Distributed safe planning for satisfying minimal temporal relaxations of TWTL specifications

Highlights

• Each agent aims to achieve a complex task while avoiding collision with other agents.

• We propose a distributed receding horizon algorithm for online trajectory planning.

• Each agent safely achieves its temporal logic task, with a minimal finite delay.

• The proposed algorithm is tested via simulations and experiments with quadrotors.

Abstract

We investigate a multi-agent planning problem, where each agent aims to achieve an individual task while avoiding collisions with other agents. Each agent’s task is expressed as a Time-Window Temporal Logic (TWTL) specification defined over a discretized environment. We propose a distributed receding horizon algorithm for online planning of agent trajectories. We show that under mild assumptions on the environment, the resulting trajectories are always safe (collision-free) and lead to the satisfaction of the TWTL specifications or a finite temporal relaxation. Accordingly, each agent is guaranteed to safely achieve its task, possibly with some minimal finite delay. Performance of the proposed algorithm is demonstrated via numerical simulations and experiments with quadrotors.

Introduction

The control and coordination of multi-agent systems for package delivery, disaster relief, warehouse logistics, smart transportation, or persistent surveillance have received significant attention in recent years (e.g., [1], [2], [3], [4], [5]). Collision avoidance is critical for the safe operation of multi-agent systems while performing such complex tasks over a shared environment. Since the complexity of joint planning grows exponentially with the number of agents, significant effort has been devoted to the design of distributed algorithms with safety and performance guarantees. In this paper, we propose a distributed algorithm for satisfying complex high-level specifications while ensuring safety.

In the literature, potential fields (e.g., [6]), sequential convex programming (e.g. [7]), priority based planning (e.g., [8]), control barrier functions (CBFs) (e.g., [9]) or buffered Voronoi cells (e.g., [10]) have been employed for distributed collision avoidance. More recently, some of these methods have been extended to account for constraints on agent dynamics as well as deadlock detection and avoidance (e.g., [11], [12]). In particular, [12] explicitly defines different types of deadlock which may occur, and how those deadlocks are accounted for through their implementation of CBFs. There also exist methods such as optimal reciprocal collision-avoidance (ORCA) [13] and a buffered Voronoi cell collision-avoidance strategy which requires less information than ORCA and can produce smooth collision-free trajectories with reduced number of deadlocks [14]. However, these methods do not guarantee deadlock avoidance. Other approaches such as distributed model predictive control (e.g. [15], [16]), differential games (e.g., [17]), and game theoretic methods (e.g., [18]) have also been used for distributed planning. The methods described thus far typically do not accommodate complex spatio-temporal requirements that can be expressed as temporal logics.

Recently, there has been a significant interest in the analysis and control of dynamical systems under temporal logic specifications. For instance, linear temporal logic (LTL) [19] has been extensively used in motion planning and control of robots (e.g., [20], [21], [22]). To ensure collision avoidance while satisfying more complex tasks defined as LTLs in a distributed manner, hybrid controllers can be employed, which can essentially combine a high-level mission planner with a local planner which enforces collision avoidance (e.g., [23], [24]). While these results are promising, LTL cannot express tasks with explicit time constraints. For example, consider an agent that is required to perform the following task: “visit $A$ for 1 time unit within 5 time units, and after this visit $B$ for 3 time units within 10 time units, and visiting $C$ must be performed after visiting both $A$ and $B$ twice”. Such tasks with time constraints can be expressed via bounded temporal logics (e.g., [25], [26], [27], [28]).

Ensuring safety under bounded temporal logics for multi-agent systems has received considerable attention in recent years (e.g., [29], [30], [31], [32]). When it is impossible to find safe paths that satisfy the original specifications, it may be desired to satisfy some relaxed versions of the specifications instead. The framework in [33] gives feedback on why the specification is not satisfiable and how to modify it. In [34], the robots slow down to give their iterative optimization algorithm more time to solve when necessary. The framework in [32] looks to minimize the violation by considering both hard and soft constraints, where the hard constraints (such as collision avoidance) must be satisfied. Alternatively, our algorithm allows for the temporal relaxation of time-window temporal logic (TWTL) specifications, and we show that, under mild assumptions on the environment topology, our algorithm ensures the completion of all TWTL specifications with finite temporal relaxation.

This work is closely related to [20], [29], [35]. Similar to this paper, a multi-agent receding horizon algorithm is proposed in [20] to generate each agent’s path independently using only local information. However, each task was defined by LTL (which cannot express explicit time constraints) and collision avoidance was not considered. In [35], collision avoidance was ensured by penalizing transitions in the centralized graph which captures the environment and TWTL specifications for all agents in the system. This centralized algorithm is not scalable as the number of agents increases. Moreover, in [35], if a safe path satisfying the TWTL cannot be found, the algorithm terminates and does not allow for relaxations of the TWTL specifications. Finally, the work in [29] considers a global task that needs to be achieved by the multi-agent system and allows for temporal relaxation with TWTL specifications. However, collision avoidance is not incorporated in path planning.

The collision avoidance procedure in this work is closely related to [36], [37]. In [36], a distributed prioritized planning algorithm is introduced, and [37] is essentially an asynchronous extension of [36] called revised priority planning. These algorithms assume complete peer-to-peer communication of trajectories while planning, and the entire path must be calculated before execution which does not lend itself well to unexpected interruptions. Our proposed collision avoidance procedure dynamically allocates priorities based on how close the agents are to completing their tasks. Our algorithm differs from previous versions of distributed prioritized planning (e.g., [36], [37]) by (1) detecting and resolving deadlocks (under mild assumptions on the environment), (2) considering partial/local information while computing agent paths in a receding horizon manner using dynamic programming, and (3) having the capability of changing goal regions of agents (allowable by the temporal logic) while avoiding collisions and resolving deadlocks (whereas state-of-the-art collision avoidance algorithms assume fixed goal regions for each agent).

This paper introduces a distributed algorithm for safe satisfaction of TWTL specifications, which can efficiently express complex time bounded tasks [28], for multi-agent systems operating in shared environments. In this paper, each agent is assigned with an individual TWTL specification and is assumed to move over a discretized environment. Agents communicate with the other agents in their local neighborhoods to plan collision-free paths in a receding horizon manner. The main contribution of this paper is a distributed algorithm for safe satisfaction of TWTL specifications by resolving conflicts via online re-planning, relaxing time windows if infeasibility occurs, and resolving possible deadlocks when detected. We theoretically show that the proposed algorithm produces individual agent paths that avoid both collisions and deadlock, and complete tasks in finite time.

Some preliminary results of this work has been presented in [38]. This paper is a significant extension of [38], and the main differences are as follows: (1) [38] shows that if the environment has $n$-connectivity where $n$ is the number of agents, there will not occur any deadlocks. Thus, [38] shows that agent paths with finite temporal relaxations can always be found under this environment assumption. In this paper, we eliminate the $n$-connectivity assumption and introduce a deadlock resolution algorithm that can resolve conflicts under an assumption that is milder than $n$-connectivity (i.e., the premise of Lemma 1). We also prove that the proposed algorithm can find paths with finite relaxations by resolving deadlocks (Theorem 1); (2) in this paper, we generalize the formulation and introduce a mapping for the conflicting transitions. Accordingly, one can flexibly define conflicting transitions depending on the problem’s assumptions and the capabilities of the low-level controller. (3) Moreover, [38] finds agent paths via exhaustive search. In this paper, we implement dynamic programming, which dramatically reduces the computational time to solve for the agent paths; (4) Finally, we present a benchmark analysis and compare our algorithm with some of the well-known methods in addition to different priority settings.

The remainder of the paper is organized as follows: Section 2 introduces TWTL, its temporal relaxation, and some graph theory preliminaries. Section 3 states the problem. The proposed algorithm and theoretical results are presented in Section 4. Experimental results on a team of quadrotors, numerical results, and benchmark analysis are shown in Section 5. The paper is concluded with a summary and possible future work in Section 6.