Chinese Journal of Electronics
Vol.23, No.3, July 2014
An Adaptive Designing Method of Broadcast
Almanac Parameters∗
WANG Luxiao, HUANG Zhigang and ZHAO Yun
(School of Electronic and Information Engineering, Beihang University, Beijing 100191, China )
Abstract — A model of the signal acquisition time
influenced by the almanac was introduced to evaluate
the almanac model. It shows that the six parameters almanac model outperforms the three parameters almanac
model. The almanac truncation error was introduced to
describe the error between the original data and the interface specification formatted terms. The parameter sensitivity and the truncation error were analyzed to validate
the necessity to choose the proper definition of the almanac
parameters, including the effective range and the scale factor. It demonstrates the parameter definition is the key
designing element while the almanac representation is determined. Based on the research above, An Adaptive almanac designing method (AADM) of broadcast almanac
parameters was presented. Simulation results reveal that
the AADM can generate an optimum set of almanac parameters with the shortest signal acquisition time to a limited bit allocation.
Key words — Satellite navigation, Almanac, Truncation error.
I. Introduction
One purpose of the GPS navigation message is to provide almanac information of all Space vehicles (SVs) to the
user. The aim of the almanacs is to provide the user with less
precise space vehicle position and clock correction information
(relative to the basic positioning ephemeris and clock parameters) to aid in the direct acquisition of the SV’s signals[1] . With
a priori knowledge of the pseudo-range to a SV, the user can
perform direct precise (P) code and clear acquisition (C/A)
signal acquisition in a specified amount of time. The navigation message shall broadcast almanac for the complete set
of SVs in the constellation using short duration of time. It
is of great significance to design a set of compact broadcast
almanac parameters in navigation message designing. The almanac data are based on a curve fit to the predicted satellite
orbit over a six days interval. The design of almanac involves
three steps:
Step 1 Analyze the influence of the almanac parameters to the signal acquisition time; innovatively propose the
∗ Manuscript
Received July 2013; Accepted Oct. 2013.
almanac evaluation indicator; signal acquisition time.
Step 2 Perform the partial derivatives and sensitivity
analysis of satellite position to six almanac parameters variations; construct the sensitivity analysis of satellite position
error to almanac parameters truncation error; and fit the almanac parameters using the six-parameter almanac set to obtain the effective range.
Step 3 Obtain the relationship between the Range error
of truncation (RET) and the total bits of the six-parameter
almanac set. Compared to the three-parameter almanac set, a
restriction on almanac model and parameter definition can be
concluded. Then the adaptive almanac designing method to a
limited bit allocation can be obtained.
Several previous studies have published the different almanac sets. A few of previous studies have discussed parameters’ definition of ephemeris and evaluated the performance of
the existed GPS almanac sets. Refs.[2] and [3] analyzed the relations between the scale factors of GPS ephemeris parameters
and the RET. However, the RET of the ephemeris parameters
needs to be known in advance. Since the range error due to
ephemeris and clock usually does not exceed several meters,
the RET of ephemeris parameters is required to be less than
0.3m[2] . While, to almanac parameters, the rationale behind
the almanac parameters’ definition is unknown. Meanwhile, all
these studies focus entirely on performances of the different almanac data sets including the visible satellite prediction, satellites’ position and the different user algorithms. The previous
published studies have not discussed the design of almanac for
GPS. Here the author describes how to design almanac for SVs
based on the combination of orbital\clock elements and their
definitions. Note that the almanac contain orbital parameters,
clock correction parameters, SV identification and SV health,
however, only the design of orbital and clock correction parameters is discussed here, and the SV identification and SV
health are outside the scope of this paper.
The remainder of this paper is organized as follows. Section II introduces some definition and notations used in this
paper. In Section III, we describe the proposed almanac algorithms and analysis. The simulation results are presented in
Chinese Journal of Electronics
640
Section IV. While conclusions are drawn in Section V.
II. Definition and Notations
Almanac can be represented by a series of parameters with
fixed bit lengths and scale factors. Denote V is the set of n
almanac parameters, including ephemeris and clock representation parameters. The High-precision almanac (HA) in NAV,
the Midi almanac (MA) and the Reduced almanac (RA) in
CNAV are three special cases of V . S1×n is the set of scale
factors to the n parameters. E1×n is the set of effective range
to the n parameters. B1×n is the set of bit length to the
n parameters. bi is obtained from si and ei , according to
whether the parameter has the sign bit (+ or –) occupying
the Most significant bit (MSB). The relationship among the
set of scale factors, the effective range, and the bit length is
log2 ei = log 2 si + bi . Denote T as the sum of the parameter
bits of the set V , and whose Range error due to truncation
(RET) is r. Given a set of source almanac parameters p ∈ V ,
almanac parameters designing is to obtain a set of destination
parameters D(S, E, B).
√
pH A/M A = [e, δi , Ω̇, A, Ω0 , ω, M0 , af0 , af1 ]T
pRA = [δA , Ω0 , Φ0 ]T
Because of the limited allocated space, truncation error
occurs when parameters are converted from the numerical values to the binary type navigation message. To a particular
parameter, the broadcast almanac parameter truncation error
is
l p m
i
(1)
δpi = pi − S · 2Si
2 i
means to truncate to integer. Parameters truncation error
vector can be expressed as δp.
The broadcast GPS almanac parameters for a given satellite are defined in Refs.[4–6]. The user algorithm is essentially
the same as that used for computing the precise ephemeris
from the parameters of MA and RA.
III. The proposed algorithms and analysis
1. The signal acquisition time model impacted by
almanac
The measured pseudo range for a given satellite to the re. The approximate pseudo range for each satellite
ceiver is ρ̃eph
i
is obtained by utilizing the almanac data when a warm start
occurs.
q
= (Xu − Xieph )2 + (Yu − Yieph )2 + (Zu − Zieph )2
ρ̃eph
i
(2)
−cδteph
Si + cδtAi + cδtu , i = 1, · · · , 4
=
ρ̃alm
i
p
(Xu − Xialm )2 + (Yu − Yialm )2 + (Zu − Zialm )2
−cδtalm
Si + cδtAi + cδtu , i = 1, · · · , 4
(3)
The range between the receiver and satellite can
be expressed to the Taylor expansion about the base
bialm ), which is the real almanac pobialm , Ybialm , Z
point (X
sition deduced from almanac parameters the user reis the pseudo range computed by
ceived. Denote ρ̂alm
i
2014
the truncated almanac parameters, which is equal to
q
(Xu − X̂ialm )2 + (Yu − Ŷialm )2 + (Zu − Ẑialm )2 . The pseudo
range can be written as follows[7] , while the high order minor
term is neglected.
(Xu − X̂ialm ) · (Xialm − X̂ialm )
ρ̂alm
i
alm
alm
(Yu − Ŷi ) · (Yi
− Ŷialm )
−
alm
ρ̂i
(Zu − Ẑialm ) · (Zialm − Ẑialm )
−
ρ̂alm
i
alm
−c(δtalm
−
δ
t̂
) + cδtAi + cδtu
i
i
= ρ̂alm
−
ρ̃alm
i
i
(4)
(Xialm , Yialm , Zialm ) denotes the satellites’ vector position computed by the almanac data without truncation. Subtracting
) from the
a reduced-precision estimate of the range (ρ̂alm
i
ephemeris forecasted range:
= ρ̃eph
− ρ̃alm
= (f
δρalm
i
i
i
g
h
−1) · e1
(5)
e1 is the error vector (δXi , δYi , δZi , δcTi )T , which denotes the
difference between ephemeris and almanac parameters. δXi =
X̃ieph − X̃ialm , δYi = Ỹieph − Ỹialm , δZi = Z̃ieph − Z̃ialm ,
δcTi = c(T̃ieph − T̃ialm ). (f, g, h) is the direction vector.
,
f = −(Xu − X̃ialm )/ρ̂alm
i
g = −(Yu − Ŷialm )/ρ̂alm
,
i
h = −(Zu − Ẑialm )/ρ̂alm
i
The code phase delay caused by satellite position error can
be given as described in Eq.(6). The constant c is the propagation speed of light.
/(1/2 · 10−3 /1023 · c)
δτ = δρalm
i
(6)
Similarly, the scalar velocity error can be written as
δνialm = (f
g
h) · e2
(7)
The Doppler frequency shift ambiguity of a single satellite
can be written as:
δfD = fT /(c · δνialm )
(8)
Then, signal acquisition time ambiguity of a single satellite
is presented as Eq.(9).
Tiacq = δτ ∗ δfD ∗ dwelling time
(9)
Therefore, the quality of almanac becomes the key factor
of the signal acquisition time. Based on analyzing the signal
acquisition time of the almanac, it is useful for seeking a tradeoff between occupied satellite’s memory space and the signal
acquisition time performance. It is indispensable for navigation
message design and convenient for users’ utilization.
2. Almanac representation model and accuracy
analysis
The final GPS almanac representation evolved through numerous trade studies. There were many criteria to be considered. The representation model for the SVs’ clock corrections
in the almanacs is chosen as an one order linear function and
that for the SVs’ ephemeris in the almanacs is the Keplerian
model. Keplerian representation of the orbit provides advantages in all criteria except for user computation time and possibly, user storage requirements. A definite advantage was the
An Adaptive Designing Method of Broadcast Almanac Parameters
fact that the Keplerian representation has a physical meaning. This made it relatively easy to size the data words, even
for off nominal orbits. However, perturbations about the Keplerian orbit require additional parameters. To the same duty
cycle, the accuracy of different almanac sets was investigated
to decide which candidate provided a more graceful degradation.
Assuming p is the parameter vector with i elements. Defining the instantaneous truncation as a vector: δp =
[δpeph δpclk ], δp = p∗ − pT . Where p∗ and pT are vectors
of ephemeris and clock parameters without and with truncation respectively. To every parameter, the truncation error
obeys an uniform distribution between -SF and SF. J, K, L
and M are nonlinear functions defined by the satellite position algorithms in GPS ICD-200E. Now consider sensitivity[8]
to parameter variations:
2
3
3 2
δxsv (t)
∂J(p, toa , t)/∂p1 · · · ∂J(p, toa , t)/∂pi
6 δysv (t) 7 6 ∂K(p, toa , t)/∂p1 · · · ∂K(p, toa , t)/∂pi 7
6
7
7 6
6 δz (t) 7 6 ∂L(p, t , t)/∂p
· · · ∂L(p, toa , t)/∂pi 7
oa
1
6 sv 7 6
7
6
7
7 6
6 δΔt(t) 7 = 6 ∂M (p, toa , t)/∂p1 · · · ∂M (p, toa , t)/∂pi 7
6
7
7 6 2
6 δ ẋsv (t) 7 6 ∂ J(p, toa , t)/∂t∂p1 · · · ∂ 2 J(p, toa , t)/∂t∂pi 7
6
7
7 6 2
4 δ ẏsv (t) 5 4 ∂ K(p, toa , t)/∂t∂p1 · · · ∂ 2 K(p, toa , t)/∂t∂pi 5
δ żsv (t)
∂ 2 L(p, toa , t)/∂t∂p1 · · · ∂ 2 L(p, toa , t)/∂t∂pi
· [δp1
···
δpi ]T
In a more compact form, we can write
[e1 ; e2 ] = A(p, toa , t)δp + o(δp)
(10)
Where A is the 7∗i sensitivity matrix, that is the matrix
which can be computed either analytically or numerically by
partial differentiation of J, K, L and M . The last equation is a
linearized expression directly relating parameter and position
variation.
To make it simple, the truncation error distribution density function is written as:
8
< 1 , −S < δp < S
i
i
i
(11)
f (δpi ) = 2 ∗ Si
: 0, else
Choose the largest truncation error SF to evaluate the worstcase influence to range. The user’s range to the satellite can
be expanded in a Taylor’s series at any given time. Then the
general sensitivity of the range error to variations is:
δr = r − ˛r ∗
˛
∂r ˛˛
∂r ˛˛
2
=
δp
+
O[δp
]
≈
δp
∂p ˛p ∗
∂p ˛p ∗
„
∂r ∂ysv
∂r ∂zsv
∂r
∂xsv
+
+
=
·
·
·
∂xsv
∂p
∂ysv ∂p
∂zsv
∂p
«˛
∂ΔtSV ˛˛
∂r
·
+
˛ ∗ · δp
∂ΔtSV
∂p
641
set of almanac parameters with specific definition to any bit
allocation. Three factors including accuracy, duty cycle and
memory were considered in order of importance during the almanac designing procedure. ICD tells that URE (User range
error) provided by a given almanac during each of the operational intervals is (1 sigma): normal operation interval (70
hours after the first valid transmission) 900; short-term extended 900-3600m; long-term extended 3600-300000m. Since
almanac is updated every day, we should find out the actual
accuracy and RET of the broadcast HA during the duty cycle at first. Then we present the realistic accuracy demand
and a heuristic that employs the concept of ephemeris parameter definition method to design the almanac parameter definition. That is: the scale factors of the almanac parameters’
LSBs (Least significant bit) can be determined through the
sensitivity analysis of the range error due to truncation. A
set of almanac definition can be obtained with a definite
RET.
Assuming the contribution to range error is uniform over
the components, the scale factor of every parameter is:
,
˛ ˛ ˛˛
˛ ∂r ˛ ˛
(13)
δpi < R N ∗ ˛˛ ˛˛ ˛
∂p p ∗ ˛
R indicates the requirement of RET. N is the number of
the ephemeris parameters. The inequality is used to determine
the most significant bit of δpi . The LSB scale factor of the
components of pi is chosen to be one bit larger. Fractions of
bits are dropped. Based on this method, an optimum almanac
definition can be obtained with a determined RET.
In order to improve the quality of finding an optimum set
of almanac parameters, an AADM to a limited bit allocation
is proposed in Table 1. To any bit allocation, the method can
draw out a set of concrete almanac parameters and give out
the corresponding RET.
Table 1. AADM description
Algorithm AADM to a limited bit allocation
Input Bit allocation (BA) for one SV’ almanac
Output A set of almanac parameters and the definition
1: With the almanac representations, obtain the variation trend
and the effective range of each parameter.
2: To the constant positive\negative parameter, choose the
mean value as the reference value.
3: for RET = 100∗i, i ∈ [1, 3000] Correlate the RET and bit
allocation with the LSB’s definition algorithm.
4: end for
(12)
p
3. AADM to a limited bit allocation
During the procedure of navigation message designing, the
bit space for almanac is unknown initially. While other styles
of parameters occupy a certain account of space, the rest of
bit space is limited for almanac. The adaptive almanac designing method aims to find a solution which can obtain a
5: while BA lies in interval [a, b] do choose a and the corresponding parameters’ definition as the final almanac definition.
IV. Simulation Results
1. The signal acquisition time model
Considering the almanac inheritance (legacy), two representations or variations thereof were investigated. Candidate
representations were:
(1) six basic Keplerian parameters plus one perturbation
642
Chinese Journal of Electronics
parameters and two clock corrections in HA and MA
(2) three Keplerian parameters in RA
In the first set of experiments, the variations of two impact
factors including the different types of almanac and age of data
versus the code phase range as well as the Doppler frequency
are tested. The results are presented from Fig.1 to Fig.2, in
2014
which the left subfigure is denoted as (a), the middle subfigure was denoted as (b), while the right subfigure is denoted as
(c). Firstly, the range error variations of X, Y , Z direction in
ECEF coordination and clock error are tested, and then the
code phase is evaluated. The tested age of data ranges from 0
hour to 168 hours without almanac updated.
Fig. 1. Code phase search bins vs. age of data
Fig. 2. Doppler frequency bins vs. age of data
Fig.1(a) and (b) show that with less bit allocation than HA,
MA double code phase search bins. Fig.2 shows that Doppler
frequency search bins have nothing to do with the bit allocation but the representation model. TTFF (Time-to-first-fix) is
the multiple of code phase bins, Doppler frequency bins and
the dwelling time for the receiver in a search bin. In the same
environment, the three-parameter almanac set has a TTFF exaggeration of more than 1000 times compared to the HA. The
MA has a TTFF exaggeration of 10 times compared to the HA,
which is purely due to the different parameter definition. Then,
the results are as follows: when URE is less than 6km, only
the code phase search bins will be degraded over time, the
Doppler frequency search bins keep the same; when URE is
larger than 50km, both of the code phase and the Doppler
frequency search bins will degrade over time.
2. Sensitivity analysis of almanac parameters
Prior to deriving sensitivity analysis algorithms, it is instructive to examine the general sensitivity of satellite position error to variations and truncation errors in the broadcast
ephemeris parameters. Using the sensitivity matrix and the
standard deviations of the daily parameter variations, it is
possible to determine 1 sigma satellite position error due to
nominal daily variations on the individual almanac parame-
ters. Fig.3 shows the satellite position error sensitivity results
for the first almanac parameter set. The toa of almanac corresponds to the beginning of the x axis in the plot. The values
computed and shown go from t = toa , to t = toa + 24h. The 1σ
satellite position error contribution due to the 1σ daily variation of each individual parameter is shown in the separate
traces in the figure.
As discussed shortly, every almanac parameter is directly
relevant to the pseudo range. The Ω dot has the least influence to the pseudo range. The result, especially the uppermost
trace in the figure, shows that satellite position error using previously validated almanac parameters is affected by variations
in the first order clock correction polynomial coefficient more
than by any other parameter. The individual 1σ errors due to
M 0 and ω variations are not directly plotted in the figure, because daily variations in these parameters are not independent,
but rather exhibit a perfect negative correlation. This correlation is due to the fact that the GPS orbits are nearly circular,
making it difficult to explicitly distinguish between M 0 and
ω in the orbit determination process[8] . With nearly circular
orbits, the argument of latitude, M 0 + ω is more convenient
to utilize because it is always well defined, and the correlation
issues between the two parameters are avoided. This conclus-
An Adaptive Designing Method of Broadcast Almanac Parameters
Fig. 3. Sensitivity of satellite position to daily
almanac parameter variations
Fig. 4. Sensitivity of range error to daily HA\MA parameter truncation error
variations
ion can be utilized to simplify the almanac model such as the
reduced almanac.
Meanwhile, utilizing the sensitivity matrix, the daily parameter values and the two sets of almanac, it is possible to
determine 1σ satellite position error variations due to nominal
daily values on the HA\MA parameter truncation.
As we can see from Fig.4, the clock correction parameters
caused larger error than the orbital parameters. It is advisable
to space more bits to the clock correction parameters. The influences of orbital parameters’ truncation to the range are on
a close level, which demonstrated the presented heuristic to
utilize the LSB definition for ephemeris to almanac is reasonable. It can be seen that RET is different for HA and
MA. To HA, the RET is about a tenth of the accuracy of
the almanac representations; while to MA, the RET is about
one percent of the accuracy of the almanac representations.
Then the conclusion is that the corresponding parameters’
definition and the RET are the bottleneck of the almanac
performance.
3. AADM to a limited bit allocation
According to the statistics data for the real-time GPS
broadcast ephemeris, the change trend and the effective range
of each almanac parameter are obtained. Then there are several almanac parameters’ definition can be improved as described in Table 2, while the other parameters remain unchanged.
Table 2. Almanac parameter changes
Original
Terms
√
A
δi
e
Ω dot
New designed
EF
Terms
√
δ A
√
δi
±2−8
Note
±2−4
iref = 0.3π
δe
±2−8
eref = 0.005
±2−28
(Ω dot)ref = −2.6 × 10−9 π/s
√
643
Aref = 5153.611m
The set with the changed almanac parameters is defined
as the new design set. The scale factors of the new design set
can also be determined through the sensitivity analysis of the
RET. Fig.5(a) reveals the relationship between the total bits
of the first almanac candidate and the RET when the age of
almanac data arrives to 86400s. These results in Fig.5(a) show
that the total bits is greatly influenced by the RET when the
RET is less than 1km. This demonstrated the results that HA
with 182 bits would outperform MA with 127 bits to the extent of several kilometers. From Fig.5(b), it can be seen the
total bits of almanac decreased to 20 bits as compared to the
left chart.
Fig. 5. Total bits vs. RET
5. Conclusions
To enhance the TTFF of GNSS navigation message, one
strategy is to improve the almanac accuracy and minimize the
almanac data space. Comparisons among the existed three sets
of almanac show that to the first 9-element almanac representation, RET is the bottleneck of URE; while to the 3-element
almanac model, the exploration error due to the simple model
is the main URE source. Meanwhile, to HA and MA, the TTFF
is proportionate to the increased URE; while to RA, the TTFF
has connection with both of the URE and the velocity error, which corresponds to the code phase search bins and the
Doppler frequency search bins respectively. The quantization
evaluation of range errors due to the short data field lengths
verified that MA introduced a larger error than HA. Through
analyzing the results, the proposed adaptive almanac designing method can reasonably obtain a set of reasonable almanac
parameters and assign almanac parameter which will occupy
less space without the performance deteriorates. Therefore this
algorithm is an attractive choice in designing navigation mes-
Chinese Journal of Electronics
644
sage parameter format.
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2014
WANG Luxiao
is in pursuit
of Ph.D. degree in Beihang University,
China. She carries out research on navigation message optimization, designing and
performance analysis. (Email: Suellen@ee.
buaa.edu.cn)
HUANG Zhigang received Ph.D.
degree in electronics from Beihang University, Beijing, China. Now he is a
professor of Electronic and Information Engineer School in Beihang University. His research interests include satellite navigation technology and application, aeronautics theory and method, etc.
(Email:[email protected])
ZHAO Yun
is a lecturer in the
School of Electronic and Information Engineering at Beihang University. Her main
research interests lie in multi-constellation
GNSS signal simulator and receiver-based
GNSS multipath mitigation. (Email: [email protected])