Safety-Critical Control for Smoothed Implicit Contact Dynamics

Summary

Abstract Summary

Smoothed implicit contact dynamics enables gradient-based planning and control for contact-rich tasks without predefined mode sequences. However, safety-critical control remains challenging because implicit contact dynamics makes safety-filter design nontrivial. The smoothing parameter $\kappa$ relaxes contact complementarity constraints, which makes the dynamics smooth but affects the contact force. This paper provides a safety-filtering framework for smoothed implicit contact dynamics. We first derive a discrete-time control barrier function (CBF) constraint using a first-order Taylor approximation of the implicitly defined contact force. We show that, although reducing $\kappa$ can improve local force-approximation accuracy, the resulting closed-loop force-constraint violations can vary non-monotonically with $\kappa$. Motivated by this observation, we introduce boundary-focused rollouts that screen candidate $\kappa$ values by comparing the predicted safety margin with the observed one-step under-prediction. We then robustly tighten the predicted CBF constraint with a fixed margin to account for residual force under-prediction. Simulations on four contact-rich systems show that the proposed method eliminates force violations observed under a standard CBF. Project website this https URL

Key Points

Smoothed implicit contact dynamics enables gradient-based planning and control for contact-rich tasks without predefined mode sequences.
However, safety-critical control remains challenging because implicit contact dynamics makes safety-filter design nontrivial.
The smoothing parameter $\kappa$ relaxes contact complementarity constraints, which makes the dynamics smooth but affects the contact force.
This paper provides a safety-filtering framework for smoothed implicit contact dynamics
We first derive a discrete-time control barrier function (CBF) constraint using a first-order Taylor approximation of the implicitly defined contact force.

Preliminaries and Problem Formulation

In this section, we briefly review the implicit contact dynamics model and discrete-time control barrier functions used in the proposed safety filter. Accurate modeling of contact forces is important when robots interact with environments. This paper adopts the complementarity-based contact dynamics model of [ 9 ] , in which contact and friction are modeled using a Nonlinear Complementarity Problem (NCP).

Let $\phi:\mathcal{Z}\to\mathbb{R}$ be the signed-distance function, where $\mathcal{Z}$ denotes the configuration space.

Preliminaries and Problem Formulation — Safety-Critical Control for Smoothed Implicit Contact Dynamics — Fig. 2 : Gradients are computed for different values of the central-path parameter $\kappa$. The dynamics involve a box of mass 0.01 kg resting on a flat surface in the XZ plane. The applied force was gradually increased from 0 N to 0.5 N.

Safety-Critical Control with Implicit Contact Dynamics

CBF frameworks typically rely on explicit control-affine system dynamics, allowing safety constraints to be written as quadratic programs (QPs). In our setting, the contact force is defined implicitly through ( 4 ), so the standard CBF-QP cannot be applied directly. We therefore use a local linear approximation of the solution to derive a QP-compatible constraint.

Experiments and Results

We evaluate the proposed safety filter on the four contact-rich systems shown in Fig. 4 : a 1D box contact, a planar push, a box pivot, and a hopper.

The box-contact input is the vertical force $u_{z}$; the planar-push and box-pivot inputs are $u=[u_{p,x},u_{p,y}]$ and $u=[u_{p,x},u_{p,z}]$, respectively; and the hopper input is $u=[M,\dot{r}]$, where $M$ is the body-pitch torque and $\dot{r}$ is the leg-length rate command. Task-specific $\kappa$ screening follows Sec. III-C .

Experiments and Results — Safety-Critical Control for Smoothed Implicit Contact Dynamics — Fig. 4 : Contact-force safety examples under smoothed implicit contact dynamics. Red circles denote independently actuated pushers in frictional contact; the bottom row shows forward hopping over a possible terrain drop.

Sources and Demonstrations

Figures are reproduced from Lee et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.