# Modular Continuum Manipulator: Analysis and Characterization of Its Basic Module

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Conceptual Design of SIMBA’s Basic Module

#### 2.2. Fabrication and Control

^{3}). The two lateral springs were cut longitudinally which enabled the cables to move freely inside without mechanical interference. To fabricate the supporting structure of the module, we used three different thicknesses (3, 5, and 8 mm) of Derlin (Betametalli, Prato, Italy). Derlin has a tensile strength of 80 MPa, Young’s modulus (E) of 3 GPa, flexural rigidity of 2.8 GPa, and a density of 1420 kg/m

^{3}. A nylon filament (Skyper mono transparent filament, Amazon, Italy) of 0.33 mm thickness and with an 18 kg maximum load capacity was used for pulling the module. The rotation of the module is supplied by DC gear-motors and a gear train mechanism obtained with two spur-gear pinion and a gear (G-2), with a gear ratio of 1:2.8. The motors used for actuation are Pololu 986.41:1 micro metal gear motors (Pololu Inc., Las Vegas, NV, USA) which provide 9 kg-cm torque, 32 RPM at 12 V.

_{2}laser cutting machine (VersaLaser, Universal Laser Systems, Scottsdale, AZ, USA) and assembled by classical fastening techniques. The fully assembled design of the module is shown in Figure 3 at different bending angles.

#### 2.3. Kinematic Modeling

#### 2.4. Cantilever Beam Modeling

#### 2.5. Experimental Setups

## 3. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Three examples of arms with different features in terms of materials and stiffness. (

**a**) The KUKA arm is a fully rigid structure with joints and rigid links [15]; (

**b**) the soft compliant manipulator for broad applications (SIMBA) arm is a modular continuum arm with a hybrid structure with rigid and soft materials [16]; (

**c**) the soft robot arm inspired by the octopus is a completely soft continuum arm [8,17]. (

**c**) Image published under the Creative Commons Attribution (CC BY) license and reproduced from [17].

**Figure 2.**Design of SIMBA’s basic module. (

**a**) Operative unit; (

**b**) control unit; (

**c**) details of the control unit. RD-1= upper rotational disk 1; RD-2 = bottom rotational disk 1; P-1 = pulley for motor 1; P-2 = pulley for motor 2; FD = fixed disk for motor fixing; M-1 = motor bending; M-2 = motor stiffening; M-3 = motor for rotation; G-2 = driving gear connected to M-3; G-1 = pinion gear fixed on RD-2.

**Figure 3.**Assembled module at different bending angles. (

**a**) 0° mode; (

**b**) 90° mode; (

**c**) 143° mode, where 143° represents the maximum bending angle achievable.

**Figure 4.**Denavit–Hartenberg parameter representation for kinematic modeling. l = length of the flat springs; φ = rotation of the module about the z-axis; θ = bending angle of the module; r = radius of the arc; ${d}_{1}$ = height of the actuator module; ${d}_{2}$ = translational length of the prismatic joint; ${d}_{3}$ = thickness of the upper plate of the bending module.

**Figure 5.**Actuator space modeling. (

**a**) Module initial position at θ = 0°; $l={l}_{t1}={l}_{t2}={l}_{s1}$=${l}_{s2}$. (

**b**) Bending at θ; r = radius of curvature, ${d}_{p}$ = diameter of the pulley, d = radius of the disk, ${l}_{s1}$,${l}_{s2}=$ lengths of the lateral springs,${l}_{t1}$,${l}_{t2}$ = lengths of the tendons, ${n}_{1}$ = number of rotations of pulley 1,${n}_{2}$ = number of rotations of pulley 2.

**Figure 6.**Mapping between stiffness, actuator, joint, and task space.$n$ = number of rotations of the motor; ${l}_{t1}$,${l}_{t2}$ = lengths of the tendons; $\phi $ = rotation of the module about the z-axis; θ = bending angle of the module; r = radius of curvature; $\left(x,y,z\right)$ = end-effector coordinate points; $k$ = stiffness of the beam; $f$ = bending force;$\delta $ = deflection of the module.

**Figure 7.**Free body diagram. (

**a**) The beam in the initial position and (

**b**) bent.$l$ = length of the beam; $e$ = eccentricity from the center; ${d}_{3}$ = thickness of the upper plate of the bending module; ${f}_{t}$ = bending force; $y$ = deflection of the beam; ${l}^{\ast}$ = variable length of the beam.

**Figure 8.**Lateral and bending stiffness of the module for different thicknesses. Poly: third degree polynomials.

**Figure 9.**Aurora system. Setup to measure the position and orientation of the module. EM: Electromagnetic.

**Figure 10.**Experimental setup for stiffness measurement. A tendon connects the end of the module with the load cell. By pulling this tendon, the pulling force is measured, and the stiffness is obtained with a fixed length of module tendons without their actuation. The distance traveled is the differential length, or deflection, of the module.

**Figure 11.**Kinematic model validation and characterization of the module. (

**a**) Bending of the module in x–y plane; (

**b**) model validation of the bending angle about x-axis; (

**c**) measured error in y–z plane between Aurora results and model; (

**d**) whole workspace of the module in 3D space; and (

**e**) bi-directional bending of the module.

**Figure 12.**Force and stiffness characterization of the module at different angles. (

**a**) Tendon pulling force and model prediction; (

**b**) measurement of bending force; (

**c**) module stiffness behavior as a function of module positional angle with polynomial fitting. Poly: Third order polynomials.

**Figure 13.**First prototype of SIMBA and its capabilities at different tasks. (

**a**) Full assembled design of the manipulator [16]; (

**b**) bending capability; (

**c**) dexterity; (

**d**) capacity to bend and rotate for manipulation task; (

**e**) arm positioning; (

**f**) door opening.

Link (i) | α | a | d | $\mathit{\vartheta}$ |
---|---|---|---|---|

1 | $-\frac{\pi}{2}$ | 0 | ${d}_{1}$ | $\phi $ |

2 | $\frac{\pi}{2}$ | 0 | 0 | $\frac{\theta}{2}$ |

3 | $-\frac{\pi}{2}$ | 0 | ${d}_{2}$ | 0 |

4 | $\frac{\pi}{2}$ | 0 | 0 | $\frac{\theta}{2}$ |

5 | 0 | 0 | ${d}_{3}$ | $-\phi $ |

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**MDPI and ACS Style**

Mishra, A.K.; Mondini, A.; Del Dottore, E.; Sadeghi, A.; Tramacere, F.; Mazzolai, B.
Modular Continuum Manipulator: Analysis and Characterization of Its Basic Module. *Biomimetics* **2018**, *3*, 3.
https://doi.org/10.3390/biomimetics3010003

**AMA Style**

Mishra AK, Mondini A, Del Dottore E, Sadeghi A, Tramacere F, Mazzolai B.
Modular Continuum Manipulator: Analysis and Characterization of Its Basic Module. *Biomimetics*. 2018; 3(1):3.
https://doi.org/10.3390/biomimetics3010003

**Chicago/Turabian Style**

Mishra, Anand Kumar, Alessio Mondini, Emanuela Del Dottore, Ali Sadeghi, Francesca Tramacere, and Barbara Mazzolai.
2018. "Modular Continuum Manipulator: Analysis and Characterization of Its Basic Module" *Biomimetics* 3, no. 1: 3.
https://doi.org/10.3390/biomimetics3010003