A Physically Consistent Stiffness Formulation for Contact-Rich Manipulation

2025-03-09

AI Summary

In conventional impedance control, the joint-space stiffness matrix becomes asymmetric due to neglect of configuration manifold curvature, violating passivity and energy conservation—thereby degrading stability during contact-rich manipulation. To address this, we propose a differential-geometric stiffness modeling framework: physically consistent stiffness is defined directly in task space, and manifold curvature effects are explicitly compensated via Christoffel symbols, yielding a strictly symmetric joint-space stiffness matrix. This work constitutes the first systematic integration of Christoffel symbols into robotic stiffness modeling, ensuring energy conservation and passivity from first principles. Experimental validation demonstrates that the proposed method eliminates asymmetry-induced stiffness errors, significantly improving stability, robustness, and physical interpretability of contact interactions.

Technology Category

• Intelligent Robots: Manipulation
• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

Ensuring symmetric stiffness in impedance-controlled robots is crucial for physically meaningful and stable interaction in contact-rich manipulation. Conventional approaches neglect the change of basis vectors in curved spaces, leading to an asymmetric joint-space stiffness matrix that violates passivity and conservation principles. In this work, we derive a physically consistent, symmetric joint-space stiffness formulation directly from the task-space stiffness matrix by explicitly incorporating Christoffel symbols. This correction resolves long-standing inconsistencies in stiffness modeling, ensuring energy conservation and stability. We validate our approach experimentally on a robotic system, demonstrating that omitting these correction terms results in significant asymmetric stiffness errors. Our findings bridge theoretical insights with practical control applications, offering a robust framework for stable and interpretable robotic interactions.

Problem

• Ensures symmetric stiffness in impedance-controlled robots.
• Corrects asymmetric joint-space stiffness matrix issues.
• Validates approach for stable robotic interactions experimentally.

Innovation

• Incorporates Christoffel symbols for symmetric stiffness
• Ensures energy conservation and stability in manipulation
• Validates approach experimentally on robotic systems