Matrix Definitions Phi and Q - Revision history

Carlos.Lopez at 11:34, 23 February 2012

2012-02-23T11:34:19Z

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Jaume.Sanz at 12:45, 23 January 2012

2012-01-23T12:45:16Z

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

	The determination of the state transition matrix <math>{\boldsymbol \Phi}</math> and Process Noise matrix <math>{\mathbf Q}</math> is usually based on physical models describing the estimation problem. For instance, for satellite tracking or orbit determination, they are derived from the orbital motion equations. Elemental formulations, i.e., for the SPP and PPP navigation, are covered by the examples given as follows:		The determination of the state transition matrix <math>{\boldsymbol \Phi}</math> and Process Noise matrix <math>{\mathbf Q}</math> is usually based on physical models describing the estimation problem. For instance, for satellite tracking or orbit determination, they are derived from the orbital motion equations. Elemental formulations, i.e., for the SPP and PPP navigation, are covered by the examples given as follows:

Carlos.Lopez at 09:59, 28 October 2011

2011-10-28T09:59:42Z

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	\|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=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
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Carlos.Lopez at 08:11, 9 August 2011

2011-08-09T08:11:47Z

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	[[Category:Fundamentals]]		[[Category:Fundamentals]]
			[[Category:Solving Navigation Equations]]

Carlos.Lopez: Created page with "{{Article Infobox2 |Category=Fundamentals |Title={{PAGENAME}} |Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain. |Level=Medium..."

2011-08-05T13:46:03Z

Created page with "{{Article Infobox2 |Category=Fundamentals |Title={{PAGENAME}} |Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain. |Level=Medium..."

New page

{{Article Infobox2
|Category=Fundamentals
|Title={{PAGENAME}}
|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
|Level=Medium
|YearOfPublication=2011
|Logo=gAGE
}}

The determination of the state transition matrix <math>{\boldsymbol \Phi}</math> and Process Noise matrix <math>{\mathbf Q}</math> is usually based on physical models describing the estimation problem. For instance, for satellite tracking or orbit determination, they are derived from the orbital motion equations. Elemental formulations, i.e., for the SPP and PPP navigation, are covered by the examples given as follows:

== Static positioning ==

The state vector to determine is given by <math>\widehat{\mathbf X}=(dx,dy,dz,\delta t)</math> where the coordinates <ref group="footnotes">We are referring to deviations from nominal values <math>(x_0,y_0,z_0)</math>, that is what it is estimated from the navigation equations.</ref> are considered as constants (because the receiver is kept fixed) and the clock offset can be modelled as a white noise with zero mean. Under these conditions matrix <math>\Phi</math> and <math>{\mathbf Q}</math> are given by:

::<math>
{\boldsymbol \Phi} (n)=\left(
\begin{array}{cccc}
1 & & & \\
& 1 & & \\
& & 1 & \\
& & & 0 \\
\end{array}
\right)
\qquad
{\mathbf Q}(n)=\left(
\begin{array}{cccc}
0 & & & \\
& 0 & & \\
& & 0 & \\
& & & \sigma _{\delta t}^2 \\
\end{array}
\right)
\qquad \mbox{(1)}
</math>

being <math>\sigma _{dt}</math> the uncertainty in the clock prediction model (for instance <math>\sigma _{dt}=1\,ms=300\,km</math> for a unknown clock i.e. 1 leap-millisecond). Notice that the prediction model for the coordinates is exact and, thence, the associated elements in matrix <math>{\mathbf Q}</math> are null.

== Kinematic positioning ==

* If it is a vehicle running at a high velocity, the coordinates can be modeled as a white noise with zero mean, the same as the clock offset:

:: <math>
{\boldsymbol \Phi} (n)=\left(
\begin{array}{cccc}
0 & & & \\
& 0 & & \\
& & 0 & \\
& & & 0 \\
\end{array}
\right) \qquad
{\mathbf Q}(n)=\left(
\begin{array}{cccc}
\sigma_{dx}^2 & & & \\
& \sigma_{dy}^2& & \\
& & \sigma_{dz}^2 & \\
& & & \sigma _{\delta t}^2 \\
\end{array}
\right)
\qquad \mbox{(2)}
</math>

* If it is a vehicle running at a low velocity, the coordinates can be modelled as a random walk process with its uncertainty growing with time:

::<math>
{\boldsymbol \Phi} (n)=\left(
\begin{array}{cccc}
1 & & & \\
& 1 & & \\
& & 1 & \\
& & & 0 \\
\end{array}
\right) \qquad
{\mathbf Q}(n)=\left(
\begin{array}{cccc}
Q^{\prime}_{dx} \Delta t & & & \\
& Q^{\prime}_{dy} \Delta t & & \\
& & Q^{\prime}_{dz} \Delta t & \\
& & & \sigma _{\delta t}^2 \\
\end{array}
\right)
\qquad \mbox{(3)}
</math>

==Notes==
<references group="footnotes"/>

[[Category:Fundamentals]]