Fast Near Time-Optimal Motion Planning for Holonomic Vehicles in Structured Environments

TL;DR

This work tackles real-time, near time-optimal motion planning for holonomic planar movers in structured environments by modeling free space as axis-aligned rectangular corridors and guiding trajectories with low-dimensional parametric motion primitives. It combines an analytical baseline solution with a constrained optimization that optimizes the remaining degrees of freedom, enabling significantly faster computation than full OCP-based methods while preserving near-optimality within a corridor sequence. The authors validate the approach through extensive simulations, benchmarking against OCP, OMG-tools, and VP-STO, and demonstrate real-world feasibility on Beckhoff’s XPlanar hardware with millimeter-level tracking accuracy. The methodology offers a practical path to real-time, collision-free motion planning for high-throughput industrial automation, while acknowledging limitations in unstructured environments and proposing future extensions to multi-agent and 3D settings.

Abstract

This paper proposes a novel and efficient optimization-based method for generating near time-optimal trajectories for holonomic vehicles navigating through complex but structured environments. The approach aims to solve the problem of motion planning for planar motion systems using magnetic levitation that can be used in assembly lines, automated laboratories or clean-rooms. In these applications, time-optimal trajectories that can be computed in real-time are required to increase productivity and allow the vehicles to be reactive if needed. The presented approach encodes the environment representation using free-space corridors and represents the motion of the vehicle through such a corridor using a motion primitive. These primitives are selected heuristically and define the trajectory with a limited number of degrees of freedom, which are determined in an optimization problem. As a result, the method achieves significantly lower computation times compared to the state-of-the-art, most notably solving a full Optimal Control Problem (OCP), OMG-tools or VP-STO without significantly compromising optimality within a fixed corridor sequence. The approach is benchmarked extensively in simulation and is validated on a real-world Beckhoff XPlanar system

Figures (13)

• Figure 1: APM4220 XPlanar Mover (without any payload) by Beckhoff Automation, measuring 113mm $\times$ 113mm, executing a time-optimal trajectory that is projected from above as a blue line. The red squares represent (virtual) obstacles
• Figure 2: Illustration of the different steps of the proposed method. The red squares indicate cells that are occupied by an obstacle. The starting and final positions are shown as well as the vehicle footprint. (a) The environment with starting position and desired destination indicated. (b) The path found by performing a graph-search, denoted by $\mathcal{P}$, is shown in light gray color. The cells in $\mathcal{P}'$ that are not in $\mathcal{P}$ are shown in dark gray. (c) The corridor sequence $\bm{C}$ is shown in light green. The overlap of two consecutive corridors is shown in dark green. (d) Black lines show 500 trajectories described using the selected motion primitives (random samples from $\bm{\Omega}_n$). The blue trajectory is the time-optimal one ($\Omega_n^* = \bm{\Phi}\left(\Pi^*\right)$ where $\Pi^* \in \mathcal{X}$)
• Figure 3: Illustration of the iterative growing strategy for an example environment. Note that corridors $\mathcal{C}_i$ are removed if $\mathcal{C}_i \subseteq \left(\mathcal{C}_{i-1} \cup \mathcal{C}_{i+1} \right)$ or if $\mathcal{C}_{i-1} \cap \mathcal{C}_{i+1}$ is nonempty, which occurs often in this environment. The corridor sequence $\bm{C}$ is shown in light green. The overlap of two consecutive corridors is shown in dark green. Obstacles are shown in red. The blue line shows the optimized trajectory.
• Figure 4: Illustration of the one-dimensional motion primitive $\phi(1, -1, p_k, v_k, \bm{\tau})$. The black dots are points on which box constraints are applied, as mentioned in Section \ref{['sec:optimization']}
• Figure 5: Illustration of heuristics to determine $\{\bm{p}_k\}_{k=1}^{n-1}$. $\bm{C}$ is shown in green and the green dots show centers of $\mathcal{O}_{k-1,k}$ and $\mathcal{O}_{k,k+1}$. The points in the set $\mathcal{W}_k$ are shown as blue circles. The bottom right blue circle will be selected as $\bm{p}_k$