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The purpose of this paper is to solve the trajectory planning problem of industrial robots in a complex environment.

A simultaneous algorithm was presented in which the trajectory was generated gradually as the robot moves. It takes into account the presence of obstacles (to avoid collisions) and differential constraints related to the dynamics of the robotic system. The method poses an optimization problem that aims at minimizing the time to perform the trajectory when several interpolation functions are used.

A new approach to solving the trajectory planning problem in which the behaviour of four operational parameters (execution time, computational time, distance travelled and number of configurations) have been analyzed when changing the interpolation functions, therefore enabling the user to choose the most efficient algorithm depending on which parameter the user is most interested in. From the examples solved the interpolation function that yields the best results has been found.

This new technique is very time consuming due to the great number of mathematical calculations that have to be made. However, it yields a solution.

The algorithm is able to obtain the solution to the trajectory planning problem for any industrial robot. Also, even mobile obstacles in the workspace could be incorporated at the same time as the robot is moving and creating the path and the time history of motion.

It gives a new tool for solving the trajectory planning problem and describes the best interpolation function.