Precise Point Positioning (PPP): A High-Accuracy Positioning Technique without the need of additional Infrastructure

1. Joint Precise Point Positioning (PPP) and Attitude Determination

Converse to RTK positioning, the method of PPP does not require data of close-by reference receivers. Even though not as accurate as RTK, PPP offers a great accuracy in the magnitude of around 1-2 dm horizontally and approximately twice the value for the vertical positioning. The basis for this precise positioning is a consideration of several models, an estimation of more unknowns, and highly accurate satellite data. The latter needs to be provided by external correction services. However, in contrast to RTK, these corrections are globally valid and starting with the Galileo High Accuracy Service (HAS), they are potentially directly received from the satellites. Thus, no external communication link is necessary for highly precise positioning.

The pseudorange measurements  and carrier phase measurements  from a satellite received by a user  on a frequency are given as follows.

1.1. Pseudorange Measurements

We start with the model for pseudorange measurements for satellite k observed by user u on frequency m given by:

\(
\newcommand{\R}[1]{{\mathrm{#1}}}
\begin{eqnarray*}
\rho_{u,m}^k &=& (\vec{e}_u^{\:k})^{\R{T}}\left(\vec{x}_u-\vec{x}^k\right) + c\left(\delta\tau_u-\delta\tau^k\right) + m_{\R{T}}(\theta_u^k)T_{\R{z},u} + q_{1m}^2I_u^k \\ && + b_{u,m} – b_m^k + \Delta\rho_{\R{MP},u,m}^k + \eta_{u,m}^k
\end{eqnarray*}
\)

with the following notations:

Notation	Description
\( \vec{e}_u^{\:k} \)	Normalized line of sight vector from satellite to receiver
\( \vec{e}_u \)	Receiver position
\( \vec{x}^k \)	Satellite position
\( c \)	Speed of light
\( \delta\tau_u \)	Receiver clock offset
\( m_{\R{T}} \)	Tropospheric mapping function
\( \theta_u^k \)	Satellite elevation angle at receiver location
\( T_{\R{z},u} \)	Tropospheric zenith delay
\( q_{1m}^2 \)	Ratio of frequencies
\( I_u^k \)	Ionospheric slant delay on reference frequency (L1/ E1)
\( b_{u,m} \)	Receiver code bias
\( b_m^k \)	Satellite code bias
\( \Delta\rho_{\R{MP},u,m}^k \)	Pseudorange multipath error
\( eta_{u,m}^k \)	Pseudorange measurement noise

1.2. Carrier Phase Measurements

The carrier phase measurement is much less noisy but ambiguous (due to the periodicity of the sinusoidal carrier signal) and affected by other biases, i.e.

\(
\begin{eqnarray*}
\lambda_m\varphi_{u,m}^k
&=& (\vec{e}_u^{\:k})^{\R{T}}\left(\vec{x}_u-\vec{x}^k\right)
+ c\left(\delta\tau_u-\delta\tau^k\right)
+ m_{\R{T}}(\theta_u^k)T_{\R{z},u} – q_{1m}^2I_u^k \\
&& + \lambda_m N_{u,m}^k + \beta_{u,m} – \beta_m^k
+ \lambda_m\Delta\varphi_{\R{MP},u,m}^k + \varepsilon_{u,m}^k
\end{eqnarray*}
\)

with the following additional parameters:

\( \lambda_m \)	Wavelength of carrier phase
\( N_{u,m}^k \)	Carrier phase integer ambiguity
\( \beta_{u,m} \)	Receiver phase bias
\( \beta_m^k \)	Satellite phase bias
\( \Delta\varphi_{\R{MP},u,m}^k \)	Carrier multipath error
\( \varepsilon_{u,m}^k \)	Carrier phase measurement noise

1.3. Doppler Measurements

The Doppler measurement can be considered as a scaled version of the time-derivative of the carrier phase measurement. It is modelled as:

\(
\begin{eqnarray*}
f_{\R{D},u,m}^k
&=& -f_m/c\cdot\left((\vec{e}_u^{\:k})^{\R{T}}\left(\vec{v}_u-\vec{v}^k\right)
+ c\left(\delta\dot{\tau}_u-\delta\dot{\tau}^k\right)\right) \\
&& + \Delta f_{\R{D},\R{MP},u,m}^k + \eta_{f_{\R{D}},u,m}^k
\end{eqnarray*}
\)

with the following notations:

\( f_m \)	Carrier frequency
\( \vec{v}_u \)	Receiver velocity
\( \vec{v}^k \)	Satellite velocity
\( \delta\dot{\tau}_u \)	Receiver clock drift
\( \delta\dot{\tau}^k \)	Satellite clock drift
\( \Delta f_{\R{D},\R{MP},u,m}^k \)	Doppler multipath error
\( \eta_{f_{\R{D}},u,m}^k \)	Doppler measurement noise

1.4. Measurement Precorrections

Utilising the concept of single differences, the terms which are common to a receiver are eliminated, i.e., the receiver clock offset as well as the receiver biases. While for RTK only the user position and the phase ambiguities need to be estimated, the atmospheric parameters for the zenith delay of the troposphere and for the ionospheric slant delays on the first frequency are also unknowns in the PPP algorithm. The ANavS positioning framework offers the possibility to also build the ionosphere-free combination of measurements to get rid of the slant delays as states of the estimating filter.

The complete procedure of all the modelling and estimation techniques can be found in the following figures.

2. Description of HAS PPP User Algorithm

This section includes a description of the proposed reference HAS PPP user algorithm. It uses a Kalman filter with an integrated partial or full integer ambiguity resolution, that processes satellite-satellite single differences of un-combined measurements as this approach is considered to be most attractive.

The following figure shows the two main blocks of the HAS PPP user algorithm that are relevant for the Detailed Processing Model (DPM) and their input and output interfaces. The first main block is the Pre-Processing of Observation, Navigation and Correction data (PPO), which includes e.g. the determination of the standard PVT solution and the forming of satellite-satellite single differences. The second main block includes the actual High Accuracy Service PVT solution (HAS) with a Kalman filter and integer ambiguity resolution.

 

2.1 Pre-Processing Software Module (PPO)

Pre-Processing steps (PPO) of HAS PPP User Algorithm:

1. Determination of satellite positions, velocities, clock offsets and clock drifts
   from broadcast ephemeris data
2. Determination of receiver position and clock offset with iterative least-squares (Standard PVT)
3. Determination of absolute receiver velocity and clock drift with least-squares (Standard PVT)
4. Determination of HAS-corrected satellite positions and clock offsets
5. Checking of measurements for outliers
6. Selection of a reference satellite for each constellation
7. Determination of satellite-satellite single difference measurements
8. 8a.   Detection and correction of cycle slips
9. 8b.   Determination of measurement noise and process noise covariance matrices

2.2 High Accuracy PPP Software Module (HAS)

Main processing steps (HAS) of HAS PPP User Algorithm:

9. Initialization of state parameters:
   Least-squares estimation of absolute receiver position, velocity, tropospheric zenith delay, ionospheric slant delays and float ambiguities using HAS-based pre-corrected SD measurements.
10. Linear prediction of state parameters
    (position, velocity, tropospheric zenith delay, ionospheric slant delays and ambiguities)

         Iterative state update (shown in orange)

11. Determination of normalized satellite-receiver direction vectors
12. Determination of geometry matrix (linearized mapping between states and measurements)
13. Calculation and application of single difference HAS corrections
14. Update of predicted states and their covariance matrix
              with pre-processed measurements and their covariance matrix
15. Determination of subset of ambiguities to fix

         Integer ambiguity fixing with LAMBDA (shown in green)

16. Integer decorrelation of set of float ambiguities to be fixed
17. Sequential tree search for determination of fixed integer vector candidates
18. Application of inverse of integer decorrelation to fixed integer vector candidates
19. Checking of reliability of best fixed integer candidate with ratio test
20. Adjustment of all other states to exploit fixing of ambiguities
21. Reset of fixed solution to float solution
22. Initialization of ionospheric and ambiguity states for newly traced satellites
23. Application of site-displacement corrections (Earth tides, polar motion and ocean loading)

4. Challenges of PPP and Potential Solutions

PPP positioning requires GNSS satellite visibility. Challenging environments with severe multipath and/ or limited satellite visibility might prevent an accurate PPP position convergency. Moreover, measurement statistics might be inaccurate for highly-dynamic users in challenging environments.

The above challenges can be addressed by a sensor fusion with inertial measurements and wheel odometry. There exist different types of coupling, where as a tight coupling provides more accurate solutions than a loose coupling. For very challenging environments (e.g. tunnels, long sections in forests, etc.), an integration of visual sensors (camera or LiDAR) and a Simultaneous Localization and Mapping (SLAM) is recommended.

5. Applications of PPP Positioning

PPP positioning is very attractive for a wide range of applications including surveying, mapping, smart farming, robotics and automation, mining, the automotive, maritime, railway and aerospace (UAS) industries.

6. PPP Products of ANavS®

ANavS® is offering several products for PPP positioning:

Multi-Sensor RTK/PPP module – our standard solution for PPP and RTK (with handover between both techniques) positioning with a tight coupling of GNSS, INS and wheel odometry.

The Integrated Sensor Platform (ISP) – our solution for the most demanding applications. It includes 3 GNSS receivers, an inertial sensor, a wheel odometry interface, 2 cameras, a 3D LiDAR and a very powerful sensor fusion. It is the ideal solution for Autonomous Driving.