Supertoroid fitting of objects with holes for robotic grasping and scene generation

TL;DR

The paper extends geometric object fitting from simple primitives to supertoroids to capture holes in objects, addressing a key limitation of classic superquadrics. It introduces the meridian radial distance as a fast, interpretable distance for fitting and derives a 12-parameter, both intrinsic and extrinsic, model for shape and pose, accompanied by a practical differential-geometric framework. The method handles partial, noisy point clouds from single views and enables grasping-oriented analyses by providing surface normals, tangents, and curvature-informed heuristics. The approach is validated on real and synthetic data, showing fast convergence and robust fits, with code available for broader use in robotic grasping and scene generation.

Abstract

One of the strategies to detect the pose and shape of unknown objects is their geometric modeling, consisting on fitting known geometric entities. Classical geometric modeling fits simple shapes such as spheres or cylinders, but often those don't cover the variety of shapes that can be encountered. For those situations, one solution is the use of superquadrics, which can adapt to a wider variety of shapes. One of the limitations of superquadrics is that they cannot model objects with holes, such as those with handles. This work aims to fit supersurfaces of degree four, in particular supertoroids, to objects with a single hole. Following the results of superquadrics, simple expressions for the major and minor radial distances are derived, which lead to the fitting of the intrinsic and extrinsic parameters of the supertoroid. The differential geometry of the surface is also studied as a function of these parameters. The result is a supergeometric modeling that can be used for symmetric objects with and without holes with a simple distance function for the fitting. The proposed algorithm expands considerably the amount of shapes that can be targeted for geometric modeling.

Figures (14)

• Figure 1: The angle $\omega$ on the $x-y$ plane and the angle $\eta$ for each cross-section.
• Figure 2: Different supertoroid shapes obtained with same $a_i$ values and increasing numerical values for the exponents $\epsilon_i$, from 0.5 to 2.5.
• Figure 3: The mean superellipse can be seen as the mean value of the section of the supertoroid with the plane $z=0$.
• Figure 4: The vector to a point $p$ is the sum of vector $R_\pi$ (intersection of the mean superellipse with the projection of $p$ on plane $x-y$) and vector $p_R$
• Figure 5: The vector $p_s$ of the closest point on the surface, according to the major and minor radial distance coefficients.