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Combined difference square observation-based ambiguity determination for ground-based positioning system

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Abstract

In the absence of a global navigation satellite system, a ground-based positioning system can provide stand-alone positioning service and has advantages in layout flexibility of terrestrial base stations that broadcast ranging signals. To realize precise point positioning (PPP) in ground-based positioning systems, the carrier phase ambiguity must be determined for the receiver. On-the-fly (OTF) ambiguity determination methods are desirable for their convenience in practice. In most existing OTF methods based on the initial position estimate obtained from code measurements or other measuring instruments, the nonlinear term representing true distances are linearized by a series expansion. However, due to the severe nonlinear effects, if the accuracy of the initial position estimate is relatively poor, such linearization will result in large errors and convergence difficulties. Moreover, the more accurate initial estimate a method requires, the more inconvenient it will be. To avoid the dependence on the initial estimate, we proposed a combined difference square (CDS) observation and it provides a framework to eliminate the nonlinear terms in the difference square observations by linear combination. Based on this, a rotational-symmetry CDS (RS-CDS) observation-based ambiguity determination method is proposed, which needs no a priori information or reliable code measurements and is especially suitable for dynamic applications. In addition, it does not require accurate time synchronization of base stations, making the deployment of the overall system easier. The numerical simulations show that geometry diversity effectively improves the performance of ambiguity determination. Two real-world experiments indicate that the proposed method enables PPP for ground-based positioning systems without accurate time synchronization.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (NSFC), under Grant 61771272. The datasets of the two experiments are available from the corresponding author on reasonable request.

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Authors and Affiliations

  1. Department of Electronic Engineering, Tsinghua University, Beijing, China

    Tengfei Wang, Zheng Yao & Mingquan Lu

Authors
  1. Tengfei Wang
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  2. Zheng Yao
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Corresponding author

Correspondence to Zheng Yao.

Appendices

Appendix A: RS-CDS observations at a same position

In the following analysis, the noise is temporarily ignored, and it will be seen that an RS-CDS observation can be linearly represented by others at the same position.

Assume \({\mathbf {u}}_{m_{1}}={\mathbf {u}}_{m_{q}}\)for \(1\le q\le M\). Then, with (1) and (4), we have

$$\begin{aligned} \phi _{m_{1}}^{i}-\phi _{m_{q}}^{i}=f_{c}\tau (m_{1}-m_{q}) \end{aligned}$$
(31)

and

$$\begin{aligned} \phi _{m_{q}}^{ij}= & {} \lambda ^{-1}(\Vert {\mathbf {s}}_{i}-{\mathbf {u}}_{m_{q}}\Vert -\Vert {\mathbf {s}}_{j}-{\mathbf {u}}_{m_{q}}\Vert )+N_{ij}-f_{c}\delta t_{ij}.\nonumber \\ \end{aligned}$$
(32)

With (32), it can be obtained that

$$\begin{aligned} \phi _{m_{1}}^{ij}=\phi _{m_{q}}^{ij},\ {\mathrm{for}}\ 1\le q\le M. \end{aligned}$$
(33)

With (31) and (33), we have

$$\begin{aligned}&\left[ (\phi _{m_{1}}^{i})^{2}-(\phi _{m_{1}}^{j})^{2}\right] -\left[ (\phi _{m_{q}}^{i})^{2}-(\phi _{m_{q}}^{j})^{2}\right] \nonumber \\&\quad = \phi _{m_{1}}^{ij}(\phi _{m_{1}}^{i}+\phi _{m_{1}}^{j})-\phi _{m_{q}}^{ij}(\phi _{m_{q}}^{i}+\phi _{m_{q}}^{j})\nonumber \\&\quad = \phi _{m_{1}}^{ij}(\phi _{m_{1}}^{i}+\phi _{m_{1}}^{j}-\phi _{m_{q}}^{i}-\phi _{m_{q}}^{j})\nonumber \\&\quad = 2\phi _{m_{1}}^{ij}f_{c}\tau (m_{1}-m_{q}). \end{aligned}$$
(34)

Then, it can be obtained that

$$\begin{aligned}&y_{m_{1}1K}^{ij}-y_{m_{q}1K}^{ij}\nonumber \\&\quad = (m_{1}-m_{q})\left\{ [(\phi _{K}^{i})^{2}-(\phi _{K}^{j})^{2}]-[(\phi _{1}^{i})^{2}-(\phi _{1}^{j})^{2}]\right\} \nonumber \\&\qquad +(1-K)\left\{ [(\phi _{m_{1}}^{i})^{2}-(\phi _{m_{1}}^{j})^{2}]-[(\phi _{m_{q}}^{i})^{2}-(\phi _{m_{q}}^{j})^{2}]\right\} \nonumber \\&\quad = (m_{1}-m_{q})\left\{ [(\phi _{K}^{i})^{2}-(\phi _{K}^{j})^{2}]-[(\phi _{1}^{i})^{2}-(\phi _{1}^{j})^{2}]\right\} \nonumber \\&\qquad +2(m_{1}-m_{q})(1-K)\phi _{m_{1}}^{ij}f_{c}\tau . \end{aligned}$$
(35)

In other words, we have

$$\begin{aligned} \frac{y_{m_{1}1K}^{ij}-y_{m_{2}1K}^{ij}}{m_{1}-m_{2}}=\frac{y_{m_{1}1K}^{ij}-y_{m_{q}1K}^{ij}}{m_{1}-m_{q}},\ {\mathrm{for}}\ 2\le q\le M. \end{aligned}$$
(36)

As a result, if the noise is neglected, the RS-CDS observations \(y_{m_{q}1K}^{ij}\) can be linearly represented by others at the same position and considered to be redundant. In fact, these observations provide no additional geometric diversity.

Appendix B: Expression of the autocovariance matrix

Denote \({\mathbf {Q}}_{kl}={\mathbb {E}}\{{\mathbf {n}}_{k}{\mathbf {n}}_{l}^{\mathrm {T}}\}\)and we have

$$\begin{aligned} {\mathbf {Q}}=&{\mathbb {E}}\{{\mathbf {n}}{\mathbf {n}}^{\mathrm {T}}\}\nonumber \\ =&\begin{bmatrix}{\mathbf {Q}}_{22}&\quad {\mathbf {Q}}_{23}&\quad \cdots&\quad {\mathbf {Q}}_{2(K-1)}\\ {\mathbf {Q}}_{32}&\quad {\mathbf {Q}}_{33}&\quad \cdots&\quad {\mathbf {Q}}_{3(K-1)}\\ \vdots&\quad \vdots&\quad \ddots&\quad \vdots \\ {\mathbf {Q}}_{(K-1)2}&\quad {\mathbf {Q}}_{(K-1)3}&\quad \cdots&\quad {\mathbf {Q}}_{(K-1)(K-1)} \end{bmatrix}. \end{aligned}$$
(37)

The expression is derived with the following assumptions:

  1. 1.

    The measurement noise \(w_{k}^{i}\) and the unmodeled clock error \(e_{k}\) are independent, that is \({\mathbb {E}}\{w_{k}^{i}e_{l}\}=0\).

  2. 2.

    Errors at different epochs are independent. In other words, for \(k\ne l\), we have \({\mathbb {E}}\{w_{k}^{i}w_{l}^{j}\}=0\) and \({\mathbb {E}}\{e_{k}e_{l}\}=0\).

  3. 3.

    The signal noise of different base stations is independent, that is \({\mathbb {E}}\{w_{k}^{i}w_{l}^{j}\}=0\) for \(i\ne j\).

According to (9), we have

$$\begin{aligned} n_{k}^{ij}= & {} \lambda ^{-1}\Vert {\mathbf {s}}_{i}-{\mathbf {u}}_{k}\Vert w_{k}^{i}-\lambda ^{-1}\Vert {\mathbf {s}}_{j}-{\mathbf {u}}_{k}\Vert w_{k}^{j}\nonumber \\&+\,\lambda ^{-1}(\Vert {\mathbf {s}}_{i}-{\mathbf {u}}_{k}\Vert -\Vert {\mathbf {s}}_{j}-{\mathbf {u}}_{k}\Vert )f_{c} e_{k}. \end{aligned}$$
(38)

As can be seen from (38), it is necessary to know the true distances \(\Vert {\mathbf {s}}_{i}-{\mathbf {u}}_{k}\Vert \) to compute the accurate statistical characteristics of noise. However, this is impossible until the positioning procedure is completed. So the following approximation is made to simplify the derivation for all k

$$\begin{aligned} {\mathbb {E}}\left\{ \left( \lambda ^{-1}\Vert {\mathbf {s}}_{i}-{\mathbf {u}}_{k}\Vert w_{k}^{i}\right) ^{2}\right\}&\approx \sigma _{w}^{2} \end{aligned}$$
(39)
$$\begin{aligned} {\mathbb {E}}\left\{ \left[ \lambda ^{-1}(\Vert {\mathbf {s}}_{i}-{\mathbf {u}}_{k}\Vert -\Vert {\mathbf {s}}_{j}-{\mathbf {u}}_{k}\Vert )f_{c}e_{k}\right] ^{2}\right\}&\approx \sigma _{e}^{2},\ {\mathrm{for}}\ i\ne j \end{aligned}$$
(40)

where \(\sigma _{w}^{2}\) and \(\sigma _{e}^{2}\) are approximate estimates. Then, it can be obtained that

$$\begin{aligned} {\mathbb {E}}\left\{ n_{k}^{i1}n_{l}^{j1}\right\} =\delta _{kl}\left( \delta _{ij}\sigma _{w}^{2} +\sigma _{w}^{2}+\sigma _{e}^{2}\right) \end{aligned}$$
(41)

where

$$\begin{aligned} \delta _{ij}={\left\{ \begin{array}{ll} 0, &{} i\ne j\\ 1, &{} i=j \end{array}\right. } \end{aligned}$$
(42)

With (41), we have

$$\begin{aligned}&{\mathbb {E}}\left\{ n_{k1K}^{i1}n_{l1K}^{j1}\right\} \nonumber \\&\quad = [(K-k)(K-l)+(k-1)(l-1)] \left( \delta _{ij}\sigma _{w}^{2}+\sigma _{w}^{2}+\sigma _{e}^{2}\right) \nonumber \\&\qquad +\delta _{kl}\delta _{ij}(1-K)^{2}\sigma _{w}^{2}+\delta _{kl}[(1-K)^{2}+(K-k)^{2}\nonumber \\&\qquad +(k-1)^{2}](\sigma _{w}^{2}+\sigma _{e}^{2}). \end{aligned}$$
(43)

Then, it can be obtained from (43) that

$$\begin{aligned} {\mathbf {Q}}_{kl}=&\, 4[(K-k)(K-l)+(k-1)(l-1)]\nonumber \\&\quad \cdot [\sigma _{w}^{2}{\mathbf {I}}_{L-1}+(\sigma _{w}^{2}+\sigma _{e}^{2}){\mathbf {1}}_{L-1}{\mathbf {1}}_{L-1}^{\mathrm {T}}]\nonumber \\&\quad +4\delta _{kl}(1-K)^{2}\sigma _{w}^{2}{\mathbf {I}}_{L-1}+4\delta _{kl}[(1-K)^{2}\nonumber \\&\quad +(K-k)^{2}+(k-1)^{2}](\sigma _{w}^{2}+\sigma _{e}^{2}){\mathbf {1}}_{L-1}{\mathbf {1}}_{L-1}^{\mathrm {T}} \end{aligned}$$
(44)

where \({\mathbf {I}}_{L-1}\) represents an \(L-1\) dimensional identity matrix, and \({\mathbf {1}}_{L-1}\) denotes a column vector of \(L-1\) elements that are all one.

By substituting (44) into (37), the approximate expression of \({\mathbf {Q}}\) can be obtained.

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Wang, T., Yao, Z. & Lu, M. Combined difference square observation-based ambiguity determination for ground-based positioning system. J Geod 93, 1867–1880 (2019). https://doi.org/10.1007/s00190-019-01288-0

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