Jerk-limited Traversal of One-dimensional Paths and its Application to Multi-dimensional Path Tracking

TL;DR

The paper tackles jerk-constrained time-optimal traversal of multi-dimensional paths by decomposing the problem into per-dimension one-dimensional analyses. It computes feasible target accelerations for intermediate waypoints through binary search and generates time-optimal trajectories using a jerk-aware solver to realize upper (fast) and lower (slow) progress, followed by an iterative scheme to align the multi-dimensional path length with a reference. Key contributions include a binary-search procedure for feasible accelerations, explicit upper and lower trajectories for fastest and slowest progress, and an iterative tracking framework validated against state-of-the-art baselines on high-dimensional references. The results demonstrate fast jerk-limited traversal with controlled tracking error, offering a practical approach for online-like path tracking on high-DOF robotic systems.

Abstract

In this paper, we present an iterative method to quickly traverse multi-dimensional paths considering jerk constraints. As a first step, we analyze the traversal of each individual path dimension. We derive a range of feasible target accelerations for each intermediate waypoint of a one-dimensional path using a binary search algorithm. Computing a trajectory from waypoint to waypoint leads to the fastest progress on the path when selecting the highest feasible target acceleration. Similarly, it is possible to calculate a trajectory that leads to minimum progress along the path. This insight allows us to control the traversal of a one-dimensional path in such a way that a reference path length of a multi-dimensional path is approximately tracked over time. In order to improve the tracking accuracy, we propose an iterative scheme to adjust the temporal course of the selected reference path length. More precisely, the temporal region causing the largest position deviation is identified and updated at each iteration. In our evaluation, we thoroughly analyze the performance of our method using seven-dimensional reference paths with different path characteristics. We show that our method manages to quickly traverse the reference paths and compare the required traversing time and the resulting path accuracy with other state-of-the-art approaches.

Figures (8)

• Figure 1: A time parameterization for an exemplary one-dimensional path with an initial position $p_0$, final position $p_{III}$ and two intermediate positions $p_I$ and $p_{II}$.
• Figure 2: The range of valid target accelerations for each waypoint $p_0$, $p_{I}$, $p_{II}$, $p_{III}$ includes zero as lower limit and $a_{\mathrm{max}_{0}}$, $a_{\mathrm{max}_{I}}$, $a_{\mathrm{max}_{II}}$ and $a_{\mathrm{max}_{III}}$ as upper limit, respectively.
• Figure 3: Binary search to find $a_{\mathrm{in}_{\mathrm{max}}}$ assuming $a_{\mathrm{out}} = 0$. After the steps (a), (b) and (c), it can be said that $a_{\mathrm{in}_{\mathrm{max}}}$ is greater than or equal to $0.75 \cdot a_{\mathrm{max}}$ but less than $a_{\mathrm{max}}$.
• Figure 4: Resulting trajectories when selecting an input acceleration of $\alpha \cdot a_{\mathrm{in}_{\mathrm{max}}}$ and a target acceleration of $\beta \cdot a_{\mathrm{out}_{\mathrm{max}}}$
• Figure 5: Traversing a path with a fixed mapping factor of $0.7$.