# Design Optimisation of a Magnetic Field Based Soft Tactile Sensor

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sensor Concept

**F**for any given change in the input

**B**, or in other words to minimise

**Γ**. This ensured a robust sensor response, with

**F**being less sensitive to noise in the measured

**B**. Sensitivity was affected by the choices made in the design of the sensor such as the geometry, material properties and component selection. In the concept developed here, the magnet and Hall effect module were chosen leaving the geometry and material properties as design variables. This decision was made because it provides the simplest means of varying parameters in the design while demonstrating the complexity of delivering an optimised sensor.

#### 2.1.1. Sensor Mechanics

**u**the distance between the Hall effect module and magnet

**Δ**was changed. The relationship between

**u**and

**Δ**was described by Equations (3) and (4),

**u**required for a given sensor design. The space defined by the ranges of t and $\mathsf{\theta}$ is continuous and bounded by $\mathrm{t}\in \left[0,1\right]$ and $\mathsf{\theta}\in \left[0,{\mathsf{\theta}}_{\mathrm{max}}\right]$.

#### 2.1.2. Magnetic Field

**B**was obtained as a function of the magnet displacement. This approach was applicable because the structural mechanics of the sensor under load does not have an effect on the magnetic field distribution. A 2D axially-symmetric cylindrical coordinate system was employed for developing the model because

**B**is independent of the orientation of the shear direction r. The solution for

**B**was described by Equation (9), and Equation (10) described the relationship between the magnetized field

**H**and magnetic scalar potential V.

_{m}in r and ± 50 H

_{m}in z), where

**n**is the surface normal vector and an axial-symmetry condition was applied through the z-axis of the magnet centre (r = 0), as shown in Figure 3. The magnetic field was not affected by the material properties of the elastomer such that the relative permeability of the elastomer and space surrounding the sensor is ${\mathsf{\mu}}_{\mathrm{r}}=1$, i.e., no shielding effect [27]. Using these operating and boundary conditions and solving Equations (9) and (10) together with Equations (11) and (12) for V produced the magnetic field $B$. This was subsequently assessed for all displacements and sensor designs by varying the distance between the magnet and assessment location.

#### 2.1.3. Structural Mechanics

#### 2.2. Design Specification

#### 2.2.1. Parameterisation

**F**were determined over the range of the design variables and displacements. The responses were subsequently characterised by Equations (14) and (15) as functions of the parameters G, H, t, and $\mathsf{\theta}$. The sensitivity

**Γ**was calculated by taking partial derivatives of

**F**with respect to $B$, leading to Equation (16).

#### 2.2.2. Loading Stability

#### 2.2.3. Design Optimisation

**Γ**over the ranges of the design variables H and G. In order to achieve this the worst sensitivity (maximum value of

**Γ**) over all stable displacements was used. The worst sensitivity corresponds to the maximum value of

**Γ**because this implies the greatest rate of change of

**F**with the measured

**B**. By minimising the worst sensitivity achieved during displacement ensures that the optimised design has the best possible sensitivity for all displacements considered. By using the stable range of displacements the optimised sensor was also ensured to produce a monotonically changing

**F**with measured

**B**.

#### 2.3. Numerical Simulations

#### 2.3.1. Magnetic Field Simulation

**B**was determined in a fixed cylindrical coordinate system according to the boundary conditions outlined in Section 2.1.2. This was undertaken using the FE method as implemented in the software COMSOL Multiphysics [31]. The magnetisation

**M**was chosen based on the characteristics specified by the manufacturer and the orientation of the magnetic poles. For the sensor concept developed by Wang et al. [9] ${\mathrm{M}}_{\mathrm{z}}=1.2\times {10}^{6}$ A/m and the remaining components were set to zero. The permeability of free space was given by ${\mathsf{\mu}}_{0}=1.257\times {10}^{-6}$ (m.kg)/(s.A)

^{2}.

**B**of less than 1.34% produced when a greater number of elements was used. The calculation took approximately 2 min to run on a 2.8 GHz 4-core CPU with 16 GB of RAM. Once the solution was achieved a post-processing stage allowed

**B**to be given as a function of t, θ, H from the fixed position solution.

#### 2.3.2. Structural Mechanics Simulations

**F**of less than 1.88% found when using smaller element sizes. The meshing procedure ensured that the smallest elements were placed in regions where high stress and strains are produced under load, as such the corner and edge regions on the domain were more densely populated than regions far from boundaries. For each value of θ, G, H a simulation was run and ${\mathrm{F}}_{\mathrm{r}},{\mathrm{F}}_{\mathrm{z}}$ extracted as a function of t from the lower (stationary) indenting surface. The time to compute varied based on the values of θ, G, H. The longest simulation occurred at the maximum for each parameter and took approximately 3 h 40 min using the same computer hardware as described in Section 2.3.1.

#### 2.3.3. Genetic Programming

**B**and

**F**must be calculated from the simulations as described in Section 2.3.1 and Section 2.3.2. For this purpose a full factorial Design of Experiments (DOE) was used to select the values of t, θ, G, and H as to ensure that the entire design space was populated in an evenly-distributed manner. The number of experiments in each dimension were chosen as: 21 in t; 5 in θ; 5 in H; and 5 in G. The total number of DOE points in the

**B**response was 525 which were obtained from 1 simulation, and the total number of DOE points in the

**F**response was 2625, obtained from 125 simulations (taking approximately 2 weeks to compute using the same computer hardware as outlined in Section 2.3.1).

^{−9}and the remaining parameters set to their default values. The solver was subsequently run for 4 h to derive each of the expressions for ${\mathrm{B}}_{\mathrm{r}},{\mathrm{B}}_{\mathrm{z}},{\mathrm{F}}_{\mathrm{r}},$ and ${\mathrm{F}}_{\mathrm{z}}$. These tolerances and time to compute has been used in previous studies to obtain sufficiently accurate relationships [25]. After calculating these expressions they were differentiated using Matlab to provide the sensitivity ${\mathrm{\Gamma}}_{\mathrm{r}},{\mathrm{\Gamma}}_{\mathrm{z}}$ (Equation (16)) and higher derivatives needed for the optimisation studies.

#### 2.3.4. Optimisation Procedure

^{−12}. For ga the population size was set to 200, and within fmincon the trust-region-reflective-algorithm was chosen to which the gradients and Hessian of

**Γ**and

**F**with respect to t and θ were supplied. Due to stability conditions, ${\mathrm{F}}_{\mathrm{r},\mathrm{min}}$ and the corresponding ${\mathrm{t}}_{\mathrm{m}}$ and ${\mathsf{\theta}}_{\mathrm{m}}$ were solved for first because ${\mathrm{t}}^{\prime}$ and ${\mathsf{\theta}}^{\prime}$ were required for the calculation of the ${\mathrm{\Gamma}}_{\mathrm{r},\mathrm{max}},{\mathrm{\Gamma}}_{\mathrm{z},\mathrm{max}}$ and ${\mathrm{F}}_{\mathrm{z},\mathrm{max}}$.

^{−12}and the population size was set to 200. For the purpose of this study the force constraints were specified as ${\mathrm{F}}_{\mathrm{r},\mathrm{c}}=-0.25\text{}\mathrm{N}$ and ${\mathrm{F}}_{\mathrm{z},\mathrm{c}}=5\text{}\mathrm{N}$, respectively. The optimisation procedure and corresponding data required for visualisation took approximately 23 h 35 min to compute.

#### 2.4. Design Validation

## 3. Results

#### 3.1. Magnetic Field

#### 3.2. Strcutural Mechanics

#### 3.3. Sensitivity

#### 3.4. Design Optimisation

#### 3.5. Design Validation

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

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**Figure 1.**Photographs of the MagOne sensor designs. (

**a**) As developed by Wang et al. [9]. (

**b**) From the optimisation of sensitivity considered in this study.

**Figure 2.**Cross-sectional sketch of the MagOne sensor. (

**a**) Unloaded. (

**b**) Loaded in both normal and shear directions.

**Figure 4.**Diagram of the structural mechanics simulation domain. (

**a**) Unloaded. (

**b**) Loaded in both normal and shear directions.

**Figure 5.**Response of ${\mathrm{F}}_{\mathrm{r}}$ in N for H = 7 mm and G = 5 kPa. Region of stable shear loading bounded in red, point placed at the location the minimum ${\mathrm{F}}_{\mathrm{r}}$. Region of negative ${\mathrm{F}}_{\mathrm{r}}$ bounded in black, point placed at the location of ${\mathrm{F}}_{\mathrm{r}}=0$ for $\mathsf{\theta}={\mathsf{\theta}}_{\mathrm{max}}$. (

**a**) ${\mathrm{F}}_{\mathrm{r}}$ shown as a function of t and $\mathsf{\theta}$. (

**b**) ${\mathrm{F}}_{\mathrm{r}}$ shown as a function of ${\mathrm{u}}_{\mathrm{r}}$ and ${\mathrm{u}}_{\mathrm{z}}$.

**Figure 6.**Magnetic field distribution in the near magnet region. (

**a**) Coloured by ${\mathrm{B}}_{\mathrm{r}}$. (

**b**) Coloured by ${\mathrm{B}}_{\mathrm{z}}$.

**Figure 7.**Response of displacement magnitude shown in m for the sensor under load at t = 1, θ = 0.423 rad, H = 7 mm and G = 5 kPa.

**Figure 8.**Sensitivity responses generated from GP showing the effect of t and θ for H = 7 mm and G = 5 kPa. (

**a**) Coloured by ${\mathrm{\Gamma}}_{\mathrm{r}}$. (

**b**) Coloured by ${\mathrm{\Gamma}}_{\mathrm{z}}$.

**Figure 9.**Response of the objectives (

**a**) and constraints (

**b**) as functions of H and G under stable shear loading conditions. (

**a**) Showing contours of ${\mathrm{\Gamma}}_{\mathrm{r},\mathrm{max}}$ in N/T (solid) and ${\mathrm{\Gamma}}_{\mathrm{z},\mathrm{max}}\text{}$ in N/T (dashed). (

**b**) Showing contours of ${\mathrm{F}}_{\mathrm{r},\mathrm{min}}$ in N (solid) and ${\mathrm{F}}_{\mathrm{z},\mathrm{max}}$ in N (dashed).

**Figure 10.**Pareto optimal set showing: (

**a**) the competing objectives of ${\mathrm{\Gamma}}_{\mathrm{r},\mathrm{max}}$ and ${\mathrm{\Gamma}}_{\mathrm{z},\mathrm{max}}$; (

**b**) the competing constraints of ${\mathrm{F}}_{\mathrm{r},\mathrm{min}}$ and ${\mathrm{F}}_{\mathrm{z},\mathrm{max}}$; and (

**c**) the design variables H and G. Result obtained under stable loading conditions with ${\mathrm{F}}_{\mathrm{c},\mathrm{r}}$ = −0.25 N and ${\mathrm{F}}_{\mathrm{c},\mathrm{z}}$ = 5 N.

**Figure 11.**Response of the magnetic field components given as a function displacement with $\mathsf{\theta}={\mathsf{\theta}}_{\mathrm{max}}$, H = 4.1 mm and G = 2.94 kPa. (

**a**) Showing ${\mathrm{B}}_{\mathrm{r}}$ as a function of ${\mathrm{u}}_{\mathrm{r}}$ as simulated and from the experimental mean. (

**b**) Showing ${\mathrm{B}}_{\mathrm{z}}$ as a function of ${\mathrm{u}}_{\mathrm{z}}$ as simulated and from the experimental mean.

**Figure 12.**Response of the force components given as a function of displacement with $\mathsf{\theta}={\mathsf{\theta}}_{\mathrm{max}}$, H = 4.1 mm and G = 2.94 kPa. (

**a**) Showing ${\mathrm{F}}_{\mathrm{r}}$ as a function of ${\mathrm{u}}_{\mathrm{r}}$ as simulated and from the experimental mean. (

**b**) Showing ${\mathrm{F}}_{\mathrm{z}}$ as a function of ${\mathrm{u}}_{\mathrm{z}}$ as simulated and from the experimental mean.

Parameter | Value (Unit) |
---|---|

${\mathrm{H}}_{g}$ | 4.97 (mm) |

${\mathrm{H}}_{m}$ | 2 (mm) |

$\mathrm{R}$ | 6 (mm) |

${\mathrm{R}}_{h}$ | 1 (mm) |

${\mathrm{R}}_{g}$ | 6 (mm) |

${\mathrm{R}}_{m}$ | 2.5 (mm) |

${\mathsf{\Delta}}_{\mathrm{z},\mathrm{sat}}$ | 2.97 (mm) |

${\mathsf{\theta}}_{\mathrm{max}}$ | 0.42 (rad) |

© 2017 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Boer, G.d.; Raske, N.; Wang, H.; Kow, J.; Alazmani, A.; Ghajari, M.; Culmer, P.; Hewson, R. Design Optimisation of a Magnetic Field Based Soft Tactile Sensor. *Sensors* **2017**, *17*, 2539.
https://doi.org/10.3390/s17112539

**AMA Style**

Boer Gd, Raske N, Wang H, Kow J, Alazmani A, Ghajari M, Culmer P, Hewson R. Design Optimisation of a Magnetic Field Based Soft Tactile Sensor. *Sensors*. 2017; 17(11):2539.
https://doi.org/10.3390/s17112539

**Chicago/Turabian Style**

Boer, Gregory de, Nicholas Raske, Hongbo Wang, Junwai Kow, Ali Alazmani, Mazdak Ghajari, Peter Culmer, and Robert Hewson. 2017. "Design Optimisation of a Magnetic Field Based Soft Tactile Sensor" *Sensors* 17, no. 11: 2539.
https://doi.org/10.3390/s17112539