# Computational Study of a Motion Sensor to Simultaneously Measure Two Physical Quantities in All Three Directions for a UAV

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**u**is the flow velocity vector field, $\nabla $ is the spatial divergence operator, p is the pressure, I is the total stress tensor, and f is the body forces acting on the fluid. The parameters C

_{p}, $\rho $, and k are the specific heat, density, and thermal conductivity of the fluid in the cavity, respectively.

_{2}) was selected as a gas medium because of its high density and low kinematic viscosity. The low viscosity of CO

_{2}enables a more efficient flow and results in greater sensitivity than gases with higher viscosities because high viscosity impedes gas flow [21].

_{max}and T

_{min}).

_{2}molecules was practically the same throughout the motion sensor cavity with a maximum velocity of 49.23 m/s

^{2}at the instant of t = 2.5 s. Furthermore, the temperature change with respect to time for all five meshes is shown in Figure 5. The maximum values in this graph were extracted and used for comparison.

_{max}values with respect to the number of mesh elements, are presented in Table 2 and illustrated in Figure 6.

_{max}value can be observed between the second and third meshes. However, from the third mesh onwards, the difference in T

_{max}values between the meshes was approximately 1%, which was minimal compared to the differences between the first three meshes. Therefore, a mesh with 166,675 elements was selected for future calculations because it provides a reliable solution with minimal computational requirements such as CPU time. The placement of the heaters and sensors is shown in Figure 7, and the meshing structure used in the computational study is shown in Figure 8. Four heaters (H1–H4) were placed on all four axes, 40 mm from the center of the cavity. Heaters H1 and H3 on the X-axis are surrounded by pairs of X-sensors: X21 and X22, and X11 and X12, respectively. Similarly, H2 and H4 on the Y-axis are bounded by Y-sensor pairs: Y21 and Y22, and Y11 and Y12, respectively. On the Z-axis, each heater constitutes a temperature sensor located 10 mm away from the heater in the Z-direction. The notation of the Z-sensors is such that the number denotes the heater number (e.g., Z1 around H1).

## 3. Results and Discussions

^{2}) and rotations ranging from 200 to 1000°/s in all three directions. As described in Section 2 and Figure 7, four heaters and four pairs of temperature sensors were positioned in all three directions. Owing to the symmetry of the structure, identical results were obtained for X11 and X21, and X12 and X22. Similarly, Y11 and Y21, and Z1 and Z3 had similarly extreme temperature values. Therefore, to obtain these extreme values, the maxima and minima of X11, Y12, and Z4 were considered. These values were then recorded for all accelerations and rotations in all three directions and correlated with the applied physical quantities of acceleration and rotation.

_{max}and T

_{min}values for all three axes are listed in Table 3, Table 4 and Table 5, and the inverse functions used to obtain acceleration and rotational speed in all three directions corresponding to the measured maximum and minimum temperature values are shown in Figure 11, Figure 12 and Figure 13. Node values indicate each data point. These inverse functions were installed in the computing unit of a real thermal motion sensor.

_{max}and Y

_{min}, and this region is represented by an ellipse in Figure 14. Therefore, the results should be verified, and the parameters that generate unique solutions should be determined. This problem can be reduced by either altering the cavity shape of the sensor or changing the positions of heaters and sensors.

_{max}and T

_{min}) extracted from Table 4. For real measurements, if the output (measured) values of the sensor can be obtained (plotted in Figure 15b (purple circle)), the input values for the output values can be approximately and geometrically calculated without using an inverse function, as shown in the blue circle in Figure 15a. If quadrilateral shapes overlap with each other for the output data, it will indicate multiple solutions. This technique can also be utilized in 3D to simultaneously measure three physical quantities.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 2.**Changes in isotherms with no acceleration (

**left**) and acceleration applied to the right side (

**right**).

**Figure 4.**Velocity (

**left**) and pressure distribution (

**right**) at t = 2.5 s for the mesh with 166,675 elements.

**Figure 5.**Temperature response at 500°/s around the Z-axis and 2g applied in the X-direction for different mesh sizes.

**Figure 9.**Schematic of obtaining three inverse functions for the measurement of acceleration and rotation in all three directions.

**Figure 11.**Graphs for X-acceleration (

**left**) and Z-rotation (

**right**) values from X

_{min}and X

_{max}measured around heater 1.

**Figure 12.**Graphs for Y-acceleration (

**left**) and X-rotation (

**right**) values from Y

_{min}and Y

_{max}measured around heater 2.

**Figure 13.**Graphs for Z-acceleration (

**left**) and X-rotation (

**right**) values from Z

_{min}and Z

_{max}measured around heater 4.

**Figure 14.**Region of multiple solutions indicated on the graph to obtain Y-acceleration using Y

_{max}and Y

_{min}.

Maximum Velocity at 3 s (m/s) | Increment Velocity (m/s) | ||
---|---|---|---|

Acceleration | 1g | 29.43 | 0.0412 |

2g | 58.86 | 0.0824 | |

3g | 88.29 | 0.1236 | |

4g | 117.72 | 0.1648 | |

Rotation (°/s) | 250 | 0.0436 | 0.0436 |

500 | 0.0873 | 0.0873 | |

750 | 0.1309 | 0.1309 | |

1000 | 0.1745 | 0.1745 |

S | No. of Mesh Elements | V_{max} (m/s) | P_X11_{min} (Pa) | P_X11_{max} (Pa) | T_{max} (K) |
---|---|---|---|---|---|

1 | 13,002 | 49.23 | −0.497392 | −0.168213 | 433.7 |

2 | 36,930 | 49.23 | −0.509235 | −0.167053 | 411.0 |

3 | 166,675 | 49.235 | −0.508066 | −0.181941 | 429.8 |

4 | 322,586 | 49.235 | −0.513125 | −0.183977 | 427.5 |

5 | 554,001 | 49.24 | −0.518651 | −0.185735 | 426.0 |

6 | 752,760 | 49.16 | −0.523673 | −0.186014 | 421.6 |

T_X11_{max} | ω | 250 | 500 | 750 | 1000 | |

a | ||||||

1g | 477.9 | 488.9 | 488.3 | 489.7 | ||

2g | 495.0 | 502.6 | 501.1 | 500.7 | ||

3g | 500.9 | 504.8 | 501.4 | 501.4 | ||

4g | 498.4 | 503.3 | 495.5 | 496.7 | ||

T_X11_{min} | ω | 250 | 500 | 750 | 1000 | |

a | ||||||

1g | 407.1 | 415.7 | 417.3 | 415.7 | ||

2g | 359.1 | 373.2 | 379.5 | 380.1 | ||

3g | 328.3 | 338.5 | 346.1 | 348.9 | ||

4g | 312.7 | 316.4 | 320.3 | 322.1 |

T_Y12_{max} | ω | 250 | 500 | 750 | 1000 | |

a | ||||||

1g | 497.3 | 507.0 | 512.6 | 514.3 | ||

2g | 499.4 | 519.3 | 529.5 | 536.2 | ||

3g | 495.6 | 516.1 | 527.1 | 534.8 | ||

4g | 496.1 | 512.2 | 523.5 | 533.2 | ||

T_Y12_{min} | ω | 250 | 500 | 750 | 1000 | |

a | ||||||

1g | 425.9 | 435.8 | 440.6 | 440.0 | ||

2g | 381.3 | 393.7 | 404.0 | 408.1 | ||

3g | 353.7 | 355.9 | 361.3 | 361.0 | ||

4g | 337.6 | 333.4 | 332.3 | 329.3 |

T_Z4_{max} | ω | 250 | 500 | 750 | 1000 | |

a | ||||||

1g | 483.1 | 490.3 | 497.3 | 500.2 | ||

2g | 471.4 | 475.5 | 486.3 | 494.3 | ||

3g | 452.8 | 454.8 | 471.9 | 486.9 | ||

4g | 438.9 | 426.8 | 444.1 | 457.3 | ||

T_Z4_{min} | ω | 250 | 500 | 750 | 1000 | |

a | ||||||

1g | 442.4 | 440.6 | 438.4 | 444.8 | ||

2g | 388.2 | 398.6 | 401.7 | 399.7 | ||

3g | 350.7 | 354.8 | 360.0 | 360.2 | ||

4g | 332.2 | 329.4 | 333.2 | 334.0 |

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**MDPI and ACS Style**

Siddique, K.; Ogami, Y.
Computational Study of a Motion Sensor to Simultaneously Measure Two Physical Quantities in All Three Directions for a UAV. *Sensors* **2023**, *23*, 5265.
https://doi.org/10.3390/s23115265

**AMA Style**

Siddique K, Ogami Y.
Computational Study of a Motion Sensor to Simultaneously Measure Two Physical Quantities in All Three Directions for a UAV. *Sensors*. 2023; 23(11):5265.
https://doi.org/10.3390/s23115265

**Chicago/Turabian Style**

Siddique, Kamran, and Yoshifumi Ogami.
2023. "Computational Study of a Motion Sensor to Simultaneously Measure Two Physical Quantities in All Three Directions for a UAV" *Sensors* 23, no. 11: 5265.
https://doi.org/10.3390/s23115265