As a feature of the biological body that is expected to be applied to robots, musculoskeletal system has garnered considerable attention in the literature. The musculoskeletal system is generally overdriven because muscles can only generate tension. This means that some of the tension in a muscle is consumed as an internal force to balance the tension in other muscles. The function of these antagonistic pairs is to alter the joints stiffness, and coordinate postural changes in response to external forces. Several musculoskeletal robots have been developed to provide robots with functionalities, i.e., bio-inspired embodied intelligence (Pfeifer et al., 2007), generated by this redundancy. In particular, the challenge of realizing a complex three-dimensional musculoskeletal system like that of living organisms in robots has been continuously pursued in this field (Kozuki et al., 2016; Kurumaya et al., 2016; Tadesse et al., 2016; Hitzmann et al., 2018). However, it is tough to realize the complex musculoskeletal robot that is practical and applicable for various tasks because several features are difficult to realize by engineering, such as multi-degree-of-freedom joints, soft tissues, and their lubrication. Therefore, to understand and apply bio-inspired embodied intelligence, which arises from the redundancy of the complex musculoskeletal system, it is necessary to take an approach that is not limited to mimicking the actual musculoskeletal system. As one of the approaches for this purpose, we leverage tensegrity in this study.

Tensegrity originally refers to a stable three-dimensional structure comprising struts that bear compressive forces and cables that bear only tensile forces, but where the struts are not connected (Burkhardt, 2008). It has begun to attract attention in architecture (Gomez-Jauregui, 2004), and is now attracting widespread attention in biology (Ingber, 2003a,b), engineering (Sultan, 2009), and information science (Caluwaerts et al., 2013). Using this broad spread across multiple disciplines, the term “tensegrity” has been utilized in recent years, even if it does not meet all of the aforementioned definitions. Specifically, even if the struts are not bar-shaped and bear bending stresses, or even if there are connections between rigid bodies, they are often called tensegrity if they are stable three-dimensional structures under cables tension. In this study, hereafter, the former will be referred to as the strict tensegrity, and the latter as the broad tensegrity.

Research and development of robots utilizing tensegrity have been actively pursued in robotics, and both strict and broad tensegrities have been leveraged. As tensegrity robots that employ the strict tensegrity, deformable and rolling spherical robots have been studied for years (Rieffel et al., 2007; Shibata and Hirai, 2009; Kim et al., 2014, 2017; Vespignani et al., 2018). The strict tensegrity comprises an insignificant number of parts, which are lightweight and robust, and can be made flexible and actively deformable by introducing springiness and actuators to some of the cables. This is a valuable feature in robots for locomotion, where interaction with the environment is inevitable. Therefore, there have been active studies on its control (Caluwaerts et al., 2014; Kim et al., 2020), and the utilization of data-driven approaches has gained significant attention in recent years (Kim et al., 2015; Zhang et al., 2017; Zhu et al., 2018; Surovik et al., 2019). However, there are also studies utilizing broad tensegrity for this type of tensegrity robot. The utilization of bent bars instead of struts to improve the shape and deformation during movement is one of the approaches utilizing broad tensegrity (Rhodes et al., 2019; Schorr et al., 2021a). In this case, the bent bar bears the bending stress, which is not entirely the strict tensegrity, but the bent bar can be regarded as a straight strut; hence, it approximately follows the definition of strict tensegrity. This means that the bent bar must be thicker and heavier for strength, but other advantages of the strict tensegrity are maintained, such as flexibility to the entire structure. In addition, as a tensegrity robot for another kind of locomotion, a tensegrity robot that stores and releases elastic energy for jumping has been developed (Schorr et al., 2021b). This robot cannot even be interpreted as the strict tensegrity, and does not benefit from its inherent advantages. Meanwhile, it is easy to design a mechanism that solely allows specific movements by limiting deformation locations. The ease of design to obtain a target mechanism is the main reason to utilize the broad tensegrity instead of strict tensegrity for tensegrity robots.

The fact that the broad tensegrity is helpful to pursue a specific function is not limited to locomotion. For instance, the broad tensegrity is utilized to realize multi-degree-of-freedom joints such as the shoulder joint (Wang and Post, 2021) and spinal structures Sabelhaus et al. (2020); Zappetti et al. (2020). A similar trend is also determined in manipulators utilizing tensegrity structures. Most of the manipulators utilizing tensegrity can be classified into two types: those that utilize multiple struts connected like rigid body parts (Lessard et al., 2016; Jung et al., 2018; Ramadoss et al., 2020), and those that utilize linkage mechanisms by connecting struts with a single degree of freedom joints (Fadeyev et al., 2019; Fasquelle et al., 2020). In these studies, it is clear that they are not utilizing the strict tensegrity, but the broad tensegrity. In addition, there are tensegrity manipulators in which a few struts are replaced with compression springs (Wei et al., 2020). This mechanism is widely utilized in continuum manipulators (Rao et al., 2021), and it cannot also be regarded as the strict tensegrity because it includes either a member that bears the bending moment, or a joint that reduces the degree of freedom.

The above indicates that the broad tensegrity has been utilized for manipulators, while the strict tensegrity has not been utilized. This is because the broad tensegrity easily achieves desired non-spherical shapes and mechanisms, while the strict tensegrity is difficult to do the same. However, to increase redundancy and flexibility, it is desirable to utilize the strict tensegrity, which reduces the number of parts, allows several actuators to be placed without interference, and provides flexibility to the entire structure. The further away from the strict tensegrity, the more the inherent features of the strict tensegrity are lost. Therefore, as the system’s flexibility and redundancy increase, similar implementation challenges are thought to eventually appear as in tendon-driven systems utilizing rigid joints (e.g., musculoskeletal systems).

This study proposes a design method for an arm-type tensegrity robot with a long shape in one direction, and can be deformed like a continuum manipulator. This method is based on the idea of utilizing flexible and straightforward strict tensegrity modules, and connecting them recursively so that they remain strict tensegrity even after being connected. The tensegrity obtained by this method strongly resists compressive forces in the longitudinal direction, but is flexible in the bending direction. Therefore, the changes in stiffness owing to internal forces, such as in musculoskeletal robots, appear more in the bending direction. This study describes this design method, then describes a developed pneumatically driven tensegrity robot arm with 20 actuators accordingly. As evaluations of the availability of the robot, the range of motion and stiffness under various driving patterns are presented.

Figure 1 illustrates the overview of the design rule proposed for tensegrity robot arms in this study. This design rule is based on the idea of connecting simple modules to obtain an overall complex tensegrity. Here, a simple module refers to a strict tensegrity that can be designed intuitively by the designer. The tensegrity to be utilized as a simple module does not require any structural complexity such as the number of struts and cables, but it requires the following two characteristics:

1) It must have stiff cables that can be observed as unchanging in length, and constitute at least one closed path.

The first characteristic is for specifying the paths that connect simple modules. Because the structure of the connected modules is an arm-like tensegrity long in one direction, this characteristic also suppresses the change in length in the longitudinal direction. The second characteristic is for allowing the deformations required when simple modules are connected. The longitudinal length change is suppressed, while the curved posture is allowed in the arm-like tensegrity. In particular, Figure 1 illustrates an example of such a simple module, comprising four struts, and having two closed paths.

Simply connecting two modules to share a closed path of stiff cables does not make the structure tensegrity. To make the connected structure have tensegrity, each end on the closed path needs an additional connection with an end belonging to the different module, not on the closed path, and does not have the opposite end neighboring. As illustrated in Figure 1, the proposed design rule utilizes actuators for those additional cables. Aforementioned, because the arm-like tensegrity is solely allowed to deform in the bending direction, because of the closed path of stiff cables, the active deformation of the tensegrity by the actuators appears as bending. To distribute the actuators uniformly, the connection of the third and subsequent modules is according to a different rule, which inserts a module into the already connected closed path. In particular, to keep the structure after inserting a tensegrity, actuated cables that connect ends on the two new closed paths but not belonging to the inserted module are introduced. Consequently, the arm-like tensegrity appears to have a similar number of layers as the connected modules, and each layer/module will have a similar number of actuators.

To validate the basic idea of the proposed design rule, we fabricated a mock-up in which the actuators were implemented with flexible cables. This model is an arm-like tensegrity comprising five modules connected by the proposed design rule, which belongs in the strict tensegrity. In this model, the stiff cables are the PE lines for fishing, and the flexible cables are implemented by attaching a spring in the middle of the PE lines. By observing the passive deformation when an external force is applied, we can verify that the tensegrity fabricated by the proposed design rule can be continuously bent, and that it will be able to bend when actuators apply internal force actively.

Figure 2 illustrates the overview of the mock-up and the result of the verification. From Figure 2, first, it can be confirmed that the proposed design rule provides an arm-like tensegrity. Because five modules are connected, we can observe five layers corresponding to them. Because the module is a tensegrity comprising four struts, as illustrated in Figure 1, four actuators per layer should be implemented, but flexible cables should replace them in this mock-up. Next, it can be observed that the arm-like tensegrity takes various continuous bending postures when an external force is applied. This indicates that the closed path comprising stiff cables play a role in suppressing the longitudinal length change as expected. Therefore, we can expect that the active posture changes by actuators will also result in a similar continuous bending behavior.

By utilizing pneumatic cylinders as actuators for the five-modules arm-like tensegrity (see Figure 2) constituted by following the proposed design rule (see Figure 1), we developed the tensegrity robot arm capable of continuous bending. Figure 3 illustrates the overview of the developed robot. The robot arm incorporates 20 double-acting pneumatic cylinders arranged according to the proposed design rule. As illustrated in the picture, the robot arm is mounted on a black aluminum frame box, which contains the pressure control valves, the built-in controller, and other devices for the control. The size of the base box is 0.6 (m) × 0.6 (m) × 0.3 (m), the height of the whole arm is 1.2 (m) without the box, and the mass of the whole arm including the air tubes is 2.9 (Kg). In the following subsections, we provide detailed information on the mechanical structure and the control system.

To verify that the developed tensegrity robot arm can be utilized as a flexible and redundant robot platform with qualitative features of musculoskeletal robots, we conducted experiments to evaluate the range of motion and stiffness. In the experiments, postures of the developed tensegrity robot arm were measured utilizing the optical motion capture system (MAC3D System, Motion Analysis).

Figure 7 illustrates the overview of the measuring system. As illustrated in the figure, 24 markers of the optical motion capture system are attached to the developed tensegrity robot arm. The robot has six closed paths of rigid cables, and the center positions of the closed paths are calculated from the four markers on each closed path. These center positions represent the posture of the entire robot arm, which is a serial link mechanism with five links connected by multi-degree-of-freedom joints. The base coordinate system of the robot is defined as a right-handed system, with the origin at the closed path’s center at the bottom. However, the hand-tip coordinate system is defined with the closed path’s center at the top as the origin. The X-Y plane is first defined as the least-squares plane of markers 1 to marker 4, to determine the normal, i.e., the Z-axis. The projection of marker 2 onto the least-squares plane next defines the orientation of the X-axis, and the remaining Y-axis is defined so that the entire system is right-handed. In this study, this tip coordinate system is utilized to quickly understand the posture and twist of the whole arm from the measurement results.

Figure 8 illustrates examples of postures and their corresponding measurements. As illustrated in this figure, it can be observed that the developed tensegrity robot arm can take a variety of continuous bending postures, and the optical motion capture system can successfully measure them. In the following subsections, we explain the evaluation of the movable area, and the variable stiffness feature of the developed tensegrity robot.

In the design rules proposed in this study, the cables of each tensegrity module are defined as stiff cables constituting closed paths and springy cables. The closed path is necessary to prevent the arm from collapsing in the longitudinal direction, while the springy cable, which is passive, can be replaced by an actuator with series elasticity such as a pneumatic cylinder. Utilizing actuators that directly deform each module is promising for increasing the range of motion, because it can be difficult to significantly deform a module by driving the cables that cross between modules, depending on the posture. This improvement also leads to increased redundancy, which is highly compatible with the focus of this study.

A more obvious approach to increase the range of motion is to increase the number of modules. However, to increase the number of modules without altering the total length of the tensegrity robot arm, it is necessary to reduce the size of the module. The main challenge in reducing the size of the module is the displacement of the actuators. The contraction ratio, which means the amount of displacement relative to the maximum total length of the actuator, is generally less than 0.5 for linear actuators, and it decreases as the total length is reduced. Therefore, if the module is reduced in size, the amount of deformation of the module must be reduced. Winding cables by rotary actuators are one possible solution, but it still poses a mounting challenge because the clearance decreases as the module is reduced in size. In summary, to significantly increase the range of motion of the tensegrity robot arm, a comprehensive approach that covers the size and structure of the module, and the number and type of actuators is necessary.

Quantitative evaluation of the stiffness and payload of the developed tensegrity robot arm is crucial future work. As a preliminary evaluation, we measured the blocking force of the hand-tip by pulling in one direction from the maximum bending position to an approximately upright position. The result shows approximately 6 (N). However, because the tensegrity robot arm has high redundancy, the blocking force measurement essentially requires a suitable jig and multi-axis force/torque sensors. In addition, the strong dependence on posture also makes measuring the blocking force difficult. In parallel with improving the tensegrity robot arm, we will establish a measurement environment for quantitative evaluation of its capability.

This study has evaluated the developed tensegrity robot arm utilizing manually determined desired pressure values, but automatically generating those inputs is essential to utilize the tensegrity robot arm. We presume that the data utilized in this study’s analysis can be directly leveraged to model the forward kinematics utilizing machine learning. However, modeling inverse kinematics will be difficult because multiple sets of desired pressure values can realize a similar posture because of the high redundancy. In the future, we plan to utilize a method that simultaneously learns kinematics and inverse kinematics, and models redundancy as well (Masuda et al., 2019) to exploit the vital feature of the developed tensegrity robot arm.

In this study, we introduced a tensegrity robot arm, developed to reproduce complex musculoskeletal structures features. Preliminarily, a design rule for a tensegrity robot arm in which an entire structure is constituted by stacking simple tensegrity modules was proposed. Based on the proposed design rule, a tensegrity robot arm comprising five four-struts tensegrity modules was developed, and the technical details were explained. This robot arm is driven by 20 pneumatic cylinders, and can bend to various postures like a continuum manipulator. Utilizing an optical motion capture system, postures of this robot arm were measured in several experimental setups. By these experiments, the range of motion and the stiffness variation were evaluated. Accordingly, the fact that the developed tensegrity robot arm has similar features to musculoskeletal robots was successfully confirmed.