A stochastic algorithm for makespan minimized multi-agent path planning in discrete space
Graphical abstract
Introduction
Multi-agent path planning (MAPP) requires finding a collision-free path or route ri for each agent Ai that navigates the agent Ai from its initial position to its unique destination. The MAPP problem arises in practical applications such as path planning of unmanned autonomous vehicles on a factory floor or route planning for a fleet of delivery trucks in a city. The traditional approaches for solving the MAPP problem can be mainly divided into coupled or decoupled approach. The coupled approaches simultaneously plan the paths for all agents and can therefore find an optimal solution. However, finding this optimal solution creates a scalability issue as the problem can become computationally intractable as the number of agents or the size of the search map increases. In contrast, the decoupled approach decomposes the MAPP problem into several sub-problems, and then solves each sub-problem separately. Though the computational efficiency of decoupled approaches can be improved by reducing the dimensionality of the search space, this often sacrifices the optimality of the solution and may make this approach incomplete.
The MAPP goals addressed in this work were motivated by the application scenario in Fig. 1, which shows a tangible interactive system for teaching children how to spell words in the English language. This system consists of a set of passive and active cubes marked with letters of the English alphabets, which the child can arrange to form words. The active cubes are intelligent autonomous mobile robots or agents that can interact and assist the child by maneuvering themselves to appropriate localities so as to provide meaningful contextual scaffold to the child while he is forming a word. Overseeing the entire workspace is a video camera that is able to track the position and orientation of each individual letter block and mobile robot, essentially providing an instantaneous map of the entire operating environment. The central host PC uses this map to plan the traversal path of relevant mobile robots so that they can quickly travel to appropriate positions in the map to form a required word. In order to quickly respond to the child, the paths should be computed with the minimal latency and meanwhile the mobile robots are required to move to their respective positions as soon as possible. As a result, the objective of this work is to quickly find a makespan minimized MAPP solution on grid maps. In the scenario where all n agents move concurrently, finding this makespan minimized MAPP solution can be effectively viewed as the problem of minimizing the time related cost of the agent that takes the maximum time to reach its destination.
Makespan minimized MAPP is essentially a minimax optimization problem. Previous approaches [1], [2], [3], [4], [5] for minimax problem approximated its optimal solution by solving a sequence of quadratic or trust-region subproblems. Instead of solving the makespan minimized MAPP problem directly, this work tries to approximate the makespan minimized MAPP solution by solving its smooth approximation problem, which is obtained by using a regularization technique in [6]. One of the contributions in this paper is the novelly proposed iterative algorithm for solving this approximation problem. The iterative algorithm approximates the optimal solution of the makespan minimized MAPP problem by solving a sequence of computationally hard MAPP minimization problems with a linear objective function. Finding an optimal MAPP solution can become intractable [7], [8] as the agent count increases. For this reason, heuristic methods are often adopted to find a good solution in a reasonable time. Heuristic approaches have been used to effectively solve complex problems like optimization of mathematical functions [9], and combinatorial optimization problem [11]. However, solving MAPP minimization problem by heuristic approaches is still a challenging task. Kala [10] adopted evolutionary algorithms for a MAPP problem in a maze-like environment. However, this approach is not efficient in a gridded based map nor is it effective in solving the collisions between agents in a crowded environment. Moreover, population based heuristic methods like evolutionary algorithm have memory issue as maintaining and improving a group of MAPP solutions is costly in terms of memory requirement, especially when the map size and agent count are large. Recently, Wang and Goh [48] proposed a guided local search algorithm called Guided Iterative Prioritized Planning (GIPP) for MAPP, however, their algorithm suffers from quickly increasing computational time with increasing agent count. The second contribution of this work is the incorporation of a novel probabilistic solution generation mechanism for MAPP into the local search method. By generating a better neighboring MAPP solution with a higher probability, the new proposed local search algorithm named Probabilistic Iterative Path Coordination (PIPC) was shown to be more computationally efficient in finding a superior MAPP solution than GIPP in [48]. Unfortunately, the proposed PIPC algorithm is sensitive to its inherent parameters. To address the problem of parameter adaptation, many nature-inspired optimization algorithms [12], [13] used fuzzy logic to dynamically adapt their important parameters. To improve the performance of the proposed PIPC algorithm, several informed parameter tuning strategies were suggested. In so doing, PIPC is able to efficiently switch between extensive and intensive search in the solution space. This final contribution was experimentally shown to produce superior performances in both the abilities to find feasible and makespan-minimized MAPP solutions.
This paper is organized as follows: Section 2 discusses some previous related work in MAPP, followed by Section 3, which provides a formal definition of the MAPP problem that is being addressed in this work. The detail of the proposed stochastic algorithm is described in Section 4. Experimental results and discussions are presented in Section 5, followed by a conclusion in Section 6.
Section snippets
Related work
Path planning for a single agent on a graph requires finding a collision-free path from an initial vertex to a destination vertex. This single agent path planning problem can be effectively solved by the A* algorithm [14] when an optimal path or solution is required. Though the search space of single-agent path planning is bounded by the size of map, the computational cost for the algorithm like A* to find an optimal path can be high when the map is large. In this case, by sacrificing the
Makespan minimized multi-agent path planning problem definition
The MAPP problem for n agents is commonly described as the problem of finding a feasible path ri for each agent Ai, where i = 1, 2, …, n, so that each agent can move from a starting point to its unique destination without colliding with other agents or static obstacles S. Since the MAPP problem is considered to be completed when all the agents reach their destinations, the additional goal addressed in this paper is to find a makespan minimized solution which has the
The stochastic algorithm for makespan minimized multi-agent path planning
The objective function F0(r) in (6) is a non-smooth function, in this work we approximate the minimax objective function by using a maximum entropy function given by
As c→ + ∞, Fe(r ; c) → F0(r). As a result, the approximate problem of the minimax problem (6) is given by
Subject to (4), (5). An incremental scheme is recommended for the control parameter c, in which c is initially set to a small value and then gradually increased. In so
Evaluation
The proposed Probabilistic Iterative Makespan Minimization algorithm for makespan minimized MAPP is evaluated in a simulation environment consisting of a 30 × 20 grid map. Each unit agent is assigned different starting and target positions. The unit time interval is set to dt, which is the time taken by the agent to traverse one grid node.
All the experimental results presented were obtained with the same 100 random initial scenarios. The simulation program was coded in the C++ programming
Conclusion
Finding an optimal solution for makespan minimized MAPP in a grid map can be intractable since the size of the search space increases rapidly with the number of agents and the map size. In addition, the non-smooth makespan minimized MAPP problem cannot be directly solved by standard optimization algorithms. This paper addressed the non-smoothness issue by approximating the optimal solution through the solving of a smooth approximation problem. An iterative algorithm called Probabilistic
Acknowledgement
This research is funded by the Singapore National Research Foundation Interactive Digital Media for Education grant NRF2008-IDM001-017.
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