# Length Modelling of Spiral Superficial Soft Strain Sensors Using Geodesics and Covering Spaces

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## Abstract

**:**

## 1. Introduction

#### 1.1. Common Soft Actuator Models

#### 1.2. Piecewise Constant Curvature Model

#### 1.3. Twist Measurement

#### 1.4. Common Reinforcement Approaches

#### 1.5. Contributions

## 2. The Torus Parameterisation

#### 2.1. Soft Sensor Modelling

#### 2.2. Length Modelling

## 3. Case Studies

#### 3.1. Case 1: A Straight Sensor at the Initial Position

#### 3.2. Case 2: A Spiral Sensor at the Initial Position

#### 3.3. Case 3: A Straight Sensor under Bending

#### 3.4. Case 4: A Spiral Sensor under Bending and Rotation

## 4. Finite Element Analysis (FEA)

#### 4.1. Braided Sleeves as Reinforcements

#### 4.2. Orthotropic Layer

#### 4.3. Multi-Muscle Actuator

#### 4.4. Centreline and Soft Sensors’ Length

`MultiMuscles_data_extract.py`) that extracted the data to MATLAB is provided in the supplementary materials.

#### 4.5. Centreline Curvature

#### 4.6. Bending, Rotation, and Twist Angles

## 5. Simulation Results and Discussion

#### 5.1. Single-Muscle Extension Test

`SingleMuscle.py`) that generated our Abaqus model shown in Figure 14a,b is provided in the supplementary materials.

#### 5.2. Multiple-Muscle Actuator

`MultiMuscle.py`) that generated our Abaqus model shown in Figure 12 and Figure 13 is provided in the supplementary materials.

#### 5.2.1. Simulation Scenarios

#### 5.2.2. Curvature Constancy

#### 5.2.3. Length of Soft Sensors

#### 5.2.4. Twist in Length Model

#### 5.2.5. Sensitivity

#### 5.3. Model Validation

`FBG_SNA2021.m`) along with the extracted data (

`FBG_SNA2021_data.m`) that were used to generate the results are provided in the supplementary materials.

#### 5.4. Limitations and Applications

## 6. Conclusions

## Supplementary Materials

`SingleMuscle.py`), multi-muscle actuator (

`MultiMuscle.py`), data extraction (

`MultiMuscle_data_extract.py`), experimental validation (

`FBG_SNA2021.m`), and (

`FBG_SNA2021_data.m`) as a single archive file. Also, the source codes can be downloaded from our GitHub repository as per the details provided in the data availability statement below.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CRM | Cosserat rod model |

DOF | Degree of freedom |

FEA | Finite element analysis |

FBG | Fibre Bragg grating |

IMU | Inertial measurement unit |

CLD | Constant centreline length (design) |

NCD | Noncompressible inner surface (design) |

PCC | Piecewise-constant curvatures |

PCS | Piecewise-constant strain |

Probability density function | |

PET | Polyethylene terephthalate |

SPA | Soft pneumatic actuator |

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**Figure 1.**Soft actuator modelling where (

**a**) represents a variable curvature actuator, (

**b**) represents a piecewise constant curvature (PCC) actuator, and (

**c**) shows the actuator basic movements.

**Figure 2.**Torus geometry in the Euclidean space ${\mathbb{R}}^{3}$ where (

**a**) shows a full torus radially symmetric about the $z-axis$, and (

**b**) shows a toric segment moved and rotated to represent a PCC soft actuator.

**Figure 3.**Torus notation where (

**a**) shows a full torus radially symmetric about the $z-axis$, and (

**b**) shows a toric segment moved and rotated to represent a PCC soft actuator.

**Figure 4.**(

**a**) A planar cover of a cylinder showing straight lines that bend as (

**b**) geodesic curves on a cylinder, and (

**c**) nongeodesic curves on a toric segment.

**Figure 7.**Case study 3, bending in the $XZ$ plane, where (

**a**) shows a 3D view, and (

**b**) shows the front and top views.

**Figure 8.**Case study 4, bending and rotation, where (

**a**) shows a 3D view, and (

**b**) shows the initial ${h}_{o}$, the centreline h, and the inner surface ${h}_{in}$ lengths.

**Figure 9.**Case study 4, where (

**a**) shows the sensor length and rotation angle $\varphi $, and (

**b**) shows the actuator curvature. Both plots are against bending angle $\theta $ and for CLD and NCD designs.

**Figure 10.**Length vs. diameter variation of a commercially available braided sleeve. (

**a**) Length and diameter at rest, fully compressed, and $20\%$ extension. (

**b**) Experimental vs. simulation results adapted from [49]. Reproduced with permission. Copyright 2019, IEEE.

**Figure 12.**Multi-muscle SPA where (

**a**) shows the cross-section details in which $D=22\phantom{\rule{0.166667em}{0ex}}$ mm and $d=8\phantom{\rule{0.166667em}{0ex}}$ mm, (

**b**) shows the SPA surface partitions for the first and third sensors, the total length of the actuator is $254\phantom{\rule{0.166667em}{0ex}}$ mm, (

**c**) shows the tracking nodes of the top end, and (

**d**) is the top view that shows the three soft sensors and the reference point. The first and second sensors are marked as dots because they are perpendicular to the XY plane.

**Figure 13.**Centreline and soft sensors of the multi-muscle actuator (SPA) where (

**a**) shows the second and third soft sensors’ nodal path, (

**b**) shows the centreline and first soft sensor nodal paths, and (

**c**) is the isometric view of a single sector from the SPA body showing the centreline edge. The second pneumatic muscle, not shown, occupies the cylindrical hole of this sector.

**Figure 14.**Comparison of simulated extension test in Abaqus vs. experimental results adapted from F. Connolly et al.’s test [43]. (

**a**) Initial length. (

**b**) Circumferential path to measure the change in diameter (if any). (

**c**) Comparison of the results.

**Figure 15.**Four simulation scenarios where (

**a1**–

**a4**) are the 3D plots of the SPA positions and their projections onto the XY plane as solid black lines, (

**b1**–

**b4**) are the curvature vs. time, (

**c1**–

**c4**) are the curvature across the SPA length at the last time frame, and its mean value, and (

**d1**–

**d4**) are the probability density function (PDF) results for the curvatures in (

**c1**–

**c4**).

**Figure 16.**Simulation results where (

**a1**–

**a4**) are the rotation ($\varphi $), bending ($\theta $), and twist ($\psi $) angles, (

**b1**–

**b4**) are the centreline length as well as the comparison of the three soft sensors’ length based on Abaqus (A) vs. theoretical (T) results, and (

**c1**–

**c4**) are the length error percentages of the theoretical model with respect to the Abaqus results. The last set of figures (

**d**–

**f**) are error comparison between including vs. excluding twist (NT) in the theoretical model for the third simulation scenario at $M=15$, 30, and 45 MPa, respectively, and (

**g**) is the top view of the SPA showing the arching effect on the angle rotation $\varphi $. The blue line represents the deviated angle, while the red line represents the theoretical angle.

**Figure 17.**Comparison between spiral and straight sensor geometries where (

**a**) is the sensitivity ratio vs. the SPA length-to-radius ratio, and (

**b**) is the error comparison between including vs. excluding the twist (NT) in the theoretical model for the fourth simulation scenario.

**Figure 18.**Bending test of the multi-core optical fibres, where (

**a**) are the samples of the straight and spiral cores, (

**b**) is a cross-sectional view showing the peripheral and core fibres as well as an arbitrary bending direction marked with a green vector, and (

**c**) are the comparisons between the theoretical and experimental curvature results for both the straight and spiral cores.

**Table 1.**Pressures ${p}_{1},{p}_{2},{p}_{3}$, and moment M values for bending (B), extension (E), rotation (R), and twist (T) simulation scenarios. Pressure is in MPa and moment is in N·mm.

Simulation | Scenario | ${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{p}}_{3}$ | M |
---|---|---|---|---|---|

1 | B, E | 0.05 | 0.2 | 0.05 | 0 |

2 | B, E, R | 0.05 | 0.2 | 0.2 | 0 |

3 | B, E, R, T | 0.05 | 0.2 | 0.2 | 15, 30, 45 |

4 | T, E | 0.2 | 0.2 | 0.2 | 45 |

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Al-Azzawi, A.; Stadler, P.; Kong, H.; Sukkarieh, S.
Length Modelling of Spiral Superficial Soft Strain Sensors Using Geodesics and Covering Spaces. *Robotics* **2023**, *12*, 164.
https://doi.org/10.3390/robotics12060164

**AMA Style**

Al-Azzawi A, Stadler P, Kong H, Sukkarieh S.
Length Modelling of Spiral Superficial Soft Strain Sensors Using Geodesics and Covering Spaces. *Robotics*. 2023; 12(6):164.
https://doi.org/10.3390/robotics12060164

**Chicago/Turabian Style**

Al-Azzawi, Abdullah, Peter Stadler, He Kong, and Salah Sukkarieh.
2023. "Length Modelling of Spiral Superficial Soft Strain Sensors Using Geodesics and Covering Spaces" *Robotics* 12, no. 6: 164.
https://doi.org/10.3390/robotics12060164