Kinematic Synthesis of a Serial Manipulator Using Gradient-Based Optimization on Lie Groups

This paper addresses a specialized kinematic synthesis problem: designing a manipulator capable of following a specific trajectory of end-effector positions and orientations with minimal actuators. This requires optimizing the robot's kinematic parameters and solving inverse kinematics to ensure its configuration aligns with the desired trajectory. This paper introduces a method for optimizing robot design by representing joint motions using Lie algebra and applying the Levenberg–Marquardt (LM) algorithm. The proposed approach integrates inverse kinematics into the optimization process, solving both problems simultaneously. To achieve this, the method computes derivatives of the end-effector’s positions and orientations with respect to both kinematic parameters and the robot's configuration, leveraging the intrinsic relationship between Lie groups and their corresponding Lie algebra. The use of Lie algebra-based derivatives eliminates the singularities inherent in traditional kinematic parameterizations, enhancing stability and smoothness in the optimization process. Experimental results on a synthetic example demonstrate the method's robustness, showing independence from initial parameter selection and superiority over approaches based on local parameterization.