Spatial motion planning with Pythagorean Hodograph curves (2209.01673v1)

Abstract: This paper presents a two-stage prediction-based control scheme for embedding the environment's geometric properties into a collision-free Pythagorean Hodograph spline, and subsequently finding the optimal path within the parameterized free space. The ingredients of this approach are twofold: First, we present a novel spatial path parameterization applicable to any arbitrary curve without prior assumptions in its adapted frame. Second, we identify the appropriateness of Pythagorean Hodograph curves for a compact and continuous definition of the path-parametric functions required by the presented spatial model. This dual-stage formulation results in a motion planning approach, where the geometric properties of the environment arise as states of the prediction model. Thus, the presented method is attractive for motion planning in dense environments. The efficacy of the approach is evaluated according to an illustrative example.

• The paper introduces a novel two-stage framework that embeds PH curves to generate collision-free, optimized paths using spatial path parameterization.
• It employs a robust spatial parameterization and RMF approximation to overcome the limitations of traditional Frenet-Serret framing in 3D trajectories.
• Simulations demonstrate efficient performance with planning stages running at 5-10 Hz and 250 Hz, ensuring dynamic obstacle avoidance in complex environments.

Overview of "Spatial Motion Planning with Pythagorean Hodograph Curves"

The paper "Spatial Motion Planning with Pythagorean Hodograph Curves" by Jon Arrizabalaga and Markus Ryll addresses the challenges associated with motion planning in cluttered and dynamic environments. The authors introduce a two-stage, prediction-based control scheme designed to embed the geometric attributes of the environment into a collision-free Pythagorean Hodograph spline, ultimately achieving an optimal path in a parameterized free space. This approach integrates spatial path parameterization applicable to any arbitrary curve and identifies the suitability of Pythagorean Hodograph (PH) curves for compactly and continuously defining path-parametric functions utilized by the presented spatial model.

Contributions and Methodology

1. Spatial Path-Parameterization: The authors develop a novel spatial path parameterization for three-dimensional Euclidean coordinates. This parameterization is not constrained to the traditional Frenet-Serret frame, enabling robust handling of complex trajectories, including those that incorporate multiple directions of curvature.
2. Pythagorean Hodograph Curves: PH curves form the backbone of this method. Unlike traditional arcs or splines, PH curves ensure that the parametric speed is polynomial and the curve can be continuously defined with a small number of parameters. Additionally, these curves provide smooth transition without singularities, making them suitable for densely packed or cluttered paths.
3. RMF Approximation: Recognizing the downfalls of the Frenet-Serret frame approach, the researchers utilize an adapted frame known as the Rotation Minimizing Frame (RMF), which reduces unnecessary rotations, ensuring that the second and third components of the frame remain stable and consistent throughout motion planning.
4. Two-Stage Motion Planning: The authors divide the motion planning task into two stages. In the first stage, a collision-free spline of PH curves is computed using environments' spatial constraints. The second stage leverages this spline in a Nonlinear Model Predictive Control (NMPC) scheme to plan motion dynamically.

Strong Numerical Results

The paper's strong numerical results are highlighted by the spline of PH nonic curves evaluated in simulations. The framework demonstrated capability in constructing highly non-convex path trajectories with efficient computational performance, running stage 1 at 5-10 Hz and stage 2 at 250 Hz on a typical notebook processor. Notably, the RMF approximation cost was minimized to a negligible magnitude, indicating a successful avoidance of unnecessary frame rotations, while effectively planning through non-convex corridors.

Theoretical and Practical Implications

Theoretically, this research offers a novel approach to integrating spatial features into motion planning without constraining to path-specific frames, which traditionally led to planning inefficiencies and inaccuracies. Practically, the control framework is substantial for autonomous navigation systems, specifically in environments that are dynamic, crowded, or intricate. The two-stage hierarchical approach efficiently utilizes environment topography to generate collision-free, optimized paths, making it suitable for autonomous drones, urban vehicle navigation, and indoor robotics systems.

Future Developments

The research opens avenues for further exploration of RMFs in spatial path-planning and could lead to the enhancement of autonomous system navigation capabilities in real-world environments. Future developments may focus on improving real-time computational efficiency using advanced optimization techniques or expanding the application to include human-robot interaction dynamics.

In summary, the presented motion planning framework utilizing PH curves offers a robust means of embedding environmental geometry into the motion planning process, leading to enhanced collision avoidance and path optimization in complex environments. The paper's results indicate a significant advancement in both the theoretical underpinnings and practical application of spatial motion planning technologies.