GPS/INS组合导航：扩展卡尔曼滤波器调优与低cost传感器校准

组合导航

GPS

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As shown in Eq.(4), the acceleration due to gravity is considered in the Z-axis velocity update equation. The

nonlinearity of the aircraft kinematic equations is caused by the coordinate transformation from the body axis to the

local Cartesian navigation as shown. Rearranging the nine navigation states into position, velocity and attitude, the

continuous time update equations are more concisely written as:

cos cos ( sin cos cos sin sin ) (sin sin cos sin cos )

sin cos (cos cos sin sin sin ) ( cos sin sin sin cos )

sin cos s

x x y z

y x y z

z x

x V

y V

z V

V a a a

V a

           

 







 

 

    

 

 

    

 

 

 

 



in cos cos

sin tan cos tan

cos sin

( sin cos )sec

y z

a a g

p q r

q r

  

   

 

  

 

 



 

 

 

 

 



 



 

 

(5)

Also, as shown in Eq. (5), the position states are updated through a direct integration of the three velocity states,

since they have already been transformed into the local navigation axes.

The application of EKF consists of linearizing the system model via a first order Taylor Series expansion, and

then applying the Kalman Filter equations. This approach was introduced shortly after the introduction of the

Kalman Filter.

. The well known Kalman Filter equations within the EKF formulation are shown below:

1 1

ˆ ˆ

( , ,0)

k k k

x f x u



 



(6)

1 1

k k k k k

P A P A Q



 

 

(7)

( )

T T T

k k k k k k k k k

K P H H P H V R V

  

 

(8)

ˆ ˆ ˆ

( ( ,0))

k k k k k

x x K z h x

 

  

(9)

( )

k k k k

P I K H P



 

(10)

where Eqs. (6-7) represent the prediction equations, and Eqs. (8-10) represent the update equations. Within Eq. (6)

f represents the discretized equivalent to the INS equations provided in Eq. (5), which are shown in Eq. (11).

1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

(cos cos ( sin cos cos sin sin ) (sin sin cos sin cos )

k k k k

k x

k y

k k z

x x k k x k k k k k y k k k k k

x TsV

y TsV

z z TsV

V V Ts a a

           







  



           



 

 



 



 

     

 



 

 

1 1 1

1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1

1 1

)

(sin cos (cos cos sin sin sin ) ( cos sin sin sin cos ) )

( sin cos sin cos cos )

(

k k k

k k

y k x k k k k k y k k k k k z

z k x k k y k k z

k k

V Ts a a a

V Ts a a a g

Ts p q

           

    





  

 

          



    



 

     

    

 

1 1 1 1 1 1

1 1 1 1 1

1 1 1 1 1 1

sin tan cos tan )

( cos sin )

( sin cos )sec

k k k k k k

k k k k k

k k k k k k

Ts q r

   

  

   



     

    

     

 

 



 



 

 

 

 

 

 

(11)

Eq. (7) is the predicted covariance equation, where Q is the process noise covariance matrix that is traditionally

parameterized offline based on static sensor error analysis. The matrix as assumed for this particular study within

the baseline EKF is provided in Eq. (12).

)516.1,512.2,548.5,436.9,447.8,429.7,0.0.0(

),,,,,,0.0.0(

222222





eeeeeediagQ

rofqofpofaofaofaofdiagQ

zyx



(12)

In order to obtain the values of the Q matrix, noise characteristics of each IMU sensor were measured over a period

of time in which the test bed aircraft was at rest on the runway. Therefore, the Q matrix values in the baseline EKF

represent the measurement variance exhibited on the sensors during static conditions. The unit of the acceleration

variances is (m/s

)

while the unit of the rate gyro variances is (rad/s)

. A time-varying Q matrix that considers

dynamic measurement conditions is implemented within the tuning method described later in this study.

Since the Kalman Filter equations only apply to linear models, the system equations provided in Eq. (5) are

linearized. This is performed using a first order Taylor series expansion leading to the generation of a 9x9 Jacobian

matrix. The linearized state relationship matrix is indicated as A within Eq. (7), and is shown in expanded form

within Eq. (13).

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