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Exact Feasibility Certification and Optimal Responsibility Allocation for Multi-Robot CBF Safety Filters
Chandan Kumar Sah, Jishnu Keshavan
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 9/16/2026, 7:27:36 AM
Summary
This paper introduces an exact feasibility certificate for multi-robot Control Barrier Function (CBF) safety filters to diagnose infeasibility in quadratic programs. The certificate decomposes feasibility into constraint demand and actuator supply, identifying responsible agents and interactions. It also proposes an algorithm to optimally allocate shared safety constraints by maximizing the worst local feasibility margin, significantly reducing infeasible control steps and safety violations in simulations.
Entities (8)
Relation Signals (6)
Control Barrier Function → canbecome → Infeasible
confidence 98% · Multi-robot Control Barrier Function (CBF) safety filters can become infeasible
Responsibility Allocation → reduces → Infeasibility
confidence 96% · the proposed allocation reduces infeasible control steps from roughly 50% to 6.2%
Feasibility Certificate → decomposes → Constraint Demand
confidence 95% · The certificate quantifies a feasibility reserve by separating the demand imposed by safety constraints from the available actuator supply.
Feasibility Certificate → decomposes → Actuator Supply
confidence 95% · The certificate quantifies a feasibility reserve by separating the demand imposed by safety constraints from the available actuator supply.
Responsibility Allocation → yields → Linear Program
confidence 94% · yielding a linear program for polyhedral input sets
Feasibility Certificate → identifies → Agents
confidence 92% · identifies the agents and interactions responsible for the conflict
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Abstract
Abstract:Multi-robot Control Barrier Function (CBF) safety filters can become infeasible, but a failed quadratic program (QP) does not indicate why the conflict occurred or how to resolve it. To address this, we develop an exact feasibility certificate for multi-agent CBF filters with heterogeneous control-affine dynamics and convex input sets. The certificate quantifies a feasibility reserve by separating the demand imposed by safety constraints from the available actuator supply. This decomposition shows when CBF gain tuning or increased actuation can and cannot resolve infeasibility, and identifies the agents and interactions responsible for the conflict. We further propose an algorithm to optimally allocate shared safety constraints by maximizing the worst local feasibility margin, yielding a linear program for polyhedral input sets. In $320$ paired closed-loop simulations, the proposed allocation reduces infeasible control steps from roughly $50\%$ to $6.2\%$, and reduces safety-violating runs from $118/160$ to $24/160$. In addition, across $52$ infeasibility events, the certificate identifies an interaction whose relaxation restores feasibility in $94\%$ of cases.
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- Source: https://arxiv.org/abs/2609.14935v1
- Canonical: https://arxiv.org/abs/2609.14935v1
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Exact Feasibility Certification and Optimal Responsibility Allocation for Multi-Robot CBF Safety Filters Chandan Kumar Sah Jishnu Keshavan †thanks: The authors are with the Department of Mechanical Engineering, Indian Institute of Science, Bengaluru 560012, India. chandanks@iisc.ac.in, kjishnu@iisc.ac.in Abstract Multi-robot Control Barrier Function (CBF) safety filters can become infeasible, but a failed quadratic program (QP) does not indicate why the conflict occurred or how to resolve it. To address this, we develop an exact feasibility certificate for multi-agent CBF filters with heterogeneous control-affine dynamics and convex input sets. The certificate quantifies a feasibility reserve by separating the demand imposed by safety constraints from the available actuator supply. This decomposition shows when CBF gain tuning or increased actuation can, and cannot, resolve infeasibility, and identifies the agents and interactions responsible for a conflict. We further propose an algorithm to optimally allocate shared safety constraints by maximizing the worst local feasibility margin, yielding a linear program for polyhedral input sets. In 320320 paired closed-loop simulations, the proposed allocation reduces infeasible control steps from roughly 50%50\% to 6.2%6.2\%, and reduces safety-violating runs from 118/160118/160 to 24/16024/160. In addition, across 5252 infeasibility events, the certificate identifies an interaction whose relaxation restores feasibility in 94%94\% of cases. [Project page] [Code] I Introduction Control barrier function (CBF) safety filters provide a principled way to enforce safety while retaining a nominal controller. In multi-robot systems, however, multiple pairwise or higher-order CBF constraints must often be satisfied simultaneously. Although each constraint may be feasible individually, their conjunction can become infeasible. A standard QP returns only a binary infeasibility flag, giving no indication of which interaction caused the conflict, whether the limitation comes from the CBF encoding or available actuation, or how the conflict should be resolved. Existing approaches address infeasibility through feasibility-guaranteeing constructions [1, 2], compatibility conditions [3], nonsmooth barrier compositions [4], actuation-aware formulations [5], or online adaptation of CBF parameters [6]. These methods provide ways to avoid or recover from infeasibility, but do not provide an exact characterization of its source. Closest to ours, [7] gives a Farkas-type feasibility test for stacked barrier constraints in LTI systems, where the normals are constant and the input set polyhedral, and [8] treats opposing HOCBF bounds for a single underactuated system. Both decide feasibility without quantifying it: neither returns a value that separates the demand a constraint imposes from the actuation available to meet it, which is what supports the diagnosis and allocation below. Neither addresses state-dependent constraint normals, arbitrary convex input sets, or the multi-agent setting. This paper develops an exact conic certificate that turns multi-robot CBF filter infeasibility into an actionable diagnosis. The certificate exactly characterizes pointwise feasibility and decomposes the feasibility reserve into constraint demand and actuator supply. This decomposition determines whether infeasibility can be addressed through the CBF encoding, increased actuation, or redistribution of constraint responsibility. It further shows that, under the standard HOCBF construction, class-K gain tuning cannot increase actuator supply, and characterizes underactuated states where increasing actuation cannot restore feasibility. Fig. 1: An infeasible multi-robot CBF safety filter provides only a binary flag without revealing the cause, the responsible agents, or remedy. The certificate also provides a principled basis for decentralized responsibility allocation. For each shared constraint, responsibility is allocated to maximize the worst local feasibility margin. For polyhedral input sets, this max-min problem reduces to a linear program, directly optimizing local feasibility rather than relying on uniform or capability-weighted heuristics [9]. The certificate additionally provides sparse conflict attribution through its dual variables and quantifies the marginal value of additional actuation. When no allocation can make all local programs feasible, the same optimization provides the input that maximizes the worst local margin. Our main contributions are: • We derive an exact conic feasibility certificate that decomposes CBF filter feasibility into constraint demand and actuator supply. • We use the certificate to characterize the limits of CBF tuning and actuation, identify responsible interactions, and quantify actuator leverage. • We develop a certificate-based responsibility allocation that maximizes the worst local feasibility margin. The paper is organized as follows: Section I formulates the problem; Section I develops the feasibility certificate; Section IV diagnoses infeasibility; and Section V presents the responsibility allocation. Sections VI–VIII cover experiments, limitations, and conclusions. I Problem Formulation I-A Agents and Constraints Consider N agents with heterogeneous control-affine dynamics ˙i=i(i)+i(i)i,i∈i⊂ℝmi, x_i= f_i( x_i)+ g_i( x_i) u_i, u_i _i ^m_i, (1) where i∈i⊂ℝni x_i _i ^n_i, and i,i f_i, g_i are locally Lipschitz and sufficiently smooth. Let =(1,…,N) x=( x_1,…, x_N), =(1,…,N) u=( u_1,…, u_N), and =∏iU= _iU_i. For a scalar function φ() ( x), define ℒfφ=∑i∇iφi,ℒgiφ=∇iφi∈ℝ1×mi.L_f = _i _ x_i \, f_i,\;L_g_i = _ x_i \, g_i ^1× m_i. We index agents by i, safety constraints by k, and stages of a barrier chain by j. For a nonempty compact convex set ⊂ℝmS ^m, its support function is σ:ℝm→ℝ _S:R^m , σ()=max∈⟨,⟩ _S( ν)= _ s ν, s , and is convex. For ∈ℝK λ ^K we write supp=k∈1,…,K:λk≠0supp λ=\k∈\1,…,K\: _k≠ 0\. Let E index a finite collection of safety constraints. Each constraint k∈Ek∈ E is specified by a barrier function hk:→ℝh_k:X with hk()≥0.h_k( x)≥ 0. Its support Sk⊆1,…,NS_k \1,…,N\ contains the agents whose states affect hkh_k, and its arity is |Sk||S_k|. Thus, |Sk|=1|S_k|=1, 22, or ≥3≥ 3 corresponds to single-agent, pairwise, or higher-order interactions, respectively. We write Ei=k∈E:i∈SkE_i=\k∈ E:i∈ S_k\ for the set of constraints incident to agent i. The joint safe set is =∈:hk()≥0,∀k∈E.C=\ x :h_k( x)≥ 0,\ ∀ k∈ E\. Assumption 1. Each iU_i is nonempty, convex, and compact. Definition 2 (Uniform relative degree). A barrier function hkh_k has uniform relative degree rk≥1r_k≥ 1 if LgiLfjhk≡0L_g_iL_f^jh_k≡ 0 for all i∈Ski∈ S_k and all j≤rk−2j≤ r_k-2, and LgiLfrk−1hk≢0L_g_iL_f^r_k-1h_k ≡ 0 for some i∈Ski∈ S_k. I-B High-Order Barrier Function (HOCBF) Chains For higher-order constraints, we enforce safety using a HOCBF chain [2]. Following [2], set ψk,0=hk _k,0=h_k and define ψk,j=ψ˙k,j−1+αk,j,j=1,…,rk, _k,j= ψ_k,j-1+ _k,j, j=1,…,r_k, (2) where αk,j _k,j are the class-K functions. The corresponding admissible set is α=:ψk,j()≥0,∀k∈E, 0≤j≤rk−1.C_α=\ x: _k,j( x)≥ 0,\ ∀ k∈ E,\ 0≤ j≤ r_k-1\. Under Definition 2, ψk,j _k,j is independent of the input for j<rkj<r_k. Consequently, the final stage of the chain is affine in the control and can be written as ∑i∈Skk,i()i≥−bk(), _i∈ S_k G_k,i( x) u_i≥-b_k( x), (3) k,i:=Lgiψk,rk−1,bk:=Lfψk,rk−1+αk,rk. G_k,i:=L_g_i _k,r_k-1,\;b_k:=L_f _k,r_k-1+ _k,r_k. (4) Because hkh_k depends only on the states of agents in SkS_k, k,i≡0 G_k,i≡ 0 for i∉Ski∉ S_k. Stacking the constraints therefore gives ()≥−(), G( x) u≥- b( x), where ∈ℝK×∑imi, G ^K× _im_i, and K=|E|.K=|E|. The pointwise feasible set is thus ℱ=:∃∈,()≥−().F=\ x:∃\, u ,\ G( x) u≥- b( x)\. Hence, the safety filter is feasible at x if and only if ∈ℱ x , and αC_α is forward invariant if and only if α⊆ℱC_α . I An Exact Feasibility Certificate This section characterizes the pointwise feasibility of the safety filter through a scalar reserve M()M( x). To construct it, let k,i:=k,i⊤∈ℝmi ν_k,i:= G_k,i ^m_i denote the input-space direction associated with constraint k for agent i. For a nonnegative weighting ∈Δ λ∈ , where Δ=∈ℝ≥0K:∑kλk=1 =\ λ ^K_≥ 0: _k _k=1\, define the combined direction i()=∑k:i∈Skλk,i. ν_i( λ)= _k:\,i∈ S_k _k ν_k,i. The maximum input contribution along this direction is given by the support function σi()=max∈i⟨,⟩. _U_i( ν)= _ u _i ν, u . Definition 3 (Feasibility reserve). The feasibility reserve at x is M():=min∈Δ[⊤()+∑i=1Nσi(i())].M( x):= _ λ∈ [ λ b( x)+ _i=1^N _U_i\! ( ν_i( λ) ) ]. The filter is feasible at x if and only if M()≥0M( x)≥ 0. In words, the minimization searches for a weighting of the constraints under which their combined demand exceeds everything the actuators can supply together. If no such weighting exists, an admissible input does. Theorem 4. Under Assumption 1, the minimization defining M()M( x) is a convex program with an attained minimum, and ∈ℱ⟺M()≥0. x \; \;M( x)≥ 0. Proof. Let ϕ():=⊤+σ(⊤)φ( λ):= λ b+ _U( G λ). Since ⊤ G λ is linear in λ and support functions are convex, ϕφ is convex. It is also continuous because U is compact. Since Δ is compact, the minimum of ϕφ over Δ is attained. We next establish the feasibility equivalence. Define :=+:∈S:=\ G u+ b: u \. By convexity and compactness of U, S is convex and compact. The safety filter is feasible if and only if ∩ℝ≥0K≠∅S ^K_≥ 0≠ . Suppose first that ∉ℱ x . Then ∩ℝ≥0K=∅S ^K_≥ 0= . By the strict separation theorem, there exist ≠0 λ≠ 0 and γ∈ℝγ such that ⊤<γ λ z<γ for all ∈ z , and ⊤>γ λ y>γ for all ∈ℝ≥0K y ^K_≥ 0. Setting = y= 0 gives γ<0γ<0. Moreover, if λk<0 _k<0 for some k, then choosing =tk y=t e_k (k e_k denotes the kkth standard basis vector in ℝKR^K) with t→∞t→∞ violates the second inequality; hence ≥0 λ≥ 0. Therefore, sup∈⊤(+)<0 _ u λ ( G u+ b)<0. By positive homogeneity, normalize λ so that ⊤=1 1 λ=1. Thus ∈Δ λ∈ and M()<0M( x)<0. Conversely, suppose that M()<0M( x)<0. Then there exists ∈Δ λ∈ such that ⊤+σ(⊤)<0 λ b+ _U( G λ)<0. For any ∈ u , ⊤(+)≤⊤+σ(⊤)<0 λ ( G u+ b)≤ λ b+ _U( G λ)<0. Since ≥0 λ≥ 0, this is incompatible with +≥0 G u+ b≥ 0. Thus ∉ℱ x . Finally, using the product structure =∏iU= _iU_i and the block decomposition ⊤=(1(),…,N()) G λ=( ν_1( λ),…, ν_N( λ)), the support function separates as σ(⊤)=∑i=1Nσi(i()) _U( G λ)= _i=1^N _U_i( ν_i( λ)). Substitution gives the stated expression for M()M( x). ∎ The above result follows from separation of a compact convex set, rather than from KKT conditions. The reserve measures demand against supply. The term ⊤ λ b represents the demand imposed by the constraints, while ∑iσi(i()) _i _U_i( ν_i( λ)) measures the available actuation capability. Feasibility holds when supply meets demand for every weighting λ. Note that the certificate is instantaneous and does not by itself imply feasibility over a future trajectory. Lemma 5. Suppose i=i+iU_i= o_i+W_i, where iW_i is symmetric, convex, and compact. Then σi()=⟨,i⟩+σi() _U_i( ν)= ν, o_i + _W_i( ν), and the feasibility reserve can be written as M()=min∈Δ[⊤′()+S(,)]M( x)= _ λ∈ [ λ b ( x)+S( x, λ)], where bk′=bk+∑i∈Sk⟨k,i,i⟩,S(,)=∑iσi(i()).b _k=b_k+ _i∈ S_k ν_k,i, o_i ,\;S( x, λ)= _i _W_i( ν_i( λ)). (5) Proof. For i=i+iU_i= o_i+W_i, σi()=max∈i⟨,i+⟩=⟨,i⟩+σi() _U_i( ν)= _ w _i ν, o_i+ w = ν, o_i + _W_i( ν). Substituting this into the definition of M and using i()=∑k:i∈Skλk,i ν_i( λ)= _k:i∈ S_k _k ν_k,i combines the offset terms, by Lemma 5, into ⊤′ λ b and leaves S(,)S( x, λ). ∎ This decomposition covers standard input sets such as balls, ellipsoids, boxes, and segments, while asymmetries such as thrust bounds [min,max][ u_ , u_ ] are absorbed into the effective demand ′ b . Define S⋆():=min∈ΔS(,)S ( x):= _ λ∈ S( x, λ) as the supply. Theorem 6. If αk,j _k,j is chain-valued for every j≤rk−1j≤ r_k-1, then for all i∈Ski∈ S_k k,i()=LgiLfrk−1hk(), G_k,i( x)=L_g_iL_f^\,r_k-1h_k( x), independently of αk,1,…,αk,rk _k,1,…, _k,r_k. Proof. By Lemma 15 (Appendix A), at j=rk−1j=r_k-1, ψk,rk−1=Lfrk−1hk+Φk,rk−1(hk,…,Lfrk−2hk) _k,r_k-1=L_f^r_k-1h_k+ _k,r_k-1(h_k,…,L_f^r_k-2h_k). Then Lgiψk,rk−1=LgiLfrk−1hk+∑m≤rk−2∂mΦLgiLfmhkL_g_i _k,r_k-1=L_g_iL_f^r_k-1h_k+ _m≤ r_k-2 _m \,L_g_iL_f^mh_k, and each term in the sum vanishes by Definition 2. The last function αk,rk _k,r_k enters only bkb_k in (4). ∎ Theorems 4 and 6 establish the key separation used throughout the paper. The reserve decomposes into a demand term ⊤ λ b and a supply term OPEN∑iσi(i())) _i _U_i( ν_i( λ))), where the latter depends only on the barriers, configuration, and dynamics. Hence, class-K tuning can modify the demand but cannot change the available input supply. The achievable feasibility margin is therefore determined by both how the constraint is encoded and by the robots’ actuation geometry. Theorem 6 has one exception: the penultimate class-K function, αk,rk−1 _k,r_k-1, may depend on states outside its own chain. This affects the constraint geometry rather than the demand term and is not used in our proposed allocation algorithm. We therefore defer it to Appendix A. IV Diagnosing Infeasibility The feasibility certificate (Theorem 4) also enables us to diagnose the source of infeasibility. In particular, the demand-supply decomposition distinguishes failures due to insufficient actuator capability from those caused by conflicting constraints. IV-A Actuation limitations This section characterizes the states where increased actuation cannot restore feasibility. Such states cannot arise under full actuation, since a strictly separating i∈i y_i _i then exists for every constraint and forces S⋆>0S >0, but they can occur for underactuated agents. Throughout, let i=i+iU_i= o_i+W_i, where iW_i is symmetric, and consider the scaled sets iρ=i+ρiU_i^ρ= o_i+ _i, where ρ>0ρ>0. Increasing ρ increases actuation magnitude while preserving the drift and input directions. The condition below says that some weighted combination of the constraint normals points in a direction no agent can push along. Scaling the input sets then changes nothing, because every support function in the supply is evaluated at a direction the agent cannot act in. Theorem 7. The supply S⋆()=0S ( x)=0 if and only if there exists ⋆∈Δ λ ∈ such that i(⋆)⟂spani,∀i. ν_i( λ ) _i, ∀ i. (6) If iW_i is full-dimensional in ℝmiR^m_i for every i, equivalently 0∈inti0 _i, then (6) reduces to i(⋆)=0 ν_i( λ )=0 for all i. Equivalently, the origin lies in the convex hull of the constraint directions, 0∈convk:k∈E,0∈ \ A e_k:k∈ E\, where :ℝK→ℝ∑imi A:R^K _im_i is defined by =(i())i A λ=( ν_i( λ))_i. Proof. Since 0∈i0 _i and iW_i is symmetric, σi()≥0 _W_i( ν)≥ 0. Moreover, σi()=0 _W_i( ν)=0 if and only if ⟨,⟩=0 ν, w =0 for every ∈i w _i, which is equivalent to ⟂spani ν _i. Hence S(,)=0S( x, λ)=0 if and only if i()⟂spani ν_i( λ) _i for every i. Since Δ is compact, the minimum defining S⋆S is attained, which proves the first claim. If iW_i is full-dimensional, then spani=ℝmi _i=R^m_i, so (6) is equivalent to i(⋆)=0 ν_i( λ )=0 for every i. Thus ⋆=0 A λ =0 with ⋆∈Δ λ ∈ , which is equivalent to 0∈convk:k∈E0∈ \ A e_k:k∈ E\. ∎ In words, when S⋆()=0S ( x)=0, some convex combination of the constraint directions is orthogonal to the agents’ available actuation subspaces. Consequently, scaling the input sets cannot increase the supply, since the corresponding support-function contributions are zero (Corollary 8). Corollary 8. Let ⋆ λ satisfy (6). Then Mρ()≤⋆⊤′()M^ρ( x)≤ λ b ( x) for every ρ>0ρ>0, where MρM^ρ is the corresponding reserve for ρU^ρ. In particular, if ⋆⊤′()<0 λ b ( x)<0 then ∉ℱρ x ^ρ for every ρ>0ρ>0, i.e., no scaling of the input sets renders x feasible. Proof. ′ b does not depend on ρ, and σρi(i(⋆))=ρσi(i(⋆))=0 _ _i( ν_i( λ ))=ρ\, _W_i( ν_i( λ ))=0. Hence Mρ()≤⋆⊤′+0M^ρ( x)≤ λ b +0, and Theorem 4 applies. ∎ The degeneracy condition is therefore a property of the collective input geometry, not of the individual input sets being degenerate. Even when each iW_i is full-dimensional in its own input space, underactuation reduces the dimension of the codomain of A, making the condition 0∈convk:k∈E0∈ \ A e_k:k∈ E\ easier to satisfy. Proposition 17 exhibits such a state for the agents of Section VI-A. IV-B Conflict Localization This subsection derives three consequences of the certificate, each obtained from the multiplier ⋆ λ . IV-B1 Marginal Value of Actuation Let each agent’s reachable set scale independently, iρi=i+ρiiU_i _i= o_i+ _iW_i with ρi>0 _i>0, and write M(,)M( x; ρ) for the reserve of the scaled family. Proposition 9. If the minimizer ⋆ λ is unique, then ∂M∂ρi(,)=σi(i(⋆))≥0. ∂ M∂ _i( x; ρ)= _W_i\! ( ν_i( λ ) )≥ 0. (7) In general, when the minimizer is not unique, the Clarke subdifferential [10] of M with respect to ρi _i is the convex hull of σi(i()) _W_i\! ( ν_i( λ) ) over all minimizers λ. Proof. By Lemma 5, ϕ(,,)=⊤′+∑iρiσi(i())φ( x, λ; ρ)= λ b + _i _i\, _W_i( ν_i( λ)), in which the offsets enter only through ′ b and are independent of ρ. So ϕφ is affine, hence smooth, in ρ, and continuous in (,)( λ, ρ) on the compact set Δ . Danskin’s theorem [11] gives ∇M=∇ϕ(,⋆,) _ ρM= _ ρφ( x, λ ; ρ), whose i-th entry is σi(i(⋆)) _W_i( ν_i( λ )). Nonnegativity follows from 0∈i0 _i. ∎ The marginal value of actuation for agent i is the support function of its reachable set evaluated at the aggregate normal at the dual optimum ⋆ λ . This value is zero for two distinct reasons: the incident normals may cancel, or their aggregate direction may lie outside the agent’s actuation directions. In either case, increasing that agent’s actuation cannot improve feasibility. IV-B2 Sparsity of the Conflict Certificate Proposition 10. There exists a minimizer ⋆ λ of M()M( x) with |supp⋆|≤rank[⊤]≤∑imi+1, λ \ ≤\ rank\! bmatrix A\\ 1 bmatrix\ ≤\ Σ _im_i+1, irrespective of |E||E|, where =(i())i A λ=( ν_i( λ))_i. Proof. A minimum-support minimizer ⋆ λ exists, the argmin set being nonempty by Theorem 4; let T=supp(⋆)T=supp( λ ). If |T|>rank[;⊤]|T|>rank[ A;1 ], there exists ≠0 z≠ 0 supported on T with =0 A z=0 and ⊤=01 z=0. Since ⋆>0 λ >0 on T, ⋆+t∈Δ λ +t z∈ for sufficiently small |t||t|. Moreover, =0 A z=0 gives ϕ(⋆+t)=ϕ(⋆)+t⊤φ( λ +t z)=φ( λ )+t b z. Minimality for both signs of t implies ⊤=0 b z=0, so ϕφ is constant along the feasible line. Since ⊤=01 z=0 with ≠0 z≠ 0, its entries take both signs, so moving along this line reaches a point where a support coordinate vanishes, giving a minimizer of strictly smaller support and contradicting the choice of ⋆ λ . Hence |T|≤rank[;⊤]|T| [ A;1 ]. The second inequality counts rows: [;⊤][ A;1 ] has ∑imi+1 _im_i+1 of them, so its rank cannot exceed that. ∎ Thus, the worst irreducible conflict contains at most one more constraint than the actuated degrees of freedom, making the bound dimensional rather than combinatorial. V Certificate-Optimal Responsibility Allocation A decentralized filter must divide each shared constraint among the agents it couples. Existing choices are heuristic [12, 9, 13, 14]: they guarantee the individual pairwise conditions but not the feasibility of the local programs that result when one agent carries several constraints, a gap [9] patches with a clipping safeguard, and [14] assumes away. The proposed certificate prices any division exactly, so the best one follows from an optimization rather than a rule. The first step is that the certificate separates across agents: if each agent can meet its own assigned share, the joint filter is feasible, and each agent can verify this using only its own constraints and its own input set. Theorem 11. Choose weights θk,i≥0 _k,i≥ 0 with ∑i∈Skθk,i=1 _i∈ S_k _k,i=1, and set for ∈Δ(Ei) μ∈ (E_i), ϕi()=∑k∈Eikθk,ibk+σi(∑k∈Eikk,i),miloc=minϕi(). _i( μ)= _k∈ E_i μ_k _k,ib_k+ _U_i ( _k∈ E_i μ_k ν_k,i ),\;m_i^loc= _ μ _i( μ). If miloc()≥0m_i^loc( x)≥ 0 for every i, then M()≥0M( x)≥ 0 and the filter is feasible at x. Each ϕi _i depends only on the constraints incident to agent i and on iU_i. Proof. Splitting b and using ∑i∈Skθk,i=1 _i∈ S_k _k,i=1 gives ⊤=∑kλkbk∑i∈Skθk,i=∑i∑k∈Eiλkθk,ibk, λ b= _k _kb_k _i∈ S_k _k,i= _i _k∈ E_i _k _k,ib_k, so the objective separates as ϕ()=∑iϕi(Ei)φ( λ)= _i _i( λ_E_i). Each ϕi _i is positively homogeneous. For ∈Δ λ∈ , let wi=∑k∈Eiλkw_i= _k∈ E_i _k. If wi>0w_i>0, then Ei/wi∈Δ(Ei) λ_E_i/w_i∈ (E_i) and ϕi(Ei)=wiϕi(Ei/wi)≥wimiloc≥0. _i( λ_E_i)=w_i _i\! ( λ_E_i/w_i )≥ w_im_i^loc≥ 0. If wi=0w_i=0, then ϕi(Ei)=0 _i( λ_E_i)=0. Hence ϕ()≥0φ( λ)≥ 0 for every ∈Δ λ∈ , and therefore M()≥0M( x)≥ 0. ∎ Proposition 12. Let θ satisfy θk,i≥0 _k,i≥ 0 and ∑i∈Skθk,i=1 _i∈ S_k _k,i=1 for every k, and suppose each agent applies an input i∈i u_i _i meeting its own share, θk,ibk()+k,i()i≥ 0,k∈Ei. _k,ib_k( x)+ G_k,i( x) u_i\ ≥\ 0, k∈ E_i. (8) Then ψk,rk(,)≥0 _k,r_k( x, u)≥ 0 for every k∈Ek∈ E. Consequently the HOCBF forward-invariance argument [2] applies for every admissible θ, including one recomputed at each state. Proof. Summing (8) over i∈Ski∈ S_k and using ∑i∈Skθk,i=1 _i∈ S_k _k,i=1 gives bk+∑i∈Skk,ii≥0b_k+ _i∈ S_k G_k,i u_i≥ 0, which is exactly ψk,rk≥0 _k,r_k≥ 0. The argument is pointwise in x and does not involve θ beyond the normalization, so it is unaffected by any dependence of θ on the state or on time. ∎ Proposition 12 separates safety from local feasibility: any valid allocation preserves the barrier condition, so θ can be optimized without compromising safety. Its effect is instead on whether the local programs (8) remain feasible, which we address next. The uniform choice θk,i=1/|Sk| _k,i=1/|S_k| recovers the standard decentralized CBF condition of [12], used as the hand-crafted baseline in [15]. Theorem 11 shows that this is one choice within a broader family, making θ a design variable. The per-agent quantity in Theorem 11 is not merely a bound. It is exactly the largest worst-case slack agent i can reach with its own input. Lemma 13. For θ admissible and iU_i convex compact, miloc(,)=maxi∈imink∈Ei(θk,ibk()+k,i()i),m_i^loc( x; θ)= _ u_i _i\ _k∈ E_i\ ( _k,ib_k( x)+ G_k,i( x) u_i ), (9) so miloc≥0m_i^loc≥ 0 if and only if agent i’s local program admits a solution. Proof. Writing σi _U_i as a maximum over iU_i, ϕi()=max∑k∈Eii∈iμk(θk,ibk+k,ii) _i( μ)= _ u_i _i _k∈ E_i _k( _k,ib_k+ G_k,i u_i). The expression is bilinear in (,i)( μ, u_i) and both Δ(Ei) (E_i) and iU_i are convex and compact, so Sion’s theorem [16] exchanges the minimum and the maximum. For fixed i u_i, the inner objective is linear in μ and its minimum over the simplex is attained at a vertex, giving mink∈Ei _k∈ E_i. The stated equivalence is then immediate, since agent i’s program is exactly the requirement that some i∈i u_i _i satisfy every θk,ibk+k,ii≥0 _k,ib_k+ G_k,i u_i≥ 0. ∎ Lemma 13 sharpens Theorem 11 by identifying each agent’s quantity as its exact feasibility margin. Thus, maximizing the minimum local margin milocm_i^loc is the natural allocation problem and is tractable. Maximizing the worst margin looks like a search over allocations. Dualizing the inner minimization collapses it to one linear program. Theorem 14. Let each iU_i be a polytope. Then maxminimiloc(,) _ θ\ _im_i^loc( x; θ) (10) equals the optimal value t⋆t of the linear program maxt,, _t, θ, u t t (11) s.t. .t. t≤θk,ibk+k,i,k∈E,i∈Sk, t≤ _k,ib_k+ G_k,i u_i, k∈ E,\ i∈ S_k, ∑i∈Skθk,i=1,≥0,i∈i. _i∈ S_k _k,i=1, θ≥ 0, u_i _i. If t⋆≥0t ≥ 0, the resulting allocation makes all local programs feasible and, by Theorem 11, certifies M()≥0M( x)≥ 0. If t⋆<0t <0, no admissible allocation can make all local programs feasible. The converse does not hold, i.e., M()≥0M( x)≥ 0 may still have t⋆<0t <0, since a jointly feasible input need not admit a feasible decomposition into local shares. Thus, M()≥0>t⋆M( x)≥ 0>t characterizes the feasibility gap introduced by decentralization. Proof. Substitute (9) into (10). The result is a maximization over θ and the per-agent inputs i u_i of the minimum over (k,i)(k,i) of terms affine in those variables. Introducing the epigraph variable t gives (11). Its constraints are linear because each iU_i is a polytope and θk,ibk _k,ib_k is linear in θ. The consequences follow from Lemma 13 and Theorem 11. ∎ Algorithm 1 Certificate-optimal responsibility allocation 1: state x; barriers hk\h_k\ with supports Sk\S_k\; input sets i\U_i\; nominal inputs nom u^nom 2: inputs u applied by the agents 3: form () G( x), () b( x) from the chains (4) ⊳ Thm. 6 4: (t⋆,⋆,⋆)←(t , θ , u )← solve (11) 5: if t⋆≥0t ≥ 0 then 6: broadcast ⋆ θ ; discard ⋆ u 7: for all agents i in parallel do 8: i←argmin∥i−inom∥2 u_i← \, u_i- u^nom_i ^2 over i∈i u_i _i 9: i← u_i .t. θk,i⋆bk+k,ii≥0θ _k,ib_k+ G_k,i u_i≥ 0 for k∈Eik∈ E_i 10: else⊳ no split can work 11: ←⋆ u← u ⊳ maximizes the worst margin over all agents 12: report supp⋆supp λ and the agents with σi(i(⋆))>0 _W_i( ν_i( λ ))>0 Algorithm 1 is the control step that Theorem 14 yields. The linear program is solved on the joint state, and only the scalars ⋆ θ are broadcast. Each agent then solves a small program over its own input and its own constraints, so execution is local even though the allocation is not, as in [15, 14]. The program also returns inputs ⋆ u , which serve two purposes. When t⋆≥0t ≥ 0, we discard them, because they maximize the margin rather than track the task; their only role is to show that a feasible input exists, and each agent instead picks the input closest to its nominal one that meets its share. When t⋆<0t <0 no split works, and ⋆ u is the input that keeps the worst margin as high as possible. Applying it is then the least infeasible action available, at no extra cost. Lemma 13 holds for any convex compact iU_i, but (11) is linear only when each iU_i is a polytope; otherwise, the same argument gives a convex program that is still solvable. Fig. 2: Certificate-based conflict localization. (a) A representative conflict and the pair identified by supp⋆supp λ . (b) Agent-wise actuation leverage ∂M/∂ρi∂M/∂ _i. (c) Conflicts resolved by relaxing the identified pair versus alternative criteria. (d) Sparsity of ⋆ λ with fleet size. VI Numerical Results We evaluate the effectiveness of the proposed attribution and allocation algorithms in closed-loop simulations. VI-A Simulation Setup Agents. Each agent is a planar underactuated vehicle with state i=(i,i,ϕi)∈ℝ5 x_i=( q_i, v_i, _i) ^5, comprising position, velocity, and heading, and a single scalar input uiu_i, so mi=1m_i=1: ˙i=i,˙i=ui[cosϕisinϕi]⊤,ϕ˙i=ωi. q_i= v_i,\; v_i=u_i bmatrix _i& _i bmatrix ,\; φ_i= _i. (12) By (12) an agent accelerates only along its heading, which is itself a state and turns at a bounded rate, |ωi|≤6| _i|≤ 6 rad s-1, with first-order tracking of a commanded heading at time constant τ=0.12τ=0.12 s. The input sets are the intervals i=[−u¯i,u¯i]U_i=[- u_i, u_i], with u¯i=1 u_i=1 for all agents in the identical-fleet case and u¯i u_i drawn uniformly from [0.5,2.0][0.5,2.0], a fourfold spread, in the heterogeneous case. Constraints. Every pair within a sensing radius of 2.42.4 m is enforced, giving hk=∥i−j∥2−(2r)2h_k= q_i- q_j ^2-(2r)^2 with agent radius r=0.3r=0.3 m. Each hkh_k has relative degree two, and the chain uses constant gains αk,1=3hk _k,1=3h_k and αk,2=3ψk,1 _k,2=3 _k,1. Task. Agents start at random positions in a 5.2×5.25.2× 5.2 m square, separated by at least 0.750.75 m, with zero initial velocity and headings pointing at their goals, where goals are drawn independently in the same square. Because start and goal assignments are independent, trajectories cross, and the fleet is driven into dense conflict rather than a structured formation. The nominal controller is a PD law, inom=−3(i−igoal)−3.6i a^nom_i=-3( q_i- q^goal_i)-3.6 v_i. Each run is integrated at 5050 Hz, with the filter and the allocation recomputed at every step. VI-B Localization of Conflicts We evaluate whether the dual multiplier identifies the interactions responsible for infeasibility. Across randomized tasks with N=6,…,16N=6,…,16 underactuated agents, we obtain 5252 conflict events spanning 88-7878 enforced pairs. All events have positive supply and are therefore not degenerate (Theorem 7). The dual support provides a finer diagnosis by identifying the interactions consuming the available actuation. Fig. 2(a) shows a conflict with N=10N=10 and 2424 enforced pairs. The dual support is a single pair, and relaxing it moves the reserve from −10.02-10.02 to +0.72+0.72, whereas relaxing the closest pair, the smallest barrier value, or the smallest ψ does not. Over all 5252 conflicts the pair named by supp⋆supp λ restores feasibility in 94%94\% of cases against 44-6%6\% for those heuristics (Fig. 2(c)). The three failures admit a second certificate once the named pair is relaxed, so the attribution gives a sufficient cause, not a unique one. The support stays small as the fleet grows, averaging 1.41.4 pairs with exactly two agents carrying nonzero leverage (7), well inside the bound of Proposition 10 (Fig. 2(d)). Fig. 3: Responsibility allocation for N=8N=8 and 2525 enforced pairs. (a) Local feasibility margins under uniform [12], capability-weighted [9], and certificate-optimal allocation (b) Agent-wise capability leverage ∂M/∂ρi∂ M/∂ _i. VI-C Allocation Mechanism Figure 3 illustrates the allocation mechanism at a fixed state. For N=8N=8 and 2525 enforced pairs, three agents have no admissible input under the uniform split and two under the capability-weighted split. The proposed optimizer redistributes responsibility from agents with slack to those with negative margins, increasing the worst local margin from −0.211-0.211 to +0.001+0.001, so that all local programs become feasible. This illustrates the max-min objective in (11). Fig. 3(b) shows the leverage ∂M/∂ρi∂ M/∂ _i at the same state. It is identical under all three allocations, because the joint certificate does not depend on θ, and it is nonzero for a single agent. The allocation, therefore, changes neither the fleet’s capability nor which agent holds it, only whether the local programs can use it. Fig. 4: Comparison of three responsibility allocations for an identical task instance (N=5N=5), with fixed initial conditions, goals, and actuation bounds. Top: four snapshots of each run; paths and goals are color-coded by agent, while edge color indicates the responsibility share θk,i _k,i. Red outlines mark agents with infeasible local programs. (d)-(f) Local margins milocm_i^loc; black dashed line shows the worst margin. Fig. 5: Responsibility allocation in a decentralized filter on heterogeneous fleets, over 160160 paired runs with identical tasks for all methods. (a) Percentage of control steps at which a program is infeasible (b) The resulting closest approach. (c) The percentage of the safety distance that is violated. TABLE I: Infeasible control steps (%) over 160160 paired runs per fleet. allocation N=6N=6 N=8N=8 N=10N=10 N=12N=12 pooled identical centralized (ref.) 5.9 5.0 8.3 8.0 6.8 uniform [12] 27.3 44.9 60.9 68.6 50.4 capability [9] 25.8 39.9 59.0 68.2 48.2 certificate (11) 5.2 4.8 7.0 7.2 6.0 heterog. centralized (ref.) 5.7 3.9 5.7 7.0 5.6 uniform [12] 28.9 46.8 59.7 68.4 51.0 capability [9] 23.9 43.9 61.2 68.0 49.2 certificate (11) 6.3 3.1 7.5 7.8 6.2 VI-D Closed-Loop Evaluation of the Allocation We evaluate the decentralized filter of Section VI-A, where each agent enforces only its allocated share θk,ibk _k,ib_k of each shared constraint. We compare uniform allocation [12], capability-weighted allocation [9], and the proposed optimizer (11). A centralized filter is included as a reference. Each method is evaluated on the same 160160 paired random tasks for each fleet type, with N∈6,8,10,12N∈\6,8,10,12\. Figure 4 illustrates a representative run under the three allocations. With N=5N=5 and identical initial conditions, local program infeasibility occurs at 45%45\% and 32%32\% of control steps under uniform and capability-weighted allocation, respectively, compared with only 0.3%0.3\% under the optimizer of (11). All three runs impose the same safety constraints, and only their allocation among agents differs. Table I and Fig. 5 show that the certificate-optimal allocation substantially reduces local infeasibility. The infeasible-step rate is 6.0%6.0\% on identical fleets and 6.2%6.2\% on heterogeneous ones, against roughly 50%50\% for both heuristics and 6.8%6.8\% and 5.6%5.6\% for the centralized reference. The proposed allocation therefore recovers most of the feasibility of a centralized filter, while both heuristics retain a large decentralization penalty. Safety follows: on heterogeneous fleets the mean closest approach is +0.017+0.017 against −0.044-0.044 and −0.038-0.038 for the heuristics, and violations occur in 24/16024/160 runs against 118/160118/160 and 121/160121/160. We additionally evaluate the t⋆<0t <0 branch of Algorithm 1, where no responsibility allocation can make all local programs feasible. Across 8787 paired runs, the proposed coordinated input reduces infeasible steps from 6.1%6.1\% to 4.3%4.3\% and improves the mean closest approach from +0.016+0.016 to +0.020+0.020, relative to independent per-agent relaxation. On the 2424 runs in which this branch is activated, the corresponding infeasible-step rates are 10.7%10.7\% and 17.3%17.3\%. These results demonstrate that the proposed allocation improves both local feasibility and the resulting closed-loop safety, while capability weighting provides little benefit over uniform allocation. VII Limitations The allocation guarantees local feasibility, not task completion. Deadlock and liveness are therefore outside the scope of this work. The local certificate of Theorem 11 is sufficient but not necessary, and although we establish that degenerate states exist, their closed-loop reachability is not characterized. Hardware validation is left for future work. VIII Conclusion We presented an exact conic certificate for multi-robot CBF filter infeasibility that separates constraint demand from actuator supply. It determines whether a conflict can be resolved by gain tuning, by more actuation, or only by redividing responsibility, and it exposes degenerate underactuated states where extra actuation cannot help at all. Its dual variables localize the conflict to a small set of interactions and grade the agents by the marginal value of actuation. 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Then ψk,j _k,j depends only on (hk,Lfhk,…,Lfjhk)(h_k,L_fh_k,…,L_f^jh_k). In particular, there exists a differentiable Φk,j _k,j such that ψk,j=Lfjhk+Φk,j(hk,Lfhk,…,Lfj−1hk), _k,j=L_f^jh_k+ _k,j\! (h_k,L_fh_k,…,L_f^j-1h_k ), (13) with Φk,0≡0 _k,0≡ 0. Moreover, Lgiψk,j≡0L_g_i _k,j≡ 0 for all i∈Ski∈ S_k whenever j≤rk−2j≤ r_k-2. Thus, the input can first appear at j=rk−1j=r_k-1, exactly as prescribed by the relative-degree assumption. Proof. Induction on j. At j=0j=0 the claim is ψk,0=hk _k,0=h_k with Lgihk=0L_g_ih_k=0. Assume it at j−1≤rk−2j-1≤ r_k-2. Differentiating, ψ˙k,j−1=Lfψk,j−1+∑iLgiψk,j−1i ψ_k,j-1=L_f _k,j-1+ _iL_g_i _k,j-1 u_i, and every Lie derivative in Lgiψk,j−1L_g_i _k,j-1 has order at most j−1≤rk−2j-1≤ r_k-2 and so vanishes by Definition 2. Hence ψ˙k,j−1=Lfψk,j−1=Lfjhk+Φ~k,j ψ_k,j-1=L_f _k,j-1=L_f^jh_k+ _k,j for a differentiable Φ~k,j _k,j of the lower derivatives. Since αk,j _k,j is chain-valued it depends only on those same derivatives, and absorbing it into Φ~k,j _k,j gives (13). Stage j=rk−1j=r_k-1 is the first at which Lgiψk,jL_g_i _k,j need not vanish. ∎ Theorem 16. Let αk,l _k,l be chain-valued for l≤rk−2l≤ r_k-2, and αk,rk−1()=ak()ψk,rk−2(),ak∈C1, _k,r_k-1( x)\;=\;a_k( x)\, _k,r_k-2( x), a_k∈ C^1, (14) with rk≥2r_k≥ 2. Then ψk,rk−1 _k,r_k-1 is input-free, the terminal condition remains affine in u, and k,i()=LgiLfrk−1hk()⏟invariant part+ψk,rk−2()Lgiak()⏟design part. G_k,i( x)\;=\; L_g_iL_f^\,r_k-1h_k( x)_invariant part\;+\; _k,r_k-2( x)\,L_g_ia_k( x)_design part. (15) Proof. Write ψ:=ψk,rk−2ψ:= _k,r_k-2, so ψk,rk−1=Lfψ+akψ _k,r_k-1=L_fψ+a_kψ (using Lgiψ=0L_g_iψ=0 from Lemma 15). This is a state function, so ψk,rk−1 _k,r_k-1 is input-free and ψk,rk _k,r_k is affine in u. Differentiating with the product rule, Lgiψk,rk−1=Lgi(Lfψ)+ψLgiak+akLgiψ.L_g_i _k,r_k-1=L_g_i(L_fψ)+ψ\,L_g_ia_k+a_k\,L_g_iψ. The third term vanishes. For the first, by (13), Lfψ=Lfrk−1hk+∑m≤rk−3∂mΦk,rk−2Lfm+1hkL_fψ=L_f^r_k-1h_k+ _m≤ r_k-3 _m _k,r_k-2L_f^m+1h_k, whose LgiL_g_i is LgiLfrk−1hkL_g_iL_f^r_k-1h_k because all remaining indices are ≤rk−2≤ r_k-2. ∎ Thus, the design can modify the supply only through (15). Three properties are notable: if LgiakL_g_ia_k is not parallel to LgiLfrk−1hkL_g_iL_f^r_k-1h_k, it rotates the constraint gradient; its magnitude scales with ψk,rk−2 _k,r_k-2 and vanishes on the corresponding boundary; and it requires aka_k to depend on states outside its own chain. If aka_k is chain-valued, Lemma 15 gives Lgiak=0L_g_ia_k=0, reducing (15) to Theorem 6. The choice (14) scales the design term with ψk,rk−2 _k,r_k-2, so it offers freedom before a conflict rather than during one, and it requires aka_k to depend on states outside its own chain: a chain-valued aka_k gives Lgiak=0L_g_ia_k=0 and recovers Theorem 6. Any bounded, strictly positive aka_k preserves forward invariance [17], so the lever carries no safety cost, but it changes both αC_α and ℱF. B. When degeneracy can and cannot occur For the agents of Section VI-A, degeneracy is not vacuous. Proposition 17. Place N agents at the vertices of a regular polygon, i=R^i q_i=R ρ_i, each at rest with its body axis tangential, ^i=^i e_i= τ_i, and let every pair be enforced. Then i()=0 ν_i( λ)=0 for every agent at the uniform multiplier λk=1/|E| _k=1/|E|, so S⋆()=0S ( x)=0 and x is degenerate. Proof. Here k,i=2⟨ij,^i⟩ G_k,i=2 p_ij, e_i with ij=i−j p_ij= q_i- q_j. Summing over the pairs incident to agent i and using ∑j=0 _j q_j=0 gives ∑j≠ij=Ni _j≠ i p_ij=N q_i, which is radial, whereas ^i e_i is tangential. Hence i()=0 ν_i( λ)=0 and S(,)=0S( x, λ)=0. ∎