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.