论文标题
圆锥形框架无限刚性
Conic Frameworks Infinitesimal Rigidity
论文作者
论文摘要
本文介绍了称为圆锥形框架及其刚性的新结构。它们由代理和一对代理之间的一组定向约束组成。当结构在保留约束时无法弯曲时，据说它是刚性的。如果仅将光滑的变形视为刚性的充分条件，则称为无限刚性。在Conic Frameworks中，每个代理$ u $具有空间位置$ x_u $，而时钟偏移为偏见$β_U$表示。如果从框架中，从代理$ u $到代理$ w $的约束是从$ u $到$ w $的伪范围，将定义为$ {\ left \ lvert {x_u -x_w} \ x_w} \ right \ rvert} +β_w -β_w -β_U$。使用期限方法测量试验间距离时，会出现伪范围。本文完全表征了锥体框架的无限刚性，其代理处于一般位置。引入了两个特征：一个用于一维框架，另一个用于多维框架。他们都依靠约束图，并在空间和偏置变量之间使用脱钩。在多维情况下，这种新的圆锥形范式急剧减少了维持相对于经典的双向范围方法所需的最小限制数量。
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.
