On the universal rigidity of generic bar frameworks

Let $V={1,ldots,n}$ be a finite set. An $r$-configuration is a mapping $p:V ightarrow R^r$, where $p^1,ldots,p^n$ are not contained in a proper hyper-plane. A framework $G(p)$ in $R^r$ is an $r$-configuration together with a graph $G=(V,E)$ such that every two points corresponding to adjacent vertices of $G$ are constrained to stay the same distance apart. A framework $G(p)$ is said to be generic if all the coordinates of $p^1,ldots, p^n$ are algebraically independent over the integers. A framework $G(p)$ in $R^r$ is said to be unique if there does not exist a framework $G(q)$ in $R^s$, for some $s$, $1 leq s leq n-1$, such that $||q^i-q^j||=||p^i-p^j||$ for all $(i,j) in E$. In this paper we present a sufficient condition for a generic framework $G(p)$ to be unique, and we conjecture that this condition is also necessary. Connections with the closely related problems of global rigidity and dimensional rigidity are also discussed.