On Fast Jerk–, Acceleration– and Velocity–Restricted Motion Functions for Online Trajectory Generation

Abstract

Finding fast motion functions to get from an initial state (distance, velocity, acceleration) to a final one has been of interest for decades. For a solution to be practically relevant, restrictions on jerk, acceleration and velocity have to be taken into account. Such solutions use optimization algorithms or try to directly construct a motion function allowing online trajectory generation. In this contribution, we follow the latter strategy and present an approach which first deals with the situation where initial and final accelerations are 0, and then relates the general case as much as possible to this situation. This leads to a classification with just four major cases. A continuity argument guarantees full coverage of all situations which is not the case or is not clear for other available algorithms. We present several examples that show the variety of different situations and, thus, the complexity of the task. We also describe an implementation in MATLAB® and results from a huge number of test runs regarding accuracy and efficiency, thus demonstrating that the algorithm is suitable for online trajectory generation.

On Fast Jerk-Continuous Motion Functions with Higher-Order Kinematic Restrictions for Online Trajectory Generation

“…Broquere et al [38] introduced the so-called velocity-acceleration plane to visualize seven-segment motion functions. This visualization has proved to be very helpful in [4] in order to systematically go through the space of all possible situations. In this contribution, we proceed analogously, but the plane has more structure to it.…”

“…Such a curve must run from left to right through the upper half-plane and from right to left through the lower one. Note that the parametric curves do not have any kinks, as is the case for only acceleration-continuous motion functions with piecewise constant jerk, as considered in [4,38].…”

“…Therefore, very often, motion functions are required to have at least a continuous acceleration. An algorithm for computing such a function with piecewise constant jerk ("seven segment profile") was presented by the author in [4], where restrictions on velocity, acceleration, and jerk and boundary conditions regarding velocity and acceleration can be prescribed. If there are stronger requirements on vibration reduction (e.g., for restricting position errors), higher-order continuity of the motion function has proven to be effective in experimental settings [5][6][7][8][9][10][11][12].…”

“…If there are stronger requirements on vibration reduction (e.g., for restricting position errors), higher-order continuity of the motion function has proven to be effective in experimental settings [5][6][7][8][9][10][11][12]. In this contribution, we present an algorithm for computing a jerk-continuous motion function (as opposed to the piecewise-constant, discontinuous jerk function designed in [4]) in one dimension, such that symmetric restrictions regarding velocity, acceleration, jerk, and snap (the derivative of the jerk, abbreviated as sn in the sequel) are fulfilled and boundary 2 of 39 conditions for position and velocity can be arbitrarily prescribed. We term this function as "fast" because, at any time, for at least one of the kinematic restrictions, the maximum or minimum is reached.…”

“…[17,35,36]), e.g., when a tool moves with constant velocity for performing a certain task and then has to quickly move to another place to again perform a task with constant velocity. Using an abbreviation from a guideline of the German Association of Engineers [37], we term this task GG in Table 1 (for more information on the tasks, see [4]). Note that the task GG comprises the task RR, since the velocities are allowed to be zero.…”