Differentiable Collision-Free Parametric Corridors

TL;DR

The paper addresses the challenge of generating collision-free corridors that are differentiable with respect to a path-parameter for path-parametric navigation. It introduces a path-parametric corridor defined by a rotating off-centered ellipse whose axes and center evolve via Chebyshev polynomials, and casts corridor volume maximization as a convex optimization solvable in real time: a 2D LP and a 3D SDP (or an LP via diagonal-dominance). A sequence of convexification steps—including discretization, replacing det by trace, a bounding wrapper, and a diagonal-dominance reformulation—yields tractable solutions, achieving real-time performance (roughly 5–20 Hz). Real-world KITTI data and toy simulations validate that the method attains comparable volumetric capacity to convex decomposition but with a continuous, differentiable representation that integrates easily with learning and optimization pipelines.

Abstract

This paper presents a method to compute differentiable collision-free parametric corridors. In contrast to existing solutions that decompose the obstacle-free space into multiple convex sets, the continuous corridors computed by our method are smooth and differentiable, making them compatible with existing numerical techniques for learning and optimization. To achieve this, we represent the collision-free corridors as a path-parametric off-centered ellipse with a polynomial basis. We show that the problem of maximizing the volume of such corridors is convex, and can be efficiently solved. To assess the effectiveness of the proposed method, we examine its performance in a synthetic case study and subsequently evaluate its applicability in a real-world scenario from the KITTI dataset.

Figures (6)

• Figure 1: Given a reference path $\Gamma$ paramaterized by path-parameter $\xi$ and a point cloud $\mathrm{P}$, we present a method that computes differentiable and smooth parametric corridors $\mathcal{C}$. The reference path $\Gamma$ is depicted by the blue line with a moving frame, the colored dots refer to the point cloud $\mathrm{P}$ and the corridor $\mathcal{C}$ is given in magenta. To generate the corridor, we optimize over a parametric ellipse $\{\mathrm{E}(\xi),\mathbf{d}(\xi)\,|\,\xi\in[\xi_0,\xi_f]\}$, where besides searching for its dimensions, we also compute the offset from the reference $\Gamma$. In other words, the center of the ellipse, shown in red, is allowed to deviate from the reference $\Gamma$. Our method can run real-time and is applicable to both, planar (2D) and spatial (3D) cases.
• Figure 2: Projection of a point $\bm{p}_i$, represented by the pink dot, from its Euclidean coordinates onto the reference path $\Gamma$ which is parameterized by $\xi$ and has a moving frame $\mathrm{R}_\Gamma(\xi)$ attached to it. The orthogonal plane $\perp_\Gamma$ defined by the second (green) and third (blue) components of the frame is depicted in gray. The coordinates resulting from the projection are decoupled into a tangent $\bm{p}_{\parallel,i}$ and an orthogonal $\bm{p}_{\perp,i}$ component.
• Figure 3: Area maximization for different cross section representations centered in the red cross within the transverse point cloud $\mathrm{P}_{\perp}$ represented by the black dots. The depicted cross sections are: a circle with $1$ degree of freedom (red), an ellipse with $3$ (green) and an offset-ed ellipse with a total $5$ (blue). The gray area represents the maximum convex polygon for the given point cloud encompassing the center. The offset for the off-centered ellipse is given by the blue dashed line. From this comparison, it is apparent that the higher the number the degrees of freedom, the bigger the area of the cross section.
• Figure 4: 2D and 3D corridors generated by our method for different polynomial degrees: $3$ in blue, $6$ in orange, and $15$ in green. In the planar case (A) the point cloud is depicted in black, while in the spatial case (B) it is colored according to the height. The black dashed line represents the reference path $\Gamma$. The correlation between the corridor's area -- applicable to (A) -- and volume -- applicable to (B) -- with respect to the polynomial degree is plotted in (C). As the degree increases, the respective parametric areas and volumes increase. This trend is confirmed in (A) and (B), where the expressiveness of the sampled corridors augments with the degree. Panel (D) shows the corridor associated with a parametric ellipse centered on reference $\Gamma$, making apparent the benefits of optimizing over the offset. Panel (E) depicts the wrapper that prevents the 3D corridors in (B) and (D) from becoming unbounded.
• Figure 5: Two different 3D corridors for polynomial degree $n=9$ generated by our method in a real-world challenging driving scenario, where the road bifurcates into two narrow lanes, with other cars and cyclists. The case study is part of the KITTI Vision Benchmark Suite Geiger2013IJRR. Panel (A) shows an RGB camera-based overview of the scene, while the respective point cloud and corridors are given in panels (B) --top view-- and (C) --isometric view--. The point cloud is color-coded based on proximity to the camera's location. To facilitate the comprehension of the environment, numbered markers in white (from 1 to 4), associated with relevant features in the scene, have been added to all panels. For a detailed analysis of the corridor circled by the dashed line in Panel (B), please refer to Fig. \ref{['fig:experiment_results_2']}.