This paper presents an optimization-based method to solve the smooth trajectory planning problem where the user knows only the start and end points of the end-effector or the via point plus the start and end target points. For the start and end target points, we use an optimization approach to determine the manipulator configurations. Having obtained the desired minimum jerk path in the Cartesian space using the minimum jerk theory and having represented each joint motion by the third-degree B-spline curve with unknown parameters (i.e., control points), an optimization approach, rather than the pseudoinverse technique for inverse kinematics, is used to calculate the control points of each joint spline curve. The objective function includes several parts: (a) dynamic effort; (b) the inconsistency function, which is the joint rate change (first derivative) and predicted overall trend from the initial point to the end point; and (c) the nonsmoothness function of the trajectory, which is the second derivative of the joint trajectory. This method can be used for robotic manipulators with any number of degrees of freedom. Minimum jerk trajectories are desirable for their similarity to human joint movements, for their amenability to limit robot vibrations, and for their control (i.e., enhancement of control performance). Illustrative examples are presented to demonstrate the method.
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