*3.1. Calculation of Tendon Lengths*

The robot inputs are one inclination value, *θ*, and one orientation value, *ψ*, and the outputs will be tendons lengths:

$$L\_i = [L\_1 \ L\_2 \ L\_3]'\tag{2}$$

Inverse kinematics was used to calculate tendon lengths for the target end position. It is important to point out that unlike works such as [27] or [25], this design does not have the tendons in the open air, but the performance of the tendons is embedded within the morphology of the soft joint itself. This makes the length of the tendons not straight, but rather the tendons project the curvature of the soft joint, thus having a curvature similar to that of the joint. Therefore, *Li*, the lengths of the tendons form an arc between both ends of the joint, Figure 10b.

Thus, tendons and joint are considered robots shaped by continuously bending actuators, such as those described by [30,31], where a pneumatic actuation is usually used, considering joint curvature and tendon curvature as a continuous curvature. The equations shown in [3] are adapted to this specific morphology case.

An angular-curved approach is used, with the inclination and orientation parameters. The lengths of the tendons *Li* depend on both inclination and orientation angles. The length of the joint, *L*, remains constant in its central fiber at all times, regardless of the curvature; and the distance, *a*, of the tendons from the center of the joint section, remains constant, too (Figure 10b). For this morphology, *a* measures 0.035 m, *L* measures 0.2 m. The actuator for tendon 1 is placed at *ν*<sup>1</sup> = *<sup>π</sup>* <sup>2</sup> radians, tendon 2 is placed at *<sup>ν</sup>*<sup>2</sup> <sup>=</sup> <sup>7</sup>·*<sup>π</sup>* <sup>6</sup> radians and tendon 3 is placed at *ν*<sup>3</sup> = <sup>10</sup>·*<sup>π</sup>* <sup>6</sup> radians.

**Figure 10.** (**a**) Base projection of the soft joint, *ψ* = 45◦, for the representation of orientation, distances and numbering of tendons. (**b**) Three-dimensional representation of the joint with *θ* = 45◦ orientation and *ψ* = 90◦ inclination. Note the different curvatures for the soft joint and for each tendon *Li*.

As previously discussed, it can be determined that *L*, the central fiber length of the soft joint, is constant independently of the inclination angle. Tendon lengths are calculated through the arc equations, due to the assumption of constant curvature. The radius *r* of the curvature *L* is determined as *L* = *r* · *θ*, where *θ* has a value in radians. As the central fiber and tendons move, they move in the direction given by the angle of orientation, and by projecting the arcs and radii, the representation in Figure 11 is obtained. Therefore, *Li* can be determined as *Li* = *ri* · *θ*, where *ri* = *r* − *a* · *cos*(*ν<sup>i</sup>* − *ψ*), resulting in the following equations:

$$L\_1 = L - \theta \cdot a \cdot \cos(\nu\_1 - \psi) \tag{3}$$

$$L\_2 = L - \theta \cdot a \cdot \cos(\nu\_2 - \psi) \tag{4}$$

$$L\_3 = L - \theta \cdot a \cdot \cos(\nu\_3 - \psi) \tag{5}$$

Hence, *φ<sup>i</sup>* is the angle between orientation, which is the plane containing the curvature, and the plane of tendon location, *i*. This angle *φ<sup>i</sup>* depends on the configuration of the orientation and the number of actuators. The relationship of each tendon with the orientation is as follows:

$$
\phi\_1 = \nu\_1 - \Psi \tag{6}
$$

$$
\phi\_2 = \nu\_2 - \psi \tag{7}
$$

$$
\phi\_3 = \nu\_3 - \Psi \tag{8}
$$

A generic equation is obtained for lengths:

$$L\_i = L - \theta \cdot a \cdot \cos(\phi\_i) \tag{9}$$

**Figure 11.** Representation in the perpendicular view of the orientation plane formed by the orientation angle *ψ* and an inclination angle *θ*. It can be seen that the projection of the radii of the constant curvature of the soft joint. The central fiber curvature *L* and its corresponding radius *r* are represented in blue. The arcs of tendons *Li* are represented by dashed black lines, and their corresponding radii *ri* by continuous black lines. The difference between *r* and *ri* is represented by a red line whose distance for each tendon is given by equation *a* · *cos*(*ν<sup>i</sup>* − *ψ*).
