Tensegrity Structures Research Articles

A multistable tensegrity structure with a gripper application

Multistable tensegrity structures are a new interesting class of compliant prestressed structures. Due to their beneficial properties, these structures are attractive for robotic applications. In this paper a gripper is introduced, which is based on a mechanical compliant, multistable tensegrity structure. The underlying tensegrity structure of the considered gripper is investigated in detail. The influence of the member parameters on the existence of multiple states of self-equilibrium and the mechanical compliance is discussed with the help of static geometric nonlinear analyses, based on the Finite Element Method. The dynamical behaviour of the structure, during the change between the equilibrium configurations, is considered. Therefor the dynamical equations of motion are derived. Then gripper arms are added to the tensegrity structure to obtain a gripper. Different actuation principles for the gripper are discussed. Additionally, a prototype of the gripper has been built and is presented, as well as, selected experimental results.

Joint Friction during Deployment of a Near-Full-Scale Tensegrity Footbridge

Most deployable structures, such as operable roofs and masts, move over one-degree of freedom. This paper describes a structure that involves loosely coupled movement over several degrees of freedom. Analysis models of these structures are typically inaccurate. A source of inaccuracy is joint friction. Static and kinetic friction are studied experimentally and analytically. Simulations have been modied to account for these effects and two methods are used to quantify friction effects. Friction has a significant effect on the movement of the tensegrity structure. Of two candidate parameters, cable tension and interior cable angle, cable angle is the factor that best characterizes friction effects. Values of static and kinetic friction coefficients are not significantly different in this context and this leads to a reduction in the complexity of the friction model for simulation. Including friction effects in analysis decreases the difference between simulations and tests. Lastly, strut elements of the tensegrity structure are most critically affected by friction.

Effects of Self-Stress Distributions on Stability of Tensegrity Structures

Tensegrity systems are composed of any given set of cables connected to a set of struts in which the cables connectivity must be able to stabilize the configuration. Self-stresses contribute to the rigidity and stability of the system. Therefore, self-stress distribution has a dominant effect on the stability behavior of these systems. In this study, the stability behavior of plane double-layer tensegrity systems considering different distributions of self-stresses is evaluated. Based on the results obtained, collapse mechanisms, load carrying capacities, stiffness of the systems and slackening of the cables are affected by self-stress distribution. Therefore, self-stress design is a key factor that plays a dominant role on the stability behavior of tensegrity systems. These results can lead to the suggestion of some guidelines on the selection of self-stress distribution for the design of tensegrity systems against instability.

Full and folded forms: a compact review of the formulation of tensegrity structures

This piece of work presents a simple and compact overview of the design problem of tensegrity structures in two and three dimensions. The main aim of this study is to present the design and calculation of tensegrity structures in their simplest form, avoiding unnecessary simplifications that can rule out solutions, as has happened up until now. As a result of the simplicity of the procedure, two types of tensegrity structures are obtained for the same initial topology: full and folded forms. Several examples are shown.

Soft Spherical Tensegrity Robot Design Using Rod-Centered Actuation and Control

This paper presents the design, analysis, and testing of a fully actuated modular spherical tensegrity robot for co-robotic and space exploration applications. Robots built from tensegrity structures (composed of pure tensile and compression elements) have many potential benefits including high robustness through redundancy, many degrees-of-freedom in movement and flexible design. However, to take full advantage of these properties, a significant fraction of the tensile elements should be active, leading to a potential increase in complexity, messy cable, and power routing systems and increased design difficulty. Here, we describe an elegant solution to a fully actuated tensegrity robot: The TT-3 (version 3) tensegrity robot, developed at UC Berkeley, in collaboration with NASA Ames, is a lightweight, low cost, modular, and rapidly prototyped spherical tensegrity robot. This robot is based on a ball-shaped six-bar tensegrity structure and features a unique modular rod-centered distributed actuation and control architecture. This paper presents the novel mechanism design, architecture, and simulations of TT-3, an untethered, fully actuated cable-driven six-bar spherical tensegrity robot. Furthermore, this paper discusses the controls and preliminary testing performed to observe the system's behavior and performance and is evaluated against previous models of tensegrity robots developed at UC Berkeley and elsewhere.

Stem cell mechanical behaviour modelling: substrate\u2019s curvature influence during adhesion

Recent experiments hint that adherent cells are sensitive to their substrate curvature. It is already well known that cells behaviour can be regulated by the mechanical properties of their environment. However, no mechanisms have been established regarding the influence of cell-scale curvature of the substrate. Using a numerical cell model, based on tensegrity structures theory and the non-smooth contact dynamics method, we propose to investigate the mechanical state of adherent cells on concave and convex hemispheres. Our mechanical cell model features a geometrical description of intracellular components, including the cell membrane, the focal adhesions, the cytoskeleton filament networks, the stress fibres, the microtubules, the nucleus membrane and the nucleoskeleton. The cell model has enabled us to analyse the evolution of the mechanical behaviour of intracellular components with varying curvature radii and with the removal of part of these components. We have observed the influence of the convexity of the substrate on the cell shape, the cytoskeletal force networks as well as on the nucleus strains. The more convex the substrate, the more tensed the stress fibres and the cell membrane, the more compressed the cytosol and the microtubules, leading to a stiffer cell. Furthermore, the more concave the substrate, the more stable and rounder the nucleus. These findings achieved using a verified virtual testing methodology, in particular regarding the nucleus stability, might be of significant importance with respect to the division and differentiation of mesenchymal stem cells. These results can also bring some hindsights on cell migration on curved substrates.

Form-finding and analysis of an alternative tensegrity dome configuration

Geiger domes are composed of cable and strut elements. This property of cable domes is the same as tensegrity structures, but in contraction to tensegrity structures, strut elements do not have a function that balances tension in cable elements with compression. In this study, a new cable dome configuration, that mimics the form of tensegrities, is proposed which is able to spread effect of an applied load into all elements of the dome and reduces its local impact. Form-finding and analysis of the Geiger and new dome configurations are performed based on the principle of minimum potential energy. Self-equilibrium forms with minimum potential energy are determined using genetic algorithms. The ability of genetic algorithm based potential energy minimization approach to perform form-finding of loaded or load free cable domes is investigated. Performance of the proposed configuration is tested and compared with the Geiger configuration under various loading conditions.

A reduced-order model for the dynamic and post-buckling behavior of tensegrity structures

Traditional approaches for modeling the behavior of tensegrity structures have their origin either on form-finding applications or on the desire to capture their quasi-static behavior. As such, they generally assume that (i) bars are perfectly rigid, (ii) cables are linear elastic, and (iii) bars experience pure compression and strings pure tension. In addition, a common design constraint is to assume that the structure would fail whenever any of its bars reaches the corresponding Euler buckling load. In reality, these assumptions tend to break down in the presence of dynamic events. In this work, we develop a physics-based reduced-order model to study aspects related to the dynamic and nonlinear response of tensegrity-based structures. With very few degrees of freedom, our model captures their buckling and post-buckling behavior as well as their dynamic response. We then adopt our model to show how, under dynamic events, buckling of individual members of a tensegrity structure does not necessarily imply structural failure. Our research suggests that efficient structural design of impact-tolerant tensegrity structures could be achieved by exploiting rather than avoiding the buckling behavior of its compression members.

On the analytical form-finding of tensegrities

We propose a computationally efficient platform for the analytical form-finding of tensegrity structures. The platform suggested utilizes the well-known Faddeev-LeVerrier algorithm to generate required relationships between force densities of elements providing explicit analytical conditions of self-stressed states. The method only requires sum and multiplications as major computational operations and bypasses complicated triangular factorizations and eigenvalue decompositions of the symbolic force density matrix. This makes our approach very efficient while enables us to employ powerful programs with symbolic math capabilities for the analytical form-finding. The viability and efficiency of the proposed method are demonstrated through examples of the analytical form-finding of some well-known tensegrities.

Računarsko projektovanje i analiza tensegriti struktura

The paper presents the application of computational tools in design and analysis of integrally tensioned (tensegrity) structures. The shaping process of this specific type of spatial structures is determined by consideration of form-structure interrelation. Therefore, the sustainable approach to their design implies the application of tools which could both represent geometry and simulate their physical behaviour. With respect to this, in the research we tested the application of computational tool Fasttens for design of tensegrity structures. Fasttens programme is based on the computational procedure for non-linear analysis of tensegrity systems originally developed by Miodrag Nestorović. In order to run the original programme written in Turbo Pascal, we performed emulation, using emulator software DOSBox. We tested the application of the tool in the design of concrete tensegrity structure. Presented design and numerical experiment confirmed the effectiveness of the proposed approach.

エアコンプレッサを内蔵した跳躍着地可能なテンセグリティ肩関節を有する四足口ボットの開発

In disaster site such as the earthquake and nuclear accident, the autonomous quadruped robots are required for the long-term rescue activities. In order to search the victims inside collapsed buildings, the robots requirements are follows; the built-in power source for the independent drive without the power cable, the jumping mobility for uneven terrains, the lightweight so that it could not promote the further collapse. In this research, we focus on the pneumatic driven robots, because these robots are the higher power-to-weight ratio compared with the electric and hydraulic driven ones. As the goal of our research, we propose the robot concept called AeroCat. The AeroCat have the desirable property such as running,jumping climbing and landing for rescue activity.However, joints will be destroyed by impact at landing. Therefore, we developed a shock absorbing mechanism based on the tensegrity structure.

A discussion on the stiffness matrices used in tensegrity structures

The expression stiffness matrices used in the study of tensegrity structures has inherited a formulation initially conceived for the shape finding problems of tension structures. Nevertheless stiffness and geometrical stiffness matrices are well known concepts of structural analysis and all the formulations which are used need to be congruent. In this paper, we present a formal discussion about the formulations used in several publications showing that they are not equivalent. A classic example is presented, which has been solved using several expressions available in the relevant literature, the results have been compared with a classical finite element software, showing the discrepancies.

身体構造のダイナミクスを利用したReservoir Computingに関する研究

Some actual creatures calculate not only on their brain but also on their body itself. This is called morphological computation. Physical reservoir computing, which uses body itself as a neural network like system, is one approach of the way of morphological computation. The purpose of this research is to understand the effect of body structure and environment around it by the experiments with Tensegrity structures. When spring constant was large, structures were hard to move and dropped its ability. When input amplitude was large, input could counteract the effect of past input and dropped its ability. In addition, the environment around the body especially how to contact with ground (coefficient of restitution) gave large effect on calculation. With ground, the convergence oscillation’s period and amplitude changed.

A Genetic Algorithm Based Form-finding of Tensegrity Structures with Multiple Self-stress States

A form-finding method of tensegrity systems is a process of finding an equilibrium configuration and a key step in the design of tensegrity. Over the past few years, several studies have been made on the form-finding methods of tensegrity systems, however, these methods are limited in the tensegrity systems with multiple self-stress states. In this study, a numerical method is presented for form-finding of tensegrity structures with multiple states of self-stress by using a force density method combined with a genetic algorithm. The proposed method can design the desired tensegrity shape through a genetic algorithm with appropriate constraints. The design variable can be uniquely defined in the case of multiple states of self-stress using only the constraint of the member types. An eigenvalue decomposition of the force density matrix and a singular value decomposition of the equilibrium matrix are performed repeatedly in order to determine a feasible solution for nodal coordinates and force densities. A genetic algorithm is then adopted to uniquely define a single integral feasible set of force densities. Several numerical examples are presented to prove efficiency in searching for self-equilibrium configurations of tensegrity structures. In all cases, the single integral feasible self-stress states can be obtained.

Geometrical nonlinear elasto-plastic analysis of tensegrity systems via the co-rotational method

An efficient finite element formulation is presented for geometrical nonlinear elasto-plastic analyses of tensegrity systems based on the co-rotational method. Large displacement of a space rod element is decomposed into a rigid body motion in the global coordinate system and a pure small deformation in the local coordinate system. A new form of tangent stiffness matrix, including elastic and elasto-plastic stages is derived based on the proposed approach. An incremental-iterative solution strategy in conjunction with the Newton-Raphson method is employed to obtain the geometrical nonlinear elasto-plastic behavior of tensegrities. Several numerical examples are given to illustrate the validity and efficiency of the proposed algorithm for geometrical nonlinear elasto-plastic analyses of tensegrity structures.

Analysis of Tensegrity Structures with Redundancies, by Implementing a Comprehensive Equilibrium Equations Method with Force Densities

A general approach is presented to analyze tensegrity structures by examining their equilibrium. It belongs to the class of equilibrium equations methods with force densities. The redundancies are treated by employing Castigliano’s second theorem, which gives the additional required equations. The partial derivatives, which appear in the additional equations, are numerically replaced by statically acceptable internal forces which are applied on the structure. For both statically determinate and indeterminate tensegrity structures, the properties of the resulting linear system of equations give an indication about structural stability. This method requires a relatively small number of computations, it is direct (there is no iteration procedure and calculation of auxiliary parameters) and is characterized by its simplicity. It is tested on both 2D and 3D tensegrity structures. Results obtained with the method compare favorably with those obtained by the Dynamic Relaxation Method or the Adaptive Force Density Method.

On Visual-Mechanical Synthesis in Shell Structures

While form-finding and optimization methods in shell structures have been widely advanced, discourses regarding an ideation process of how a designer or engineer may synthesize a precedent into novel designs have been less explored, as such ideation mainly relies on his/her intuition in representing structural problems. In this paper, we present a computational method to synthesize shell designs through the interplay between mechanical idealization and visual computation using traditional craft as a reference. Examples of how to generate a folding structure from weaving and tensegrity structure from folding show not only the way to convert craft logic from one type to another in shell structures but also the method to explicitly interpret the designer’s ideation process in generating a certain structural design.

Initial prestress design and optimization of tensegrity systems based on symmetry and stiffness

This study mainly focuses on initial prestress design of tensegrity structures with given shapes. For symmetric structures with multiple self-stress modes, two efficient methods, respectively applying singular value decomposition on the constrained equilibrium matrix and performing eigenvalue analysis without manual classifications of member types, are proposed to compute the independent states of self-stress retaining full symmetry. For structures with multiple symmetric states of self-stress, a novel optimization model for the purpose of maximizing the global rigidity of the structure is presented to determine the combination coefficients of the independent symmetric self-stress states. A penalty function is employed to consider both hard and soft constraints including stability conditions, unilateral property of members and prestress distribution evenness. The fitness-ranking-based genetic algorithms with and without incorporating isolation niche technique are developed to solve such optimization problem. Several numerical examples are tested to demonstrate the efficiency of the proposed methods.

The self-equilibrium problem of the Class-Theta tetrahedral tensegrity module

Abstract The relationship between forms and forces is one of the main topics o structural morphology. This harmonious coexisting relationship is very strong for systems in tensegrity state, commonly called tensegrity systems. There are different methods to find the equilibrium state attained by a tensegrity system for given connectivity of the constituent cables and bars. In paper is investigate the tensegrity tetrahedron, characteristically shaped and free-standing; i.e. with no external loads. It is currently apparent that among the tensegrity systems also exist cable-bar cells with a discontinuous network of cables. It is possible to design a separate set of cables inside the cable-bar elementary cell and to establish a self-stress state of equilibrium. Each of the basic tensegrity systems termed Class Θ possesses an external and internal set of tension components. The shape of Greek capital letter Θ (Theta) reflects two sets of such components (two sets of tendons, cables etc.). This notation corresponds to Skelton's Class k tensegrity structure. The advent of digital form-finding technologies give designers a new toolbox. Form finding of the Class Θ tensegrity tetrahedron starts from the mother configuration and develops into a structurally efficient, but only under some circumstances, expanded configurations. An original approach permits to solve the self-equilibrium problem independently of material choices and external loads.

Mechanical response of 3-dimensional tensegrity lattices

Abstract Most available techniques for the design of tensegrity structures can be grouped in two categories. On the one hand, methods that rely on the systematic application of topological and geometric rules to regular polyhedrons have been applied to the generation of tensegrity elementary cells. On the other hand, efforts have been made to either combine elementary cells or apply rules of self-similarity in order to generate complex structures of engineering interest, for example, columns, beams, and plates. However, perhaps due to the lack of adequate symmetries on traditional tensegrity elementary cells, the design of 3-dimensional tensegrity lattices has remained an elusive goal. In this work, we first develop a method to construct 3-dimensional tensegrity lattices from truncated octahedron elementary cells. The required space-tiling translational symmetry is achieved by performing recursive reflection operations on the elementary cells. We then analyze the mechanical response of the resulting lattices in the fully nonlinear regime via two distinctive approaches: we first adopt a discrete reduced-order model that explicitly accounts for the deformation of individual tensegrity members, and we then utilize this model as the basis for the development of a continuum approximation for the tensegrity lattices. Using this homogenization method, we study tensegrity lattices under a wide range of loading conditions and prestressed configurations. We present Ashby charts for yield strength to density ratio to illustrate how our tensegrity lattices can potentially achieve superior performance when compared to other lattices available in the literature. Finally, using the discrete model, we analyze the dynamic response of a finite tensegrity lattice impacting an elastic wall, where a strong asymmetry in the dispersion of tensile and compressive stress-waves propagating through the medium is observed.