Satellite Coordinates Computation - Revision history

Jaume.Sanz at 11:53, 13 January 2013

2013-01-13T11:53:00Z

← Older revision		Revision as of 11:53, 13 January 2013
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	\right ] \qquad		\right ] \qquad
	~~ and ~~ \qquad		~~ and ~~ \qquad
	\Delta t = \frac{\sqrt{(~~x_{0_{rcv}}-~~\tilde{x}^{sat}~~)^2+(y_~~{0_{rcv}}-\tilde{y}^{sat}~~)^2+(z_~~{0_{rcv}}-\tilde{z}^{sat})^2}}{c}		\Delta t = \frac{\sqrt{(\tilde{x}^{sat}-x_{0_{rcv}})^2+(\tilde{y}^{sat}-y_{0_{rcv}})^2+(\tilde{z}^{sat}-z_{0_{rcv}})^2}}{c}
	\qquad \mbox{(3)}</math>		\qquad \mbox{(3)}</math>

Jaume.Sanz at 11:15, 13 January 2013

2013-01-13T11:15:19Z

← Older revision		Revision as of 11:15, 13 January 2013
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	\right ] \qquad		\right ] \qquad
	~~ and ~~ \qquad		~~ and ~~ \qquad
	\Delta t = \frac{\sqrt{(x_{0_{rcv}}-x^{sat})^2+(y_{0_{rcv}}-y^{sat})^2+(z_{0_{rcv}}-z^{sat})^2}}{c}		\Delta t = \frac{\sqrt{(x_{0_{rcv}}-\tilde{x}^{sat})^2+(y_{0_{rcv}}-\tilde{y}^{sat})^2+(z_{0_{rcv}}-\tilde{z}^{sat})^2}}{c}
	\qquad \mbox{(3)}</math>		\qquad \mbox{(3)}</math>

Jaume.Sanz at 10:21, 1 February 2012

2012-02-01T10:21:03Z

← Older revision		Revision as of 10:21, 1 February 2012
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	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}
	Once the signal transmission time is known, the satellite coordinates may be calculated at that epoch in a given reference frame (whatever, inertial or ECEF rotating frame). Algorithms ~~(and the associated software code programs)~~ to compute such coordinates in a ECEF reference frame, from the broadcast navigation message, were given in [[GPS and Galileo Satellite Coordinates Computation]], and in [[GLONASS Satellite Coordinates Computation]]. Algorithms ~~(and software routines)~~ using precise orbits can be found in [[Precise GNSS Satellite Coordinates Computation]].		Once the signal transmission time is known, the satellite coordinates may be calculated at that epoch in a given reference frame (whatever, inertial or ECEF rotating frame). Algorithms to compute such coordinates in a ECEF reference frame, from the broadcast navigation message, were given in [[GPS and Galileo Satellite Coordinates Computation]], and in [[GLONASS Satellite Coordinates Computation]]. Algorithms using precise orbits can be found in [[Precise GNSS Satellite Coordinates Computation]].

Jaume.Sanz at 12:56, 30 January 2012

2012-01-30T12:56:29Z

← Older revision		Revision as of 12:56, 30 January 2012
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	{{Article Infobox2		{{Article Infobox2
	\|Category=Fundamentals		\|Category=Fundamentals
	~~\|Title={{PAGENAME}}~~		\|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
	\|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. ~~Hernandez~~-Pajares, University of ~~Catalunia~~, Spain.
	\|Level=Intermediate		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011
	\|Logo=gAGE		\|Logo=gAGE
			\|Title={{PAGENAME}}
	}}		}}

	Once the signal transmission time is known, the satellite coordinates may be calculated at that epoch in a given reference frame (whatever, inertial or ECEF rotating frame). Algorithms (and the associated software code programs) to compute such coordinates in a ECEF reference frame, from the broadcast navigation message, were given in [[GPS and Galileo Satellite Coordinates Computation]], and in [[GLONASS Satellite Coordinates Computation]]. Algorithms (and software routines) using precise orbits can be found in [[Precise GNSS Satellite Coordinates Computation]].		Once the signal transmission time is known, the satellite coordinates may be calculated at that epoch in a given reference frame (whatever, inertial or ECEF rotating frame). Algorithms (and the associated software code programs) to compute such coordinates in a ECEF reference frame, from the broadcast navigation message, were given in [[GPS and Galileo Satellite Coordinates Computation]], and in [[GLONASS Satellite Coordinates Computation]]. Algorithms (and software routines) using precise orbits can be found in [[Precise GNSS Satellite Coordinates Computation]].

Timo.Kouwenhoven: Text replace - "|Level=Medium" to "|Level=Intermediate"

2011-11-06T07:46:49Z

Text replace - "|Level=Medium" to "|Level=Intermediate"

← Older revision		Revision as of 07:46, 6 November 2011
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	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	\|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.		\|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
	\|Level=~~Medium~~		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011
	\|Logo=gAGE		\|Logo=gAGE

Carlos.Lopez at 09:13, 28 October 2011

2011-10-28T09:13:24Z

← Older revision		Revision as of 09:13, 28 October 2011
Line 3:		Line 3:
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	\|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.		\|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
	\|Level=~~Advanced~~		\|Level=Medium
	\|YearOfPublication=2011		\|YearOfPublication=2011
	\|Logo=gAGE		\|Logo=gAGE

Carlos.Lopez: moved Geometric Range Modelling: Satellite Coordinates Computation to Satellite Coordinates Computation

2011-08-09T09:05:28Z

moved Geometric Range Modelling: Satellite Coordinates Computation to Satellite Coordinates Computation

← Older revision	Revision as of 09:05, 9 August 2011
(No difference)

Carlos.Lopez at 08:22, 9 August 2011

2011-08-09T08:22:33Z

← Older revision		Revision as of 08:22, 9 August 2011
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	[[Category:Fundamentals]]		[[Category:Fundamentals]]
			[[Category:GNSS Measurements Modelling]]

Carlos.Lopez at 09:16, 8 August 2011

2011-08-08T09:16:14Z

← Older revision		Revision as of 09:16, 8 August 2011
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	Once the signal transmission time is known, the satellite coordinates may be calculated at that epoch in a given reference frame (whatever, inertial or ECEF rotating frame). Algorithms (and the associated software code programs) to compute such coordinates in a ECEF reference frame, from the broadcast navigation message, were given in [[GPS and Galileo Satellite Coordinates Computation]], and in [[GLONASS Satellite Coordinates Computation]]. Algorithms (and software routines) using precise orbits can be found in [[Precise GNSS Satellite Coordinates Computation]].		Once the signal transmission time is known, the satellite coordinates may be calculated at that epoch in a given reference frame (whatever, inertial or ECEF rotating frame). Algorithms (and the associated software code programs) to compute such coordinates in a ECEF reference frame, from the broadcast navigation message, were given in [[GPS and Galileo Satellite Coordinates Computation]], and in [[GLONASS Satellite Coordinates Computation]]. Algorithms (and software routines) using precise orbits can be found in [[Precise GNSS Satellite Coordinates Computation]].


	It must be taken into account that an ECEF frame is an earth-fixed system and, thence, rotates with the earth. This system must be taken at the reception time (in the GNSS time scale), because it is a common reference for all measurements (see [[Emission Time Computation]]).		It must be taken into account that an ECEF frame is an earth-fixed system and, thence, rotates with the earth. This system must be taken at the reception time (in the GNSS time scale), because it is a common reference for all measurements (see [[Emission Time Computation]]).


	An algorithm to compute the satellite coordinates at the transmission time, but referred to the ECEF tied to the earth at the reception time is given as follows:		An algorithm to compute the satellite coordinates at the transmission time, but referred to the ECEF tied to the earth at the reception time is given as follows:


	: 1. Calculate satellite coordinates at the emission time in the associated ECEF reference frame (i.e., tied to the emission time).		: 1. Calculate satellite coordinates at the emission time in the associated ECEF reference frame (i.e., tied to the emission time).
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	0 & 0 & 1\\		0 & 0 & 1\\
	\end{array}		\end{array}
	\right ]		\right ] \qquad
	~~</math>~~ and~~: <math>~~		~~ and ~~ \qquad
	\Delta t = \frac{\sqrt{(x_{0_{rcv}}-x^{sat})^2+(y_{0_{rcv}}-y^{sat})^2+(z_{0_{rcv}}-z^{sat})^2}}{c}		\Delta t = \frac{\sqrt{(x_{0_{rcv}}-x^{sat})^2+(y_{0_{rcv}}-y^{sat})^2+(z_{0_{rcv}}-z^{sat})^2}}{c}
	\qquad \mbox{(3)}</math>		\qquad \mbox{(3)}</math>
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	: Note: It is advisable to calculate <math>\Delta t</math> using the former expression, even when pseudorange method is used to figure out signal propagation time. The reason for this is that "<math>R/c</math>" includes other delays (clock offsets, ...) besides just the purely geometric part <math>\rho/c</math>. In other words, <math>R/c</math> establish a very precise link between the reception time (in the receiver time tags) and the transmission time (in the satellite clock). Nevertheless, as a geometric distance it can be biased (mainly by the satellite and receiver clock offsets) by hundreds of kilometres.		: Note: It is advisable to calculate <math>\Delta t</math> using the former expression, even when pseudorange method is used to figure out signal propagation time. The reason for this is that "<math>R/c</math>" includes other delays (clock offsets, ...) besides just the purely geometric part <math>\rho/c</math>. In other words, <math>R/c</math> establish a very precise link between the reception time (in the receiver time tags) and the transmission time (in the satellite clock). Nevertheless, as a geometric distance it can be biased (mainly by the satellite and receiver clock offsets) by hundreds of kilometres.



	Figure 1 illustrates the effect of neglecting the earth rotation (equation 2) during the travelling time from the satellite to receiver. The effect on range is up to <math>20</math> meters. As expected, it is seen in the position, basically, as a rotation in the east direction, moving the receiver about <math>25</math> meters (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>).		Figure 1 illustrates the effect of neglecting the earth rotation (equation 2) during the travelling time from the satellite to receiver. The effect on range is up to <math>20</math> meters. As expected, it is seen in the position, basically, as a rotation in the east direction, moving the receiver about <math>25</math> meters (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>).


	{\|		{\|

Carlos.Lopez at 10:26, 1 April 2011

2011-04-01T10:26:06Z

← Older revision		Revision as of 10:26, 1 April 2011
Line 16:		Line 16:
	: 1. Calculate satellite coordinates at the emission time in the associated ECEF reference frame (i.e., tied to the emission time).		: 1. Calculate satellite coordinates at the emission time in the associated ECEF reference frame (i.e., tied to the emission time).

	<math>		:: <math>
	T[emission] \Longrightarrow \mathbf {\tilde{r}}^{sat} \qquad \mbox{(1)}		T[emission] \Longrightarrow \mathbf {\tilde{r}}^{sat} \qquad \mbox{(1)}
	</math>		</math>
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	: 2. Transform satellite coordinates from the system tied to the earth at "emission time" to the system tied to the earth at "reception time" (which is common for all measurements). In order to do so, one must consider the earth rotation during the time interval <math>\Delta t</math> that the signal takes to propagate from the satellite to the receiver:		: 2. Transform satellite coordinates from the system tied to the earth at "emission time" to the system tied to the earth at "reception time" (which is common for all measurements). In order to do so, one must consider the earth rotation during the time interval <math>\Delta t</math> that the signal takes to propagate from the satellite to the receiver:

	<math>		::<math>
	{\mathbf r}^{sat} = {\mathbf R}_3\left( \omega _E\Delta t\right) \cdot \tilde{\mathbf r}^{sat}\qquad \mbox{(2)}		{\mathbf r}^{sat} = {\mathbf R}_3\left( \omega _E\Delta t\right) \cdot \tilde{\mathbf r}^{sat}\qquad \mbox{(2)}
	</math>		</math>
	[

	: where <math>R_3[\theta] </math> is a matrix defining a rotation of angle <math> \theta </math> around the <math>z</math>-axis:		: where <math>R_3[\theta] </math> is a matrix defining a rotation of angle <math> \theta </math> around the <math>z</math>-axis:

	<math>		:: <math>
	\mathbb{\mathbf R}_3[\theta]=\left [		\mathbb{\mathbf R}_3[\theta]=\left [
	\begin{array}{ccc}		\begin{array}{ccc}
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	\end{array}		\end{array}
	\right ]		\right ]
	</math>		</math> and: <math>
	and:
	<math>
	\Delta t = \frac{\sqrt{(x_{0_{rcv}}-x^{sat})^2+(y_{0_{rcv}}-y^{sat})^2+(z_{0_{rcv}}-z^{sat})^2}}{c}		\Delta t = \frac{\sqrt{(x_{0_{rcv}}-x^{sat})^2+(y_{0_{rcv}}-y^{sat})^2+(z_{0_{rcv}}-z^{sat})^2}}{c}
	\qquad \mbox{(3)}</math>		\qquad \mbox{(3)}</math>
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	Figure 1 illustrates the effect of neglecting the earth rotation (equation 2) during the travelling time from the satellite to receiver. The effect on range is up to <math>20</math> meters. As expected, it is seen in the position, basically, as a rotation in the east direction, moving the receiver about <math>25</math> meters (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>).		Figure 1 illustrates the effect of neglecting the earth rotation (equation 2) during the travelling time from the satellite to receiver. The effect on range is up to <math>20</math> meters. As expected, it is seen in the position, basically, as a rotation in the east direction, moving the receiver about <math>25</math> meters (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>).

	~~\begin{tabular}~~{\|c\|c\|~~}\hline~~		{\|
	~~\includegraphics~~[~~scale=0~~.~~21,angle=270~~]~~{fig/ems2rec_01.png}&~~		\|+align="bottom"\|''Figure 1:Earth rotation effect on positioning''
	~~\includegraphics~~[~~scale=0~~.~~21,angle=270~~]~~{fig/range_rotation.png}\\ \hline~~		\| [[File:Geom_Range_Modell_Sat_Coord_Computation_ems2rec_01.png‎ \|none\|thumb\|400px\|frameless]]
	~~\includegraphics~~[~~scale=0~~.~~21,angle=270~~]~~{fig/H_Nov_rot.png}&~~		\| [[File:Geom_Range_Modell_Sat_Coord_Computation_range_rotation.png\|none\|thumb\|400px\|frameless]]
	~~\includegraphics~~[~~scale=0~~.~~21,angle=270~~]~~{fig/V_Nov_rot.png}\\\hline~~		\|-
	~~\caption{Earth rotation effect on positioning.~~}~~\label{fig:earth_rot}~~		\| [[File:Geom_Range_Modell_Sat_Coord_Computation_H_Nov_rot.png\|none\|thumb\|400px\|frameless]]
			\| [[File:Geom_Range_Modell_Sat_Coord_Computation_V_Nov_rot.png\|none\|thumb\|400px\|frameless]]
			\|}


	[[Category:Fundamentals]]		[[Category:Fundamentals]]