International Review of Mechanical Engineering (I.RE.M.E.), Vol. 1, n. 2 Generation of Point to Point Trajectories for Robotic Manipulators under Electro-Mechanical Constraints T. CHETTIBI 1 , P. LEMOINE 2 Abstract – A simple direct method is applied to solve the problem of optimal trajectory generation for serial manipulators under electro-mechanical constraints. The goal is to increase the robot productivity by using its electric motors outside of their continuous operating range. This is possible only if dynamics of actuators is considered and inherent constraints are included. For this purpose, a general electro-mechanical model for serial robots is first presented. Then, the problem of trajectory generation is cast as a non-linear optimization program using an approximation of joint position variables by means of algebraic polynomial splines which interpolate a set of control points. Finally, the optimization problem is solved using a sequential quadratic programming method for the unknown transfer time and the unknown position of control points, while minimizing a cost function and respecting elecro-mechanical constraints. Numerical and experimental results are presented to illustrate the efficiency of the proposed approach. Copyright © 2007 Praise Worthy Prize - All rights reserved. Keywords: Robot, DC Motor, Dynamics, Trajectory, Optimization Nomenclature m Γ [n  1] vector of motors torque N [n  n] matrix of gear transmission ratio Γ [n  1] vector of joints torque q q  , [n  1] vectors of joints position and velocity q q      , [n  1] vectors of joints acceleration and jerk q m [n  1] vector of motor configuration variables M [n  n] robot inertia matrix C [n  1] vector of Coriolis forces G [n  1] vector of gravity forces J m [n  n] motors inertia matrix U nom [n  1] vector of motors feeding voltage K m [n  n] matrix of motors constant torque qi, qf [n  1] vector of initial and final configurations J cost function T f transfer time i Γ bounds on the i th joint torque l number of spline segments L i inductance of the i th motor R i resistance of the i th motor K bemf,i back electromotive force constant of the i th motor K i torque constant of the i th motor I current in the motor armature C I maximum continuous armature current p I maximum continuous armature pulse current A I maximum continuous amplifier current  i a spline control point a ji the j th coefficient of the i th spline segment n number of the robot degrees of freedom f i polynomial function of the i th spline segment i q , i q lower and upper bounds on the i th joint position F c,i , F v,i coulomb and viscous friction coefficients of the i th joint Kp, Kd, Ki adjustable gains of the controller I. Introduction The exploitation of robotic manipulators is based on two fundamental steps, namely: trajectory generation and control design. Trajectory generation can be defined as the process of selecting a motion and the associated optimal input controls from the set of admissible motions and controls while verifying all constraints and minimizing a performance index. This phase is expected to provide a complete and precise description of the robot motion using a suitable robot and environment models. Controls are supposed to carry out the execution of programmed motions despite inevitable modeling errors and existing perturbations. Steps of robot modeling and motion generation may occupy a large part of the effort required in the system groundwork. Effectiveness of the proposed trajectory planning algorithms depends largely on the accuracy of employed models. By accurate model we mean two things: a model respecting physics laws and a model using precise parameters. The knowledge of the exact robot parameters is very important for both simulation and control. Calibration and identification are the two main procedures which are commonly used in robotics Manuscript received January 2006, revised January 2006, accepted February 2006 Copyright © 2006 Praise Worthy Prize - All rights reserved hal-00362618, version 1 - 19 Feb 2009 Author manuscript, published in "International Review of Mechanical Engineering, IREME, ISSN 1970-8734 1, 2 (2007) pp. 131-143"