Geometric Phase Self-Stability and Zero-Velocity Topological Pinning in Nonholonomic Monorail Systems via Artificial Gauge Field Injection

Abstract The self-stability of riderless bicycles and the stabilization of monorail robots at zero velocity remain profound challenges in nonholonomic mechanics and control theory. Classical explanations relying on gyroscopic precession and caster trail fail to capture the fundamental geometric nature of the system, while zero-velocity stabilization traditionally requires bulky control moment gyroscopes (CMGs) and high-frequency brute-force actuation. Here, utilizing the Unified Vacuum Medium Model (UVMM), we present a rigorous topological framework for nonholonomic dynamics. We mathematically prove that the pure-rolling constraints define a non-trivial fiber bundle whose curvature acts as an Abelian gauge field. The natural self-stability at forward velocities is rigorously derived as a macroscopic manifestation of the geometric phase (Berry/Hannay phase), with the stable speed range topologically protected by a non-zero winding number in the complex eigenvalue plane. To conquer the zero-velocity topological collapse, we propose anArtificial Gauge Field Injection via an internal orthogonal momentum wheel. This generates a pure anti-symmetric topological damping matrix that induces a topological repulsion in the phase space, strictly pinning all Jacobian eigenvalues in the left-half plane. This work transitions nonholonomic control from empirical force-balance to exact topological gauge theory, fundamentally subverting traditional analytical mechanics and modern nonlinear control paradigms. 摘要 无骑手自行车自稳定、单轨机器人零速稳态控制,是非完整力学与非线性控制领域的核心难题。传统理论依托陀螺进动、前叉尾迹效应解释自行车自稳定,未能触及系统底层几何本质;工程领域实现零速稳定,普遍依赖笨重的控制力矩陀螺(CMG)与高频强制驱动,存在功耗高、载荷大、鲁棒性差的固有缺陷。本文基于统一真空介质模型(UVMM),构建完备的非完整动力学拓扑规范理论框架。 通过严格数学推导证明:单轨系统纯滚动约束可构建非平庸纤维丛结构,其内在曲率等效为阿贝尔规范场;系统运动过程中产生的几何相位(Berry/Hannay相位)是前进速度下自然自稳定的核心物理机制,其稳定速度区间由复特征值平面非零拓扑缠绕数全域保护。针对零速工况下系统拓扑崩溃难题,本文提出基于内置正交动量轮的人工规范场注入方法,可生成纯反对称拓扑阻尼矩阵,在相空间诱发拓扑排斥效应,将系统雅可比特征值严格钉扎于复平面左半区域。本研究彻底摒弃传统经验化力平衡控制逻辑,将非完整系统控制升级为精准的拓扑规范场理论,实现了对经典分析力学与现代控制理论的维度革新。 Keywords: Nonholonomic System; Monorail Robot; Geometric Phase; Artificial Gauge Field; Topological Eigenvalue Pinning; UVMM Theory; Nonlinear Control 关键词: 非完整系统;单轨机器人;几何相位;人工规范场;特征值拓扑钉扎;UVMM理论;非线性控制