Shape abstraction, simplifying shape representation into a set of primitives, is a fundamental topic in computer vision. The choice of primitives shapes the structure of world understanding, yet achieving both high abstraction accuracy and versatility remains challenging. In this paper, we introduce a novel framework for shape abstraction utilizing a differentiable support function (DSF), which offers unique advantages in representing a wide range of convex shapes with fewer parameters, providing smooth surface approximation and enabling differentiable contact features (gap, point, normal) essential for downstream applications involving contact-related problems. To tackle the associated optimization and combinatorial challenges, we introduce two techniques: differentiable shape parameterization and hyperplane-based marching to enhance accuracy and reduce DSF requirements. We validate our method through experiments demonstrating superior accuracy and efficiency, and showcase its applicability in tasks requiring differentiable contact information.

This paper presents a framework designed to tackle a range of planning problems arise in manipulation, which typically involve complex geometric-physical reasoning related to contact and dynamic constraints. We introduce the Contact Factor Graph (CFG) to graphically model these diverse factors, enabling us to perform inference on the graphs to approximate the distribution and sample appropriate solutions. We propose a novel approach that can incorporate various phenomena of contact manipulation as differentiable factors, and develop an efficient inference algorithm for CFG that leverages this differentiability along with the conditional probabilities arising from the structured nature of contact. Our results demonstrate the capability of our framework in generating viable samples and approximating posterior distributions for various manipulation scenarios.

The multi-contact nonlinear complementarity problem (NCP) is a naturally arising challenge in robotic simulations. Achieving high performance in terms of both accuracy and efficiency remains a significant challenge, particularly in scenarios involving intensive contacts and stiff interactions. In this article, we introduce a new class of multi-contact NCP solvers based on the theory of the Augmented Lagrangian (AL). We detail how the standard derivation of AL in convex optimization can be adapted to handle multi-contact NCP through the iteration of surrogate problem solutions and the subsequent update of primal-dual variables. Specifically, we present two tailored variations of AL for robotic simulations: the Cascaded Newton-based Augmented Lagrangian (CANAL) and the Subsystem-based Alternating Direction Method of Multipliers (SubADMM). We demonstrate how CANAL can manage multi-contact NCP in an accurate and robust manner, while SubADMM offers superior computational speed, scalability, and parallelizability for high degrees-of-freedom multibody systems with numerous contacts. Our results showcase the effectiveness of the proposed solver framework, illustrating its advantages in various robotic manipulation scenarios.

Collision detection is a fundamental problem across various fields such as robotics, physical simulation, and computer graphics. While numerous studies have provided efficient solutions, based on the well-known Gilbert, Johnson, and Keerthi (GJK) algorithm and Expanding Polytope Algorithm (EPA), existing methods utilizing GJK-EPA often struggle with smooth strictly convex shapes like ellipsoids. This paper proposes a novel approach to the collision detection problem converting it to a problem compatible with an unconstrained Riemannian optimization problem. Moreover, we presents a specific method of solving the problem based on twice differentiable support functions and the Riemannian trust region (RTR) method. The method exhibits fast and robust convergence rate, leveraging the well-established theory of Riemannian optimization. The evaluation studies comparing our method to GJK-EPA method are done with pre-defined primitive shapes. Additionally, a test result with several more complex shapes is demonstrated exhibiting the method’s effectiveness and applicability.