Stable Walking for Bipedal Locomotion under Foot-Slip via Virtual Nonholonomic Constraints

Foot slip is a major source of instability in bipedal locomotion on low-friction or uncertain terrain. Standard control approaches typically assume no-slip contact and therefore degrade when slip occurs. We propose a control framework that explicitly incorporates slip into the locomotion model through virtual nonholonomic constraints, which regulate the tangential stance-foot velocity while remaining compatible with the virtual holonomic constraints used to generate the walking gait. The resulting closed-loop system is formulated as a hybrid dynamical system with continuous swing dynamics and discrete impact events. A nonlinear feedback law enforces both classes of constraints and yields a slip-compatible hybrid zero dynamics manifold for the reduced-order locomotion dynamics. Stability of periodic walking gaits is characterized through the associated Poincaré map, and numerical results illustrate stabilization under slip conditions.

The Asymptotic Behavior of Attention in Transformers

A key component of transformers is the attention mechanism orchestrating how each token influences the propagation of every other token through a transformer. In this paper we provide a rigorous, mathematical analysis of the asymptotic properties of attention in transformers. Although we present several results based on different assumptions, all of them point to the same conclusion, all tokens asymptotically converge to each other, a phenomenon that has been empirically reported in the literature. Our findings are carefully compared with existing theoretical results and illustrated by simulations and experimental studies using the GPT-2 model.