]> A Practical Guide to Tensegrity Design: 4.2 Diamond Structures

4.2 Diamond Structures

4.2.1 Diamond Structures: Descriptive Geometry

line drawing of dowel-and-fishing-line tensegrity tetrahedron in diamond configuration
Figure 4.1: 2ν Diamond T-Tetrahedron

As described in Section 2.3, diamond structures are characterized by the fact that each tendon triangle is connected to adjacent tendon triangles via one strut and two interconnecting tendons. This section examines a diamond configuration of the tensegrity tetrahedron. The zig-zag configuration of the 2ν1 t-tetrahedron was examined in Section 2.4. The diamond configuration of the 2ν t-tetrahedron is illustrated in Figure 4.1. It is topologically identical to the t-icosahedron (Figure 2.5 of Section 2.3). The only difference is that the tendon triangles of the 2ν diamond t-tetrahedron are two different sizes. The t-icosahedron is actually a special case of the 2ν diamond t-tetrahedron with all tendons the same length.

To review the contrast between the diamond and zig-zag configurations presented in Section 2.3, it is most productive to focus on the group of four small triangles from the 2ν diamond t-tetrahedron. These correspond to the 2ν zig-zag t-tetrahedron's four tendon triangles. If two tendon triangles from this group are considered to be facing each other nose-to-nose, the strut can be seen to connect the right ear of one triangle with the right ear of the other triangle as it did in the zig-zag t-tetrahedron. However, there are now two tendons interconnecting the two tendon triangles instead of just one. Each connects the right ear of one tendon triangle with the nose of the other.

These two tendons are symmetrical to each other, so the problem still consists of minimizing one length as it did in the original zig-zag problem, and even the same geometrical model as was used to solve that problem could be used here. However, the general case is more complex than this and is not amenable to treatment with models such as were used to examine the simple zig-zag t-tetrahedron. So to illustrate the general procedure, calculations are done for a frequency-four (or 4ν for short) diamond t-tetrahedron.

triangular grid embedded in triangle embedded in circle
Figure 4.2: 4ν Breakdown of Tetrahedron Face Triangle


triangular grid embedded in triangle projected onto sphere represented by circle
Figure 4.3: 4ν Tetrahedron Face Triangle Projected on to a Sphere

It is called a 4ν structure because its geometry derives from the 4ν geodesic subdivision of the tetrahedron.2 Only even-frequency subdivisions are used in tensegrity designs. Figure 4.2 shows a 4ν breakdown of a triangle, in this case the face of a tetrahedron. The labels a and b indicate which triangles are symmetrically equivalent. The heavy lines represent the lines of the geodesic breakdown used in the tensegrity design. Kenner's procedure is followed and these triangles are projected onto a sphere circumscribing the tetrahedron (see Figure 4.3). Notice that, considering symmetry transformations, there are two types of tendon triangles composing the system, an equilateral tendon triangle and an isosceles one.

three triangles connected by two solid struts with point labels
Figure 4.4: 4ν Diamond T-Tetrahedron: Representative Struts


three triangles connected by four dashed tendons with point labels
Figure 4.5: 4ν Diamond T-Tetrahedron: Representative Tendons

Next the interconnecting struts and tendons are introduced. Figure 4.4 shows representative examples of the interconnecting struts. There are two types of strut. One type connects adjacent isosceles triangles, the other type connects isosceles with equilateral triangles. Figure 4.5 shows the corresponding interconnecting tendons. There are a pair of tendons corresponding to each strut type. Note that in both the figures, the triangles are skewed toward their final positions for clarity's sake. In the tensegrity programming problem, the sum of second powers of the lengths of the four diamond tendons are minimized, while the lengths of the struts and triangle tendons are considered constraints.

4.2.2 Diamond Structures: Mathematical Model

isometric view of labeled triangular grid embedded in xyz space
Figure 4.6: 4ν Diamond T-Tetrahedron: Coordinate Model (Face View)


axial view of labeled triangular grid embedded in xyz space
Figure 4.7: 4ν Diamond T-Tetrahedron: Coordinate Model (Edge View)

Figures 4.6 and 4.7 show a tetrahedron inscribed within Cartesian coordinate space in a convenient orientation. With this orientation, any symmetry transformation of the tetrahedron can be accomplished merely by permuting the coordinate axes. On the tetrahedral face which falls in the positive quadrant (but extends into three others as well), the elements of the 4ν geodesic subdivisioning relevant to tensegrities have been inscribed. On this triangle, there are four points labeled P1, P2, P3 and P4. P1, P2 and P3 represent the vertices of the isosceles triangle (or at least it will be isosceles when these points are projected onto a sphere); P4 is a point on the equilateral triangle.

With these four points, all of the other points of the 4ν subdivisioning can be generated by using the symmetry transforms of the tetrahedron. Notice that, although geodesic structures exhibit mirror symmetry frequently, tensegrity structures generally do not. So P2 cannot be generated from P1 using a mirroring operation. Also, initially P3 and P4 coincide since initially the vertices of the isosceles and the equilateral triangle are in contact. When the computations start though, they part company.

The four points, P1, P2, P3 and P4, can be generated from the three vertex points, V1, V2 and V3, of the triangular tetrahedron face as follows:

P1= 34 V1+ 04 V2+ 14 V3
P2= 34 V1+ 14 V2+ 04 V3
P3= 24 V1+ 14 V2+ 14 V3
P4= 24 V1+ 14 V2+ 14 V3

Thus, the coordinates of V1, V2 and V3 summarized in Table 4.1 imply the coordinate values of P1, P2, P3 and P4 summarized in Table 4.2.

Coordinates
 Vertex    x     y     z  
V1   1.0     -1.0      1.0  
V2   -1.0      1.0     1.0  
V3   1.0     1.0     -1.0   
 
Table 4.1: Tetrahedron Face: Vertex Coordinate Values


Coordinates
 Point    x     y     z  
P1   1.0     -0.5      0.5  
P2   0.5     -0.5      1.0  
P3   0.5     0.0     0.5  
P4   0.5     0.0     0.5  
 
Table 4.2: 4ν Diamond T-Tetrahedron: Unprojected Point Coordinates

When the values for P1, P2, P3 and P4 are projected onto the unit sphere, Table 4.3 is obtained. These coordinates serve as the initial values for the computation process. From them, the initial values of all member lengths are computed.

Coordinates
 Point    x     y     z  
P1   23      - 16      16   
P2   16      - 16      23   
P3   12      0     12   
P4   12      0     12   
 
Table 4.3: 4ν Diamond T-Tetrahedron: Projected Point Coordinates

In order to express all the members of the tensegrity, three more points are needed, P5, P6 and P7. These points are symmetry transforms of P2, P3 and P4 respectively. P5 and P6 are obtained from P2 and P3 by a 120° left-hand rotation of the tetrahedron about the vector from the origin to V1. In this coordinate system, this is achieved by taking the x axis into the -y axis, the -y axis into the z axis, and the z axis into the x axis, so that P5 and P6 can be expressed respectively as z2 -x2 -y2 and z3 -x3 -y3 .3 P7 is obtained from P4 by a 120° left-hand rotation of the tetrahedron about the vector from the origin to the point 1.0-1.0 -1.0. This is achieved by taking the x axis into the z axis, the y axis into the x axis, and the z axis into the y axis, so that P7 can be expressed as y4 z4x4 .

So whenever coordinates for P5, P6 or P7 are required, these transformed versions of P2, P3 or P4 are used. Thus the symmetry constraints of the programming problem are implicitly subsumed in these expressions for P5, P6 and P7. The variables of the programming problem are still limited to the x yz coordinates of the original four points, and no new constraints need to be added to take into account symmetry.

 Member 
# ID
 End Points 
Length
 Comments 
1 t12
2 t13
3 t23
4 t47
P1 P2
P1 P3
P2 P3
P4 P7
 0.577350 
 0.517638 
 0.517638 
 1.0
 Constraint 
 Constraint 
 Constraint 
 Constraint 
5 sab
6 sbb
P1 P7
P2 P6
 1.414214 
 0.919401 
 Constraint 
 Constraint 
7 tab1
8 tab2
9 tbb1
10 tbb2
P3 P7
P1 P4
P1 P6
P2 P5
 1.0
 0.517638 
 0.517638 
 0.577350 
 To be minimized 
 To be minimized 
 To be minimized 
 To be minimized 
 
Table 4.4: 4ν Diamond T-Tetrahedron: Initial Member Lengths

Table 4.4 summarizes the initial lengths for the constrained members obtained using these coordinate values. The relevant mathematical programming problem is:

minimize o P3 -P72 + P1 -P42 + P1 -P62 + P2 -P52 P1, P2, P3, P4 subject to Tendon constraints: 13 P1 -P22 tan π12 P1 -P32 tan π12 P2 -P32 1 P4 -P72 Strut constraints: -2 - P1 -P72 -0.84529946 - P2 -P62

This completely specifies the problem. Again, only the coordinates of P1, P2, P3 and P4 are variables in the minimization process since the coordinates of P5, P6 and P7 are specified to be symmetry transforms of the coordinates of these points.

This is a very formal statement of the problem, and, as stated in Section 3.2, to solve it the inequality constraints are assumed to be met with equality.

4.2.3 Diamond Structures: Solution

As described in Section 3.2, the partials of the member equations and the non-member constraint equations can be conceived as a matrix, Ψ, which has as many rows as their are equations (10 in this case) and as many columns as there are coordinate values (12 in this case). The ijth element of this matrix, ψ ij, is the derivative of the ith equation with respect to the jth coordinate value. The coordinate values are numbered in the order they appear, so for example, ψ 4,11 is the partial derivative of the second power of the length of the t 47 tendon with respect to y4. Its value is 2 y4- x4+ 2 y4- z4. This partial is unusual in that it has two terms. Most of the member-equation partials are either zero or consist of a single difference.

The first step is to reformulate this as an unconstrained minimization problem by choosing a subset of the coordinates to be dependent coordinates whose values are obtained by solving the constraints given the values for the independently specified coordinates. Since there are six constraints, there are six dependent coordinates. This leaves six 12-6 independent coordinates. By coincidence, the number of independent coordinates is equal to the number of dependent coordinates in this problem. Using Gaussian elimination with double pivoting on the partial derivative matrix for the system resulted in x1, x2, x3, z3, x4 and z4 being used as the initial independent coordinates. So, given the values for these coordinates, the constraints were solved for the remaining dependent coordinates, y1, z1, y2, z2, y3 and y4.

 Coordinate   Derivative 
x1   -0.875117 
x2   -0.160155 
x3  1.38037   
z3  0.345092 
x4   -0.345093 
z4  0.597720 
 
Table 4.5: 4ν Diamond T-Tetrahedron: Initial Objective Function Derivatives

The initial derivatives of the objective function with respect to the independent coordinates are summarized in Table 4.5. At a minimum point, the values of all these derivatives will be as close to zero as the accuracy of the computations permits. Instead of constantly looking at this whole list of derivatives (which can be very long for a complex structure) to assess how close to a minimum the system is, two summary statistics can be examined, the geometric average of the absolute values of these derivatives, and the variance of the natural logarithm of (the absolute value of) these derivatives. The variance is an important statistic, since if the system starts going singular, one or more of the derivatives starts to diverge from the rest. This singularity is a signal that the partitioning of variables between independent and dependent variables needs to be redone.

The value of the objective function is initially 1.86923. The system is solved using the Parallel Tangents technique which results in an objective function value of 1.65453. Table 4.6 summarizes the corresponding point values, and Table 4.7 summarizes the lengths of the members in the objective function thus obtained.

Coordinates
 Point    x     y     z  
P1  0.887555   -0.438450    0.455646 
P2  0.677306   -0.505030    0.989215 
P3  0.614181   -0.076748    0.705421 
P4  0.710900   -0.048791    0.590190 
 
Table 4.6: 4ν Diamond T-Tetrahedron: Preliminary Coordinate Values


 Member 
ID
 Length 
tab1  0.940409 
tab2  0.448489 
tbb1  0.455651 
tbb2  0.601166 
 
Table 4.7: 4ν Diamond T-Tetrahedron: Preliminary Objective Member Lengths

This would be the end of the calculations, except that when the endogenous member forces are calculated, they indicate that "tendon" t12 is marginally in compression (see Table 7.1). This problem stems from the substitution of equalities for inequalities in the constraints. If inequalities had been used, this particular constraint would be found to be not effective. At this point the problem is dealt with by eliminating the member from the constraints which means the tendon doesn't appear in the final structure.4 Eliminating this constraint also means a new selection of independent variables needs to be made since seven are now needed. Repartitioning results in z1 being added to the independent variables. Using the Parallel Tangents technique on this problem resulted in a final objective-function value of 1.65174. Table 4.8 summarizes the corresponding point values; Table 4.9 summarizes the objective function member lengths, and Figure 4.8 shows the final design where the location of the omitted tendon is indicated by a dashed line.

Coordinates
 Point    x     y     z  
P1  0.874928   -0.442843    0.484207 
P2  0.675644   -0.506061    0.981906 
P3  0.602311   -0.068420    0.715369 
P4  0.699892   -0.049794    0.605188 
 
Table 4.8: 4ν Diamond T-Tetrahedron: Final Coordinate Values


 Member 
ID
 Length 
tab1  0.937671 
tab2  0.446946 
tbb1  0.473042 
tbb2  0.590748 
 
Table 4.9: 4ν Diamond T-Tetrahedron: Final Objective Member Lengths


view of final design for 4v diamond t-tetrahedron
Figure 4.8: 4ν Diamond T-Tetrahedron: Final Design

1 The qualifier "2ν" is explained below.

2 Kenner76, Chapter 5.

3 xn, yn and zn represent the Cartesian coordinates of Pn.

4 Alternatively, its length could be shortened until it is effective.