#### definition

From a basic physics perspective what is tensegrity? Is it just a variation of torque; some sort of esoteric inner-see-saw? Why is it not found in physics books? A fundamental idea of tensegrity is the compression:tension ratio. In this writeup, the mathematical framework and idea of tensegrity will focus only on the range of optimal configurations a shape can take, from full compression(flattened) to full tension(expansion).

#### compression-tension

• The spokes in a spoked-wheel provide outward tension; thus, a structure, which is lighter and more flexible than a solid wheel, but provides equivalent compressive strength can be achieved.
• A vertical length of column can hold up a compressive load. Based on material properties, there is an inverse relationship between length and compression described by a force density (f/l). For an unloaded column, there exists a maximal length beyond which tension cannot expand.
• Geodesic domes can be formulated at planetary size, thousands(km) in diameter. This planetary sized leverage results from smaller changes in tension over the outer diameters leading to greater compressive strength centrally (ie the roof does not collapse).

#### applications

• Joints, such as ankle, pelvis, and spine, are pre-stressed with tensive loads leading to increase compressive strength and mobility
• A manta ray’s wing can be modeled with several lines of tensegrity, described as a learning surface providing nonlinear fluid control, capable of firmly flexing to quickly glide away, or rippling in-place to hover.
• shape of cells and DNA

#### models

• Given overhanging blocks, tension exists in the regions where blocks overlay. The blocks will not fall over if each subsequent block’s overhang is half the one below it; this allows the combined center of mass to be supported. This can be described using a harmonic series sum, H(n).
• H(2), case of 2 bricks:
• $$X2 \cdot 2M = X1 \cdot M \to (x1=1/2)$$
• H(4), span of total overhang greater than length of a block:
• x1=1/2, x2=1/4, x3=1/6, x4=1/8
• $$\left(\sum x1+..+x4 = \frac{25}{24} \right) > 1$$
• H(n), can it span any distance?:
• $$X_n\cdot(nM) = X_1 \cdot M$$
• $$\sum\limits_{k=1}^n \frac{1}{2k}$$ ; $$\int \frac{1}{k} \, dk$$
• H(k) - ln(k) log-approx undercounts the series_sum $$\to Euler_constant( \gamma )$$

• template shape
• struts (compression)
• a centered cube with sides 2d
• len = 2l, in each plane of cube
• strut(0,d,-l)(0,d,l) $\parallel$ strut(0,-d,l)(0,-d,-l)
• permute rotational symmetry over cube's diagonal: strut_coords $$(0,d,l) \rightarrow (d,l,0) \rightarrow (l,0,d) , .. )$$
• tendons (tension)
• $$s \cong d/l$$ where s is len of tendon connect two vertices
• for len s : $% $

• torque:

• interactive equilibrium modeling:

runga-kutta

1. constrain force density $\hat{q}^0$ from network geometry

2. calc nodal residuals $r$ and gradient $\triangle \hat{q}, x,y,z$

3. update force densities and network geometry

• terminate $\| r \| > \epsilon$
linear least square: $r=\|\underbrace{A}_\textrm{3nx3m}q^0 - \underbrace{p}_\textrm{load}\|$

$q^0 = (A^TA)^{-1}A^Tp$

Fundamental Linear Algebra

• space
• rowspace C($A^T$)
• colspace C(A)
• nullspace N(A) contains all solution to $Ax=0$
• dimensionality
• dim r
• rank == dim C(A) == dim of C($A^T$)
• dim m
• dim C(A) + dim $N(A^T)$ = m
• dim n
• dim $C(A^T)$ + dim N(A) = n
• nullspace
• kernel = dim N(A) + rank = n
• dim $N(A^T)$ = m
• orthogonality
• $$N(A) \cdot C(A^T) = 0$$
• $$N(A^T) \cdot C(A) = 0$$
• transform
• basis: n-linear-indpt vectors; minimal spannning set; spans n dimension
• reducible if A(mxn) some rows linear-indpt
• invertible (unique solution) if A(mxm) rows linear-indpt
• overdetermined m>n, all m rows lin indpt, (usual case, sys not solve exactly, LS)
• underdetermined m<n, infinite solutions

## refs

prestressed ankle: http://www.jbiomech.com/article/S0021-9290%2809%2900355-8/abstract

http://www.magicalrobot.org/BeingHuman/2012/10/tensegrity-snake-robot https://github.com/kcaluwae/tensegrity-el-simulator/blob/master/examples/cartesian_coordinates_simulator/show_6bar_18springs_minimal.py

http://graphicallinearalgebra.net/2015/06/09/matrices-diagrammatically/ http://bl.ocks.org/mbostock/7782500

Interactive Equilibrium Modeling, Lauchauer