**2. Design and Prototype of the Soft Joint**

This section presents in detail the design and prototype of the soft joint.

#### *2.1. Geometry*

The soft joint has an asymmetrical morphology that allows its end tip to be positioned in the three-dimensional environment, robustly supporting high loads during its performance. Its design provides greater flexibility and a wider range of movement than a rigid joint. It consists of a series of links with asymmetrical prism morphology and circular section pitch. A triangular morphology is represented in Figure 1.

The small section and soft nature of the central axis of action, allow a greater bending capacity in all directions. The asymmetrical prismatic section provides the property of blocking and a natural protection, as well as the routing of the tendons for their action.

**Figure 1.** *Cont*.

**Figure 1.** Triangular asymmetric geometry of the soft joint with two links in different views. (**a**) Top view showing the 120◦ angle relationship between the different tendon routing points. (**b**) Front view. (**c**) Perspective view, showing d1 and d2 distances defining asymmetry and holes for routing tendons.

The design performance is achieved by tendons that are routed through the asymmetric prismatic sections, as shown in Figure 2. It is possible to change the morphology of the prism and route the tendons through different points of these sections. This change would cause the variation of the forces and moments the joint is subjected to, therefore obtaining different kinematics and dynamics. By acting on the tendons, the joint can flex and orientate with two DOF.

**Figure 2.** Conceptual design of the joint with its components: base, continuous soft axis, tendons for performance and tip (mobile base) of the soft joint.

One of the novel characteristics of this design is the natural morphological protection of the joint against large loads provided by the proposed asymmetrical morphology. An example of the triangular morphology are the two different configurations of extreme load:


In configuration 1, protection when turning in the direction of one of the vertices is the most restrictive, as shown in Figure 3a. In the case of excessive bending, caused by high loads at the end of the joint or by control failures, the vertices contact each other. This produces a blocking curve of the structure that protects the joint from possible breakage due to wear or due to exceeding its elastic limit. This protection allows the joint to act with robustness and safety, especially in the regions of maximum flexion. In this configuration, the action is achieved by a single tendon, which is routed through the vertices that form the bending curve.

Configuration 2 allows larger flexion of the joint, compared to Configuration 1, while also maintaining the natural protection of the joint. When the flexion is towards one of the edges of the triangle, the blocking curve has a smaller radius, as shown in Figure 3b. This is because the edges are closer to the central axis of rotation, as can be seen from the distance ratio *d*1 < *d*2 in Figure 1c. A larger bending occurs due to the fact that a larger bending angle is necessary before these edges contact each other and lock the joint structure. In this configuration, performance is achieved by the action of the two tendons that form the edge of the triangle where bending occurs.

**Figure 3.** Different bending configurations. Relationship between bending angles: *α* < *β*. (**a**) Flexion in configuration 1 has the lowest maximum bending angle. (**b**) Flexion in configuration 2 has the higher maximum bending angle.

#### *2.2. Actuation*

As mentioned above, there are several ways to operate soft robots. This paper focuses on operation by tendons of variable length using a winch coupled to a motor shaft. Tendon lengths must be translated into motor angular positions. *Lo* = 0.2 m is the length of the tendons when the joint is at rest position, and *Li* is the target tendon length. The linear displacement is transformed into an angular displacement by the length of the arc formed by the circumference of the winch for a certain angle (Figure 4), following the equation below:

$$
\Omega = \frac{(L\_{\rm 0} - L\_{\rm i})}{R} \tag{1}
$$

*R* is the radius of the winch where the tendon is wound or unwound, in this case 9.3 mm, and Ω is the angle that provides that displacement.

(**a**)

**Figure 4.** Diagram depicting winch winding based on radius and angle. *Lo* − *Li* is the distance for tendon winding, and Ω and *R* are the angle and the radius, respectively. (**a**) Tendon and soft joint prior winch actuation. (**b**) Tendon and soft joint when the winch is operating, with the radius R, and the angle Ω.
