p. 569-576In this paper we propose an optimization-based approach to finding a tensegrity structure
based on the ground structure method. We first solve a problem which maximizes the number of struts over the self-equilibrium condition and the discontinuity condition of struts.
Subsequently we solve the minimization problem of the number of cables in order to remove redundant self-equilibrium modes. The optimization problem at each step can be formulated as a mixed integer programming (MIP) problem. The method does not require any connectivity information of cables and struts to be known in advance, while the obtained tensegrity structure is guaranteed to satisfy the discontinuity condition of struts rigorously.Ehara, S.; Kanno, Y. (2009). A mixed 0-1 programing approach to topology-finding of tensegrity structures. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/654

EHARA, Shintaro

KANNO, Yoshihiro

English

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Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, ValenciaEvolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures28 September - 2 October 2009, Universidad Politecnica de Valencia, SpainAlberto DOMINGO and Carlos LAZARO (eds.)A mixed 0–1 programming approach totopology-finding of tensegrity structuresShintaro EHARA a and Yoshihiro KANNO Department of Mathematical Informatics,University of Tokyo, Bunkyo, Tokyo 113-8656, Japan.e-mail: kanno@mist.i.u-tokyo.ac.jpa Bank of Tokyo-Mitsubishi UFJ, Ltd.AbstractIn this paper we propose an optimization-based approach to finding a tensegrity structurebased on the ground structure method. We first solve a problem which maximizes the num-ber of struts over the self-equilibrium condition and the discontinuity condition of struts.Subsequently we solve the minimization problem of the number of cables in order to removeredundant self-equilibrium modes. The optimization problem at each step can be formulatedas a mixed integer programming (MIP) problem. The method does not require any connec-tivity information of cables and struts to be known in advance, while the obtained tensegritystructure is guaranteed to satisfy the discontinuity condition of struts rigorously.Keywords: Tensegrity; Shape design; Self-equilibrated configuration; Mixed integer pro-gram; Topology optimization.1. IntroductionTensegrity structure is a class of tension structures, which consists of pin-jointed memberstransmitting only axial forces. According to the definition given by Fuller [3], a tensegritystructure is a prestressed pin-jointed structure consisting of continuous tensile members (ca-bles) and discontinuous compressive members (struts). Later, the concept of tensegrity hasbeen generalized extensively; see, e.g., [7] and the references therein.In this paper we propose an optimization-based approach to find a tensegrity structure whichrigorously satisfies the discontinuity condition of struts. It is emphasized that our method569Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, ValenciaEvolution and Trends in Design, Analysis and Construction of Shell and Spatial Structuresdoesnot require any information of the connectivity of cables and struts to be known inadvance.Therehave been many studies on form-finding of tensegrity structures; see, e.g., the reviewpapers [5, 9] and the references therein. However, as input data, those methods require tospecify the connectivity of members as well as the labeling indicating whether each memberis to be a cable or strut. Based on the group representation theory, a systematic approach waspresented to enumerate topologies, i.e. connectivities and labelings of members, of tenseg-rity structures which share a common group-theoretic symmetry property [1]. Particularly,for tensegrity structures with a rotational symmetry property, form-finding methods utilizingsuch a symmetry property have been proposed [6, 8, 10]. However, these methods basedon the group theory assume that the group symmetry underlying a family of tensegrities isknown in advance, i.e. it is necessary to specify a symmetry property of tensegrity structuresbefore the form-finding process. Thus, it still remains as a challenging problem to find acompletely new pair of the connectivity and the labeling of members of a tensegrity structuresatisfying its definition rigorously.We propose an approach to find a tensegrity structure based on the ground structure method.Given a pin-jointed structure with the specified locations of nodes and sufficiently manycandidate members, our problem is to find a labeling of members which indicates whethereach member is to be a cable, a strut, or removed, so that the resulting structure becomesa tensegrity structure. It is emphasized that our approach does not require any labeling ofmembers or any underlying group symmetry property to be known in advance.Our approach consists of two parts; at the first step we find a self-equilibrium mode of axialforces satisfying the discontinuous condition of struts, while at the second step we removeredundant cables from the structure obtained at the first step. At each step we solve amixedinteger programming(MIP) problem.2. Maximization of number of strutsBased on the conventional ground structure method, consider a pin-jointed structure in thethree-dimensional space, which consists of the nodes with the specified locations and suffi-ciently many members that can exist. LetV andE denote the set of nodes and the set ofmembers, respectively. We denote byq = (qi) ∈ R|E| the vector of the member axial forces.Note that a cable and a strut transmit only compressive and tensile forces, respectively, andhence we haveqi > 0 for a cable andqi < 0 for a strut.Although there exist various definitions of tensegrity structures [7], we employ the onewhich consists of the self-equilibrium condition and the discontinuity condition of struts.Let E(n j) ⊂ E denote the set of indices of the members which are connected to the noden j ∈V. We denote byH ∈R3|V|×|E| the equilibrium matrix. Since any two struts do not share570Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, ValenciaEvolution and Trends in Design, Analysis and Construction of Shell and Spatial Structuresacommon node, the self-equilibrium mode,q̂, should satisfyq̂ 6= 0 : Hq̂ = 0, (1)|{i ∈ E(n j) | q̂i < 0}| ≤1, ∀n j ∈V. (2)We next introduce a binaryvariable,xi ∈ {0,1}, for each member in order to indicate whetherthe memberi is a strut or not, i.e.xi = 1 implies that the memberi is to be a cable, whilexi = 0 implies that the memberi is to be either a cable or a removed member. LetM andεbe positive constants, whereM is sufficiently large, i.e. 0< ε M. It is easy to see that thecondition (2) is rewritten as−Mxi ≤ qi ≤M(1−xi)− ε, ∀i ∈ E, (3)∑i∈E(n j )xi ≤ 1, ∀n j ∈V, (4)wherexi ∈ {0,1} (∀i ∈ E).Note that the total number of struts is given by∑i∈E xi . Since it is natural to attempt to chooseas many struts as possible from the given ground structure, we consider the maximizationproblem of the number of struts. From (3) and (4), the maximization problem of the numberof struts of a tensegrity structure is formulated as(MIP-1) : maxq,x ∑i∈Exi (5a)s.t. Hq = 0, (5b)−Mxi ≤ qi ≤M(1−xi)− ε, ∀i ∈ E, (5c)∑i∈E(n j )xi ≤ 1, ∀n j ∈V, (5d)x ∈ {0,1}|E|. (5e)Note that the problem (5) is a 0–1 mixed integer programming problem. Let(q̂, x̂) denotethe optimal solution of (5). Observe thatq̂ = 0 implies x̂ = 0. Since we attempt to maximize∑i∈E xi ≥ 0 in the problem (5), at its optimal solution the condition (1) is satisfied, unlessq = x = 0 is a unique feasible solution of (5).The characteristics of members are determined byq̂i > 0 ⇒ the memberi is to be a cable;q̂i < 0 ⇒ the memberi is to be a strut;q̂i = 0 ⇒ the memberi is to be removed.(6)571Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, ValenciaEvolution and Trends in Design, Analysis and Construction of Shell and Spatial StructuresThuswe can obtain a tensegrity structure satisfying the discontinuity condition of struts.3. Minimization of number of cablesWehave shown in section 2 that a tensegrity structure satisfying the discontinuity conditionsof struts, as well as the self-equilibrium condition, can be obtained by solving (MIP-1) in (5).Since (MIP-1) requires only the existence of self-equilibrium mode of axial forces which sat-isfies the discontinuity condition of struts, the self-equilibrium mode of the obtained tenseg-rity structure is not unique in general. In such a case, there may exist some cables whichcan be removed from the tensegrity structure without changing the locations of struts. Thismotivates us to consider in this section the minimal tensegrity for the given set of struts.Let (q̂, x̂) denote the optimal solution of (MIP-1). DefineEcable, Estrut⊂ E byEcable= {i ∈ E | q̂i > 0}, (7)Estrut = {i ∈ E | q̂i < 0}. (8)Let Ēcable be the set of candidates of cables, whereĒcable⊇ Ecable and Ēcable∪Estrut = /0.Consider a ground structure, which consists of the set of nodesV and set of membersEstrut∪Ēcable. We say that a tensegrity structure satisfying the discontinuity condition of struts is aminimal tensegrity if it includes no redundant cable. This condition is rigorously stated asq∣∣∣∣∣∣∣∣Hq = 0qi′ = 0qi ≥ 0 (∀i ∈ Ēcable\{i′})q j < 0 (∀ j ∈ Estrut)= /0, ∀i′ ∈ Ēcable. (9)For eachi ∈ Ēcable, we introduce a binary variableyi ∈ {0,1} which indicates whether thecablei can be removed or not. Consider the linear inequalities0≤ qi ≤Myi , (10)whereM is a sufficiently large positive constant. Sinceyi ∈ {0,1}, we see that (10) is equiv-alent toqi ≥ 0, (11){qi > 0 ⇒ yi = 1,qi = 0 ⇐ yi = 0.(12)We remove the memberi if yi = 0. From (12) it follows that by minimizingyi over (10),yibecomes equal to one if and only ifqi > 0. Consequently, the minimization problem of the572Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, ValenciaEvolution and Trends in Design, Analysis and Construction of Shell and Spatial Structuresnumberof cables with the specifiedEstrut is formulated as(MIP-2) : minq,y ∑i∈Ēcableyi (13a)s.t. Hq = 0, (13b)qi ≤−ε, ∀i ∈ Estrut, (13c)0≤ qi ≤Myi , ∀i ∈ Ēcable, (13d)y ∈ {0,1}|Ēcable|, (13e)where 0< ε M. Note that (13) is a 0–1 mixed integer programming problem.Let (q∗,y∗) denote the optimal solution of the problem (13). Observe that in (13) we attemptto minimize the sum ofyi , from which and (12) we obtain{q∗i > 0 ⇔ y∗i = 1,q∗i = 0 ⇔ y∗i = 0,at the optimal solution. Hence, the optimal value of the problem (13) is equal to the numberof remaining cables. By removing cables corresponding toy∗i = 0, we can obtain the minimaltensegrity which does not include any redundant cables.4. Numerical examplesConsider a ground structure shown in Figure 1, where|V|= 10 and|E|= 41. This structureconsists of three layers, where the top and bottom layers are in triangular shapes and themiddle one in a rectangular shape. Note that the configuration of this structure is symmetricby the reflection with respect to theyz-plane and the rotation around thex-axis with the angleπ. It is often that the symmetry of a tensegrity configuration causes the rank-deficiency of theequilibrium matrix, and hence the conventional Maxwell counting rule does not necessarilyhold [2].We solve (MIP-1) and (MIP-2) by using CPLEX Ver.11.2 [4] with the default settings. Theoptimal solution of (MIP-1) is shown in Figure 2(a), which consists of 5 struts and 18 cables.For finding the tensegrity with the minimum number of cables, we next solve (MIP-2), wherethe ground structure for (MIP-2) is given by Figure 1, i.e.Ēcable := E \Estrut in (13). Theoptimal solution of (MIP-2) is illustrated in Figure 2(b), which has 5 struts and 16 cables. Wesee that the tensegrity in Figure 2(b) satisfies 21= 5+16< 3|V|−6 = 24, which implies thatthe conventional Maxwell rule does not hold. However, this tensegrity can be stabilized byintroducing prestresses, which is verified from an actually constructed model.5. ConclusionsIn this paper we have presented a numerical method for finding a tensegrity structure based573Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, ValenciaEvolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures  	XXXXz60xyzFigure1: A ground structure with 10 nodes and 41 members.on the ground structure method. In our method we solve the two MIPs (mixed integer pro-gramming problems) sequentially in order to find a tensegrity structure which satisfies theself-equilibrium condition as well as the discontinuity condition of struts.At the first step we solve an MIP which maximizes the number of struts over the self-equilibrium condition and the discontinuity condition of struts. Note that it is very difficultto deal with the discontinuity condition of struts rigorously by existing methods for designof tensegrities. We have shown that this condition can be written as a system of linear in-equalities in terms of the axial forces and some additional binary variables. Since the optimalsolution obtained at the first step has some self-equilibrium modes in general, we solve anMIP which minimizes the number of cables as the second step.AcknowledgmentThe second author was supported in part by Global COE Program “The research and trainingcenter for new development in mathematics”, MEXT, Japan.574Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, ValenciaEvolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures(a)The optimal solution of (MIP-1). (b) The optimal solution of (MIP-2).Figure 2: The optimal solutions of a 10-node example.References[1] Connelly, R., Back,A.: Mathematics and tensegrity.American Scientist,86, 142–151(1998).[2] Connelly, R., Fowler, P.W., Guest, S.D., Schulze, B., Whiteley, W.J.: When is a sym-metric pin-jointed framework isostatic?International Journal of Solids and Structures,46, 762–773 (2009).[3] Fuller, R.B.:Synergetics: Explorations in the Geometry of Thinking. Collier McMillian,London (1975).[4] ILOG: ILOG CPLEX 11.2 User’s Manual and Reference Manual. ILOG, S.A. (2008).http://www.ilog.com.[5] Juan, S.H., Mirats Tur, J.M.: Tensegrity frameworks: Static analysis review.M chanismand Machine Theory,43, 859–881 (2008).[6] Masic, M., Skelton, R.E., Gill, P.E.: Algebraic tensegrity form-finding.InternationalJournal of Solids and Structures,42, 4833–4858 (2005).[7] Motro, R.: Tensegrity. Kogan Page Science, London (2003).[8] Sultan, C., Corless, M., Skelton, R.: Symmetrical reconfiguration of tensegrity struc-tures.International Journal of Solids and Structures.39, 2215–2234 (2002).575Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, ValenciaEvolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures[9] Tilbert, A.G., Pellegrino. S., Review of form-finding methods for tensegrity structures.International Journal ofSpaceStructures, 18, 209–223 (2003).[10] Zhang, J.Y., Ohsaki, M., Kanno, Y.: A direct approach to design of geometry and forcesof tensegrity systems.International Journal of Solids and Structures, 43, 2260–2278(2006).576

A mixed 0-1 programing approach to topology-finding of tensegrity structures

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A mixed 0-1 programing approach to topology-finding of tensegrity structures

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