Single Point Positioning with GPS Pseudoranges

Mass market GNSS receivers tend to only use pseudorange measurements for position and time offset computation due to the difficulty to use carrier phase measurements (low robustness of tracking mechanism, cycle slipping, ambiguous measurement, etc.). As a consequence, in this section, the estimation of a stand-alone user’s position using only GPS pseudorange measurements is illustrated to provide an overview of GNSS navigating. The performance of a single-frequency standard positioning GPS user can observe better than 10 m, 95% positioning and 20-ns, 95% timing accuracy worldwide autonomously using broadcast orbits and clocks [2], [5], [39]. In this section, the basic technique of Leas-Squares estimation (LSE) is implemented.

Least Squares Estimation

Let us define a generic linear system in the form of 𝑌_𝑘= 𝐻_𝑘𝑋_𝑘+ 𝜖_𝑘 where

 𝑌𝑘 represents the observables vector at instant k,

 𝑋𝑘 represents the states vector to be estimated at instant k,

 𝐻𝑘 is the design matrix linking 𝑌𝑘 and 𝑋𝑘,

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 𝜖𝑘 represents the Gaussian white observation noise, with 𝑅𝑘 the variance-covariance (VC-) matrix.

LSE is an optimal estimation process to find the best solution which fits the above linear system when the observation noise is Gaussian [37]. It uses the criterion of minimizing the sum of squared residuals:

𝑚𝑖𝑛𝑋̂_𝑘 ‖𝑌𝑘− 𝑌̂_𝑘‖² (2-33)

Inside the equation, 𝑌̂𝑘= 𝐻𝑘𝑋̂𝑘 is the estimation of observables based on states estimates 𝑋̂𝑘. The weighted version of LSE gives the best solution in the form:

𝑋̂_𝑘= (𝐻_𝑘^𝑇𝑊_𝑘𝐻_𝑘)⁻¹𝐻_𝑘^𝑇𝑊_𝑘𝑌_𝑘 (2-34) where 𝑊𝑘 is the weighting matrix which modulates the importance of each observation.

A common option is to use the inverse of the VC-matrix 𝑅𝑘 of measurements as weights: 𝑊𝑘= 𝑅_𝑘⁻¹. Measurements with lower noise level should be highly weighted. In such case, the VC-matrix of the solution 𝑋̂𝑘 is finally given by

𝑃_𝑘= (𝐻_𝑘^𝑇𝑊_𝑘𝐻_𝑘)⁻¹ (2-35)

The measurements residuals, 𝑟𝑘, are thus formulated as:

𝑟𝑘= 𝑌𝑘− 𝑌̂𝑘 = 𝑌𝑘− 𝐻𝑘𝑋̂𝑘 (2-36) In the quality assessment of an estimation, the measurement residuals play an important role.

Estimation of Position and Time

The previous estimation theory is based on the construction of a linear system. However, the measurement models between GNSS observables and the desired states, i.e. the user position and clock, is not a linear system. Therefore, an extended form of LSE should be applied in which a linearization is needed.

We denote 𝑃𝑢𝑗 the measured pseudorange between the 𝑗-th satellite and the receiver 𝑢, with 𝑗 ∈ [1, 𝐾].

𝐾 is the total number of satellites visible in view of the receiver 𝑢 . Let vectors 𝑿𝒖(𝑥, 𝑦, 𝑧) and 𝑿^𝒋(𝑥^𝑗, 𝑦^𝑗, 𝑧^𝑗), represent respectively the ECEF coordinates of the receiver’s phase center and the ECEF coordinates of the 𝑗-th satellite in orbit. The true user-to-satellite geometric range is thus

𝜌_𝑢^𝑗 = √(𝑥 − 𝑥^𝑗)² + (𝑦 − 𝑦^𝑗)² + (𝑧 − 𝑧^𝑗)² = ‖𝑿_𝒖− 𝑿^𝒋‖ (2-37) After applying the correction of the atmosphere effects and the satellite clock delays, the simplified form of the pseudorange then becomes

𝑃_𝑢^𝑗 = 𝜌_𝑢^𝑗 + 𝑐𝑙𝑘_𝑢+ 𝜖_𝑃

𝑢𝑗 (2-38)

with

 𝑐𝑙𝑘𝑢 the receiver clock delay in meter with respect to the time reference system and

 𝜖_P

𝑢

𝑗 the combined effect of the residual errors including the residual satellite clock delays, the residual atmosphere delays, the potential multipath errors, etc. Note that due to the model now

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𝑗 and 𝑿𝒖 is non-linear. The basic idea is to linearize the system around an approximate value of the state parameters, solve for the system, and then adjust the system in an iterative way. As a consequence, let us define 𝑿𝟎(𝑥₀, 𝑦₀, 𝑧₀) and 𝑐𝑙𝑘0 the first state guesses. The difference between the true and the approximated values are denoted as

𝜹𝑿_𝒖,𝟎= 𝑿_𝒖− 𝑿_𝟎 (2-39)

𝛿𝑐𝑙𝑘0= 𝑐𝑙𝑘 − 𝑐𝑙𝑘0 (2-40)

The position bias 𝜹𝑿𝒖,𝟎 is generally negligible compared to the distance ‖𝑿𝒖− 𝑿^𝒋‖ between the satellite and the rover on the ground which is usually of order 1000 km. According to the first-order approximation, the residual measurement 𝛿P_𝑢,0^𝑗 is expressed as

𝛿P_𝑢,0^𝑗 = P_𝑢^𝑗− P_𝑢,0^𝑗

= (‖𝑿𝟎+ 𝜹𝑿𝒖,𝟎 − 𝑿^𝒋‖ + 𝑐𝑙𝑘) − (‖𝑿𝟎− 𝑿^𝒋‖ + 𝑐𝑙𝑘0) + 𝜀̃_𝑃,𝑢^𝑗

≈ − 𝑿^𝒋−𝑿𝟎

‖𝑿^𝒋−𝑿_𝟎‖. 𝜹𝑿𝒖,𝟎+ 𝛿𝑐𝑙𝑘0+ 𝜀̃_𝑃,𝑢^𝑗

= −𝒆^𝒋. 𝜹𝑿𝒖,𝟎+ 𝛿𝑐𝑙𝑘0+ 𝜀̃_𝑃,𝑢^𝑗

where 𝒆_𝟎^𝒋 is the estimated line-of-sight (LOS) unit vector along the direction pointing from the initial estimate user position to the satellite, 𝒆_𝟎^𝒋 =_{‖𝑿𝒋−𝑿}^𝟏

𝟎‖(𝑥^𝑗−𝑥₀, 𝑦^𝑗−𝑦₀, 𝑧^𝑗−𝑧₀).

A set of 𝐾 linear equations, representing the K measurements, can be presented as:

𝜹𝑷𝟎=

The matrix 𝑯 is referred to be the geometry matrix describing the user-satellite geometry. The bias state vector [𝜹𝑿𝒖,𝟎, 𝛿𝑐𝑙𝑘₀]^𝑇 is then resolved according to Eq.(2-34) and added to the first state guess [𝑿_𝟎, 𝑐𝑙𝑘0]^𝑇 to refine new state estimate [𝑿𝟏, 𝑐𝑙𝑘1]^𝑇. The whole estimation process iterates with the new state estimate until the solution converges, e.g. norm([𝜹𝑿𝒖,𝟎, 𝛿𝑐𝑙𝑘0]) ≤ 1𝑒-5.

The stochastic performance of the GNSS positioning highly depends on two factors:

 The number and distribution of the tracked satellites in the sky with respect to the user position, which is referred to as the dilution of precision (DOP);

 The quality of the generated GNSS measurements, which is referred to as user equivalent range error (UERE). UERE is the total error budget representing the measurement model errors with respect to this system..

Consider a simplified case that the measurement VC-matric 𝑅𝑘 has only diagonal components identical to the square of the satellite UERE, noted 𝜎²𝑈𝐸𝑅𝐸. The VC-matrix of states 𝑃𝑋 is then expressed as stated in Eq.(2-35):

𝑃_𝑋= 𝜎²_{𝑈𝐸𝑅𝐸}(𝑯^𝑻𝑯)⁻¹

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Note that under the consumption that 𝜎²𝑈𝐸𝑅𝐸 related to satellites is invariant, the state quality is determined by the matrix 𝑷 = (𝑯^𝑻𝑯)⁻¹= (𝑃_𝑖𝑗)𝑖,𝑗∈{𝑥,𝑦,𝑧,𝑡}. 𝑷 only depends on the number of satellites and the relative geometry between the satellites and the user. To describe the translation of measurement noise level 𝜎𝑈𝐸𝑅𝐸 to the state noise level, the construction of the DOP terms, i.e. the geometric DOP (GDOP), the position DOP (PDOP) and the time DOP (TDOP), are given by

𝐺𝐷𝑂𝑃 = √𝑃𝑥,𝑥+ 𝑃𝑦,𝑦+ 𝑃𝑧,𝑧+ 𝑃𝑡,𝑡

𝑃𝐷𝑂𝑃 = √𝑃𝑥,𝑥+ 𝑃_𝑦,𝑦+ 𝑃_𝑧,𝑧 𝑇𝐷𝑂𝑃 = √𝑃𝑡,𝑡

Intuitively with the same number of visible satellites, a lower PDOP value and consequently a more accurate solution are expected with a good satellite distribution. In Figure 2-13, an illustration of a good distribution of satellites on the left side and a bad one on the right wide with satellites clustered in one side is provided.

When the position is resolved in a local ENU frame, the horizontal DOP (HDOP) and the vertical DOP (VDOP) describing respectively the precision in the horizontal plane and the vertical plane are introduced,

𝐻𝐷𝑂𝑃 = √𝑃𝑒,𝑒+ 𝑃_𝑛,𝑛; 𝑉𝐷𝑂𝑃 = √𝑃𝑢,𝑢

Figure 2-13. Good (left) and bad (right) GDOP cases (adopted from [43])

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Precise Positioning and the Problem Raised by Urban