9.2.1  Kalman Filter Model

9.2.1.1  The System Dynamics Process

The system dynamics process is the model of how the error state vector transitions over

time.  The total navigation state is defined here to mean position, velocity, and perhaps

attitude.  The rate of change of the total navigation state will then be a nonlinear function of

the state, and is assumed to be driven by white noise.  Let  y(t) denote the total navigation

state and 

y

(t) its estimate.  The error in the estimated total navigation state is defined to be

x(t)   

y

(t)   y(t).  The nonlinear differential equation representing the time rate of change of

the estimated navigation state is expanded in a Taylor s series and differenced with the

equation for the true state.  By ignoring higher order terms, a linear differential equation for

the time rate of change of the navigation error state is obtained.

It is natural to consider the behavior of the error state vector at discrete instants of time

since a computer is used to implement the Kalman filter.  Let 

x

k

 = x(t

k

)

 denote the error state

vector at time

 t

k

.  Then the discrete form of the continuous error state differential equation is

x    =   

  x   +  

  w

k

k 1

k 1

G

k 1

K  1

The matrix F

k 1

 is the state transition matrix and describes how the error state vector

changes with time.  The sequence { w

k

} is a white, zero mean Gaussian noise sequence

called the process noise or plant noise.  The expected value of the outer product of the

vector w

k

 with itself is a matrix of the second central moments of the components of the

noise vector.  This covariance matrix has the variances of the components of  w

k

 on the

diagonal and the covariances of the components on the off diagonal, and is defined to be

E[w

T

k

 w

k

] = Q

k

 where E[*] is the expectation operator.

9.2.1.2  The Measurement Process

The measurement model defines how the error state vector is related to measurements

provided by sensor(s).  Some examples of sensors are doppler velocimeter providing

line of sight velocity, radar altimeter used to form terrain based measurements of

position, such as for TERCOM, or GPS considered as a sensor giving position and

velocity or raw pseudorange and deltarange measurements.  Similar to the total

navigation state differential equation, the measurement is often a nonlinear function of

the total navigation state.  By expanding the measurement equation for the estimated

navigation state in terms of its error state and neglecting higher order terms, a linear

measurement equation is obtained for the error state vector.  The measurement

equation is written in discrete form as

z    =   

  x   +   v

k

H

k

k

k

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