1. INTRODUCTION
Development of improved navigation
techniques which utilize radiometric
( Ranging and Doppler ) data acquired from
ISASfs station and NASAfs DSN have
received considerable study in several years,
as these data types are routinely collected in
tracking, telemetry, and command operations.
A sequential data filtering strategy currently
under study is the orbit estimator, in which
most if not all of the major systematic ground
system calibration error sources are treated as
estimated parameters, along with the
spacecraft trajectory parameters. This
strategy differs from current practice, in
which the ground system calibration error
sources are represented as unestimated bias
parameters, accounted for only when
computing the error covariance of the filter
( estimator ) parameters.
This article reviews the fundamental concepts
of reduced-order filtering theory, which are
essential for sensitivity analysis and error
budget development. The theory is then
applied to the development of an error budget
for a Venus mission cruise scenario in which
enhanced orbit estimation is used to reduce
X-band Doppler and ranging data. The filter
model is described and error budgets are
given for two different strategies: X-band
Doppler only, X-band Doppler plus ranging.
For this study, the filter model is assumed to
be correct representation of the physical
world.
2. REDUCED-ORDER FILTER
In some navigation applications, it is not
practical to implement a full-order or the
optimal filter when system model, with all
major error and noise sources, is of high
order.
Use of reduced-order filter allows the analyst
to obtain estimates of key parameters of
interest, with reduced computational burden
and with moderate complexity in the filter
model. Thus, reduced-order, or , suboptimal,
filters are results of design trade-offs in
which sources of error are most critical to
over all system performance. Nevertheless,
there are reasons for not always using a
full-order optimal filter for spacecraft orbit
estimation.
Some of reasons includes : (1) there may be a
lack of adequate models for an actual
physical effect; (2) certain parameters, such
as the station location, may be held fixed in
order to define reference frame and/or length
scale; (3) if estimated, the computed
uncertainty in model parameters would be
reduced far below the level warrandted by
model accuracy.
2.1 Estimator evaluation
There are a number of error analysis
methods which can be used to evaluate
estimator ( filter ) models and predict filter
performance. Reduced-order error analysis
techniques enable an analyst to study the
effects of using incorrect a priori statistics,
data-noise/data-weight assumptions, or
process noise model on the filter design.
If the filter is optimal, then the filter and
truth models coincide. If the filter is
suboptimal, then the filter model is of equal
or lower order (i.e., reduced-order) than the
truth model and possibly represents a subset
of the states of the truth model. In practice, a
fully detailed truth model may be difficult to
develop and thus one typically evaluates a
range of ereasonablef truth models to assess
whether the filter results are especially
sensitive to a particular elements of the
filtering strategy being used. The objective is
to design a filter model to achieve the best
possible accuracy, but which is also robust,
so that its performance will not be adversely
affected by the use of slightly incorrect filter
parameters.
In a special case of reduced-order error
analysis, various systematic error sources are
treated as unmodeled parameters which are
not estimated, but whose effects are
accounted for in computing the error
covariance of the estimated parameters. In a
consider state analysis, the sensitivity of the
estimated parameter set to various unmodeled
consider parameters can be computed via
partial derivatives of the state estimate with
respect to the consider parameters set. The
filter has no knowledge about the
contribution the unmodeled parameters to the
uncertainty in the state estimate since the
modified covariance, which includes effects
from both the estimated and consider
parameters, is not fed back to the filter.
2.1 Optimal and suboptimal estimator
Restricting the discussion to the filter
measurement up-date equations, the
mathematical model presented here is the
estimator form of the measurement up-date.
Let
!
x represent the state estimate and
P represent the error covariance matrix.
Using the convention that e(-)f denotes a
pre-observation up-date value and e(+)f
denotes a post-observation up-date value, the
filter observation up-date equations for
Extended Kalman typefs estimator are given
by
State!estimate : ?xk
(+) = ?xk
(!) +
!
Kk zk ! Hxk ?xk
(!) "#
$%
(1)
Error!covariance!:!P
k
(+) = I !
!
K
k
H
xk
"#
$%
P
k
(!) (2)
Gain!matrix!!:!!!
!
K
k
= !
k
"1P
k
(")H
xk
T (3)
where zk is the observation vector defined
by the measurement model, H
xk is
observation matrix of measurement partial
derivatives, I is simply the unit matrix, and
!
k
= H
xk
P
k
(")H
xk
T +W
k
"1 is the innovation
covariance. W
k represents the weighting
matrix, the inverse of which is taken to be the
diagonal observation covariance V
k ; thus
for
i = 1,c,m observations,
Wk
!1 " Vk = diag v1,!!, vm [ ] for observation
variances v
i . The filter equations described
by Eqs. (1) through (3) can be employed
without loss of generality, since whitening
procedures can be used to statistically
decouple the measurements in the presence
of correlated observation noise and obtain a
diagonal V
k . The gain matrix K
k is used to
up-date estimates of the filter parameters as
each measurement is processed. And denote
that Eq.(2) is valid only for the optimal gain
!
K
k
The use of Eq.(2) to compute the error
covariance matrix has historically been
suspect due to finit computer word length
limitations. As a result, a utilized alternative
is the stabilized form of the up-date,
expressed as
P
k
(+) = (I ! K
k
H
xk )P
k
(!) (I ! K
k
H
xk )T + K
k
W
k
!1K
k
T
(4)
Although this form of the covariance
observation up-date is more stable
numerically than Eq.(2), it requires a greater
number of computations; however, a further
advantage is that it is valid for arbitary gain
matrices; therefore, K
k in Eq.(4) need not
be optimal.
In some cases, the observation up-date
equation may also be deficient numerically.
As a result, factorization methods have been
developed to help alleviate the numerical
deficiencies of the up-date algorithms. The
details of the factorization procedures will
not be discussed here; hoever, an important
observation from the literature and critical to
the general evaluation mode of the filter is
the observation that Eq.(4) can be written in
an equivalent form as
P
k
(+) = (I ! K
k
H
xk )P
k
(!) +"
k (K
k
! ?K
k )(K
k
! ?K
k )T
(5)
where K
k is an suboptimal gain matrix and
?K
k is the optimal gain matrix. This equation
of the error covariance observation up-date is
referred to as the suboptimal observation
up-date since it includes a correlation based
on the gain difference between the filter
evaluation run and the original estimation run.
In the general evaluation mode, the estimator
uses suboptimal gains saved in an evaluation
filter from an earlier filter which is run
purposely with what is believed to be an
incorrect model, in order to generate
suboptimal gains. It is this strategy of the
suboptimal observation up-date which will be
critical to the error described in the following
section. It is important to note that the time in
the filter evaluation mode takes the same
style as the original estimator time up-date,
except that in the presence of process noise
modeling parameters, the original estimator
stochastic time constants and process noise
(system noise) uncertainties are replaced with
evaluation mode time constants and process
noise terms.
3. OBSERVATION STRATEGY AND
THE ESTIMATOR
3.1 Observation strategy
Observation data acquisition plan is
assumed, containing several passes of
two-way Doppler and ranging data per week.
And also, the data schedule consisted of
about 6 hours tracking pass of two-way
Doppler and of about 2 hours tracking pass of
two-way range from USUDA station basis
from VE (Venus encounter) ? 30 days to
VE-10 days.
To account for observation noise, an assumed
one-sigma random measurement uncertainty
of 0.02 mm/sec was chosen for two-way
Doppler, and for two-way ranging, the
one-sigma random measurement uncertainty
was assumed to be about 5 m. It should be
noted that the data weights quoted here are
for the round trip range-rate and range,
respectively. Both data types were collected
at a rate of one point every 10 min., and the
noise variances were adjusted by an
elevation-dependent function for USUDA
station, to reduce the weight of the low
elevation data; furthermore, no data were
acquired at elevations of less than 13 deg.
3.2 The estimator
Table 1 summarizes the parameters which
make up the filter model, along with a priori
statistics, steady state uncertainties for the
Gauss-Markov parameters, and noise
densities for the random-walk parameters.
All of the parameters were treated as filter
( estimator ) parameters and grouped into
three categories: spacecraft epoch state,
spacecraft nongravitational force model, and
ground system error model. Effects of
uncertainty in the ephemeris and mass of
Venus were believed to be relatively small in
this scenario.
The simplified spacecraft nongravitational
force model was used. There were filter
parameters representing solar radiation
pressure (SRP) forces as well as small
anomalous forces due to gas leaks and
attitude control thruster misalignments, and
so on.
Table 1 Estimation parameters (Assumed)
For processing the two-way range data, the
filter model included a stochastic bias
parameter associated with each ranging pass
from the station, in order to approximate the
slowly varying, nongeometric delays in
ranging observations that are caused
principally by station delay calibration
errors and uncalibrated solar-plasma effects.
The station location covariance represents
the uncertainty in the station location.
4. THE ERROR VALUES
The purpose of developing an error budget
is to determine the contribution of individual
error sources, or groups of error sources, to
the total navigational uncertainty. In general,
an error budget is a catalog of the
contributions of the error sources which
contribute to errors in the filter estimate at a
particular point in time, whether explicitly
modeled in the filter or not. For the first
analysis, it is assumed that filter is optimal,
that estimator model is an accurate
representation of the physical world.
In order to establish an error budget, it is
necessary to compute a time history of the
filter gain matrix for the complete filter and
to subsequently use these gains in the
sensitivity calculations ( Eq.(4)) during
repeated filter evaluation mode runs, in
which only selected error sources or groups
of error sources are eturned onf in each
particular run. In this way, the individual
contributions of each error sources or group
of error sources to the total statistics
uncertainty obtained for all of the filter
parameters for given radiometric data set can
be established.
Using the reduced observation data schedule
and the filter model derived for Venus
mission scenario, orbit estimation error
statistics were computed for Doppler-only
and Doppler-plus-ranging observation data
sets. The orbit estimation were propagated to
the nominal time of Venus encounter and
expressed as dispersions in a Venus centered
aiming plane, or B-plane, coordinate system;
specifically, the one-sigma magnitude
uncertainty of the miss vector, resolved into
respective miss components B!T (parallel to
planetary equatorial plane) and B!R (normal
to planetary equatorial plane. This plane
definite Fig. 1.
In the B-plane ellipse, there are semimejor
(SMAA) axis and semiminor (SMIA) axis.
Fig. 1 Definition of B-plane ellipse.
Where ! is the orientation angle of
semimajor axis measured positive clockwise
from T to R .
Additional, the one-sigma uncertainty on
the linaerized time of flight (LTF). The LTF
defines the time from encounter ( point of
closest approach ) and specifies what the time
of flight to encounter would be if the
magnitude of the miss vector were zero. In
the case, the errors were expressed as
dispersion ellipses in the B-plane to
graphically significant groups of error
sources.
Estimate parameter Uncertainty (one-sigma)
State vector
Position element
Velocity element 1 km/s
Nongaravitational force
SRP !"#
Anoumalous accelerations
Range biase 5 m
Station location (USUDA)
Spin radious 0.05 m
Z-hight 0.05 m
Longitudu
??
1! 107 km
1!10"12 !!!km/sec2
3 !10"9 !!!deg
Ellipse
(1Ð)
SMAA
SMIA axis
axis
Á
4.2 In the case of 2-way Doppler only
With the reduced-filter, the 2-way Dopppler
data allowed determination of the B!T
component of the miss vector to about 50 km
and the B!R component of miss vector to
about 25 km, with the LTF determined
approximately 8 sec. These results
summarized in Fig.2, which gives the
magnitude of the B-plane error ellipse around
the nominal aim point for the groups of the
filter model error sources to the total
statistical uncertainty, in a root-sum-square.
The most dominant error source groups were
the random nongravitational acceleration,
followed by solar radiation pressure
coefficient uncertainty, and ground system
calibration error. For this encounter phase,
the direction of the Earth-spacecraft range is
closely aligned with semmimajor (SMMA)
axis of the B-plane error ellipse. The
Doppler data alone were able to determine
this component of the solution to only about
55 km.
Fig. 2 The error ellipse on the B-plane for
X-band 2-way Doppler only at closest
approach
4.3 In the case of 2-way Doppler plus
ranging
More one case in which both the 2-way
Doppler plus ranging data were used, the
B!T component of the miss vector was
determined to about 6 km. And the
B!R component to about 5 km, with the LTF
determined approximately 5 sec.
Similar to the results for the Doppler-only
data strategies (See 4.2), random
nongravitational accelerations were the
dominant error source group.
In additional ranging data to Doppler data,
the dispersion is reduced compare with only
Doppler observation strategy. B-plane error
ellipse are also provided ( see Fig. 3),
illustrating the contributions of the major
source groups to the total root-sum-square
error and the orientation of the ellipses in the
aiming plane. In this case, the accuracy with
which the Earth-spacecraft range component
at encounter was determined was roughly
12 km.
Fig. 3 The error ellipse on the B-plane for
X-band 2-way Doppler plus ranging at
closest approach
5. SENSITIVITY ANALYSIS
The results of the linearity assumptions used
to develop error budgets is that sensitivity
values can readily bee generated. These
values graphically illustrate the effects of
using different prescribed values of the error
source statistics on the estimation errors, with
the assumption that the reduced-order filter.
The procedure for sensitivity values
development is repeated here for
completeness:
(1) Subtract the contribution of the error
source under consideration from total
mean-square navigation error.
(2) To compute the effect of changing the
error source by a preset scale factor, multiply
its contributions to the mean-square errors by
the square of the scale factor value.
(3) Replace the original contribution to
mean-square error by the one computed in
the previous step.
(4) Take the square root of the newly
computed mean-square error to obtain the
total root-sum square navigation error.
Several cases were used to generate
sensitivity curves for the major groups of
error sources in the filter ( estimator ). Fig. 4
and Fig 5. Give the sensitivity curves for the
random nongarvitational accelerations and
illustrate the sensitivity of this error source
group to various scale factor values. Random
nongravitationnal acceleration dominated the
error budget in two data strategy cases
considered.
As seen from the figures, a quadratic
growth in the sensitivity is evident for scale
factor values ranging from 1 to 3, and a
nearly linear growth is exhibited for scale
factor values ranging from 4 to 10. On
average, for two data strategies considered,
an order of magnitude increase in the preset
scale factor resulted in about a factor of three
to six increase in the root-mean-square
estimation errors.
6. CONCLUSIONS
A sensitivity analysis was conducted for a
reduced-order filter referred to as the
enhanced orbit estimation filter. In practice,
the enhanced filter attempts to represent all or
nearly all of the principal ground system
error sources affecting radiometric data types
as filter parameters. A reduced-order filter
technique were reviewed and utilized to
perform the sensitivity analysis, and, in
particular, to develop navigation error
budgets for two different data acquisition
strategies.
Fig. 4 Sensitivity of he estimation error to
perturbation of random nongravitational
accelerations
(X-band two-way Doppler only)
Fig. 5 Sensitivity of he estimation error to
perturbation of random nongravitational
accelerations
(X-band two-way Doppler plus ranging)
REFERENCES
[1]Jordan, J.F., et al, gThe Effects of major
Error sources on Planetary Spacecraft
Navigation Accuraciesh, New York, 1971.
[2]T.Ichikawa, gStudy of spacecraft
navigation-guidance problemsh, International
symposium on Space Flight Dynamics Proc.,
Germany1997
[3]T.Ichikawa, A study of navigation
accuracy at encounter to planetary orbiter
mission, ISTS., Japan, 2002
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