* Corresponding author. Tel.: 33 240 371654; fax: 32 2407 47406; e-mail: Jean-Francois.Petiot@lan.ec-nantes.fr. Robotics and Computer-Integrated Manufacturing 14 (1998) 237 — 251 Contribution to the scheduling of trajectories in robotics Jean-Franiois Petiot*, Patrick Chedmail, Jean-Yves Hascoe¨t Equipe CMAO et Productique, IRCyN, UMR C.N.R.S. 6597, Ecole Centrale de Nantes. 1, rue de la Noe ¨ , BP 92101, 44321 Nantes Cedex 3, France Received 22 October 1996; received in revised form 10 September 1997; accepted 28 September 1997 Abstract In this paper, we intend to propose a method for minimizing the cycle time of robotic tasks by using an optimal scheduling of trajectory points: we consider functional points (welding or laser-cutting points on a car-body for example) that a robot has to reach. The problem is to order these points in such a way that the global cycle time is minimum. Originally, this scheduling problem is similar to the well-known travelling salesman problem. Among the algorithms used in combinatorial optimization, we have taken an interest in an original connectionist method called ‘‘the elastic net method’’, initially presented in an euclidian two-dimension space. The method described in this article is adapted for robotic purpose. It has been tested in industrial cases and compared with a classical method in combinatorial optimization: the Little’s algorithm. The elastic net method is generalized to the specific case of robotics applications. The elastic net method is generalized in the case of non-redundant and redundant robots; we develop a simultaneous research of the optimal scheduling and of the optimal choice of the configurations of the robot for each functional point. The optimal scheduling of the points of a trajectory in the case of redundant robots is presented. We consider it to be the original contribution of this work. In the case of cluttered environments, the above algorithm is adapted by introducing a repulsion potential between the obstacles and the ‘‘elastic’’. This leads to the simultaneous research of the optimal scheduling and of the free path planning. The method, using a new kind of algorithm, leads to original and reliable results for minimizing cycle time in robotics.  1998 Elsevier Science Ltd. All rights reserved. Keywords: Robotics; Cycle time optimisation; Scheduling; Travelling salesman problem; Connectionist algorithm; Path planning; Configurations optimisation; Redundant robots 1. Introduction For higher productivity purposes, the cycle time of industrial robots must be optimized. The layout design of robotized cells, the choice of the morphology and the positioning of the robots, the scheduling of trajectory points and path planning are fundamental problems for robotized cell designers. We distinguish three different classes among the trajectory optimization problems:  the optimal scheduling of robot tasks,  the research of collision free optimal trajectories (i.e. the path planning problem),  optimization of control (which gives an optimal motion of the robot along a given trajectory). These themes are subjected to considerable research in CAD-robotics and control. We are interested here in the problem of the minimization of the trajectory run time of the robot. This criterion depends on many design para- meters such as:  the scheduling of functional points/frames (the func- tional points/frames related to a task),  the trajectory between the points/frames,  the control law of the robot,  the choice of the configuration of the robot for per- forming a task,  the location of the robot,  the morphological choice of the robot. Classically [1—3], the approach in CAD systems in robotics consists in a ‘‘try and test process’’, by simula- ting the task of the robot. The designer modifies the parameters of the robotized cell in such a way that all the constraints of his problem are met (no collision, accessi- bility to the functional frames). New approaches in research propose to optimize auto- matically the criterion in the space of the design para- meters subjected or not to the constraints of non collision 0736-5845/98/$19.00  1998 Elsevier Science Ltd. All rights reserved. PII: S0736-5845(97)00032-X