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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.

Composite materials with engineered band gaps are promising solutions for wave control and vibration mitigation at various frequency scales. Despite recent advances in the design of phononic crystals and acoustic metamaterials, the generation of wide low-frequency band gaps in practically feasible configurations remains a challenge. Here, we present a class of lightweight metamaterials capable of strongly attenuating low-frequency elastic waves, and investigate this behavior by numerical simulations. For their realization, tensegrity prisms are alternated with solid discs in periodic arrangements that we call 'accordion-like' meta-structures. They are characterized by extremely wide band gaps and uniform wave attenuation at low frequencies that distinguish them from existing designs with limited performance at low-frequencies or excessively large sizes. To achieve these properties, the meta-structures exploit Bragg and local resonance mechanisms together with decoupling of translational and bending modes. This combination allows one to implement selective control of the pass and gap frequencies and to reduce the number of structural modes. We demonstrate that the meta-structural attenuation performance is insensitive to variations of geometric and material properties and can be tuned by varying the level of prestress in the tensegrity units. The developed design concept is an elegant solution that could be of use in impact protection, vibration mitigation, or noise control under strict weight limitations.

We construct infinite periodic versions of the stress matrix and establish sufficient conditions for periodic tensegrity frameworks to be globally rigid in \\(R^d\\) in the cases when the lattice is either fixed, fully flexible, or flexible with a volume constraint for the fundamental domain. For the fixed and fully flexible lattice variants, we also establish necessary and sufficient conditions for generic infinite periodic bar-joint frameworks to be globally rigid in \\(R^d\\). These results provide periodic versions of the fundamental results of Connelly, as well as Gortler, Healy and Thurston on the global rigidity of generic finite bar-joint frameworks.

This article tackles a central problem in soft and tensegrity robotics: designing effective control strategies for adaptive behavior. According to the principle of morphological computation, this challenge can be addressed by “offloading” computation from the brain (controller) to the body, leveraging the agent’s physical properties. I propose a conceptual reconsideration grounded in ecological psychology, whose focus on affordances highlights how soft devices can exploit body-environment interactions to enhance their adaptability to ever-changing conditions. Contributions: (i) a diagnosis of the limitations of morphological computation as a control paradigm for soft robotics; (ii) the articulation of the Design Principle of Agent-Environment Duality, which could reframe control as the regulation of affordance relations at the agent-environment scale.

Clustered tensegrity structures, which retain the advantages of the classical tensegrity structure, are better controllable, and provide more possibilities for a wide range of application scenarios. However, the dynamic modeling and analysis of clustered tensegrity structures with the significant characteristic of rigid-flexible-soft coupling have great challenges: (1) Rigid components (rods) and flexible components (classical cables) lead to a dynamic model that has the characteristic of rigid-flexible coupling, which severely limits the efficiency of dynamic simulation; (2) The \"soft\" characteristic of sliding cable slacking, which makes sliding cables frequently switch between tense and slack states in the actuation process, results in non-smooth dynamic behavior. In this paper, a model smoothing method based on the positional finite element method is proposed for dynamic analysis of clustered tensegrity structures with the rigid-flexible coupling characteristic, which can effectively and controllably filter the elastic high-frequency oscillations in the system without losing the accuracy of structural macroscopic deformation and greatly improve the computational efficiency than the model without smoothing. Then, for the non-smooth dynamics caused by soft characteristic, a nonlinear complementary method for sliding cable slacking is proposed, which makes the numerical simulation of sliding cable slacking and tensioning more stable than the traditional method. Finally, the model smoothing method for rigid-flexible coupling is integrated with the nonlinear complementary method for soft sliding cables, and a rigid-flexible-soft coupling framework that combines the implicit discrete scheme is formed for dynamic modeling and analysis of clustered tensegrity structures. The numerical simulation results show that the rigid-flexible-soft coupling framework can realize efficient, stable, and accurate dynamic analysis of clustered tensegrity structures and provide support for the design of clustered tensegrity structures.

This paper presents a novel flexible tensegrity-based jumping robot that achieves multifunctional locomotion control solely through its deformable structure. The proposed design enables three essential capabilities, directional steering, take-off angle modulation, and autonomous self-righting, using only the inherent compliance of its flexible tensegrity architecture, eliminating the need for additional dedicated mechanisms. The energy storing and releasing mechanism is implemented through sets of planetary gears. Experiments show that the constructed robot can jump more than 420 mm high at a take-off angle of 85 degrees.

This paper introduces a data-driven approach to address the long-standing challenge of modeling complex tensegrity systems. The proposed approach focuses on approximating unknown black box systems and estimating their output error covariance using input/output (IO) information. First, an approximation system that mirrors the input–output relation of the black box system is obtained. Next, output error covariance between the approximation and the black box system is calculated, which evaluates the accuracy of the identified model. This two-step approach relies exclusively on the black box system’s Markov parameter sequence, eliminating the need for dynamics knowledge of the system. Nonlinear examples of a NACA 2412 tensegrity morphing airfoil and a 3D tensegrity prism are studied for validation. The proposed approach successfully identified approximation systems in state space realization in both cases with insignificant output error covariances. Compared to the widely-used Mode Displacement Method (MDM), the proposed approach exhibits an advantage in identifying velocity outputs for tensegrity systems. The developed approach in this paper applies to other tensegrity structures and structural identification problems.

Tensegrity robotic fish is an emerging solution in the field of robotic fish, and its dynamic model plays a crucial role in guiding the improvement of swimming performance. However, there is a lack of in-depth research in this area. This paper proposes a planar dynamic model for the tensegrity robotic fish, which can be used to simulate and investigate the influence of body stiffness on swimming performance. An equivalent modeling approach is introduced for the soft fish skin and tail fin to consider their effects of flexibility on the robotic fish’s swimming motion. A hydrodynamic discretization model based on virtual nodes is proposed to incorporate hydrodynamic effects such as drag force, lift force, and added mass into the fish’s vertebral column. By integrating the above models, the dynamic model for the tensegrity robotic fish is established after embedding the tail joint constraints. The method of reconstructing fish body waves in the simulation is provided to calculate fish swimming characteristics. The parameter identification method for the drag coefficient and torque coefficient is proposed. The dynamic model of the tensegrity robotic fish is verified using swimming experimental data. By conducting numerous numerical simulations, the effects of the body stiffness distribution, tail fin stiffness, and fish skin stiffness on the swimming performance are analyzed. The simulation results reveal the significant role of adjusting body stiffness in enhancing the swimming characteristics of the robotic fish.

Tensegrity continuum robots (TCRs) are an emerging class of flexible continuum robots that employ the tensegrity concept to achieve a natural balance of compliance and strength. The dynamics of a TCR with slack cables is a high-dimensional non-smooth nonlinear system. Motion control of such a system is challenging, especially when the TCR is placed horizontally. In this work, a differential–algebraic equations (DAEs) model-based instantaneous optimal control (IOC) approach for the tip’s position and orientation collaborative tracking of a horizontal TCR with slack stiffening cables is proposed. Based on the complementarity theory and the Fischer–Burmeister function, the non-smooth problem of cable relaxation can be described as a continuous differentiable algebraic equation. Then, by combining the dynamic differential equations, a DAEs model of the TCR with slack cables can be obtained. Subsequently, the original continuous tip tracking problem is approximated to a series of IOC problems at each discrete time slot. Finally, considering the control input saturation constraints, a suboptimal control law can be obtained just by solving a small-scale optimization problem. The proposed IOC approach provides a novel and unified control framework to deal with the tip tracking problem of the horizontal TCRs, and the non-smooth property of cable relaxation is naturally involved in the IOC controller. Numerical experiments on the tip tracking of a horizontal TCR are conducted to demonstrate the effectiveness and advantages of the proposed IOC method.

This paper presents an optimization approach for design of tensegrity structures based on graph theory. The formulation obtains tensegrities from ground structures, through force maximization using mixed integer linear programming. The method seeks a topology of the tensegrity that is within a given geometry, which provides insight into the tensegrity design from a geometric point of view. Although not explicitly enforced, the tensegrities obtained using this approach tend to be both stable and symmetric. Borrowing ideas from computer graphics, we allow “restriction zones” (i.e., passive regions in which no geometric entity should intersect) to be specified in the underlying ground structure. Such feature allows the design of tensegrities for actual engineering applications, such as robotics, in which the volume of the payload needs to be protected. To demonstrate the effectiveness of our proposed design method, we show that it is effective at extracting both well-known tensegrities and new tensegrities from the ground structure network, some of which are prototyped with the aid of additive manufacturing.