On the reconstructibility of frameworks from self-stresses

Let $\mathfrak P$ be the set of all frameworks in $\mathbb R^d$ consisting of rods connected by universal hinges with a given junction scheme and with given point of fastening of some hinges. The problem is to find conditions for a framework $\mathbf p\in\mathfrak P$ to be reconstructible from the space $W(\mathbf p)$ of its self-stresses. In other words, under what conditions is $\mathbf p$ the unique framework in $\mathfrak P$ with the given space $W(\mathbf p)$ of self-stresses? A complete answer to this question is obtained only for frameworks in the line. We also investigate geometric properties of the image of the rigidity map which are related to the study of frameworks admitting self-stresses.