Tensegrity Prisms Research Articles

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.

Stiffness properties of regular p-bars tensegrity prisms under the compressive loading

The regular tensegrity prisms are a kind of fundamental unit, which can be formed into different configurations by different splicing approaches to achieve specific structural functions. Taking regular prisms as the research object, this article presents an analytical method for structural deformations under compressive loading. The structural parameters after loading can be obtained directly by solving the nonlinear force density equation, which simplifies the form-finding process. This article also studies the structural stiffness under different connection modes. By defining the twist angle change rate and internal force utilization rate, we can compare the relative deformation of different tensegrity prisms. The results show that the relative deformation of the three-bar prism is minimal, and it has the best stiffness.

Generalized prismatic tensegrity derived by dihedral symmetric lines

Classic prismatic tensegrity structures, characterized by dihedral symmetry with one orbit of nodes, are among the simplest and possibly the earliest spatial tensegrity structures invented. This paper introduces a generalized form of the prismatic tensegrity structures by converting a single-loop linkage into truss, in which the lines of joint axes rather than the nodes have dihedral symmetry. Since the vector space formed by the line coordinates of these joints has rank degeneracy one, the generated tensegrity structures are kinematically and statically indeterminate. These tensegrity structures are further proved to be prestress-stable, generally, for the total or partial parameter space of lines based on a second-order analysis of screws, and are called dihedral-line tensegrity structures in this paper. Specifically, this paper focuses on symmetric dihedral-line tensegrity structures, in which the nodes also have dihedral symmetry but in two orbits and members in seven orbits, and are called two-orbit dihedral-line tensegrity structures. It is found that there are at least N struts for the generated tensegrity with DN symmetry. And the classic prismatic tensegrity structures can be recovered from these dihedral-line tensegrity structures by removing certain zero-force members. Symmetric-adapted force density matrices are also provided as well as the relation to that of the classic prismatic tensegrity. Given 4N+6 dimensional parameters inherent to these tensegrity structures, a rich variety of tensegrity structure family is presented.

Self-Equilibrium Analysis and Minimal Mass Design of Tensegrity Prism Units

Abstract This paper provides a specific analysis strategy for tensegrity prism units with different complexities and connectivity. Through the nodal coordinate matrix and connectivity matrix, we can establish the equilibrium equation of the structure in the self-equilibrium state, and the equilibrium matrix can be obtained. The Singular Value Decomposition (SVD) method can find the self-equilibrium configuration. The torsional angle formula between the upper and bottom surfaces of the prismatic tensegrity structure, which includes complexity and connectivity, can be obtained through the SVD form-finding method. According to the torsional angle formula of the self-equilibrium configuration, we carry out the mechanical analysis of the single node, and the force density relationship between elements is gained. As one of the standards, the mass is used to evaluate the light structure. This paper also studied the minimal mass of the self-equilibrium tensegrity structure with the same complexity in different connectivity and got the minimal mass calculation formula. A six-bar tensegrity prism unit is investigated in this paper, which shows the feasibility of systematic analysis of prismatic structures. This paper provides a theoretical reference for prismatic tensegrity units.

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.

A novel torque application method for tensegrity structures

The torsional resistance of the beam-like tensegrity structure is an essential mechanical property, but there is currently no systematic approach to applying torque to the structure using external force. This article presents a novel torque application strategy to analyze beam-like tensegrity structures. First, the torque that needs to be applied and the initial self-stable configuration are given. Second, the number of iteration steps is set, and the torque is gradually increased by using an arithmetic sequence, and the torque of each step is evenly distributed to the nodes of the acting surface. Third, the current external forces can be calculated by the nodal coordinates and torque, and the stable configuration under the obtained external force will be found based on the form-finding method. This process will be cycled until the specified torque is achieved. Finally, the equilibrium configuration under the set torque will be obtained, and the deformation analysis of the configuration will be carried out to study its mechanical properties. The elastic potential energy errors of the three-bar tensegrity prism unit and the class 2 tensegrity tower are corrected by 2.8198% and 66.9135%, respectively. This method has been proven to improve calculation precision.

A continuum model for the tensegrity Maxwell chain

<abstract><p>A recent study has presented a Maxwell mass–spring model for a chain formed by two different types of tensegrity prisms alternating with lumped masses. Such a model shows tensegrity theta prisms arranged in parallel with minimal regular prisms acting as resonant substructures. It features a tunable frequency bandgap response, due to the possibility of adjusting the width of the bandgap regions by playing with internal resonance effects in addition to mass and spring contrasts. This paper expands such research by presenting a continuum modeling of the tensegrity Maxwell chain, which is useful to conduct analytic studies and to develop finite element models of the plane wave dynamics of the investigated system. In correspondence to the high wave-length limit, i.e., in the low wave number regime, it is shown that the dispersion relations of the discrete and continuum models provide similar results. Analytic solutions to the wave dynamics of physical systems are presented, which validate the predictions of the bandgap response offered by the dispersion relation of the continuum model.</p></abstract>

Design of Tensegrity Torus Based on Bilevel Optimization Model

This paper presents a design method for novel tensegrity torus based on a bilevel optimization model. The novel tensegrity torus is assembled by a plurality of deformed three-rod prismatic tensegrity units, which can be uniquely determined by 5 geometric parameters. Taking the five parameters as optimization variable, the uniformity of both geometry and prestress as optimization objective, a bilevel optimization model is innovatively established. This model can integrate multiple optimization objectives and address nested optimization problems. Three numerical examples with different equal divisions are established and the optimization results indicate good effect of optimization model. Further parameter analysis of the bilevel optimization model demonstrates that reasonable constraints can enhance the optimization effect of the model. Compared to other equal divisions, the prestress of 18-division torus is the most uniform.

Design of a Fully Pulley-Guided Wire-Driven Prismatic Tensegrity Robot: Friction Impact to Robot Payload Capacity

Design of a Fully Pulley-Guided Wire-Driven Prismatic Tensegrity Robot: Friction Impact to Robot Payload Capacity

Form-Finding and Free Vibration Analysis of a Class-One Triplex Tensegrity Prism

Form-Finding and Free Vibration Analysis of a Class-One Triplex Tensegrity Prism

Minimal mass of prismatic tensegrity structures

PurposeThis paper is contributed to find the minimal mass prismatic tensegrity structures.Design/methodology/approachIn the stable state of the structure with any given external force, the internal forces of the structure members are taken as the critical force to calculate the cross-sectional area, and the total mass of the structure can be obtained. Firstly, the mathematical model of prismatic tensegrity was built. Secondly, the stability of the tensegrity was analyzed based on the force equilibrium of one node, the force density relationship of elements was obtained. The deformation of p-bar tensegrity prism unit was studied with the same mass. The force of the structure under external force was analyzed.Findings(1) The length of bar and the structural radius are almost invariant, and the mechanical properties of 3-bar tensegrity prism is more outstanding; (2) theoretically, the mass of the structure is minimal while the projection of bar passes the center of the circle. Under the circumstances, the force of diagonal cable is 0 N, the vertical force component of bar cancels the axial external force.Originality/value(1) By analyzing the deformation of p-bar tensegrity prism with the same mass, the length of bar and the structural radius are proved be almost invariant and the mechanical properties of 3-bar tensegrity prism is more outstanding; (2) theoretically, the mass of the structure is minimal while the projection of bar passes the center of the circle. Under the circumstances, the force of diagonal cable is 0 N, the vertical force component of bar cancels the axial external force.

A Simplified Kinematics and Kinetics Formulation for Prismatic Tensegrity Robots: Simulation and Experiments

Tensegrity robots offer several advantageous features, such as being hyper-redundant, lightweight, shock-resistant, and incorporating wire-driven structures. Despite these benefits, tensegrity structures are also recognized for their complexity, which presents a challenge when addressing the kinematics and dynamics of tensegrity robots. Therefore, this research paper proposes a new kinematic/kinetic formulation for tensegrity structures that differs from the classical matrix differential equation framework. The main contribution of this research paper is a new formulation, based on vector differential equations, which can be advantageous when it is convenient to use a smaller number of state variables. The limitation of the proposed kinematics and kinetic formulation is that it is only applicable for tensegrity robots with prismatic structures. Moreover, this research paper presents experimentally validated results of the proposed mathematical formulation for a six-bar tensegrity robot. Furthermore, this paper offers an empirical explanation of the calibration features required for successful experiments with tensegrity robots.

Self-equilibrium, stability, and accuracy degradation of imperfect prismatic tensegrities with additional cable nets

Prismatic tensegrity is an attractive structure due to the unique advantages of self-equilibrium, lightweight, small compression volume, specific super stability, ease of deployment, and functional extension. However, it is subject to geometric large deformation when suffering from imperfect factors like manufacturing errors, thermal deformation, slips of tensioning knots, and mechanical relaxation of members, which will eventually reduce the accuracy and reliability. We develop a kind of prismatic tensegrities with additional cable nets for space deployable constructions. For the specific requirements of high accuracy and high reliability, we fully consider the influence of imperfect factors, and explore their environment- and time-dependent characteristics from the view of self-equilibrium, stability, and accuracy. First, a preload optimization method is established for high-accuracy determination of the shape and uniform distribution of preload. Then, a high-fault-tolerance accuracy estimation method is proposed for accuracy degradation research, for which nonlinear force density equations and time-dependent tangential stiffness matrix are formulated to obtain deformation shapes and stability criteria beforehand. Eventually, the characteristics of the prismatic tensegrities with additional cable nets, under the influence of imperfect factors, are fully revealed and forecasted. The methods and some conclusions of this study can also be applied to other complex tensegrities or prestressed pin-jointed systems.

An analytic study on the properties of solitary waves traveling on tensegrity-like lattices

This paper develops an analytic study on the existence and properties of solitary waves on 1D chains of lumped masses and nonlinear springs, which exhibit a mechanical response similar to that of tensegrity prisms with locking-type response under axial loading. Making use of the Weierstrass’ theory of 1D Lagrangian conservative systems, we show that such waves exist and that their shapes depend on the wave speed. A progressive localization of the traveling pulses in narrow regions of space is observed as the wave speed increases up to a limit value. A comparative analysis illustrates that the presented study is able to capture the wave dynamics observed in previous numerical studies on tensegrity mass–spring chains.

Fabrication and experimental characterisation of a bistable tensegrity-like unit for lattice metamaterials

The peculiar nonlinear mechanical behaviour of tensegrity structures and their engineering applications have attracted considerable interest in the last two decades. However, the difficulties in their traditional fabrication and assembling methods represent current limitations to their widespread use. This paper presents a novel design and fabrication procedure for bistable tensegrity-like units. Starting from the classical triangular tensegrity prism and using stereolithography technology, a double tensegrity-like unit was designed and realised monolithically as a compliant mechanism. High repeatability of compression tests confirmed the activation of the designed bistable twisting mechanism in large displacements, proving that the bistability of a tensegrity-like unit with null selfstress and no cables can dependably be achieved. Numerical simulations showed that a reduced-order stick-and-spring model is able to provide predictions on the nonlinear mechanical behaviour of the unit in close agreement with experimental results. Low relative density and bistable characteristics make this type of tensegrity-like unit suitable to manufacture highly-customisable multistable metamaterials. The proposed procedure could be applied to transform and additively manufacture other types of tensegrity structures with different nonlinear responses into corresponding tensegrity-like versions. • Stereolithography fabrication of tensegrity-like units without pin-joints or cables. • Experimental investigation on the compressive behaviour of a tensegrity-like unit. • Bistable response of the designed unit confirmed by experimental results. • Design procedure suitable to lightweight metamaterials with nonlinear behaviour. • Tessellation of units for multistable metamaterials and control of compact waves.

A tensegrity-based morphing module for assembling various deployable structures

Deployable structures have a wide range of applications in, for instance, soft robotics, bionic machinery, aviation equipment, and programmable metamaterials. Tensegrity, based on the equilibrium of axially stretched and compressed elements, can be an attractive candidate for designing deployable structures. However, existing tensegrity-based deployable structures mostly hold symmetrical deformation modes, which greatly limits their applicable scenarios. To this end, we here propose a tensegrity-based morphing module that can exhibit not only symmetrical configurations, but also a series of asymmetrical configurations. This module is composed by a prismatic tensegrity and two end plates, in which the tensegrity contributes to the deformability of the module and the plates allow the module rather flexible to construct various deployable structures with abundant configurations. Our theoretical and experimental results both confirm that the proposed module is capable of symmetrical/asymmetrical deployment through a variety of active actuations. Finally, several quasi-one-dimensional structures are assembled by a number of the morphing modules as examples, and their flexible deformability helps to broaden the application prospects of tensegrity-based deployable structures.

Experimental Investigation on the Vibration Attenuation of Tensegrity Prisms Integrated with Particle Dampers

Experimental Investigation on the Vibration Attenuation of Tensegrity Prisms Integrated with Particle Dampers

Nonlinear Mechanics of a Smart Biotensegrity Human Foot Prosthesis

This work deals with the nonlinear mechanics of smart bioinspired tensegrity structures. A minimal regular tensegrity prism actuated by shape memory alloy (SMA) elements is investigated to represent a human foot. A formulation considering the force density matrix approach is used to model the equilibrium equations of the tensegrity structure based on node mapping. Lagrange multipliers are employed to represent constraints. The SMA thermomechanical behavior is described by considering a modified polynomial constitutive model. Numerical simulations are developed from an optimization procedure employing the Levenberg–Marquardt method. An investigation of the tensegrity capability to model a human foot is carried out analyzing either mechanical or physiological aspects of the tensegrity prosthesis. The mechanical performance is compared with high performance prostheses available on the market, showing that it is an interesting alternative with respect to mechanical resistance. Regarding physiology, foot movements are properly mimicked from SMA actuation.

Directional snapping instability in a bistable tensegrity under uniaxial loads

Multistable structures possess fascinating shape reconfigurability and property programmability, which have been applied in various scenarios. Multistable tensegrities are a novel class of compliant prestressed structures with different stable states. However, how structural transitions occur among these states remains an interesting challenge. Here we investigate a continuous configuration transition of bistable quadrangular prismatic tensegrity under uniaxial loads. A theoretical model is proposed to determine the deformation behavior of the structure starting from different stable states clamped by structural height or axial forces. The obtained structural mechanical responses exhibit multi-value curves, leading to the occurrence of snapping instability and bistable states. Interestingly, the transition of these two stable states is directional and one-way irreversibility, i.e., the structure can only jump from one stable state to another, but not vice versa. Finally, displacement and force-clamped loading experiments are carried out under these two stable states, and the test results agree well with our theoretical predictions. This work shows that applying uniaxial loads can directionally trigger a specific stable-state transition of the tensegrity, which holds great applications in the precise configuration programming of metamaterials and advanced devices.

Form-finding of tessellated tensegrity structures

Many large-scale tensegrity structures are formed by assembling repeating units; a tensegrity tower formed by stacking tensegrity prisms is one example. Tessellated tensegrity structures tend to feature a repeating topology, but the geometry and self-stress vary across the assembly. In this work we design tensegrity structures by imposing periodic boundary conditions on a single unit cell to enforce repetitive topology, geometry and self-stress. A tessellated tensegrity tower is designed by using a stiffness matrix form-finding technique in combination with nodal position constraints. Since the tessellated tensegrity is not globally in equilibrium, a transition structure is required to balance the tessellated tensegrity tower on one side and to connect to a fixed boundary on the other. The challenge in designing the transition structure is that both coordinates and residual forces are simultaneously predefined at specific nodes. In this paper, we propose an adapted stiffness form-finding method to design the transition structure as an alternative method to the conventional ground structure method.