Satellite navigation solution for the receiver's position (
geopositioning) involves an algorithm. In essence, a
GNSS receiver measures the transmitting time of GNSS signals emitted from four or more GNSS satellites (giving the
pseudorange) and these measurements are used to obtain its position (i.e.,
spatial coordinates) and reception time.
The following are expressed in
inertial-frame coordinates.
The solution illustrated
Essentially, the solution shown in orange, , is the intersection of
light cones.
The
posterior distribution of the solution is derived from the product of the distribution of propagating spherical surfaces. (See
animation.)
Calculation steps
A
global-navigation-satellite-system (GNSS) receiver measures the apparent transmitting time, , or "phase", of GNSS signals emitted from four or more GNSS
satellites ( ), simultaneously.[1]
The transmitting time of GNSS satellite signals, , is thus derived from the non-
closed-formequations and , where is the
relativistic clock bias, periodically risen from the satellite's
orbital eccentricity and Earth's
gravity field.[2] The satellite's position and velocity are determined by as follows: and .
The receiver's position, , and reception time, , satisfy the
light-cone equation of in
inertial frame, where is the
speed of light. The signal time of flight from satellite to receiver is .
The above can be solved by using the
bivariateNewton–Raphson method on and . Two times of iteration will be necessary and sufficient in most cases. Its iterative update will be described by using the approximated
inverse of
Jacobian matrix as follows:
In the field of GNSS, is called
pseudorange, where is a provisional reception time of the receiver. is called receiver's clock bias (i.e., clock advance).[1]
Standard GNSS receivers output and per an observation
epoch.
The temporal variation in the relativistic clock bias of satellite is linear if its orbit is circular (and thus its velocity is uniform in inertial frame).
The signal time of flight from satellite to receiver is expressed as , whose right side is
round-off-error resistive during calculation.
The geometric range is calculated as , where the
Earth-centred, Earth-fixed (ECEF) rotating frame (e.g.,
WGS84 or
ITRF) is used in the right side and is the Earth rotating matrix with the argument of the signal
transit time.[2] The matrix can be factorized as .
The line-of-sight unit vector of satellite observed at is described as: .
The above notation is different from that in the Wikipedia articles, 'Position calculation introduction' and 'Position calculation advanced', of
Global Positioning System (GPS).
Satellite navigation solution for the receiver's position (
geopositioning) involves an algorithm. In essence, a
GNSS receiver measures the transmitting time of GNSS signals emitted from four or more GNSS satellites (giving the
pseudorange) and these measurements are used to obtain its position (i.e.,
spatial coordinates) and reception time.
The following are expressed in
inertial-frame coordinates.
The solution illustrated
Essentially, the solution shown in orange, , is the intersection of
light cones.
The
posterior distribution of the solution is derived from the product of the distribution of propagating spherical surfaces. (See
animation.)
Calculation steps
A
global-navigation-satellite-system (GNSS) receiver measures the apparent transmitting time, , or "phase", of GNSS signals emitted from four or more GNSS
satellites ( ), simultaneously.[1]
The transmitting time of GNSS satellite signals, , is thus derived from the non-
closed-formequations and , where is the
relativistic clock bias, periodically risen from the satellite's
orbital eccentricity and Earth's
gravity field.[2] The satellite's position and velocity are determined by as follows: and .
The receiver's position, , and reception time, , satisfy the
light-cone equation of in
inertial frame, where is the
speed of light. The signal time of flight from satellite to receiver is .
The above can be solved by using the
bivariateNewton–Raphson method on and . Two times of iteration will be necessary and sufficient in most cases. Its iterative update will be described by using the approximated
inverse of
Jacobian matrix as follows:
In the field of GNSS, is called
pseudorange, where is a provisional reception time of the receiver. is called receiver's clock bias (i.e., clock advance).[1]
Standard GNSS receivers output and per an observation
epoch.
The temporal variation in the relativistic clock bias of satellite is linear if its orbit is circular (and thus its velocity is uniform in inertial frame).
The signal time of flight from satellite to receiver is expressed as , whose right side is
round-off-error resistive during calculation.
The geometric range is calculated as , where the
Earth-centred, Earth-fixed (ECEF) rotating frame (e.g.,
WGS84 or
ITRF) is used in the right side and is the Earth rotating matrix with the argument of the signal
transit time.[2] The matrix can be factorized as .
The line-of-sight unit vector of satellite observed at is described as: .
The above notation is different from that in the Wikipedia articles, 'Position calculation introduction' and 'Position calculation advanced', of
Global Positioning System (GPS).