A unifying framework for form-finding and topology-finding of tensegrity structures
Introduction
Tensegrity is a novel structural system invented by Fuller [1]. The word ‘tensegrity’ is a contraction of the words “tensile” and “integrity”, which is referred to by Fuller as a continuous network of tension. Tensegrity is also referred to as “floating compression structures” by Snelson [2]. In the general definition, tensegrity is a kind of self-equilibrium system consisting of continuous tensile components and stabilized by introducing proper prestress. Since invented, tensegrity structures have been widely applied in art [2], [3], civil engineering [4], [5], active and deployable structures [6], [7], [8], robotics [9], [10], [11], and biomechanics fields [12], [13], etc. due to their unique advantages, such as aesthetics, large stiffness-to-mass ratio, deployability, reliability, and controllability.
From a structural design point of view, the first and foremost step for an application of a tensegrity structure is to determine a specific self-equilibrium configuration and feasible prestress distribution of the members, which is usually called form-finding [14]. Therefore, the form-finding of tensegrity structures has drawn great attention from researchers in different engineering and scientific fields. Concerning different application scenarios, many types of form-finding methods have been proposed. As the pioneers of tensegrity, Fuller [15] and Snelson [16] carried out early studies on the form-finding of regular tensegrity structures through analytical methods that are suitable for relatively simple tensegrity structures with high symmetry. To find more general tensegrity structures, other numerical form-finding methods were suggested. As a pioneering work of numerical form-finding, Schek [17] firstly proposed the so-called force density method for the form-finding of tensile structures; then Vassart and Motro [18] adopted force density method for the form-finding of tensegrity structures. Motro et al. [3] presented the dynamic relaxation method that has been reliably applied to solve nonlinear problems and then extend it to the form-finding of tensegrity structures. Based on the affine transformations of nodal coordinates, Masic et al. [19] proposed an algebraic method for the form-finding of tensegrity structures with shape constraints. Additionally, mathematical programming or optimization-based form-finding methods were also suggested by some [20], [21].
Following the pioneering and fundamental work mentioned above, many advanced form-finding methods were suggested in recent years. For example, Zhang and Ohaski [22] developed an iterative form-finding method based on the force density formulation, in which the rank deficiency condition of the force density matrix is enforced through an iterative process; Estrada et al. [23] and Tran and Lee [24] proposed an iterative method for the form-finding of tensegrity structures with a single self-stress state, in which the force density and equilibrium matrices are repeatedly computed and updated to determine a self-equilibrium configuration; further, Tran and Lee [25] proposed a method for the form-finding of tensegrity structures with multiple self-stress states by grouping members. Li et al. [26] developed a Monte Carlo form-finding method for large-scale tensegrity structures. Koohestani [27] developed a new formulation for the form-finding of tensegrity structures in which the Cartesian components of element lengths are taken as primary variables. Ohsaki and Zhang [28] studied the form-finding and folding analysis of tensegrity structures through a nonlinear programming approach. Furthermore, metaheuristics [29] have also been applied for the form-finding of tensegrity structures, such as the genetic algorithm [30], [31], [32], [33] and the ant colony algorithm [34]. More recently, Yuan et al. [35] used the Levenberg–Marquardt method to solve the nonlinear equilibrium equation to realize form-finding of tensegrity structures; Koohestani [36] and Zhang et al. [37] proposed new analytical methods to address form-finding problems of tensegrity structures through Faddeev-LeVerrier algorithm and the determinant of force density matrix, respectively; Xu et al. [38] proposed a form-finding method combining the force density method and the mixed-integer nonlinear programming, in which both the topology (i.e., member connectivities) and force densities are taken as design variables; Chen and Feng [39] applied group theory in the form-finding to reduce the computational complexity in designing symmetric tensegrity structures; Xue et al. [72] transformed the form-finding problem to a stationary point problem which was then solved by an intermediate function; Wang et al. [73] presented a general computational framework for the form-finding of tensegrity structures based on the rank minimization of force density matrix. The form-finding methods summarized above have shown their robustness and efficacy on the form-finding of symmetric or regular tensegrity structures with single or a small number of self-stress states. However, for the form-finding of complex or irregular tensegrity structures, the unilaterality properties of the member forces (i.e., struts in compression and cables in tension) and the required rank deficiency conditions of certain matrices are often difficult to be guaranteed, especially for tensegrity structures with many self-stress states. Furthermore, even though lots of form-finding methods have been proposed, different form-finding methods are usually designed to deal with different special types of form-finding problems. For examples, methods for the form-finding of tensegrity structures with single self-stress state [23], [24], and with multiple self-stress states [25], [40]; methods for the form-finding of regular tensegrity structures [36], [37], [41], irregular tensegrity structures [26], [30], [42], and symmetric tensegrity structures [43], [44]. However, even though different form-finding methods have shown their effectiveness to the corresponding specified types of problems, to the best of the authors’ knowledge, no method exists until now that is capable of addressing all types of the form-finding problems mentioned above. Also, in practical design it might be difficult for the designer to choose a suitable method when he or she does not have too much understanding of the problem. Therefore, developing a unifying and versatile approach that can handle all the types of the form-finding problems is meaningful and necessary. This is one of the motivations and objectives of this work.
On the other hand, topology-finding (i.e., the nodal positions are specified beforehand and fixed during the design process, and the member topology is taken as the primary design variable) has drawn more and more attention in recent years in designing tensegrity structures with specified shapes [45], [46], [47], [48], [49], [50], [51][64], [65], [70], [71]. However, due to the complexity and difficulty of the topology-finding problem, the number of available and effective methods for the topology-finding of tensegrity structures is still limited. Estrada et al. [23] ever pointed out that the application of form-finding methods to topology design problems would be an interesting and meaningful research direction. Unfortunately, even though numerous studies have been carried out on the form-finding of tensegrity structures, seldom work until now has ever investigated the extension of form-finding methods to the topology-finding of tensegrity structures. In fact, topology-finding can be seen as the inverse problem of form-finding [52] but all the existing studies treat them independently and use different schemes to deal with them. Building a bridge between form-finding and topology-finding of tensegrity structures and developing a unifying approach to handle these two problems is also part of the motivation of this work.
In this paper, based on the force density formulation and the stability condition of tensegrity structures, the form-finding and topology-finding of tensegrity structures are transformed into linear matrix inequalities (LMIs) formulations with matrix rank constraints. A Newton-based algorithm is employed to solve the rank-constrained LMIs efficiently. Various examples are carried out to verify the efficiency, accuracy, and versatility of the proposed approach.
The paper is organized as follows: Section 2 introduces the basic formulations of the force density method and necessary rank deficiency conditions for the force density matrix of tensegrity structures. Section 3 reformulates the form-finding problem into a rank constrained linear matrix inequalities (LMI) problem. Section 4 gives the algorithm to solve the rank constrained LMI-based problem and proposes two methods to determine the nodal coordinates. Section 5 reformulates the topology-finding of tensegrity structures into a rank constrained LMI-based problem. Section 6 gives various numerical examples to illustrate the efficiency, accuracy, and versatility of the proposed approach. Finally, Section 7 summarizes and concludes the paper.
Section snippets
Basic formulations of tensegrity structures
Tensegrity structures are pin-jointed systems consisting of members that can only carry axial forces (struts in compression and cables in tension). Without loss of generality, considering a tensegrity structure with n nodes and m members in three-dimensional space, use x, y, and z to denote the nodal coordinate vectors and use and to denote the internal force and length of member i respectively. The force density of member i is defined as , and use to denote
Problem definition
A tensegrity structure is mainly determined by three factors: topology (i.e., member connectivities), geometry (i.e., nodal coordinates: x, y, and z), and force densities (i.e., member forces divided by their lengths: q). In a form-finding process, the topology is given advance and does not change throughout the process, and the geometry and force densities are unknown and thus are treated as design parameters to achieve a stable tensegrity structure. Note that in form-finding, which members
Algorithm for determining force densities
Firstly, some basic concepts and notation definitions that will be used in the algorithm are introduced as follows.
Let denote the set of real symmetric matrix pairs as
Let denote the set of real symmetric positive-semidefinite matrices, i.e.,and, for each integer s, let denote the set of real symmetric positive-semidefinite matrices with a rank of s, i.e.,
Define
Then is
Reformulating topology-finding as rank-constrained linear matrix inequalities
Topology-finding can be seen as the inverse problem of form-finding. In topology-finding, the nodal coordinates are given in advance and do not change in the design process, the topology (i.e., member connectivities) are the primary design variables. In the existing studies on topology-finding [45], [46], [47], [48], [49], [64], [65], which are based on the mixed-integer programming framework, additional binary variables should be introduced to describe the topology. Here, we show that the
Tensegrity structures with a single self-stress state
In this example, two tensegrity structures are presented to illustrate the effectiveness of the proposed method for the form-finding of tensegrity structures with a single self-stress state.
- (a)
Two-dimensional six-node tensegrity structure
A two-dimensional system consisting of six nodes and nine members is considered. The topology is given in Fig. 2. The thick lines and thin lines act as struts and cables, respectively, and this representation will be used throughout the paper.
The algorithm in the
Conclusions and future work
This work proposes a unifying framework for the form-finding and topology-finding of tensegrity structures. The main conclusions and contributions can be summarized as follows:
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Both the form-finding and topology-finding of tensegrity structures have been appropriately formulated into rank-constrained LMI problems and thus can be solved through a uniform way.
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The Newton-like algorithm employed is an efficient and robust method to solve the generated rank-constrained LMI problems.
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The proposed
Data reproduction
All data used in the study have been given in the paper. The source code is available upon request from the corresponding author.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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). The EPFL Applied Computing and Mechanics Laboratory (IMAC) is thankfully acknowledged for its support during the review process of this article.
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