Stable Equilibria of Tensegrity Frameworks

Assume we are given a tensegrity framework given by nodes and edges together with an embedding . The edges are partitioned into rigid bars with lengths and elastic cables with corresponding elasticity coefficients and restings lenngths . With Hooke's Law and the introduction of slack variables for the elastic cables, a polynomial system consisting of the polynomials

,

arises. Using the corresponding potential energy function enables us to create the Lagrange multiplier . Its derivative contains the critical points of the energy function subject to the variety generated by the polynomial system . We are interested in its real, local minima. The sufficient condition of local minima is verified by looking at the Lagrange multiplier 's Hessian and checking whether it is positive definite. This process is more thoroughly described in the underlying paper Catastrophe in Elastic Tensegrity Frameworks (Heaton and Timme, 2020).

julia> using Pkg
julia> Pkg.add(url="https://github.com/matthiashimmelmann/TensegrityEquilibria.jl.git")

The main method of this program is stableEquilibria. It is exported in the module functionsForStableEquilibria and can be used from the main module TensegrityEquilibria. The former also exports a test suite start_demo() that is automatically executed when importing the main module. The main method of this package stableEquilibria has the following inputs with corresponding expected format:

• vertices: This array consists of the embedded nodes of the tensegrity framework, given by arrays. If the nodes either contain internal variables or control parameters, they are included here as @var types from the numerical algebraic geometry package HomotopyContinuation. The expected input format in this case is [[p_11, ..., p_1d], ..., [p_m1, ..., p_md]]
• unknownBars: This array consists of the rigid bars of the framework that are not yet determined by the choice of vertices. The expected input format is [[i, j, l_ij], ...] for a bar between the embedded vertices p(i) and p(j).
• unknownCables: This array consists of the elastic cables of the framework that are not yet determined by the choice of vertices. The expected input format is [[i, j, r_ij, e_ij], ...] for a cable between the embedded vertices p(i) and p(j).
• listOfInternalVariables is a flat list of all the internal parameters of the framework (X in the paper).
• listOfControlParameters is a flat list of all the control parameters of the framework (\Omega in the paper).
• targetParams is a flat initial configuration of the variables in listOfControlParameters of the same length. It contains entries of type Float64.
• knownBars is a list of the already determined bars for plotting purposes. The expected input format is [[i,j], ...] for a bar between the embedded vertices and .
• knownCables is a list of the already determined cables for plotting purposes. The expected input format is [[i,j], ...] for a cable between the embedded vertices and .
• (optional) timestamps: If an animation of parameters moving on a curve is desired, samples from a curve can be entered as points in a list: [[q_11, q_12, ...], ...]

The output of this functions is primarily an interactive plot created using Makie.jl and a configuration of the involved in stable equilibrium. Interacting with the plot window is possible by moving a slider corresponding to each control parameter that was given as an argument to the method via listOfControlParameters. When changing the position of one of the sliders, a parameter homotopy of the underlying polynomial system is solved in real time. Depending on the system's size, this might take up to a few seconds. Nevertheless, this method is significantly faster than recalculating the solutions from scratch each time a slider is moved. Solving the polynomial system is made possible by the library HomotopyContinuation.jl.

As an example, consider a triangular bipyramid framework with an equilateral triangel as base two unknown nodes and a rigid bar of unknown length between them. This tensegrity framework can be realized by the input

@var p[1:6] ell
stableEquilibria([p[1:3], p[4:6], [0,1,0], [sin(2*pi/3),cos(2*pi/3),0], [sin(4*pi/3),cos(4*pi/3),0]],
    [[1,2,ell]],
    [[1,3,1,1], [1,4,1,1], [1,5,1,1], [2,3,1,1], [2,4,1,1], [2,5,1,1]],
    p, [ell], [1.0],
    [[3,4], [3,5], [4,5]], 
    []
)

The input let's us deduce that and that the target value of is also . The output of the plotting routine would then be the following image:

The vertices of this framework are displayed in red, the rigid bars are black and the elastic cables are portrayed in blue. If there were any visibly different solutions to the polynomial system, in this case obtained by mirroring the unknown vertices and in the top and bottom of the image respectively, they would be transparently plotted in grey as "shadow vertices".

Finally, the catastrophe set corresponding to our choice of control parameters is samples and plotted. This hypersurface depicts the region, where discontinuous changes of the framework might happen. However, this does not yet work for larger or 3D frameworks.