# A Physically Consistent Stiffness Formulation for Contact-Rich Manipulation

arXiv: https://arxiv.org/abs/2503.06802 · HTML: https://mlsanity.com/paper/2503.06802/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2025.

## Built on by (most cited first, 1 of 1)

- [Diffusion-Based Impedance Learning for Contact-Rich Manipulation Tasks](https://mlsanity.com/paper/2509.19696/) (2025)

## Abstract (arXiv)

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.