Simultaneous algorithm to solve the trajectory planning problem Francisco J. Rubio * , Francisco J. Valero, Jose Luis Suñer, Vicente Mata Department of Mechanical Engineering, U.P.V., Valencia 46022, Spain article info Article history: Received 18 January 2008 Received in revised form 8 April 2009 Accepted 19 April 2009 Available online 17 May 2009 Keywords: Robotics Trajectory planning Direct method Obstacle avoidance abstract This paper addresses the solution of smooth trajectory planning for industrial robots in environments with obstacles using a direct method, creating the trajectory gradually as the robot moves. The method presented deals with the uncertainties associated with the lack of knowledge of kinematic properties of intermediate via-points since they are generated as the algorithm evolves looking for the solution. Several cost functions are also proposed, which use the time that has been calculated, to guide the robot motion. The method has been applied successfully to a PUMA 560 robot and four operational parame- ters (execution time, computational time, distance travelled and number of configurations) have been computed to study the properties and influence of each cost function on the trajectory obtained. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction For industrial robots, the problem of minimum time trajectory planning has been addressed by numerous researchers motivated by the direct relation between executing the tasks in minimum time and ensuring productivity in manufacturing systems. Traditionally, the trajectory planning problem has been analysed using two different approaches: direct and indirect methods. Indirect methods usually involve planning the trajectory in two (decoupled) stages: in the first, a path is searched for in the configuration space and in the second a minimum time trajectory is adjusted to the path obtained previously and subjected to the torque constraints of the actuators (see [1,2]). Some other authors have directly taken a specified path with a number of intermediate via-points and have calculated a trajectory for it [1,5,11]. Most of the existing methods belong to one or other of these types, although the indirect methods are the most widely used (see some examples in [1–5,9–11]). A characteristic of indirect methods is that the path is known previously, either because it depends on the activity to be undertaken by the industrial robot or because it has been generated by a path planner. Generally speaking, indirect methods combine path planning with the obtaining of the time history of motion, usually in a sequential way. The method of Saramago and Steffen is worth mentioning at this point. In [1], the trajectory must pass through a num- ber of given points. The trajectories between points are interpolated by means of cubic polynomial functions. An objective function is considered with two terms, one corresponding to the optimum time of motion and the other to the minimum mechanical energy of the actuators. In [3], Piazzi and Visioli introduced a global method to solve the trajectory planning for industrial robots in which cubic polynomial functions are also used to interpolate joint trajectories. Another interesting approach is described by Piazzi and Visioli in [4], where the interpolation is done through trigonometric splines that ensure the continuity of the jerk. In [5], Gasparetto introduces a new trajectory planning technique which assumes that 0094-114X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2009.04.005 * Corresponding author. Tel.: +34 652972454; fax: +34 963877629. E-mail addresses: frubio@mcm.upv.es (F.J. Rubio), fvalero@mcm.upv.es (F.J. Valero), josuner@mcm.upv.es (J.L. Suñer), vmata@mcm.upv.es (V. Mata). Mechanism and Machine Theory 44 (2009) 1910–1922 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt