Explicit Optimization-Free CBFs

• The paper introduces an analytic, optimization-free CBF for polygonal collision avoidance that employs log-sum-exp smoothing to ensure a nonconservative safety boundary.
• The paper derives explicit state feedback controllers via offline multiparametric programming, replacing online quadratic programs with piecewise continuous control laws.
• The paper guarantees forward invariance and safety in control-affine systems while delivering 5–10× computational speedups for real-time applications.

Explicit optimization-free control barrier functions (CBFs) are methods for safe control synthesis in dynamical systems that avoid repeated online numerical optimization, instead relying on either closed-form analytic expressions or explicit offline characterizations of feasible domains and controller mappings. These approaches are motivated by scenarios where computational efficiency, deterministic timing, or embedded implementation preclude online optimization. Two central lines of work are the construction of analytic, nonconservative, differentiable CBFs for nonsmooth geometric safety tasks, exemplified by polygonal collision avoidance, and explicit, piecewise formulae for CBF-based safety filters in control-affine systems, synthesized via offline multiparametric programming.

1. Analytic Construction of Optimization-Free CBFs for Polygonal Collision Avoidance

Polygonal collision avoidance (PCA) addresses the problem of guaranteeing safety between planar convex polytopes governed by their own dynamic equations. Classical CBF designs for PCA have relied on optimization-defined distance measures, such as the signed distance field (SDF). However, SDF’s evaluation and differentiation are not analytically tractable and typically require online optimization or heuristics through boundary sampling. The optimization-free approach developed in "Optimization-free Smooth Control Barrier Function for Polygonal Collision Avoidance" (Wu et al., 22 Feb 2025) constructs an explicit, smooth, and nonconservative CBF for PCA through the following steps:

1. Signed Distance Field and Boolean Lower Bound: Define the Minkowski difference $P(x) = P^j(x^j) - P^i(x^i)$ for convex polygons and let the SDF be $h_s(x) = sd(P^i, P^j) = sd(0, P(x))$. By decomposing $P(x)$ into intersections of half-space differences and expressing the SDF as a maximum over minimal affine combinations, a finite-union lower bound $h_a(x)$ is obtained (Eq. (3)), given as a three-layer Boolean formula in terms of atomic affine functions.
2. Exactness and Nonconservatism: The zero-superlevel set of $h_a(x)$ coincides exactly with that of $h_s(x)$ for all $x$ where $h_s(x) \leq 0$, ensuring that no artificial conservatism is introduced around the safety boundary (Eq. (6), Theorem 1).
3. Log-Sum-Exp Smoothing: The nested min and max operators in $h_a$ are smoothed via log-sum-exp approximations, yielding $\hat{h}_a(x; b, \kappa)$ as a differentiable surrogate (Eq. (7)), where buffer parameter $b$ and smoothing factor $\kappa$ control the approximation’s tightness. Error bounds between $h_a$ and $\hat{h}_a$ scale as $\mathcal{O}(1/\kappa)$, and setting $b$ to exceed $\ln(r^i + r^j)$ (with $r^i, r^j$ the polygon face counts) ensures $\hat{h}_a(x) \leq h_a(x)$, preserving safety.
4. Final CBF and Safety Condition: The smooth, explicit CBF is $h(x) = \hat{h}_a(x; b, \kappa)$, which is $C^1$ and forward-invariant near the true SDF safe set. The classical CBF safety condition $\nabla h(x)^\top u \geq -\alpha(h(x))$ is imposed (Eq. (11)), certifying forward invariance of $\{ h \geq 0 \}$.

2. Explicit Solution of CBF-Based Safety Filters via Offline Parameterized Programming

For general nonlinear control-affine systems $\dot{x} = f(x) + g(x)u$, the standard CBF framework realizes safe feedback via an online quadratic program (QP) minimizing control deviation subject to the CBF constraint. In "Explicit Solutions for Safety Problems Using Control Barrier Functions" (Wang et al., 2022), the online QP is transformed into an explicit, piecewise-continuous state feedback law through offline parameterization:

1. Multi-parametric QP Formulation: Viewing the state $x$ as a parameter, one reformulates the CBF-QP as a convex program where feasible constraint sets partition the state space into polyhedral regions.
2. Explicit Controller Derivation:

The KKT conditions identify three exhaustive cases: - (C1) CBF constraint inactive: controller is the nominal $u^{\text{des}}(x)$; - (C2) CBF active, no input bound active: explicit affine correction in direction of the CBF gradient (Eq. (C2-u)); - (C3) CBF and some input bounds active: controller is a closed-form function of both the CBF and active input constraints (Eq. (C3-u)). State-space is thus partitioned so that within each region, the explicit safe control law is a fixed closed-form function.

1. Piecewise Lipschitz Continuity and Feasibility: The controller $x \mapsto u^*(x)$ is piecewise Lipschitz continuous within the feasible set, with branch transitions handled without loss of continuity due to complementarity conditions. Infeasibility due to saturation can be resolved by incorporating slack variables, always yielding a valid (though possibly less conservative) safe input.

3. Theoretical Guarantees: Safety, Invariance, and Nonconservatism

Both approaches provide formal guarantees known from standard CBF theory:

• The superlevel set $\{ x \mid h(x) \geq 0 \}$ is forward-invariant under the closed-loop dynamics (CBF-based invariance theorem).
• For the polygonal PCA construction, nonconservatism is guaranteed at the safety boundary: soft-smoothing induces only a quantifiable, vanishing inner approximation error as $\kappa \to \infty$ (Theorem 1, (Wu et al., 22 Feb 2025)).
• For explicit QP-derived controllers, uniqueness, continuity, and formal safety are maintained in all regions, with the global law $u^*(x)$ always being piecewise-defined but explicit and deterministic (Wang et al., 2022).

4. Computational and Practical Implications

Optimization-free CBFs offer significant computational advantages. For the polygonal approach, all CBF and gradient evaluations are explicit sums of linear and exponential terms. No per-step inner optimization or sampling is required—contrasting with GJK or boundary-sampled SDF/CBF methods. The only runtime optimization is a low-dimensional (often 2×2) QP per agent for the final safety filter. This results in 5–10× speedups in simulation benchmarks for vehicular and crane systems (Wu et al., 22 Feb 2025).

In multi-parametric explicit CBFs, the entire online QP computation is replaced by table-lookup or piecewise function evaluation—dramatically reducing computational overhead and enabling implementation in platforms with tight real-time constraints.

Method	Run-Time Optimization	Smoothness	Conservatism at Boundary
Polygonal analytic CBF (Wu et al., 22 Feb 2025)	Tiny QP, explicit CBF	Yes (via log-sum-exp)	None with large $\kappa$/$b$
Explicit QP-based CBF (Wang et al., 2022)	None; piecewise function	Piecewise Lipchitz	None (exact by construction)

5. Applications and Empirical Evaluations

The analytic, optimization-free CBF for PCA has been demonstrated in:

• Distributed collision avoidance of nonholonomic vehicles: Each vehicle models its shape as a polygon (triangle, trapezoid), and the CBF filter successfully averts collisions, outperforms SDF-linearized QPs with dense boundary sampling, and is computationally efficient (Wu et al., 22 Feb 2025).
• Underactuated container crane collision avoidance: The CBF is used atop an energy-shaped barrier for a 3-DOF crane system, ensuring safe transit with moving obstacles, and is tractable for real-time deployment.

Multiparametric explicit CBF design is applicable to any control-affine system with polyhedral input bounds, as confirmed by extensive simulation in (Wang et al., 2022).

6. Historical Context and Relation to Optimization-Based CBFs

Traditional CBF synthesis has relied on online convex optimization, especially for systems with complex geometry (SDF, GJK, etc.) or multiple constraints. The shift towards explicit, optimization-free realization emerges from both the analytic tractability of geometric primitives (as in PCA) and advances in multiparametric programming enabling explicit structuring of state-control maps. Central to these advances are the exploitation of Boolean formulas with softmax smoothings and the enumeration of feasible active sets, enabling deterministic and certifiable safe control.

7. Limitations and Future Directions

Optimization-free CBFs are contingent on the analytic tractability of the safety set geometry or on the prefabrication of the state partition for multiparametric QPs. As dimensionality and the number of constraints grow, storage and explicit offline computation can become challenging. A plausible implication is that hybrid approaches—combining analytic smoothing and online optimization, or coarser explicit approximations—may be advantageous in high-complexity domains. Extensions to non-polyhedral geometry and general underactuated systems remain active areas of research.

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References:

• "Optimization-free Smooth Control Barrier Function for Polygonal Collision Avoidance" (Wu et al., 22 Feb 2025)
• "Explicit Solutions for Safety Problems Using Control Barrier Functions" (Wang et al., 2022)