Automatically Synthesized Controllers for Distributed Assembly

We consider the task of assembling a large number of self controlled parts (or robots) into copies of a prescribed assembly (or formation). In particular, we describe a computationally tractable way to synthesize, from a specification of the desired assembly, local controllers to be used by each part, which when taken together, have the global effect of assembling the parts. We then prove that the controlled discrete dynamics of the system are correct with respect to a simplified model of the dynamics— meaning that a maximal number of parts are correctly assembled into copies of the desired assembly. Keywords: Controller synthesis, distributed control, self-assembly. kml nporqts uwvyxyz q{n uwo We consider the problem of controlling hundreds or thousands of robots to perform a task in concert. This problem presents many fundamental issues to robotics, control theory and computer science. With a great number of robots, decentralization is critical due to the cost of communication and the need for fault tolerance. In decentralized control, each robot should act based only on information local to it. It then becomes difficult, however, to guarantee or even derive the behavior of the entire system given the behaviors of the individual | This research is supported in part by the DARPA SEC program under grant number F33615-98-C-3613 and by AFOSR grant number F49620-01-1-0361. components. In this paper we address this difficulty in a novel way: We begin with a specification of an assembly and develop methods that allow us to automatically synthesize individual behaviors so that they are guaranteed to produce the desired global behavior. Specifically, we consider the task of assembling many disk-shaped parts in the plane into copies of a prescribed assembly (formation), which is specified by a graph with n vertices. We do not allow the parts to collide, making the task more difficult due to the non-trivial topology of the resulting })~ dimensional configuration space. As shown in Figure 1.1 we suppose that each part can move itself and can play any role in an assembly, which makes the task particularly rich. We first demonstrate a means of synthesizing from the specified assembly, a set of identical controllers for the parts to run which have the net effect of moving the parts to form copies of the specified assembly without colliding. The idea is that parts should join together into subassemblies which should in turn join together to make larger assemblies and so on. To achieve this, some theory is developed along with algorithms that compile a specified assembly into a list of allowable subassemblies. Next we show how to produce a lookup table from the list which can be used as a discrete event controller (Figure 1.2) that guides parts through a “soup” of other parts and subassemblies. Then we add a continuous motion controller based on the assembly rules represented by the lookup table and a (provably correct) method for deadlock avoidance. Finally, we show formally that the discrete dynamics given by the lookup table and the deadlock avoidance mechanism (and employed in the control of each part) are correct. The proof assumes a certain logical model of the dynamics which accounts for the discrete interactions between parts (forming neighbor relationships) but neglects the continuous dynamics. A formal analysis of the complete, highly nonlinear hybrid dynamics is not yet forthcoming. kmlPkml sw€ƒ‚„q{…v s†…‡ …‚ˆs z ‰ We are most strongly inspired by the work of Whitesides and his group ( Bowden et al., 1999; Breen et al., 1999) in meso-scale self-assembly. In this work, small, regular plastic tiles with hydrophobic or hydrophylic edges are placed on the surface of some liquid and gently shaken. Tiles with hydrophobic edges are attracted along those edges while hydrophylic edges repel. Striking “crystals” emerge as larger structures self assemble. By using different shapes and edge types, different gross structures can be created. A similar idea is used on a much smaller scale in (Mirkin, 2000) where strands of DNA are attached to tiny gold balls in solution. Complementary strands attract and a gross structure is revealed. By choosing which strands go where, the “programmer” has some control over the resulting emergent structure. At least two next steps are apparent. First, these and similar (Bonabeau et al., 2000) methods generally produce arrays or lattices of parts, meaning that there is no general way to terminate a regular pattern at, say, a ŠŒ‹Š array of parts (There has been work on changing the function of parts as they combine (Saitou, 1999). Second, there is no known formal method of starting with a specification of the desired emergent structure and devising the structure of the individual parts. In this paper we address both of these issues by supposing that each part can run a program that tells it when to join with another part, and when to repel it, based on some state information. Of course, this is a far way away from the reality of small plastic parts or gold balls, but our ideas could easily be implemented with teams of robots and may even, when developed further, present the chemist with new tools. The motivation for considering disk shaped parts in the plane and for the potential field construction in Section 4 comes from the work of Koditschek and others (Koditschek and Bozma, 2000; Karagoz et al., 2000) in assembly. There, a global artificial potential function over the configuration space of ~ disk shaped parts is used to guide the parts to their assembled state, corresponding to the unique minimum of the potential function. The approach is not distributed, however, because it requires that each part knows the full state of the system to act. Other work has applied similar ideas, in a distributed fashion (Reif and Wang, 1995; Balch and Hybinette, 2000), although without a means of synthesizing the desired behavior. Still other approaches to the control of a group of robots (Desai et al., 1999) assume a leader. In contrast, the present paper commits to the synthesis idea and to a strong degree of decentralization, using decentralized potential fields merely as a primitive in a more sophisticated hybrid control scheme. The ideas in this paper also grow from our own work in controller synthesis in manufacturing systems (Klavins and Koditschek, 2000; Klavins, 2000). Our approach to manufacturing has been to synthesize a decentralized automated factory description from a description of a product. The description includes the layout of the factory and the control programs the robots should run to produce the product. In that sense, the present work is an extension of the idea, although it assumes fewer constraints on the topology of the workspace. Ž l q{‰r‘s uw’“€ƒ…” We consider a simple form of assembly process by assuming that parts are programmable and able to sense the position and state of other nearby parts. We start with • disk-shaped parts (of radius – ) confined to move in — . Denote the position of part ˜ by the vector ™›š . We desire that each part move smoothly, without colliding with other parts, so that all parts eventually take some role in an assembly or formation. This is shown graphically in Figure 1.1. For simplicity, we assume that the dynamics of each disk are given by œ ™ šž Ÿ š . We believe that control of parts with more complicated dynamics can be based on the control algorithms we develop for this simple situation. In this section we describe the goal of assuming a role in a formation formally. Let ¡ £¢¥¤ž¦+§© ̈ be a finite undirected, acyclic graph. Thus, ¤ is a finite set of nodes (in this paper, ¤a¬«Z­Z¦+® ® ® ¦ ~m ̄ ) and § is a collection of edges of ° 3GNP(+-P,±W)23W)2 The goal of the assembly problem. Each disk shaped part must move from its initial position (a) to a position in a a copy of the specified assembly (b). Dashed lines show the resulting adjacency relationship ́ . There may be leftover parts. the form «Zμ7¦+¶ ̄ with μ ¦+¶Œ· ¤ and μw ̧ 1¶ . In this paper, we will call such a graph an assembly and only consider the case where ¡ is a tree (i.e., contains no cycles). There are technical details, which are solvable but not addressed in this paper except briefly, that prevent the direct application of the methods in this paper to general graphs. Given an assembly ¡ o¢¥¤D¦+§ ̈ with » ¤ »  ~ , consider the case where