Distributed Control of Autonomous Swarms by Using Parallel Simulated Annealing Algorithm Wei Xi and John S. Baras Abstract— In early work of the authors, it was shown that Gibbs sampler based sequential annealing algorithm could be used to achieve self-organization in swarm vehicles based only on local information. However, long travelling time presents barriers to implement the algorithm in practice. In this paper we study a popular acceleration approach, the parallel annealing algorithm, and its convergence properties. We first study the convergence and equilibrium properties of the synchronous parallel sampling algorithm. A special example based on a battle field scenario is then studied. Sufficient conditions that the synchronous algorithm leads to desired configurations (global minimizers) are derived. While the synchronized algorithm reduces travelling time, it also raises delay and communication cost dramatically, in order to synchronize moves of a large group of vehicles. An asynchronous version of the parallel sampling algorithm is then proposed to solve the problem. Convergence properties of the asynchronous algorithm are also investigated. I. I NTRODUCTION In recent years, with the rapid advances in sensing, com- munication, computation, and actuation capabilities, groups (or swarms) of autonomous unmanned vehicles (AUVs) are expected to cooperatively perform dangerous or explorative tasks in a broad range of potential applications [1]. Due to the large scales of vehicle networks and bandwidth constraints on communication, distributed control and co- ordination methods are especially appealing [2], [3], [4], [5]. A popular distributed approach is based on artificial potential functions (APF), which encode desired vehicle behaviors such as inter-vehicle interactions, obstacle avoid- ance, and target approaching [6], [7], [8], [9]. Despite its simple, local, and elegant nature, this approach suffers from the local minima entrapment problem [10]. Researchers attempted to address this problem by designing potential functions that have no other local minima [11], [12], or escaping from local minima using ad hoc techniques, e.g., random walk [13]. An alternative approach to dealing with the local minima problem was explored using the concept of Markov Random Fields (MRFs) and simulated annealing (SA) approach by Baras and Tan [14]. Traditionally used in statistical This research was supported by the Army Research Office under the ODDR&E MURI01 Program Grant No. DAAD19-01-1-0465 to the Center for Networked Communicating Control Systems (through Boston University), and under ARO Grant No. DAAD190210319. W. Xi and J. S. Baras are with the Institute for Systems Research and the Department of Electrical & Computer Engi- neering, University of Maryland, College Park, MD 20742, USA. {wxi,baras}@isr.umd.edu mechanics and in image processing [15], MRFs were pro- posed to model swarms of vehicles. Similar to the APF approach, global objectives and constraints (e.g., obstacles) are reflected through the design of potential functions. The movement of vehicles is then decided using a Gibbs sampler based SA approach. The SA algorithm has also been adopted for UAV preposition in [16]. Theoretical studies and simulations have shown that, with a special sequential sampling, the global goals can be achieved despite the presence of local minima in the potentials [17], [18]. However, the maintainance of global indices, which is required for sequential sampling, in large vehicle networks, is difficult when there exist node failures. Moreover, long maneuvering time, which is due to the fact that only one vehicle moves at each time instance, presents difficulties in practice. The above problems can be resolved using parallel sam- pling [14], i.e., each node in the vehicle swarm executes the local Gibbs sampling in parallel. Parallel sampling techniques have been studied for many years in order to accelerate the slow convergence rate of the sequential simulated annealing algorithm [19]. It is usually required that nodes update their locations at the same time clock (synchronously). However, synchronization causes commu- nication cost and delay, which degrade performance. This can be resolved by using asynchronous parallel sampling, i.e., each vehicle uses its own clock to do the local sampling. In this paper, we first investigate the convergence prop- erties of a synchronous parallel sampling algorithm. In the analysis of the asynchronous parallel algorithm, the fact that there is a “time-varying” number of active nodes presents challenges. Fortunately, by applying a partially parallel model in [20], the asynchronous algorithm could be described by a homogeneous Markov chain. The conver- gence of the asynchronous parallel algorithm then follows. Finally, a special example based on a battle field scenario was investigated. Sufficient conditions that guarantee the optimality of the parallel sampling algorithm were analyzed. II. REVIEW OF GIBBS SAMPLER BASED ALGORITHM A. MRFs and Gibbs Sampler One can refer to, e.g., [15], [21], for a review of MRFs. Let S be a finite set of cardinality σ, with elements indexed by s and called sites. For s ∈ S, let Λ s be a finite set called the phase space for site s.A random field on S is a collection X = {X s } s∈S of random variables X s taking values in Λ s .A configuration of the system is x = {x s ,s ∈ S}, where x s ∈ Λ s , ∀s. The product space Λ 1 ×···× Λ σ is