Algorithmic Approaches to Kinematic Modeling of Continuum Manipulators and Complex Flexible Mechanisms

Abstract

The kinematic modeling of continuum manipulators and complex flexible mechanisms remains a fundamental challenge in robotics and computational mechanics due to their infinite degrees of freedom and inherently nonlinear behavior. Unlike traditional rigid-link robots governed by standard Denavit-Hartenberg parameters, continuum manipulators require advanced mathematical and computational frameworks to accurately map actuator space to task space. This paper presents a comprehensive computational approach to the kinematic modeling of such systems, bridging traditional physics-based models with modern machine learning techniques. We investigate the limitations of the Piecewise Constant Curvature assumption and extend the analysis to discrete Cosserat rod models, integrating a neural-augmented solver to resolve real-time computational bottlenecks. Through extensive algorithmic development and simulation-based empirical validation, we demonstrate how data-driven models can approximate complex integrals of the Cosserat formulation while maintaining high spatial accuracy. The proposed hybrid framework reduces computational latency, making it highly suitable for real-time control applications in surgical robotics and industrial inspection. Experimental results validate the superiority of our algorithmic approach in handling external disturbances and unmodeled non-linearities compared to baseline numerical methods. The findings underscore the critical role of hybrid computational architectures in advancing the capabilities of next-generation soft and continuum robotic systems.