Safety-Critical Control for Smoothed Implicit Contact Dynamics

Abstract: 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 $κ$ relaxes contact complementarity constraints, which makes the dynamics smooth but affects the contact force. This paper provides a method for bounding the actual contact force despite the use of relaxed complementarity constraints. We show that constraint violations can be non-monotonic in $κ$. Smaller $κ$ reduces force-approximation error, but it does not necessarily improve safety performance. To address this issue, we introduce boundary-focused rollouts to screen $κ$ by comparing the safety margin with the approximation error. We then develop a discrete-time control barrier function (CBF) framework based on a first-order Taylor approximation of the implicitly defined contact force. To account for possible force under-prediction, we augment the resulting safety constraint with a fixed robust margin. Simulations on four contact-rich systems show that the proposed method eliminates force violations observed under a standard CBF.

• The paper develops a QP-compatible safety filter that uses implicit differentiation and first-order contact-force prediction to apply control barrier functions to non-control-affine contact dynamics.
• The paper shows that the smoothing parameter κ affects safety non-monotonically, so boundary-focused screening is needed instead of simply selecting the smallest value.
• The paper introduces a robustly tightened barrier constraint that eliminated observed force violations across four deterministic simulations, while preserving task feasibility with calibrated error margins.

Overview

This paper addresses safety-critical control for robotic systems whose dynamics are governed by smoothed implicit contact models, specifically the complementarity-based formulation used in differentiable simulators such as Dojo (Howell et al., 2022). The central difficulty is that the contact force is defined implicitly through a nonlinear complementarity problem (NCP), so the dynamics are not control-affine and standard discrete-time control barrier function (CBF) quadratic programs cannot be applied directly. The authors develop a QP-compatible safety filter based on a first-order Taylor approximation of the implicitly defined contact force, and they identify and address two sources of error: linearization error in the local force prediction, and mismatch introduced by the smoothing (central-path) parameter $\kappa$.

The paper's key empirical finding is that reducing $\kappa$—which improves local model accuracy—does not monotonically improve closed-loop safety. In a 1D box-contact example with $\gamma_{\max}=0.25$ N, only $\kappa = 10^{-4}$ among four tested values remained within the force limit; both smaller and larger values produced violations. This non-monotonicity motivates both a screening procedure for selecting $\kappa$ and a robust tightening of the CBF constraint.

Problem setup

The contact model relaxes hard complementarity via a central-path condition: $\phi(z)\ge 0$, $\gamma\ge 0$, and $\gamma\,\phi(z)=\kappa$, where $\phi$ is the signed-distance function. Friction follows from the maximum dissipation principle subject to the friction cone $\|\beta\|_2 \le \mu\gamma$. Given problem data $\kappa$0, the relaxed NCP is written as $\kappa$1, and sensitivities of the solution—including the contact force input derivative $\kappa$2—are obtained by implicit differentiation.

The barrier function is defined directly on the force constraint: $\kappa$3. Since enforcing the discrete-time CBF condition $\kappa$4 on the true implicit dynamics would require solving a nonlinear program at each step, the authors instead linearize the next-step contact force around a nominal input $\kappa$5:

$\kappa$6

and impose the CBF condition on the predicted margin $\kappa$7, yielding an affine constraint solvable as a standard CBF-QP that minimally deviates from the nominal controller.

Non-monotonic dependence of safety on κ

Writing the true margin as $\kappa$8 where $\kappa$9 is the one-step approximation error, a violation occurs whenever $\gamma_{\max}=0.25$0. Although $\gamma_{\max}=0.25$1 decreases as $\gamma_{\max}=0.25$2 shrinks, the predicted margin itself also varies with $\gamma_{\max}=0.25$3. The authors derive a local identity under the assumption that the closed-loop quantities are differentiable in $\gamma_{\max}=0.25$4: log-differentiating the central-path relation gives

$\gamma_{\max}=0.25$5

where $\gamma_{\max}=0.25$6 is the logarithmic sensitivity of the closed-loop contact gap to $\gamma_{\max}=0.25$7. Since the gap changes with $\gamma_{\max}=0.25$8 through the state, filtered input, and smoothed contact update, the sign of this sensitivity is not fixed: decreasing $\gamma_{\max}=0.25$9 improves the margin only when $\kappa = 10^{-4}$0. This formally establishes that $\kappa = 10^{-4}$1 cannot be treated merely as a numerical smoothing parameter—it is a safety-relevant design choice, and the safe region in $\kappa = 10^{-4}$2 may be an interval rather than a single point.

Boundary-focused κ screening

To select $\kappa = 10^{-4}$3, the paper proposes Algorithm 1, which runs short-horizon CBF rollouts on representative boundary-approaching scenarios for each candidate $\kappa = 10^{-4}$4 and compares two statistics: $\kappa = 10^{-4}$5, the upper-tail quantile of the observed one-step under-prediction $\kappa = 10^{-4}$6, and $\kappa = 10^{-4}$7, the lower-tail quantile of the predicted margin minus $\kappa = 10^{-4}$8. A candidate is compatible if it has zero implicit-solver failures and $\kappa = 10^{-4}$9; among compatible candidates, the largest score wins, with ties broken toward smaller $\kappa$0. This procedure operationalizes the insight that a good $\kappa$1 must balance approximation accuracy against margin preservation near the boundary.

Robust barrier tightening

Residual linearization error remains after screening. Because under-prediction of the contact force is the safety-critical case—an unsafe input can appear feasible—the authors assume a lower bound $\kappa$2 over the operating region and prove a simple proposition: enforcing the tightened condition $\kappa$3 guarantees $\kappa$4, hence forward invariance of the safe set when $\kappa$5. In practice $\kappa$6 is chosen conservatively from rollout statistics (upper quantile or maximum of $\kappa$7).

The framework extends conditionally to the unsmoothed ($\kappa$8) model by decomposing its margin error into linearization error plus smoothing mismatch, each requiring its own bound. The smoothing mismatch admits a physical interpretation through impulse-momentum conservation:

$\kappa$9

bounding the force difference by the effective mass and the discrepancy in post-contact normal velocities. Safety for the unsmoothed system therefore requires bounding both terms simultaneously—a stronger requirement than safety for the smoothed model alone.

Experimental results

Four systems were evaluated—1D box contact, planar push, box pivot, and hopper—with $\phi(z)\ge 0$0 fixed throughout and reference trajectories from trajectory optimization tracked by nominal PD controllers. A $\phi(z)\ge 0$1-sweep over 20 log-spaced values from $\phi(z)\ge 0$2 to $\phi(z)\ge 0$3 confirmed substantial cross-system variability: box contact and box pivot exhibited violations at intermediate $\phi(z)\ge 0$4 (near $\phi(z)\ge 0$5) but not at extremes, planar push showed strongly non-monotonic margins, while the hopper was monotonic in $\phi(z)\ge 0$6.

With $\phi(z)\ge 0$7 and $\phi(z)\ge 0$8 fixed before evaluation, the robust CBF (rCBF) eliminated all observed force violations across all four deterministic rollouts, whereas the standard CBF left residual violations (e.g., violation rates of 0.345 on box contact and 0.053 on box pivot). Notably, rCBF also restored task feasibility on box pivot and hopper, where the standard CBF failed to complete the task. A sensitivity study on $\phi(z)\ge 0$9 revealed a conservatism trade-off: on the hopper the 85th-percentile margin sufficed, while planar push required the 90th percentile to become violation-free, with increasing $\gamma\ge 0$0 raising mean control deviation from the nominal input.

Limitations and open questions

Several caveats bear directly on the strength of the results. First, the guarantee in Proposition 1 holds only when the bound $\gamma\ge 0$1 is valid over the operating region; since no closed-form bound exists, $\gamma\ge 0$2 is calibrated empirically from offline rollouts, so the guarantee is conditional on rollout coverage being representative. Second, extension to the unsmoothed model requires bounding the smoothing mismatch, which relies on the impulse-momentum argument and knowledge of the effective mass; this bound is stated but not validated experimentally. Third, all experiments are deterministic simulations without model uncertainty or state estimation error, so robustness to disturbances remains untested. Fourth, the screening procedure depends on a set of boundary-approaching rollout scenarios, and the paper does not characterize how sensitive the selected $\gamma\ge 0$3 is to that scenario set or to the quantile parameters $\gamma\ge 0$4 and $\gamma\ge 0$5. Finally, whether the non-monotonicity phenomenon generalizes to higher-dimensional contact configurations, or admits a predictive characterization beyond the local sensitivity identity, is left open.

Conclusion

The paper formulates discrete-time CBF-based force-safety filtering for smoothed implicit contact dynamics via local Taylor approximation, proves that tightening the predicted CBF condition by a calibrated margin restores the nominal barrier guarantee despite linearization error, and demonstrates—both analytically and across four simulated contact-rich systems—that the smoothing parameter $\gamma\ge 0$6 affects closed-loop safety non-monotonically and must therefore be screened rather than simply minimized. The proposed rCBF eliminates force violations in all tested deterministic simulations while preserving task feasibility, establishing contact smoothing as a safety-relevant design parameter rather than a purely numerical device.