# Form-Finding Analysis of a Class 2 Tensegrity Robot
## 1 Introduction  
A tensegrity system is established when a discontinuous set of compression components interacts with a continuous set of tension components to define a stable volume in space [1]. The biggest advantage of tensegrity systems is their low weight and high stiffness, which makes them particularly good for many applications from architecture to robotics. It should be mentioned that the components of tensegrity systems are subject only to axial loads. Due to their characteristics, tensegrities can be considered as biomimetic structures and have shown the capability to create lightweight, strong and fault tolerant robots with latent potential for applications such as a spine-like tensegrity moving robot [2] and a bio-inspired tensegrity manipulator [3], which resembles the biomechanics of a human arm. Researchers consider this as an opportunity to include tensegrity robots in the field of soft robotics due to their biological inspiration and inherent ability to combine stiff elements with soft materials to create deformable robots [4; 5]. Some examples found in the literature are the soft robotic muscle by [PERSON] et al. [6] and the soft robotic system by [PERSON] et al. [7], both of which use a combination of springs and electric muscles to modify their shape. There exists a way to distinguish between various types of tensegrity systems that fit in the general tensegrity definition established by [PERSON] and [PERSON]: \"A tensegrity configuration that has no contacts between its rigid bodies is a class 1 tensegrity system, and a tensegrity system with as many as k rigid bodies in contact is a class k tensegrity system\" [8]. Tensegrity prisms are the simplest topological family of tensegrity systems where the basic arrangement consists of two equal polygons in parallel planes joined by members at their vertices; the polygons do not necessarily have to keep the same size or remain in parallel planes [9]. However, if it is assumed that the shapes of the upper and lower polygons do not change, platforms can be coupled at the end of the system. As a result, this device is a parallel device [10]. The structure proposed by [PERSON] et al. [11] possesses these characteristics, including a system of combined flexible elements formed by a spring attached to a cable with a fixed length, where the length of its cables can be adjusted using a system of pulleys coupled to two servomotors. This mechanism was used as a case study, giving continuity to previous work where the study of the kinematic analysis of the mechanism's position was obtained using the geometric parameters of Denavit-Hartenberg. These new type of robots cannot be studied in the same way as conventional manipulators; new strategies for their analyses are being proposed. For example, [PERSON] et al. [12] presented a mechanism composed of four pneumatic muscles, and they obtained its kinematic analysis with geometric parameters. [PERSON] et al. [13] presented a mechanism that combines pneumatic muscle, a spring and a wire, the motion of which is similar to that presented in [6], and its kinematic analysis is based on geometric parameters and arc length. Regarding the kinematic analysis of tensegrity mechanisms, due to the presence of flexible components, its kinematics has been treated differently. In conventional mechanisms, kinematics includes only the geometric properties of motion. The joint variables of the mechanism are related to the position and orientation of the end-effector and these relationships are the points of interest in the study of mechanisms kinematics. By definition, in tensegrity systems, the equilibrium configuration must be maintained by the effect of the internal forces of the arrangement of its components. Therefore, various authors, e.g., [PERSON] and [PERSON] [14] and [PERSON] et al. [15], considered that, in tensegrity mechanisms, the static and kinematic analysis must be considered simultaneously.