Pergamon Robotics & Computer-lntegratcdManufacturing, Vol. 13, No. 2, pp. 131-143, 1997 O 1997 Elsevier Science Ltd. All rights reserved Print~l in Great Britain 0736-5845/97 $17.00 + 0.00 PII:S0736-5845(96)00036-1 • Paper ASSEMBLY SCHEDULING FOR AN INTEGRATED TWO-ROBOT WORKCELL K. JIANG, L. D. SENEVIRATNE and S. W. E. EARLES Department of Mechanical Engineering, King's College London, Strand, London, U.K. An assembly sequence planning scheme, giving the time optimal solution for a two-robot cell, is presented. The scheme is based on dynamic programming, and the time optimum assembly sequence is generated in two stages. In stage one, an initial candidate sequence is derived. The stage-one algorithm is computationally efficient, the complexity being O(log n/k), where n is the number of elements in k groups. It also gives near-optimal results. Expression for the lower bounds for the total assembly time are derived in order to assess the departure from optimality of the stage-one sequence. In stage two, the initial candidate sequence is optimized using an iterative technique to yield the time optimal sequence. The scheme is extended to account for precedence constraints. It can also be extended for assembly cells with more than two robots. The scheme is tested using computer simulations, and some test results are presented. For the problems solved (typically involving 20 elements), the computational times were less than 5 sec in an Apollo 300 workstation. © 1997 Elsevier Science Ltd. 1. PROBLEM STATEMENT This paper presents a study of an integrated workcell, where two robots operate simultaneously on a product, say assembling a printed circuit board (Fig. 1). When the intersecting area of the robot operation domain is small compared with the size of the robot grippers, collision avoidance can be achieved by requiring only one robot to occupy the intersecting area at any instant, while the other robot is operating outside. The objective of this study is to provide algorithms for finding the optimum or near- optimum task sequence for each robot, taking collision avoidance into consideration. The problem is one of constrained operation scheduling. If the area shared by the robots is considered as a "machine" and the operations of the robots as "jobs", then the problem is one of n job single-machine scheduling. The algorithms ensure that collisions between the robots are avoided by allowing only one robot to be present in the work area at any given instant of time. The operations research-based approach presented uses dynamic programming to solve the scheduling problem for a two-robot workcell. Collision avoidance is consid- ered as a constraint to the optimization problem. The solution to this problem has applications in other fields, e.g. aircraft queuing-up for landing, the treatment of patients in a hospital, scheduling different programmes on a computer, processing different batches of crude oil at a refinery, and repairing cars in a garage. 131 2. LITERATURE REVIEW The science of operational research was actively developed in conjunction with the availability of high-speed digital computing devices after 1950. Linear programming I and its numerous descendants provided the foundation of operations research methodology (see Ref. 2 for a detailed treatment). These include Monte Carlo simulation techniques, stochastic optimization, queuing theory, integer programming, 3 dynamic programming, 4 combinator- ial methods, 5 and branch-and-bound methods. 6 Owing to the computational complexity of combinatorial deterministic optimization, many in- vestigations have sought tractable methods, which generally only find near-optimal solutions. This class of methods comprises several related algorithms, such as simulated annealing, 7 Tabu search, s-l° and genetic algorithms, ~'t2 all having been applied to scheduling. Around 1960, the symbolic processing capabilities of digital computers were clearly recognized, and have become a hallmark of artificial intelligence. Consequently, artificial intelligence-based scheduling methods have been developed, where problems are represented analogously instead of mapping informa- tion to numbers and subjected to numerical computations. This is particularly relevant to constraint-based approaches. 3'13A4 Currently, there are a multitude of research lines being followed in scheduling, and two particular trends can be identified: the introduction of new