robust motion control are essential.