Grigoriy Blekherman

Grigoriy Blekherman, Mario Kummer, Cordian Riener et al.

A quadrature rule of a measure $μ$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $μ$ for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric determinantal formulas for this polynomial, which translate the problem of finding the nodes to solving a generalized eigenvalue problem.

Hangjun Liu, Jiarui Geng, Jinxuan Ding et al.

Centipede-like robots offer unique locomotion advantages due to their small cross-sectional area for accessing confined spaces, and their redundant legs enhance robustness in cluttered environments such as search-and-rescue and pipe inspection. However, elongated robots are particularly vulnerable to tipping over when climbing large obstacles, making reliable self-righting essential for field deployment. Self-righting strategies for elongate, multi-legged systems remain poorly understood. In this study, we conduct a comparative biomechanics and robophysical investigation to address three key questions: (1) What self-righting strategies are effective for elongate, many-legged systems? (2) How should these strategies depend on morphological parameters such as leg length and leg number? (3) Is there a morphological limit beyond which reliable self-righting becomes infeasible? We compare two biological exemplars: Scolopendra subspinipes (short legs) and Scutigera coleoptrata (house centipedes with long legs). Scolopendra subspinipes reliably self-rights both during aerial phases and through ground-assisted self-righting, whereas house centipedes rely predominantly on aerial reorientation and struggle to generate effective self-righting torques during ground contact. Motivated by these observations, we construct a parameterized space of bio-inspired self-righting strategies and develop an elongate robot with adjustable leg lengths. Systematic experiments reveal that increasing leg length necessitates a shift in control strategy to prevent torque over-concentration in mid-body actuators, and we identify a critical limb-length threshold above which robust self-righting becomes challenging. These results establish morphology-strategy coupling principles for self-righting in elongate robots and provide design guidelines for centipede-like systems operating in uncertain terrain.