Prestress Design of Tensegrity Structures Using Semidefinite Programming

Wang, Yafeng; Xu, Xian

doi:https://doi.org/10.1155/2019/5081463

Advances in Civil Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 5081463 | https://doi.org/10.1155/2019/5081463

Prestress Design of Tensegrity Structures Using Semidefinite Programming

Yafeng Wang¹and Xian Xu²

Academic Editor: Andreas Lampropoulos

Received16 May 2018

Revised11 Dec 2018

Accepted19 Dec 2018

Published16 Jan 2019

Abstract

Finding appropriate prestresses which can stabilize the system is a key step in the design of tensegrity structures. A semidefinite programming- (SDP-) based approach is developed in this paper to determine appropriate prestresses for tensegrity structures. Three different stability criteria of tensegrity structures are considered in the proposed approach. Besides, the unilateral property of members and the evenness of internal forces are taken into account. The stiffness of the whole system can also be optimized by maximizing the minimum eigenvalue of the tangent stiffness matrix. Deterministic algorithms are used to solve the semidefinite programming problem in polynomial time. The applicability of the proposed approach is verified by three typical examples. Compared to previous stochastic-based approaches, the global optimality of the solution of the proposed approach is theoretically guaranteed and the solution is exactly reproducible.

1. Introduction

Tensegrity structures are stable self-stressed pin-jointed structures. Because of the unique features and remarkable appearances of tensegrity structures, they are attracting more and more interest in the fields of science and engineering [1–3]. In the view of mathematics, in any stable prestressed pin-jointed systems such as prestressed trusses, cable net structures can be seen as tensegrity systems [4]. The above general definition of tensegrity systems is adopted in this study; that is, the term tensegrity structure is the same as the stable prestressed pin-jointed structure.

A statically and kinematically indeterminate pin-jointed assembly can be converted to a stable tensegrity structure if appropriate prestresses and gain adequate stiffness could be introduced. Therefore, finding appropriate prestresses which can stabilize the system is a key step in generating tensegrity structures from statically and kinematically indeterminate pin-jointed assemblies. There are two main stability criteria, i.e., super stability and prestress stability, of tensegrity structures [5, 6]. Prestress stability of a prestressed pin-jointed structure can be tested by using the Geometry force method proposed by Calladine and Pellegrino [7]. Super stability is a stronger stability criterion for prestressed pin-jointed structures compared with prestress stability because it is irrespective of selection of materials and level of self-stresses [5, 8].

If a statically indeterminate pin-jointed assembly has only one independent self-stress state, the real prestress state introduced into the system must be in accordance with the self-stress state and only the level of prestresses needs to be designed. However, if a statically indeterminate pin-jointed assembly has more than one independent self-stress state, how to find a feasible self-stress state that can stabilize the system becomes a challenging problem, especially when the unilateral features of some specified members (i.e., they are appointed to be in tension or in compression in advance) are considered. Many studies have been carried out towards this problem. Tran and Lee [9] and Zhou et al. [10] used iterative methods to design initial prestress of tensegrity structures, but these iterative methods can only find a feasible solution but not an optimized solution for a specified objective function. In order to find optimal initial prestress for a tensegrity system, Ohsaki et al. [11, 12] used an optimization method by enumeration of the vertices of feasible region of the prestresses in the self-equilibrium state. However, not all the three kinds of stability criteria of tensegrity structures were taken into consideration in their approach and the enumeration method may be not very effective when the structure systems become more complex. Besides these deterministic algorithms, many stochastic or heuristic algorithms have also been used to solve this problem. El-Lishani et al. [13] and Lee et al. [14] used a genetic algorithm (GA) to find feasible self-stresses of pin-jointed structures. Xu and Luo [15], Zhang and Ohsaki [16] both used a simulated annealing algorithm (SAA) to find optimal prestress of tensegrity structures. Chen et al. [17, 18] converted this problem into a modified traveling salesman problem (TSP) and used a new heuristic search method, ant colony system (ACS), to search feasible solutions. Although these approaches have been verified capable to deal with the problem, they all only considered the prestress stability criterion and did not take super stability criterion into consideration. Moreover, they are all heuristic algorithms with randomness. There is no mathematical convergence theory for these algorithms and thus no assurance of global optimality of the final solution found [19]. And due to their stochastic basis, it is difficult to reproduce results from the method. Running the same algorithm, on the same problem, may result in widely different solutions [19].

It is also worth mentioning that recently the topology optimization of tensegrity structures has been intensively studied by using mixed integer linear programming (MILP) [20–23]. The prestress design problem is also a critical subproblem in the topology optimization of tensegrity structures. But the stability test cannot be incorporated into the current MILP-based formulation due to its nonlinear formation, not to mention incorporating the heuristic algorithms with randomness into the MILP-based formulation.

Semidefinite programming (SDP) problems are an important class of mathematical optimization problems which are applicable to a wide range of areas such as control theory, circuit design, sensor network localization, and principal component analysis [24]. In the field of graph theory, SDP was used by So and Ye [25] to realize 3-realizable graphs based on a tensegrity framework. In their approach, SDP is only used as a tool to find 3-realizable graphs which can be treated as a kind of unyielding tensegrity system, so stability conditions and evenness of prestresses of members in the tensegrity systems are not considered. Therefore, the approach is not suitable for the application of prestress design and optimization of tensegrity systems in the field of engineering application. In this paper, SDP is also adopted to solve the prestress design problem of tensegrity structures which includes both stability condition and evenness of internal forces of members. Because SDP is an optimization-based method, it can be used to find optimized solution corresponding to a specified objective function. Moreover, SDP problems can be solved efficiently in polynomial time by implementing a suitable algorithm. Compared with heuristic search methods with randomness, there is no randomness in the process of solving SDP problems; therefore, no repetitive computation is needed. Another advantage of using SDP formulation for the prestress design problem of tensegrity structures is that it has the potential to be incorporated into or to be extended to incorporate the MILP formulation of the topology optimization of tensegrity structures.

The motivation of this study includes (a) providing an alternative approach for prestress design of tensegrity structures, (b) incorporating the different stability criteria into the prestress design approach, and (c) formulating the stability test within a frame of mathematical programming and thus potentially being able to incorporate with the topology optimization formulation.

2. Formulations for Tensegrity Structures

Equilibrium condition and stability condition are necessary conditions for a tensegrity structure. Considering a d-dimensional tensegrity structure with m members and n nodes, the equilibrium condition and stability condition of it can be derived as follows.

2.1. Equilibrium Condition

The topology of the tensegrity structure can be expressed by a connectivity matrix defined in the field of graph theory [26]. Suppose that member k connects node i and node j (i < j), then the ith and jth elements of the kth row of are set to 1 and −1, respectively, as

Take partitions of matrix so thatwhere relate to the free DOFs and relate to the fixed DOFs in x direction, y direction, and z direction, respectively [27].

If all the DOFs of a node are restrained, then assume that r_x = r_y = r_z = r and s_x = s_y = s_z = s and that and , thus equation (2) can be simplified as

Let and denote the internal force and length of member k, respectively. The member force vector and member length matrix are defined as and . Then, the coordinate difference matrices are given aswhere are the nodal coordinate vectors of the free DOFs in the x direction, y direction, and z direction, respectively and are the nodal coordinate vectors of the fixed DOFs in the x direction, y direction, and z direction, respectively. , , and represent the number of free DOFs in the x direction, y direction, and z direction, respectively. Similarly, , , and represent the number of fixed DOFs in the x direction, y direction, and z direction, respectively. There exists .

The equilibrium matrix can be written as

The equilibrium equations for a tensegrity structure without external loads can be written as [28]

Note that the internal forces of a tensegrity system can also be expressed as the combination of its independent self-stress modes [28]. For symmetrical systems, symmetry representations can also be used to simplify the equilibrium equation [29]. In this paper, the symmetry of systems is not considered; therefore, only equation (6) is used to express equilibrium equations.

2.2. Stability Condition

2.2.1. Stiffness Matrices

The tangent stiffness matrix K of a pin-jointed structure can be described aswhere and are the linear elastic stiffness matrix and the geometrical stiffness matrix, respectively. and are defined aswhere is the diagonal matrix, is the identity matrix, is the tensor product, and is the force density matrix. Diagonal matrix and force density matrix in equation (8) are defined aswhere and are Young’s modulus and the cross-sectional area of member , respectively. More details on stiffness matrices can be found in the literature [6, 30–32].

2.2.2. General Stability Criterion Based on Tangent Stiffness Matrix

Based on the general theory of elastic stability [33], if the tangent stiffness matrix K is positive definite, then the structure is stable. That is, holds for any nonzero vector d. In order to simplify the formulation expressions, is used to denote that matrix N is positive definite in the following content. Similarly, is used to denote that matrix N is positive semidefinite. Therefore, positive definiteness of tangent stiffness matrix K can be simply expressed as

Note that if the structure is free standing, the global rigid body motions should be eliminated beforehand.

It can be seen from equations (7)–(9) that this general stability condition for a given tensegrity structure not only depends on the material properties of members but also depends on the level of prestress.

2.2.3. Prestress Stability

If the material properties of members and the level of prestress are not considered, a reduced stability criterion, prestress stability, for prestressed pin-jointed structures was presented by Connelly [5]. For a prestress stable system, the stiffness degradation can be induced by a relatively high level of prestress [34, 35]. Prestress stability can be tested by using the Geometry force method proposed by Calladine and Pellegrino [7]. Let M denotes the inextensional mechanism modes of a statically and kinematically indeterminate pin-jointed structure. It has been proved that the matrix used by Calladine and Pellegrino [7] for the stability test is equivalent to , which can be seen as a reduced form of geometrical stiffness matrix K^G [30]. Note that if the structure is free standing, the rigid body motions are excluded from M beforehand. Therefore, from the definition of prestress stability, if is positive definite, then the structure is prestress stable. This condition can be written as

Note that a structure may be unstable in practice even if it satisfies the prestress stability condition because the material properties and the level of prestress are not considered in the prestress stability condition. That is to say that the positive definiteness of is not a sufficient condition but only a necessary condition for the stability of kinematically indeterminate structures when the material properties of members are taken into consideration. More details on this issue can be found in the literature [36].

2.2.4. Super Stability

Super stability is a stronger stability criterion for prestressed pin-jointed structures compared with prestress stability [5]. The stability of a super stable structure is ensured irrespective of the material properties and the level of prestresses, if yielding and buckling of members are excluded. Therefore, super stability systems may be more preferred in practical applications.

Based on the super stability criterion [5], a pin-jointed structure is super stable if the geometrical stiffness matrix K^G satisfies equation (11) and meanwhile is positive semidefinite. The positive semidefinite condition can be written as

Note that the three stability criterions mentioned above are not independent of each other. The relationship of them can be described by a diagram shown in Figure 1. The shaded part which is the intersection region of the three parts denotes super stability condition.

3. Mathematical Optimization Model

The problem of finding appropriate prestresses which can satisfy the equilibrium condition and stability condition for a pin-jointed assembly can be formulated into an optimization model. The optimization variables of the optimization model are internal forces of members, and the constraints of the optimization model are equilibrium constraint, stability constraint, and additional unilateralism constraint of member forces.

3.1. Equilibrium Constraint

By using optimization variables t and the equilibrium matrix A, the equilibrium constraint can be expressed as equation (6).

3.2. Stability Constraint

Depending on which kind of stability is expected to be satisfied by the tensegrity structure under design, equations (10), (11), or (12) can be selected as the stability constraint.

Note that the strict inequalities cannot be dealt in continuous numerical optimization. Hence, if equation (10) is adopted, a transformed expression is used, that is,where is the small positive value as a margin and is the identity matrix whose dimensionality is the same as the matrix . Note that if the minimum eigenvalue of tangent stiffness matrix is needed to be optimized, can be also treated as an optimization variable.

Similarly, if equation (11) is adopted, a transformed expression as follows is used:where is the small positive value as a margin and is the identity matrix whose dimensionality is the same as the matrix .

3.3. Unilateralism Constraints of Member Forces

If there are some members that are appointed in advance to be in tension or compression, unilateralism constraints should be added. Suppose that the set of labels of members appointed to be in tension is denoted as P, the set of labels of members appointed to be in compression is denoted as Q, and the set of labels of the rest of members is denoted as N. Then, the unilateralism constraints can be expressed aswhere is the small positive value as a margin.

In practical design, the evenness of internal forces of members in tension or compression is sometimes needed to be considered. For this purpose, equation (15) can be converted to another form by using additional optimization variables:where , , , and are the continuous variables. In this way, smaller or means more even tension forces or more even compression forces, respectively.

3.4. Objective Function

If the objective function is not adopted in the optimization model, the solution which satisfies above constraints is a feasible solution. If optimization objective is considered, different objective functions can be used to realize different design intentions. For example, if the evenness of the member internal forces is required to be optimized, the corresponding objective function f₁ can be expressed as

If the minimum eigenvalue of the tangent stiffness matrix K is needed to be optimized, it is equivalent to maximize variable in equation (13) in the optimization process. To realize this purpose, in equation (13) can be treated as an optimization variable, and the objective function f₂ can be expressed as

3.5. Optimization Model

The optimization model of the problem can be expressed as

Stability constraint,where denotes the set of optimization variables, denotes objective function which can be chosen as or by designers according to actual needs and constraints considered, and stability constraint denotes constraint on stability which is chosen from equations (10)–(12) by designers. If , then constraints (19c) and (19e) are not considered. Similarly, constraints (19d) and (19f) are not considered if .

4. Semidefinite Programming

Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a symmetric matrix variable Y is optimized subject to linear constraints on the elements of Y and the additional constraint that Y must be positive semidefinite. The linear programming (LP) problems can be deemed as a special case of SDP, namely, when all the matrices involved are diagonal. Similar to LP problems, SDP problems also come in pairs. One of the problems is referred to as the primal problem, and the other one is the dual problem. Let R·S denotes the standard inner product of matrices R and S in the linear space Sⁿ, i.e., . The most common standard formulation of the SDP primal problem is given as follows:where is the variable vector, and are the symmetric matrices, and is the column vector. The dual problem of P_SDP is formulated in variables as

More details on the primal problem and the dual problem can be found in the literature [24].

SDP has been applied in a wide range of fields [37–39]. The primal-dual interior-point methods, which were developed to solve linear programming problems [40] at first, have been widely used to solve SDP problems [41]. It is theoretically guaranteed that the pair of SDP problems (P_SDP) and (D_SDP) can be solved to the optimal solutions in polynomial time [38, 41].

Both linear constraints and matrix semidefinite constraints are included in equation (19); therefore, it is a SDP model, and primal-dual interior-point method can be applied to solve it.

5. Numerical Examples

In this paper, the SDP problems are solved by using SeDuMi [42], which implements the primal-dual interior-point method. Computations are carried out on a Dual-Processor Intel(R) Xeon(R) CPU (2.80 GHz with 32 GB RAM) computer. The parameters μ and ε are set as μ = ε = 1 × 10⁻³. Without loss of generality, the range of prestresses is set as [−1, 1], and the elastic modulus and cross-sectional areas of members are both set as unit.

5.1. Example 1: Planar Pin-Jointed Assembly I

A planar pin-jointed assembly, as shown in Figure 2, is considered in this example. The system consists of eight nodes and seven members and has three mechanism modes and two self-stress states. It has been analytically studied by Calladine and Pellegrino [7] and later used as an example in numerical prestress design by the heuristic algorithms [13, 15, 17].

The objective function f₂ is adopted in this example. Two computations using the prestress stability condition and the super stability condition, respectively, are conducted. In addition, (i.e., ) is used in both computations.

The normalized computation solutions together with the analytical result given by Calladine and Pellegrino [7] are listed in Table 1. It is worth noting that no repetitive runs are needed in both computations because the primal-dual interior-point method is a deterministic algorithm rather than a stochastic algorithm.

Let C1 and C2 denote the two computation cases separately. It is found that computations C1 and C2 have the same solution, which means that the solution is not only prestress stable but also super stable. It is also shown that the solution obtained by SDP exactly agrees with the analytical result reported by Calladine and Pellegrino [7].

5.2. Example 2: Planar Pin-Jointed Assembly II

In order to demonstrate that the positive definiteness of the reduced form of the geometrical stiffness matrix is not a sufficient condition for the stability of kinematically indeterminate structures, a planar pin-jointed assembly (Figure 3) used by Ohsaki and Zhang [36] is considered here. This system has one self-stress state and one mechanism mode.

No objective function was used in the computations carried out in this example. At first, the prestress stability condition was adopted in computation C1. Then, the general stability condition instead of the prestress stability condition was adopted in computation C2. The results of the two computations are listed in Table 2 where and denote the minimum eigenvalue of matrices and , respectively.

It can be seen that the system found in computation C1 is prestress stable () but does not satisfy the general stability condition (). This result demonstrates that a prestress stable system does not necessarily lead to a general stable system, and positive definiteness of is not a sufficient condition for the stability of kinematically indeterminate structures. When the general stability condition is adopted (computation C2), the found system is a general stable system () and of course also a prestress stable system ().

5.3. Example 3: Three-Dimensional Pin-Jointed Assembly

A three-dimensional pin-jointed assembly, as shown in Figure 4, is considered in this example. This system consists of 40 nodes and 132 members and has one mechanism mode and nineteen self-stress states. In this assembly, , , and , which means that all the members are divided into two groups: members 1–96 in tension and members 97–132 in compression.

(a)

(b)

(c)

This assembly consists of nine identical four-strut prism tensegrity units. The prism tensegrity unit has one self-stress state which can be analytically determined [9]. The analytical solution of the unit can be extended to the whole assembly due to the symmetry of the whole system. The solution obtained based on the analytical solution of the tensegrity unit is a feasible solution and is listed in Table 3 where T_s and T_c are the number of magnitudes of compression prestress and the number of magnitudes of tension prestress, respectively. Because there is more than one self-stress state in the whole assembly, there may be other feasible prestresses besides the analytical one.

This pin-jointed assembly has been used as an example by Chen et al. [17] to search stable prestresses by using ACS algorithms. Chen et al. carried out 60 individual runs, and all of them converge to the analytical solution. The average runtime for the runs was 1.981 s. Because of the stochastic basis of ACS algorithms, the computation efficiency and the solution quality of ACS algorithms are sensitive to parameter settings such as the number of ants used [17]. Four computations using different stability conditions and objective functions are carried out in this example, as shown in Table 4 where “Yes” means that the corresponding item is adopted by the computation and “No” means that the corresponding item is not adopted by the computation. No repetitive runs are needed because the algorithm used in this paper is a deterministic algorithm. Table 5 gives the results of the four computations. , , and denote minimum eigenvalue of matrices , , and , respectively. The runtime used by each computation is also given in the table.

When the prestress stability constraint is used (computation C1), the found system is a prestress stable system () but not a super stable system (). When super stability constraint is used (computation C2), the minimum eigenvalue of geometrical stiffness matrix equals to zero, which means that is positive semidefinite, and thus, the corresponding system is a super stable system. In both solutions of computation C1 and computation C2, the numbers of prestress magnitudes in members (i.e., T_s and T_c) are large. This observation indicates that the final prestresses in members are very irregular. When objective function f₁ is used (computation C3), the found prestresses become more regular. The results of member prestresses corresponding to this computation case are listed in Table 6. It can be seen from Table 6 that the normalized solution of computation C3 is the same as the analytical result given in Table 3. The purpose of computation C4 is to maximize the minimum eigenvalue of tangent stiffness matrix K. It can be seen that although the found prestresses in computation C4 is very irregular, the value of increases a lot compared with other computations, which means that the corresponding system has higher stiffness.

It is also shown that the runtime used in each computation is less than 0.5 s. In particular, the runtime for the computation C3 which obtains the solution same to the analytical solution is 0.299 s which is considerably less than the average runtime 1.981 s needed by the ACS algorithms [17]. Noting that different computers have been used in this paper and in the previous study [17], the above comparison on the runtimes is not rigorous. But emphasis is also given to the fact that both runtime needed and solution obtained by the proposed approach are assured and reproducible.

6. Conclusions

Finding appropriate prestresses which are able to stabilize the system is an important step in design of a tensegrity structure. In this paper, this problem has been formulated into a semidefinite programming (SDP) model. In the SDP model, different stability criteria and different objective functions can be adopted to realize specific design intentions. The proposed approach is verified by three typical examples.

As an optimization-based framework, the SDP model can be used to get more even prestress distribution and larger system stiffness. Besides, the SDP model is solvable in polynomial time, and the optimality of the solution is theoretically guaranteed. Compared to the previous heuristic and stochastic algorithm-based approaches, the approach proposed in this paper is based on a deterministic algorithm. For a given prestress design problem, the solution obtained by the proposed approach is unique and is totally reproducible. Hence, no trial or repeated computations are needed in the solving process. It is also worth noting that the SDP formulation is compatible with the MILP formulation used for topology optimization problem of tensegrity structures. It is believed that this work will be helpful in incorporating the stability test into the topology optimization model within the frame of mathematical programming. This will be the interest of the authors’ future work.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

Acknowledgments

This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFC0806100) and Natural Science Foundation of Zhejiang Province (Grant No. LR17E080001).

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Copyright © 2019 Yafeng Wang and Xian Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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