Global navigation sattelite system
Download 0.87 Mb. Pdf ko'rish
|
- Bu sahifa navigatsiya:
- A 3.2.3 the Instance of calculation of co-ordinates and components of velocity vector SV according to system GLONASS almanac
|
Т is a time from the epoch 5 January 1900 (GMT) to time reference t e of
ephemeris parameters (in Julian centuries of 36525 ephemeris days); 27392.375 is a number of days from the epoch 5 January 1900 to the epoch 0 January 1975 (Moscow Time or MT) taking into account the three-hour offset between MT and GMT when re-computing t e into GMT; Σ days
- sum of days from the epoch at 00 hours MT on 0 January 1975 to the epoch at 00 hours MT of current date within which the instant t e is.
Coordinates X(t e ), Y(t
e ), Z(t
e ) and velocity vector components V x (t
) , V y (t e ),
V z (t e ) are initial conditions for integration of the system (1); they are taken from a navigation message and then re-computed from Greenwich coordinate system (PZ- 90.02) to an absolute coordinate system OX a Y
Z a using the following formulae: X o (t е ) = x(t е ) cosS(t
е ) - y(t
е ) sinS(t
е ),
Y o (t е ) = x(t
е ) sinS(t
е ) + y(t
е ) cosS(t
е ),
Z o (t е ) = z(t
е ),
Vx o (t е ) = Vx(t
е ) cosS(t
е ) - Vy(t
е ) sinS(t
е ) - ω
з Y o (t е ), Vy o (t е ) = Vx(t
е ) sinS(t
е ) + Vy(t
е ) cosS(t
е ) + ω
з X o (t е ), Vz o (t е ) = Vz(t
е ),
S(t е ) = s + ω з ( t
е – 3
h )
Where:
ω E - Earth's rotation rate (0.7292115 ∗ 10
-4 s -1 ); Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
e is
specified.
After integration received in an absolute system of units of co-ordinates OX 0 Y 0 Z 0 of co-ordinate X o (t i ), Y o (t i ), Z o (t i ) and components of velocity vector of space vehicle Vx o (t i ), Vy o (t i ), Vz o (t i ) can be translated in an earth-referenced Greenwich geocentric conception of co-ordinates ПЗ-90-02 Oxyz under formulas:
x(t
i ) = X
o (t i ) cosS(t i ) + Y o (t i ) sinS(t i ), y(t i ) = -X o (t i ) sinS(t i ) + Y o (t i ) cosS(t i ), z(t i ) = Z o (t i ), Vx(t
i ) = Vx
o (t i ) cosS(t i ) + Vy o (t i ) sinS(t i ) + ω з Y(t
i ),
Vy(t i ) =-Vx o (t i ) sinS(t i ) + Vy o (t i ) cosS(t i ) - ω з X(t
i ),
Vz(t i ) = Vz o (t i ), S(t
i ) = s + ω з ( t
i – 3
h ).
Notes: Accelerations Jx a s, Jx
a m, Jy
a s, Jy
a m, Jz
a s, Jz
a m in equation (1) can be either adopted constant and computed once per an instant te using the formulae (2) or excluded from (1) and then added the results of integration of corrections:
Δ
a m + JX
a s ) *
τ 2 /2, Δ Y = ( Jy a m + Jy
a s ) *
τ 2 /2, Δ Z = ( Jz a m
Jz a s ) τ 2 /2 , Δ Vx = ( JX a m + JX
a s ) *
τ ,
Δ Vy = ( Jy a m + Jy
a s ) *
τ ,
Δ Vz =( Jz
a m + Jz a s ) τ ,
where τ = t i
e .
2. Directive cosines ξ k , η k , ζ k can be computed using the formulae (3) or taken from an external source.
The origin of Greenwich (right-hand) coordinate system is in the center of Earth's body; OZ-axis is directed to northern pole along Earth's rotation axis; OX- axis is directed to the point of intersection of Greenwich meridian and equatorial plane.
If to exclude lunar-solar accelerations when integrating system (1) and take into them account by addition of them to the results of integration
Δ X = ( JX a m + JX a s ) *
τ 2 /2, Δ Y = ( Jy a m + Jy
a s ) *
τ 2 /2, Δ Z = ( Jz a m
Jz a s ) τ 2 /2 , Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
Vx = ( JX a m + JX a s ) *
τ ,
Δ Vy = ( Jy a m + Jy
a s ) *
τ ,
Δ Vz = ( Jz a m
Jz a s ) τ ,
then increasing, due to this, of ephemeris extrapolation errors does not exceed 10%. Here (JX a m + JX a s), (Jy
a m + Jy
a s) , (Jz
a m + Jz a s) are projection of lunar-solar accelerations to axes of OX a Y a Z a system at instant t e to which ephemeris parameters are referenced, they are computed using the formulae (2).
To calculate ephemeris parameters at instant t j the projections of lunar-solar accelerations to axes of Greenwich geocentric coordinate system X ″(t
e ), Y
″(t e ), Z ″(t e ) can be used; they are transmitted within navigation message. Prior to the integration of the system (1) these accelerations should be transformed into an absolute Cartesian geocentric coordinate system OX a Y a Z a using the following formulae:
(JX a m + JX
a s) = X
″(t e ) ∗ cos S - Y ″(t e
∗ sin S , (Jy a
a s) = X
″(t e ) ∗ sin S + Y ″(t e
∗ cos S , (Jz a
+ Jz a s) = Z ″(t
e )
An accuracy of ephemeris data multiplication is given in the following table: Step of integration, minutes Interval of integration
5 minutes 10 minutes 15 minutes 1 0.42
0.56 0.77
2.5 0.42
0.56 0.77
5 0.45
0.61 0.83
7.5 - - 1.21
Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
Re-calculation of ephemeris within the interval of measurement is performed using technique of numerical integration of differential equations that describe motion of the satellites in coordinate system PZ -90.02: dx dt Vx / = dy dt Vy
/ =
dz dt Vz / = dV dt x J a r x z r x V x x r e y / && = − − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + + + μ μ ω ω 3 3 2 1 5 2 0 2 2 5 2 2 2
dt y J a r y z r y V y y r e x / && = − − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + + + μ μ ω ω 3 3 2 1 5 2 0 2 2 5 2 2 2
dt z J a r z z r z z r e / && = − − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + μ μ 3 3 2 1 5 0 2 2 5 2 2
where:
r x y z = + + 2 2 2 ;
μ= 398600.44*109 m3 / s2 - Gravitational constant; ae= 6 378 136 m- Semi-major axis of Earth ; J02= 1082625.7 10 –9 – Second zonal harmonic of the geopotential; ω= 7.292115 10 -5 radian/s - Earth rotation rate. Initial conditions of integration of reduced equations set are co-ordinates and components of velocity vector of n th SV x n (t b ), y n (t b ), z n (t b ), x’ n (t b ) = Vx, y’ n (t b ) = Vy, z’ n (t b ) = Vz.
Accelerations due to lunar-solar gravitational perturbation && ( ),&& ( ),&& ( ) x t
y t z t
n b n b n b are constant in the integration interval ±15 minutes.
A.3.1.3. Transformation of GLONASS-M current data information into common form
Satellite navigation message contains current data information in N T parameter. It could be transformed into the common form by the following algorithm: 1). Current year number J in the four-year interval is calculated: If 1 ≤ N
T
≤ 366; J = 1; Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
≤ N T
≤ 731; J = 2; If 732
≤ N T
≤ 1096; J = 3;
If 1097 ≤ N
T
≤ 1461; J = 4. 2). Current year in common form is calculated by the following formula: Y = 1996 + 4*(N 4 –1) + (
J – 1
stored in user equipment ROM. The table interrelates NT parameter and common form dates.
For example, meaning N Т = 839 then according to algorithm point 1 we discover meaning J, it will be equal 3. Further from a navigational frame we take meaning N 4 , we will accept it equally 2.
And now we compute a value Y - current year in the conventional form: Y = 1996 + 4 * ( 2 – 1 ) + ( 3 – 1 ) = 1006 + 4 * 1 + 2 = 1996 + 4 + 2 = 2002 Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
A.3.2 Algorithm of calculation of satellite motion parameters using almanac
The algorithm is used when selecting optimal constellation, calculating satellite position to provide acquisition and tracking the selected satellite. The algorithm allows calculating the coordinates and velocity vector components of a satellite at instant of acquisition t i
A.3.2.1 Almanac data
GLONASS almanac contains orbital parameters specified for each satellite at an instant t λ j A list of the parameters for each satellite is as indicated below:
N Aj
- Calendar number of a day within four-year interval starting from latest leap year; almanac data for j-satellite are referenced to N Aj ; λ j
- Greenwich longitude of ascending node of orbit of j-satellite at instant t λ j
(in radians); t λ j
- An instant of a first ascending node passage of j-satellite within N Aj –
day (in seconds); Δ i j
- Correction to the mean value of inclination of j-satellite at instant t λ j (mean value of inclination is equal to 63 °); Δ Tj - Correction to the mean value of Draconian period of j-satellite at instant t λ j
seconds); Δ Т ′ j
- Rate of change of orbital period for j-satellite; ε j - Eccentricity of j-satellite orbit at instant t λ j
ω j
- Argument of perigee of j-satellite orbit at instant t λ j (in radians). λ - Index of an accessory of parameters АС By time of passing of an ascending node of an orbit t λ
, and j - number SV (j = 1......, 24). Further the index j is omitted. Average values of obliquity of orbital plane SV GLONASS system i ср a period of revolution Т
. Make 63 ° and 43200 with, accordingly. The gang of orbit parameters for everyone SV is set in the Greenwich geocentric conception of co-ordinates OXYZ "frozen" during the moment t
. The system beginning is combined with a centre of mass of the Earth. The Z- axis is routed to a mean northern pole for a mean epoch of 1900-1905 of, shaft OX lies in a plane of terrestrial equator of an epoch of 1900-1905 of, plane XOZ is thus parallel to mean Greenwich meridian and determines a rule of a zero-mark of the count system of longitudes, shaft OY adds system to the right.
A.3.2.2 Algorithm of calculation Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
Calculation of satellite and velocity vector components at instant t i (MT) of a day N 0 within four-year interval, and in absolute geocentric coordinate system OX a Y a Z a (which origin and Z-axis coincide with origin and Z-axis of OXYZ system, offset between XOZ-plane and X a OZ
is equal to true sidereal time, and OY a - axis completes the system to the right-handed one) is performed in two steps. At the first step the time t k of ascending node passage at k-orbital period and corresponding longitude λ k are calculated using the almanac parameters Δ Т, Δ Т' and
λ . Here the specified instant t i is within the following interval: (t i - t
k < T
mean +
Δ Т).
Other parameters are assumed constant and equal to the corresponding parameters of almanac. Then osculating elements are re-computed from the instant t k to the instant t i
using analytic formulae and taking into account secular and periodic perturbations of the orbital elements caused by second zonal harmonic C 20 . Then the osculating elements at instant t i are transformed into kinematic parameters, as indicated below.
semi-major axis "a" of orbit is calculated using technique of successive approximations:
3 2 ) 1 ( ) 1 ( 2 μ π ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = + + n оск n T а , ( ) ( ) ( ) 1 2 3 2 2 3 2 2 2 ) ( 20 ) 1 ( 1 cos 1 cos 1 1 sin 2 5 2 2 3 1 − + ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ − ⋅ + + ⋅ + − ⋅ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ + ⋅ = e e e e i p a С T T n e др n оск υ ω , p (n) =
⋅ ) (n a ( 1 – e
2 ) , n = 0, 1, 2,…,
where
υ = -
ω , i= i
ср + Δ i and Т др = Т ср +
Δ Т .
An initial approximation ) 0 ( a = 3 2 2 μ π ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ др T . The process of approximation ends when fulfilling the following condition: км a a n n 3 ) ( ) 1 ( 10 − + < − . Usually it is enough to make three iterations for it.
The time t k of ascending node passage on k-orbital period (within which the instant t i is located) and respective longitude λ k are calculated: t λκ = [ λκ t ] mod 86400 , , ' 2 W T W T t t др ⋅ Δ + ⋅ + = λ λκ W k = др T t * , W – hole part W k ,
Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
( 86400
0 A i N N t t t − ⋅ + − = ∗ λ , ( ) ( ) 2 ' ' W T W T др з k ⋅ Δ + ⋅ ⋅ − Ω + = ω λ λ , ( ) 2 2 2 20 1 cos 2 3 ' − − ⋅ ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ ⋅ = Ω e i a a n C e ,
T n π 2 = ,
S k + = Ω λ , ) 10800
( 0 − ⋅ + = k З t S S λ ω .
Where: С 20 – Second zonal harmonic of geopotential (-1082.63 * 10 -6 );
a е – Equatorial radius of Earth (6378.136 km); S 0 – True sidereal time at Greenwich midnight on day N 0 , within which the instant t i is located; ω з – Earth's rotation rate (0.7392115 * 10 -4 s -1 ); μ - Gravitational constant (398600.44 km 3 / s
2 ).
3) Constant parameters of integration at the instant t λ k
a a m) ( δ = 2J 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛
a е ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − i 2 sin 2 3 1 ( ) λ λ sin
cos ⋅ + ⋅ h l + J
2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ a a е ⋅
2 sin
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ + ⋅ + + ⋅ − ⋅ λ λ λ λ λ 3 sin 2 7 3 cos
2 7 2 cos cos
2 1 sin 2 1
l l h , − ⎥⎦ ⎤ ⎢⎣ ⎡ ⋅ − ⋅ + + ⋅ ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = λ λ λ τ δ 2 cos
2 3 2 sin 2 3 sin sin
2 3 1 2 2 ) ( h l n l i a a J h е m
+ ⎥⎦ ⎤ ⎢⎣ ⎡ ⋅ + ⋅ + ⋅ − ⋅ + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ λ λ λ λ λ 2 cos 4 cos
2 17 4 sin 2 17 2 sin
5 3 sin 3 7 sin sin 4 1 2 2
h l l i a a J е
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ − ⋅ ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ τ 2 sin 2 1 cos 2 2 l n l i a a J е , ) (m l δ =J 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ a a е ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − i 2 sin 2 3 1 − ⎥⎦ ⎤ ⎢⎣ ⎡ ⋅ + ⋅ + + ⋅ ⋅ − λ λ λ τ 2 sin 2 3 2 cos
2 3 cos h l n h
+ ⎥⎦ ⎤ ⎢⎣ ⎡ ⋅ + ⋅ − ⋅ − ⋅ − − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ λ λ λ λ λ 2 cos 4 sin 2 17 4 cos 2 17 2 sin
5 3 cos 3 7 cos sin 4 1 2 2
h l h i a a J е
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ + ⋅ ⋅ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ τ 2 sin
2 1 cos 2 2
n h i a a J е
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ + ⋅ − − ⋅ − ⋅ + ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = Ω λ λ λ λ λ τ δ 3 cos 6 7 3 sin 6 7 2 sin
2 1 cos 2 5 sin 2 7 cos 2 ) ( h l h l n i a a J е m , ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ + ⋅ + + ⋅ + ⋅ − ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = λ λ λ λ λ δ 3 sin 3 7 3 cos 3 7 2 cos
sin cos
cos sin
2 1 2 ) (
l h l i i a a J i e m ,
Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ − ⋅ + ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = i a a J h l n i a a J e e m 2 2 2 2 ) ( sin
3 cos
4 7 sin 4 7 sin 2 3 1 2 λ λ τ λ δ ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⋅ + ⋅ − ⋅ − ⋅ − i a a J l h l h e 2 2 cos 2 sin 4 1 3 sin 72 49 3 cos
72 49 sin 24 7 cos 24 7 λ λ λ λ λ
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ + ⋅ − − ⋅ − ⋅ + ⋅ λ λ λ λ λ τ 3 cos
6 7 3 sin 6 7 2 sin
2 1 cos 2 5 sin 2 7
l h l n , (1)
where: λ = M +
ω , M = E – e sinE , tg 2
= 2 1 1 υ tg e e + − , h = e sin ω , l= e cos ω , m=1, τ = 0, J = - 2 3
20 , a = a (n) (1).
4) Corrections to orbital elements at instant t i due to effect of C 20 are computed:
)
( ) 2 ( a a a δ δ δ − = , ) 1 ( ) 2 ( Ω − Ω = Ω δ δ δ , ) 1 ( ) 2 ( h h h δ δ δ − = , δ i = δ i ) 2 ( - δ i ) 1 ( , ) 1 ( ) 2 ( l l l δ δ δ − = , ) 1 ( ) 2 ( * λ δ λ δ δλ − = .
Parameters ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( , , , , λ δ δ δ δ δ δ
i l h a Ω are computed for τ = t i - t λ k and m =2 using the formulae (1), where τ ω λ ⋅ + + =
M .
Perturbing orbital elements of satellites at instant t i are computed: h h h i δ + = ,
l l i δ + = , 2 2 i i i l h + = ε , ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛
i l h arctg , and ε i
≠ 0 and
0 ≠
l , 0 , and ε i =0 , = i ω
2 π , and ε i
≠ 0 and h
i = ε i , - 2 π , if ε i
≠ 0 and h
i = -
ε i ,
a i = a + a δ , i i = i + δ i ,
Ω + Ω = Ω δ i , M i =
i ω λ − * , * * ) ( δλ ω λ λκ + − ⋅ + + =
t n M i .
Here "i" indicates reference to instant t i , Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
Coordinates and velocity vector components at instant ti in OXaYaZa coordinate system are computed:
)
( ) ( sin − + = n i i i n i E M E ε
, i i M E = ) 0 ( , 8 ) 1 ( ) ( 10 − − < −
i n i E E ,
2 1 1 2 ) (n i i i i E tg tg ⋅ − + = ε ε υ , u i =
i i ω υ + , r i = a
i (1 –
ε i cos ) (n i E ),
2 1 sin i i i i i a Vr ε υ ε μ − ⋅ = , 2 1 cos 1 i i i i i a Vu ε υ ε μ − + ⋅ = ,
( ) i i i i i i oi i u u r X cos
sin sin
cos cos
⋅ Ω ⋅ − Ω ⋅ = , ( ) i i i i i i i i u u r Y o cos
cos sin
sin cos
⋅ Ω ⋅ + Ω ⋅ = ,
i i i i u r Z sin
sin ⋅ ⋅ = ,
( ) ( ) i i i i i i i i i i i i i i u u Vu i u u Vr Vx o cos
sin cos
cos sin
cos sin
sin cos
cos ⋅ Ω ⋅ + Ω ⋅ − ⋅ Ω ⋅ − Ω ⋅ = , ( ) ( )
i i i i i i i i i i i i i u u Vu i u u Vr Vy o cos
cos cos
sin sin
cos cos
sin sin
cos ⋅ Ω ⋅ − Ω ⋅ − ⋅ Ω ⋅ + Ω ⋅ = , i i i i i i i i u Vu i u Vr Vz o sin
cos sin
sin ⋅ ⋅ + ⋅ ⋅ = .
Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
of velocity vector SV according to system GLONASS almanac 1) AC SVof system GLONASS Is set:
N
= 615
Date 06.09.2001 λ j
Half cycle
t λ j = 27122.09375 seconds Δ i j =
0.011929512 Half
cycle Δ T j = -2655.76171875 seconds Δ Т ′ j = 0.000549316 Secjnds/cycle 2
ε j = 0.001482010
ω j =
0.440277100 Half
cycle
It is necessary to calculate co-ordinates and components of velocity vector in co-ordinate system OX o Y o Z o on an instant:
N
= 615
date
06.09.2001
t λ j = 33300. seconds
S 0 = 6.02401539573 rad
Outcome: Coordinates and components of velocity vector SV in co-ordinate system OX o Y o Z o on an instant t λ
dates
:
X oi = 10947.021572 кm Y o i = 13078.978287 кm Z o i = 18922.063362 кm Vx o
m/s Vy o i = -0.161453 Кm/s
Vz o i = 2.060844 Кm/s Edition 5.1 2008 ICD L1, L2 GLONASS Russian Institute of Space Device Engineering
Numbers of pages / numbers of partitions changed substitute d new except ed In total pages in doc.
number deed
Entering № the Covering note doc. And date Signat ure
Date Partitions: Signature lists;
Section 2; Section
3.3.1.2; Section
3.3.3; Section
3.3.4; 3 application 75
On all problems linked with ICD of GLONASS system, you can revert in the Russian institute of space device engineering. e-mail: contact@rniikp.ru Internet: http://www.rniikp.ru
© 2008 Russian institute of space devise engineering Download 0.87 Mb. Do'stlaringiz bilan baham: 
|