Guaranteed Rejection-free Sampling Method Using Past Behaviours for Motion Planning of Autonomous Systems

TL;DR

The paper addresses motion planning for autonomous systems under changing obstacle configurations by leveraging past trajectories to inform sampling. It introduces a kernel density estimation framework with a finite-support kernel and bandwidth-driven obstacle inflation to produce a non-parametric description of the free space, guaranteeing that samples remain inside the originally defined feasible region. The method supports both biased and approximately uniform sampling around historical data, with rejection-free guarantees by construction. Two real-world case studies, involving a 2D autonomous surface vessel and a 3D drone, along with Monte Carlo simulations, demonstrate improved planning efficiency and lower-cost solutions, while highlighting the approach's dependence on representative historical data.

Abstract

The paper presents a novel learning-based sampling strategy that guarantees rejection-free sampling of the free space under both biased and approximately uniform conditions, leveraging multivariate kernel densities. Historical data from a given autonomous system is leveraged to estimate a non-parametric probabilistic description of the domain, which also describes the free space where feasible solutions of the motion planning problem are likely to be found. The tuning parameters of the kernel density estimator, the bandwidth and the kernel, are used to alter the description of the free space so that no samples can fall outside the originally defined space.The proposed method is demonstrated in two real-life case studies: An autonomous surface vessel (2D) and an autonomous drone (3D). Two planning problems are solved, showing that the proposed approximately uniform sampling scheme is capable of guaranteeing rejection-free samples of the considered workspace. Furthermore, the effectiveness of the proposed method is statistically validated using Monte Carlo simulations.

Figures (6)

• Figure 1: One dimensional resampling of the weighted KDE, such that uniform samples of the KDE domain are generated.
• Figure 2: Example of closed and bounded free space $\mathcal{W}_\mathrm{free}$.
• Figure 3: Visual representation of each step of the proposed method, as described in Section \ref{['sec:propsed_method']}. Once the space $\mathcal{W}_{\text{free}}^2$ (inner blue polygon) has been generated based on the selected bandwidth and the corresponding bounded KDE, one guarantees the ability to sample the $\mathcal{W}_{\text{free}}^1$ space without rejection sampling, within the domain covered by the historical data. Fig. \ref{['fig:toy_biased_sampling']} and Fig. \ref{['fig:toy_uniform_sampling']} demonstrates the ability to generate rejection-free samples in both a biased and approximately uniform manner.
• Figure 4: Outcomes from the selected case studies. Fig. \ref{['fig:scenario_ASV']} shows a path where the vessel maintains the specified safety distance to shallow waters. The proposed method ensures that only safe samples are generated, thereby increasing the efficiency of computing low cost solutions, as evident in Table \ref{['tab:stopping_costs_ASV']} and Fig. \ref{['fig:acces_box_plots']}. Details regarding safe sampling-based motion planning for ASVs can be found in enevoldsen2021kde and enevoldsen2022oe. Fig. \ref{['fig:drone_scenario_path']} details a similar scenario, but instead for the drone. This particular case study mimics an inspection task, and therefore the drone must also maintain some safety distance from the obstacles (see inspectdrone_data for more details). Note that due to limitations with the 3D engine used for plotting, the location of the data points within the figure may be deceptive, see therefore instead the 2D projection in Fig. \ref{['fig:drone_scenario_path_2d']}.
• Figure 5: Application of the proposed method detailed in Section \ref{['sec:propsed_method']}, where the generated sampling schemes are created based on real historical data from vessels passing through the Little belt area of Denmark, such that the resulting samples ensure feasibility with respect to the available water depth. The white regions in Fig. \ref{['fig:marine_raw']} and \ref{['fig:marine_raw_data']} are infeasible regions for the chosen vessel. For more details regarding this particular case study and associated data, see enevoldsen2021kde.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Theorem 1
• proof
• Remark 3