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.
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Notes
The examples in this paper are solved by the optimization software Gurobi 6.5 (Gurobi Optimization 2014) executed by a MATLAB code. The code is operating on a desktop with an 8-core 3.0 GHz Intel Xeon CPU. It is also possible to use other solvers such as the MATLAB built-in function “intlinprog” to solve the problem.
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Acknowledgements
The authors would like to extend their appreciation to Dr. Tomas Zegard and Ms. Emily D. Sanders for helpful discussions which contributed to improve the present work and to Mr. Rob Felt for taking photos of the physical models.
Funding
This study received support from the US NSF (National Science Foundation) through Grants 1538830 and 1321661. In addition, Ke Liu received support from the China Scholarship Council (CSC). We are grateful to the support provided by the Raymond Allen Jones Chair at the Georgia Institute of Technology.
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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.
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).
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.
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:
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 Gp reads:
As discussed before, the coincident intersection point is split into three fictitious intersection points.
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 Gpx2 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 KG 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 Ai, Li, and Ei, respectively. The coordinates of node j are stored in the vector pj. 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:
The size of the modified incidence matrix C is NE × NV. Then, the augmented incidence matrix that connects the degrees of freedom to the members is defined as:
where ⊗ means the Kronecker product, so that Caug has size NE × 3NV. The vector 1 is a vector of ones. The total number of degrees of freedom in the structure is 3NV because we are considering three-dimensional space. We assemble all the nodal coordinates (i.e., pj’s) in a vector p by blocks of 3 components. We obtain the equilibrium matrix as:
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 NE × NE, such that
Then, the linear stiffness matrix KE of a tensegrity is given as:
which is a symmetric matrix with 3NV rows and 3NV columns. By assuming that the prestress force in member i is Fi, we define a diagonal matrix Q as:
The ratio Fi/Li 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:
which is of size NV × NV. Then, the geometrical stiffness matrix is constructed by:
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:
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
- hi, ki:
-
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
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Liu, K., Paulino, G.H. Tensegrity topology optimization by force maximization on arbitrary ground structures. Struct Multidisc Optim 59, 2041–2062 (2019). https://doi.org/10.1007/s00158-018-2172-3
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DOI: https://doi.org/10.1007/s00158-018-2172-3