The performance of MEMSSINS/GPS integrated system degrades evidently during GPS outage due to the poor error characteristics of lowcost IMU sensors. The normal EKF is unable to estimate SINS error accurately after GPS outage owing to the large nonlinear error caused by MEMSIMU. Aiming to solve this problem, a hybrid KFUKF algorithm for realtime SINS/GPS integration is presented in this paper. The linear and nonlinear SINS error models are discussed, respectively. When GPS works well, we fuse SINS and GPS with KF with linear SINS error model using normal EKF. In the case of GPS outage, we implement Unscented Transform to predict SINS error covariance with nonlinear SINS error model until GPS signal recovers. In the simulation test that we designed, an evident accuracy improvement in attitude and velocity could be noticed compared to the normal EKF method after the GPS signal recovered.
Strapdown Inertial Navigation System (SINS) is a highly selfcontained navigation system, utilizing Inertial Measurement Units (IMU) fixed to the vehicle to determine its attitude, velocity, and position by calculating the integral of the angular rates and accelerations that IMU measures. Global Positioning System (GPS) is a satellite based radio navigation system that can provide accurate velocity and position information for a vehicle equipped with a GPS receiver [
In recent years, with the development of Microelectronical Mechanical System (MEMS), IMU can be manufactured quite small with very low costs. So MEMSSINS/GPS integrated navigation systems have been widely used in many areas, such as landvehicle navigation, Unmanned Aerial Vehicle (UAV) control, and tactical missile guidance [
The first problem is that the performance of SINS degrades very fast since MEMSIMU has low accuracy and no measurement information can be used for KF to correct the errors. And when GPS signal recovers, another problem occurs. KF or the extended KF (EKF) only works when the system is linear or slightly nonlinear so that it can be approximated by linearization. However, the SINS error model may have changed and became strongly nonlinear during GPS outages since SINS errors have been growing very fast. Even after GPS signal recovery, the KF cannot estimate the system error correctly.
The first problem has been studied for many years and several methods have been proposed to solve it. In [
In order to overcome the limitation of KF, Julier and Uhlmann proposed the Unscented Kalman Filter (UKF) in 1995 [
In this research, we aimed at improving the MEMSSINS/GPS integrated navigation system performance after GPS outage. A hybrid KFUKF algorithm for realtime MEMSSINS/GPS integration is presented in the paper. First, we will discuss two kinds of SINS error models which are the nonlinear model and the approximately linear model. Then we will present the hybrid KFUKF algorithm and its calculation progress. When GPS works well, we fuse SINS and GPS with KF as usual. In the case of GPS outage, we implement UT to predict SINS error covariance until GPS signal recovers. Finally, a simulation and an incar experiment were designed to test the algorithm and the results are compared with common KF method.
Usually, the integrated navigation system filter is designed as indirect filtering, which means the system state is selected as the error of the system parameters instead of the parameters themselves so that the model is less complex. The navigation system model is a function of SINS error and the errors of SINS are selected as the system state. They are a vector of the errors of position, attitude, velocity, gyroscopes, and accelerometers:
The attitude error is defined as the Euler angle
Denoted by attitude transition matrix, the attitude error is expressed as
KF is only able to estimate the state when the system is linear. Fortunately, when SINS integrated with GPS, the SINS errors can be corrected every filtering period. And SINS errors accumulated in this period are pretty small. So we can neglect the higher order terms of the SINS error function and the model is approximately linear.
The attitude error is approximated as
The attitude, the velocity, and the position error equations, respectively, are
And the SINS error model can be written as
So it is able to predict the system state by using a transform matrix. And KF is available when GPS works well.
Although the linear SINS model has been proved to be efficient in SINS/GPS integration, it may be not accurate enough if the SINS errors accumulate with time when the navigation system works in SINS alone mode. In this situation, we cannot neglect the nonlinear parts of the SINS error function and it is necessary to develop the nonlinear SINS error model.
Now let us review (
Without linearization, the attitude error differential equation can be derived from (
Based on (
And the SINS error model is written as
Now we cannot predict the SINS errors by transform matrix. As we presented in the next part, the UT method is used to predict the system state.
Developed in 1960s by Kalman, KF has been proved to be a powerful optimal estimation theory for linear systems. In SINS/GPS integration, KF is triggered in every GPS update epoch using the measurement information, that is, the difference of the velocity and position between SINS and GPS. Then the errors of SINS could be estimated and corrected. When GPS works well, no SINS error is accumulated and the navigation errors are bounded. Thus the linear SINS error model is accurate enough since the nonlinear parts could be ignored.
The linear SINS error transition matrix is discretized as follows [
For a loosely coupled SINS/GNSS system, the measurement equation is [
KF algorithm is committed by using such formulas [
In SINS/GPS integration, a closeloop configuration is used. In every KF epoch when the SINS errors are corrected, the state
As we mentioned above, KF is only available when the state equation is linear because it is not feasible to predict the state by transition matrix for a nonlinear system. If we neglect the higher order terms, KF will introduce errors which cannot be ignored for a strongly nonlinear system [
In UT, the state is predicted in three steps [
Second, the states are predicted with every sigma point. This step is executed by solving the error state differential equations with fourorder RungeKutta method:
Finally, the states are weighted and summarized so we obtain the predicted state
In loosely coupled SIN/GPS, the measurement equation is linear as we saw in (
In this part we discussed the architecture of the hybrid KFUKF algorithm. As shown in Figure
The architecture of the hybrid KFUKF algorithm.
If GPS information is available, SINS and GPS can be integrated by normal KF method, as shown in Figure
Filter updates procedure when GPS is functioning well.
During GPS signal blockage, the GPS information is unavailable and onestep state prediction is executed without measurement update. Now the nonlinear SINS error equation and UT are applied to predict the state and error covariance. As shown in Figure
Filter updates procedure during GPS outage.
For the sake of testing the hybrid KFUKF algorithm, we design a simulation comparing this algorithm with normal EKF algorithm. The schematic diagram of the simulation is shown in Figure
Main specifications of experiment instruments in the simulation test.
Equipment  Parameters  

GPS receiver  Velocity accuracy  0.1 m/s (1 
Position accuracy  5 m (horizontal)  
Output frequency  10 Hz  


MEMSIMU  Gyroscope  
Bias  50°/h (1  
White noise  35°/h/  
Accelerometer  
Bias  2 mg (1  
White noise  0.3 mg/  
Output frequency  200 Hz 
Schematic diagram of the simulation.
In the simulation test, the aircraft has done a series of maneuvers which are listed in Table
Aircraft maneuvers in the test.
Stage  Period  Aircraft maneuver  Parameters 

1  0~30 s  Stationary 

2  30~60 s  Acceleration 

3  60~75 s  Pitching 

4  75~95 s  Straight flight 

5  95~120 s  Pitching 

6  120~140 s  Straight flight 

7  140~165 s  Coordinate turn 

8  165~190 s  Straight flight 

9  190~215 s  Coordinate turn 

10  215~300 s  Straight flight 

Comparison of navigation error between EKF and hybrid KFUKF after GPS outage.
Outages length  Adopted approach  RMS attitude error (°)  RMS velocity error (m/s)  RMS position error (m)  

Roll  Pitch  Yaw  North  East  Down  Latitude  Longitude  Height  
50 s  EKF  0.0451  0.0566  0.2605  0.0256  0.0233  0.0331  1.1990  0.8245  3.3450 
Hybrid KFUKF  0.0438  0.0549  0.2499  0.0255  0.0233  0.0331  1.1984  0.8199  3.2310  


100 s  EKF  0.0553  0.1050  0.2630  0.0286  0.0273  0.0367  1.1832  3.7463  3.0911 
Hybrid KFUKF  0.0508  0.0767  0.1706  0.0240  0.0259  0.0339  1.1701  3.7434  3.9175  


150 s  EKF  0.0616  0.0590  0.1282  0.0363  0.0341  0.0479  1.3913  2.2053  3.1303 
Hybrid KFUKF  0.0489  0.0367  0.1140  0.0219  0.0237  0.0333  1.3876  2.1949  3.9078  


200 s  EKF  0.0969  0.0560  1.7882  0.0431  0.0544  0.0619  1.5559  3.9014  3.6364 
Hybrid KFUKF  0.0586  0.0403  1.7131  0.0231  0.0318  0.0360  1.5552  3.8998  3.5416 
Comparison of attitude error between EKF and hybrid KFUKF.
Figure
Comparison of velocity error between EKF and hybrid KFUKF.
Table
In Table
Comparison of computational cost of EKF and UKF.
EKF  UKF (for linear measurement equation)  

Step  Flops  Step  Flops  
Prediction update  Generate sigma points 


State prediction 

State prediction 


Error covariance prediction 

Error covariance prediction 


Subtotal 

Subtotal 




Measurement update  Filter gain 

Filter gain 

State update 

State update 


Error covariance update 

Error covariance update 


Subtotal 

Subtotal 




In total 

In total 





30801 

42336 
This paper presents a hybrid KFUKF algorithm for realtime MEMSSINS/GPS integration. The linear and nonlinear SINS models are discussed. The flowchart of the hybrid KFUKF algorithm is described. The SINS error is estimated and corrected with linear SINS model when GPS is functioning well while it is predicted with nonlinear SINS model by applying Unscented Transform during GPS outage. The simulation results indicate that, compared to normal EKF algorithm, the hybrid KFUKF algorithm is able to predict the SINS error more accurately if GPS suffers from longtime GPS outage and the navigation error is lower after GPS signal was regained. The results also show that the hybrid KFUKF algorithm is more efficient for attitude error prediction but the effect on position error is not evident. Compared with normal UKF, the hybrid KFUKF algorithm has low calculation amount. In this paper, the remaining problem which we have not solved is that SINS errors still grow fast during GPS outage. So, in our future work, we may combine the UKF with ANN or SVM to create a new algorithm to reduce the SINS errors during GPS outage.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the 3rd Innovation Fund of Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP) and the UAV semiphysical simulation platform research project, Chinese Science Academy (CSA).