In case of over damped system, that has a damping ratio is C>0.707, SF has a maximum value at the frequency
ω_{m} = 2π/t.
. Therefore, W should be evaluated at the appropriate frequency for given PSD estimation error bound based on damping ratio of the second-order linear system.
- 2.4 Estimation of the PSD and Transfer Function
In static condition, the position error dynamic model is identified as a first/second order transfer function, and the velocity error model is identified as a band-limited Gaussian white noise via non-parametric method of a PSD estimation in continuous time domain. Based on quick identification of the position error model of the second order transfer function and its damping ratio
w_{m}
using one hour recoding data analysis, we obtained
n_{d}
=25 and
T_{r}
=45 minutes that satisfy PSD estimation error boundary
and
Fig. 2
. (a) shows effectiveness of ensemble average of the PSD in frequency domain. In this case, 25 data set of
Transfer Function Parameter and Required Recoding Time: (a) First Order System SR with respect to Time Constant, (b) First Order SystemRequired Recoding Time with respect to Bias Error, (c) Second Order System SR with respect to Damping Ratio, (d) Second Order SystemRequired Recoding Time with respect to Bias Error.
position error PSD are averaged.
Fig. 2
. (b) shows fitting of the averaged PSD estimation using nonlinear least square regression method.
Fig. 2
. (c) bode diagram of the identified transfer function.
Fig. 2
. (d) shows comparison of the real experimental data and linear simulation of the identified transfer function. In fact, the GPS position error model can be approximated as a first order transfer function within low frequencies with
-20dB/dec
roll-off ratio of the magnitude as shown in
Fig. 2
. (c).
The Identified GPS position error model in NEU coordinate system is summarized in the
Table 1
.
As shown in the
Fig. 3
, the velocity measurement error PSD has relatively low roll-off ratio and flat magnitude within Nyquist frequency. Therefore it is reasonable to assume that velocity measurement noise is band limited white.
A Kalman filter can be applied to the state-space equation transformed from the error transfer function. The first order transfer function can be obtained as Eq.(11). σ and δ of the first order transfer function corresponding to each axis on NEU coordinate system is summarized in
Table 2
.
Eq.(11) can be discretized for the sampling time △
t
yielding
3. Kalman Filtering Methodology
The Kalman filtering method is applied for one measurement noise is first order Gauss-Markov process and the other measurement noise is white. Consider the discrete state and measurement equation with both colored measurement noise and white noise.
Identification of the Easting Position Error Transfer Function: (a) Ensemble Average, (b) Nonlinear Least Square Fitting with respect to PSD, (c)Bode Diagram of the Identified Transfer Function, (d) Comparison Real Data and Linear Simulation of ID
Summary of the GPS Position Error Model
Summary of the GPS Position Error Model
Summary of the Model Parameters of 1st Order Transfer Function
Summary of the Model Parameters of 1st Order Transfer Function
Ensemble Average of 25 Data Set of Velocity Error PSD.
where
w, d, c
denotes states, measurements, state of time-time correlated measurement error, process noise, white noise measurement and white noise for shaping time-correlated measurement error, respectively. These white noises are mutually un-correlated. State variables can be classified as the three groups:(i)
x_{c}
, which are disturbed by the time-correlated measurement noise, (ii)
x_{n}
, which are disturbed by white measurement noise, and (iii)
x_{s}
, which can be estimated. Each state variables have rows
n_{c}
,
n_{n}
and
n_{s}
. It is also assumed that the time-correlated measurement noise can be represented by the first order transfer function having
n_{c}
state variables. Eq.(19) shows the size of each variable vectors with appropriate subsystems.
State and measurement equations can be rewritten using Eq.(19) as
After augmenting time-correlated measurement error state into state vector, state and measurement equations can be rearranged as follows.
where,
As can be seen in Eq.(24), time-correlated measurement for augmented system model does not contain the white noise. A numerical difference method is used in this paper to avoid the 'perfect measurement condition',which is often the cause of numerical instability. In order to do this, a new measurement vector can be defined as the linear difference of the measurement
z
between
t(k-1)
and
t(k)
as follows.
Combination of the augmented state equation and the new measurement equation yields
Using Eqs(28)-(29) Eq.(30) is obtained.
Eq.(27) can be rearranged as,
Eqs.(30)-(31) yields,
Using Eq.(26) and Eq(32), expression on
u_{c}(k)
can be given by
Time delay of the update on state estimation can be avoided when the measurement can be rewritten using the inverse of the state transition matrix of the augmented state equations:
Now, new parameters
N
and
U
are introduced to combine Eqs.(35)-(36),
Eq.(22) and Eq.(37) can be simplified as,
where the new measurement noise is white, and
w_{a}(k-1)
,
w_{c}(k-1), d(k)
are not time-correlated. However, the process noise and measurement noise is correlated because both state equation and measurement equation have the term
w_{a}(k-1)
. Measurement noise covariance matrix
F
and process-measurement cross covariance matrix
S
can be defined as,
In general, Kalman filter cannot be applied for the case process noise and measurement noise is correlated. Therefore, a generalized Kalman filter is used in this paper in order to obtain optimal estimation that considers both time-correlated measurement noise and white noise. State equation and measurement equation with updates for a generalized Kalman filter are shown in Eq.(44)-(53)
[12]
.
Simulation Results: (a) Position Estimation, (b) Velocity Estimation, (c) Acceleration Estimation, (d) Kalman Gain.
Time Update
Measurement Update
4. Numerical Simulation
A simple kinematic CWNJM(Continuous White Noise Jerk Model) is considered for numerical simulation. The states are position, velocity and acceleration. It is assumed that the position measurement noise is first-order Gauss- Markov process and velocity measurement noise is white as mentioned in chapter II. Discretized linear dynamic model of CWNJM as follows.
where,
State equation for augmented system can be expressed as,
Discretized version of measurement equation can be given by the following equations
Now the state equation and measurement equation are reconstructed so that the generalized Kalman filter can be applied. Parameters for process noise and white noise are set as
q
=0.001
^{2}
,
D
=0.01
^{2}
,
V
=0.03
^{2}
.
ᴪ
=0.999925 for Northing position error model from the discretization of the GPS position error model with △
t
=0.05 in Eq.(12). Numerical simulation is performed using Eqs.(44)-(53).
Fig. 4
. (a) and (b) shows the proposed filter effectively estimates both position corrupted by time-correlated measurement error and velocity corrupted by white noise. Also, the acceleration estimation is appropriate as shown in
Fig. 4
(c). In this example, the process noise covariance is set relatively small value to show the estimate with respect to noise magnitude during small time window. It should be noted that the initial states are assumed to be known. Actually, this proposed Kalman filter de-correlates the time correlated measurement error of the position using velocity information which is not corrupted by time correlated noise, but by white noise. Moreover, using the velocity information and simple kinematics, we can get acceleration estimation without additional measurements.
5. Conclusions
In this paper, a dynamic modeling method for the velocity and position information of a single frequency stand-alone GPS receiver is described. The relationship between ensemble averages, required data recoding time and PSD estimation error that satisfy the given error bound is described. Also, analysis on transfer function parameters of a first and a second order linear system with respect to PSD error is described. A Kalman filter is proposed that consider both correlated/white measurements noise based on identified GPS error model. The proposed Kalman filter is derived from the fusion of the state augmentation approach and measurement differencing approach. The performance of the proposed Kalman filter is verified via numerical simulation. Using this filter, the time correlated position error of the GPS measurement is effectively decorrelated via its own GPS velocity information without any additional sensors. In near future, the proposed Kalman filtering method will be formulated for more general cases and applied to the precise navigation of moving vehicles.
Acknowledgements
This study was financially supported by research fund ofChungnam National University in 2010.
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