GLONASS Satellite Coordinates Computation - Revision history

Gema.Cueto at 10:49, 16 April 2026

2026-04-16T10:49:47Z

← Older revision		Revision as of 10:49, 16 April 2026
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	In order to compute PZ-90 [[GLONASS General Introduction\|GLONASS]] satellite coordinates from the navigation message, the following algorithm must be used [GLONASS ICD, 1998] <ref> [GLONASS ICD, 1998] ~~GLONASS ICD, 1998. Technical report. v.4.0.~~</ref>.		In order to compute PZ-90 [[GLONASS General Introduction\|GLONASS]] satellite coordinates from the navigation message, the following algorithm must be used [GLONASS ICD, 1998] <ref> [https://www.unavco.org/help/glossary/docs/ICD_GLONASS_4.0_(1998)_en.pdf GLONASS ICD, 1998]</ref>.

Carlos.Lopez at 10:16, 23 February 2012

2012-02-23T10:16:17Z

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	\|Level=Intermediate		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
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Jaume.Sanz at 08:10, 21 February 2012

2012-02-21T08:10:45Z

← Older revision		Revision as of 08:10, 21 February 2012
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	:::where <math>\theta_G</math> is the sidereal time at Greenwich meridian at time <math>t</math>:		:::where <math>\theta_G</math> is the sidereal time at Greenwich meridian at time <math>t</math>, where <math>t</math> is in GLONASS time, see equation (4) :

	:::::<math>		:::::<math>

Jaume.Sanz at 08:07, 21 February 2012

2012-02-21T08:07:47Z

← Older revision		Revision as of 08:07, 21 February 2012
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	:::Where <math>(\theta_{G_e})</math> is the sidereal time at epoch <math>t_e</math>, to which are referred the initial conditions, in Greenwich meridian:		:::Where <math>(\theta_{G_e})</math> is the sidereal time at epoch <math>t_e</math>, to which are referred the initial conditions, in Greenwich meridian:
	:::::<math>		:::::<math>
	\theta_{G_e}= \theta_{G_0} + \omega_E (t_e+3\, hours) \qquad \mbox{(4)}		\theta_{G_e}= \theta_{G_0} + \omega_E (t_e-3\, hours) \qquad \mbox{(4)}
	</math>		</math>

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	:::- <math>\omega_E</math>: earth's rotation rate (<math>0.7292115\, 10^{-4}\; rad/s</math>)).		:::- <math>\omega_E</math>: earth's rotation rate (<math>0.7292115\, 10^{-4}\; rad/s</math>)).
	:::- <math>\theta_{G_0}</math>: the sidereal time in Greenwich at midnight GMT of a date at which the epoch <math>t_e</math> is specified. (Notice: GLONASS_time = UTC(SU) - <math>3</math> hours).		:::- <math>\theta_{G_0}</math>: the sidereal time in Greenwich at midnight GMT of a date at which the epoch <math>t_e</math> is specified. (Notice: GLONASS_time = UTC(SU) + <math>3</math> hours).

Jaume.Sanz at 08:05, 21 February 2012

2012-02-21T08:05:59Z

← Older revision		Revision as of 08:05, 21 February 2012
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	:::Where <math>(\theta_{G_e})</math> is the sidereal time at epoch <math>t_e</math>, to which are referred the initial conditions, in Greenwich meridian:		:::Where <math>(\theta_{G_e})</math> is the sidereal time at epoch <math>t_e</math>, to which are referred the initial conditions, in Greenwich meridian:
	:::::<math>		:::::<math>
	\theta_{G_e}= \theta_{G_0} + \omega_E (t_e-3\, hours) \qquad \mbox{(4)}		\theta_{G_e}= \theta_{G_0} + \omega_E (t_e+3\, hours) \qquad \mbox{(4)}
	</math>		</math>

Rui.Pereira at 15:07, 24 January 2012

2012-01-24T15:07:11Z

← Older revision		Revision as of 15:07, 24 January 2012
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	}}		}}
	The [[GLONASS]] satellite coordinates shall be computed according to the specifications in the [[GLONASS]]-ICD document. An accuracy level of about three meters can be reached using the algorithm provided by this ICD.		The [[GLONASS General Introduction\|GLONASS]] satellite coordinates shall be computed according to the specifications in the [[GLONASS General Introduction\|GLONASS]]-ICD document. An accuracy level of about three meters can be reached using the algorithm provided by this ICD.

	In table 1 are listed the broadcast ephemeris parameters which are used to compute [[GLONASS]] satellites coordinates. Essentially, the ephemeris contain the initial conditions of position and velocity to perform the numerical integration of the [[GLONASS]] orbit within the interval of measurement <math>\|t - t_e\| < 15</math> minutes. The accelerations due solar and lunar gravitational perturbations are also given.		In table 1 are listed the broadcast ephemeris parameters which are used to compute [[GLONASS General Introduction\|GLONASS]] satellites coordinates. Essentially, the ephemeris contain the initial conditions of position and velocity to perform the numerical integration of the [[GLONASS General Introduction\|GLONASS]] orbit within the interval of measurement <math>\|t - t_e\| < 15</math> minutes. The accelerations due solar and lunar gravitational perturbations are also given.

	::[[File: GLONASS_SV_Coord_Comp.png\|none\|thumb\|400px\|'''''Table 1:''''' GLONASS broadcast ephemeris and clock message parameters.]]		::[[File: GLONASS_SV_Coord_Comp.png\|none\|thumb\|400px\|'''''Table 1:''''' GLONASS broadcast ephemeris and clock message parameters.]]


	In order to compute PZ-90 [[GLONASS]] satellite coordinates from the navigation message, the following algorithm must be used [GLONASS ICD, 1998] <ref> [GLONASS ICD, 1998] GLONASS ICD, 1998. Technical report. v.4.0.</ref>.		In order to compute PZ-90 [[GLONASS General Introduction\|GLONASS]] satellite coordinates from the navigation message, the following algorithm must be used [GLONASS ICD, 1998] <ref> [GLONASS ICD, 1998] GLONASS ICD, 1998. Technical report. v.4.0.</ref>.


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	::*''' 1. Coordinates transformation to an inertial reference frame:'''		::*''' 1. Coordinates transformation to an inertial reference frame:'''

	:::- The initial conditions <math>(x(t_e),y(t_e),z(t_e),v_x(t_e),v_y(t_e),v_z(t_e))</math>, as broadcast in the [[GLONASS]] navigation message, are in the ECEF Greenwich coordinate system PZ-90. Therefore, and previous to orbit integration, they must be transformed to an absolute (inertial) coordinate system using the following expressions <ref group="footnotes"> Note: Over a small integration intervals, a simple rotation of <math>\theta_{G_e}</math> angle around Z-axis is enough to perform this transformation. Nutation and precession of the earth and polar motion are a very slow processes and will not introduce significant deviations on such short integration time intervals (see [[Transformation between Celestial and Terrestrial Frames]]).</ref>:		:::- The initial conditions <math>(x(t_e),y(t_e),z(t_e),v_x(t_e),v_y(t_e),v_z(t_e))</math>, as broadcast in the [[GLONASS General Introduction\|GLONASS]] navigation message, are in the ECEF Greenwich coordinate system PZ-90. Therefore, and previous to orbit integration, they must be transformed to an absolute (inertial) coordinate system using the following expressions <ref group="footnotes"> Note: Over a small integration intervals, a simple rotation of <math>\theta_{G_e}</math> angle around Z-axis is enough to perform this transformation. Nutation and precession of the earth and polar motion are a very slow processes and will not introduce significant deviations on such short integration time intervals (see [[Transformation between Celestial and Terrestrial Frames]]).</ref>:

	::::'''Position:'''		::::'''Position:'''
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	::*''' 2. Numerical integration of differential equations that describe the motion of the satellites.'''		::*''' 2. Numerical integration of differential equations that describe the motion of the satellites.'''

	:::According to [[GLONASS]]-ICD, the re-calculation of ephemeris from epoch <math>t_e</math> to epoch <math>t_i</math> within the measurement interval (<math>\|t_i-t_e\|<15 min</math>) shall be performed by a numerical integration of the differential equations (5) describing the motion of the satellites. These equations shall be integrated in a direct absolute geocentric coordinate system OXa, OYa, OZa, connected with current equator and vernal equinox, using the 4th order Runge-Kutta technique:		:::According to [[GLONASS General Introduction\|GLONASS]]-ICD, the re-calculation of ephemeris from epoch <math>t_e</math> to epoch <math>t_i</math> within the measurement interval (<math>\|t_i-t_e\|<15 min</math>) shall be performed by a numerical integration of the differential equations (5) describing the motion of the satellites. These equations shall be integrated in a direct absolute geocentric coordinate system OXa, OYa, OZa, connected with current equator and vernal equinox, using the 4th order Runge-Kutta technique:

	:::::<math>		:::::<math>
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	:::''Note'': In the above differential equations system (5), the term <math>C_{20}=-J_2=+\sqrt{5}\bar{C}_{20}</math> is used instead of <math>J_2</math> in equations <math>		:::''Note'': In the above differential equations system (5), the term <math>C_{20}=-J_2=+\sqrt{5}\bar{C}_{20}</math> is used instead of <math>J_2</math> in equations <math>
	V(r,\phi,\lambda)=\frac{\mu}{r}\left[1+\frac{1}{2}\left(\frac{a_e}{r}\right)^2 J_2\;\;(1-3\sin^2 \phi) \right] </math> and <math> \mathbb{\mathbf {\ddot r}}=\nabla V+\mathbb{\mathbf k}_{sun\_moon} </math> to keep the same expressions as in the [[GLONASS]]-ICD (please refer to [[Perturbed Motion]] and [[GNSS Broadcast Orbits]])		V(r,\phi,\lambda)=\frac{\mu}{r}\left[1+\frac{1}{2}\left(\frac{a_e}{r}\right)^2 J_2\;\;(1-3\sin^2 \phi) \right] </math> and <math> \mathbb{\mathbf {\ddot r}}=\nabla V+\mathbb{\mathbf k}_{sun\_moon} </math> to keep the same expressions as in the [[GLONASS General Introduction\|GLONASS]]-ICD (please refer to [[Perturbed Motion]] and [[GNSS Broadcast Orbits]])


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	</math>		</math>

	:::The method is initialised with the initial conditions <math>\mathbb{\mathbf Y}(t_0)</math> and <math>\mathbb{\mathbf Y}'(t_0)</math>. For the numerical integration of [[GLONASS]] satellite orbits, the function <math>\mathbb{\mathbf F}(t,\mathbb{\mathbf Y})</math> is given by (7).		:::The method is initialised with the initial conditions <math>\mathbb{\mathbf Y}(t_0)</math> and <math>\mathbb{\mathbf Y}'(t_0)</math>. For the numerical integration of [[GLONASS General Introduction\|GLONASS]] satellite orbits, the function <math>\mathbb{\mathbf F}(t,\mathbb{\mathbf Y})</math> is given by (7).


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	:::Note that [[GLONASS]] satellite coordinates are computed in PZ-90 reference system, instead of WGS-84 where the GPS coordinates have been calculated. To bring the PZ-90 coordinate system in coincidence with WGS-84 the transformation given by equation (11) must be applied (see [[Reference Frames in GNSS]]):		:::Note that [[GLONASS General Introduction\|GLONASS]] satellite coordinates are computed in PZ-90 reference system, instead of WGS-84 where the GPS coordinates have been calculated. To bring the PZ-90 coordinate system in coincidence with WGS-84 the transformation given by equation (11) must be applied (see [[Reference Frames in GNSS]]):

	:::::<math>		:::::<math>

Jaume.Sanz at 21:29, 18 January 2012

2012-01-18T21:29:37Z

← Older revision		Revision as of 21:29, 18 January 2012
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	The [[GLONASS]] satellite coordinates shall be computed according to the specifications in the [[GLONASS]]-ICD document. An accuracy level of about three meters can be reached using the algorithm provided by this ICD.		The [[GLONASS]] satellite coordinates shall be computed according to the specifications in the [[GLONASS]]-ICD document. An accuracy level of about three meters can be reached using the algorithm provided by this ICD.

	In table 1 the broadcast ephemeris parameters, which are used to compute [[GLONASS]] satellites coordinates~~, are listed~~. Essentially, the ephemeris contain the initial conditions of position and velocity to perform the numerical integration of the [[GLONASS]] orbit within the measurement ~~interval~~ <math>\|t - t_e\| < 15</math> minutes. The accelerations due to solar and lunar gravitational perturbations are also given.		In table 1 are listed the broadcast ephemeris parameters which are used to compute [[GLONASS]] satellites coordinates. Essentially, the ephemeris contain the initial conditions of position and velocity to perform the numerical integration of the [[GLONASS]] orbit within the interval of measurement <math>\|t - t_e\| < 15</math> minutes. The accelerations due solar and lunar gravitational perturbations are also given.

	::[[File: GLONASS_SV_Coord_Comp.png\|none\|thumb\|400px\|'''''Table 1:''''' GLONASS broadcast ephemeris and clock message parameters.]]		::[[File: GLONASS_SV_Coord_Comp.png\|none\|thumb\|400px\|'''''Table 1:''''' GLONASS broadcast ephemeris and clock message parameters.]]
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	::*''' 1. Coordinates transformation to an inertial reference frame:'''		::*''' 1. Coordinates transformation to an inertial reference frame:'''

	:::- The initial conditions <math>(x(t_e),y(t_e),z(t_e),v_x(t_e),v_y(t_e),v_z(t_e))</math>, as broadcast in the [[GLONASS]] navigation message, are in the ECEF Greenwich coordinate system PZ-90. Therefore, and previous to orbit integration, they must be transformed to an absolute (inertial) coordinate system using the following expressions <ref group="footnotes"> Note: Over small integration intervals, a simple rotation of <math>\theta_{G_e}</math> angle around Z-axis is enough to perform this transformation. Nutation and precession of the earth and polar motion are very slow processes and will not introduce significant deviations on such short integration time intervals (see [[Transformation between Celestial and Terrestrial Frames]]).</ref>:		:::- The initial conditions <math>(x(t_e),y(t_e),z(t_e),v_x(t_e),v_y(t_e),v_z(t_e))</math>, as broadcast in the [[GLONASS]] navigation message, are in the ECEF Greenwich coordinate system PZ-90. Therefore, and previous to orbit integration, they must be transformed to an absolute (inertial) coordinate system using the following expressions <ref group="footnotes"> Note: Over a small integration intervals, a simple rotation of <math>\theta_{G_e}</math> angle around Z-axis is enough to perform this transformation. Nutation and precession of the earth and polar motion are a very slow processes and will not introduce significant deviations on such short integration time intervals (see [[Transformation between Celestial and Terrestrial Frames]]).</ref>:

	::::'''Position:'''		::::'''Position:'''
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	:::Where <math>(\theta_{G_e})</math> is the sidereal time at epoch <math>t_e</math>, to which the initial conditions, in Greenwich meridian~~, are referred~~:		:::Where <math>(\theta_{G_e})</math> is the sidereal time at epoch <math>t_e</math>, to which are referred the initial conditions, in Greenwich meridian:
	:::::<math>		:::::<math>
	\theta_{G_e}= \theta_{G_0} + \omega_E (t_e-3\, hours) \qquad \mbox{(4)}		\theta_{G_e}= \theta_{G_0} + \omega_E (t_e-3\, hours) \qquad \mbox{(4)}
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	:::being:		:::being:

	:::- <math>\omega_E</math>: ~~Earth~~'s rotation rate (<math>0.7292115\, 10^{-4}\; rad/s</math>)).		:::- <math>\omega_E</math>: earth's rotation rate (<math>0.7292115\, 10^{-4}\; rad/s</math>)).
	:::- <math>\theta_{G_0}</math>: the sidereal time in Greenwich at midnight GMT of a date at which the epoch <math>t_e</math> is specified. (Notice: GLONASS_time = UTC(SU) - <math>3</math> hours).		:::- <math>\theta_{G_0}</math>: the sidereal time in Greenwich at midnight GMT of a date at which the epoch <math>t_e</math> is specified. (Notice: GLONASS_time = UTC(SU) - <math>3</math> hours).

Jaume.Sanz at 12:56, 17 January 2012

2012-01-17T12:56:22Z

← Older revision		Revision as of 12:56, 17 January 2012
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	:::::<math>\mu= 398\,600.44\; km^3/s^2</math> Gravitational constant (PZ-90).		:::::<math>\mu= 398\,600.44\; km^3/s^2</math> Gravitational constant (PZ-90).

	:::::<math>C_{20}=-1\,082.63\cdot 10^{-6}</math> Second zonal coefficient of spherical & harmonic expression.		:::::<math>C_{20}=-1\,082.63\cdot 10^{-6}</math> Second zonal coefficient of spherical harmonic expression.

Jaume.Sanz at 12:55, 17 January 2012

2012-01-17T12:55:38Z

← Older revision		Revision as of 12:55, 17 January 2012
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	:::::<math>\bar{\mu}=\frac{\mu}{r^2}</math>,<math>\bar{x_a}=\frac{x_a}{r}</math>, <math>\bar{y_a}=\frac{y_a}{r}</math>, <math>\bar{z_a}=\frac{x_a}{r}</math>, <math>\bar{\rho}=\frac{a_E}{r}</math>, <math>r=\sqrt{x_a^2+y_a^2+z_a^2}</math>		:::::<math>\bar{\mu}=\frac{\mu}{r^2}</math>,<math>\bar{x_a}=\frac{x_a}{r}</math>, <math>\bar{y_a}=\frac{y_a}{r}</math>, <math>\bar{z_a}=\frac{x_a}{r}</math>, <math>\bar{\rho}=\frac{a_E}{r}</math>, <math>r=\sqrt{x_a^2+y_a^2+z_a^2}</math>

	:::::<math>a_E= 6\,378.136\; km</math> & Equatorial radius of the Earth (PZ-90).		:::::<math>a_E= 6\,378.136\; km</math> Equatorial radius of the Earth (PZ-90).

	:::::<math>\mu= 398\,600.44\; km^3/s^2</math>& Gravitational constant (PZ-90).		:::::<math>\mu= 398\,600.44\; km^3/s^2</math> Gravitational constant (PZ-90).

	:::::<math>C_{20}=-1\,082.63\cdot 10^{-6}</math> & Second zonal coefficient of spherical & harmonic expression.		:::::<math>C_{20}=-1\,082.63\cdot 10^{-6}</math> Second zonal coefficient of spherical & harmonic expression.

Jaume.Sanz at 07:00, 17 January 2012

2012-01-17T07:00:22Z

← Older revision		Revision as of 07:00, 17 January 2012
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	:::The transformation from PZ-90.02 to WGS-84 (actually ITRF2000) is given by <math>\Delta x= -0.36\,m</math>, <math>\Delta y= +0.08\, m</math>, <math>\Delta z= +0.18\, m</math>, with no rotation, i.e., equation (12) was implemented in September 20th, 2007 at 18:00. (refer to [[Reference Frames in GNSS]]):		:::The transformation from PZ-90.02 to WGS-84 (actually ITRF2000) is given by <math>\Delta x= -0.36\,m</math>, <math>\Delta y= +0.08\, m</math>, <math>\Delta z= +0.18\, m</math>, with no rotation, i.e., equation (12)<ref group="footnotes"> The PZ-90.02 was implemented in September 20th, 2007 at 18:00. (refer to [[Reference Frames in GNSS]]).</ref>:

	:::::<math>		:::::<math>