Each element in tensioned structural networks -- such as tensegrity, architectural fabrics, or medical braces/meshes -- requires a specific tension level to achieve and maintain the desired shape, stability, and compliance. These structures are challenging to manufacture, 3D print, or assemble because flattening the network during fabrication introduces multiplicative inaccuracies in the network's final tension gradients. This study overcomes this challenge by offering a fabrication algorithm for direct 3D printing of such networks with programmed tension gradients, an approach analogous to the spinning of spiderwebs. The algorithm: (i) defines the desired network and prescribes its tension gradients using the force density method; (ii) converts the network into an unstretched counterpart by numerically optimizing vertex locations toward target element lengths and converting straight elements into arcs to resolve any remaining error; and (iii) decomposes the network into printable toolpaths; Optional additional steps are: (iv) flattening curved 2D networks or 3D networks to ensure 3D printing compatibility; and (v) automatically resolving any unwanted crossings introduced by the flattening process. The proposed method is experimentally validated using 2D unit cells of viscoelastic filaments, where accurate tension gradients are achieved with an average element strain error of less than 1.0\%. The method remains effective for networks with element minimum length and maximum stress of 5.8 mm and 7.3 MPa, respectively. The method is used to demonstrate the fabrication of three complex cases: a flat spiderweb, a curved mesh, and a tensegrity system. The programmable tension gradient algorithm can be utilized to produce compact, integrated cable networks, enabling novel applications such as moment-exerting structures in medical braces and splints.

ACCEPTED MARCH, 2025 1 Real-Time Shape Estimation of Tensegrity Structures Using Strut Inclination Angles Tufail Ahmad Bhat 1, Y uhei Y oshimitsu 1, Kazuki Wada 1, Shuhei Ikemoto 1 Abstract --T ensegrity structures are becoming widely used in robotics, such as continuously bending soft manipulators and mobile robots to explore unknown and uneven environments dynamically. Estimating their shape, which is the foundation of their state, is essential for establishing control. However, on-board sensor-based shape estimation remains difficult despite its importance, because tensegrity structures lack well-defined joint structures, which makes it challenging to use conventional angle sensors such as potentiometers or encoders for shape estimation. T o our knowledge, no existing work has successfully achieved shape estimation using only onboard sensors such as Inertial Measurement Units (IMUs). This study addresses this issue by proposing a novel approach that uses energy minimization to estimate the shape. We validated our method through experiments on a simple Class 1 tensegrity structure, and the results show that the proposed algorithm can estimate the real-time shape of the structure using onboard sensors, even in the presence of external disturbances. I NTRODUCTION T HE concept of "tensegrity" is coined by the iconoclastic architect R. Buckminster Fuller. It describes structures that achieve stability through a balance of forces: specific components, known as "cables" are always in tension, while others, known as "struts" are constantly under compression [1]. In tensegrity, the cables of the structure are always under continuous tension, a condition known as "prestress".

Biomimicry has played a pivotal role in robotics. In contrast to rigid robots, bio-inspired robots exhibit an inherent compliance, facilitating versatile movements and operations in constrained spaces. The robot implementation in fabrication, however, has posed technical challenges and mechanical complexity, thereby underscoring a noticeable gap between research and practice. To address the limitation, the research draws inspiration from the unique musculoskeletal feature of vertebrate physiology, which displays significant capabilities for sophisticated locomotion. The research converts the biological paradigm into a tensegrity-based robotic system, which is formed by the design of rigid-flex coupling and a compliant mechanism. This integrated technique enables the robot to achieve a wide range of motions with variable stiffness and adaptability, holding great potential for advanced performance in ill-defined environments. In summation, the research aims to provide a robust foundation for tensegrity-based biomimetic robots in practice, enhancing the feasibility of undertaking intricate robotic constructions.

In the design of tensegrity structures, traditional form-finding methods utilize kinematic and static approaches to identify geometric configurations that achieve equilibrium. However, these methods often fall short when applied to actual physical models due to imperfections in the manufacturing of structural elements, assembly errors, and material non-linearities. In this work, we develop a deep neural network (DNN) approach to predict the geometric configurations and physical properties-such as nodal coordinates, member forces, and natural frequencies-of any tensegrity structures in equilibrium states. First, we outline the analytical governing equations for tensegrity structures, covering statics involving nodal coordinates and member forces, as well as modal information. Next, we propose a data-driven framework for training an appropriate DNN model capable of simultaneously predicting tensegrity forms and physical properties, thereby circumventing the need to solve equilibrium equations. For validation, we analyze three tensegrity structures, including a tensegrity D-bar, prism, and lander, demonstrating that our approach can identify approximation systems with relatively very small output errors. This technique is applicable to a wide range of tensegrity structures, particularly in real-world construction, and can be extended to address additional challenges in identifying structural physics information.

Abstract-- A tensegrity-based system is a promising approach for dynamic exploration of uneven and unpredictable environments, particularly, space exploration. However, implementing such systems presents challenges in terms of intelligent aspects: state recognition, wireless monitoring, human interaction, and smart analyzing and advising function. Here, we introduce a 6-strut tensegrity integrate with 24 multimodal strain sensors by leveraging both deep learning model and large language models to realize smart tensegrity. Using conductive flexible tendons assisted by long short-term memory model, the tensegrity achieves the self-shape reconstruction without extern sensors. Finally, human interaction system of the tensegrity helps human obtain necessary information of tensegrity from the aspect of human language. The concept of using tensegrity structures in space exploration is an innovative approach that offers several advantages due to the unique properties of tensegrity systems. One famous example is the "Super Ball Bot" developed by NASA (National Aeronautics and Space Administration) [1][2]. Tensegrity structures are composed of solid compression components (rods/struts) connected by tension elements (cables/strings).

We introduce collision-resilient aerial vehicles with icosahedron tensegrity structures, capable of surviving high-speed impacts and resuming operations post-collision. We present a model-based design approach, which guides the selection of the tensegrity components by predicting structural stresses through a dynamics simulation. Furthermore, we develop an autonomous re-orientation controller that facilitates post-collision flight resumption. The controller enables the vehicles to rotate from an arbitrary orientation on the ground for takeoff. With collision resilience and re-orientation ability, the tensegrity aerial vehicles can operate in cluttered environments without complex collision-avoidance strategies. These capabilities are validated by a test of an experimental vehicle operating autonomously in a previously-unknown forest environment.

Lagrangian and Hamiltonian neural networks (LNNs and HNNs, respectively) encode strong inductive biases that allow them to outperform other models of physical systems significantly. However, these models have, thus far, mostly been limited to simple systems such as pendulums and springs or a single rigid body such as a gyroscope or a rigid rotor. Here, we present a Lagrangian graph neural network (LGNN) that can learn the dynamics of articulated rigid bodies by exploiting their topology. We demonstrate the performance of LGNN by learning the dynamics of ropes, chains, and trusses with the bars modeled as rigid bodies. LGNN also exhibits generalizability -- LGNN trained on chains with a few segments exhibits generalizability to simulate a chain with large number of links and arbitrary link length. We also show that the LGNN can simulate unseen hybrid systems including bars and chains, on which they have not been trained on. Specifically, we show that the LGNN can be used to model the dynamics of complex real-world structures such as the stability of tensegrity structures. Finally, we discuss the non-diagonal nature of the mass matrix and its ability to generalize in complex systems.

While the ultimate goal is to produce a tensegrity more than 6 struts, e.g. a 15-bar tensegrity, past experience has demonstrated that we must first develop an innovative system that will facilitate the assembly of a general n-bar tensegrity. To be successful, we believe the development of the new assembly methodology must encompass not only the design of the clamping system but also the design of the tensegrity itself, including the struts, the springs and the spring-to-strut connectors. We therefore propose to develop the 15-bar in two phases: Phase I will be the development of an innovative assembly method, and Phase II will focus on the design and manufacture of a 15-bar tensegrity, with a new strut design probably being part of this. Longer term goals will be aimed at repackaging the wireless electronics on the new struts and adding encoders to control the phase of the motors shafts.

Mobile soft robots offer compelling applications in fields ranging from urban search and rescue to planetary exploration. A critical challenge of soft robotic control is that the nonlinear dynamics imposed by soft materials often result in complex behaviors that are counterintuitive and hard to model or predict. As a consequence, most behaviors for mobile soft robots are discovered through empirical trial and error and hand-tuning. A second challenge is that soft materials are difficult to simulate with high fidelity -- leading to a significant reality gap when trying to discover or optimize new behaviors. In this work we employ a Quality Diversity Algorithm running model-free on a physical soft tensegrity robot that autonomously generates a behavioral repertoire with no a priori knowledge of the robot dynamics, and minimal human intervention. The resulting behavior repertoire displays a diversity of unique locomotive gaits useful for a variety of tasks. These results help provide a road map for increasing the behavioral capabilities of mobile soft robots through real-world automation.