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Bancroft Method

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FundamentalsFundamentals
Title Bancroft Method
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Advanced
Year of Publication 2011

The Bancroft method allows obtaining a direct solution of the receiver position and the clock offset, without requesting any "a priori" knowledge for the receiver location.


Raising and resolution

Let [math]PR^j[/math] the prefit-residual of satellite-[math]j[/math], computed from equation (1)

[math] R^j=\rho^j+c(\delta t-\delta t^j)+T^j+\hat{\alpha}\, I^j+TGD^j+\mathcal{M}^j+{\boldsymbol \varepsilon}^j \qquad \mbox{(1)}[/math]


after removing all model terms not needing the a priory knowledge of the receiver position:[footnotes 1]

[math] PR^j\equiv R^j +c\,\delta t^j-TGD^j \qquad \mbox{(2)}[/math]


Thence, neglecting the tropospheric and ionospheric terms, as well as the multipath and receiver noise, the equation (3)

[math] \begin{array}{r} R^j-D^j\simeq \sqrt{(x^j-x)^2+(y^j-y)^2+(z^j-z)^2}+c\,\delta t\\[0.3cm] j=1,2,...,n~~~~ (n \geq 4)\\ \end{array} \qquad \mbox{(3)}[/math]

can be written as:

[math] PR^j = \sqrt{(x^j-x)^2+(y^j-y)^2+(z^j-z)^2}+c \, \delta t \qquad \mbox{(4)}[/math]


Developing the previous equation (4), it follows:

[math]\left[{x^j}^2+{y^j}^2+{z^j}^2-{PR^j}^2 \right]-2 \left[x^j x+y^j y+z^j z-{PR^jc\,\delta t} \; \right] + \left[x^2+y^2+z^2-(c\,\delta t)^2 \right]=0 \qquad \mbox{(5)}[/math]


Then, calling [math]{\mathbf r}=[x,y,z]^T[/math] and considering the inner product of Lorentz [footnotes 2] the previous equation (5) can be expressed in a more compact way as:

[math]\frac{1}{2} \left \langle \left[ \begin{array}{c} {\mathbf r}^j\\ PR^j\\ \end{array} \right], \left[ \begin{array}{c} {\mathbf r}^j\\ PR^j\\ \end{array} \right] \right \rangle - \left \langle \left[ \begin{array}{c} {\mathbf r}^j\\ PR^j\\ \end{array} \right], \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right] \right \rangle + \frac{1}{2} \left \langle \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right], \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right] \right \rangle =0 \qquad \mbox{(6)}[/math]


The former equation can be raised for every satellite (or prefit-residual [math]PR^j[/math]).


If four measurements are available, thence, the following matrix can be written, containing all the available information on satellite coordinates and pseudoranges (every row corresponds to a satellite):

[math] {\mathbf B}= \left( \begin{array}{cccc} x^1&y^1&z^1&PR^1\\ x^2&y^2&z^2&PR^2\\ x^3&y^3&z^3&PR^3\\ x^4&y^4&z^4&PR^4\\ \end{array} \right) \qquad \mbox{(7)}[/math]


Then, calling:

[math]\Lambda= \frac{1}{2} \left \langle \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right], \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right] \right \rangle \; , \; {\mathbf 1}= \left[ \begin{array}{c} 1\\ 1\\ 1\\ 1\\ \end{array} \right] \; , \; {\mathbf a}= \left[ \begin{array}{c} a_1\\ a_2\\ a_3\\ a_4\\ \end{array} \right] \; \mbox{being} \; \; a_j= \frac{1}{2} \left \langle \left[ \begin{array}{c} {\mathbf r}^j\\ PR^j\\ \end{array} \right], \left[ \begin{array}{c} {\mathbf r}^j\\ PR^j\\ \end{array} \right] \right \rangle \qquad \mbox{(8)}[/math]


The four equations for pseudorange can be expressed as:

[math]{\mathbf a} -{\mathbf B}\,{\mathbf M} \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right] +\Lambda \; {\mathbf 1}=0\;\;,\;\;\;\; \mbox{being} \;\;\;\;\;\; {\mathbf M}=\left( \begin{array}{cccc} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&-1\\ \end{array} \right) \qquad \mbox{(9)}[/math]


from where:

[math] \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right] ={\mathbf M} {\mathbf B}^{-1} (\Lambda \; {\mathbf 1} + {\mathbf a}) \qquad \mbox{(10)}[/math]


Then, taking into account the following equality

[math]\langle {\mathbf M}{\mathbf g},{\mathbf M}{\mathbf h} \rangle=\langle {\mathbf g},{\mathbf h} \rangle \qquad \mbox{(11)}[/math],


and that

[math] \Lambda= \frac{1}{2} \left \langle \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right], \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right] \right \rangle \qquad \mbox{(12)}[/math],


from the former expression (10), one obtains:

[math] \left \langle {\mathbf B}^{-1} {\mathbf 1}, {\mathbf B}^{-1} {\mathbf 1} \right \rangle \Lambda^2+ 2\left [ \left \langle {\mathbf B}^{-1} {\mathbf 1}, {\mathbf B}^{-1} {\mathbf a} \right \rangle -1 \right ] \Lambda + \left \langle {\mathbf B}^{-1} {\mathbf a}, {\mathbf B}^{-1} {\mathbf a} \right \rangle =0 \qquad \mbox{(13)}[/math]


The previous expression (13) is a quadratic equation in [math]\Lambda[/math] (note that matrix [math]{\mathbf B}[/math] and the vector [math]\mathbf a [/math] are also known) and provides two solutions, that introduced in expression (10) provides the searched solution:

[math] \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right] \qquad \mbox{(14)}[/math].

The other solution is far from the earth surface.

Generalisation to the case of [math]n[/math]-measurements:

If more than four observations are available, the matrix [math]{\mathbf B}[/math] is not square. However, multiplying by [math]{\mathbf B}^T[/math], one obtains (Least Squares solution):

[math] {\mathbf B}^T{\mathbf a} -{\mathbf B}^T {\mathbf B}\,{\mathbf M} \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right] +\Lambda \; {\mathbf B}^T {\mathbf 1}=0 \qquad \mbox{(15)}[/math]


where:

[math] \left[ \begin{array}{c} {\mathbf r}\\ c\,\delta t\\ \end{array} \right] ={\mathbf M} ({\mathbf B}^T {\mathbf B})^{-1}{\mathbf B}^T(\Lambda \; {\mathbf 1} + {\mathbf a}) \qquad \mbox{(16)}[/math]


and then:

[math] \begin{array}{r} \left \langle ({\mathbf B}^T {\mathbf B})^{-1} {\mathbf B}^T{\mathbf 1}, ({\mathbf B}^T {\mathbf B})^{-1} {\mathbf B}^T{\mathbf 1} \right \rangle \Lambda^2+ 2\left [ \left \langle ({\mathbf B}^T {\mathbf B})^{-1} {\mathbf B}^T{\mathbf 1}, ({\mathbf B}^T {\mathbf B})^{-1} {\mathbf B}^T{\mathbf a} \right \rangle -1 \right ] \Lambda +\\[0.3cm] + \left \langle ({\mathbf B}^T {\mathbf B})^{-1} {\mathbf B}^T{\mathbf a}, ({\mathbf B}^T {\mathbf B})^{-1} {\mathbf B}^T{\mathbf a} \right \rangle =0 \end{array} \qquad \mbox{(17)}[/math]


Notes

  1. ^ The tropospheric and ionospheric terms, [math]T^j[/math] and [math]\hat{\alpha} \,I^j[/math], can not be included, because the need to consider the satellite-receiver ray. Off course, after an initial computation of the receiver coordinates, the method could be iterated using the ionospheric and tropospheric corrections to improve the solution.
  2. ^ [math] \left \langle{\mathbf a},{\mathbf b}\right \rangle={\mathbf a}^{t} \; {\mathbf M} \; {\mathbf b}= \left[ \begin{array}{c} a_1,a_2,a_3,a_4 \end{array} \right] \left( \begin{array}{cccc} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&-1\\ \end{array} \right) \left[ \begin{array}{c} b_1\\ b_2\\ b_3\\ b_4 \end{array} \right] [/math]