3连杆蛇形机器人的调和曲率
The harmonic curvature of 3-link snake robots
- Masaryk University（马萨里克大学）
- University of Hradec Králové（赫拉德茨-克拉洛韦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究3连杆蛇形机器人的(2,3,5)几何，通过简化外微分计算得到显式二元四次不变量，证明其不消失，从而否定局部平坦模型的存在。
AI中文摘要：
3连杆蛇形机器人是一个非完整力学系统的例子，其具有在5维构型空间中的秩2分布。它是(2,3,5)几何之一，因此，它允许通过类型为(G_2,P)的抛物几何来描述。(2,3,5)几何的另一个例子多年前已被充分研究，已知对于无滑动且无扭转地相互滚动的球，如果球半径之比为1:3，则它在(G_2,P)抛物几何意义下局部同构于平坦模型。为了回答P. Nurowski的问题，我们寻找使得(2,3,5)几何局部平坦的3连杆蛇形机器人的参数。我们扩展了先前一篇论文的观察，即蛇形机器人的分布包含生成有限维李代数的基。我们利用这一观察来简化机器人几何的外微分计算。这使我们能够实施有效的规范化程序，并获得机器人的显式二元四次不变量。最后，我们证明该不变量对于任何参数都不为零。因此，对于这些类型的蛇形机器人，无法实现局部平坦模型。
英文摘要：
The $3$-link snake robot is an example of a non-holonomic mechanical system with rank $2$ distribution in a $5$-dimensional configuration space. It is one of $(2,3,5)$-geometries, and as such, it admits a description by a parabolic geometry of type $(G_2,P)$. Another example of $(2,3,5)$-geometry was well-studied years ago, and it is known that for balls rolling one over the other without slipping or twisting, if the ratio of ball radii is $1:3$, then it is locally isomorphic to the flat model in sense of $(G_2,P)$ parabolic geometries. Answering a question by P. Nurowski, we are looking for parameters of the $3$-link snake robots yielding a locally flat $(2,3,5)$-geometry. We extend the observation of a previous paper that the distributions of the snake robots contain bases generating finite dimensional Lie algebras. We exploit this observation to simplify the exterior calculus of the robots' geometry. This allows us to implement an effective normalization procedure and we obtain an explicit binary quartic invariant of the robot. Finally, we show that it does not vanish for any of the parameters. Therefore, the locally flat model cannot be achieved for these types of snake robots.
