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An analysis of the results of an algorithm for optimal trajectory planning of robot manipulators is described in this paper. The objective function to be minimized is a weighted sum of the integral squared jerk and the execution time. Two possible primitives for building the trajectory are considered: cubic splines or fifth-order B-splines. The proposed technique allows to set constraints on the robot motion, expressed as upper bounds on the absolute values of velocity, acceleration and jerk. The described method is then applied to a 6-d.o.f. robot (a Cartesian gantry manipulator with a spherical wrist); the results obtained using the two different primitives are presented and discussed.

The trajectory planning problem in Robotics is thus defined: find a motion law along a given geometric path, taking into account some predefined requirements, so as to be able to generate suitable reference inputs for the control system of the manipulator. The inputs of the trajectory planning are: the geometric path, the kinematic and dynamic constraints, while the output is the trajectory of the joints (or of the end-effector), expressed as a time sequence of position, velocity and acceleration values. The trajectory generated by the planner must not excite the mechanical resonances of the manipulator: this is the case if it is smooth, i.e. the trajectory itself and some of its derivatives are continue functions. In particular, it would be desirable to obtain trajectories with continuous joint accelerations, so that the absolute value of the jerk (i.e. of the derivative of the acceleration) keeps bounded. Limiting the jerk is very important, because high jerk values can wear out the robot structure, and heavily excite its resonance frequencies. Vibrations induced by non-smooth trajectories can damage the robot actuators, and introduce large errors while the robot is performing tasks such as trajectory tracking. Moreover, low-jerk trajectories can be executed more rapidly and accurately. Most of the proposed trajectory planning algorithms are based on a minimization of an objective function, that depends on execution time, actuator effort, absolute value of the jerk, or a combination of these variables.

An analysis of the results yielded by a technique for trajectory planning of robotic manipulators has been described in this paper. The technique is based on minimization of an objective function that takes into account both the execution time and the integral of the squared jerk along the whole trajectory. The algorithm has been applied by taking both cubic splines or fifth-order B-splines as primitives of the trajectory to be generated. Application to the kinematics of a 6-DoF Cartesian gantry robot with a spherical wrist has been carried out. The results obtained in the two cases have been compared and discussed.