**1. Introduction**

Soft robotics is a growing research area that has shown advantages over conventional robotics. In this area highly adaptive robots have been developed for soft interactions, providing greater security such as safe human-machine interaction. Compliance and adaptability of the soft structures are used for better efficiency and ability to interact with the environment [1]. Soft robotics is a new solution that covers the unmet need to perform tasks in unstructured and poorly defined environments, where conventional rigid robotics mainly seeks to be fast and accurate.

The advantages of soft robots allow for a wide variety of applications. However, this requires a paradigm shift in the methods of modeling, operation, control, materials and new designs to develop soft robots. The deformation property of soft robots is a restrictive element when using many of the most common conventional rigid sensors or other conventional control techniques [2].

Soft robotics is a subdomain of what is known as continuum robotics, it is defined by [3] as those robots with an elastic, continuously flexing structure and an infinite degree of freedom (DOF); and which are related to (but distinct from) hyperredundant robots, consisting of a finite number of many short, rigid links [4,5]. These models are usually more complex than traditional robot models, which have a small number of rigid links.

The incorporation of soft robotics into robotic systems comes mainly with two types of approaches [6]. One approach involves the use of compliant joints between different rigid links of the robot, while in another approach continuous soft robots are used, such as those mentioned above. This article explores this last type of design. Continuum soft robotic arms show features of soft robotics such as adaptability, high dexterity, and conformability to the external environment. However, they often cannot achieve the high rigidity and robustness required to handle objects or higher loads. Therefore, it is necessary to find a solution capable of providing the robustness of rigid arms and the versatility of soft

**Citation:** Nagua, L.; Relaño, C.; Monje, C.A.; Balaguer, C. A New Approach of Soft Joint Based on a Cable-Driven Parallel Mechanism for Robotic Applications. *Mathematics* **2021**, *9*, 1468. https://doi.org/ 10.3390/math9131468

Academic Editors: Mikhail Posypkin, Andrey Gorshenin and Vladimir Titarev

Received: 10 May 2021 Accepted: 18 June 2021 Published: 23 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

arms [7], which is one of the points addressed in this paper. From now own the term robustness will refer to the ability to cope with the action of external disturbances or loads that cause unwanted deformations in soft bodies, while providing sufficient stiffness.

Robots made of soft materials can generate complex behavior with simpler actuation by partially outsourcing control to their morphological properties and materials. That improves the active coupling of control, body and environment. Soft robots can be actuated in different ways, but the most common actuators are variable length tendons in the form of tension cables or shape memory alloys (SMA) [8,9], fluidic actuators such as pneumatic or hydraulic ones, and electro-active polymers (EAP) [2]. Other kinds of actuations focus on the morphological structure of the soft robot itself, as in [10], where the topological architecture of low-density soft robots is presented; Ref. [11], where a three-dimensionalprinted soft origami rotary actuator is studied; or [12], which presents a soft origami tendon-driven actuator. For continuum robotic arms, pneumatic actuators are most used as they can continuously change their shape with a few DOF actuated [13]. However, arms with pneumatic actuators are usually less precise and difficult to control; less portable, since they rely on an external energy source such as a pneumatic compressor; and more expensive to maintain.

Many works have been carried out on the design and control of soft robots in recent years [14,15], but the state of the art shows that there are few approaches to soft robotic arms, either for integration into a manipulator robot or for use as a stand-alone manipulator. Some designs are based on soft silicone tentacles, as in [16] or [17]. Generally, this type of joint performs an instinctive gripping function that is actuated pneumatically or by cables, and its morphology does not allow its use in handling tasks that require greater precision and robustness. Nevertheless, other works such as [18–20] present soft robotic joint designs that combine a light weight and a high load-weight ratio. Others such as [21,22] present joints with an inflatable structure that can move through highly restricted environments by changing their three-dimensional structure.

Within the framework of soft servo-mechanical actuation, there are several examples, such as a cable-driven soft robot for cardiothoracic endoscopic surgery [23] or a practical 3D-printed soft robotic prosthetic hand [24]. In addition, servomechanically actuated soft limbs, which are closer to the proposal presented in this document, have been developed. An example is the neck developed by the DLR [25] and the soft robotic manipulator applying an adaptive algorithm [26] which includes a continuous silicone-based and tendon actuated mechanism. The RoboticsLab at the University Carlos III of Madrid has also developed a soft robotic neck [27,28] within the Humasoft project, with orientation and inclination capacity in the three-dimensional space and a large load capacity (with only 14 gr of weight, it can support up to 1 kg loads). Work has also been carried out on model identification of this robotic neck using different methods such as neural networks [29].

However, the use of those soft neck link designs cannot be generalized for soft robotic joints, as they do not meet the necessary robustness criteria. When working with different joint positions and orientations in 3D space, gravity comes into play depending on the orientation, and handling large loads can cause the joint to bend or break. Similarly, high stresses exerted on the tendons may seriously compromise the integrity of the actuation system.

These problems have motivated the approach presented in this paper, which is inspired by the soft robotic neck previously design by the authors [27,28]. Differently from that design, this new solution consists of a novel three-dimensional soft joint morphology based on asymmetric links. The joint is composed of a soft material that is flexible but robust. This material allows deformation to achieve bending movements, but prevents too complex deformations and undesired gravity effects. When the designed soft joint is bending, from a certain inclination angle and at certain orientation angles given by model measurements, a natural protection is provided by its own morphology, which limits maximum bending. Therefore, sufficient robustness is achieved to support different loads throughout its positioning range in 3D space, while maintaining the advantages of its soft nature.

Furthermore, the proposed joint is scalable and adaptable to operational requirements in a modular and simple way. Therefore, joint properties, such as maximum bending angle or blocking bending, can be configured by modifying the morphological design and number of the links in the joint, or the distance between them, as well as increasing the number of DOF by concatenating joints.

Finally, this proposal is a low-cost construction, primarily designed by 3D printing and actuated by three motors that vary the length of tendons. Tendons are integrated within the morphology itself, which favors constant curvature and simplification of the model. Electromechanical action is proposed for the articulation, as opposed to other energy sources such as pneumatics or hydraulics. This feature allows the portability of the prototype and a greater integrability in any system (a robot, a humanoid, etc.), as well as more precise control and easier maintenance.

The rest of the paper is organized as follows: Section 2 introduces the soft joint design and prototype. It also shows its geometric design and includes the analysis of its characteristics and configurations. The section also shows the performance and assembly of the prototype and examines the properties of the material chosen for the joint morphology. Section 3 introduces the description of the mathematical model developed for the soft link, considering its workspace and the tendon length ratio that enables performance. The experimental tests carried out with the platform are described in Section 4, where the behavior of the soft joint is analyzed against different inputs and movements using two different tests. The discussion of the experimental results is presented in Section 5, and Section 6 concludes by highlighting the main achievements. This work is under a licensing process and the patent details are given in Section 7.
