receiverposition

Estimate GNSS receiver position and velocity

Syntax

recPos = receiverposition(p,satPos)

[recPos,recVel] = receiverposition(___,pdot,satVel)

recPos = receiverposition(gnssmeas)

[recPos,recVel] = receiverposition(___,gnssopts)

[recPos,recVel,hdop,vdop] = receiverposition(___)

[recPos,recVel,hdop,vdop,info] = receiverposition(___)

Description

recPos = receiverposition(p,satPos) returns the receiver position estimated from the pseudoranges and satellite positions.

[recPos,recVel] = receiverposition(___,pdot,satVel) also returns the receiver velocity estimated from the pseudorange rates pdot and satellite velocities satVel.

recPos = receiverposition(gnssmeas) returns the receiver position estimated raw GNSS measurements.

[recPos,recVel] = receiverposition(___,gnssopts) specifies options such as atmospheric models.

[recPos,recVel,hdop,vdop] = receiverposition(___) also returns the horizontal dilution of precision hdop and vertical dilution of precision vdop associated with the position estimate.

[recPos,recVel,hdop,vdop,info] = receiverposition(___) returns information about the clock bias, clock drift, and time dilution of precision.

Examples

Estimate Receiver Position Using GNSS Measurements from RINEX Files

Read observation and navigation data from the RINEX files.

[obsdata,obshdr] = rinexread("GODS00USA_R_20211750000_01H_30S_MO.rnx");
[navmsg,navhdr] = rinexread("GODS00USA_R_20211750000_01D_GN.rnx");

Get GNSS measurements from the observation and navigation data.

meas = gnssmeasurements(obsdata.GPS,navmsg.GPS);

Estimate the receiver position from the GNSS measurements and display the receiver position for the first time stamp of the RINEX file.

lla = receiverposition(meas);
lla(1,:)

ans = 1×3

   39.0205  -76.8273   36.6238

Estimate Receiver Position from Pseudoranges and Satellite Positions

Use the receiverposition function to estimate a GNSS receiver position. Get the satellite positions and velocities using the gnssconstellation function. Generate pseudoranges from these positions using the pseudoranges function.

Specify a receiver position in geodetic coordinates (latitude, longitude, altitude) and a receiver velocity in the local navigation frame.

recPos = [42 -71 50];
recVel = [1 2 3];

Get the satellite positions for the current time.

t = datetime('now');
[gpsSatPos,gpsSatVel] = gnssconstellation(t);

Get the pseudoranges and pseudorange rates between the GNSS receiver and the satellites.

[p,pdot] = pseudoranges(recPos,gpsSatPos,recVel,gpsSatVel);

Use the pseudoranges to estimate the receiver position and velocity. The values close to your original receiver position and velocity used to generate the satellite position and pseudoranges.

[lla,gnssVel] = receiverposition(p,gpsSatPos,pdot,gpsSatVel)

lla = 1×3

   42.0000  -71.0000   49.5625

gnssVel = 1×3

    1.0000    2.0140    2.9972

Calculate Receiver Positions with Ionospheric and Tropospheric Delays

Read the data from a RINEX navigation and observation messages. These RINEX files contain GNSS data for March 29, 2025.

navFileName = "GODE00USA_R_20250880000_01D_GN.rnx";
obsFileName = "GODE00USA_R_20250880800_01H_30S_MO.rnx";
[rnxnavdata,rnxnavhdr] = rinexread(navFileName);
[rnxobsdata,rnxobshdr] = rinexread(obsFileName);

Create ionosphere and troposphere delay models. Use the Klobuchar model for the ionosphere delay model with the GPS alpha and beta coefficients from the header file.

ionoParams = rnxnavhdr.IonosphericCorrections;
alpha = ionoParams(1).Parameters;
beta = ionoParams(2).Parameters;
ionoDelayModel = gnssIonosphere("klobuchar",alpha,beta);
tropoDelayModel = gnssTroposphere("saastamoinen");

Create the GNSS options for estimating the receiver position.

opts = gnssoptions(Ionosphere=ionoDelayModel,Troposphere=tropoDelayModel);

Get the GNSS measurements for the GPS satellites using the RINEX navigation and observation messages. Show the data for the first three time steps.

gpsData = gnssmeasurements(rnxobsdata.GPS,rnxnavdata.GPS);
gpsData(1:3,:)

ans=3×8 timetable
            Time            SatelliteID    Pseudorange                SatellitePosition                SatelliteClockBias    PseudorangeRate          SatelliteVelocity          SatelliteClockDrift    EphemerisAccuracy
    ____________________    ___________    ___________    _________________________________________    __________________    _______________    _____________________________    ___________________    _________________

    29-Mar-2025 08:00:00         1         2.4375e+07      -2.126e+07    -1.4117e+07     7.4018e+06        0.00019738              NaN          -471.47    -855.01      -2983        1.8076e-11                 2        
    29-Mar-2025 08:00:00         2          2.566e+07     -2.0524e+07    -1.6951e+07    -2.4589e+06       -0.00020789              NaN           429.94     23.954    -3140.9         9.436e-12                 2        
    29-Mar-2025 08:00:00         3         2.2235e+07     -1.2415e+07    -9.5047e+06      2.132e+07         0.0006808              NaN           578.88    -2675.8    -837.42        3.5243e-12                 2        

Calculate the receiver positions by using the GPS measurements and the options specified by the gnssoptions object. Show the receiver positions for the first three time steps.

[recPos,recVel,hdop,vdop,info] = receiverposition(gpsData,opts);
recPos(1:3,:)

ans = 3×3

   39.0217  -76.8268   13.7420
   39.0217  -76.8269   13.0003
   39.0217  -76.8268   12.2894

Input Arguments

p — Pseudoranges between satellites and receiver

N-element vector

Pseudoranges between the satellites and receiver, specified as an N-element vector in meters. N is the number of satellites in the constellation.

Data Types: single | double

satPos — Satellite positions

N-by-3 matrix of scalars

Satellite positions in the Earth-centered Earth-fixed (ECEF) coordinate system in meters, specified as an N-by-3 matrix of scalars. N is the number of satellites in the constellation.

Data Types: single | double

pdot — Pseudorange rates between satellites and receiver

N-element vector

Pseudorange rates between the satellites and receiver, specified as an N-element vector in meters per second. N is the number of satellites in the constellation.

Data Types: single | double

satVel — Velocity readings in Earth-centered Earth-fixed (ECEF) coordinate system (m/s)

N-by-3 matrix of scalar

Velocity readings of the GNSS receiver in the Earth-centered Earth-fixed (ECEF) coordinate system in meters per second, specified as a N-by-3 matrix of scalars. N is the number of satellites in the constellation.

Data Types: single | double

gnssmeas — Raw GNSS measurements

timetable

Raw GNSS measurements, specified as a timetable.

Use the gnssmeasurements function to get these measurements.

gnssopts — Options for calculating receiver position from GNSS measurements

gnssoptions object

Options for calculating receiver position from GNSS measurements, specified as a gnssoptions object.

Output Arguments

recPos — Receiver position

three-element vector of the form [lat lon alt]

Receiver position in geodetic coordinates, returned as a three-element vector of the form [latitude longitude altitude]

Data Types: single | double

recVel — Receiver velocity

three-element vector of the form [vx vy vz]

Receiver velocity in the local navigation frame using north-east-down (NED) coordinates, returned as a three-element vector of the form [vx vy vz].

Data Types: single | double

hdop — Horizontal dilution of precision

scalar

Horizontal dilution of precision, returned as a scalar.

Data Types: double

vdop — Vertical dilution of precision

scalar

Vertical dilution of precision, returned as a scalar.

Data Types: double

info — Information about clock bias, clock drift, and TDOP

structure

Information about clock bias, clock drift, and time dilution of precision (TDOP), returned as a structure containing these fields:

• ClockBias — Estimated bias error in receiver clock, in seconds.

• ClockDrift — Estimated drift error in receiver clock, in seconds per second.

• TDOP — Time dilution of precision.

Algorithms

Position Estimation Using Weighted Least Squares

For single-point positioning, the receiverposition function estimates a the position of a GNSS receiver using pseudorange measurements from multiple satellites. It uses an iterated weighted least squares (WLS) approach to solve for the receiver position and clock bias.

At each iteration i, the estimated state vector is updated as:

$${\widehat{x}}_{i+1}={\widehat{x}}_i+{(H^{T}WH)}^{-1}{H}^{T}W(y-h({\widehat{x}}_i))$$

• x — Receiver position r_r and receiver clock bias Δt_r.

  $$x={(r_r^T,c\Delta t_r)}^T$$

  c is the speed of light in meters per second.

• H — Jacobian matrix containing the line-of-sight (LOS) vectors e^s_r between satellites s and the receiver r for m total satellites.

  $$H=\left[\begin{array}{cc}-e_r^{1T}& 1\\ -e_r^{2T}& 1\\ ⋮& ⋮\\ -e_r^{mT}& 1\end{array}\right]$$

• y — Measurement vector containing the pseudorange measurements from all m satellites visible to the receiver.

  $$y={(P_r^1,P_r^2,\mathrm{...},P_r^m)}^T$$

  $$P_r^s$$ is a pseudorange measurement from a satellite s with respect to receiver r.

• W — Weight matrix.

• h(x) — Predicted pseudorange measurement for estimated state.

For more information about the weights and predicted pseudorange measurement equations, see the Pseudorange Measurement and Weights section.

Pseudorange Measurement and Weights

The predicted pseudorange measurement equation that the receiverposition function uses is defined as:

$$h(x)={\rho }_r^s+c\Delta t_r-c\Delta t^s+I_r^s+T_r^s$$

• $${\rho }_r^s$$ — Geometric range of satellite s with respect to receiver r. For more information about calculating the geometric range, see Geometric Range Between Satellite and Receiver

• c — Speed of light in meters per second.

• $$\Delta t_r$$ — Clock bias of receiver r in seconds.

• $$\Delta t^s$$ — Clock bias of satellite s in seconds.

• $$I_r^s$$ — Ionospheric delay of satellite s with respect to receiver r in meters.

• $$T_r^s$$ — Tropospheric delay of satellite s with respect to receiver r in meters.

And the weight matrix weighs each measurement according to its expected error, so less reliable measurements have less influence on the estimated position.

$$W=diag({\sigma }_1^{-2},{\sigma }_2^{-2},\mathrm{...},{\sigma }_m^{-2})$$

The total variance $${\sigma }_s{}^2$$ for a satellite s is defined as:

$${\sigma }_s{}^2=F^s{R}_r(a_{\sigma }^2+b_{\sigma }^2/sin(e{l}_r^s))+{\sigma }_{eph}^2+{\sigma }_{ion}^2+{\sigma }_{trop}^2+{\sigma }_{bias}^2$$

• $$F^s$$ — Satellite system error factor of satellite s. This table shows the factor for each satellite system.

  Satellite System	Factor
  GPS, Galileo, BeiDou	1
  GLONASS	1.5
  SBAS	3

• $$R_r$$ — Code or carrier-phase error ratio of receiver r. The receiverposition function assumes a value of 100.

• $$a_{\sigma },b_{\sigma }$$ — Carrier-phase error factors in meters. The receiverposition function assumes a value of 0.003 meters for both factors.

• $$e{l}_r^s$$ — Elevation angle of satellite s with respect to receiver r, in meters.

• $${\sigma }_{eph}$$ — Standard deviation of ephemeris and clock error, in meters. This value is determined by indexing into this vector with the user range accuracy (URA) index:

  [2.4 3.4 4.85 6.85 9.65 13.65 24 48 96 192 384 768 1536 3072 6144]

  Note that this index is zero-based.

• $${\sigma }_{iono}$$ — Standard deviation of ionosphere correction model error, in meters.

  The value depends on the ionosphere correction model you specify in the Ionosphere property of gnssopts.

  $${\sigma }_{iono}\{\begin{array}{ll}5\hfill & \text{no correction model}\hfill \\ 0.5I_r^s\hfill & \text{Klobuchar correction model}\hfill \end{array}$$

  $$I_r^s$$ is the ionospheric delay, in meters.

  For more information about the Klobuchar model, see Klobuchar Delay Model.

• $${\sigma }_{trop}$$ — Standard deviation of troposphere correction model error, in meters.

  The value depends on the troposphere correction model you specify in the Troposphere property of gnssopts.

  $${\sigma }_{trop}\{\begin{array}{ll}3\hfill & \text{no correction model}\hfill \\ 0.3/(0.1+sin(e{l}_r^s))\hfill & \text{Saastamoinen correction model}\hfill \end{array}$$

  $$e{l}_r^s$$ is the elevation angle of satellite s with respect to receiver r, in meters.

  For more information about the Saastamoinen model, see Saastamoinen Delay Model.

• $${\sigma }_{bias}$$ — Standard deviation of code bias error, in meters. The receiverposition function assumes a value of 0.3 meters.

Note

Prior to R2026a, the pseudorange measurements and weight equations are:

$$\begin{array}{l}h(x)={\rho }_r^s+c\Delta t_r\\ W=I_m\end{array}$$

I_m is an identity matrix, which weighs all m satellite measurements equally for receiver position estimation.

Satellite Clock Bias

The satellite clock bias is the difference between the time kept by the onboard clock of a satellite and the actual GNSS system time.

The satellite clock bias of satellite s at transmission time t^s is given as:

$$\Delta t^s(t^s)=f_0+f_1(t^s-t_{oe})+f_2{(t^s-t_{oe})}^2-\frac{2\sqrt{\mu }}{c^2}e\sqrt{A}\sin E-b{T}_{GD}$$

The equation has three components:

1. Satellite clock parameters

   $$f_0+f_1(t^s-t_{oe})+f_2{(t^s-t_{oe})}^2$$

   • $$f_0$$ — Satellite clock bias parameter

   • $$f_1$$ — Satellite clock drift parameter

   • $$f_2$$ — Satellite clock drift rate parameter

   • t_oe — Time of ephemeris

2. Relativistic correction

   $$\frac{2\sqrt{\mu }}{c^2}e\sqrt{A}\sin E$$

   • μ — Earth gravitational constant

   • c — Speed of light in meters per second

   • e — Eccentricity

   • A — Eccentric anomaly in radians

3. Group delay parameter

   $$b{T}_{GD}$$

   • b — $$\frac{f_1^2}{f_1^2}$$ for L_i pseudorange

   • T_GD — Group delay parameter, in seconds

Geometric Range Between Satellite and Receiver

The geometric range is the Euclidean between the position of a satellite and the position of a receiver at an instance of time.

The geometric range $${\rho }_r^s$$ between the satellite s and the receiver r is defined as:

$${\rho }_r^s\approx \Vert r_r(t_r)-r^s(t^s)\Vert +\frac{{\omega }_e}{c}(x^s{y}_r-y^s{x}_r)$$

• r_r(t_r) — Receiver position in ECEF at reception time t_r. Units in meters.

• r^s(t^s) — Satellite position in ECEF at reception time t^s. Units in meters.

• ω_e — Earth’s rotation rate in radians per second from the WGS84 model. This corrects for the Earth's rotation during the travel of the signal.

• c — Speed of light in meters per second.

• x^s(t^s) — Satellite x-position in ECEF at transmission time t^s. Units in meters.

• y^s(t^s) — Satellite y-position in ECEF at transmission time t^s. Units in meters.

• x_r(t_r) — Receiver x-position in ECEF at transmission time t_r. Units in meters.

• x_r(t_r) — Receiver y-position in ECEF at transmission time t_r. Units in meters.

The satellite transmission time t^s is defined as:

$$t^s=t_r-P_r^s/c-\Delta t^s(t^s)$$

• P^s_r — Pseudorange measurement, in meters, of satellite s with respect to receiver r.

• $$\Delta t^s(t^s)$$ — Satellite clock bias, in seconds, of satellite sat transmission time t^s.

The satellite clock bias that the receiverposition function uses for geometric range calculation is derived using only the clock bias parameters part of the clock bias equation:

$$\Delta t^s(t^s)=f_0+f_1(t^s-t_{oe})+f_2{(t^s-t_{oe})}^2$$

For more information about satellite clock bias, see the Satellite Clock Bias section.

References

[1] Groves, Paul D. Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems. 2nd ed. GNSS Technology and Application Series. Boston: Artech House, 2013.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2021a

R2026a: Calculate receiver position from raw GNSS measurements

The receiverposition function now calculate receiver position from raw GNSS measurements. Use the gnssmeasurements function to generate the measurements to specify the receiverposition function, and use the gnssoptions object to specify biases and atmospheric delay models.