Generalized Multi-Speed Dubins Motion Model

TL;DR

This work introduces the Generalized Multi-Speed Dubins Motion Model (GMDM), which extends the classic Dubins framework by allowing per-segment speeds on the three-segment CSC/CCC path types to enable sharper turns and reduced collision risk. GMDM retains closed-form forward and inverse solutions, guarantees full reachability of the SE(2) space, and subsumes the Dubins model as the constant-speed special case, while enabling real-time computation suitable for high-level planners. Through extensive simulations and comparisons with Dubins and Wolek's models, GMDM delivers near time-optimal performance in obstacle-free settings and significantly safer, time-risk-optimized trajectories in obstacle-rich environments; it also markedly accelerates planning when integrated with sampling-based planners like RRT$^*$ and high-level time-risk optimization (T$^*$). The results demonstrate that multi-speed per segment provides a practical balance between travel time and safety, offering a scalable, easy-to-implement improvement for kinodynamic motion planning in 2D environments and potential extensions to 3D. The approach holds promise for real-time robotics applications requiring fast planning with tunable risk-awareness in cluttered spaces.

Abstract

The paper develops a novel motion model, called Generalized Multi-Speed Dubins Motion Model (GMDM), which extends the Dubins model by considering multiple speeds. While the Dubins model produces time-optimal paths under a constant speed constraint, these paths could be suboptimal if this constraint is relaxed to include multiple speeds. This is because a constant speed results in a large minimum turning radius, thus producing paths with longer maneuvers and larger travel times. In contrast, multi-speed relaxation allows for slower speed sharp turns, thus producing more direct paths with shorter maneuvers and smaller travel times. Furthermore, the inability of the Dubins model to reduce speed could result in fast maneuvers near obstacles, thus producing paths with high collision risks. In this regard, GMDM provides the motion planners the ability to jointly optimize time and risk by allowing the change of speed along the path. GMDM is built upon the six Dubins path types considering the change of speed on path segments. It is theoretically established that GMDM provides full reachability of the configuration space for any speed selections. Furthermore, it is shown that the Dubins model is a specific case of GMDM for constant speeds. The solutions of GMDM are analytical and suitable for real-time applications. The performance of GMDM in terms of solution quality (i.e., time/time-risk cost) and computation time is comparatively evaluated against the existing motion models in obstacle-free as well as obstacle-rich environments via extensive Monte Carlo simulations. The results show that in obstacle-free environments, GMDM produces near time-optimal paths with significantly lower travel times than the Dubins model while having similar computation times. In obstacle-rich environments, GMDM produces time-risk optimized paths with substantially lower collision risks.

Figures (12)

• Figure 1: Comparison of GMDM with the Dubins paths.
• Figure 2: Motion primitives of GMDM.
• Figure 3: GMDM paths with different control inputs on each segment.
• Figure 4: Visualization of the reachable (white) and unreachable (colored) sets from $\textbf{p}_0=(0,0,0)$ for the (a) CSC and (b) CCC paths. The top row shows the reachability in the SE(2) space. The bottom three rows show the cross-sections of $x_f$-$y_f$ planes for different $\theta_f$. The plots are drawn for the control inputs $(v_1,v_2,v_3)=(0.1,0.5,1)$ m/s and $|\omega_i|\in\{0,1\}$ rad/s, which correspond to $(|r_1|,|r_3|)=(0.1,1.0)$ m for all paths and $|r_2|=0.5$ m for the CCC paths.
• Figure 5: Visualization of the unreachable regions of LSL and RSR path types. Since these regions are disjoint, full reachability is achieved by GMDM as per Theorem \ref{['Th:GMDM_full_reachability']}. The plots are drawn for $\textbf{p}_0=(0,0,0)$ and $(|r_1|, |r_3|) = (0.1,1.0)$.

Theorems & Definitions (38)

• Definition 3.1: Forward problem
• Proposition 3.1: CSC forward
• proof
• Proposition 3.2: CCC forward
• proof
• Definition 3.2: Inverse problem
• Proposition 3.3: CSC inverse
• proof
• Proposition 3.4: CCC inverse
• proof