# An Optimal Design of an Electromagnetic Actuation System towards a Large Homogeneous Magnetic Field and Accessible Workspace for Magnetic Manipulation

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## Abstract

**:**

## 1. Introduction

## 2. Design of the Magnetic Manipulation System

#### 2.1. Motivations

#### 2.2. Design and Optimization

^{3}located at the center of the coil configuration. This volume of the workspace depends on the separation distance of three coil pairs. Next, in the bending process, we bend the x and y coil pair to shape the curve coils with a purpose of expanding an area of the workspace on the x-y-plane, exhibited in Figure 3c. When the coils are bent, the original rectangular workspace about $7.5\times 6.3$ cm

^{2}is transformed into a circular workspace about $r7.5$ cm

^{2}.

^{2}, and then it transforms into the circular area of about $r7.5$ cm

^{2}. The workspace in the z component increases about 180%, from the original length of about 8.8 cm to the final length of about 16 cm. Therefore, the final form of the coil configuration provides a cylindrical workspace of about $r7.5\times 16$ cm

^{3}, and the overall number of coils changes from six to seven.

#### 2.3. Mathematic Models of Magnetic Field Generation

^{2}) at the point p experiences magnetic torque, ${\overrightarrow{\mathit{T}}}_{p}=\left[\begin{array}{ccc}{T}_{x}& {T}_{y}& {T}_{z}\end{array}\right]$, (N·m) and force, ${\overrightarrow{\mathit{F}}}_{p}=\left[\begin{array}{ccc}{F}_{x}& {F}_{y}& {F}_{z}\end{array}\right]$, (N), exerted by magnetic field, ${\overrightarrow{\mathit{B}}}_{p}$, which are expressed by

_{1}, w

_{2}, w

_{3}. In this work, the system is capable of both non-uniform and uniform field generation, so the mathematical models are divided into uniform and non-uniform magnetic field generation.

#### 2.3.1. Uniform Field Generation

_{5}, I

_{6}and I

_{7}are current passed into the front, middle and rear coil with the same flowing direction respectively. Relation of them adopts Merritt, et al. [30].

#### 2.3.2. Non-Uniform Field Generation

#### 2.4. Conclusion of Magnetic Field Generation Investigated by Numerical Simulation Results

#### 2.5. Conclusion of Homogeneous Region of Uniform Field

## 3. System Building and Implementation

#### 3.1. Coils and Control Hardware Setup

#### 3.2. Microrobots

## 4. System Demonstrations

#### 4.1. Three-D-Helical Propulsion in the Large Workspace by Rotating Magnetic Field

#### 4.2. Translation by Pulling Force of Gradient-Based Field

#### 4.3. Sweeping-Slip Locomotion by Oscillating Field

#### 4.4. Rocking-Slip Locomotion by Gradient-Based Field

#### 4.5. Helical Propulsion Following the Complex Network Path

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Analyses of the Coil Separation Distance

**Figure A1.**Field distribution of the 45°-y-bent coil. Field strength of four values of the coil separation distance, d, is plotted against the y-coordinate. Red circle displays a r6 × 15 cm

^{3}cylindrical workspace, hidden-line box defines a boundary equal to diameter of the workspace over which the field is distributed. Highlighted areas indicate the best result when comparing to the Others.

$\mathbf{Coil}\mathbf{Separation}\mathbf{Distance},\mathit{d}$ | ^{[I]} Field Difference | Coordinate Range Defined by Homogeneity, (cm) | The Biggest Available Workspace | ||
---|---|---|---|---|---|

$\mathbf{(}\mathbf{\le}\mathbf{0.1}\mathbf{\%}\mathbf{)}$ | $\mathbf{(}\mathbf{\le}\mathbf{1}\mathbf{\%}\mathbf{)}$ | $\mathbf{(}\mathbf{\le}\mathbf{5}\mathbf{\%}\mathbf{)}$ | |||

$0.45w$ | 19% | −1.6 to 1.6 | −3.3 to 3.3 | −4.1 to 4.1 | $r6\times 15$ |

$0.5w$ | 14% | −1.0 to 1.0 | −4.0 to 4.0 | −4.8 to 4.8 | $r6.3\times 15$ |

$0.6w$ | 5% | −0.3 to 0.3 | −2.6 to 2.6 | −5.1 to 5.1 | $r7.5\times 15$ |

$0.7w$ | 23% | −0.1 to 0.1 | −0.7 to 0.7 | −2.0 to 2.0 | $r8\times 15$ |

^{[I]}Field difference is the field variation over the workspace which is determined by the percent difference between the maximum and minimum field of the workspace.

#### Appendix A.2. Analyses of the Coil Separation Distance

**p**to a curve wire are

**p**to the straight coil are

#### Appendix A.3. Investigation into the Influence of Other Field Components to Homogeneous Region

^{3}in which we create the 8 axes parallel to the z-axis within the boundary of the workspace, such as the line of x = 3, y = 0, the line of x = 3, y = −3, etc., detailed in the legend box of the plot. The lines are created to ensure that the generated field is covering the whole workspace. The plot of magnetic field generated by the z-coil depicts clearly the uniformity of magnetic field around the center of the workspace. Moreover, it obviously reports that magnetic field on all of the lines have the uniform distribution around the center as well.

**Figure A2.**Magnetic field generated by the z-coils with magnitude of on and off-z-axis. (

**a**) Magnetic field in the z-direction. The hidden-line red box emphasizes 1%-variation of magnetic field. (

**b**) Magnetic field in the y direction. (

**c**) Magnetic field in the x direction. (

**d**) 3D graphic of Homogeneous region of magnetic field generated by the z-coils over the whole workspace, represented by the blue color.

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**Figure 1.**The HyBrid system. (

**a**) a large bore. (

**b**) and (

**c**) the front and side view of a scheme to manipulate a helical microswimmer to swim in a large cylinder containing 350-cst. silicon oil which is inserted into the bore.

**Figure 2.**Schemes of biomedical applications by the HyBrid system. (

**a**) A. deployment of a helical microswimmer to CSF by injection of a spinal needle, then swim to access brain. (

**b**) Magnetic manipulation in a human head inserted into the bore space of the system about r 12 × 30 cm

^{3}. (

**c**) An in vivo experiment in a 12-cm-sized mouse inserted into the bore space of the system about r 7.5 × 18 cm

^{3}.

**Figure 3.**The optimization process from the prototype to the HyBrid system. l. Pre-bending and adding process; (

**a**,

**b**) Tri-axial nested Helmholtz square coil consists of three pairs of six square coils perpendicular to each other (the coil square size: 150, 175, 200 mm) with an available bore about 7.5 × 6.3 × 8.8 cms represented by the grey cube. 2. Bending process; (

**c**) the x and y coil pair are bent to increase volume ofthe workspace. Three bending angles are considered, 60°, 45° and 30°. Each of the bending angles is simulated to investigate uniform field generation to the cylindrical workspace about r7.5 × 16 cm

^{3}. Result of (

**d**) 60° angle reports 15 mT with a homogeneous region about 8% of the workspace, (

**e**) 45° angle reports 12.2 mT with a homogeneous region about 35% of the workspace, (

**f**) 30° reports 9.5 mT on its homogeneous region about 15% of the workspace. After bending, space on the x–y-plane expands about 370%. 3. Adding process; one more coil is added into the z-coil group. Then, space along the z-direction is prolonged about 180%. Field magnitude about 13.5 mT has a homogeneous region about 70% ofthe workspace. Uniform magnetic field exhibited by the plots and the homogeneous region depicted by the blue area on the cross-sectional area of the cylindrical workspace. 4. Post-bending and adding process; (

**h**,

**i**) finally, the prototype, Tri-axial nested Helmholtz square coil becomes the HyBrid system. The original cubic workspace turns to be the cylindrical workspace with 680% larger.

**Figure 4.**The models Of the Hybrid system with parameters. (

**a**) The configuration of three coil groups. (

**b**) The y-coil group. (

**c**) The x-coil group. (

**d**) the z-coil group. Bending angle is 45°, so α = 90°.

**Figure 5.**Double bent coils with parameters of magnetic fields generated by four curve and straight wires (α = 90°). Mathematical development and analysis are in the Appendix A.

**Figure 6.**Numerical simulations of magnetic field generation by the HyBrid system. (

**a**) uniform field distribution in the x, y, and Z component forms a homogeneous region at the center of the cylindrical workspace. (

**b**) non-uniiOrm field of each Component is at a minimum of about 9 mT with a gradient of about 100 mT/ m.

**Figure 7.**Homogeneous region of field uniformity field determined by Homogeneity in the x, y, z component on the defined workspace about r7.5 × 16 cm

^{3}. Variability is set at about 1%. Magnetic field of each component at the center, B

_{0}, with 1% variance covers a distribution area. (

**a**) x: 12.2–12.32 mT wide about 50 mm, (

**b**) y: 12.5–12.62 mT wide about 50 mm, (

**c**) z: 13.36–13.5 mT wide about 80 mm. The blue color areas represent the homogeneous region of magnetic field.

**Figure 8.**Helical propulsion in the large workspace (r6 × 15-cm-eylinder). (

**a**) a sample of 5 Hz rotation frequency for a hover swimming toward the x-direction. (

**b**) rotation frequency is variant from 0 to 5 Hz by the velocity control algorithm to drive the swimmer. (

**c**) swimming of the helical robot against time over the whole journey, displayed by the top and front view.

**Figure 9.**Translation locomotion by gradient based field. (

**a**) the double-layer cylinder containing 100-cst.-silicone Oil. (

**b**) a micro-cylindrical robot is manipulated by both force and torque to move in the arena. Red arrows represent the moving path of the robot.

**Figure 10.**Sweeping-slip locomotion by oscillating field. (

**a**) oscillating signal produces the superposition of the x- and z-magnetic field,

**B**and

_{x}**B**, sampled by 2.5 Hz. (

_{z}**b**) the motion path of the microcube which sweeps from the left to right side rapidly to slip forward.

**Figure 11.**Rocking-slip locomotion by gradient based field. (

**a**) a sample of 10Hz frequency produces the superposition of the horizontal and vertical field,

**B**and

_{h}**B**. (

_{v}**b**) the motion path of the microcube which is wrenched by the actuating force to rock up and down to slip forward.

**Figure 12.**Helical propulsion following the complex network path. (

**a**) the model of the path built by using Ø10-mm rubber tubes, and fully filled by 350-cst. silicone oil. (

**b**) the path is inserted into the bore of the system. (

**c**) under rotating magnetic field, the helical microswimmer swims along the complex network path, black arrows define the swimming path of the swimmer, and red circle indicates the swimmer, the start and finish position.

The Coil Group | Homogeneity, H (%) | Coordinate Range on the Axis | Covered Area (% of the Workspace) |
---|---|---|---|

x | $\le 0.1$ | −0.3 to 0.3 | 4% |

$\le 0$.5 | −1.0 to 1.0 | 13% | |

$\le 1.0$ | −2.5 to 2.5 | 34% | |

$\le 3.0$ | −3.8 to 3.8 | 51% | |

$\le 5.0$ | −5.0 to 5.0 | 67% | |

y | $\le 0.1$ | −0.3 to 0.3 | 4% |

$\le 0$.5 | −1.1 to 1.1 | 15% | |

$\le 1.0$ | −2.6 to 2.6 | 35% | |

$\le 3.0$ | −4.0 to 4.0 | 53% | |

$\le 5.0$ | −5.1 to 5.1 | 68% | |

z | $\le 0.1$ | −1.0 to 1.0 | 14% |

$\le 0$.5 | −2.0 to 2.0 | 28% | |

$\le 1.0$ | −3.0 to 3.0 | 43% | |

$\le 3.0$ | −4.0 to 4.0 | 57% | |

$\le 5.0$ | −6.5 to 6.5 | 73% |

The Coil Group | Coil Parameters | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

A ^{[I]} | B | C | D | E ^{[II]} | F | G ^{[III]} | H ^{[IV]} | I | J | |

x | 200 | $1.8\times 1.8$ | 13.05 | 3.6 | 3.63 | 17.5 | 12.72/12.2 | 76 | Cylinder: $\mathrm{r}7.5\times 1$8 | $35\times 35\times 3$5 |

y | 170 | $1.6\times 1.6$ | 7.82 | 2.7 | 2.9 | 15 | 12.65/12.5 | 72 | ||

z | 200 | $2.2\times 2.2$ | 15.83 | 3.9 | 4.06 | 20 | 12.91/13.5 | 78 |

^{2}), C is Inductance (L: mH), D is Resistance (R: Ω), E is Response time (t: ms), F is Square size of coils (cm), G is of magnetic field that is from the $\mathrm{actual}$ system and the simulation model (mT), H is Coil surface temperature (°C), I is Bore dimension available volume of the bore (cm

^{3}), and J is Overall Dimension of the system (cm

^{3}),

^{[I]}Enamel-insulated copper wire with 1.2-mm diameter.

^{[II]}Response time estimated by $t=\frac{L}{R}$.

^{[III]}Magnetic field of the actual coil configuration and the model is compared.

^{[IV]}Operating temperature is measured after operating the system about 15 min to generate 3D-rotating field to drive a helical microswimmer, similar to the experiment of sub-section IV-A.

Microrobots | Materials | Dimension | Actuation Methods, Field Magnitude and Frequency | Environment Setup | Locomotion Types and Details | |
---|---|---|---|---|---|---|

Helical microswimmers | PVA/ PEG double-network hydrogel embedded by Fe_{3}O_{4} | 45$\xb0$ pitch angle, 0.6-mm helical radius | (a) 300-µm ribbon stripe, 3.5 turns, 9-mm long | 3D-Rotating uniform field for torque, 2.5-7.5Hz, 12 mT of the x, y and z field | A $r6\times 15$ cm^{3} cylinder filled by 350-cst. silicone oil | Helical propulsion-Rotating body caused by alignment with the direction of rotating field -Transforming the rotating body to forward or backward propulsion -Able to propel in various viscosity of fluid -The actuation needs the velocity control to balance between the body weight and swimming direction of the swimmer [31] -Velocity depends on rotating frequency |

(b) 500-µm ribbon stripe, 2.5 turns, 6-mm long | 3D-Rotating uniform field for torque, 3-5Hz, 12 mT of the x, y and z field | The complex network path ($\varnothing 10$ mm diameter) filled by 350-cst. silicone oil | ||||

Micro-cylindrical robot | CoNi | (c) $r500\mu \mathrm{m}\times 1$ mm | 3D-Gradient-based field: force by 12 mT of the x and y field and 16 mT of the z field. Rotating uniform field for torque by 12 mT | a double-layer cylinder containing 100-cst. silicone oil | Translation and rotation locomotion-3D-translation caused by the pulling magnetic force, but torque is applied to rotate the robot -Velocity depends on field magnitude to vary the pulling force | |

Micro-cubic robot | NdFeB | (d) 500-µm cube | Oscillating uniform field, 12 mT of the x and z field (the planar field), 2.5Hz and 10Hz | A 500-mL cylinder containing 100-cst. silicone oil | Sweeping-slip locomotion-Side-to-side sweeping to slip forward, caused by alignment with the direction of oscillating field -Velocity depends on oscillating frequency | |

NdFeB | (d) 500-µm cube | Periodical gradient-based field, 10 mT of the superposition of the vertical and horizontal field, 10Hz | A 500-mL cylinder containing 100-cst. silicone oil | Rocking-slip locomotion-The robot is wrenched by magnetic force to slip forward.-The actuation method is the switching between on- and off-field rapidly -Velocity depends on the actuating frequency |

^{2}/s.

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**MDPI and ACS Style**

Manamanchaiyaporn, L.; Xu, T.; Wu, X.
An Optimal Design of an Electromagnetic Actuation System towards a Large Homogeneous Magnetic Field and Accessible Workspace for Magnetic Manipulation. *Energies* **2020**, *13*, 911.
https://doi.org/10.3390/en13040911

**AMA Style**

Manamanchaiyaporn L, Xu T, Wu X.
An Optimal Design of an Electromagnetic Actuation System towards a Large Homogeneous Magnetic Field and Accessible Workspace for Magnetic Manipulation. *Energies*. 2020; 13(4):911.
https://doi.org/10.3390/en13040911

**Chicago/Turabian Style**

Manamanchaiyaporn, Laliphat, Tiantian Xu, and Xinyu Wu.
2020. "An Optimal Design of an Electromagnetic Actuation System towards a Large Homogeneous Magnetic Field and Accessible Workspace for Magnetic Manipulation" *Energies* 13, no. 4: 911.
https://doi.org/10.3390/en13040911