Solving positioning problems with minimal data

Abstract

Global and local positioning systems (LPS) make use of nonlinear equations systems to calculate coordinates of unknown points. There exist several methods, such as Sturmfels’ resultant, Groebner bases and least squares, for dealing with this kind of equations. We introduce two methods for solving this problem with the aid of symbolic techniques relying on closed-form solutions for the solution set of a system of linear equations. We suppose the receiver just detects or chooses minimal data, i.e., four satellites in global positioning systems (GPS) or three stations in LPS. Both methods proceed by parameterizing the line joining two solution points to later solve a nonlinear univariate equation, either quadratic or with degree smaller than 6. The first one uses the Generalized Cramer Identities, which is a different presentation of the generalized Moore–Penrose inverse, and ends with a degree 6 univariate equation for GPS and a degree 4 univariate equation for LPS. The second one solves the system by dealing with a more geometric way, ending with a quadratic equation. Our approach covers all possible cases with a finite number of solutions, while Bancroft’s method cannot be applied when the four satellites, taking the clock bias as fourth coordinate, and the origin lay in the same hyperplane of ${\mathbb{R}}^{4}$, and the method by Grafarend and Shan fails when the four satellites are in the same plane in ${\mathbb{R}}^{3}$. The two proposed methods fail only when the pseudorange 4 point problem has infinite solutions.

Appendix: Generalized Cramer Identities

In this appendix we deduce the Generalized Cramer Identities which will be used for solving the linear systems of equations in (3) and (5). All the results here are taken from Diaz-Toca et al. (2005). Let $A \in {\mathbb{R}}^{m \times n}$ be any matrix. Let us introduce $A^\circ \in {\mathbb{R}}\left( t \right)^{m \times n}$ as the product

$$A^\circ = Q_{n}^{ - 1} A^{t} Q_{m}$$

where $Q_{n} = {\text{diagonal}}\left( {1,t,t^{2} , \ldots ,t^{n - 1} } \right)$, and $Q_{m} = {\text{diagonal}}\left( {1,t,t^{2} , \ldots ,t^{m - 1} } \right)$. In practice, if $A = \left( {a_{ij} } \right)$ then $A^\circ = \left( {t^{i - j} a_{j,i} } \right)$. For instance, we would have

$$\begin{aligned} &A = \left[ {\begin{array}{*{20}c} {a_{11} } & {\begin{array}{*{20}c} {a_{12} } & {a_{13} } & {a_{14} } \\ \end{array} } \\ {\begin{array}{*{20}c} {a_{21} } \\ {a_{31} } \\ \end{array} } & {\begin{array}{*{20}c} {a_{22} } & {a_{23} } & {a_{24} } \\ {a_{32} } & {a_{33} } & {a_{34} } \\ \end{array} } \\ \end{array} } \right], \\ \quad &A^\circ = \left[ {\begin{array}{*{20}c} {a_{11} } & {\begin{array}{*{20}c} {ta_{21} } & {t^{2} a_{31} } \\ \end{array} } \\ {\begin{array}{*{20}c} {t^{ - 1} a_{12} } \\ {t^{ - 2} a_{13} } \\ {t^{ - 3} a_{14} } \\ \end{array} } & {\begin{array}{*{20}c} {a_{22} } & {ta_{32} } \\ {t^{ - 1} a_{23} } & {a_{33} } \\ {t^{ - 2} a_{24} } & {t^{ - 1} a_{34} } \\ \end{array} } \\ \end{array} } \right] \end{aligned}$$

(A 1)

when $n = 3$ and $m = 4$. With this notation, we can introduce the following definition:

Definition

(Generalized Gram polynomials and coefficients) The generalized Gram polynomials, ${\mathcal{G}}_{k}^{'} \left( A \right)\left( t \right) = a_{k} \left( t \right) \in {\mathbb{R}}\left[ {t,\frac{1}{t}} \right]$, and the generalized Gram coefficients, ${\mathcal{G}}_{kl}^{'} \left( A \right) = a_{kl} \in {\mathbb{R}}$, are given by:

$$\left\{ {\begin{array}{*{20}l} \begin{aligned} \det \left( {I_{m} + zAA^\circ } \right) = 1 + a_{1} \left( t \right)z + \cdots a_{m} \left( t \right)z^{m} , \hfill \\ a_{k} \left( t \right) = t^{{ - k\left( {n - k} \right)}} \left( {\mathop \sum \limits_{i = 0}^{{k\left( {m + n - 2k} \right)}} a_{k,l} t^{l} } \right) \hfill \\ \end{aligned} \hfill \\ \end{array} } \right.$$

(A 2)

We also define ${\mathcal{G}}_{0}^{'} \left( A \right)\left( t \right) = 1$ and ${\mathcal{G}}_{l}^{'} \left( A \right)\left( t \right) = 0$ for $l > m$.

Generalized Gram coefficients provide the rank of the given matrix A. Next result is Lemma 2.4 of Diaz-Toca et al. (2005).

Lemma

Generalized Gram conditions for rank:

• 1. ${\rm{rank}}\left(A \right) \le r \ \Leftrightarrow \ {\mathcal{G}}_k^{\prime}\left(A \right)\left(t \right) = 0, \ \forall \,k \ge r,$

• 2. ${\rm{rank}}\left(A \right) = r \, \Leftrightarrow \,{\cal G}_r^{\prime}\left(A \right)\left(t \right) \ne 0$ and ${\mathcal{G}}_{k}^{\prime} \left(A \right)\left(t \right) = 0,\quad \forall \, k > r$.

The solution of a linear system of equations is found via the Generalized Moore–Penrose inverse of A, as can be seen in the following theorem.

Theorem

(Theorem 2.7 of Diaz-Toca et al. 2005). Let $a_{k\left( t \right)} = {\mathcal{G}}_{k}^{'} \left( A \right), v \in F$ and ${\text{rank}}\left( A \right) = k$ . Then

• 1. The generalized Moore–Penrose inverse of $A \in {\mathcal{M}}_{m \times n} \left( {\mathbb{R}} \right) \subset {\mathcal{M}}_{m \times n} \left( {{\mathbb{R}} ( {{t)}}} \right)$ is given by

  $$A^{{\dag_{r,t} }} = a_{r}^{ - 1} \left( {a_{r - 1} I_{n} - a_{r - 2} A^{ \circ } A + \cdots + \left( { - 1} \right)^{r - 1} \left( {A^{ \circ } A} \right)^{r - 1} } \right)A^{ \circ };$$

• 2. The linear system of equations Ax = v has a solution if and only if ${\mathcal{G}}_{r}^{'} \left( {A|v} \right) = 0$.

• 3. If $v \in Im A$ then $x = A^{{\dag_{r,t} }} v$ is a solution of the linear system of equations Ax = v. Moreover, it is the unique solution in the linear space generated by the columns of A.

Now, we are able to introduce the Generalized Cramer Identities. First of all, we need to write Gram coefficients of A as sums of squares of minors.

Lemma

(Lemma 2.9 in Diaz-Toca et al. 2005) Let $A \in {\mathcal{M}}_{n \times n} \left( {\mathbb{R}} \right) \subset {\mathcal{M}}_{n \times n} \left( {{\mathbb{R}}\left( t \right)} \right)$ and $\mu_{\alpha ,\beta }$ be the k-minor where the rows and columns retained are given by subscripts $\alpha = \left\{ {\alpha_{1} , \ldots , \alpha_{k} } \right\} \subset \left\{ {1, \ldots ,m} \right\}$ and $\beta = \left\{ {\beta_{1} , \ldots ,\beta_{k} } \right\} \subset \left\{ {1, \ldots ,n} \right\}$ . If $\left| \alpha \right| = \mathop \sum \nolimits_{i} \alpha_{i}$ and $\left| \beta \right| = \mathop \sum \nolimits_{i} \beta_{i}$ , define $S_{m,n,k,l}$ as

$$S_{m,n,k,l} = \left\{ {\left( {\alpha ,\beta } \right):\left| \alpha \right| = \left| \beta \right| = l} \right\}$$

Then,

$${\mathcal{G}}_{k,l}^{'} \left( A \right) = \mathop \sum \limits_{{\left( {\alpha ,\beta } \right) \in S_{m,n,k,l} }} \mu_{\alpha ,\beta }^{2}$$

(A 3)

is the formula for the Generalized Gram coefficient $a_{k,l} = {\mathcal{G}}_{k,l}^{'} \left( A \right)$.

Remark

The generalized Gram coefficients, ${\mathcal{G}}_{k,l}^{'} \left( A \right)$, are always non-negative.

Given two subscripts $\alpha = \left\{ {\alpha_{1} , \ldots ,\alpha_{k} } \right\} \subset \left\{ {1, \ldots ,m} \right\}$ and $\beta = \left\{ {\beta_{1} , \ldots ,\beta_{k} } \right\} \subset \left\{ {1, \ldots ,n} \right\},$ we define ${\text{Adj}}_{\alpha ,\beta } = \left( {I_{n} } \right)_{1:n,\beta } {\text{Adj}}\left( {A_{\alpha ,\beta } } \right)\left( {I_{m} } \right)_{\alpha ,1:m}$, where ${\text{Adj}}\left( {A_{\alpha ,\beta } } \right)$ is the adjoint of the submatrix of A where the rows and the columns retained are given by subscripts α and β, respectively. In a similar way, the submatrices $\left( {I_{n} } \right)_{1:n,\beta }$ and $\left( {I_{m} } \right)_{\alpha ,1:m}$ are obtained retaining the rows labeled by α and the columns labeled by β of the identity matrices $I_{m}$ and $I_{n}$, respectively.

With this notation, it is possible to prove that the following Cramer identity holds

$$\mu_{\alpha ,\beta } v = A {\text{Adj}}_{\alpha ,\beta } \left( A \right)v$$

(A 4)

for every subscript pair $\left( {\alpha ,\beta } \right)$ and every vector $v \in Im A$. Notice that if $A \in {\mathcal{M}}_{n \times n} \left( {\mathbb{R}} \right)$ and ${\text{rank}}\left( A \right) = n$, (A4) reduces to

$$v = \frac{1}{\left| A \right|}A {\text{Adj}}\left( A \right)v$$

(A 5)

which is the classical Cramer identity for representing the unique solution of a linear system of n equations with n unknowns when the rank of the coefficients matrix is equal to n.

Generalized Moore–Penrose pseudo-inverse $A^{{\dag_{r,t} }}$ can be written as a sum of weighted Cramer identities in the following form (Diaz-Toca et al. 2005, Proposition 2.10).

$$A^{{\dag_{r,t} }} = {\mathcal{G}}_{r}^{'} \left( A \right)\left( t \right)^{ - 1} \mathop \sum \limits_{\alpha ,\beta } \mu_{\alpha ,\beta } t^{\left| \alpha \right| - \left| \beta \right|} \mathrm{Adj}_{\alpha ,\beta } \left( A \right)$$

(A 6)

where the sum is taken over all $\left( {\alpha ,\beta } \right) \in S_{mnkl}$, with $0 \le l \le k\left( {m + n - 2k} \right)$.