# A Dual-Axis Rotation Scheme for Redundant Rotational Inertial Navigation System

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## Abstract

**:**

## 1. Introduction

## 2. Error Model of RIMU and Error Variation in Rotation

#### 2.1. General Error Model of RIMU

#### 2.1.1. Inertial Sensor Error Model

#### 2.1.2. Installation Error Model

#### 2.1.3. Vector Representation of RIMU Error Model

#### 2.2. Gyroscope Error Variation in Rotation

#### 2.2.1. Gyroscope Bias Variation

#### 2.2.2. Gyroscope MESSF Variation

#### 2.2.3. Gyroscope MEASF Variation

#### 2.2.4. Gyroscope MEIE Variation

#### 2.3. Accelerometer Error Variation in Rotation

#### 2.3.1. Accelerometer Bias Variation

#### 2.3.2. Accelerometer MESSF Variation

#### 2.3.3. Accelerometer MEASF Variation

#### 2.3.4. Accelerometer MEIE Variation

## 3. Dual-Axis Rotation Scheme

#### 3.1. Rotation Axis Switching Principle

#### 3.2. Reciprocating Rotation Principle

#### 3.3. Rotation Scheme Design

## 4. RRINS Prototype and Experiment

#### 4.1. RRINS Prototype

#### 4.2. Simulation of Error Variation in Rotation

- (1)
- Bias: ${b}_{3}^{g}={0.1}^{\circ}/\mathrm{h}$, ${b}_{3}^{a}=50\mathsf{\mu}\mathrm{g}$.
- (2)
- Symmetric scale factor error: ${\lambda}_{3}^{g}=50\mathrm{p}\mathrm{p}\mathrm{m}$, ${\lambda}_{3}^{a}=30\mathrm{p}\mathrm{p}\mathrm{m}$.
- (3)
- Asymmetric scale factor error: ${\mu}_{3}^{g}=50\mathrm{p}\mathrm{p}\mathrm{m}$, ${\mu}_{3}^{a}=30\mathrm{p}\mathrm{p}\mathrm{m}$.
- (4)
- Installation error: ${\delta}_{u3}^{g}={\delta}_{u3}^{a}=1{0}^{\u2033}$, ${\delta}_{v3}^{g}={\delta}_{v3}^{a}=1{0}^{\u2033}$.

#### 4.3. RIMU-Based and Traditional IMU-Based Navigation Simulation

#### 4.3.1. Strapdown Navigation Simulation

- (1)
- All gyroscopes in IMU have a bias: 0.1°/h.
- (2)
- All accelerometers in IMU have a bias: 50 μg.

- (1)
- The biases of four gyroscopes in RIMU: 0.1°/h, 0.11°/h, 0.12°/h, 0.13°/h.
- (2)
- The biases of four accelerometers in RIMU: 50 μg, 55 μg, 60 μg, 65 μg.

#### 4.3.2. Rotational Navigation Simulation

#### 4.4. Static Experiment

- 1
- The RRINS remained stationary throughout the experiment.
- 2
- Initial longitude: 116.668° E, latitude: 40.3554° N, height: 40 m.
- 3
- Initial velocity: 0.
- 4
- Dual-axis rotation parameters: angular rate of rotation: 2°/s, one position period: 100 s (rotating 90 s and standing 10 s), 16-position period: 1600 s.

#### 4.5. Dynamic Semi-Physical Simulation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The installation direction of any single inertial sensor in RIMU. Here, $o{x}_{s}{y}_{s}{z}_{s}$ represents the s-frame, ${s}_{i}$ represents the ith gyroscope or accelerometer, ${\alpha}_{i}$ indicates the angle between the ${s}_{i}$ axis and the ${z}_{s}$ axis, and ${\beta}_{i}$ indicates the angle between the projection of the ${s}_{i}$ axis in the $o{x}_{s}{y}_{s}$ plane and the ${x}_{s}$ axis.

**Figure 2.**The RIMU installation error model by using rotation vector. Here, $\mathit{h}$ represents the designed installation vector of the sensor (gyroscope or accelerometer), $\mathit{u}$ and $\mathit{v}$ are unit vectors that are perpendicular to $\mathit{h}$, $\overline{\mathit{h}}$ represents the actual installation vector, $\delta $ represents the rotation vector from $\mathit{h}$ to $\overline{\mathit{h}}$, and $\mathit{\delta}={\delta}_{u}^{}\mathit{u}-{\delta}_{v}^{}\mathit{v}$.

**Figure 3.**The frames in RRINS. Here, $o{x}_{b}{y}_{b}{z}_{b}$ represents the b-frame, ${\alpha}_{I}$ represents the rotation angle of the inner gimbal, and ${\alpha}_{O}$ represents the rotation angle of the outer gimbal.

**Figure 4.**The 8-position dual-axis rotation scheme. Here, 1, 2, …, and 8 represent the 1st, 2nd, … , and 8th rotation order respectively.

**Figure 6.**Configuration structure of RIMU. Here, si indicates the ith group sensors (i = 1,2,3,4). The group sensors include a gyroscope and accelerometer.

**Figure 9.**Strapdown navigation error comparison between IMU and RIMU. Here, $\delta {v}_{e}$ is the eastward velocity error, $\delta {v}_{n}$ is the northward velocity error, $\delta \lambda $ is the longitude direction position error, and $\delta L$ is the latitude direction position error.

Position | Rotation of RIMU | Control of Turntable | ||
---|---|---|---|---|

Axis | Angle | Axis | Angle | |

1 | z | +180° | Inner | +180° |

2 | x | +180° | Outer | −180° |

3 | z | +180° | Inner | +180° |

4 | x | +180° | Outer | +180° |

5 | x | +180° | Outer | +180° |

6 | z | +180° | Inner | +180° |

7 | x | +180° | Outer | −180° |

8 | z | +180° | Inner | +180° |

9 | z | −180° | Inner | −180° |

10 | x | −180° | Outer | +180° |

11 | z | −180° | Inner | −180° |

12 | x | −180° | Outer | −180° |

13 | x | −180° | Outer | −180° |

14 | z | −180° | Inner | −180° |

15 | x | −180° | Outer | +180° |

16 | z | −180° | Inner | −180° |

Parameter | Gyroscope | Accelerometer |
---|---|---|

Range | −300 to + 300 °/s | −20 to + 20 g |

Bias | −10 to + 10 °/h | ≤3 mg |

Stability (4 in order) | [0.0502, 0.0303, 0.0355, 0.0517] °/h | [12.4, 8.3, 27.7, 11.7] μg |

Repeatability (4 in order) | [0.0831, 0.0109, 0.0720, 0.0770] °/h | [5.7, 5.8, 7.4, 5.1] μg |

Scale Factor Repeatability | ≤50 ppm | ≤30 ppm |

Object | Static Simulation | Static Experiment | Dynamic Semi-Physical Simulation | ||
---|---|---|---|---|---|

$\mathrm{max}\left|\delta \lambda \right|$ | $\mathrm{max}\left|\delta L\right|$ | $\mathrm{max}\left|\delta \lambda \right|$ | $\mathrm{max}\left|\delta L\right|$ | End Point Error | |

Strapdown IMU | 9 km | 9 km | - | - | - |

Rotation IMU | 86 m | 69 m | - | - | - |

Strapdown RIMU | 0.95 km | 0.55 km | 69 km | 192 km | 195 m |

Rotation RIMU | 5 m | 2 m | 9.6 km | 21.3 km | 30 m |

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**MDPI and ACS Style**

Zhu, T.; Wang, L.; Zou, T.; Peng, G.
A Dual-Axis Rotation Scheme for Redundant Rotational Inertial Navigation System. *Micromachines* **2023**, *14*, 351.
https://doi.org/10.3390/mi14020351

**AMA Style**

Zhu T, Wang L, Zou T, Peng G.
A Dual-Axis Rotation Scheme for Redundant Rotational Inertial Navigation System. *Micromachines*. 2023; 14(2):351.
https://doi.org/10.3390/mi14020351

**Chicago/Turabian Style**

Zhu, Ting, Lifen Wang, Tao Zou, and Gao Peng.
2023. "A Dual-Axis Rotation Scheme for Redundant Rotational Inertial Navigation System" *Micromachines* 14, no. 2: 351.
https://doi.org/10.3390/mi14020351