Yaoyu Cheng | Searcharxiv

arXiv2026

Transpose Symmetry of Injectivity over Commutative Semirings

Let R be a commutative semiring, not necessarily with a multiplicative identity, and let A be an element of Mn(R). We prove that the map x to Ax on Rn is injective if and only if x to AT x is injective. Equivalently, the left- and right-cancellative elements of the multiplicative semigroup Mn(R) coincide. The proof splits formal determinant expansions into their even and odd halves; it uses no subtraction, additive cancellation, group completion, inverse, or multiplicative identity. As consequences we recover the stable-finiteness theorem for matrices over unital commutative semirings. We also prove that surjectivity is invariant under transpose. In fact, the existence of a surjective square matrix of positive size forces R to have a multiplicative identity.

math.RA↗

arXiv2025

AcL: Action Learner for Fault-Tolerant Quadruped Locomotion Control

Quadrupedal robots can learn versatile locomotion skills but remain vulnerable when one or more joints lose power. In contrast, dogs and cats can adopt limping gaits when injured, demonstrating their remarkable ability to adapt to physical conditions. Inspired by such adaptability, this paper presents Action Learner (AcL), a novel teacher-student reinforcement learning framework that enables quadrupeds to autonomously adapt their gait for stable walking under multiple joint faults. Unlike conventional teacher-student approaches that enforce strict imitation, AcL leverages teacher policies to generate style rewards, guiding the student policy without requiring precise replication. We train multiple teacher policies, each corresponding to a different fault condition, and subsequently distill them into a single student policy with an encoder-decoder architecture. While prior works primarily address single-joint faults, AcL enables quadrupeds to walk with up to four faulty joints across one or two legs, autonomously switching between different limping gaits when faults occur. We validate AcL on a real Go2 quadruped robot under single- and double-joint faults, demonstrating fault-tolerant, stable walking, smooth gait transitions between normal and lamb gaits, and robustness against external disturbances.

cs.RO↗