Robert Connelly is a Professor of Mathematics at Cornell University. He received his Ph.D. in 1969 from University of Michigan. His research areas are discrete geometry, computational geometry and the rigidity of discrete structures.
In his 2009 paper "Tensegrities and Global Rigidity" Connelly proposed a model that describes tensegrity stability in the constructions of Snelson, Fuller and others. His model relies on a stress-energy function, that is really a quadratic equation. The paper outlined a self-contained elementary set of principles relying on the properties of the stress matrix. he then applies the properties of the stress matrix to generic configurations of bar tensegrities (usually called bar frameworks).
Working with Robert Terrell, Connelly maintains an online catalog of several hundred tensegrity structures , "Highly Symmetric Tensegrity Structures"
Mathematical definition of a tensegrity
The mathematical definition of a tensegrity, according to Connelly and Terrell, is a finite configuration of points, the nodes, in space or the plane where some pairs of the nodes are designated cables, constrained not to get further apart, and some pairs are designated struts, constrained not to get closer together. Note with this definition cables and struts are allowed to intersect and cross.It is a finite configuration of points, the nodes, in space or the plane where some pairs of the nodes are designated cables, constrained not to get further apart, and some pairs are designated struts, constrained not to get closer together. Note with this definition cables and struts are allowed to intersect and cross. See http://www.math.cornell.edu/~tens/faq.html
Comparison of loose packed particles with tensegrity
Connelly asserts that if you have a jammed packing of spherical, frictionless particles, in any sort of reasonable container, the number of contacts must at least match the number of free variables. This is what he calls the "canonical push". "But this argument fails when the particals are frictionless but not round, and indeed for all but the most well-ordered packings, the number of contacts is significantly less than the number of free variables. This is called a hypostatic configuration, since it is not statically (or infinitesimally) rigid, behaving like an underconstrained tensegrity that is prestress stable." His paper with Aleks Donev, Sal Torquato, and Frank Stillinger appears here .
A Selection of Articles by Connelly
Below, a partial list of articles by Connelly.
Mathematics and Tensegrity
Mathematics and Tensegrity by Robert Connelly and Allen Back
Tensegrities and Global Rigidity
Globally Rigid Symmetric Tensegrities
Infinitesimally Locked Self-Touching Linkages
Tensegrity Structures Why Are They Stable
Tensegrlty Structures: Why Are They Stable? June 14, 1998, A particular definition of stability for tensegrity structures is presented, super stability. This is a stronger case of prestress stability that applies to many examples of tensegrities found in nature. 1. Introduction: The Basic Object. A basic question in dealing with tensegrity structures is: What are they? Here several definitions are briefly described.
Globally Rigid Symmetric Tensegrities
The authors address the question, "if one builds a tensegrity structure with cables and struts, when will it be globally rigid in the sense that there is no other non-congruent configuration of the points satisfying the cable and strut constraints?" They report on a family of such structures that have dihedral symmetry and are globally rigid. To define the structures the authors apply some stress-energy techniques that in turn require showing that a certain symmetric matrix is positive semi-definite with the right rank. The work proceeds with three three constraints regarding distance defined on the struts, cables and bars. For any cable, its end points or vertices are at maximum distance, meaning they cannot move further apart. For any strut, its end points are at minimum distance, meaning that they cannot get closer together. Lastly, they define a bar as a pair of vertices that is constrained as-is, meaning that their distance apart cannot vary. A graph notation is used whose vertices correspond to the joints of the framework and whose edges record which pairs of vertices are cables, bars, or struts. The authors then pose and prove a series of theorems and share several conjectures.
Packings of Circles and Spheres
Packings of circles and spheres, Lectures III and IV, Session on Granular Matter Institut Henri Poincaré, R. Connelly, Cornell University Department of mathematics
Second Order Rigidity and Pre-stress Stability
Infinitesimally Locked Self-Touching Linkages With Applications to Locked Trees
Abstract. Recently there has been much interest in linkages (bar-and-joint frameworks) that are locked or stuck in the sense that they cannot be moved into some other conﬁguration while preserving the bar lengths and not crossing any bars. We propose a new algorithmic approach for analyzing whether planar linkages are locked in...
Connelly worked on the polygon unwrapping problem with Eric Demaine and Gunther Rote. Connelly found that the problem resonates with tensegrity, as the transformation of 'unwrapping' resembles the activity of a strut (or bar length) in a tensegrity. The abstract: Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the well-studied carpenter's rule conjecture.
Links and References
Online Catalog of Highly Symmetric Tensegrity Structures, https://robertconnelly.github.io/symmetric-tensegrity/ Cornell math department page: http://www.math.cornell.edu/People/Faculty/connelly.html