Structured Jacobian Construction for Motion Optimization with High-Order Time Derivatives in Multi-Link Systems

2026-05-15

AI Summary

This work addresses the inefficiency and numerical instability of existing trajectory optimization methods, which typically rely on numerical or automatic differentiation to compute Jacobians of high-order time derivatives—such as jerk and rate of force change—while neglecting the structural properties of multibody systems. The authors propose a novel analytical framework that explicitly models physical quantities and their higher-order derivatives by leveraging the inherent structure of multibody dynamics. For the first time, they derive structured Jacobian matrices with respect to generalized coordinates and their higher-order time derivatives. By integrating analytical differentiation with multibody dynamics, the method enables efficient and scalable forward and inverse optimization. It significantly improves computational efficiency and numerical stability compared to conventional approaches and demonstrates success in accurately recovering cost function weights from motion data in inverse optimization tasks.

Abstract

This paper presents a novel framework for Jacobian computation in motion optimization problems involving multi-link systems, where physical quantities are represented using higher-order time derivatives. In motion optimization of robots and humans, cost functions may incorporate higher-order time derivatives, such as jerk or the time variation of forces, to capture smoothness and perceptual characteristics, particularly in motion skill analysis and expressive behaviors, thereby necessitating Jacobian computations involving these quantities. However, such Jacobians are typically computed using numerical or automatic differentiation without explicitly exploiting the underlying multi-link structure, which can lead to increased computational cost and numerical instability. To address this limitation, we propose a structured Jacobian formulation for motion optimization, based on the comprehensive motion computation framework, in which physical quantities and their higher-order time derivatives are systematically represented along the multi-link structure. The proposed method systematically derives analytical expressions for Jacobians of kinematic and dynamic quantities, including momentum, forces, and joint torques, with respect to generalized coordinates and their higher-order derivatives. The resulting framework is applicable to both direct and inverse optimization. Through numerical experiments, we demonstrate that the proposed method improves computational efficiency compared to numerical and automatic differentiation, while achieving comparable accuracy. Furthermore, we demonstrate its effectiveness in inverse optimization by recovering cost function weights from motion data. Together, these results indicate that the proposed formulation provides a scalable and structured computational foundation for motion optimization involving higher-order time derivatives in multi-link systems.

Problem

• Jacobian computation
• motion optimization
• multi-link systems
• higher-order time derivatives
• computational efficiency

Innovation

• structured Jacobian
• high-order time derivatives
• multi-link systems
• motion optimization
• analytical differentiation