Tensegrity Systems Research Articles

Stability Conditions and Stiffness Variability of General Tensegrity Systems with Kinematic Joints

Abstract Tensegrity systems represent promising candidate mechanisms with in-situ stiffness variability through changing the cables' prestress levels. However, prestress-based stiffness behaviors of tensegrity systems with arbitrary kinematic joints have not been analyzed systematically. This paper adopts the natural absolute coordinates for static modeling of tensegrity systems consisting of rigid members and tension elements. Then, a generic stiffness analysis method is developed to formulate the reduced-basis tangent stiffness matrix, which is found to include three parts: positive semi-definite material and geometric stiffness matrices, and an indefinite constraint stiffness matrix. Based on these findings, a systematic stability-checking procedure is derived to determine prestress and super stability, which are qualitative indicators of the softening and stiffening effects in different tensegrity systems. Then, we proceed to quantify the range of prestress-based stiffness variability by formulating semidefinite programming problems that numerically pinpoint the maximum and zero stiffness points. Furthermore, this paper reveals the composable nature of multiple self-stress states, enabling the composability of stiffness properties in mechanism designs. Several numerical examples verify the efficacy and versatility of the proposed method and demonstrate interesting stiffness behaviors of tensegrity systems with kinematic joints.

Morphogenesis analysis of icosahedral and prismatic tensegrity modules

Morphogenesis is a critical aspect of tensegrity system design. This paper explores icosahedral and prismatic tensegrity modules' pattern formation and foundational theories. First, a material consumption functional and self-strain energy functional are introduced from an engineering perspective, along with a variational principle involving independent dual-variable in the morphogenesis process of tensegrity structures. Next, a Symbol Parsing Method is proposed and successfully applied to the optimal morphology analysis of the icosahedral tensegrity module. Due to memory limitations in personal computers, the Singular Value Decomposition of the symbolic equilibrium matrix for the prismatic tensegrity module is constrained. As an alternative, a discrete numerical solution, the semi-symbolic parsing method, is obtained by directly utilizing a numerical equilibrium matrix. This method was recently employed to analyze and optimize a practical tensegrity footbridge designed by the authors. Lastly, the paper delves into the stability of self-equilibrium states in icosahedral and classical prismatic tensegrity modules by employing a second-order variational analysis of the self-strain energy functional. The findings presented are expected to contribute to future research on the morphogenesis of tensegrity structures.

Tensegrity FlaxSeat: Exploring the Application of Unidirectional Natural Fiber Biocomposite Profiles in a Tensegrity Configuration as a Concept for Architectural Applications

Material selection is crucial for advancing sustainability in the building sector. While composites have become popular, biocomposites play a pivotal role in raising awareness of materials deriving from biomass resources. This study presents a new linear biocomposite profile, fabricated using pultrusion technology, a continuous process for producing endless fiber-reinforced composites with consistent cross-sections. The developed profiles are made from flax fibers and a plant-based resin. This paper focuses on the application of these profiles in tensegrity systems, which combine compression and tension elements to achieve equilibrium. In this study, the biocomposite profiles were used as compression elements, leveraging their properties. The methods include geometrical development using physical and digital models to optimize the geometry based on material properties and dimensions. A parametric algorithm including physics simulations was developed for this purpose. Further investigations explore material options for tension members and connections, as well as assembly processes. The results include several prototypes on different scales. Initially, the basic tensegrity principle was built and explored. The lessons learned were applied in a final prototype of 1.5 m on a furniture scale, specifically a chair, integrating a hanging membrane serving as a seat. This structure validates the developed system, proving the feasibility of employing biocomposite profiles in tensegrity configurations. Furthermore, considerations for scaling up the systems to an architectural level are discussed, highlighting the potential to enhance sustainability through the use of renewable and eco-friendly building materials, while promoting tensegrity design applications.

Experimental investigations on mechanical propertiesof 3D-printed tensegrity-inspired metamaterialsbased on 4-strut simplex module

The present study is focused on experimental investigations of mechanical properties of 3D-printed tensegrity-inspired metamaterials. Tensegrity systems have many advantageous features, such as: light weight, high stiffness-to-mass ratio, controllability, inherent attributes of smart structures, and unique mechanical behaviour. They may be applied not only in macro-scale, but they can also be used to create cellular mechanical metamaterials and lattices in various scales. Metamaterials are understood here as human-designed artificial materials, which do not exist in nature, and whose mechanical properties result from the morphology of the inner structure rather than from chemical or phase composition. Experimental studies on tensegrity metamaterials manufactured using 3D printing techniques are hardly present in the literature. This paper presents results of uniaxial compression tests carried out on a number of 3D-printed tensegrity-based modules corresponding to the metamaterial cells, differing in the manufacturing technology, parent material, and size. The following observations were made during the tests: one of the most important parameters that has a direct impact on the results is the elongation at break of the parent material; any inaccuracies at the production stage greatly affect the mechanical behaviour of the structure; it is crucial to ensure a free deformation consistent with the infinitesimal mechanism mode of tensegrity; a post-critical behaviour of the struts was clearly visible in the performed tests.

Bandgap structure of tensegrity mass–spring chains equipped with internal resonators

This work studies the dispersion relation of a Maxwell type mass–spring chain formed by lumped masses and the parallel arrangement of two different types of tensegrity prisms. Use is made of the Bloch–Floquet theory of discrete systems in association with a linearized model of the response of the tensegrity units under compression loading. Such a modeling is aimed at studying the propagation of compression waves under small perturbations of the initial equilibrium state of the system. For a given value of the cable’s prestress, the tensegrity systems connecting the lumped masses react as elastic springs, which exhibit axial deformations accompanied by relative twisting rotations of the terminal bases. The twisting motion of the chain affects the expression of the kinetic energy, and is accounted for by introducing a suitable definition of equivalent masses. The bandgap structure of the analyzed system is analytically determined and numerical results are obtained for a chain formed by physical models tensegrity θ=1 prisms aligned in parallel with minimal tensegrity prisms. The given results highlight the highly tunable frequency bandgap properties of tensegrity mass–spring chains exhibiting internal resonance capabilities.

Configuration Design and Verification of Shear Compliant Border in Space Membrane Structure.

To solve the non-uniformity of stress in space membrane structure and the lack of shear compliant border configuration design method, shear compliant borders are designed, optimized, and verified in terms of configuration. Firstly, an orthotropic model of the borders is built by combining Hill and Christensen-Lo composite material models. Secondly, a finite element form-finding method is put forward by establishing rectangular and cylindrical coordinates in different areas. The configuration of borders is obtained and the influence of the borders on the edge of the membrane is 0.23%, which means that the borders are compatible with the existing tensegrity systems, especially the tensioning components and the cable sleeves. Thirdly, simulation verifies that borders can cut the spread of shear stress and improve the stress uniformity in membrane structure. The maximum stress in the membrane effective area is decreased by 35.6% and the stress uniformity is improved by 30.5%. Finally, a membrane extension experiment is committed to compare the flatness of membrane surface under shear stress with and without shear compliant borders. The borders decrease the increment speed of flatness by 58.1%, which verifies the amelioration of stress uniformity. The shear compliant border configuration design method provides a reference for space membrane structure stress-uniform design.

Form-finding for tensegrity structures based on the equilibrium equation

Finding form is a critical step in designing tensegrity structures. On the condition that the partial node coordinates, topology, and a bar/cable attribute (the force density of bar is -1 and the force density of cable is 1.) are known, a form-finding method, which is used to find the remaining node coordinates and the force density relation between elements, is proposed in this paper. Firstly, the equilibrium conditions of the tensegrity system are analyzed, and the equilibrium equation is established. Secondly, the variables that must be solved are set and substituted into the equilibrium equation, and the target equation with the variables is built. The Levenberg-Marquardt method with a damping parameter updating strategy is introduced to solve the least squares problem by transforming the equilibrium equation problem into the least squares problem. The form-finding process is performed by solving the least squares formula. Three examples demonstrate the efficiency and accuracy of searching for self-equilibrium configurations.

A Dynamical Model for the Control of a Guyed Tensegrity Beam Under Large Displacements

Abstract Most studies regarding models of tensegrity systems miss the possibility of large static deformations or provide elaborate and lengthy solutions to determine the system dynamics. Contrarily, this work presents a straightforward methodology to find the dynamic characteristics of a guyed tensegrity beam structure, allowing the application of vibration control strategies in conditions of large deformations. The methodology is based on a low-order, adaptive, nonlinear finite element model with pre-stressed components. The method is applied to numerical and experimental models of a class 2 tensegrity structure with a high length-to-width aspect ratio. Image processing and accelerometer data are combined to extract the experimental natural frequencies of the structure, which are compared to numerical results. Prony’s method is applied to estimate damping, and a numerical control strategy is employed using the dynamical model of the structure.

A metaheuristic‐based method for analysis of tensegrity structures

SummaryTensegrity systems are construction system that offers solutions for lighter designs compared with a standard truss system consisting of tension and compression elements. Due to their structure, they are recommended in the design of space structures due to the use of lighter and easy‐to‐assemble structural elements such as cables, ropes, and similar to provide the desired durability. In this study, energy minimization was carried out to solve the structural element displacements. It is expected to reduce the total potential energy of the system with minimization. By minimizing the total potential energy of the tensegrity system models, the displacement of each building element is found for the equilibrium condition. Tensegrity models were analyzed by minimizing energy via the adaptive harmony search (AHS) algorithm. In this research, two distinct tensegrity structure specimens were employed. One comprised a cantilever beam, while the other adopted a cyclic model, forming eight equal octagons from eight nodes at both the base and the top. Additionally, an examination was conducted on a two‐layer iteration of the cyclic model. The method is robust for both space and planar tensegrity structures, allowing the determination of deformed shapes under various loads without design assumptions. The TPO/MA method demonstrates superiority in handling nonlinear and barely stable systems, as evidenced by examples illustrating its efficacy in maintaining structural form under increasing loads and challenging conditions.

Functionally antagonistic polyelectrolyte for electro-ionic soft actuator

Electro-active ionic soft actuators have been intensively investigated as an artificial muscle for soft robotics due to their large bending deformations at low voltages, small electric power consumption, superior energy density, high safety and biomimetic self-sensing actuation. However, their slow responses, poor durability and low bandwidth, mainly resulting from improper distribution of ionic conducting phase in polyelectrolyte membranes, hinder practical applications to real fields. We report a procedure to synthesize efficient polyelectrolyte membranes that have continuous conducting network suitable for electro-ionic artificial muscles. This functionally antagonistic solvent procedure makes amphiphilic Nafion molecules to assemble into micelles with ionic surfaces enclosing non-conducting cores. Especially, the ionic surfaces of these micelles combine together during casting process and form a continuous ionic conducting phase needed for high ionic conductivity, which boosts the performance of electro-ionic soft actuators by 10-time faster response and 36-time higher bending displacement. Furthermore, the developed muscle shows exceptional durability over 40 days under continuous actuation and broad bandwidth below 10 Hz, and is successfully applied to demonstrate an inchworm-mimetic soft robot and a kinetic tensegrity system.

Harmonic Response of CFRP Tensegrity System in a Suspen-dome

Abstract Wind-induced excitation causes structure to vibrate which leads to instability. This paper focuses on the performance of CFRP tensegrity system in a suspen-dome subjected to wind load by assuming such load as harmonic. A comparison is made with the traditional steel tensegrity system, in order to justify the integrity of CFRP cable application in a suspen-dome system. A finite element software, namely ANSYS, was implemented for the simulation by analyzing a physical model of 4 m span and 0.4 m height. Results show that CFRP tensegrity system has similar performance as steel tensegrity system and can be used as a substitute for steel.

An upright life, the postural stability of birds: a tensegrity system.

Birds are so stable that they can rest and even sleep standing up. We propose that stable static balance is achieved by tensegrity. The rigid bones can be held together by tension in the tendons, allowing the system to stabilize under the action of gravity. We used the proportions of the bird's osteomuscular system to create a mathematical model. First, the extensor muscles and tendons of the leg are replaced by a single cable that follows the leg and is guided by joint pulleys. Analysis of the model shows that it can achieve balance. However, it does not match the biomechanical characteristics of the bird's body and is not stable. We then replaced the single cable with four cables, roughly corresponding to the extensor groups, and added a ligament loop at the knee. The model is then able to reach a stable equilibrium and the biomechanical characteristics are satisfied. Some of the anatomical features used in our model correspond to innovations unique to the avian lineage. We propose that tensegrity, which allows light and stable mechanical systems, is fundamental to the evolution of the avian body plan. It can also be used as an alternative model for bipedal robots.

Stability Conditions for General Tensegrity with Rigid Bodies

This study develops stability conditions for general tensegrity regarding the three stability criteria (e.g., prestress stability, general stability, and super stability) based on an energy approach. The coupling relations of specific nodal degrees of freedom caused by the rigid bodies are described by a set of constraint functions and incorporated into the potential energy function by the Lagrangian multiplier method. Stiffness matrices of general tensegrity systems are derived from the augmented potential energy function. It shows that the stability of general tensegrity can be evaluated through the stiffness matrices expressed in the constrained motion space generated by the constraint functions. Elegant formulations are developed for the evaluation of the three types of stability conditions for general tensegrity systems. Moreover, it turns out that the formulations developed in this study will degenerate into those for the stability analysis of classic tensegrity if no rigid bodies exist in the system, which indicates that the proposed approach is a general framework for the stability analysis of any type of tensegrity systems.

Statics of integrated origami and tensegrity systems

This study presents an explicit form of the static equilibrium equations of integrated origami and tensegrity systems. The analytical approach allows one to model and analyze the isolated origami and tensegrity paradigms as a whole system. The tensegrity and origami members are described by the nodal coordinates and hinge angels between the origami panels. The nonlinear static equations of the integrated system are derived by the Lagrangian method. By Taylor’s expansion theory, we also presented its linearized form. The developed approach is capable of conducting the following comprehensive statics studies for any integrated origami and tensegrity systems: (1) Performing loading analysis, where bars and strings can have elastic or plastic deformations. (2) Conducting infinitesimal and large deformation analysis, which is helpful in understanding the stress in structural members and actuation strategies. (3) Dealing with various kinds of boundary conditions, for example, fixing or applying static loads at any nodes in any direction (i.e., gravitational force, some specified forces, or moments). (4) Conducting stiffness analysis, including eigenvalues and their modes. Three examples, a Miura origami unit, an integrated tensegrity origami shelter, and a cable-driven Kresling structure, are carefully selected and studied. This study provides a deep insight into structures, materials, as well as performances. The integration idea also promotes our ability to design and build deployable structures at large.

Self-Equilibrium, Mechanism Stiffness, and Self-Stress Design of General Tensegrity With Rigid Bodies or Supports: A Unified Analysis Approach

Abstract The use of general tensegrity systems that incorporate rigid bodies beyond axially loaded members has garnered increasing attention in practical applications. Recent preliminary studies have been conducted on the analysis and form design of general tensegrity systems with disconnecting rigid bodies. However, existing methods cannot account for connections between different rigid bodies. In practical applications, general tensegrity systems may have interconnected rigid bodies, rendering the analysis method proposed in previous studies inapplicable. To address this issue, this work proposes a comprehensive and unified analysis method for general tensegrity systems. The proposed formulation allows for the incorporation of connections between rigid bodies and general tensegrity systems with supports into the developed framework, enabling uniform analysis. Equilibrium and compatibility equations are derived through an energy approach combined with the Lagrange multiplier method. Self-stress states and mechanism modes are then computed based on these formulations. The stiffness of the mechanism mode is analyzed and validated using both the product force method and the reduced geometric stiffness matrix method. Furthermore, a self-stress design approach based on semi-definite programming (SDP) is proposed to determine feasible member forces that can stabilize general tensegrity systems. Illustrative examples are presented to verify the effectiveness of the proposed approach. This study expands the scope of the analysis theory for tensegrity systems and provides a fundamental and unified analysis approach that can be applied to any type of tensegrity system.

Structural design and integral assembly procedure of rigid-flexible tensegrity airship structure

A rigid-flexible tensegrity airship structure is proposed through the systematic research and design of the wheel spoke–shaped stiffening ring, carbon fiber–reinforced polymer slender cable, central longitudinal compression truss, and spherical multi-part integration assemblage. This structure aimed to fully utilize the structural performance advantages of a semi-rigid airship and the self-equilibrating characteristics of a tensegrity structure. The design of the spherical multi-part integration assemblage facilitates the rotation of a wheel spoke–shaped stiffening ring and enhances the installation efficiency of longitudinal cables. The tensegrity system is applied to the rigid-flexible integrated airship structure to make the main compression-bearing shaft bear the compressive stress and the longitudinal cables bear the tensile stress, thereby creating a rigid-flexible tensegrity airship structure with the self-equilibrating system. The feasibility of the structural design, component preparation, and integrated assembly technology of the rigid-flexible tensegrity airship has been effectively verified and evaluated by the integral assembly procedure, which ensures the assembly safety and reliability of the airship structure. The rigid-flexible tensegrity airship structure exhibits a clear load transfer path, high stiffness, bearing capacity, and strong shape retention capacity under low pressure conditions, which provides an ideal structural form for the stratospheric airship.

An improved numerical model for locomotive tensegrity systems based on vector form intrinsic finite element

Locomotive robot based on tensegrity has recently drawn much attention due to its lightweight and flexibility. This article presents an improved numerical model for locomotive tensegrities. The previously used bar element for struts is replaced by beam element, and rigid joint element is used to consider more details of the tendon–strut connections. The vector form intrinsic finite element (VFIFE) method is adopted to formulate the numerical model and carry out the simulation. The improvement of the proposed model on the prediction of feasible rolling gaits is quantitatively verified by experiments on a six-strut locomotive tensegrity. Mann–Whitney U test is adopted, and the p value between the experimental success rates of the gait primitives generated by the improved model and the rates of the gait primitives generated by the previous model is [Formula: see text]. It is shown that the improved model is more consistent with the experiment by considering the details of the tendon–strut connection.

Equilibrium and stiffness study of clustered tensegrity structures with the consideration of pulley sizes

This paper presents the equilibrium and stiffness study of planar clustered tensegrity structures (CTS) considering pulley sizes. We first derive the geometric relationship between clustered strings and pulleys, where the nodal vector is chosen as the generalized coordinate. Then, the equilibrium equations of the planar clustered tensegrity structure with pulleys based on the Lagrangian method are given. Since the stiffness of a structure is usually weakened when using clustering strings, we formulate the tangent stiffness matrix equations for analysis. It is also shown that as pulley sizes go to zero, the governing equations of the planar clustered tensegrity system with pulleys yield to the classical clustered tensegrity structure without pulleys, which is consistent with the existing literature. Three examples are demonstrated to validate the given theory. The proposed method allows one to conduct equilibrium, stiffness, and robustness studies of planar clustered tensegrity structures with pulleys. Nevertheless, the approach developed in this paper is not limited to tensegrity structures. It can also be applied to a wide range of applications with pulley-rope systems, such as drilling rigs, ocean platform anchors, and cargo cranes.

Simplified form-finding for tensegrity structures through reference joints of symmetry orbits

Tensegrity structures are a class of free-standing and prestressed cable-strut structures. Different from traditional elastic structures, both geometric shapes and initial prestress distributions of tensegrity systems are indeterminate. Thus, it is of great significance to develop a robust and effective form-finding method for tensegrity systems. Based on reference joints of symmetry orbits and force density theory, this study proposes a simplified form-finding method for tensegrity structures. This method makes full use of the inherent symmetry of structural systems. It introduces a series of orthogonal transformation matrices for certain reference joints to represent independent force equilibriums of similar joints from the same symmetry orbit. Then, the original form-finding problem is simplified into finding null spaces for several small-sized force density matrices about the reference joints. Thus, the force density of each type of components and the nodal coordinate of each joint can be effectively obtained. Illustrative examples show that the proposed method can significantly simplify the computational complexity of form-finding for tensegrity systems with symmetry orbits.

Automatic construction of tensegrity structures with complex surfaces based on circle packing

Tensegrities have attracted significant attention in many disciplines as a distinct type of truss network. However, with the increase in size and complexity, building complex tensegrities remains challenging. This study presents a systematic method to automatically construct large-scale tensegrity structures with complex surfaces by ingeniously combining circle packing. This method enables direct identification of the topology and configuration of tensegrities with specific shapes. First, circle packings on target surfaces are accomplished in a practical manner. On this basis, the triangulation connection is established as the basic assembly pattern, and its mobility is further analyzed using the screw theory to inspect the assembly rules. Then, the connection requirements of the elementary units are investigated, and a novel tensegrity torus is exemplarily utilized for assembly. To evaluate the tensegrity properties, the self-equilibrium and stability of the assembled structures are respectively examined. For illustration, four representative examples are produced to verify the effectiveness of the proposed method. In this study, an alternative method to automatically manufacture tensegrities satisfying the demands of both aesthetics and practicality is developed, which will significantly expand the application scenarios for tensegrity systems.