A Physically Consistent Stiffness Formulation for Contact-Rich Manipulation

TL;DR

The paper addresses stiffness asymmetry in impedance-controlled robots that arises from ignoring curvature-induced basis changes in task-space. It introduces a physically consistent mapping from task-space stiffness to joint-space stiffness by adding the Christoffel correction, yielding the joint-space stiffness form $\mathcal{K}_{\alpha\beta} = \frac{\partial b_{\beta}^{k}}{\partial q^{\alpha}} F_k + b_{\beta}^{k} a_{\alpha}^{l} \Gamma^{m}_{kl} F_m + b_{\beta}^{k} a^{l}_{\alpha} K_{kl}$, and shows this preserves energy conservation and passivity. Theoretical derivation is complemented by practical validation on a 7-DOF KUKA LBR iiwa, demonstrating symmetry of the joint-space stiffness under external wrenches and highlighting the importance of the correction terms; without them, a sizable antisymmetric component can arise. The work includes open-source code for practitioners to implement the correction in impedance control, offering a robust framework for stable, interpretable contact-rich manipulation.

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.

Figures (6)

• Figure 1: Chart map of a toroidal manifold patch (highlighted in purple). The coordinate-induced basis vectors $\frac{\partial}{\partial \xi^1}$ and $\frac{\partial}{\partial \xi^2}$ at different locations illustrate the rotation of the basis due to the manifold’s curvature. This necessitates correction terms (Christoffel symbols) to properly account for the basis change when computing derivatives.
• Figure 2: (A) Quadratic potential function $\mathcal{U}(\xi^1, \xi^2)$ with coordinate-induced basis vectors $\frac{\partial}{\partial \xi^1}$ and $\frac{\partial}{\partial \xi^2}$ at different locations. The global minimum is marked by the black point. (B) At the green point, the presence of an external force (negative gradient $\frac{\partial}{\partial \xi^k}(\mathcal{U})$) induces a basis shift and rotation. This basis change, due to the curvature of the potential field, requires correction terms (Christoffel symbols).
• Figure 3: Robot movement, visualized as overlaid robot configurations and directions during the bowl-wiping task.
• Figure 4: Recorded robot joint configurations during both experimental trials. The solid line represents the trial with correction terms (Christoffel symbols), and the dashed line without.
• Figure 5: External forces and moments from the ATI force-torque sensor, expressed in the base frame. The solid line represents the trial with correction terms (Christoffel symbols), and the dashed line without.