Tensegrity topology optimization by force maximization on arbitrary ground structures

Abstract

This paper presents an optimization approach for design of tensegrity structures based on graph theory. The formulation obtains tensegrities from ground structures, through force maximization using mixed integer linear programming. The method seeks a topology of the tensegrity that is within a given geometry, which provides insight into the tensegrity design from a geometric point of view. Although not explicitly enforced, the tensegrities obtained using this approach tend to be both stable and symmetric. Borrowing ideas from computer graphics, we allow “restriction zones” (i.e., passive regions in which no geometric entity should intersect) to be specified in the underlying ground structure. Such feature allows the design of tensegrities for actual engineering applications, such as robotics, in which the volume of the payload needs to be protected. To demonstrate the effectiveness of our proposed design method, we show that it is effective at extracting both well-known tensegrities and new tensegrities from the ground structure network, some of which are prototyped with the aid of additive manufacturing.

Appendices

Appendix 1: On reproducing known tensegrities: a verification study

One of the most well-known types of tensegrity is the prismatic tensegrity. This type of tensegrity has various configurations, but all of them obey the dihedral symmetry. A systematic study of the configuration and stability of the prismatic tensegrity can be found in the literature (Ohsaki and Zhang 2015; Zhang et al. 2009). The nodes of a prismatic tensegrity are located on the vertices of a twisted prism with each base face being a regular N-gon. The N vertices of the N-gon are incident on a circle. The twisting angle between the two parallel base faces is denoted as α, as shown in Fig. 18.

Fig. 18

The geometry and generation of a twisted prism. a The base polygon laid on the top circle B is obtained by rotating the same polygon on the bottom circle A with an angle α. b The top view of the twisting of the hexagonal base

We first generate the ground structure based on the twisted prism with full connectivity between nodes. The prism has a height of 1.0 (i.e., h = 1), and the radius of the outline circle of the base polygon is also 1 (i.e., r = 1). The results are shown in Fig. 19 for different geometries of the twisted prism. All of the results are super-stable, which has been proved analytically by Ohsaki and Zhang (2015) and Zhang et al. (2009).

Fig. 19

Examples of prismatic tensegrities that are reproduced using the proposed method. Different base polygons are used to generate the twisted prism geometries: a–f For N-gon-based twisted prism, α = π/N if N is even, and α = π/2N if N is odd. Quantitative data is provided in Table 4

Table 4 Computational results for the designs shown in Fig. 19

There is another family of tensegrity that has similar configurations to the prismatic tensegrities, namely the symmetric star-shaped tensegrity (Zhang and Ohsaki 2015), which also satisfies the dihedral symmetry. The difference is that a star-shaped tensegrity structure has two additional nodes lying on the centroids of the base faces. Therefore, in a prismatic tensegrity structure, there is essentially only one type of node, but in a star-shaped tensegrity, there are two types of nodes. To reproduce the known star-shaped tensegrities, we generate the ground structure using the nodes on the vertices of the twisted prism and the two additional nodes at the centroids of the top and bottom faces. Figure 20 shows a few examples of reproduced star-shaped tensegrities.

Fig. 20

Examples of star-shaped tensegrities that are reproduced using the proposed method. Different base polygons are used to generate the twisted prism geometries: a–c Compared to the prismatic tensegrities, the initial ground structures have two more nodes that are located at the centroids of the two base polygons. Quantitative data is provided in Table 5

Table 5 Computational results for designs shown in Fig. 20

Appendix 2: An illustrative example of the topological constraints

We use the following example to illustrate how the topological constraint and physical constraint work. Suppose we have a ground structure as shown in Fig. 21a. Label the vertices from A to F and edges from 1 to 9. Based on the given topology, we can construct the topological constraint matrix G as:

$$ \mathbf{G} = \left[\begin{array}{llllllllll} 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \end{array}\right]_{6 \times 9} $$

(12)

The rows of the matrix correspond to the connectivity information at nodes A to F. The columns contain the connectivity information of members 1 through 9. For example, the third row shows that members 2, 3, and 9 are connected to node C. Furthermore, since members 7, 8, and 9 intersect at one point, we have the physical constraint matrix G_p reads:

$$ \mathbf{G}_{p} = \left[\begin{array}{llllllllll} 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{array}\right]_{3 \times 9} $$

(13)

As discussed before, the coincident intersection point is split into three fictitious intersection points.

Fig. 21

a A simple ground structure with 6 nodes and 9 members. b The collection of members 1 and 2 in the ground structure. The two members are connected at node B. c Members 7 and 8 are contacting each other in the middle, indicating a conflict in space

Suppose we have a collection of members from the ground structure represented by the binary vector x whose k th entry reflects the presence of member k in the collection. Let x₁ = [1, 1, 0, 0, 0, 0, 0, 0, 0]^T meaning that members 1 and 2 are in the collection as shown in Fig. 21b. The matrix vector multiplication Gx₁ gives [1, 2, 1, 0, 0, 0]^T which clearly shows that there are two members in the collection connected to node B. If the constraint is set for Class-1 tensegrity and the collection x₁ represents the struts, x₁ will violate the topological discontinuity constraint. To show how the physical constraint works, we set x₂ = [0, 0, 0, 0, 0, 0, 1, 1, 0]^T, which contains members 7 and 8, as shown in Fig. 21c. The linear operation G_px₂ produces [2, 1, 1] with first component larger than 1 indicating a violation. Thus, the physical constraint successfully shows that members 7 and 8 cannot exist at the same time.

Appendix 3: Basic structural analysis of tensegrity structures

The construction of the stiffness matrix for a tensegrity structure is different from a normal truss due to the presence of prestress forces. Detailed derivations and discussions can be found in Guest (2006, 2011) and Zhang and Ohsaki (2015). Here, we briefly summarize the key ideas. The basic assumptions here are that both struts and cables are rectilinear members made of materials that have linear elastic constitutive relationships, and the strains in the members are always small. The geometric stiffness matrix formulation adopted here is an incomplete version that is accurate for small strain analysis. Indeed, the present expression for K_G is called the “stress matrix” (Connelly 1999), which is part of the complete geometric stiffness matrix (Guest 2011).

Assume the cross-sectional area, length, and Young’s modulus of member i are A_i, L_i, and E_i, respectively. The coordinates of node j are stored in the vector p_j. First, let us define the modified incidence matrix C. In graph theory, the incidence matrix is binary (like the matrix G in (5c)), but here, the modified matrix is composed of 0’s, 1’s, and − 1’s. Suppose member i links nodes a and b. Then, C is defined as:

$$ C_{ij} \!= \!\left\{\begin{array}{llllllllll} 1, & \text{if member} \text{i is connected to node} j, \text{and} j=a\\ -1, & \text{if member} i \text{is connected to node} j, \text{and} j=b\\ 0, & \text{otherwise} \end{array}\right. $$

(14)

The size of the modified incidence matrix C is N_E × N_V. Then, the augmented incidence matrix that connects the degrees of freedom to the members is defined as:

$$ \mathbf{C}_{aug} = \mathbf{C} \otimes \mathbf{1}_{1 \times 3} $$

(15)

where ⊗ means the Kronecker product, so that C_aug has size N_E × 3N_V. The vector 1 is a vector of ones. The total number of degrees of freedom in the structure is 3N_V because we are considering three-dimensional space. We assemble all the nodal coordinates (i.e., p_j’s) in a vector p by blocks of 3 components. We obtain the equilibrium matrix as:

$$ \mathbf{B} = \mathbf{P} \mathbf{C}_{\text{aug}}^{\mathrm{T}} \mathbf{L}^{-1} $$

(16)

where P = diag(p), with its diagonal entries containing all the nodal coordinates, and L being a diagonal matrix of member lengths. We define another diagonal matrix D of size N_E × N_E, such that

$$ D_{ii} = \frac{E_{i} A_{i}}{L_{i}} $$

(17)

Then, the linear stiffness matrix K_E of a tensegrity is given as:

$$ \mathbf{K}_{E} = \mathbf{B} \mathbf{D} \mathbf{B}^{\mathrm{T}} $$

(18)

which is a symmetric matrix with 3N_V rows and 3N_V columns. By assuming that the prestress force in member i is F_i, we define a diagonal matrix Q as:

$$ Q_{ii} = \frac{F_{i}}{L_{i}} $$

(19)

The ratio F_i/L_i is known as the force density. The so-called force density matrix (Zhang and Ohsaki 2015) (or reduced stress matrix; Connelly 1999; Schenk et al. 2007) is then formed by:

$$ \mathbf{E} = \mathbf{C}^{\mathrm{T}} \mathbf{Q} \mathbf{C} $$

(20)

which is of size N_V × N_V. Then, the geometrical stiffness matrix is constructed by:

$$ \mathbf{K}_{G} = \mathbf{E} \otimes \mathbf{I}_{3 \times 3} $$

(21)

where I is the identity matrix. Finally, the tangent stiffness matrix of a tensegrity is the summation of the linear stiffness matrix and the geometrical stiffness matrix:

$$ \mathbf{K} = \mathbf{K}_{E} + \mathbf{K}_{G} $$

(22)

Appendix 4: Nomenclature

1 :

Vector of ones

B :

Equilibrium matrix

F :

Member forces

G :

Incidence matrix

G _p :

Physical contact matrix

K :

Tangent stiffness matrix

K _E :

Linear elastic stiffness matrix

K _G :

Geometrical stiffness matrix

p :

Coordinates of nodes, 3N × 1 vector

s :

A binary vector for the presence of struts

u :

Displacement of nodes, a perturbation on p

u _M :

First-order mechanisms of the dual truss of a tensegrity

E:

Edges of a graph

e:

An edge (member)

E _g :

Edges of the ground structure

G:

A graph

H, K:

Symmetry groups

h_i, k_i:

Symmetry operations

n, n:

Level of discontinuity of struts

N _I :

Number of active integer variables

n _r :

Number of rigid-body motions

$N_{E_{g}}$ :

Number of members in the ground structure

N _E :

Number of members in the obtained tensegrity

$N_{V_{g}}$ :

Number of nodes in the ground structure

N _V :

Number of nodes in the obtained tensegrity

T _{o p t} :

Running time of the optimization

V:

Vertices of a graph

v:

A vertex (node)

V _g :

Vertices of the ground structure

KI:

Kinematic indeterminacy