SQ-CBF: Signed Distance Functions for Reliable Superquadric-Based Safety Filtering

Haocheng Zhao; Lukas Brunke; Oliver Lagerquist; Siqi Zhou; Angela P. Schoellig

cs.ROFeb 11, 2026

SQ-CBF: Signed Distance Functions for Reliable Superquadric-Based Safety Filtering

Authors: Haocheng Zhao, Lukas Brunke, Oliver Lagerquist, Siqi Zhou, Angela P. Schoellig

Organizations: Learning Systems and Robotics Lab, Technical University of Munich, 80333 Munich, Germany, and also with Simon Fraser University, Burnaby, BC V5A 1S6, Canada

Abstract

Ensuring safe robot operation in cluttered and dynamic environments remains a fundamental challenge. While control barrier functions (CBFs) provide an effective framework for real-time safety filtering, their performance critically depends on the underlying geometric representation, which is often simplified, leading to either overly conservative behavior or insufficient collision coverage. Superquadrics (SQs) offer an expressive way to model complex shapes using a few primitives and are increasingly used for robot safety. To integrate this representation into CBF, most existing approaches directly use their implicit functions as barrier candidates. However, we identify a critical but overlooked issue in this practice: the gradients of the implicit SQ function can become severely ill-conditioned, potentially rendering the optimization infeasible and undermining reliable safety filtering. To address this issue, we formulate an SQ-based safety filter that uses signed distance functions (SDFs) as barrier candidates. Since analytical SDFs are unavailable for general SQs, we compute the distance using the efficient Gilbert-Johnson-Keerthi algorithm and derive its gradient in closed form. Extensive simulation and real-world experiments demonstrate consistent collision-free manipulation in cluttered scenes, showing robustness to challenging geometries, sensing noise, and dynamic disturbances, while improving task efficiency in teleoperation tasks. These results highlight a pathway toward safety filters that remain precise and reliable under the geometric complexity of real-world environments.

Figures & tables

Fig. 1 : Overview of the proposed SQ-CBF safety filter. The framework represents both the robot (blue) and the obstacles (red) using compact superquadrics and enforces collision avoidance through an SDF-based CBF. SDFs are computed from superquadric geometries using GJK/EPA, while well-behaved distance gradients are estimated online and incorporated into the safety filter. Given an unverified control command (e.g., from teleoperation or a planner), the safety filter modifies the command online by solving a CBF-QP, intervening only when safety is at risk. A video demonstrating the safety filter’s performance can be found here: http://tiny.cc/sq-cbf .

Panel of Figure 2 — Fig. 2 : Evaluating the implicit function $f_{\text{sq}}^{*}(x)=\min_{\bm{p}\in\mathbb{R}^{3}}$ $f_{\text{sq}}(\bm{R}_{z}(\pi/3)^{\top}(\bm{p}-\bar{\bm{p}}_{1}),\bm{s}_{1})~\text{s.t.}~f_{\text{sq}}(\bm{R}_{z}(-\pi/4)^{\top}(\bm{p}-\bar{\bm{p}}_{2}(x)),\bm{s}_{2})\leq 1$ from [ 24 ] and the SDF $d(x)$ and their gradients for two superquadrics with parameters $\bm{s}_{1}=[0.5,1.5,1.0,0.2,0.2]^{\top}$ and $\bm{s}_{2}=[1.0,0.5,1.0,0.2,0.2]^{\top}$ while varying their relative $x$ position with $\bar{\bm{p}}_{1}-\bar{\bm{p}}_{2}(x)=[0,0,0]^{\top}-[x,3,0]^{\top}$ . The implicit function and its gradient are large (greater than $10^{4}$ ), leading to ill-conditioned matrices in the CBF-QP. In practice, this may render the optimization infeasible, which is further demonstrated by the open-loop comparison presented in Fig. 7 .

Panel of Table 3 — TABLE I : Comparison of geometric collision models. Bold entries are from our proposed method, while notable limitations are highlighted in red.

Fig. 3 : Cycle time (mean $\pm$ std.) of the proposed SDF and gradient evaluation pipeline as a function of constraint number under different CPU parallelization levels. Multi-core execution significantly improves scalability.

Fig. 4 : Trade-off between efficiency and accuracy with increasing resolution

n_{u}=n_{v}

. The blue curve shows the relative cycle time normalized w.r.t. the

200\times 200

resolution, while the red dashed curve (log-scale) shows the empirical SDF error. Increasing the resolution significantly reduces discretization error by approximately two orders of magnitude, at the cost of slightly increased computation time.

Fig. 5 : Simulation experiment of a teleoperated insertion task under progressively tighter geometric constraints. The results show that the proposed superquadric-based safety filter consistently prevents collisions and reduces the task completion time by $47.5\%$ on average compared with the baseline without a safety filter, and by $26.8\%$ on average compared with the cylinder/box-based safety filter. In contrast, the coarse spherical model from [ 14 ] may be insufficient to guarantee safety.

Fig. 6 : Real-world teleoperated manipulation experiments demonstrating the efficacy of our proposed safety filter on three representative tasks. Across all tasks, the safety filter consistently enables collision-free execution with minimal intervention, despite tight geometric constraints, unstructured and cluttered obstacles, and dynamic disturbances. A video demonstrating the safety filter’s performance can be found here: http://tiny.cc/sq-cbf .

Fig. 7 : Comparison of the implicit function-based and the SDF-based formulations along a replayed trajectory from Experiment 2. The implicit function value exhibits extreme variations spanning

10^{5\sim 17}

, and occasionally returns numerically saturated values (displayed as

\mathrm{Inf}

) due to the ill-conditioned optimization. These effects lead to infeasible QP instances. In contrast, the SDF remains well-behaved and maintains feasibility throughout.

Task	Safety Filter	SR ( $\uparrow$ )	Minimum SDF ( $\geq 0$ )	Infeasible Freq. ( $\downarrow$ )	Completion Time † ( $\downarrow$ )
Exp.1	w/o SF	$0\%$	$-82.9\pm 7.7$ $\mathrm{mm}$	–	–
	DCOL-based [ 16 ]	$40\%$	$21.2\pm 2.3$ $\mathrm{mm}$	$0\%$	$29.1\pm 2.4s$
	Impl.-based ‡ [ 11 ]	–	–	$27.3\pm 5.3\%$	–
	SDF-based (ours)	$\mathbf{100\%}$	$\mathbf{15.1\pm 4.3}$ $\mathrm{mm}$	$\mathbf{0\%}$	$\mathbf{17.9\pm 2.2s}$
Exp.2	w/o SF	$20\%$	$-23.2\pm 10.8$ $\mathrm{mm}$	–	$35.1\pm 2.1s$
	DCOL-based [ 16 ]	$90\%$	$17.4\pm 2.5$ $\mathrm{mm}$	$0\%$	$38.5\pm 3.7s$

TABLE II : Evaluation statistics for Experiments 1 and 2 over

10

trials each. Success rate (SR) denotes the fraction of successful executions. The minimum SDF is reported as mean

\pm

std., with positive values indicating clearance and negative values indicating penetration. Infeasible Frequency denotes the average percentage of CBF-QP infeasible instances during each trial. The best value for each metric is highlighted in blue.

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Open paper AskPDF arXiv

We present a dual-barrier control barrier function (CBF) safety filter for real-time, safety-critical velocity control of holonomic robots operating in incrementally built occupancy grid maps. As a robot explores an unknown environment, unmapped regions introduce irreducible uncertainty, since obstacle geometry beyond the explored frontier is unknown, making entry into such regions a source of collision risk, especially with front-facing sensors. To address this, we enforce two constraints: avoidance of mapped obstacles and restriction from unexplored regions. Both constraints are derived analytically from the occupancy grid's signed distance field, yielding a closed-form safety filter that requires only a small linear system solve per cycle. On resource-constrained platforms such as the Raspberry Pi, where SLAM and planning already consume significant compute, the low overhead of the proposed filter preserves resources. An adaptive gain schedule relaxes the frontier constraint in information-rich regions and tightens it in well-mapped areas, improving exploration efficiency while maintaining safety. The filter operates in velocity space as a minimally invasive correction and composes with arbitrary nominal controllers, including learning-based methods. Hardware flight experiments on a PX4-controlled quadrotor demonstrate zero collisions across multiple indoor runs.

Abstract

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