圆锥框架的无穷小刚度
Conic Frameworks Infinitesimal Rigidity
摘要：
本文介绍了一种新的结构--圆锥框架及其刚度。它们由代理和代理对之间的一组有向约束组成。当结构在保持约束的同时不能弯曲时，它被称为刚性的。如果只考虑光滑变形，则刚性的充分条件称为无穷小刚性。在圆锥框架中，每个代理$u$具有空间位置$x_u$和由偏置$β_u$表示的时钟偏移。如果从代理$u$到代理$w$的约束在框架中，则设置从$u$到$w$的伪范围（定义为${\left\lVert{x_u - x_w}\right\rVert} + β_w - β_u$）。使用到达时间方法测量座席间距离时会出现伪距。本文完整地刻画了主体处于一般位置的锥框架的无穷小刚性。引入了两个刻画：一个用于一维框架，另一个用于多维框架。它们都依赖于约束图，并使用空间和偏差变量之间的解耦。在多维的情况下，这种新的圆锥范例大幅减少了所需的最小数量的约束，以保持相对于经典的双向测距方法的形成。
英文摘要：
This paper introduces new structures called conic frameworks and their rigidity. They are composed by agents and a set of directed constraints between pairs of agents. When the structure cannot be flexed while preserving the constraints, it is said to be rigid. If only smooth deformations are considered a sufficient condition for rigidity is called infinitesimal rigidity. In conic frameworks, each agent $u$ has a spatial position $x_u$ and a clock offset represented by a bias $β_u$. If the constraint from Agent $u$ to Agent $w$ is in the framework, the pseudo-range from $u$ to $w$, defined as ${\left\lVert{x_u - x_w}\right\rVert} + β_w - β_u$, is set. Pseudo-ranges appear when measuring inter-agent distances using a Time-of-Arrival method. This paper completely characterizes infinitesimal rigidity of conic frameworks whose agents are in general position. Two characterizations are introduced: one for unidimensional frameworks, the other for multidimensional frameworks. They both rely on the graph of constraints and use a decoupling between space and bias variables. In multidimensional cases, this new conic paradigm sharply reduces the minimal number of constraints required to maintain a formation with respect to classical Two-Way Ranging methods.
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