Neural Operator for Lie Group-Based Kinematic Modeling of Serial Robots

This paper introduces a novel theoretical framework for forward kinematics (FK) in robotic manipulators, diverging from traditional matrix-based methods by operating directly within the tangent space of the Special Euclidean group SE(3). Leveraging the algebraic structure of the Lie algebra se(3), we propose a neural operator designed to approximate the composition of rigid body motions as defined by the Baker-Campbell-Hausdorff (BCH) formula. Key theoretical contributions include a detailed analysis of the algebraic properties of so(3), revealing analogies to complex number fields crucial for rotation representation, and the introduction of a specific tangent space transformation $F$ inspired by Cayley-Rodrigues parameters aimed at regularizing the learning problem. Furthermore, a data-driven analysis of the BCH coefficient distributions motivates a hybrid neural architecture combining SVM classification and regime-specific MLP regression for robust coefficient prediction. This methodology avoids sequential matrix multiplications and exponential map computations during the iterative kinematic chain evaluation. While experimental validation on standard manipulators shows current limitations in computational speed and positional accuracy compared to Denavit-Hartenberg (DH) and Product of Exponentials (PoE), the approach demonstrates competitive orientation accuracy and stable convergence. The primary significance lies in the novel algebraic formulation and the learning-based methodology for approximating Lie group operations, offering a distinct perspective on kinematic modeling with potential for adaptive robotics and non-Euclidean optimization.