Synthesis of tensegrity structures of desired shape using constrained minimization

There are two types of problems in tensegrity design: (i) form-finding when the tensegrity shape is not specified and (ii) synthesis when the tensegrity shape is specified. We address synthesis problems in this paper. We first formulated and solved an optimization problem to synthesize tensegrity structures of specified shape when the connectivity of the elements (bars and cables) is known a priori. We minimize the error in force-balance at the vertices in the desired equilibrium configuration by using force densities as the design variables. This constrained minimization problem enabled us to synthesize a known asymmetric tensegrity arch and a hitherto unknown tensegrity of biconcave shape similar to that of a healthy human red blood cell. We also extend the above method to a reduced order optimization problem for synthesizing complex symmetric tensegrity structures. Using this approach, we synthesized a truncated dodecahedron inside another truncated dodecahedron to emulate a nucleus inside a cell. We use a restricted global structure on an already available two-step mixed integer linear programming (MILP) topology optimization formulation to synthesize a non-convex tensegrity structure when only the coordinates are provided. We further improve this two-step MILP to a single-step MILP. We also present static analysis of a tensegrity structure by minimizing the potential energy with unilateral constraints on the lengths of the cables that cannot take compressive loads. Furthermore, we use this method to synthesize a tensegrity table of desired height and area under a predefined load. The prototypes of three synthesized tensegrities were made and validated.     

Exploring new tensegrity structures via mixed integer programming

A tensegrity structure is a prestressed pin-jointed structure consisting of continuously connected tensile members (cables) and disjoint compressive members (struts). Many classical tensegrity structures are prestress stable, i.e., they are kinematically indeterminate but stabilized by introducing prestresses. This paper presents a procedure for generating various prestress stable tensegrity structures. This method is based on truss topology optimization and does not require connectivity relation of cables and struts of a tensegrity structure to be known in advance. Unlike the conventional form-finding methods, the locations of nodes are fixed throughout optimization. The optimization problem with the constraints expressing the definition of tensegrity structure, kinematical indeterminacy, and symmetry of configurations is formulated as a mixed integer linear programming (MILP) problem. Numerical experiments demonstrate that various tensegrity structures can be generated from one given initial structure by solving the presented MILP problems by using a few control parameters.     

Design and control of tensegrity robots for locomotion

The static properties of tensegrity structures have been widely appreciated in civil engineering as the basis of extremely lightweight yet strong mechanical structures. However, the dynamic properties and their potential utility in the design of robots have been relatively unexplored. This paper introduces robots based on tensegrity structures, which demonstrate that the dynamics of such structures can be utilized for locomotion. Two tensegrity robots are presented: TR3, based on a triangular tensegrity prism with three struts, and TR4, based on a quadrilateral tensegrity prism with four struts. For each of these robots, simulation models are designed, and automatic design of controllers for forward locomotion are performed in simulation using evolutionary algorithms. The evolved controllers are shown to be able to produce static and dynamic gaits in both robots. A real-world tensegrity robot is then developed based on one of the simulation models as a proof of concept. The results demonstrate that tensegrity structures can provide the basis for lightweight, strong, and fault-tolerant robots with a potential for a variety of locomotor gaits.     

Developing intelligent tensegrity structures with stochastic search

Since structural control of civil structures was first proposed in 1972, most research and applications have focused on enhancing the safety of structures under extreme conditions. This paper introduces a new direction in structural control: the use of computational methods and explicitly defined knowledge to improve serviceability and maintenance of civil structures. The objectives of such structures, called intelligent structures, are to maintain and improve structural performance by recognizing changes in behaviors and loads, adapting the structural geometry to meet defined goals, and using past events to improve future performance. A computational framework based on intelligent control methodology is presented that combines reasoning from explicit knowledge, search, learning and planning to illustrate a vision for intelligently controlled civil structures. This is applied to enable global shape control of an adjustable tensegrity structure using a combination of simulated annealing search, dynamic relaxation analysis and structural measurements. Active tensegrity structures are modular, reusable cable structures that do not require expensive anchorages or steep slopes. This provides an innovative solution for temporary structures. Particularly, when used for exhibitions, they could become part of an exhibition, rather than just house one. As part of a multi-stage project, results presented in this paper demonstrate the feasibility of building an intelligent tensegrity structure.     

Kinematic and Static Analyses of Statically Balanced Spatial Tensegrity Mechanism with Active Compliant Components

Tensegrity mechanisms are new type of mechanisms whose analysis is different from that of the conventional ones. This article chooses a six degree of freedom tensegrity mechanism with active compliant limbs and presents kinematic and static analysis of it. In this regard, two types of kinematic problems, the inverse and forward problems, are considered and solved. Also, this article shows that by using compliant components of tensegrity mechanism as active components, static balancing of the mechanism is achieved. This point can be considered as a new optimum approach for static balancing of the mechanisms by using tensegrity system concepts.     

Form-finding of tensegrity structures using double singular value decomposition

A numerical form-finding procedure of tensegrity structures is developed. The only required information is the topology and the types of members. The singular value decompositions of the force density and equilibrium matrices are performed iteratively to find the feasible sets of nodal coordinates and force densities which satisfy the minimum required deficiencies of these two matrices, respectively. An approach of defining a unique configuration of tensegrity structure by specifying an independent set of nodal coordinates is provided. An explanation is given for the preservation in self-equilibrium status of the tensegrity structures under affine transformation. Two- and three-dimensional examples are illustrated to demonstrate the efficiency and robustness of the proposed method in searching stable self-equilibrium configurations of tensegrity structures.     

Advanced automatic grouping for form-finding of tensegrity structures

An advanced automatic grouping method for form-finding of tensegrity structures is presented. In the proposed method, properties of self-equilibrium and stability in tensegrity structures can be obtained by using the force density method combined with a genetic algorithm. A constrained minimization problem is formulated using the standard deviation of the force density in the cables. As a result, the minimum number of member groups for tensegrity structures with automatic grouping can be obtained. This elicited regular tensegrity structures with uniform force density values. Moreover, the geometrical and mechanical parameters of tensegrity structures with multiple states of self-stress can be easily obtained by using the proposed method.     

基于动力学的张拉整体柔性臂数字孪生系统

连续型机器人因其具有柔顺大变形、灵巧运动等特点,已成为未来提升机器人安全性和交互性的发展趋势,而数字孪生是实现机器人-环境-人之间共融共存的重要技术保障.本文以张拉整体连续型柔性臂为研究对象,结合数字孪生和虚拟仿真等技术,让张拉整体柔性臂在虚拟空间和实际物理空间中得以深度融合.搭建数据通讯架构实现数据实时传输和驱动,以提升柔性臂与人的协同工作效率,并可在复杂的环境中通过碰撞检测反馈实现动态避障.进一步,开发了一款基于动力学的张拉整体柔性臂数字孪生系统,并通过虚实双向操控验证了所建系统的有效性,为机器人远程智能监测与控制提供了参考.     

A practical approach for nonlinear analysis of tensegrity systems

Tensegrity systems are lightweight structures composed of cables and struts. The nonlinear behavior of tensegrity systems is critical; therefore, the design of these types of structures is relatively complex. In the present study, a practical and efficient approach for geometrical nonlinear analysis of tensegrity systems is proposed. The approach is based on the point iterative method. Static equilibrium equations are given in nodes for subsystems, thus the maximum unknown displacement number in each step is three. Pre-stress forces in the system are taken into account in a tangent stiffness matrix, while similar calculations are carried out for each node in the system which has a minimum of one degree of freedom. In each iteration step, the values found in previous steps are used. When it reaches permissible tolerance of calculation, final displacements and internal forces are obtained. The structural behavior of the tensegrity systems were evaluated by the proposed method. The results show that the method can be used effectively for tensegrity systems.     

A minimal mass deployable structure for solar energy harvesting on water canals

This paper produces a design for a minimal mass, deployable support structure for a solar panel covering of water canals. The results are based upon the minimal mass properties of tensegrity structures. The efficient structure is a tensegrity system which has an optimal complexity (i.e. an optimal number of members) for minimal mass. This optimal complexity is derived in this paper, along with deployable schemes which are useful for construction, repairs, for Sun following, and for servicing. It is shown that the minimal structure naturally has deployable features so that extra mass is not needed to add the multifunctional features. The design of bridge structures with tensegrity architecture will show an optimal complexity depending only on material choices and external loads. The minimization problem considers a distributed load (from weight of solar panels and wind loads), subject to buckling and yielding constraints. The result is shown to be a Class 1 Tensegrity substructure (support structure only below the deck). These structures, composed of axially-loaded members (tension and compressive elements), can be easily deployable and have many port-able applications for small spans. The focus of this paper is an application of these minimal mass tensegrity concepts to design shading devices to prevent or reduce evaporation loss, while generating electric power with solar panels as the cover. While the economics of the proposed designs are far from finalized, this paper shows a technical solution that uses the smallest material resources, and shows the technical feasibility of the concept.     

Control‐oriented modeling and deployment of tensegrity–membrane systems

This study addresses control‐oriented modeling and control design of tensegrity–membrane systems. Lagrange's method is used to develop a control‐oriented model for a generic system. The equations of motion are expressed as a set of differential‐algebraic equations (DAEs). For control design, the DAEs are converted into second‐order ordinary differential equations (ODEs) based on coordinate partitioning and coordinate mapping. Because the number of inputs is less than the number of state variables, the system belongs to the class of underactuated nonlinear systems. A nonlinear adaptive controller based on the collocated partial feedback linearization (PFL) technique is designed for system deployment. The stability of the closed‐loop system for the actuated coordinates is studied using the Lyapunov stability theory. Because of system complexity, numerical tests are used to conduct stability analysis for the dynamics of the underactuated coordinates, which represents the system's zero dynamics. For the tensegrity–membrane systems studied in this work, analytical proof of zero dynamics stability remains an open theoretical problem. An H_∞ controller is implemented for rapid stabilization of the system at the final deployed configuration. Simulations are conducted to test the performance of the two controllers. The simulation results are presented and discussed in detail. Copyright © 2016 John Wiley & Sons, Ltd.     

Design of tensegrity structures using parametric analysis and stochastic search

Tensegrity structures are lightweight structures composed of cables in tension and struts in compression. Since tensegrity systems exhibit geometrically nonlinear behavior, finding optimal structural designs is difficult. This paper focuses on the use of stochastic search for the design of tensegrity systems. A pedestrian bridge made of square hollow-rope tensegrity ring modules is studied. Two design methods are compared in this paper. Both methods aim to find the minimal cost solution. The first method approximates current practice in design offices. More specifically, parametric analysis that is similar to a gradient-based optimization is used to identify good designs. Parametric studies are executed for each system parameter in order to identify its influence on response. The second method uses a stochastic search strategy called probabilistic global search Lausanne. Both methods provide feasible configurations that meet civil engineering criteria of safety and serviceability. Parametric studies also help in defining search parameters such as appropriate penalty costs to enforce constraints while optimizing using stochastic search. Traditional design methods are useful to gain an understanding of structural behavior. However, due to the many local minima in the solution space, stochastic search strategies find better solutions than parametric studies.     

Modelling and control of class NSP tensegrity structures

The method of constrained particle dynamics is used to develop a dynamic model of order 12 N for a general class of tensegrity structures consisting of N compression members (i.e. bars) and tensile members (i.e. cables). This model is then used as the basis for the design of a feedback control system which adjusts the lengths of the bars to regulate the shape of the structure with respect to a given equilibrium shape. A detailed design is provided for a 3-bar structure.     

Domain decomposition approach for non-smooth discrete problems, example of a tensegrity structure

Dealing with the interaction between numerous bodies, as in granular media, requires larger and larger computational resources. To this end, we develop a domain-decomposition-like method suited to discrete systems with diffuse non-smoothness. A multiscale enrichment completes the numerical strategy with the extra hope of bridging the gap between discrete and continuum models. The equilibrium of a tensegrity structure, closer to the continuous media case, is chosen to test this approach.     

基于张拉整体结构的连续型弯曲机械臂设计与研究

为实现对目标物体的缠绕捕获，利用张拉整体结构质量轻、变形大等特点，提出一种基于张拉整体结构的连续型机械臂的设计．本文首先设计连续型机械臂的结构，建立其力学模型．通过准静态和动态分析，对不同驱动形式下的连续型机械臂运动进行仿真，并在实验平台上验证所建力学模型的准确性，最后分析了其工作空间及奇异位姿．实验结果表明本文设计的连续型机械臂可以实现弯曲缠绕变形，满足对不同大小物体进行缠绕捕获的需求．     

Automated discovery and optimization of large irregular tensegrity structures

Tensegrities consist of disjoint struts connected by tensile strings which maintain shape due to pre-stress stability. Because of their rigidity, foldability and deployability, tensegrities are becoming increasingly popular in engineering. Unfortunately few effective analytical methods for discovering tensegrity geometries exist. We introduce an evolutionary algorithm which produces large tensegrity structures, and demonstrate its efficacy and scalability relative to previous methods. A generative representation allows the discovery of underlying structural patterns. These techniques have produced the largest and most complex irregular tensegrities known in the field, paving the way toward novel solutions ranging from space antennas to soft robotics.     

A modified symbiotic organisms search (mSOS) algorithm for optimization of pin-jointed structures

The paper introduces a modified symbiotic organisms search (mSOS) algorithm to optimization of pin-jointed structures including truss and tensegrity ones. This approach is refined from the original SOS with five modifications in the following three phases: mutualism, commensalism and parasitism. In the mutualism one, benefit factors are suggested as 1 to equally represent the level of benefit to each organism, whilst the best organism is replaced by a randomly selected one to increase the global search capability. With the aim of improving the convergence speed, randomly created coefficients in the commensalism phase are restricted in the range [0.4, 0.9]. Additionally, an elitist technique is applied to this phase to filter the best organisms for the next generation as well. Finally, the parasitism phase is eliminated to simplify the implementation and reduce the time-consuming process. To verify the effectiveness and robustness of the proposed algorithm, five examples relating to truss weight minimization with discrete design variables are performed. Additionally, two examples regarding minimization a function of eigenvalues and force densities of tensegrity structures with continuous design variables are considered further. Optimal results acquired in all illustrated examples reveal that the proposed method requires fewer number of analyses than the original SOS and the DE, but still gaining high-quality solutions. Furthermore, the mSOS also outperforms numerous other algorithms in available literature in terms of optimal solutions, especially for problems with a large number of design variables.     

Control and simulation of a tensegrity-based mobile robot

Tensegrity structures can provide a new approach to the construction of mobile robots with different shapes and properties that usual robots, wheeled or legged, do not have. Tensegrity are light, deformable structures that may be able to adapt their form to unconstrained environments. The main issue of this paper is twofold, first, to derive appropriate and general dynamic equations of motion to study the movement of such structures in the space; second to demonstrate, by means of simulation, that a tensegrity structure can execute any desired trajectory path by actuating some or all of its elements.