A New Integrated Method to Improve the Computation of the Robotics’ Inverse Kinematics in a Simple and Unified Formula

Abstract

Formulating mathematical models and deriving efficient algorithms are crucial for meeting the requirements of future robotics applications. This paper proposes a novel approach for analyzing kinematic systems and computing inverse kinematics (IK) solutions for serial robotic arms. The aim is to reduce modeling complexity and the computational cost of IK solution algorithms, while enhancing accuracy and efficiency by reformulating the kinematic equations using simplified constraints. This is achieved by integrating the rotation matrix and the unit quaternion to represent kinematic equations in a simple and unified form without compromising the degrees of freedom or raising the order of the kinematic equations, as in traditional approaches. The method combines analytical and numerical techniques to obtain an exact IK solution in two steps: first, the wrist joint variables are substituted into the position equations, resulting in a modified position vector equation obtained analytically; second, numerical iteration is applied to compensate for the error between the current and desired positions, leading to the ultimate exact inverse solution. The method is tested on a 5R robot and a 6R (UR-10) robot with an offset wrist to demonstrate the mathematical process and real-time algorithm performance. The results demonstrate that the absolute position error is less than ${10}^{-15}$ m, with no orientation error, and the mean calculation time for the IK solution is less than 5 ms. Furthermore, the results indicate higher accuracy and reduced computational time compared to other common IK methods. Moreover, the algorithm’s improved performance in processing continuous paths demonstrates its advantages in both simulation and practical applications. Finally, the proposed methodology is expected to advance further research in kinematic modeling and enhance polynomial-based numerical iterative algorithms.