Tensegrity Structures Research Articles

Dynamic analysis of a spherical mobile robot based on a tensegrity structure with two curved compressed members

The use of mechanically compliant tensegrity structures in mobile robotics is an attractive research topic. The mechanical properties and therefore the locomotion performance of mobile robots based on these structures can be adjusted reversibly during locomotion. In the present work, a rolling mobile robot, based on a simple tensegrity structure, consisting of two rigid disconnected curved members connected to a continuous net of eight prestressed tensioned members with pronounced elasticity, is considered. Pure rolling uniaxial locomotion and also planar locomotion can be realized with small control effort, induced by the movement of two internal masses. After kinematic considerations, the nonlinear equations of motion are derived and transient dynamic analyses are performed, to study the system behavior. Also the dependency of the rolling movement behavior on structural and actuation parameters is discussed. The uniaxial and planar locomotion performance of the system are verified experimentally.

Force Analytic Method for Rolling Gaits of Tensegrity Robots

This paper proposes an analytic method based on finite-element method (FEM) for the rolling gaits of tensegrity robots. By analyzing forces during the rolling motions, the following problems can be solved. First, the relation between the actuator combinations (AC) and the rolling directions is discovered for controlling the rolling motions. Second, the influence of the driving parameters on the deformations is found out for obtaining the optimized control strategy. Third, the influence of the material parameters on the rolling motions is achieved as guidance to design physical robots. Tensegrity robots were proposed as novel mobile robots based on tensegrity structures that consist of discrete rigid struts and continuous elastic cables. The tensegrity structures process the shock resistance ability and high deformation capacity, and can be used to build lightweight mobile robots. The tensegrity robots can generate multiple gaits and multiple deformations. Rolling gaits are more stable than jumping and more efficient than crawling. Aiming at rolling gaits, to solve the three problems mentioned earlier, force analysis is required. The internal forces of the tensegrity structures always keep balance to maintain the shape of the structures, however, as the structures are highly coupled, the variation of one force will change the rest of the forces and generates feedback effects to this force. As the number of the forces increase with the growing number of struts, the force analysis becomes more and more complicated. The existing exhaustive experimental method can obtain the relation between the AC and the rolling directions through massive experiments, but this method cannot tell the influence of the driving parameters on the deformations and the experiment results based on certain robots are not suitable for other robots without analyzing the influence of the material parameters. In this paper, an analytic method based on FEM is introduced. A six-strut tensegrity model is used as an example to test the feasibility of the method. Experiments on a physical robot are performed to verify the reliability of the computation results.

Composite solar façades and wind generators with tensegrity architecture

The urgent need for sustainable buildings calls for the adoption of active building façades that harvest wind and solar energy through on-site wind power generators and solar panels. Particularly interesting is the use of tensegrity structures for the construction of renewable energy supplies, due to their easy integration with solar and acoustical panels, which can form special rigid members of the structure. The present study deals with the design of active façades based on tensegrity units, which supports shading devices and/or solar panels. The tensegrity units are foldable and deployable and are controlled by stretching or relaxing selected cables. Wind generators to convert the strain energy stored in the cables of wind-excited units into electrical power are also designed. The proposed structures offer portable applications for small spans and are easy to assemble using prefabricated component parts in the case of large spans.

Comparison of the Performance of Hexagonal Grid and Half-cuboctahedron Grid Tensegrity Systems in Roof Structures

Objective: To study and compare the behaviour of half-cuboctahedron grid and hexagonal grid tensegrity system. Methods/ Statistical Analysis: Ever since the discovery of tensegrity, most works have been concentrated on the classification and form finding techniques but very less on the mechanics of these structures. This study focus on the behaviour of two basic tensegrity modules, the half-cuboctahedron and hexagonal. Findings: Later, the study has been extended to tensegrity grids of spans 2x2, 4x4 and 8x8 of both the configurations (hexagonal modules and half-cuboctahedron modules) which were developed using an FEM based software. The grids were compared on the basis of nodal displacements and member forces, resulting in the half-cuboctahedron grid to be a more feasible configuration for large span roof structures. Applications/ Improvements: The application of the tensegrity structure is to applied in the large span roof structures.

Vibration health monitoring for tensegrity structures

Tensegrities are assembly structures, getting their equilibrium from the interaction between tension in cables and compression in bars. During their service life, slacking in their cables and nearness to buckling in their bars need to be monitored to avoid a sudden collapse. This paper discusses how to design the tensegrities to make them feasible for vibrational health monitoring methods. Four topics are discussed; suitable finite elements formulation, pre-measurements analysis to find the locations of excitation and sensors for the interesting modes, the effects from some environmental conditions, and the pre-understanding of the effects from different slacking scenarios.

A fully automatic group selection for form-finding process of truncated tetrahedral tensegrity structures via a double-loop genetic algorithm

Numerous studies have developed novel form-finding methods of tensegrity structures using various techniques. Previous works have been limited by the lack of a fully automatic grouping technique in the form-finding process. In this study, an advanced form-finding method of tensegrity structures using a fully automatic labeling that can specify the member types. Constraint optimization problems were then used in the form-finding processes to minimize fitness functions by using a double-loop genetic algorithm. A well-known truncated tetrahedral tensegrity is presented to demonstrate the accuracy and efficiency of the proposed method. In conclusion, a very good performance of presented method has been shown in the results.

Tensegrity configuration method for connecting tensegrity units along their axes

Tensegrity structure is a kind of truss structures which are composed of bars under axial compression and strings under axial tension. It has become a hot spot in the field of sculpture, architecture, aviation and other fields in recent years. A method for building tensegrity structures through connecting tensegrity units is presented and researched systematically here. At first, through analysis for forces of the 3-bar tensegrity structure, the method for adding the components to keep the 3-bar tensegrity unit in equilibrium when changing its rotating angles in the units is researched. The principle for connecting tensegrity units axially follows to be presented. Then the node matrix and connectivity matrix are applied to set up common mathematic models of the tensegrity structures. With the axially-connecting principle and the mathematic models, tensegrity structures with common space shape are set up. Axially-connecting methods for same and different tensegrity units are analyzed. Besides, simulation and physical models are applied to prove that configurations of the obtained tensegrity structures are stable and the axially-connecting method is available. Through research in the paper, a method for configuring the tensegrity structure is summarized systematically, which provides theoretical support for the configuration of more complex structures.

Parallel kinematic mechanisms for distributed actuation of future structures

Future machines will require distributed actuation integrated with load-bearing structures, so that they are lighter, move faster, use less energy, and are more adaptable. Good examples are shape-changing aircraft wings which can adapt precisely to the ideal aerodynamic form for current flying conditions, and light but powerful robotic manipulators which can interact safely with human co-workers. A 'tensegrity structure' is a good candidate for this application due to its potentially excellent stiffness and strength-to-weight ratio and a multi-element structure into which actuators could be embedded. This paper presents results of an analysis of an example practical actuated tensegrity structure consisting of 3 ‘unit cells’. A numerical method is used to determine the stability of the structure with varying actuator length, showing how four actuators can be used to control movement in three degrees of freedom as well as simultaneously maintaining the structural pre-load. An experimental prototype has been built, in which 4 pneumatic artificial muscles (PAMs) are embedded in one unit cell. The PAMs are controlled antagonistically, by high speed switching of on-off valves, to achieve control of position and structure pre-load. Experimental and simulation results are presented, and future prospects for the approach are discussed.

A modified differential evolution algorithm for tensegrity structures

In this paper, a novel modified differential evolution (mDE) algorithm for advanced form-finding of tensegrity structures is proposed to define an appropriate candidate for strut members. The form-finding process only requires topology and member type of a tensegrity based on the force density method. The proposed algorithm improved from original differential evolution (DE) is performed to reduce significant computational cost. In the mDE, scale factor F and crossover rate c are adjusted as well as the mutation and selection phases of the original DE are also replaced by the best individual-based mutation and elitist selection techniques. The objective function of the product of α and β related to eigenvalues and force densities is minimized. Since force density values are considered as continuous design variables, optimal solutions obtained by mDE are more accurate than those solved from discrete design variables of GA. Several benchmark numerical examples of two- and three-dimensional tensegrity structures are investigated to verify the effectiveness and robustness of the proposed algorithm by comparing obtained results with those of other methods in the literature.

The Performance of Half-Cuboctahedron Grid Tensegrity Systems in Roof Structures

Background/Objectives: Sustainability in the use of material and economy are the watchwords of the modern engineering fraternity. Many structural systems have been evolved for this purpose. At present, the qualities of the tensegrity structures, which make the technology attractive for human use, are their resilience and ability to use the material in a very economical way. Thus, the construction of the structures using tensegrity principle will make them highly resilient and reasonably economical at the same time. Methods: Though a lot of research has been carried out on the behaviour of tensegrity systems, the area requires further research for a complete understanding. Hence, this project is aimed at analysis of a tensegrity based roof structure. The prototype module used is half-cuboctahedron. Findings: First, a single module measuring 1 m x 1 m in size, is modelled and analysed using an FEM based software. The investigations are then extended to a 2 x 2 grid, measuring 2 m x 2 m in size. The behavior of the grid is studied under the application of external loads varying the prestressing forces on the members. Later, a 4 x 4 grid, measuring 4 m x 4 m in size and an 8 x 8 grid, measuring 8 m x 8 m in size are modelled and analysed based on the 2 x 2 grid system. From the analysis, it is found that the displacements of the system satisfy the permissible limits as per previous research values. The member forces have also been analysed. Applications/Improvements: The performance of the half-cuboctahedron tensegrity grid structure proves to be more suitable for larger span structures.

On a connectivity matrix formula for tensegrity prism plates

This paper considers tensegrity structures constructed from repetition of simple fundamental units. The tensegrity prism is chosen as a fundamental unit, which allows us to build plates, columns, towers, and their variations. The connectivity matrix plays a central role in analysis and design of tensegrity structures. This paper provides a systematic way to construct connectivity matrices for tensegrity structures constructed from repetition of tensegrity prisms. The number of units and node locations (shape) can be chosen arbitrarily. As an application of the connectivity matrix, a minimal-mass design subjected to force equilibrium (force balance) and yield and buckling stress constraints is shown.

Inter-trabecular angle: A parameter of trabecular bone architecture in the human proximal femur that reveals underlying topological motifs.

Trabecular bone is an intricate 3D network of struts and plates. Although the structure-function relations in trabecular bone have been studied since the time of Julius Wolff, controversy still exists regarding the architectural parameters responsible for its stability and resilience. We present a parameter that measures the angle between two connected trabeculae - the Inter-Trabecular Angle (ITA). We studied the ITA values derived from μCT scans of different regions of the proximal femora of 5 individuals of different age and sex. We show that the ITA angle distribution of nodes with 3 connecting trabeculae has a mean close to 120°, nodes with 4 connecting trabeculae has a mean close to 109° and nodes of higher connectivity have mean ITA values around 100°. This tendency to spread the ITAs around geometrically symmetrical motifs is highly conserved. The implication is that the ITAs are optimized such that the smallest amount of material spans the maximal 3D volume, and possibly by so doing trabecular bone might be better adapted to multidirectional loading. We also draw a parallel between trabecular bone and tensegrity structures - where lightweight, resilient and stable tetrahedron-based shapes contribute to strain redistribution amongst all the elements and to collective impact dampening. The Inter-Trabecular Angle (ITA) is a new topological parameter of trabecular bone. The ITA characterizes the way trabeculae connect with each other at nodes, regardless of their thickness and shape. The mean ITA value of nodes with 3 trabeculae is close to 120°, of nodes with 4 trabeculae is just below 109°, and the mean ITA of nodes with 5 and more trabeculae is around 100°. Thus the connections of trabeculae trend towards adopting symmetrical shapes. This implies that trabeculae can maximally span 3D space using the minimal amount of material. We draw a parallel between this motif and the concept of tensegrity - an engineering premise to which many living creatures conform at multiple levels of organization.

Form-finding of free-form tensegrity structures by genetic algorithm–based total potential energy minimization

Free-form tensegrities are composed of randomly connected cable and strut elements. The complexity of these structures causes determination of their self-equilibrium form to be a formidable task. There can be an infinite number of solutions with different forms, but it is difficult to identify the best form in terms of stability. Based on the fact that stability of structures is inversely proportional to potential energy, a genetic algorithm minimization process is developed to determine the self-equilibrium form of free-form regular tensegrity structures. The capability of the form-finding process on determination of the most stable form with minimum potential energy is investigated using two main parameters of free-form regular tensegrities which are cable–strut length ratio at rest and number of strut elements. The computational performance of the proposed method is also tested using free-form tensegrities with different number of structural elements.

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.

Pole-Placement for Collocated Control of Flexible Structures

Pole assignment is one of the central problems in most control systems designs. In relation to low-authority active flexible structural systems where robustness is difficult to achieve primarily due to many closely-spaced low-frequency lightly damped modes, pole-placement techniques are commonly used to design constant gain collocated controllers that will ensure that the transient phenomenon of the structure dies down sufficiently fast. In this paper, the active control of flexible structures is presented for collocated control design using a modified pole-placement approach that reduces the control efforts significantly. The effectiveness of employing the proposed approach is demonstrated using tensegrity structures.

A shape memory alloy rod element based on the co-rotational formulation for nonlinear static analysis of tensegrity structures

In this article, a shape memory alloy rod element is derived based on the co-rotational formulation. In the co-rotational approach, the rigid body modes are removed from the total deformations by employing a local coordinate system at element level, and hence, the major part of geometric nonlinearity is isolated. The linear shape memory alloy rod element is developed using a shape memory alloy constitutive model together with the small strain framework employed by the co-rotational approach. The one-dimensional shape memory alloy model is adopted to calculate both the pseudo-elastic response and the shape memory effects. The new formulation is exploited to perform static analysis of tensegrity structures in order to study the accuracy and robustness of the proposed element and its capability to describe the structural response of shape memory alloy devices.

Form-finding analysis for a new type of cable–strut tensile structures generated by semi-regular tensegrity

A tensegrity structure is a type of self-balancing tensile structure, which consists of tension cables surrounding compression struts. Based on the geometry and topology of the classic half-octahedron tensegrity, this article presents a form-finding analysis of semi-regular tensegrity units using singular value decomposition of the equilibrium matrix. We propose the design formulas for the unit geometric transformation, obtain its internal self-stress modes and inextensional mechanism modes, and verify its geometric stability. Then, we devise a design method and compute the overall feasible self-stress of a tensegrity torus. A novel cable–strut tensile structural system is generated through combining a tensegrity torus and a Levy-type cable dome. Finally, a physical model is constructed to verify the feasibility of this structural system. This work enriches existing forms of tensegrity structures and contributes to further practical applications of tensegrity systems.

Form-Finding of Tensegrity Structures based on Force Density Method

Background/Objectives: Determining equilibrium configuration of a tensegrity structure is known as form-finding. This paper critically reviews various form-finding techniques based on force density method. Methods/ Statistical Analysis: The most appropriate method out of all the form-finding methods, to ascertain new formation of tensegrity structures is Force density method. The techniques based on Force density method such as Advanced form-finding method, Adaptive force density method, Form-finding via Genetic algorithm and Algebraic tensegrity form-finding are reviewed in detail along with scope, limitation and suitability of each method. Findings: This paper lists out the input and output parameters of each technique and comparing the effectiveness of each. Most of these methods require only type of the member (tension or compression) and topology of the tensegrity structure provided by connectivity matrix as initial parameters. Formfinding of regular shaped structures such as cable domes is best performed by Advanced form-finding method. Adaptive form-finding method is found to be efficient for irregular shaped structures and also to search novel configurations. Form finding by means of genetic algorithm provides solutions for regular shaped tensegrity structures. Algebraic method is found to be highly efficient and automated general method among all the form-finding methods discussed in this paper. A structure’s evolving geometry can be better regulated by this method. Applications/ Improvements: This paper presents the techniques which require minimum initial parameters for form-finding of tensegrity structures which also helps in choice of methods based on known parameters of the structure.

Form-finding Tensegrity Models Approach with Reverse Engineering

Background/Objectives:After recent achievements in the field of Tensegrity structure, many Tensegrity models have been investigated and evaluated. Tensegrity models have been used as symbolic or covering vast area such as fuller’s dome and other stuffs. These usages do not have sufficient attention to synthesis of architectural and structural space together.Methods/Statistical analysis: The method of this article, based on simulation and modeling of a sample structure by analyzing flow of internal forces, is adaptive methode. Restriction of exited structures to a hammock and then analysis its force flow, and consequently classify it to tensile and compressive members is the base of manner. By gathering information about Tensegrity structures and their behavior according to several definition of structural engineers and also architects, we commence combination of facts based on adaptation of existed structures with Tensegrity rules. Then, by finding a Tensegrity model and creating a replica of hammock Tensegrity, it shows the ability of structure specially in term of statistic. This outcomes can help us to develop new system of form - finding models.Findings: This system can make us able to develop modeling of Tensegrity. In the recent years, form - finding method almost base on symmetric models to expand as cover structure for vast spans. By this manner, we can design asymmetric models consist of synthesis of structural and architectural space.Application/Improvements: Form - Finding method can be developed in order to increase quality of building in term of weight of structure, flexibility, decreasing proportion of used material to its resistance and so on. In addition, we can produce asymmetric models which contain architectural space into Tensegrity structure.

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.