Letter Early-Awareness Collision Avoidance in Optimal Multi- Agent Path Planning With Temporal Logic Specifications Yiwei Zheng, Aiwen Lai, Xiao Yu, Member, IEEE, and Weiyao Lan, Senior Member, IEEE    Dear Editor, This letter investigates a multi-agent path planning problem in a road network with the requirement of avoiding collisions among all agents in the partitioned environment. We first abstract the agents to a set of transition systems, and construct a team transition system from these individual systems. A mechanism is designed for the team transition system to detect all collisions within the synthesized run. Then, “wait” and “go-back” transitions are added to the individual transition systems, while removing all possible collisions in the team transition system. Finally, two examples are given to illustrate the effectiveness of the proposed approach. Path planning is a classic problem aiming to synthesize feasible trajectories for unmanned ground mobile robots [1], aerial vehicles [2] and surface vessels [3] to navigate from one place to the other [4]. Formal method is an accurate and user-friendly way to solve path planning problem of MASs in an intuitive but mathematically pre- cise manner [5], [6]. Recently, path planning satisfying particular mission specifications has been widely studied, while optimizing a certain performance index, such as time, efficiency [7], correctness [8], or some other relevant metric [9]. Although these methods are effective in finding trajectories satisfying the temporal logic specifi- cations, yet collision avoidance is not considered in these methods. Considering the MASs in practical situations, one single collision could lead to the failure of the whole system. Thus, the synthesized trajectories need to avoid all collisions among agents and all static obstacles along the way. Various methods have been proposed for finding collision-free paths. In [10], an overall solution was pre- sented to solve both inter-agent collisions and deadlocks with low- speed obstacles. In [11], the possible motion conflict between robots is resolved by reinforcement learning. Although the aforementioned methods appear effective in collision avoidance, yet the essence of these methods is still to react to incoming collisions from which the performance of synthesized trajectories may suffer. This letter is motivated to develop a multi-agent path planning method which guarantees synthesized runs collision-free while maxi- mizing efficiency. The main contributions of this letter can be sum- marized as follows. First, a novel offline planning method is put for- ward based on formal method. Compared with the algorithm in [12] where the requirement of collision avoidance was not considered, our proposed algorithm can synthesize the optimal individual runs for MASs which are guaranteed collision-free among all agents in the partitioned environments. Second, the conflict-based searching approaches [13] and the variants of safe interval path planning [14] were adopted for finding the optimal collision-free path to a desig- nated position in a dynamic environment, but the application to lin- ear temporal logic (LTL) specifications was not considered, while our proposed method is able to synthesize a set of optimal trajecto- ries under LTL specifications for MASs. {T 1 ,..., T m } T i Problem formulation: A team of m controllable agents are deplo- yed in a road network with intersections. The behavior of these agents can be modeled as weighted transition systems (TSs) , and TS is a tuple T i := (Q i , q 0 i ,δ i , Π i , L i , w i ) Q i q i,0 ∈Q i δ i ⊆Q i ×Q i Π i L i : Q i → 2 Π i w i : δ i → N + T i δ i Q i w i T i r i = q i,0 , q i,1 , ..., q i,k , ..., k ∈ N L i (r i ) = L i (q i,0 ), L i (q i,1 ), ..., L i (q i,k ), ..., k ∈ N L i (q i,k ) ⊆ 2 Π i Π i ⊆ Π T i where is a finite set of states, is the initial state, is the transition relation, is a finite set of atomic propositions (APs), is a map giving the set of APs satis- fied in a state, is a map assigning a positive integer weight to each transition. For agent , the transition relation is abstracted from the road network, the set of states are abstracted from the intersections, and the weight corresponds to the end-to- end traveling times. The run of the agent is an infinite sequence . The trace of a run is an infinite sequence with , and . With all m controllable agents deployed in the same road network, if we denote the set of APs of the road work as Π, then we have for agent . {T 1 , ..., T m } The high-level mission specifications of agents are described by LTL formulas. An LTL formula consisting of a set of APs Π, boolean operators and temporal operators, is formed accord- ing to the following syntax: φ := TRUE | α | φ 1 ∧ φ 2 |¬φ |⃝φ | φ 1 U φ 2 α ∈ Π ⃝ U ∨ ⋄ φ 1 ∨ φ 2 := ¬(¬φ 1 ∧¬φ 2 ) ⋄φ := TRUE U φ □φ := ¬⋄¬φ φ 1 → φ 2 = ¬φ 1 U φ 2 r i T i r i L i (r i ) where is an AP, and temporal operators and mean “next” and “until”, respectively. The above definitions also induce temporal operators such as (conjunction), (eventually), □ (always), and → (implication), in which , , , and . Given a run of TS , we say satisfies an LTL formula ϕ if the trace satisfies ϕ. In this letter, it is assumed that there exist n uncontrollable agents moving along the fixed trajectories in this road network. Collisions between any two of the agents may lead to the asynchronization of the system, or even damage to the agent or the cargo. We first give the definitions of collisions and then the problem statement as fol- lows. Definition 1 (Collision): A collision happens when any two of the agents are at the same intersection of the road network, or in the same road and move toward each other. φ := ϕ ∧ (□ ⋄ π) {T 1 ,... T m } m + n Problem 1: Given a road network with intersections, a LTL for- mula over the atomic proposition set Π, a team of m controllable agents modeled as TSs , , and a group of n uncontrollable agents moving along a set of fixed trajectories, if all of the end-to-end traveling time among agents are identical, synthesize individual runs for the m controllable agents such that the cost function J(T π ) = lim sup k→+∞ (T π (k + 1) − T π (k)) (1) m + n is minimized and collisions among all agents are avoided. φ := ϕ ∧ (□ ⋄ π) m + n q k , q l ∈Q i ∩Q j i, j , k, l ∈ N + 1 ≤ i, j ≤ m + n w i ((q k , q l )) = w j ((q k , q l )) T j := (Q j , q 0 j ,δ j , Π j , L j , w j ) m + 1 ≤ j ≤ m + n j ∈ N Q j ⊆∪ m k=1 Q k δ j ⊆∪ m k=1 δ k Π j = ∅ ∀q j ∈Q j ∥Post(q j )∥ = 1 r(k, i) r = (q 1,0 ,..., q i,0 ,...), ..., (q 1,k ,..., q i,k ,...), ... r(k, i) = q i,k In Problem 1, for the LTL formula , the proposition π must be satisfied infinitely many times, and the maximum time between successive satisfactions of π must be minimized subject to (1). The assumption that all agents shares identical end-to-end traveling time means that for all , , , . With the trajectories of the uncontrollable agents fixed, we use TSs to model these uncontrollable agents, where , , , , , and , and the number successor states . With a slight abuse of notations, we use to denote the ith element in the kth state of the team run r, i.e., for , . Then, according to Definition 1, we categorize the defined collisions into two groups: Corresponding author: Xiao Yu. Citation: Y. W. Zheng, A. W. Lai, X. Yu, and W. Y. Lan, “Early-awareness collision avoidance in optimal multi-agent path planning with temporal logic specifications,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 5, pp. 1346–1348, May 2023. Y. W. Zheng, A. W. Lai, and W. Y. Lan are with the Department of Automation, Xiamen University, Xiamen 361005, and also with the Key Laboratory of Control and Navigation (Xiamen University), Fujian Province University, Xiamen 361005, China (e-mail: ywzheng1201@stu.xmu.edu.cn; aiwenlai@xmu.edu.cn; wylan@xmu.edu.cn). X. Yu is with the Department of Automation, Xiamen University, Xiamen 361005, and also with the Innovation Laboratory for Sciences and Technologies of Energy Materials of Fujian Province (IKKEM), Xiamen 361005, China (e-mail: xiaoyu@xmu.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2022.106043 1346 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 10, NO. 5, MAY 2023