**1. Introduction**

In this paper we propose the design and construction of a modular robot composed by several links which are reconfigurable in shape and size in order to generate different adaptable configurations of the robotic platform. The primary characteristics for this design are the scalability and modularity of the robot. Two designs will be proposed:


A Modular Robotic System (MRS) can be defined as link arrays joined together to form a modular structure [3]. MRSs have four interdependent components: Module, Information, Task, and Environment (MITE). MITE allows for the extraction of the characteristics of the MRS, the module properties being the most special features, as other MRS components are transversal to every kind of robot. The module component includes two useful properties for the design of an MRS, these are: class and architecture. The class refers to the different ways in which modularity can be achieved, such as fixed-configuration, manually-reconfigurable [2,4], self-reconfigurable [5,6], and self-replicable [7]. On the other hand, architecture is the hardware categories of configuration: Chain [4,5], Lattice [6,8,9], Mobile [10–12], Hybrid [13,14], Truss [15–17], and Free form [18,19].

Many MRSs change their shape and size through nesting between each module; however, in this case study our goal is to allow the resizing of each module before its connection. This configuration is not often used in the literature. For instance, the closest example is ShapeBots, which is an individual shape-changing link [20] but with no modularity capabilities. A reference work closer to our approach is the Extendable Arm by Matsuo et al. [21], where modular links are connected and each one is scalable. Compared to our approach, the orientation of this platform is limited and its operation is manual. Thanks to the introduction of a three degrees of freedom (DoF) joint connecting each link, our design allows a wider orientation range of the modular robot. Besides, the platform performs automatically thanks to the use of a control system, which avoids the manual operation of the robot and improves its usability.

**Citation:** Mena, L.; Muñoz, J.; Monje, C.A.; Balaguer, C. Modular and Self-Scalable Origami Robot: A First Approach. *Mathematics* **2021**, *9*, 1324. https://doi.org/10.3390/math9121324

Academic Editors: Mikhail Posypkin, Vladimir Titarev and Andrey Gorshenin

Received: 30 April 2021 Accepted: 1 June 2021 Published: 9 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/).

The initial idea about a flexible structure deforming by sections led us to origami-type constructions. Origami is an ancient technique for paper folding [22], which provides deployable structures that can be reconfigured and change in size and shape. Reconfigurable 3D structures, obtained from a rigid geometric 2D pattern, are booming in designs for engineering applications such as the solar panel surfaces proposed by Miura [23], the modular origami continuous manipulator by Santoso et al. [24], which supports a 1 kg mass at its tip, a gripper with multiple grasp modes [25], and other applications such as biomechanical [26], space [27], or soft robotics [28]. The wide number and type of applications of these designs has motivated us to research this topic and present a different approach in the terms described next.

Our design proposal consists of a modular robot that uses an origami-inspired linkbased resizing mechanism. The links can be assembled using a rigid connector or a 3 DoF joint as proposed below. In this study we use the Kresling pattern for the origami structure of the basic link. This pattern is formed by the folding of a thin-walled cylinder when subject to twist buckling under a torsional load. It is characterized by alternating mountain and valley folds angled along the direction of the twist [29,30].

Figure 1a represents an *n* faces polygon Kresling pattern in planar state, which forms a polyhedral cylinder when assembled. The triangulated polyhedron geometry is resolved by *LAB* = *a*, *LBC* = *a* · *sin*(*α*)/*sin*(*β*), *LAC* = *a* · *sin*(*α* + *β*)/*sinβ* at the planar state, where *a*, *α*, and *β* are constant values. The angles *α* and *β* are the main design criteria to create the cylinder, because the strength of the structure depends on them. Zhai et al. suggest that, for small angles around 30◦, the structure is easy to deploy and easy to collapse, and for greater angles around 50◦, the structure is hard to deploy and hard to collapse [31]. For large angles, the structure is stronger and able to support loads. To obtain a symmetric structure we consider *a* = *r* = 30 mm; therefore, *n* = 2*π* according to Hunt [32], where *a* = 2*πr*/*n* and *β* = *π*/*n*. The angle *α* has been obtained from the geometric resolution proposed by Jianguo et al. [33]. Here *h* is considered to be known, *α* = *asin*(*d*/*a*) and *d* can be obtained by *ah* <sup>=</sup> *<sup>d</sup>*(*<sup>d</sup>* · *cot*(*β*) + <sup>√</sup>*a*<sup>2</sup> <sup>−</sup> *<sup>d</sup>*2). Consequently, our prototype has been designed with *a* = 30 mm, *h* = 34.25 mm, *β* = 30◦ and *α* = 38◦, thus a flexible deformable link is created, and the unitary ABC triangle angle is >90◦ to achieve continuous strain at each member tension or compression in the deployed and collapsed states.

The folded cylinder link state generates a twist angle *θ* with radius *r* while height *h* is compressed (Figure 1b). This bistable behavior is due to the change of the lines length during folding (Equation (1)).

$$\begin{aligned} l\_{AB} &= 2r \sin(\pi/n) \\ l\_{BC} &= \sqrt{h^2 - 2r^2 \cos \theta + 2r^2} \\ l\_{AC} &= \sqrt{h^2 - 2r^2 \cos(2\pi/n + \theta) + 2r^2} \end{aligned} \tag{1}$$

$$r = \frac{\frac{q}{2}}{\sin\left(\frac{\pi}{n}\right)}\tag{2}$$

$$\theta = \frac{2\pi}{n} - 2a\sin\left(\frac{l\_{\rm BC}\cos\delta}{2r}\right) \tag{3}$$

$$h = l\_{BC} \cdot \sin(\delta) \tag{4}$$

The height value *h* changes during folding and this change is related to *δ* angle change, given by Equation (4), as illustrated in Figure 1c. The variable height and bistability allow the self-scaling of the simple link.

**Figure 1.** Origamipolyhedron Kresling pattern. (**a**) 2D Kresling pattern. (**b**) Folded cylinder link state. (**c**) Biestable behavior.

The main contributions of this work with respect to the state of the art are the following:

