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Time-To-Reach Separation and Safety Filtering for Safe, Fair, and Efficient Multi-Agent Coordination
Matthew Low, Jasmine Jerry Aloor, Victoria Marie Tuck, Pierluigi Nuzzo, Jason J. Choi
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This paper addresses multi-agent coordination for Advanced Air Mobility (AAM) by proposing a framework that uses minimum time-to-reach (TTR) as a unifying metric for priority assignment, temporal separation, and safety filtering. The method combines a strategic guidance layer that regulates TTR for fair, evenly spaced arrivals with a tactical safety layer using Hamilton-Jacobi reachability-based control barrier functions to ensure collision avoidance. Simulations in congested corridor merging scenarios demonstrate improved safety, fairness, and efficiency over existing time-optimal and priority-agnostic approaches.
Entities (8)
Relation Signals (10)
Proposed Framework → improves → Safety, Fairness, and Efficiency
confidence 96% · Simulation results in a highly congested corridor merging scenario show that the proposed method improves safety, fairness, and efficiency
Time-To-Reach (TTR) → usedfor → Priority Assignment
confidence 96% · Vehicles are assigned arrival-consistent priority based on TTR
Safety Filtering → ensures → Collision Avoidance
confidence 95% · ensures collision avoidance while minimally modifying the reference guidance.
Time-To-Reach (TTR) → usedfor → Temporal Separation
confidence 95% · target TTR values are used to enforce temporal spacing that induces spatial separation.
Hamilton-Jacobi Reachability → computes → Value Functions
confidence 94% · The value functions can be computed numerically via the level set method
Control Barrier Function (CBF) → implements → Safety Filtering
confidence 94% · We improve safety by using the computed BRT value function V as control barrier functions (CBFs), to filter a potentially unsafe input
Advanced Air Mobility (AAM) → requires → Autonomous Traffic Management
confidence 93% · requiring autonomous traffic management systems to ensure collision-free operations in highly congested environments.
Value Functions → →
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Abstract
Abstract:Advanced Air Mobility (AAM) operations are expected to significantly increase aerial traffic in urban airspace, requiring autonomous traffic management systems to ensure collision-free operations in highly congested environments. In this paper, we propose a multi-agent coordination framework that uses minimum time-to-reach (TTR) as a unifying metric for priority assignment, temporal separation, and safety filtering. We focus on the problem of coordinating multiple aerial vehicles merging into an air corridor while maintaining safe separation between vehicles. Vehicles are assigned arrival-consistent priority based on TTR, and target TTR values are used to enforce temporal spacing that induces spatial separation. A priority-consistent safety filtering layer based on Hamilton-Jacobi reachability value functions ensures collision avoidance while minimally modifying the reference guidance. Simulation results in a highly congested corridor merging scenario show that the proposed method improves safety, fairness, and efficiency compared to time-optimal guidance and priority-agnostic safety filtering.
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- Source: https://arxiv.org/abs/2605.20625v1
- Canonical: https://arxiv.org/abs/2605.20625v1
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Time-To-Reach Separation and Safety Filtering for Safe, Fair, and Efficient Multi-Agent Coordination Matthew Low1, Jasmine Jerry Aloor2, Victoria Marie Tuck3, Pierluigi Nuzzo1 and Jason J. Choi4 1^1\;M. Low and P. Nuzzo are with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley. 2^2\;J.J. Aloor is with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology. 3^3\;V. Tuck is with the GRASP Laboratory, University of Pennsylvania. 4^4\;J.J. Choi is with the Department of Electrical and Computer Engineering, University of California, Los Angeles. AI tools (OpenAI ChatGPT, Anthropic Claude, Google Gemini) have been used to improve the clarity of the manuscript. m.low@berkeley.edu, jjhchoi@ucla.edu Abstract Advanced Air Mobility (AAM) operations are expected to significantly increase aerial traffic in urban airspace, requiring autonomous traffic management systems to ensure collision-free operations in highly congested environments. In this paper, we propose a multi-agent coordination framework that uses minimum time-to-reach (TTR) as a unifying metric for priority assignment, temporal separation, and safety filtering. We focus on the problem of coordinating multiple aerial vehicles merging into an air corridor while maintaining safe separation between vehicles. Vehicles are assigned arrival-consistent priority based on TTR, and target TTR values are used to enforce temporal spacing that induces spatial separation. A priority-consistent safety filtering layer based on Hamilton-Jacobi reachability value functions ensures collision avoidance while minimally modifying the reference guidance. Simulation results in a highly congested corridor merging scenario show that the proposed method improves safety, fairness, and efficiency compared to time-optimal guidance and priority-agnostic safety filtering. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. I Introduction Advanced Air Mobility (AAM) is projected to introduce tens of thousands of daily flights in urban areas, pushing beyond the capacity of current air traffic management [1, 2]. Dedicated air corridors have been proposed as a structured approach to organizing traffic flows in dense airspace [3, 4, 5]. However, when multiple vehicles converge to the same corridor, coordinating their arrival while maintaining safety and efficiency becomes a challenging multi-agent control problem [6, 7]. Existing approaches to multi-vehicle traffic management and control can broadly be categorized into strategic flow management and tactical conflict resolution. At the strategic level, flow management and scheduling methods assign arrival times or sequencing orders prior to flight. These approaches operate at the operational planning level and are effective in managing throughput and scheduling fairness [8]. At the tactical level, reactive deconfliction systems such as TCAS [9] and ACAS-X [10] provide decentralized pairwise collision avoidance with formal safety guarantees. These systems are designed primarily for imminent conflict resolution rather than coordinated multi-vehicle sequencing. These approaches remain limited for the highly congested air-corridor merging problem considered in this work. Strategic scheduling typically does not account for vehicle dynamics and local and transient airspace congestion. Therefore, the assigned schedules may be dynamically infeasible or require large safety-related deviations that reduce efficiency. Moreover, fairness metrics used in scheduling are often not aligned with safety objectives. On the other hand, purely reactive safety mechanisms are inherently myopic: a collision avoidance maneuver for one pair of vehicles alters the geometry for downstream vehicles, potentially triggering new conflicts. This domino effect has been identified as a fundamental failure mode of decentralized deconfliction in high-density airspace [6, 11, 12]. During corridor merging, traffic naturally compresses in space, and reactive pairwise avoidance alone can lead to simultaneous conflicts. In this paper, we propose a framework that uses the minimum time-to-reach (TTR) [13] as a unifying metric for priority assignment, temporal separation, and integration with safety filtering. Specifically, TTR is used (1) to assign arrival-consistent priority in a fair manner; (2) to create temporal separation by assigning target TTR values, which naturally induces spatial separation as a strategic safety mechanism; and (3) to integrate with a safety filtering layer that enhances the final safe coordination. The key idea is that TTR naturally defines a fair and dynamically feasible arrival ordering: vehicles with smaller TTR can arrive at the corridor sooner and are therefore assigned the right-of-way. This priority rule not only embodies the principle of first-come-first-served fairness [14], but it also naturally arises from the actual vehicle dynamics via reachability analysis. While TTR has been adopted for single-vehicle guidance [15], and has also been used as a reward signal in reinforcement learning [16, 12], it has not been directly used as a coordination variable for safe multi-agent coordination. Prior work that integrates both strategic coordination and tactical safety enforcement for multi-agent coordination is most related to our approach. System-level coordination using formal methods has been studied, integrating ground infrastructure and centralized planning [17]. Other work employs reinforcement learning-based approaches for strategic coordination, combined with safety filters [12, 18]. Responsibility allocation for safety filtering has also been formulated as a centralized joint optimization problem [19]. These efforts share a similar hierarchical structure that combines strategic coordination and tactical safety mechanisms. Differently from them, our approach uses instead a dynamically consistent metric, namely, TTR, to jointly determine priority, temporal spacing, and safety. I Background Consider the nonlinear system dynamics x˙(t)=f(x(t),u(t))fort≥0,x(0)=x, x(t)\!=\!f(x(t),u(t))\;\;for\;t≥ 0, (0)=x, (1) with initial state x, state x(t)∈ℝnx(t) ^n, control u(t)∈U⊂ℝmu(t)∈ U ^m, where U is the compact control input set. We assume that f is Lipschitz continuous in the state and u(⋅)u(·) is Lebesgue measurable for the forward completeness of x(⋅)x(·) [20]. I-A Safety & Reachability Analysis In this work, we consider two types of reachability problems: the minimal backward reachable tube (BRT) problem, which is used to compute the value function for deconfliction safety control, and the backward reach-avoid tube (BRAT) problem, which is used to compute the guidance control for the air corridor. We denote the constraint set as C and denote the target set as T. The minimal BRT of the target set T is the set of initial states from which the system reaches the target set T over the time horizon t, under any control signal uu: ℬℛ(t;)≔x∈ℝn∣∀u(⋅),∃s∈[0,t],s.t.x(s)∈.BRT(t;T)\! \!\x\!∈\!R^n (·),\;\;∃ s\!∈\![0,t],\; s.t.\;x(s)\!∈\!T\. -2.5pt If we consider an indefinite time horizon, we get the (minimal) time-invariant BRT, ℬℛ():=limt→∞ℬℛ(t;)BRT(T):= _t→∞BRT(t;T). Given the safety constraint set C, consider the BRT of the failure region cC^c. Then, ℬℛ(c)c\BRT(C^c)\^c is the maximal control invariant set contained in C. For any trajectory exiting this set, the trajectory will inevitably reach cC^c and violate the constraint [21]. Thus, by solving the BRT problem, we can solve the maximal safe set within the constraint set. We denote this set as afe()S -1.0pta\!f\!e(C)111More formally, afe():=ℬℛ(c)cS -1.0pta\!f\!e(C)\!\!:=\!\!\ -1.0ptBRT(C^c) -1.0pt\^c is the viability kernel of C [22].. The BRAT is the set of initial states from which the system can reach a target set, T, while avoiding the failure region cC^c (or equivalently, while staying within the constraint C): ℬℛ(t;,c)≔x∈ℝn∣∃u(⋅)s.t. (t;T,C^c)\! \! \x ^n (·)\;s.t. . (2) ∃s∈[0,t],x(s)∈&∀τ∈[0,s],x(τ)∈. .∃ s∈[0,t],x(s) \;\&\;∀τ∈[0,s],x(τ) \. The maximum safe set afe()S -1.0pta\!f\!e(C) and the BRAT ℬℛ(t;,c)BRAT(t;T,C^c) can be solved by defining the corresponding Hamilton-Jacobi (HJ) reachability optimal control problems. We first represent the constraint and the target sets as level sets of Lipschitz functions, c(⋅)c(·) and l(⋅)l(·), such that =x∣c(x)≥0,=x∣l(x)≥0.C=\x c(x)≥ 0\, =\x l(x)≥ 0\. -2.5pt (3) Then, we can define the value functions whose zero-superlevel sets represent afe()S -1.0pta\!f\!e(C) and ℬℛ(t;,c)BRAT(t;T,C^c), respectively, as: Value function for afe()S -1.0pta\!f\!e(C) and BRT: V(x,t): V(x,t): =maxu(⋅)mins∈[0,t]c(x(s)) \!\!= _u(·) _s∈[0,t]c(x(s)) (4) ⇒ℬℛ(t;c)=x∣V(x,t)<0, (t;C^c)=\x V(x,t)<0\, V(x): V(x): =limt→∞V(x,t) \!\!= _t→∞V(x,t) \;\;\;\;~\; (5) ⇒afe()=x∣V(x)≥0. -1.0pta\!f\!e(C)=\x V(x)≥ 0\. Value function for BRAT: W(x,t): W(x,t): =maxu(⋅)maxs∈[0,t]minl(x(s)),minτ∈[0,s]c(x(τ)) \!\!= _u(·) _s∈[0,t] \! \l(x(s)), _τ∈[0,s]c(x(τ))\! \\!\!\! (6) ⇒ℬℛ(t;,c)=x∣W(x,t)≥0. (t;T,C^c)=\x W(x,t)≥ 0\. The optimal control in (4) maximizes the closest distance to the failure boundary. In (6), the optimal control minimizes the distance to the target set while avoiding the failure region. By applying the dynamic programming principle, the value functions become viscosity solutions [23] to the HJ partial differential equations (PDEs), derived in [24]. The value functions can be computed numerically via the level set method [13], or they can be approximated as neural value functions by using physics-informed neural networks [25]. I-B Control Barrier Function-based Safety Filtering We improve safety by using the computed BRT value function V as control barrier functions (CBFs) [26], to filter a potentially unsafe input u with an appropriate safe input and keep the system within the constraint set C. To achieve this safety filtering mechanism, we consider the following min-norm safety filter [21, 26]: CBF Min-norm Safety Filter: ucbf= u_cbf=\; argminu∈U‖u−uref‖2 u∈ U u-u_ref ^2 (7a) s.t. ∇V(x)⋅f(x,u)+α(V(x))≥0, ∇ V(x)· f(x,u)+α (V(x) )≥ 0, (7b) where α is a class κ function. For control-affine dynamics, this becomes the CBF quadratic program (CBF-QP). If V is almost-everywhere differentiable, the barrier constraint in (7b) is feasible almost-everywhere. Such usage of the reachability value function as the CBF is a prominent method of designing CBFs [27, 28, 29]. For certain dynamics, the value function can be discontinuous without introducing a discount factor in time to the cost function in (4) [30]. Remark 1. In the multi-agent setting, each pair of agents introduces a safety constraint, and these constraints may conflict with each other. A brute-force solution would require computing the value function (5) for the joint N-agent dynamics, which does not scale well and is difficult to generalize. Therefore, careful treatment of these potentially conflicting constraints is required, which is nontrivial [12]. Various architectures of extending the safety filter (7) to our multi-agent problem will be discussed in Section IV-D. I-C Time-To-Reach in Reachability Problems We consider the time-to-reach of the backward reach-avoid tube problem, given by T[u(⋅)]≔∞if >∃t0,s.t.x(t)∈Cc&∀s∈[0,t],∉x(s)T ∞else if∀t>0,x(t)∉,mint∣x(t)∈else.T[u(·)] cases∞ 310.04001pt$ ∃ t>0,\;s.t.\;x(t) ^c\;\&\;∀ s∈[0,t],x(s) $\\ ∞ if\;∀ t>0,x(t) ,\\ \t (t) \ . cases In words, the T[⋅]T[·] of a given control signal is the time the trajectory first hits the target set without entering the failure region cC^c. Next, we define the following optimal control problem, where the control tries to minimize T: Tmin(x):=infu(⋅)T[u(⋅)],T_ (x):= _u(·)T[u(·)], -5.0pt (8) Tmin(x)T_ (x) is called the minimum time-to-reach (TTR) value function [13, Ch.2.3.1], in this case, for the backward reach-avoid tube problem. Proposition 1. The BRAT of the time horizon t and the TTR satisfies ℬℛ(t;,c)=x∣Tmin(x)≤tBRAT(t;T,C^c)=\x T_ (x)≤ t\ [31, Prop.1]. The optimal control law of this problem is given as: πopt(x)=minu∈U∇Tmin(x)⋅f(x,u),π^opt(x)= _u∈ U∇ T_ (x)· f(x,u), -2.5pt (9) derived from the Hamiltonian associated with TminT_ [31]. Note that the trajectory under this control law will be the time-optimal trajectory to T while avoiding cC^c. Such a time-optimal trajectory always exists when Tmin(x)T_ (x) is finite. This minimum TTR value function-based control law will be used to design the control input that guides the vehicles to the air corridor in Section IV-C. Remark 2. Various numerical techniques are developed for the computation of the minimum TTR [13, 15, 31]. Directly solving the stationary HJ-PDE associated with TminT_ is possible by carefully treating the discontinuity of TminT_ [13, Ch.2.3.1], [31]. In our work, we first compute the value function W(x,t)W(x,t) and reconstruct TminT_ from W, as in [32]. I Problem Formulation I-A Problem Setting Figure 1: Computed minimum reach-avoid time-to-reach (TTR) in colormap (red-blue), for vehicles navigating to the air corridor (blue), and the simulated optimal trajectories (green). The corridor consists of the entry region T (black dashed), and the downstream region D (white background). (a) TTR colormap and optimal trajectories with initial heading θ=0∘θ=0 (eastbound). (b, c) TTR colormap and optimal trajectories with initial heading θ=60∘,180∘θ=60 ,180 , respectively. (d, e) TTR colormap and optimal trajectories with initial heading θ=0∘θ=0 , under constant wind blowing toward north and southeast, respectively. We consider N homogeneous aircraft approaching an air corridor where the i-th aircraft has dynamics x˙i(t)=f(xi(t),ui(t)) x^i(t)=f(x^i(t),u^i(t)) where xi(t)∈ℝnx^i(t) ^n is the vehicle’s state and ui(t)∈Uu^i(t)∈ U is their control. We define [N]:=1,…,N[N]:=\1,…,N\. The air corridor includes an entry region, which defines the target set T that each vehicle’s state xi(t)x^i(t) aims to reach, and a downstream region D that vehicles must follow to remain within the corridor (Figure 1 (a)). This downstream region D is treated as an avoid set for any vehicle outside of the corridor. This prevents vehicles entering the corridor from an unintended direction, e.g., perpendicular to the flow of traffic. We assume that once a vehicle reaches the entry region, it proceeds along the corridor under a provided in-corridor guidance controller. Therefore, in this work, we focus only on the safe coordination of multiple vehicles from outside the corridor to the corridor entry region. I-B Problem Statement For vehicles to safely enter the corridor, they must (1) reach the entry region T, while avoiding (2) (unintended) entry into the downstream region D and (3) conflicts with other vehicles. The deconfliction objective is encoded as the constraint ij:=(xi,xj)|dist(xi,xj)≥δC_ij:=\(x^i,x^j)\;|\;dist(x^i,x^j)≥δ\ for each vehicle pair i and j, where dist(xi,xj)dist(x^i,x^j) is the distance between the two vehicles. We assume the initial configuration of vehicles is feasible, meaning there exists a set of compatible trajectories such that every vehicle reaches the corridor without violating safety: ∀i ∀ i ∈[N],∃ui(⋅)s.t.∃Ti>0, ∈[N],\ ^i(·)\;\;s.t.\;\;∃ T^i>0, xi(Ti)∈,and∀τ∈[0,Ti),xi(τ)∈c,xi(τ)∈ij∀j≠i. 450.96794pt$ ^i(T^i) ,and\;∀τ\!∈\![0,T^i),x^i(τ) ^c,x^i(τ) _ij\;∀ j\!≠\!i.$ Note that if we consider only a single vehicle (without considering deconfliction against other vehicles), each vehicle has to be initialized within ℬℛ(t;,)BRAT(t;T,D). This requires Tmin(x0i)<∞T_ (x_0^i)<∞, the initial minimum TTR of vehicle i. Our main objective is finding a valid joint control sequence ×i=1Nui(⋅)∈[N] _i=1^Nu^i(·) ^[N] that minimizes the corridor entry time of the last vehicle, TlastT^last, while maintaining safety. This goal is captured in the following optimization problem: minui(⋅)∈∀i∈[N]Tlast \;\; _u^i(·) \ ∀ i∈[N] T^last (10a) s.t. x˙i(t)=f(xi(t),ui(t)), x^i(t)=f(x^i(t),u^i(t)), (10b) Ti≤Tlast,Ti:=minτs.t.xi(τ)∈, T^i≤ T^last, T^i:= τ\;\;s.t.\ x^i(τ) , (10c) xi(τ)∈c,∀τ∈[0,Ti),∀i∈[N], ^i(τ) ^c, ∀τ∈[0,T^i), ∀ i∈[N], (10d) dist(xi(τ),xj(τ))≥δ,∀τ∈[0,min(Ti,Tj)) (x^i(τ),x^j(τ))\!≥\!δ,\;∀τ\!∈\![0, (T^i,T^j)) ∀i,j∈[N],i≠j. ∀ i,j∈[N],\ i≠ j. (10e) However, as noted in Remark 1, directly solving this problem suffers from the complexity of multi-agency, making it challenging to find a solution that satisfies the large number of inter-agent safety constraints. Further, joint minimization of (10a) over all N vehicles is intractable, so we instead rely only on decentralized optimal controllers with pairwise safety filters. Therefore, we allow slack in the safety constraint (10e) and aim to minimize the safety violation rate as captured in the metric below: νunsafe=1∑k=1NTk∑i=1N∫0Timaxj≠i(xi(t),xj(t))∉ij(t)t. _unsafe= 1 _k=1^NT^k _i=1^N _0^T^i _j≠ i 1\(x^i(t),x^j(t)) _ij(t)\dt. This measures the fraction of time that vehicles spend in a safety-violating configuration relative to the total travel time of all vehicles. The indicator (xi,xj)∉ij 1\(x^i,x^j) _ij\ is 1 when the vehicles i and j are in conflict and 0 otherwise. We consider additional desiderata to represent notions of fairness. We desire that vehicles reach the corridor in the order corresponding to increasing initial TTR Tmini:=Tmin(x0i)T_ ^i:=T_ (x_0^i), exactly capturing first-come-first-served fairness. The “closest” vehicle with the lowest initial TTR is granted the highest priority, and therefore it has the right-of-way unless safety is compromised. Therefore, the initial TTR values naturally induce a desired priority order. Without loss of generality, assume that vehicle indices are assigned such that Tmini≤TminjT_ ^i≤ T_ ^j for all i<ji<j, i.e., i has higher priority. By considering the order of the actual arrival sequence, we can calculate deviation from the desired priority order: τK=2N(N−1)∑i,j∈[N],i<jsgn(i−j)sgn(Ti−Tj), _K= 2N(N-1) _i,j∈[N],i<j\!\!\!\!sgn (i-j )\;sgn (T^i-T^j ), -5.0pt which is the closed form of Kendall’s coefficient [33], a common statistic to capture variation from a desired ordering. IV Our Method & Results IV-A Overview The key insight behind our proposed method is that, under ideal coordination, vehicles would arrive at the corridor entry evenly spaced in time, and consequently, evenly spaced in distance along the approach path. This observation suggests that, rather than directly coordinating vehicle positions, one can coordinate their TTR values. By regulating TTR values to be evenly distributed rather than allowing vehicles to greedily minimize their individual TTRs, vehicles can naturally form a well-structured and safe arrival sequence. Building on this intuition, the proposed framework uses a layered structure: (1) a strategic guidance layer based on TTR separation, and (2) a tactical safety layer based on CBF safety filtering with priority awareness. The overall method consists of the precomputation of the necessary value functions, as well as the strategic guidance layer and the tactical safety layer for control: 1. Offline computation. We first compute HJ reachability value functions offline. In particular, we compute: • The BRAT value function W(t,x)W(t,x) and the associated TTR value function Tmin(x)T_ (x) for ℬℛ(t;,)BRAT(t;T,D). The TTR function is used to evaluate the priority order and to design the strategic guidance controller. • A pairwise safety value function V(x)V(x) defined over the relative dynamics of a pair of vehicles, which is used in the safety filter that maintains separation. 2. Strategic Layer—TTR separation guidance. At the strategic level, each vehicle is assigned a reference TTR function based on the priority order to evenly distribute vehicle trajectories across the TTR domain. The guidance controller then regulates each vehicle’s motion to increase or decrease its TTR to achieve the desired temporal spacing. 3. Tactical Layer—Safety filtering. At the tactical level, we employ CBF-based safety filtering to ensure separation when the strategic guidance commands would otherwise fail. This safety filter acts as a reactive mechanism that minimally modifies the guidance command only when necessary to maintain safety. We propose multiple architectures with varying levels of centralization and priority coordination. IV-B Offline Reachability Computation Modeling. In this study, we assume that other altitude ranges are reserved for different air operations or for the last resort collision avoidance [9, 10]; therefore, we focus on horizontal deconfliction. For simplicity, we consider a planar (2D) top-down model and leave the extension to full 3D dynamics for future work. We consider a reduced-order dynamic model of an autonomous air taxi given by p˙x=vcosθ+w p_x=v θ+w cosϕ,p˙y=vsinθ+wsinϕ, φ, p_y=v θ+w φ, (11) θ˙=ω,v˙=a, θ=ω, v=a, where the state is defined as x=[px;py;θ;v]x=[p_x;\,p_y;\,θ;\,v], representing position, heading, and speed. The control input is defined as u=[ω;a]u=[ω;\,a], corresponding to the turn rate and longitudinal acceleration, respectively. The speed is constrained to the range [vmin,vmax][v_ ,v_ ], and the control input bounds are defined as =[−ωmax,ωmax]×[amin,amax]U=[- _ , _ ]×[a_ ,a_ ]. Finally, we also incorporate a wind effect, characterized by the magnitude w and the wind direction ϕφ. Note that the vehicle cannot stop in midair for maintaining wing-borne flight (vmin>0v_ >0), and the dynamics are nonholonomic. Figure 2: Five vehicles initialized at states with TTR(xi)≈40TTR(x_i)≈ 40s and with initial θ=0∘,v=vnominalθ=0 ,v=v_ nominal. The first row shows the trajectories and the second row shows the TTR values over time (dashed line indicates target TTR values.) (a) Vehicles under TTR optimal control, leading to near-collision (red) near the corridor entry. (b, c) Vehicles under the TTR separation guidance with target separation time ΔT=10,20 T=10,20s, respectively. The numbers indicate the assigned priority order. Note that in this scenario, the TTR separation guidance keeps the vehicle safe even without safety filtering. Without loss of generality, we assume that the corridor center is aligned with the pxp_x-axis and the origin is fixed at its entry. Let the width of the corridor be d. The corridor in the position space is defined as (px,py)∣px≥0,|py|≤0.5d\(p_x,p_y) p_x\!≥\!0,|p_y|\!≤\!0.5d\. We design the entry region of the air corridor as the target set in the vehicle state space, given by =x∣0≤px≤0.5d,|py|≤0.5d,|θ|≤Δθ,|v−vnom|≤Δv. \!=\!\x 0\!≤\!p_x\!≤\!0.5d,\;|p_y|\!≤ 0.5d,\;|θ|\!≤ θ,\;|v\!-\!v_nom|\!≤ v\. The downstream region is defined as =x∣px≥0.5dD\!=\!\x p_x\!≥\!0.5d, |py|≤0.5d|p_y|\!≤\!0.5d\. The parameters used in this work are listed in Table IV in the Appendix, which are consistent with industry specifications [34, 35, 36, 37]. To define the relative dynamics for pairwise safety analysis, we consider the relative state x(ij)=[px(ij);py(ij);x^(ij)\!=\![p_x^(ij);\,p_y^(ij); θ(ij);vi;vj]\,θ^(ij);\,v^i;\,v^j], which includes the relative position and relative heading of vehicle j with respect to vehicle i, expressed in a coordinate frame where the px(ij)p_x^(ij)-axis is aligned with the heading of vehicle i. The relationship between the relative state and (xi,xj)(x^i,x^j), the relative dynamics, as well as visualizations of the computed V(x(ij))V(x^(ij)), are detailed in [12]. Results. The minimum TTR Tmin(x)T_ (x) is computed per Remark 2 and is presented in Fig. 1, along with the simulated time-optimal trajectories under the control law (9). In (a)-(c), we show slices of the value function at different heading angles. In (d) and (e), we introduce wind with speed w=10w\!=\!10 m/s, and directions ϕφ =90∘,−45∘,=\!90 ,-45 , respectively. Note that the presence of crosswind makes the TTR asymmetric with respect to the corridor centerline (pxp_x-axis), causing vehicles flying against the wind to arrive later than those flying with the wind. The reachability computation for the value functions V(x(ij))V(x^(ij)) and W(x,t)W(x,t) was completed within 30 minutes on an Nvidia RTX 5090 GPU. Remark 3. Due to numerical discretization and approximation errors in the offline reachability computation, the computed TTR may differ from the actual TTR. In practice, we observe this gap is less than approximately 1010 seconds. We find that our proposed method is robust to these slight discrepancies. IV-C TTR Separation Guidance The objective of the TTR separation guidance is to coordinate multiple vehicles so that they arrive at the corridor entry temporally evenly separated. Priority assignment. As specified in Section I-B, vehicles are assigned priority according to their initial TTR values. Once vehicles reach the corridor, they are removed from the priority ordering. Target TTR assignment. The target TTR of the highest priority vehicle (i.e., index 1, the vehicle with the smallest TTR) is set equal to its current TTR: Ttarget1(0):=Tmin1(0)T_target^1(0):=T_ ^1(0), and for all future times, Ttarget1(t)≔Tmin(x1(t))T_target^1(t) T_ (x^1(t)). For the downstream vehicles with priority order i>1i>1, the target TTR values are assigned with respect to this TTR of the highest priority vehicle: Ttargeti(t)=Ttarget1(t)+(i−1)ΔT,T_target^i(t)=T_target^1(t)+(i-1) T, -2.5pt (12) where ΔT T is the desired temporal separation between consecutive vehicles. Control law. Each vehicle then regulates its motion to track its assigned target TTR value. The reference control for vehicle i is defined as π π (xi(t),t)sep,i=^sep,i(x^i(t),t)= (13) maxu∈U∇Tmin(xi(t))⋅f(xi(t),u),ifTmin(xi(t))≤Ttargeti(t),minu∈U∇Tmin(xi(t))⋅f(xi(t),u),else. cases u∈ U \;∇ T_ (x^i(t))\!·\!f(x^i(t),u),&\;\;if\;\;T_ (x^i(t))\!≤\!T_target^i(t),\\ u∈ U \;∇ T_ (x^i(t))\!·\!f(x^i(t),u),&\;\;else. cases Vehicles with smaller-than-assigned TTR are guided to increase their TTR to maintain the target temporal separation, and vehicles with larger-than-assigned TTR are guided to reduce their TTR to follow the time-optimal trajectory to the corridor. This results in an even temporal spacing of vehicles. Results. Figure 2 illustrates the behavior of the proposed TTR separation guidance (without safety filtering), compared with greedy time-optimal guidance. In this scenario, five vehicles are initialized with TTR values near 40 seconds. Because TTR separation is enforced from the beginning of the maneuver, lower-priority vehicles intentionally deviate from their time-optimal trajectories, creating space for higher-priority vehicles to approach the corridor without interference. As a result, all vehicles enter the corridor in the correct order and without safety violations. This type of prescriptive coordination behavior is difficult to achieve using safety filtering alone, which primarily provides reactive conflict resolution. See Appendix for additional remarks. TABLE I: Safety filter modes Mode Filter mechanism Communication None No safety filtering – Dec Pairwise CBF-QP; relaxed fallback Neighbor states Prio Multi-stage priority; seq. propagation Priority + Resolved u Cent Joint CBF-QP over all vehicles All states & actions IV-D Safety Filtering Aligning safety priority with TTR priority is especially important under the guidance law in IV-C since unnecessary intervention on a higher-priority vehicle can create downstream delay and congestion for lower-priority vehicles. We consider safety filters with varying levels of coordination, summarized in Table I. The decentralized and centralized architectures have been considered in prior work [12, 38]. In this paper, we consider their combination with the greedy time-optimal reference control in (9) as baseline methods for comparison in the results. Decentralized (Dec): Each vehicle independently solves a local pairwise CBF-QP against its most critical neighbor. Only neighbor states are shared, and resolved control inputs are not propagated to other vehicles. For a vehicle pair (i,j)(i,j), the most critical neighbor for vehicle i is j∗(i)=argminj∈iV(x(ij))j^*(i)= _j _iV(x^(ij)), where iN_i is the set of vehicles within the communication range of vehicle i. If the QP becomes infeasible, it finds a fallback solution with a relaxed CBF-QP, detailed in the Appendix. Priority-Coordinated (Prio): Each vehicle solves a local pairwise CBF-QP incorporating the static priority order. Resolved controls are broadcast sequentially in priority order and used as reference inputs in subsequent vehicles’ CBF-QPs. When determining the control input for each vehicle, we follow a multi-stage logic that is detailed below. Let ℛR denote the set of vehicles whose controls have been resolved in the current timestep. Let i be the vehicle index with the highest priority among the vehicles with unresolved inputs, i.e., i=min([N]∖ℛ)i= ([N] ). Stage 1 (priority-assigned): Vehicle i solves the safe control input against its most critical neighbor, j=j∗(i)j=j^*(i). When doing so, the higher-priority vehicle’s control is fixed to its reference input, thereby assigning the burden of conflict resolution to the lower-priority vehicle: Priority-assigned CBF-QP: minui∈U _u^i∈ U ‖ui−urefi‖2 \|u^i-u^i_ref\|^2 (14) s.t. .t. ∇V(x(ij))⋅f(x(ij),ui,urefj)+αV(x(ij))≥0, ∇ V(x^(ij))· f\! (x^(ij),\,u^i,u_ref^j )+α\,V(x^(ij))≥ 0, ui=urefiifi<j, u^i=u^i_ref\;if\;i<j, where if vehicle i has the priority, this QP merely checks whether urefiu^i_ref is feasible. If feasible, the resolved control is broadcast and ℛ←ℛ∪iR ∪ i. Stage 2 (symmetric): If Stage 1 was infeasible, solve (7) with both control inputs of i and j=j∗(i)j=j^*(i) as decision variables. If feasible, ℛ←ℛ∪iR ∪\i\. Stage 3 (relaxed symmetric): If Stage 2 was infeasible, solve the relaxed QP in Appendix. This stage is always feasible. The last two stages are similar to the decentralized case, but the resolved control input is carried forward to inform the control selection for subsequent vehicles. This design does not require a centralized coordinator: vehicles publish their TTR values to neighboring agents, determine the priority order locally, and resolve the safe control input accordingly. Centralized (Cent): A single centralized QP in the Appendix jointly optimizes all vehicles’ controls with pairwise CBF constraints coupling their decision variables. This safety filter requires the greatest degree of centralized infrastructure and vehicle coordination to simultaneously determine corrective actions for all vehicles. IV-E Results We evaluate different combinations of guidance and safety filtering across 100 randomly sampled N=8N\!=\!8 vehicle scenarios. Two guidance strategies are considered: (1) TTR min, where each vehicle greedily minimizes its TTR under the optimal control (9), and (2) TTR sep, which applies the TTR separation guidance (13) with ΔT=10 T=10 s. Each is combined with the safety filter modes in Table I. Figure 3: Trajectories for each combination of reference guidance and safety filters on a representative scenario with eight vehicles. Red indicates safety violations. (a) Guidance without safety filtering: greedy TTR minimization leads to conflicts, while the proposed TTR separation reduces conflicts through temporal spacing. (b) Greedy TTR minimization with safety filtering, which still incurs severe safety violations in this scenario. (c) Proposed TTR separation combined with safety filtering: decentralized and centralized filtering achieve zero safety violations while maintaining the assigned arrival order. The animated trajectories of this scenario are available online at: https://w.youtube.com/watch?v=fPNJK-leRx8 IV-E1 Evaluation Metrics The main metrics for efficiency, safety, and fairness are defined in Section I-B; they are the travel time of the last vehicle arriving the corridor TlastT^last, the safety violation rate νunsafe _unsafe, and Kendall’s coefficient τKτ_K, respectively. For each simulation run, any safety violation is considered a significant failure. Thus, we consider a run to be successful if all vehicles reach the goal with no safety violations. We record the rate of successful runs under a given guidance and filter combination as PpassP_pass. We also record the rate Ppass+P_pass 0.7$+$, which additionally requires that arrival order perfectly conforms to the initial priority order. These rates encode both efficiency (goal-reaching) and safety objectives (with Ppass+P_pass 0.7$+$ further measuring fairness). They are interpreted as overall metrics for success. We summarize the full evaluation criteria in Table I. TABLE I: Evaluation metrics Symbol Definition νunsafe _unsafe System level safety (Sec I-B). τK _K Kendall’s tau (Sec I-B). std(Tdelay/Ttarget)std( T_delayT_target) Std. of normalized TdelayiT_delay^i normalized by TtargetiT_target^i. Tdelay + avgT^avg_delay 0.5$+$ Avg. non-negative TdelayiT_delay^i. TlastT^last Actual arrival time of final vehicle. PpassP_pass Success rate of νunsafe=0 _unsafe=0 and Tlast<∞T^last<∞. Ppass+P_pass 0.7$+$ PpassP_pass criteria and order maintained (τK=1 _K=1). QPtime Safety filter wall-clock solve time per vehicle per timestep. TABLE I: Evaluation of Proposed TTR Prioritization Control Scheme Safety Fairness Efficiency Overall urefu_ref Filter ↓ νunsafe _unsafe (%\%) ↑ τK _K ↓ std(Tdelay/Ttarget)std( T_delayT_target) ↓ Tdelay + avgT^avg_delay 0.5$+$ (s) ↓ TlastT^last ↑ PpassP_pass ↑ Ppass+P_pass 0.7$+$ ↓ QPtime (ms) (a) TTR min None 5.964±6.5985.964± 6.598 0.97±0.040.97± 0.04 0.000±0.0000.000± 0.000 0.00±0.000.00± 0.00 78.03±7.0278.03± 7.02 23.6123.61 11.1111.11 – TTR sep None 1.255±1.9791.255± 1.979 0.99±0.020.99± 0.02 0.062±0.0620.062± 0.062 3.64±4.193.64± 4.19 90.22±8.7090.22± 8.70 56.9456.94 56.9456.94 – (b) TTR min Dec 4.688±5.3804.688± 5.380 0.96±0.070.96± 0.07 0.011±0.0500.011± 0.050 0.20±0.990.20± 0.99 79.67±10.7579.67± 10.75 25.3525.35 11.2711.27 26.6±3.126.6± 3.1 TTR min Prio 4.786±5.1364.786± 5.136 0.97±0.050.97± 0.05 0.006±0.0350.006± 0.035 0.12±0.760.12± 0.76 78.67±7.8278.67± 7.82 23.6123.61 11.1111.11 23.1±3.423.1± 3.4 TTR min Cent 2.564±5.4402.564± 5.440 0.94±0.110.94± 0.11 0.035±0.1380.035± 0.138 0.45±1.490.45± 1.49 78.98±8.3578.98± 8.35 51.3951.39 30.5630.56 29.4±5.829.4± 5.8 (c) TTR sep Dec 1.067±2.1371.067± 2.137 0.99±0.020.99± 0.02 0.068±0.0640.068± 0.064 3.99±4.463.99± 4.46 90.84±8.9690.84± 8.96 59.7259.72 59.7259.72 25.1±2.725.1± 2.7 TTR sep Prio 1.059±1.7061.059± 1.706 0.99±0.020.99± 0.02 0.059±0.0320.059± 0.032 3.52±2.803.52± 2.80 89.94±6.0689.94± 6.06 56.9456.94 56.9456.94 24.3±1.224.3± 1.2 TTR sep Cent 0.089±0.5210.089± 0.521 0.99±0.040.99± 0.04 0.089±0.1120.089± 0.112 5.78±9.035.78± 9.03 94.38±17.1794.38± 17.17 93.0693.06 83.3383.33 28.8±3.328.8± 3.3 Metrics from (72/100) scenarios after discarding (28) unsolvable scenarios. The best value for each metric is shown in bold. The first two rows do not treat safety and are excluded from this comparison. The solve time QPtime is summed (per vehicle per timestep) over QPs in multi-stage when multiple stages were attempted. Experiments were run on an AMD 9800x3D CPU, OSQP [39] was used to solve the QP. Because we primarily focus on the corridor merging problem, our metrics are evaluated only outside the corridor. However, we note that TTR sep halves in-corridor safety violations by better aligning vehicles (temporally and spatially) at the corridor entrance. The “zippering” behavior often arises as vehicles merge. Within the corridor, an LQR controller with a centralized safety filter (over vehicles within the corridor) is applied. IV-E2 Comparison Setup Table I presents the results for different combinations of guidance and safety filtering strategies. In (a), we evaluate guidance strategies without safety filtering, where TTR min serves as an uncoordinated baseline and TTR sep demonstrates the effect of temporal separation alone. In (b), we evaluate the three safety filtering architectures combined with TTR min guidance, which does not have strategic coordination. In (c), we combine the proposed TTR sep with the safety filtering architectures, representing the full proposed framework, and compare it against the baseline methods in (b). This structure allows us to isolate the effect of temporal separation, safety filtering, and their combination. Among the 100 scenarios, we exclude cases in which all methods fail to safely coordinate the vehicles, as these scenarios likely violate the feasibility assumption stated in Section I-B. IV-E3 Interpretation of Results & Discussion In Table I, we observe that TTR separation combined with any of the safety filtering architectures in (c) outperforms the baseline combinations in terms of safety (νunsafe _unsafe), fairness (τK _K), and missions completed safety and fairly (Ppass+)(P_pass 0.7$+$). Although the baseline methods in (b) achieve better efficiency, this comes at the cost of worse safety. Most notably, we find that even without safety filtering, TTR separation guidance alone in (a) achieves better safety than all baseline methods in (b) with any safety filter. In our considered experiments, every run with TTR separation guidance eventually terminates with all vehicles arriving at the corridor. Among the proposed safety filters in (c), the centralized filter achieves the lowest safety violation rate and the highest safe mission success rate, at the cost of the greatest QP solve time. The decentralized and priority-coordinated filters also improve the safety of the reference control urefu_ref, while minimally impacting fairness and efficiency. We highlight a representative scenario in Fig. 3, in which all baseline methods, including TTR separation guidance without safety filtering, fail to maintain safety. In contrast, two of the proposed methods in (c) successfully deconflict the vehicles and maintain priority order. Our results demonstrate that TTR separation is an extremely effective mechanism for safe coordination of multi-agent systems. It highlights the importance of strategic temporal coordination in preventing conflicts before reactive safety intervention becomes necessary, and moreover, the importance of aligning priority orders with individual agents’ control and safety filtering mechanisms for the best results. 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Appendix V-A Parameters Used in Simulation Study TABLE IV: Parameters of Air Taxi Dynamics & Air Corridor Parameter Air taxi (Sim) Groundspeed Minimum speed vminv_ 60 knot (30 m/s) Maximum speed vmaxv_ 175 knot (90 m/s) Nominal cruise speed vnomv_nom 136 knot (70 m/s) Acceleration amina_ -3.3 ft/s2 (-1.0 m/s2) amaxa_ 6.6 ft/s2 (2.0 m/s2) Angular Rate (ωmax _ ) 0.1 rad/s Sampling Time 0.1 s Corridor Geometry Width (d) 0.62 miles (1 km) Entry heading threshold (Δθ θ) 45∘45 Entry speed threshold (Δv v) 38.9 knot (20 m/s) Separation Distance (δ) 1640 ft (500 m) V-B Additional remarks on TTR Separation Guidance Remark 4 (Selection of ΔT T). The minimum feasible value of ΔT T for safety can be approximated by the required separation distance divided by the nominal vehicle speed. Under the parameters in Table IV, this corresponds to approximately 7.17.1s. In practice, ΔT T can be adjusted based on operational requirements, such as regulatory separation rules or vertiport throughput limits, and it can be used to handle numerical errors in the TTR computation. Remark 5 (Protocols for Implementation). For decentralized implementation, vehicles share their TTR values with neighboring agents and determine priority ordering within their communication range. For centralized implementation, a central traffic controller issues the target TTR values to each vehicle, and each vehicle executes the control law (13). Remark 6. Maximizing TTR may cause a vehicle to move toward the downstream region D (where Tmin(x)=∞T_ (x)\!=\!∞). To prevent this behavior, an additional safety filter is applied to enforce state constraints and prevent entry into D before reaching the corridor entry. V-C Relaxed QPs in Decentralized / Centralized Safety Filters Decentralized Relaxed CBF-QP for vehicle i: minui,ξ _u^i,\,ξ ‖ui−urefi‖2+λξ+ε2ξ2 \|u^i-u_ref^i\|^2+λ\,ξ+ 2\,ξ^2 s.t. .t. ∇V(x(ij))⊤f(x(ij),ui,uj)+αV(x(ij))+ξ≥0,ξ≥0, 408.68454pt$ ∇ V(x^(ij)) f\! (x^(ij),\,u^i,u^j )+α\,V(x^(ij))+ξ≥ 0, ξ≥ 0$, where α>0α>0 is a linear class-κ function parameter consistent with (7b). This relaxed QP is always feasible since ξ is unbounded above. Centralized CBF-QP: minu[N],ξ[N] _u^[N],\,ξ^[N] ∑i=1N‖ui−urefi‖2+λ∑(i,j)∈ξij+ε2∑(i,j)∈ξij2 _i=1^N\|u^i-u^i_ref\|^2+λ\! _(i,j) _ij+ 2\! _(i,j) _ij^2 s.t. .t. ∀i,ui∈U∀(i,j)∈,ξij≥0,∇V(x(ij))⊤f(x(ij),u(ij))+αV(x(ij))+ξij≥0, aligned &∀\,i, u^i∈ U&&\\ &∀\,(i,j) ,\\ & _ij≥ 0,&&\\ & 352.31624pt$ ∇ V(x^(ij)) f\! (x^(ij),\,u^(ij) )+α\,V(x^(ij))+ _ij≥ 0,$ aligned where we denote the set of active vehicle pairs as :=(i,j)∣i<j,j∈iP:=\(i,j) i<j,\;j _i\. u[N]∈ℝNmu^[N] ^Nm is the stacked control vector and ξ[N]∈ℝ||ξ^[N] ^|P| contains independent per-pair slack variables. The QP is relaxed for feasibility.