Fast Iterative Region Inflation for Computing Large 2-D/3-D Convex Regions of Obstacle-Free Space

TL;DR

FIRI tackles the problem of efficiently generating large obstacle-free convex polytopes that contain a seed while excluding obstacles. It combines Restrictive Inflation (RsI) to guarantee manageability with a monotone Maximum Volume Inscribed Ellipsoid (MVIE) pipeline, supported by specialized solvers: SDMN for small-dimensional, massively constrained minimum-norm problems, a SOCP reformulation with Affine Scaling for MVIE, and a linear-time analytic 2-D MVIE algorithm. The approach yields substantial speedups over SDP-based IRIS and related methods, while delivering higher-quality polytopes and robust seed containment validated across 2-D and 3-D real-world scenarios, including dense corridors and cluttered environments. The work provides practical tools for fast, reliable region generation in robotics, enabling safer, more flexible trajectory and whole-body planning with real-time applicability. Overall, FIRI advances convex-region generation by achieving high quality, efficiency, and seed manageability in a unified framework with strong theoretical and empirical support.

Abstract

Convex polytopes have compact representations and exhibit convexity, which makes them suitable for abstracting obstacle-free spaces from various environments. Existing generation methods struggle with balancing high-quality output and efficiency. Moreover, another crucial requirement for convex polytopes to accurately contain certain seed point sets, such as a robot or a front-end path, is proposed in various tasks, which we refer to as manageability. In this paper, we propose Fast Iterative Regional Inflation (FIRI) to generate high-quality convex polytope while ensuring efficiency and manageability simultaneously. FIRI consists of two iteratively executed submodules: Restrictive Inflation (RsI) and Maximum Volume Inscribed Ellipsoid (MVIE) computation. By explicitly incorporating constraints that include the seed point set, RsI guarantees manageability. Meanwhile, iterative MVIE optimization ensures high-quality result through monotonic volume bound improvement.In terms of efficiency, we design methods tailored to the low-dimensional and multi-constrained nature of both modules, resulting in orders of magnitude improvement compared to generic solvers. Notably, in 2-D MVIE, we present the first linear-complexity analytical algorithm for maximum area inscribed ellipse, further enhancing the performance in 2-D cases. Extensive benchmarks conducted against state-of-the-art methods validate the superior performance of FIRI in terms of quality, manageability, and efficiency. Furthermore, various real-world applications showcase the generality and practicality of FIRI.

Figures (13)

• Figure 1: Illustration of quality and manageability. The gray polytopes represent obstacles, and the green polytopes represent the generated free polytopes.
• Figure 2: Overview of the computation process of FIRI, corresponding to the iterative modules RsI and MVIE calculation depicted in Algorithm. \ref{['alg:FIRI']}. The left diagram of (b) illustrates a specific example of halfspace computation w.r.t. $\mathcal{O}_i$ in the transformed space, comparing scenarios with and without manageability. ①-⑥ of (b) provide a visualization combined the process in Line \ref{['algl:second_start']}-\ref{['algl:BackTransformHPolytope']}. The increasing size of the inflated ellipsoid in (b) corresponds to the iterative search for the nearest halfspace in Line \ref{['algl:second_start']}. The increasing number of halfspaces corresponds to Line \ref{['algl:IntersectNewHalfspace']}, and the decreasing obstacles correspond to Line \ref{['algl:second_end']}.
• Figure 3: Illustration of a specific instance of Algorithm \ref{['alg:SDMN']} in 2-D for Sec. \ref{['sec::SDMN Algorithm Outline']} and Sec. \ref{['sec::Recursive Problem Construction']}. For Sec. \ref{['sec::SDMN Algorithm Outline']}: (a): Both the inequality constraints $h_1$ and $h_2$ have been checked, which means $\mathcal{I}=\{h_1, h_2\}$. $y_{old}$ is the solution of the 2-D $L_2$-norm minimization under $\mathcal{I}$. (a)$\Rightarrow$(b): When $y_{old}$ does not violate the newly added constraint $h$, $y_{new}=y_{old}$. (a)$\Rightarrow$(c): When $y_{old}$ violates the new constraint $h$, we need to find a new solution $y_{new}$ on the constraint plane corresponding to $h$, implying equation constraint (\ref{['eq:exampleL2_active']}). For Sec. \ref{['sec::Recursive Problem Construction']}: The vector $v$ is normal to the constraint plane, and $H^{\mathrm{T}} e_1$, $H^{\mathrm{T}} e_2$ are a set of orthogonal basis of the 2-D space, where $H^{\mathrm{T}} e_2 \bot v$. (d): We establish a new coordinate for the 1-D subspace on the constraint plane with $v$ as the origin and $H^{\mathrm{T}} e_2$ as the orthogonal basis, transform the checked constraints $h_1, h_2$ to this coordinate as $h'_1, h'_2$. Finally, we transforms (c) into a 1-D $L_2$-norm minimization with only inequality constraints (\ref{['eq:exampleL2_active_n_1']}).
• Figure 4: Illustration of a specific instance of the bottom-up strategy. The large orange box specifies the process described by (\ref{['eq:newMVIE example']}) for calculating the MVIE of $\bar{\mathcal{P}}^4_1$ based on the MVIE of its subsets $\bar{\mathcal{P}}^3_j,1\leq j \leq 4$ and the MENN of itself. The red ellipses indicate MVIE $\mathcal{E}^*$ or MENN $\bar{\mathcal{E}}$, depending on the equation beneath each subset. Using the subset in the lower left corner as an example, the solid blue lines indicate elements in the subset $\bar{\mathcal{P}}^3_1$, the black dashed lines indicate elements in $\bar{\mathcal{P}}^4_1$ that are not included in the subset $\bar{\mathcal{P}}^3_1$. Similarly, the blue box illustrates the process of calculating the MVIE of $\bar{\mathcal{P}}^5$.
• Figure 5: Left: Illustration of the calculation of the MENN of a convex quadrilateral. The blue and red ellipses represent the ellipses obtained by sampling different points $\hat{p}_\lambda$ on the coinciding edge $l_\lambda$. Right: Illustration of the calculation of the MENN of a triangle.

Theorems & Definitions (2)

• proof
• proof