Emission Time Computation - Revision history

Carlos.Lopez at 11:28, 23 February 2012

2012-02-23T11:28:29Z

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	\|Level=Advanced		\|Level=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}

Jaume.Sanz at 14:05, 6 February 2012

2012-02-06T14:05:28Z

← Older revision		Revision as of 14:05, 6 February 2012
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	::1. Calculate the position <math>\displaystyle {\mathbf r}^{sat}</math> of the satellite at signal reception time <math>\displaystyle t_{rcv}</math>.		::1. Calculate the position <math>\displaystyle {\mathbf r}^{sat}</math> of the satellite at signal reception time <math>\displaystyle t_{rcv}</math>.

	::2. Calculate the geometric distance between satellite coordinates obtained previously and receiver position <ref group="footnotes">Again, notice that satellite and receiver coordinates must be given in the same reference system, because satellite-receiver ray must be generated in a common reference system.</ref>, and from it, calculate the signal ~~traveling~~ time between both points: <math>		::2. Calculate the geometric distance between satellite coordinates obtained previously and receiver position <ref group="footnotes">Again, notice that satellite and receiver coordinates must be given in the same reference system, because satellite-receiver ray must be generated in a common reference system.</ref>, and from it, calculate the signal travelling time between both points: <math>
	\Delta t=\frac{\left\\| {\mathbf r}^{sat}-{\mathbf r}_{0_{rcv}}\right\\|}{c} \qquad \mbox{(6)}</math>		\Delta t=\frac{\left\\| {\mathbf r}^{sat}-{\mathbf r}_{0_{rcv}}\right\\|}{c} \qquad \mbox{(6)}</math>

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	Figure 1 shows an example to illustrating the effect of neglecting the travelling time in the satellite coordinates computation for positioning. It corresponds to a receiver located in Barcelona, Spain (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>). During the <math>70</math> to <math>90</math> milliseconds of ~~traveling~~ time, the satellite moves about <math>200-250</math> meters, which leads to <math>+/-60</math> meters in range. The effect on the user position is up to <math>50</math> meters or more in horizontal and vertical components.		Figure 1 shows an example to illustrating the effect of neglecting the travelling time in the satellite coordinates computation for positioning. It corresponds to a receiver located in Barcelona, Spain (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>). During the <math>70</math> to <math>90</math> milliseconds of travelling time, the satellite moves about <math>200-250</math> meters, which leads to <math>+/-60</math> meters in range. The effect on the user position is up to <math>50</math> meters or more in horizontal and vertical components.

Jaume.Sanz at 14:03, 6 February 2012

2012-02-06T14:03:24Z

← Older revision		Revision as of 14:03, 6 February 2012
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	If the receiver clock offset is small <ref group="footnotes">Some receivers apply clock-steering adjusting their clocks epoch-by-epoch and providing offsets of few nanoseconds. However, in many cases the receiver wait until gathering an offset of <math>1</math> millisecond to adjust the clock.</ref>, thence the <math>\delta t_{rcv}</math> may be neglected. On the other hand, the receiver clock estimates from the navigation solution can be used (extrapolated from the previous epoch). In any case, it must be taken into account that neglecting this term when <math>\delta t_{rcv}</math> reaches large values (e.g., 1 millisecond) may introduce errors of about one meter in satellite coordinates, and this must be taken in account when building the navigation model<ref group="footnotes">In the "design matrix" or Jacobian matrix, obtained when linearising the model with respect to coordinates and receiver clock errors, see [[Code Based Positioning (SPS)]].</ref>; or more precisely, in the partial derivative respect to receiver clock of design matrix.		If the receiver clock offset is small <ref group="footnotes">Some receivers apply clock-steering adjusting their clocks epoch-by-epoch and providing offsets of few nanoseconds. However, in many cases the receiver wait until gathering an offset of <math>1</math> millisecond to adjust the clock.</ref>, thence the <math>displaystyle \delta t_{rcv}</math> may be neglected. On the other hand, the receiver clock estimates from the navigation solution can be used (extrapolated from the previous epoch). In any case, it must be taken into account that neglecting this term when <math>displaystyle \delta t_{rcv}</math> reaches large values (e.g., 1 millisecond) may introduce errors of about one meter in satellite coordinates, and this must be taken in account when building the navigation model<ref group="footnotes">In the "design matrix" or Jacobian matrix, obtained when linearising the model with respect to coordinates and receiver clock errors, see [[Code Based Positioning (SPS)]].</ref>; or more precisely, in the partial derivative respect to receiver clock of design matrix.

	::{\|		::{\|
	\|+align="bottom"\|''Figure 1:Effect of using the satellite coordinates at the reception time instead of transmission time in positioning''		\|+align="bottom"\|''Figure 1:Effect of using the satellite coordinates at the reception time instead of transmission time in positioning.''

	\| [[File:Emission_Time_Computation_Distance_reception.png‎ \|none\|thumb\|400px\|frameless]]		\| [[File:Emission_Time_Computation_Distance_reception.png‎ \|none\|thumb\|400px\|frameless]]
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	Figure 1 shows an example to illustrating the effect of neglecting the ~~traveling~~ time in the satellite coordinates computation for positioning. It corresponds to a receiver located in Barcelona, Spain (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>). During the <math>70</math> to <math>90</math> milliseconds of traveling time, the satellite moves about <math>200-250</math> meters, which leads to <math>+/-60</math> meters in range. The effect on the user position is up to <math>50</math> meters or more in horizontal and vertical components.		Figure 1 shows an example to illustrating the effect of neglecting the travelling time in the satellite coordinates computation for positioning. It corresponds to a receiver located in Barcelona, Spain (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>). During the <math>70</math> to <math>90</math> milliseconds of traveling time, the satellite moves about <math>200-250</math> meters, which leads to <math>+/-60</math> meters in range. The effect on the user position is up to <math>50</math> meters or more in horizontal and vertical components.

Jaume.Sanz at 14:01, 6 February 2012

2012-02-06T14:01:58Z

← Older revision		Revision as of 14:01, 6 February 2012
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	Figure (1) shows an example to illustrating the effect of neglecting the traveling time in the satellite coordinates computation for positioning. It corresponds to a receiver located in Barcelona, Spain (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>). During the <math>70</math> to <math>90</math> milliseconds of traveling time, the satellite moves about <math>200-250</math> meters, which leads to <math>+/-60</math> meters in range. The effect on the user position is up to <math>50</math> meters or more in horizontal and vertical components.		Figure 1 shows an example to illustrating the effect of neglecting the traveling time in the satellite coordinates computation for positioning. It corresponds to a receiver located in Barcelona, Spain (receiver coordinates <math>\lambda\simeq 2^o</math> <math>\phi \simeq 41^o</math>). During the <math>70</math> to <math>90</math> milliseconds of traveling time, the satellite moves about <math>200-250</math> meters, which leads to <math>+/-60</math> meters in range. The effect on the user position is up to <math>50</math> meters or more in horizontal and vertical components.

Jaume.Sanz at 13:59, 6 February 2012

2012-02-06T13:59:53Z

← Older revision		Revision as of 13:59, 6 February 2012
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	where <math>\delta t_{rcv}</math> is receiver clock offset referred to the system time, that may be obtained from navigation solution (although "a posteriori").		where <math>displaystyle \delta t_{rcv}</math> is receiver clock offset referred to the system time, that may be obtained from navigation solution (although "a posteriori").

Jaume.Sanz at 13:55, 6 February 2012

2012-02-06T13:55:14Z

← Older revision		Revision as of 13:55, 6 February 2012
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	\Delta t=\frac{\left\\| {\mathbf r}^{sat}-{\mathbf r}_{0_{rcv}}\right\\|}{c} \qquad \mbox{(6)}</math>		\Delta t=\frac{\left\\| {\mathbf r}^{sat}-{\mathbf r}_{0_{rcv}}\right\\|}{c} \qquad \mbox{(6)}</math>

	::3. Calculate satellite position at the time: <math>t = t_{rcv} - \Delta t \Longrightarrow r_{sat}</math>.		::3. Calculate satellite position at the time: <math>t = t_{rcv} - \Delta t \Longrightarrow r^{sat}</math>.

	::4. Compare the new position <math>\displaystyle r^{sat}</math> with the former position. If they differ more than certain threshold value, iterate the procedure starting from 2.		::4. Compare the new position <math>\displaystyle r^{sat}</math> with the former position. If they differ more than certain threshold value, iterate the procedure starting from 2.

Jaume.Sanz at 13:53, 6 February 2012

2012-02-06T13:53:26Z

← Older revision		Revision as of 13:53, 6 February 2012
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	==A purely geometric algorithm==		==A purely geometric algorithm==

	The former algorithm (equation 2) provides the signal emission time tied to satellite clock (<math>t^{sat}</math>). The next algorithm ties this epoch to receiver clock (<math>t_{rcv}</math>):		The former algorithm (equation 2) provides the signal emission time tied to satellite clock (<math>\displaystyle t^{sat}</math>). The next algorithm ties this epoch to receiver clock (<math>\displaystyle t_{rcv}</math>):

	::<math>		::<math>
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	where <math>\Delta t</math> is now calculated by iteration assuming that an approximate receptor position <math>r_{0_{rcv}}</math> is known (it converges very fast):		where <math>\displaystyle \Delta t</math> is now calculated by iteration assuming that an approximate receptor position <math>\displaystyle r_{0_{rcv}}</math> is known (it converges very fast):


	The algorithm is based in the following steps:		The algorithm is based in the following steps:

	::1. Calculate the position <math>{\mathbf r}^{sat}</math> of the satellite at signal reception time <math>t_{rcv}</math>.		::1. Calculate the position <math>\displaystyle {\mathbf r}^{sat}</math> of the satellite at signal reception time <math>\displaystyle t_{rcv}</math>.

	::2. Calculate the geometric distance between satellite coordinates obtained previously and receiver position <ref group="footnotes">Again, notice that satellite and receiver coordinates must be given in the same reference system, because satellite-receiver ray must be generated in a common reference system.</ref>, and from it, calculate the signal traveling time between both points: <math>		::2. Calculate the geometric distance between satellite coordinates obtained previously and receiver position <ref group="footnotes">Again, notice that satellite and receiver coordinates must be given in the same reference system, because satellite-receiver ray must be generated in a common reference system.</ref>, and from it, calculate the signal traveling time between both points: <math>
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	::3. Calculate satellite position at the time: <math>t = t_{rcv} - \Delta t \Longrightarrow r_{sat}</math>.		::3. Calculate satellite position at the time: <math>t = t_{rcv} - \Delta t \Longrightarrow r_{sat}</math>.

	::4. Compare the new position <math>r^{sat}</math> with the former position. If they differ more than certain threshold value, iterate the procedure starting from 2.		::4. Compare the new position <math>\displaystyle r^{sat}</math> with the former position. If they differ more than certain threshold value, iterate the procedure starting from 2.

Jaume.Sanz at 13:50, 6 February 2012

2012-02-06T13:50:50Z

← Older revision		Revision as of 13:50, 6 February 2012
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	Thence, if the <math>\displaystyle \delta t^{sat}</math> is the satellite clock offset, regarding to the GNSS ([[GPS General Introduction\|GPS]], [[GLONASS General Introduction\|GLONASS]], [[GALILEO General Introduction\|Galileo]]...) system time scale (see [[Clock Modelling]]) the transmission time <math>T[emission]</math> in this system time scale can be computed from the receiver measurement time tags (<math>\displaystyle t_{rcv}</math>) as:		Thence, if the <math>\displaystyle \delta t^{sat}</math> is the satellite clock offset, regarding to the GNSS ([[GPS General Introduction\|GPS]], [[GLONASS General Introduction\|GLONASS]], [[GALILEO General Introduction\|Galileo]]...) system time scale (see [[Clock Modelling]]) the transmission time <math>\displaystyle T[emission]</math> in this system time scale can be computed from the receiver measurement time tags (<math>\displaystyle t_{rcv}</math>) as:

	::<math>		::<math>
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	The accuracy in determination of <math>T[emission]</math> is very high, and essentially depends on <math>\delta t^{sat}</math> error. For instance, in the case of the [[GPS General Introduction\|GPS]] system it is less than <math>10</math> or <math>100</math> nanoseconds with S/A=off and S/A=on, respectively. This allows calculating satellite coordinates with errors below one tenth of millimetre in both cases <ref group="footnotes">GPS, GLONASS or Galileo satellites speed is of few km/s.</ref>.		The accuracy in determination of <math>\displaystyle T[emission]</math> is very high, and essentially depends on <math>\displaystyle \delta t^{sat}</math> error. For instance, in the case of the [[GPS General Introduction\|GPS]] system it is less than <math>10</math> or <math>100</math> nanoseconds with S/A=off and S/A=on, respectively. This allows calculating satellite coordinates with errors below one tenth of millimetre in both cases <ref group="footnotes">GPS, GLONASS or Galileo satellites speed is of few km/s.</ref>.

Jaume.Sanz at 13:47, 6 February 2012

2012-02-06T13:47:47Z

← Older revision		Revision as of 13:47, 6 February 2012
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	Thence, if the <math>\displaystyle \delta t^{sat}</math> is the satellite clock offset, regarding to the GNSS ([[GPS General Introduction\|GPS]], [[GLONASS General Introduction\|GLONASS]], [[GALILEO General Introduction\|Galileo]]...) system time scale (see [[Clock Modelling]] the transmission time <math>T[emission]</math> in this system time scale can be computed from the receiver measurement time tags (<math>\displaystyle t_{rcv}</math>) as:		Thence, if the <math>\displaystyle \delta t^{sat}</math> is the satellite clock offset, regarding to the GNSS ([[GPS General Introduction\|GPS]], [[GLONASS General Introduction\|GLONASS]], [[GALILEO General Introduction\|Galileo]]...) system time scale (see [[Clock Modelling]]) the transmission time <math>T[emission]</math> in this system time scale can be computed from the receiver measurement time tags (<math>\displaystyle t_{rcv}</math>) as:

	::<math>		::<math>

Jaume.Sanz at 13:46, 6 February 2012

2012-02-06T13:46:49Z

← Older revision		Revision as of 13:46, 6 February 2012
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	Thence, if the <math>\delta t^{sat}</math> is the satellite clock offset, regarding to the GNSS ([[GPS General Introduction\|GPS]], [[GLONASS General Introduction\|GLONASS]], [[GALILEO General Introduction\|Galileo]]...) system time scale (see [[Clock Modelling]] the transmission time <math>T[emission]</math> in this system time scale can be computed from the receiver measurement time tags (<math>t_{rcv}</math>) as:		Thence, if the <math>\displaystyle \delta t^{sat}</math> is the satellite clock offset, regarding to the GNSS ([[GPS General Introduction\|GPS]], [[GLONASS General Introduction\|GLONASS]], [[GALILEO General Introduction\|Galileo]]...) system time scale (see [[Clock Modelling]] the transmission time <math>T[emission]</math> in this system time scale can be computed from the receiver measurement time tags (<math>\displaystyle t_{rcv}</math>) as:

	::<math>		::<math>