An intuitive approach to the GNSS positioning - Revision history

Gema.Cueto: /* A deeper analysis of a 2-D pseudorange based positioning */

2024-04-09T05:46:24Z

A deeper analysis of a 2-D pseudorange based positioning

← Older revision		Revision as of 05:46, 9 April 2024
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	==A deeper analysis of a 2-D pseudorange based positioning==		==A deeper analysis of a 2-D pseudorange based positioning==

	Right now, a perfect synchronism between lighthouses and ship clocks has been assumed, but in fact this is very difficult to assure. Notice that a synchronism error between these clocks will produce an erroneous measure of signal propagation time (because it is linked to such clocks) and, in consequence, an error in the range measurements.Let's assume that the ship clock is biased by an offset <math>d \tau</math> regarding the lighthouses clocks (which are supposed to be fully synchronised). Thence, the measured ranges, <math>R_1\, and\, R_2</math>, will be shifted by an amount <math>dt\, =\, v.d \tau</math>:		Right now, a perfect synchronism between lighthouses and ship clocks has been assumed, but in fact this is very difficult to assure. Notice that a synchronism error between these clocks will produce an erroneous measure of signal propagation time (because it is linked to such clocks) and, in consequence, an error in the range measurements.Let's assume that the ship clock is biased by an offset <math>d \tau</math> regarding the lighthouses clocks (which are supposed to be fully synchronised). Thence, the measured ranges, <math>R_1\, and\, R_2</math>, will be shifted by an amount <math>dt\, =\, v*d \tau</math>:

	::<math>R_1 = \rho_1 + dt,\, R_2 = \rho_2 + dt \qquad \mbox{(1)}</math>		::<math>R_1 = \rho_1 + dt,\, R_2 = \rho_2 + dt \qquad \mbox{(1)}</math>
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	:[[File:Unknown Offset4.png\|none\|thumb\|400px\|'''''Figure 5:''''' The measurement noise <math>\varepsilon</math> is translated to the position estimate as an uncertainty region.]]		:[[File:Unknown Offset4.png\|none\|thumb\|400px\|'''''Figure 5:''''' The measurement noise <math>\varepsilon</math> is translated to the position estimate as an uncertainty region.]]


	==Translation to the 3-D GNSS positioning==		==Translation to the 3-D GNSS positioning==

Filipe.Pelica: corrected typos.

2014-05-29T16:03:38Z

corrected typos.

← Older revision		Revision as of 16:03, 29 May 2014
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	Other comments:		Other comments:
	* '''Clocks synchronisation''': The satellite clocks are one of the most critical components of a GNSS system. In order to assure the stability of such clocks, GNSS satellites are equipped with atomic oscillators with high daily stabilities <math>\Delta f\, / \, f\, \simeq\, 10^{-13}\, -\, 10^{-14}</math>. However, despite this high stability satellite clocks accumulate some offsets along time. The satellite clock offsets are continuously estimated by the Ground Segment and transmitted to the users to correct the measurements<ref group="footnotes">A perfect synchronism was assumed between lighthouses clocks in the previous example.</ref> (see [[GNSS Basic Observables]] and [[Combination of GNSS Measurements]]). The receivers, on the other hand, are equipped with quartz-based clocks, much more economical but with a poorer stability (about 10<sup>-9</sup>). This inconvenience is overcome by estimating its clock offset together with the receiver coordinates, as in the previous example.		* '''Clocks synchronisation''': The satellite clocks are one of the most critical components of a GNSS system. In order to assure the stability of such clocks, GNSS satellites are equipped with atomic oscillators with high daily stabilities <math>\Delta f\, / \, f\, \simeq\, 10^{-13}\, -\, 10^{-14}</math>. However, despite this high stability satellite clocks accumulate some offsets along time. The satellite clock offsets are continuously estimated by the Ground Segment and transmitted to the users to correct the measurements<ref group="footnotes">A perfect synchronism was assumed between lighthouses clocks in the previous example.</ref> (see [[GNSS Basic Observables]] and [[Combination of GNSS Measurements]]). The receivers, on the other hand, are equipped with quartz-based clocks, much more economical but with a poorer stability (about 10<sup>-9</sup>). This inconvenience is overcome by estimating its clock offset together with the receiver coordinates, as in the previous example.
	* '''From 2-D to 3-D positioning''': It is not difficult to extend the previous 2-D geometrical construction to 3-D case of GNSS positioning, and to show that at least 4 satellites are needed to compute the three receiver coordinates plus clock. In this case, the previous circles and hyperbolas are generalised to spheres and hyperboloids, which intersect in two possible solutions. For a ground receiver, one of such solutions is on the earth's surface and the other far away in the space. ~~An algebraic method to compute~~ these two solutions (the [[Bancroft Method\|Bancroft's method]]). Nevertheless, the usual way to solve this non linear problem is to linearise the equations around an approximate user position and solve iteratively (see [[Solving navigation equations]]).		* '''From 2-D to 3-D positioning''': It is not difficult to extend the previous 2-D geometrical construction to 3-D case of GNSS positioning, and to show that at least 4 satellites are needed to compute the three receiver coordinates plus clock. In this case, the previous circles and hyperbolas are generalised to spheres and hyperboloids, which intersect in two possible solutions. For a ground receiver, one of such solutions is on the earth's surface and the other far away in the space. From algebric point of view these two solutions can be obtained through the [[Bancroft Method\|Bancroft's method]]. Nevertheless, the usual way to solve this non linear problem is to linearise the equations around an approximate user position and solve iteratively (see [[Solving navigation equations]]).
	* '''Dilution of precision (DOP)''': The geometry of the satellites, i.e., how the user sees them, affects the positioning error. This is illustrated in figure 1.6, where the size and shape of the region change depending on their relative position. This effect is called Dilution Of Precision (DOP) and it is studied in [[Positioning Error\|Predicted Accuracy: Dilution of Precision]].		* '''Dilution of precision (DOP)''': The geometry of the satellites, i.e., how the user sees them, affects the positioning error. This is illustrated in figure 1.6, where the size and shape of the region change depending on their relative position. This effect is called Dilution Of Precision (DOP) and it is studied in [[Positioning Error\|Predicted Accuracy: Dilution of Precision]].

Filipe.Pelica: rewording.

2014-05-29T15:57:13Z

rewording.

← Older revision		Revision as of 15:57, 29 May 2014
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	These 20 seconds will correspond to the propagation time of sound from the lighthouse (transmitter) to the ship (receiver). The distance ρ between them can be obtained multiplying this value by the speed of sound <math>v \simeq 340 m/s</math>. That is, ρ = 20 s × 340 m/s = 6.8 km.		These 20 seconds will correspond to the propagation time of sound from the lighthouse (transmitter) to the ship (receiver). The distance ρ between them can be obtained multiplying this value by the speed of sound <math>v \simeq 340 m/s</math>. That is, ρ = 20 s × 340 m/s = 6.8 km.

	Obviously, with a single lighthouse it is only possible to determine a single measure of distance. So, the ship could be at any point over a ~~circle~~ of radius ρ, see figure 1.		Obviously, with a single lighthouse it is only possible to determine a single measure of distance. So, the ship could be at any point over a circumference of radius ρ, see figure 1.


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	That is, the radius of the ~~circles~~ of figure 1 will vary by an unknown amount <math>d t</math>, see figure 2. From hereafter we will call <math>R_i</math> as pseudorange, because it contains an unknown error <math>d t</math>. At first glance, it might seem that the intersection of these ~~circles~~ (with an undefined radius <math>R_i</math>) could reach any point on the plane (for an arbitrary <math>d \tau</math> value). However, they will only intersect on the branches of a hyperbola, whose foci are located at the two lighthouses, see figure 2. Indeed, as the clock offset cancels when differencing the pseudoranges, the possible ship locations must verify the following equation (which defines an hyperbola, see figure 2):		That is, the radius of the circumferences of figure 1 will vary by an unknown amount <math>d t</math>, see figure 2. From hereafter we will call <math>R_i</math> as pseudorange, because it contains an unknown error <math>d t</math>. At first glance, it might seem that the intersection of these circumferences (with an undefined radius <math>R_i</math>) could reach any point on the plane (for an arbitrary <math>d \tau</math> value). However, they will only intersect on the branches of a hyperbola, whose foci are located at the two lighthouses, see figure 2. Indeed, as the clock offset cancels when differencing the pseudoranges, the possible ship locations must verify the following equation (which defines an hyperbola, see figure 2):

	::<math>R_1\, - \,R_2 = \rho_1\, - \,\rho_2\, = \,constant. \qquad \mbox{(2)}</math>		::<math>R_1\, - \,R_2 = \rho_1\, - \,\rho_2\, = \,constant. \qquad \mbox{(2)}</math>
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	To complete this analysis, the figure 1.4 shows another geometrical construction where the solution is in the centre of a ~~circle~~, with radius equal to the clock offset <math>d r=v \cdot d \tau</math>, and which is tangent to the three ~~circles~~ of ~~radii~~ <math>\rho_i</math> and centred in the lighthouses.		To complete this analysis, the figure 1.4 shows another geometrical construction where the solution is in the centre of a circumference, with radius equal to the clock offset <math>d r=v \cdot d \tau</math>, and which is tangent to the three circumferences of radius <math>\rho_i</math> and centred in the lighthouses.

	:[[File:Unknown Offset3.png\|none\|thumb\|400px\|'''''Figure 4:''''' Geometrical view 2-D positioning, complementing the figure 3.]]		:[[File:Unknown Offset3.png\|none\|thumb\|400px\|'''''Figure 4:''''' Geometrical view 2-D positioning, complementing the figure 3.]]


	Finally, and in order to simplify the explanation, let's go back to the situation of figure 1 where the ship and lighthouses clocks are assumed fully synchronised. If the range measurements were perfect, the sailor could determine his position as the intersection point of the two ~~circles~~ centred at F1 and F2 lighthouses. However, the measurements are not exact, having some measurement error <math>\varepsilon</math>. Figures 5 and 6 illustrate how this measurement error is translated to the coordinates estimate as an uncertainty region, which depends on the geometry defined by the ship and lighthouses relative positions.		Finally, and in order to simplify the explanation, let's go back to the situation of figure 1 where the ship and lighthouses clocks are assumed fully synchronised. If the range measurements were perfect, the sailor could determine his position as the intersection point of the two circumferences centred at F1 and F2 lighthouses. However, the measurements are not exact, having some measurement error <math>\varepsilon</math>. Figures 5 and 6 illustrate how this measurement error is translated to the coordinates estimate as an uncertainty region, which depends on the geometry defined by the ship and lighthouses relative positions.

	:[[File:Unknown Offset4.png\|none\|thumb\|400px\|'''''Figure 5:''''' The measurement noise <math>\varepsilon</math> is translated to the position estimate as an uncertainty region.]]		:[[File:Unknown Offset4.png\|none\|thumb\|400px\|'''''Figure 5:''''' The measurement noise <math>\varepsilon</math> is translated to the position estimate as an uncertainty region.]]

Teresa.Ferreira at 16:59, 3 December 2013

2013-12-03T16:59:10Z

← Older revision		Revision as of 16:59, 3 December 2013
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	\|Level=Basic		\|Level=Basic
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=ESA~~
	}}		}}

	The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.		The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.

Filipe.Pelica: Fixed article form

2013-12-03T11:51:03Z

Fixed article form

← Older revision		Revision as of 11:51, 3 December 2013
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			{{Article Infobox2
			\|Category=Fundamentals
			\|Level=Basic
			\|YearOfPublication=2011
			\|Logo=ESA
			}}

	The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.		The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.

Jaume.Sanz at 09:08, 17 February 2013

2013-02-17T09:08:03Z

← Older revision		Revision as of 09:08, 17 February 2013
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	~~{{Article Infobox2~~
	~~\|Category=Fundamentals~~
	~~\|Level=Basic~~
	~~\|YearOfPublication=2011~~
	~~\|Logo=ESA~~
	}}
	The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.		The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.

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	That is, the radius of the circles of figure 1 will vary by an unknown amount <math>d t</math>, see figure 2. From hereafter we will call <math>R_i</math> as pseudorange, because it contains an unknown error <math>d t</math>. At first glance, it might seem that the intersection of these circles (with an undefined radius <math>R_i</math>) could reach any point on the plane (for an arbitrary <math>d t</math> value). However, they will only intersect on the branches of a hyperbola, whose foci are located at the two lighthouses, see figure 2. Indeed, as the clock offset ~~<math>dt</math>~~ cancels when differencing the pseudoranges, the possible ship locations must verify the following equation (which defines an hyperbola, see figure 2):		That is, the radius of the circles of figure 1 will vary by an unknown amount <math>d t</math>, see figure 2. From hereafter we will call <math>R_i</math> as pseudorange, because it contains an unknown error <math>d t</math>. At first glance, it might seem that the intersection of these circles (with an undefined radius <math>R_i</math>) could reach any point on the plane (for an arbitrary <math>d \tau</math> value). However, they will only intersect on the branches of a hyperbola, whose foci are located at the two lighthouses, see figure 2. Indeed, as the clock offset cancels when differencing the pseudoranges, the possible ship locations must verify the following equation (which defines an hyperbola, see figure 2):

	::<math>R_1\, - \,R_2 = \rho_1\, - \,\rho_2\, = \,constant. \qquad \mbox{(2)}</math>		::<math>R_1\, - \,R_2 = \rho_1\, - \,\rho_2\, = \,constant. \qquad \mbox{(2)}</math>


	:[[File:Unknown Offset.png\|none\|thumb\|400px\|'''''Figure 2:''''' An (unknown) offset <math>d t</math> in the ship's clock produces a shift in the measured ranges (<math>R_i\, = \,\rho_i\, +\, dt</math>) (or pseudoranges), varying the circumference radius. But, as both pseudoranges <math>R_1\, and\, R_2</math> have been measured with the same clock, this offset cancels on the difference of ranges <math>R_1\, - \,R_2 = \rho_1\, - \,\rho_2\, = \,constant.</math> Thence, the ship is located at a branch of the hyperbola <math>R_1\, - \,R_2\, =\, constant.</math>]]		:[[File:Unknown Offset.png\|none\|thumb\|400px\|'''''Figure 2:''''' An (unknown) offset <math>d \tau</math> in the ship's clock produces a shift in the measured ranges (<math>R_i\, = \,\rho_i\, +\, dr</math>) (or pseudoranges), varying the circumference radius. But, as both pseudoranges <math>R_1\, and\, R_2</math> have been measured with the same clock, this offset cancels on the difference of ranges <math>R_1\, - \,R_2 = \rho_1\, - \,\rho_2\, = \,constant.</math> Thence, the ship is located at a branch of the hyperbola <math>R_1\, - \,R_2\, =\, constant.</math>]]


	A third lighthouse will reduce the uncertainty in the ship position to only two possible solutions. Such solutions are given by the intersection of two hyperbolas as it is illustrated in figure 3. Notice that, after estimating the ship coordinates, its clock offset ~~<math>d t</math>~~ can be found from equation (1).		A third lighthouse will reduce the uncertainty in the ship position to only two possible solutions. Such solutions are given by the intersection of two hyperbolas as it is illustrated in figure 3. Notice that, after estimating the ship coordinates, its clock offset can be found from equation (1).

	:[[File:Unknown Offset 2.png\|none\|thumb\|400px\|'''''Figure 3:''''' 2-D positioning with an unknown user clock offset.]]		:[[File:Unknown Offset 2.png\|none\|thumb\|400px\|'''''Figure 3:''''' 2-D positioning with an unknown user clock offset.]]


	To complete this analysis, the figure 1.4 shows another geometrical construction where the solution is in the centre of a circle, with radius equal to the clock offset <math>d t</math>, and which is tangent to the three circles of radii <math>\rho_i</math> and centred in the lighthouses.		To complete this analysis, the figure 1.4 shows another geometrical construction where the solution is in the centre of a circle, with radius equal to the clock offset <math>d r=v \cdot d \tau</math>, and which is tangent to the three circles of radii <math>\rho_i</math> and centred in the lighthouses.

	:[[File:Unknown Offset3.png\|none\|thumb\|400px\|'''''Figure 4:''''' Geometrical view 2-D positioning, complementing the figure 3.]]		:[[File:Unknown Offset3.png\|none\|thumb\|400px\|'''''Figure 4:''''' Geometrical view 2-D positioning, complementing the figure 3.]]

Timo.Kouwenhoven at 18:08, 6 November 2012

2012-11-06T18:08:13Z

← Older revision		Revision as of 18:08, 6 November 2012
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			{{Article Infobox2
			\|Category=Fundamentals
			\|Level=Basic
			\|YearOfPublication=2011
			\|Logo=ESA
			}}
	The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.		The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.

Jaume.Sanz at 10:14, 1 February 2012

2012-02-01T10:14:42Z

← Older revision		Revision as of 10:14, 1 February 2012
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	example, 20 seconds later <math>\left (t = n \times 10^m + 20^s \right)</math>.		example, 20 seconds later <math>\left (t = n \times 10^m + 20^s \right)</math>.

	These 20 seconds will correspond to the propagation time of sound from the lighthouse (transmitter) to the ship (receiver). The distance ρ between them can be obtained multiplying this value by the speed of sound <math>v \simeq ~~342~~ m/s</math>. That is, ρ = 20 s × 340 m/s = 6.8 km.		These 20 seconds will correspond to the propagation time of sound from the lighthouse (transmitter) to the ship (receiver). The distance ρ between them can be obtained multiplying this value by the speed of sound <math>v \simeq 340 m/s</math>. That is, ρ = 20 s × 340 m/s = 6.8 km.

	Obviously, with a single lighthouse it is only possible to determine a single measure of distance. So, the ship could be at any point over a circle of radius ρ, see figure 1.		Obviously, with a single lighthouse it is only possible to determine a single measure of distance. So, the ship could be at any point over a circle of radius ρ, see figure 1.

Jaume.Sanz at 10:14, 1 February 2012

2012-02-01T10:14:07Z

← Older revision		Revision as of 10:14, 1 February 2012
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	~~{{Article Infobox2~~
	~~\|Category=Fundamentals~~
	~~\|Authors=J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.~~
	~~\|Level=Basic~~
	~~\|YearOfPublication=2011~~
	~~\|Logo=gAGE~~
	}}
	The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.		The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.

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	example, 20 seconds later <math>\left (t = n \times 10^m + 20^s \right)</math>.		example, 20 seconds later <math>\left (t = n \times 10^m + 20^s \right)</math>.

	These 20 seconds will correspond to the propagation time of sound from the lighthouse (transmitter) to the ship (receiver). The distance ρ between them can be obtained multiplying this value by the speed of sound <math>v \simeq ~~340~~ m/s</math>. That is, ρ = 20 s × 340 m/s = 6.8 km.		These 20 seconds will correspond to the propagation time of sound from the lighthouse (transmitter) to the ship (receiver). The distance ρ between them can be obtained multiplying this value by the speed of sound <math>v \simeq 342 m/s</math>. That is, ρ = 20 s × 340 m/s = 6.8 km.

	Obviously, with a single lighthouse it is only possible to determine a single measure of distance. So, the ship could be at any point over a circle of radius ρ, see figure 1.		Obviously, with a single lighthouse it is only possible to determine a single measure of distance. So, the ship could be at any point over a circle of radius ρ, see figure 1.

Carlos.Lopez at 08:13, 23 January 2012

2012-01-23T08:13:57Z

← Older revision		Revision as of 08:13, 23 January 2012
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	{{Article Infobox2}}		{{Article Infobox2
			\|Category=Fundamentals
			\|Authors=J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
			\|Level=Basic
			\|YearOfPublication=2011
			\|Logo=gAGE
			}}
	The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.		The basic observable in a GNSS system is the time required for a signal to travel from the satellite (transmitter) to the receiver. This travelling time, multiplied by the speed of light, provides a measure of the apparent distance (pseudorange) between them.