AI 中文总结
该研究针对Purcell三连杆微游泳体，揭示其奇弹性仅影响动作能量、不改变控制几何，证明其全局可控且幂零模型为Cartan结构，模拟还发现其在各向同性阻力下可纯旋转。
AI 中文摘要
我们研究一种Purcell三连杆微游泳体，其关节具有“奇弹性”：扭转刚度为非厄米矩阵，其反对称部分$k_o$会注入机械功，使得内部弹性漂移为非保守的。我们的主要发现是几何与代价之间存在明确的分离——奇弹性对游泳体的控制几何是不可见的，仅在动作的能量中可见。该机制源于一个代数事实：漂移位于两个控制向量场的张成空间中，因此该系统与无漂移系统是反馈等价的，且奇模量仅通过标量$\boldsymbol{K}$的行列式$\text{det}\boldsymbol{K}=k^2+k_o^2$进入括号结构。由此我们推导出能量问题的异常极值不随$k_o$变化，游泳体对每一种非互易性都具有全局可控性，不存在阈值，且其幂零模型是Cartan(2,3,5)次黎曼结构，仅通过度量缩放$g_\boldsymbol{\text{χ}}=(1+\boldsymbol{\text{χ}}^2)g_0$发生变形。奇模量仅作用于代价：将最优转向问题转化为次芬斯勒（Randers）形式，我们证明对于$k_o$的任意符号，规定的重新定向代价都严格更低。完整的阻力理论模拟验证了该分析，并揭示在各向同性阻力下，该游泳体变为纯旋转体，仅旋转而不平移。
英文摘要
We study a Purcell three-link microswimmer whose joints are \emph{odd-elastic}: the torsional stiffness is non-Hermitian, its antisymmetric part $k_o$ injecting mechanical work so that the internal elastic drift is non-conservative. Our main finding is a sharp separation between geometry and cost --- odd elasticity is invisible to the control geometry of the swimmer and visible only in the energy of a manoeuvre. The mechanism is a single algebraic fact: the drift lies in the span of the two control vector fields, so the system is feedback-equivalent to a driftless one and the odd modulus enters the bracket structure only through the scalar $\det\textbf{K}=k^2+k_o^2$. From this we deduce that the abnormal extremals of the energy problem are unchanged by $k_o$, that the swimmer is globally controllable for every non-reciprocity with no threshold, and that its nilpotent model is the Cartan $(2,3,5)$ sub-Riemannian structure, deformed only by the metric scaling $g_χ=(1+χ^2)g_0$. The odd modulus acts solely on the cost: casting the optimal-steering problem in sub-Finsler (Randers) form, we prove that a prescribed reorientation is strictly cheaper for either sign of $k_o$. Full Resistive-Force-Theory simulations confirm the analysis and reveal that at isotropic drag the swimmer becomes a pure rotator, turning without translating.
Comments26 pages, 4 figures
