Manipulator trajectory planning using a MOEA E.J. Solteiro Pires a, * , P.B. de Moura Oliveira a,b , J.A. Tenreiro Machado c a Univ. Tra ´s-os-Montes e Alto Douro, Dep. de Engenharia Electrote ´cnica, 5000-911 Vila Real, Portugal b Centro de Estudos Tecnolo ´gicos, do Ambiente e da Vida, 5000-911 Vila Real, Portugal c Instituto Superior de Engenharia do Porto, Dep. de Engenharia Electrote ´cnica, Rua Dr. Anto ´nio Bernadino de Almeida, 4200-072 Porto, Portugal Received 2 February 2005; received in revised form 21 June 2005; accepted 26 June 2005 Available online 21 February 2006 Abstract Generating manipulator trajectories considering multiple objectives and obstacle avoidance is a non-trivial optimization problem. In this paper a multi-objective genetic algorithm based technique is proposed to address this problem. Multiple criteria are optimized considering up to five simultaneous objectives. Simulation results are presented for robots with two and three degrees of freedom, considering two and five objectives optimization. A subsequent analysis of the spread and solutions distribution along the converged non-dominated Pareto front is carried out, in terms of the achieved diversity. # 2006 Elsevier B.V. All rights reserved. Keywords: Genetic algorithms; Multi-objective optimization; Robotic manipulators; Trajectory planning 1. Introduction In the last 20 years genetic algorithms (GAs) have been applied in a plethora of fields such as: control, system identification, robotics, planning and scheduling, image proces- sing, pattern recognition and speech recognition [1]. This paper addresses the planning of trajectories, meaning the development of an algorithm to find a continuous motion that takes the robotic manipulator from a given starting configuration to a desired end position in the workspace without colliding with any obstacle. Several single-objective methods for trajectory planning, collision avoidance and manipulator structure definition have been proposed. A possible approach for generating the manipulator trajectories [2,3] consists in adopting the differential inverse kinematics, using the Jacobian matrix. However, these techniques must take into account the kinematic singularities, which may be hard to tackle. To avoid this problem, other algorithms for the trajectory generation are based on the direct kinematics [4–8]. Chen and Zalzala [2] proposed a GA method to generate the position and the configuration of a mobile manipulator. In this report the inverse kinematics scheme is applied to optimize the least torque norm, the manipulability, the torque distribution and the obstacle avoidance. Davidor [3] also applied GAs to the trajectory generation by searching the inverse kinematics solutions to pre-defined end effector robot paths. Kubota et al. [4] studied a hierarchical trajectory planning method for a redundant manipulator with a virus-evolutionary GA, running simultaneously two processes. One process calculates some manipulator collision-free positions and the other generates a collision free trajectory by combining these intermediate positions. Rana and Zalzala [5] developed a method to plan a near time-optimal, collision-free, motion in the case of multi- arm manipulators. The planning is carried out in the joint space and the path is represented as a string of via-points connected through cubic splines. Gaco ˆgne [9] presented a problem involving obstacle avoidance. The proposed techni- que looks for the emergence of system rules for a mobile robot to obtain a good road-holding behavior in different play- grounds. A multi-objective genetic algorithm is used to find short and readable solutions for every concrete problem. Indeed, multi-objective techniques using GAs have been increasing in relevance as a research area. In 1989, Goldberg [10] suggested the use of a GA to solve multi-objective problems and since then other investigators have been developing new methods, such as multi-objective genetic www.elsevier.com/locate/asoc Applied Soft Computing 7 (2007) 659–667 * Corresponding author. Tel.: +351 969 02 96 35; fax: +351 259 35 04 80. E-mail addresses: epires@utad.pt (E.J. Solteiro Pires), oliveira@utad.pt (P.B. de Moura Oliveira), jtm@dee.isep.ipp.pt (J.A. Tenreiro Machado). 1568-4946/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2005.06.009