# Performance Enhancement of a USV INS/CNS/DVL Integration Navigation System Based on an Adaptive Information Sharing Factor Federated Filter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principle of the Integration Subsystem and Error Analysis

#### 2.1. Principle of INS and System Error Model

#### 2.1.1. Principle of INS

**f**

^{b}denotes the accelerometer output, ${\mathsf{\omega}}_{ib}^{b}$ denotes the gyro output. ${\mathsf{\omega}}_{xy}^{z}$ (x = i,e,n; y = e,n,b; z = n,b) denotes the rotation rate along z between x and y.

**q**

_{nb}denotes the rotating quaternion between b and n.

**P**= [ϕ λ]

^{T}and

**R**= diag([1/R

_{m}1/(R

_{t}cosϕ)]. v

_{x}and v

_{y}denote the east and north velocity along n frame calculated by INS. ϕ and λ denote the latitude and the longitude calculated by INS.

**g**

^{n}denotes gravity along n frame. R

_{m}denotes the semi major axis of the Earth, R

_{t}denotes the semi minor axis of the Earth.

#### 2.1.2. Error Analysis for INS

**P**= [δϕ δλ]

^{T}denotes the position error; δ

**v**[δv

_{x}δv

_{y}]

^{T}denotes the velocity error; φ = [φ

_{x},φ

_{v},φ

_{z}]

^{T}denotes pitch, roll, yaw misalignment angles, respectively, and the initial value of the misalignment angles is stationary, but considering the influence of IMU errors, the misalignment angles varies with time, oscillating during the navigation process 14; ∇ = [∇

_{x},∇

_{v},∇

_{z}]

^{T}denotes accelerometers bias,

**ε**= [ε

_{x},ε

_{v},ε

_{z}]

^{T}denotes the gyro drift.

**O**

_{3×3}denotes the matrix with three rows and three columns, and all of the elements of the matrix are zero;

**A**ij (i =

**P**,

**v**,φ, j =

**P**,

**v**,φ,∇,

**ε**) denotes the transformation matrix between j and i.

_{m}denotes the radius of the Earth, ${\omega}_{iey}^{n}={\omega}_{ie}\mathrm{cos}\phi $, ${\omega}_{iez}^{n}={\omega}_{ie}\mathrm{sin}\phi $ denotes the component of angular velocity of earth rotation along the oy axis, oz axis of the navigation frame respectively.

#### 2.2. Principle of CNS and Error Analysis

#### 2.2.1. Principle of CNS

_{i}with direction vector

**v**

_{i}in the celestial coordinate system can be detected by the star sensor, whereas the vector of its image can be expressed as

**w**

_{i}in the star sensor coordinate system.

_{0},y

_{0}). The position of the image point of navigation star S

_{i}on the image plane is (x

_{i},y

_{i}). The focal length of the star sensor is L

_{f}. Vector

**w**

_{i}can be expressed using Equation (3) [48,49]:

**w**

_{i}is the vector projection of the image point of navigation star S

_{i}on the image plane.

**w**

_{i}and

**v**

_{i}under ideal conditions, where

**A**is the attitude matrix of the star sensor:

#### 2.2.2. Error Analysis for CNS

_{FOV}is the total number of stars on the FOV. M is the visual magnitude. It is assumed that the FOV is circular and is A degrees wide.

#### 2.3. Principle of DVL and Error Analysis

#### 2.3.1. Principle of DVL

_{0}denotes the frequency of the DVL emission signal. f

_{1}denotes the frequency of the signal at point P. f

_{2}denotes the frequency of the reflected signal. c

_{0}is the velocity of the signal, v

_{x}

_{0}denotes the vehicle’s velocity along its heading. The wave source moves from point O to O’ when the reflected signal is received by DVL.

#### 2.3.2. Error Analysis for DVL

**V**

^{b}is the DVL residue error; Δ

**V**

^{b}is the constant error;

**u**

_{0}is the measurement noise.

## 3. Federated Filter Based on the Adaptive Information Sharing Factor

#### 3.1. The Principle of Federated Filter

_{i}should be adjusted and increased, which means that the estimation error of the subsystem has a big error. Therefore, the mean squared error matrix of subsystem is updated by ${P}_{i}={\beta}_{i}^{-1}{P}_{g}$.

_{1}is updated by the equation as [53]:

_{1}is computed independently, it can be obtained that:

_{1}is computed by global output, it can be obtained that:

_{1}is computed independently, P

_{1}is only influenced by the local filter 1 itself. It means that when the information accuracy of one of the integrated navigation subsystems is decreased, P

_{i}should not be adjusted, which means that the estimating error of the subsystem is a big error. If P

_{1}is computed by global output, it can be adjusted by two local filters. However, the adjustment process is changeless, and the system fault cannot be checked and separated by the Federated Filter.

_{i}is a new parameter, and the proportion of every subsystem in the main filter is adjusted by β

_{i}. It means that the proportion of every subsystem in the main filter increased with the increase of β

_{i}; otherwise the proportion is decreased. Therefore, the information sharing factor of β

_{i}in the breakdown subsystem should be adjusted lower, so that the influence of the breakdown subsystem to the main system is decreased and avoided and the accuracy of the whole filter is stable. It is obtained that setting the suitable value of β

_{i}according to the statement of subsystem is of importance.

_{i}is the key to improve the fault-tolerance performance of Federated Filter.

#### 3.2. Adaptive Information Sharing Factor for Federated Filter

_{ik}, which is the “contribution” of the local filter to the main filter, is an important factor for the Federated Filter. Especially for the multi-sensor integrated navigation system based on INS/CNS/DVL, the reference information accuracy from DVL and CNS is the main factor and can influence the navigation accuracy. Hence, the DVL and CNS information accuracy must be shown by ISF, which means that the ISF must be adjusted adaptively. The principle of setting the ISF of β

_{ik}is shown as follows.

**R**

_{ik}are innovations, a predicted of the Kalman Filter respectively.

**C**

_{ik}is referred to as the calculated innovation covariance in this paper.

**C**

_{ik}, but sometimes, the exact dynamic equation is not available. Then, an estimation error and a predicted error covariance may increase by the effect of the unknown information. Similarly, the exact measurement equation is not available. Then, an innovation covariance

**C**

_{ik}may be increased by the effect of unknown information. In this case, an innovation covariance

**C**

_{ik}is increased by an increased measurement covariance

**R**

_{ik}. The change of an innovation covariance can be used for an adaptive filter. The increased innovation covariance can be estimated as:

**C**

_{ik}and the estimated innovation covariance ${\overline{C}}_{ik}$:

_{ik}denotes the math relationship between the innovation covariance ${\overline{C}}_{ik}$ and the estimated innovation covariance ${\overline{C}}_{ik}$.

**C**

_{ik}is the estimated result of observation at current time point, and ${\overline{C}}_{ik}$ is the mean value of observation estimated result in a period of time. Therefore, the factor α

_{ik}shows the stability of the observation estimated result at the current time point. When the measurement noise is known exactly, the numerical value of

**C**

_{ik}has been checked against the value of ${\overline{C}}_{ik}$, and the value of α

_{ik}is approximate to zero; when the measurement noise make a sudden change and the observation accuracy is decreased at time point k, the innovation covariance

**C**

_{ik}at time point k is deviated from the estimated innovation covariance ${\overline{C}}_{ik}$ in a last period of time, and the value of α

_{ik}is increased. Therefore, the value of α

_{ik}has a direct ratio relation with the measurement noise. Hence, the ISF can be designed by α

_{ik}, the calculation method is shown as follows:

_{ik}in Equation (14) is the inverse matrix. If there is a big difference between the measurement noise at current time point and the setting value, it is obtained that the observation is poor and the “contribution” of this local filter is decreasing to the main filter; conversely, the difference between the measurement noise at current time point and the setting value is small, it is obtained that the observation is good and the “contribution” of this local filter is increasing to the main filter. Thus, the ISF is adjusting adaptively in real time and improving the accuracy and the stability of the Federated Filter.

_{m}= 0.

## 4. The Multi-Sensor Integrated Navigation Method Using the Adaptive ISF Federated Filter

**w**(t) and

**v**

_{i}(t) denote the state noise matrix and the measurement noise respectively;

**X**denotes the state variable and the form is:

**v**

_{DVL}denotes the DVL velocity constant error, and the form is δ

**v**

_{DVL}= [δ

**v**

_{xDVL}δ

**v**

_{yDVL}]

^{T}. δ

**v**

_{xDVL}and δ

**v**

_{yDVL}denote the velocity constant error along east and north of the navigation frame.

**A**is the state transition matrix and the form is:

**A**

_{0(15×15)}is the transformation matrix and the form is shown in Equation (2).

**H**

_{1}is the measurement matrix of Kalman filter 1 and the form is:

**Z**

_{1}is the observations of the measurement equation in Kalman filter 1, which is the difference between the INS position and the CNS position, it is:

_{CNS}are the latitude information from INS and CNS respectively, λ and λ

_{CNS}are the longitude information from INS and CNS, respectively.

**H**

_{2}is the measurement matrix of Kalman filter 2 and the form is:

**Z**

_{2}is the observations of the measurement equation of Kalman filter 2, which is the difference between the INS velocity and the DVL velocity, it is:

_{x}and v

_{y}are the velocity calculated by INS along east and north of the navigation frame. v

_{xDVL}and v

_{yDVL}are the velocity from DVL along east and north of the navigation frame.

**P**

_{i}(i = 1,2) of these two filters are the input information of the main filter. Then, the final estimation results, which are the INS error and the DVL error, are obtained by integrating the input information from two Kalman Filters. The estimation results are fed back and compensated to correct the INS navigation information. The better navigation information is obtained.

**P**

_{1}and

**P**

_{2}from two Kalman Filters and the ISF updating method proposed in Section 3. For example, it is assumed that the DVL velocity accuracy is decreased due to the unknown measurement noise at time point k, which means that the setting value of

**R**

_{k}is inaccurate. Hence,

**P**

_{2}, which is called mean squared error matrixes and reflects the estimation accuracy of Kalman Filter 2, is increased. Then, the innovation covariance

**C**

_{ik}at time point k is increased compared with the average innovation covariance ${\overline{C}}_{ik}$, which is the mean value of the innovation covariance from time point k-M to time point k. The ISF of β

_{2}is increased based on the calculation method of Equation (14), and the feedback information ${\beta}_{2}^{-1}{P}_{g}$ is decreased. Then, the mean squared error matrix of Kalman Filter 2 is adjusted and corrected, and the information updating in the main filter part is also adjusted.

**v**

_{DVL}serves as the state variable. A centralized filter allows observability of the constant errors included in the state vector, and it can be estimated by local filter 1 when the DVL accuracy is not influenced by measured noise. However, if observability is kept with the federated solutions which develop independent computations of state vectors, there is some possible limitation of federated filter in this type of applications, which means that the paradoxical relationship is inevitably produced between the constant error estimating and the DVL measured noise inhibition. The principle of ISF adjusting the Kalman Filter 1 is the same as Kalman Filter 2. The difference between these adjustment processes is that DVL accuracy is decreased by measured noise and CNS accuracy is decreased by the reduction of observation stars. However, the problem is the observation accuracy is decreased for DVL or CNS. Hence, the ISF adjustment principles of these two Kalman Filters are the same. Therefore, the INS navigation error can be estimated and compensated based on the multi-sensor integrated navigation method of INS/CNS/DVL using AISFF. Hence, introducing the DVL velocity constant error as the state variable and taking the ISF adjusting the Kalman Filter are the better method to balance the influence from these two factors and improve the navigation accuracy.

**C**

_{ik}at time point k The setting value of M is 300 (300 = 5 min × 60 s). Hence, the principle of how to set the value of M depends on the sensors and environment.

## 5. Analysis of the Simulation and the Experimental Study

#### 5.1. The Simulation Analysis

#### 5.1.1. Simulation Conditions

#### Motion State of the Vehicle

#### IMU, CNS and DVL Error

^{−5}g, and the measurement noise is Gaussian white noise with amplitude of 10

^{−5}g.

#### 5.1.2. Simulation Results

#### 5.2. Experiments and Results

#### 5.2.1. Experiment Conditions

#### 5.2.2. Experiment Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Star sensor navigation result in the poor weather condition. (

**a**) The number of stars available; (

**b**) star sensor based CNS positioning error.

**Figure 6.**DVL velocity errors in both time and frequency domain. (

**a**) DVL velocity error; (

**b**) amplitude-frequency along east and north.

**Figure 11.**Simulation result with Method 1 (centralized Kalman Filter). (

**a**) Position error; (

**b**) Velocity error; (

**c**) Attitude error.

**Figure 12.**Simulation result with Method 2 (Federated Filter). (

**a**) Position error; (

**b**) Velocity error; (

**c**) Attitude error.

**Figure 13.**Simulation result with Method 3 (AISFF). (

**a**) Position error; (

**b**) Velocity error; (

**c**) Attitude error.

**Figure 14.**Simulation result with Method 4 (NF). (

**a**) Position error; (

**b**) Velocity error; (

**c**) Attitude error.

**Figure 20.**Experiment result with three integrated navigation methods: (

**a**) Latitude error; (

**b**) Longitude error.

No. | State | Value | Duration |
---|---|---|---|

1 | Moving forward with constant speed | 5 m/s | 30 min |

2 | Acceleration motion with constant | 0.1 m/s^{2} | 10 s |

3 | Moving forward with constant speed | 6 m/s | 20 min |

4 | Left turning | 1°/s | 90 s |

5 | Moving forward with constant speed | 6 m/s | 10 min |

6 | Right turning | 1°/s | 90 s |

7 | Moving forward with constant speed | 6 m/s | 20 min |

8 | Left turning | 2°/s | 45 s |

9 | Moving forward with constant speed | 6 m/s | 10 min |

10 | Right turning | 1°/s | 45 s |

11 | Moving forward with constant speed | 6 m/s | 10 min |

12 | Decelerated motion with constant | 0.1 m/s^{2} | 10 s |

13 | Moving forward with constant speed | 5 m/s | 20 min |

Parameter Item | Index | |
---|---|---|

Gyro | Dynamic Range | ±100 °/s |

Bias Stability | ≤0.05 °/h | |

Random Walk | ≤0.005 °/$\sqrt{\mathrm{h}}$ | |

Nonlinear Degree of Scale Factor | ≤20 ppm | |

Accelerometers | Bias Stability | 100 μg |

Nonlinear Degree of Scale Factor | ≤20 ppm | |

Star Sensor | Field of view | 24° |

Level of star observation | no less than +7 level | |

Data update rate | 20 Hz | |

Attitude accuracy | 5″ | |

Dynamic Range | 20°/s |

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**MDPI and ACS Style**

Wang, Q.; Cui, X.; Li, Y.; Ye, F.
Performance Enhancement of a USV INS/CNS/DVL Integration Navigation System Based on an Adaptive Information Sharing Factor Federated Filter. *Sensors* **2017**, *17*, 239.
https://doi.org/10.3390/s17020239

**AMA Style**

Wang Q, Cui X, Li Y, Ye F.
Performance Enhancement of a USV INS/CNS/DVL Integration Navigation System Based on an Adaptive Information Sharing Factor Federated Filter. *Sensors*. 2017; 17(2):239.
https://doi.org/10.3390/s17020239

**Chicago/Turabian Style**

Wang, Qiuying, Xufei Cui, Yibing Li, and Fang Ye.
2017. "Performance Enhancement of a USV INS/CNS/DVL Integration Navigation System Based on an Adaptive Information Sharing Factor Federated Filter" *Sensors* 17, no. 2: 239.
https://doi.org/10.3390/s17020239