Closed-form Minkowski sums of convex bodies with smooth positively curved boundaries

This paper studies the narrow phase collision detection problem for two general unions of convex bodies encapsulated by smooth surfaces. The approach, namely CFC (Closed-Form Contact space), is based on parameterizing their contact space in closed-form. The first body is dilated to form the contact space while the second is shrunk to a point. Then, the collision detection is formulated as finding the closest point on the parametric contact space with the center of the second body. Numerical solutions are proposed based on the point-to-surface distance as well as the common-normal concept. Furthermore, when the two bodies are moving or under linear deformations, their first time of contact is solved continuously along the timeparameterized trajectories. Benchmark studies are conducted for the proposed algorithms in terms of solution stability and computational cost. Applications of the sampling-based motion planning for robot manipulators are demonstrated.

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“…In the case of a superquadric, [11]. A nice property of a superquadric is the dual relationships between its surface point and surface gradient.…”

Section: Gradient Parameterization Of Smooth Surfaces Enclosing Conve...mentioning

confidence: 99%

“…Recently, the exact contact space between two convex bodies with smooth boundaries has been computed in closed form [11]. The solution provides parametric expressions of the center of one body when touching the other externally.…”

Section: Introductionmentioning

confidence: 99%

Collision Detection for Unions of Convex Bodies With Smooth Boundaries Using Closed-Form Contact Space Parameterization

Ruan

Wang

Chirikjian

2022

IEEE Robot. Autom. Lett.

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“…By replacing the non-negativity constraints in ( 9), (10) with the more restrictive condition that the expressions be SOS polynomials, we obtain a semidefinite program which is readily solved.…”

Section: Definition 6 (Sos-convex) a Polynomial P(x) Is Sosconvex If ...mentioning

confidence: 99%

“…In general, this does not exist as the sets are semialgebraic, involving multiple polynomial (in)equalities. A notable exception is the case of bodies whose boundary surface are smooth and admit both implicit and parametric representations [10], which includes ellipsoids and convex superquadrics [11]. However, many implicit surfaces do not admit a parametric representation and for others obtaining one is an open problem [12].…”

Section: Introductionmentioning

confidence: 99%

Closed-Form Minkowski Sum Approximations for Efficient Optimization-Based Collision Avoidance

Guthrie¹,

Kobilarov²,

Mallada³

2022

Preprint

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“…1), but only encoded by 5 parameters. In the recent few years, superquadrics have raised considerable attentions in the community and are widely applied in robotics and computer vision tasks, e.g., object modeling [20,21], collision detection and motion planning [26,27], pose estimation [4], and grasping [25,33,34]. A single superquadric is already expressive enough to reasonably model many everyday objects [13,28].…”

Section: Introductionmentioning

confidence: 99%

Robust and Accurate Superquadric Recovery: a Probabilistic Approach

Liu¹,

Wu²,

Ruan³

et al. 2021

Preprint

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Interpreting objects with basic geometric primitives has long been studied in computer vision. Among geometric primitives, superquadrics are well known for their simple implicit expressions and capability of representing a wide range of shapes with few parameters. However, as the first and foremost step, recovering superquadrics accurately and robustly from 3D data still remains challenging. The existing methods are subject to local optima and are sensitive to noise and outliers in real-world scenarios, resulting in frequent failure in capturing geometric shapes. In this paper, we propose the first probabilistic method to recover superquadrics from point clouds. Our method builds a Gaussian-uniform mixture model (GUM) on the parametric surface of a superquadric, which explicitly models the generation of outliers and noise. The superquadric recovery is formulated as a Maximum Likelihood Estimation (MLE) problem. We propose an algorithm, Expectation, Maximization, and Switching (EMS), to solve this problem, where: (1) outliers are predicted from the posterior perspective; (2) the superquadric parameter is optimized by the trust-region reflective algorithm; and (3) local optima are avoided by globally searching and switching among parameters encoding similar superquadrics. We show that our method can be extended to the multi-superquadrics recovery for complex objects. The proposed method outperforms the state-of-theart in terms of accuracy, efficiency, and robustness on both synthetic and real-world datasets. Codes will be released.

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Closed-form Minkowski sums of convex bodies with smooth positively curved boundaries

Cited by 19 publications

References 29 publications

Collision Detection for Unions of Convex Bodies With Smooth Boundaries Using Closed-Form Contact Space Parameterization

Collision Detection for Unions of Convex Bodies With Smooth Boundaries Using Closed-Form Contact Space Parameterization

Closed-Form Minkowski Sum Approximations for Efficient Optimization-Based Collision Avoidance

Robust and Accurate Superquadric Recovery: a Probabilistic Approach

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