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In this paper, we describe a new procedure for constructing robot trajectories. The procedure uses the theory of B-splines. The motion of the robot, as specified by a time sequence of position and orientation knots of the end effector, is first transformed into sets of joint displacements. B-splines are then used to fit these sequences for each joint. The procedure consists of a simple recursive algorithm and allows constraints to be imposed on the magnitudes of the joint velocities and accelerations as well as on their initial and final values. A trajectory constructed by this method has the property that a local modification can be made quickly and easily without recomputing the entire trajectory. The method has been used for generating trajectories for a PUMA 560 robot.