# Layout Design of Strapdown Array Seeker and Extraction Method of Guidance Information

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The layout of the multi-sensor was explored;
- (2)
- An MIEKF was constructed to estimate guidance information;
- (3)
- Based on the 6-DOF trajectory simulation model, the performance of the monocular seeker and the multi-eye seeker were compared and analyzed, and the performance of the EKF, Iterated EKF (IEKF), and MIEKF were also compared.

## 2. Layout Design of the Sensors

- The head of the missile is ellipsoidal, and the sensor mounting plane is tangent to the shell;
- Each sensor is qualified, and thus, the measurement error distribution is known;
- Each sensor has its own independent signal processor and can output LOS angle information;
- Through installation and adjustment, the focuses of each sensor are at the same point as the body axes of the missile.

#### 2.1. Circular FOV Sensors

#### 2.2. Rectangular FOV Sensor

## 3. Strapdown Array Seeker Guidance Information Extraction Model

#### 3.1. Normalization Processing

#### 3.2. System State Equation

#### 3.3. System Observation Equation

## 4. Filter Design

#### 4.1. Extended Kalman Filters

- State forecast

- 2.
- Filter correction

- 3.
- Measurement update

#### 4.2. Iterated Extended Kalman Filter

#### 4.3. Multivariate Iterated Extended Kalman Filter

## 5. Numerical Simulations

#### 5.1. Generation of Measurement Data

#### 5.2. Results of IKEF and EKF

#### 5.3. Results of MIKEF and EKF

## 6. Conclusions

- (1)
- The field distribution of the circular FOV sensor and the rectangular FOV sensor in the +-shaped layout and the X-shaped layout were explored. The FOV after the array superposition was characterized by the FOV of the central sensor, and the equivalent FOV of the monocular seeker was obtained. The results show that the FOV angle is enlarged to 75.1° and 63.8° in +-shape and X-shape layout when five circular FOV sensors with 45° × 45° FOV angle are used. The FOV angle is enlarged to 70.5° × 90° and 72° × 75° in +-shape and X-shape layout when five rectangular FOV sensors with 45°FOV angle are used. Although the +-shape layout has a larger FOV angle, the X-shape layout has better FOV coverage. In an X-shape layout, there are at least two sensors around the y-axis to provide coverage. In the central area, coverage can be up to five sensors, which is an ideal sensor layout.
- (2)
- In order to solve the problem that the number of observations changes during the observation process, the measurement results ${q}_{e}^{i}$, ${q}_{a}^{i}$ and noise errors ${v}_{1i}$, ${v}_{2i}$ of surface array sensors were normalized and characterized in the FOV of the central sensor. The equivalent observations ${q}_{ei}$, ${q}_{ai}$ and the corresponding error distributions ${w}_{1i}$, ${w}_{2i}$ were obtained. A model of guidance information extraction (28) and (34) was established based on the observations and error distributions.
- (3)
- For the established continuous nonlinear model, the EKF was used for processing. The IEKF was adopted for comparative analysis to overcome the nonlinear error in the filtering process. However, the simulation results show that, compared with EKF, IEKF improves the accuracy of the LOS angle and LOS angle rate by less than 5%. This means that the nonlinear error was not the main error source. In order to make full use of the observed values of the array sensors, improve the filtering quality, and reduce the noise error, the MIEKF was proposed. The simulation results show that the MIEKF can improve the estimation accuracy of LOS angle ε and η by at least 30% and the estimation accuracy of LOS angle rate dε and dη by nearly 80% compared with EKF. So, the MIEKF proposed in this paper is helpful for improving the accuracy of guidance information.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

FOV | Field-of-view |

LOS | Line-of-sight |

UKF | Unscented Kalman filter |

EKF | Extended Kalman filter |

IEKF | Iterated extended Kalman filter |

MIEKF | Multivariate iterated extended Kalman filter |

SCS I | Sensor coordinate system |

BCS B | Body fixed coordinate system |

BLCS L | Body LOS coordinate system |

LCS S | LOS coordinate system |

LaunchCS A | Launch coordinate system |

$\phi /\psi /\gamma $ | Attitude Angle pitch/yaw/roll |

i | Sensor number |

I_{B} | Transformation matrix from B to I |

${\lambda}_{a}$/${\lambda}_{e}$ | Install azimuth/altitude angle |

${q}_{ai}$$/{q}_{ei}$ | Body-LOS-azimuth/altitude angle (in BCS B) |

${q}_{a}^{i}$$/{q}_{e}^{i}$ | Body-LOS-azimuth/altitude angle (in SCS I) |

${q}_{c}$ | LOS-transfer angle |

${q}_{edg}^{a}$$/{q}_{edg}^{e}$ | Sensor’s FOV boundary (azimuth, altitude) |

${x}_{I},{y}_{I},{z}_{I}$ | The coordinates of the target in coordinate frame I |

r | The distance between missile and target |

${v}_{1i}/{v}_{2i}$ | $\mathrm{Measurement}\text{}\mathrm{noises}\text{}\mathrm{of}\text{}{q}_{e}^{i}$$\text{}\mathrm{and}\text{}{q}_{a}^{i}$ |

${w}_{1i}/{w}_{2i}$ | $\mathrm{Measurement}\text{}\mathrm{noises}\text{}\mathrm{of}\text{}{q}_{ai}$$\text{}\mathrm{and}\text{}{q}_{ei}$ |

$\epsilon /\eta $ | LOS-azimuth/altitude angle |

X_{k} | State variable at the moment k |

Z_{k} | Observations at the moment k |

${\mathrm{\Phi}}_{k}$ | State transition matrix at the moment k |

R_{k} | Observed noise matrix at the moment k |

P_{k} | Variance matrix at the moment k |

K_{k} | Kalman gain matrix at the moment k |

H_{k} | Measurement matrix at the moment k |

Q_{k} | System noise matrix at the moment k |

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**Figure 1.**Sensor layout diagrams. (

**a**) Left view of +-shaped layout. (

**b**) Front view of +-shaped layout. (

**c**) Left view of X-shaped layout. (

**d**) Front view of X-shaped layout.

**Figure 9.**The result of sensors measurement. (

**a**) Sensor 1. (

**b**) Sensor 2. (

**c**) Sensor 3. (

**d**) Sensor 4. (

**e**) Sensor 5.

**Figure 10.**The results of EKF & IEKF. (

**a**) Elevation of line-of-sight. (

**b**) The rate of elevation of line-of-sight. (

**c**) Azimuth of line-of-sight. (

**d**) The rate of azimuth of line-of-sight.

**Figure 12.**The results of EKF & MIEKF. (

**a**) Elevation of line-of-sight. (

**b**) The rate of elevation of line-of-sight. (

**c**) Azimuth of line-of-sight. (

**d**) The rate of azimuth of line-of-sight.

Sensor No. | +-Shaped Layout | X-Shaped Layout | ||
---|---|---|---|---|

${\mathit{\lambda}}_{\mathit{e}}\text{}\left[\text{\xb0}\right]$ | ${\mathit{\lambda}}_{\mathit{a}}\text{}\left[\text{\xb0}\right]$ | ${\mathit{\lambda}}_{\mathit{e}}\text{}\left[\text{\xb0}\right]$ | ${\mathit{\lambda}}_{\mathit{a}}\text{}\left[\text{\xb0}\right]$ | |

1 | 0 | 0 | 0 | 0 |

2 | 45 | 0 | 45/$\sqrt{2}$ | −45/$\sqrt{2}$ |

3 | 0 | −45 | −45/$\sqrt{2}$ | −45/$\sqrt{2}$ |

4 | −45 | 0 | 45/$\sqrt{2}$ | −45/$\sqrt{2}$ |

5 | 0 | 45 | 45/$\sqrt{2}$ | 45/$\sqrt{2}$ |

Filter | Error Mean | Standard Deviation of Error | ||||||
---|---|---|---|---|---|---|---|---|

ε [°] | η [°] | dε [°/s] | dη [°/s] | ε [°] | η [°] | dε [°/s] | dη [°/s] | |

EKF | 0.1685 | 0.1833 | 1.4103 | 1.4102 | 0.2148 | 0.2471 | 1.7671 | 1.8347 |

IEKF | 0.1662 | 0.1818 | 1.3667 | 1.3704 | 0.2121 | 0.2446 | 1.7103 | 1.7808 |

Filter | Error Mean | Standard Deviation of Error | ||||||
---|---|---|---|---|---|---|---|---|

ε [°] | η [°] | dε [°/s] | dη [°/s] | ε [°] | η [°] | dε [°/s] | dη [°/s] | |

EKF-1 | 0.1685 | 0.1833 | 1.4103 | 1.4102 | 0.2148 | 0.2471 | 1.7671 | 1.8347 |

EKF-5 | 0.0990 | 0.1132 | 0.7928 | 0.9491 | 0.1276 | 0.1439 | 1.0115 | 1.2013 |

MIEKF | 0.1039 | 0.1256 | 0.2652 | 0.2883 | 0.1325 | 0.1572 | 0.3295 | 0.3612 |

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**MDPI and ACS Style**

Yang, H.; Bai, X.; Zhang, S.
Layout Design of Strapdown Array Seeker and Extraction Method of Guidance Information. *Aerospace* **2022**, *9*, 373.
https://doi.org/10.3390/aerospace9070373

**AMA Style**

Yang H, Bai X, Zhang S.
Layout Design of Strapdown Array Seeker and Extraction Method of Guidance Information. *Aerospace*. 2022; 9(7):373.
https://doi.org/10.3390/aerospace9070373

**Chicago/Turabian Style**

Yang, Hao, Xibin Bai, and Shifeng Zhang.
2022. "Layout Design of Strapdown Array Seeker and Extraction Method of Guidance Information" *Aerospace* 9, no. 7: 373.
https://doi.org/10.3390/aerospace9070373