# Research on In-Flight Alignment for Micro Inertial Navigation System Based on Changing Acceleration using Exponential Function

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## Abstract

**:**

## 1. Introduction

## 2. Double-vector Observations and Regressive QUEST Algorithm

#### 2.1. Double-Vector Observations

**V**(i) and

**W**(i) are the vectors in the ib and in frames, and are given by:

**V**(i) and

**W**(i) are difficult to be parallel [17]. Thus, the attitude matrix ${C}_{in}^{ib}$ can be encoded according to the vectors of

**V**(i) and

**W**(i).

#### 2.2. Regressive QUEST Algorithm

## 3. Error Analysis of the Alignment Algorithm

**W**(i) and

**V**(i) at any time, then the true attitude matrix ${C}_{in}^{ib}$ can be satisfied by:

**W**(i) and

**V**(i) at any time, and we suppose, $W=\left[\begin{array}{c}W{(1)}^{T}\\ W{(2)}^{T}\\ \vdots \\ W{(j)}^{T}\end{array}\right]$ is the vector matrix of

**W**(i) in each period and its calculation error of it is $\delta \mathit{W}=\left[\begin{array}{c}\delta \mathit{W}{(1)}^{T}\\ \delta \mathit{W}{(2)}^{T}\\ \vdots \\ \delta \mathit{W}{(j)}^{T}\end{array}\right]$, and $\mathit{V}=\left[\begin{array}{c}\mathit{V}{(1)}^{T}\\ \mathit{V}{(2)}^{T}\\ \vdots \\ \mathit{V}{(j)}^{T}\end{array}\right]$ is the vector matrix of

**V**(i) in each period and its calculation error of it is $\delta \mathit{V}=\left[\begin{array}{c}\delta \mathit{V}{(1)}^{T}\\ \delta \mathit{V}{(2)}^{T}\\ \vdots \\ \delta \mathit{V}{(j)}^{T}\end{array}\right]$. Suppose the true attitude matrix ${C}_{in}^{ib}$ is

**X**, and its estimation error for it is δ

**X**. According to Equation (8), we can establish the following equation:

**V**(i) can be obtained by the observations from the global navigation satellite system (GNSS). Thus, it can be exactly calculated and the errors of δ

**V**can be approximately 0, then Equation (13) can be written by:

**W**(i) can be calculated according to Equation (7) by the observations from gyroscopes and accelerometer. Thus, the error of

**X**will be largely influenced by the parameter of κ. If κ is relatively small, then the estimation error of

**X**will be small. Conversely, if κ is relatively large, then the small δ

**W**will cause a great error of

**X**.

**X**, κ should be largely decreased. As for κ, the smaller, the better. According to the literature [22], κ is suggested to be in the range of 1 to 100.

**W**and

**V**.

**W**consists of the vector

**W**(i), which can be calculated by the observations from gyroscopes and accelerometers. For the real application, gyroscopes and accelerometers will measure the angular rate and acceleration information of the guided projectiles. Generally, the flight angular rate and acceleration of the projectile relate to its flight features; thus it cannot be easily changed. Then, the vector matrix

**W**, which consists of

**W**(i), cannot be directly changed. As a result, in order to largely decrease κ, we should try to change the vector matrix of

**V**.

**V**consists of the vector

**V**(i), in Equation (7), and

**V**(i) can be changed to the following form:

**V**(i). However, as we know, ${g}^{n}$ is a constant vector for the gravity acceleration and cannot be changed. ${C}_{n(t)}^{in}$ and ${\omega}_{ie}^{n}$ can be calculated by:

**V**, we should try to change the vector of ${\dot{v}}^{n}$ or ${v}^{n}$. ${v}^{n}$ originates from the integral of ${\dot{v}}^{n}$, thus, changing the vector of ${\dot{v}}^{n}$ will be an effective way to ensure the change of V(i) in every moment, which means to change the acceleration of the projectile.

**V**.

**V**, the acceleration using the exponential function is applied [26]. A simulation and semi-physical experiment were performed to verify the effectiveness.

## 4. Simulations and Semi-physical Experiment

#### 4.1. Simulation

_{0}= 15°, θ

_{0}= 35°, ψ

_{0}= 45°, and the initial rolling rate is ω = 10°/s. Because the shooting angle will influence the alignment result, it is set from 0° to 90°.

^{2}, and for the next 15 s the acceleration is −25exp(0.08t) m/s

^{2}. After the first 30 s, the projectile will fly in the attacking trajectory using the initial alignment attitude.

_{0}is near 90°, the alignment results for the rolling error, pitch error and heading error are not effective, and are extremely high. This is inconsistent with the analysis of κ. According to the attitude definition for the navigation, when the shooting angle is 90°, it is not easy to recognize the rolling angle and heading angle. Therefore, we should avoid an initial shooting angle of 90° for the actual application.

#### 4.2. Semiphysical Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Simulation results for the proposed alignment method. (

**a**) Alignment result for rolling error; (

**b**) Alignment result for pitch error; (

**c**) Alignment result for heading error.

**Figure 3.**Equipment used for semi-physical experiment. (

**a**) The designed micro inertial measurement unit (MIMU); (

**b**) three-axis flight attitude simulation turntable.

Steps | Calculations |
---|---|

1 | Initialization, [q_{0}, q_{1}, q_{2}, q_{3}] = [1 0 0 0]^{T}, K_{0} = 0_{4×4}, m_{0} = 0 |

2 | In each estimation period, calculate W(i) and V(i) as shown in Equation (7) and normalize W(i) and V(i) to be w(i) and v(i) |

3 | Calculate $\delta \sigma ={\displaystyle \sum _{i=1}^{j}{\alpha}_{i}w{(i)}^{T}v(i)}$, $\delta S=\delta B+\delta {B}^{T}$, $\delta B={\displaystyle \sum _{i=1}^{j}{\alpha}_{i}w(i)v{(i)}^{T}}$, $\delta Z={\displaystyle \sum _{i=1}^{j}{\alpha}_{i}w(i)\times v(i)}$, $\delta K=\left[\begin{array}{cc}\delta S-\delta \sigma I& \delta Z\\ \delta {Z}^{T}& \delta \sigma \end{array}\right]$, $\delta m={\displaystyle \sum _{i=1}^{j}{\alpha}_{i}}.$ |

4 | Regressive calculation of ${K}_{k+1}\text{}\mathrm{by}:\text{}{K}_{k+1}=\frac{{m}_{k}}{{m}_{k}+\delta m}{K}_{k}+\frac{1}{{m}_{k}+\delta m}\delta K$ |

5 | $\mathrm{Calculate}\text{}\mathrm{the}\text{}\mathrm{eigenvectors}\text{}\mathrm{of}\text{}{K}_{k+1}\text{}\mathrm{and}\text{}\mathrm{use}\text{}\mathrm{them}\text{}\mathrm{to}\text{}\mathrm{encode}\text{}\mathrm{the}\text{}\mathrm{attitude}\text{}\mathrm{matrix}\text{}{C}_{in}^{ib}.$ |

6 | Return to Step 2 for the next period estimation. |

Shooting angles | Rolling angle error | Pitch angle error | Heading angle error |
---|---|---|---|

15° | 0.232° | 0.106° | 0.530° |

30° | 0.279° | 0.112° | 0.542° |

45° | 0.346° | 0.114° | 0.554° |

55° | 0.354° | 0.113° | 0.573° |

60° | 0.463° | 0.113° | 0.599° |

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**MDPI and ACS Style**

Xu, Y.; Zhou, T.
Research on In-Flight Alignment for Micro Inertial Navigation System Based on Changing Acceleration using Exponential Function. *Micromachines* **2019**, *10*, 24.
https://doi.org/10.3390/mi10010024

**AMA Style**

Xu Y, Zhou T.
Research on In-Flight Alignment for Micro Inertial Navigation System Based on Changing Acceleration using Exponential Function. *Micromachines*. 2019; 10(1):24.
https://doi.org/10.3390/mi10010024

**Chicago/Turabian Style**

Xu, Yun, and Tong Zhou.
2019. "Research on In-Flight Alignment for Micro Inertial Navigation System Based on Changing Acceleration using Exponential Function" *Micromachines* 10, no. 1: 24.
https://doi.org/10.3390/mi10010024