Satellite navigation
A satellite navigation or satnav system is a system that uses satellites to provide autonomous geo-spatial positioning. It allows satellite navigation devices to determine their location (longitude, latitude, and altitude/elevation) to high pr ...
solution for the receiver's position (
geopositioning
Geopositioning, also known as geotracking, geolocalization, geolocating, geolocation, or geoposition fixing, is the process of determining or estimating the geographic position of an object.
Geopositioning yields a set of geographic coordinates ...
) involves an algorithm. In essence, a
GNSS
A satellite navigation or satnav system is a system that uses satellites to provide autonomous geo-spatial positioning. It allows satellite navigation devices to determine their location (longitude, latitude, and altitude/elevation) to high pre ...
receiver measures the transmitting time of GNSS signals emitted from four or more GNSS satellites (giving the
pseudorange
The pseudorange (from pseudo- and range) is the ''pseudo'' distance between a satellite and a navigation satellite receiver (see GNSS positioning calculation), for instance Global Positioning System (GPS) receivers.
To determine its position, a ...
) and these measurements are used to obtain its position (i.e.,
spatial coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
) and reception time.
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
satellite
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotope ...
s (
), simultaneously.
[Misra, P. and Enge, P., Global Positioning System: Signals, Measurements, and Performance, 2nd, Ganga-Jamuna Press, 2006.]
# GNSS satellites broadcast the messages of satellites'
ephemeris
In astronomy and celestial navigation, an ephemeris (pl. ephemerides; ) is a book with tables that gives the trajectory of naturally occurring astronomical objects as well as artificial satellites in the sky, i.e., the position (and possibly vel ...
,
, and intrinsic clock bias (i.e., clock advance),
as the functions of (
atomic)
standard time
Standard time is the synchronisation of clocks within a geographical region to a single time standard, rather than a local mean time standard. Generally, standard time agrees with the local mean time at some meridian that passes through the r ...
, e.g.,
GPST.
[The interface specification of NAVSTAR GLOBAL POSITIONING SYSTEM](_blank)
/ref>
# The transmitting time of GNSS satellite signals, , is thus derived from the non- closed-form equations
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
and , where is the relativistic clock bias, periodically risen from the satellite's orbital eccentricity
In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values betwee ...
and Earth's gravity field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
. The satellite's position and velocity are determined by as follows: and .
# In the field of GNSS, "geometric range", , is defined as straight range, or 3-dimensional distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
,[3-dimensional ]distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
is given by where and represented in inertial frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
. from to in inertial frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
(e.g., Earth-centered inertial
Earth-centered inertial (ECI) coordinate frames have their origins at the center of mass of Earth and are fixed with respect to the stars. "I" in "ECI" stands for inertial (i.e. "not accelerating"), in contrast to the "Earth-centered - Earth- ...
(ECI) one), not in rotating frame
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. (This article considers onl ...
.
# The receiver's position, , and reception time, , satisfy the light-cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
equation of in inertial frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
, where is the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. The signal time of flight from satellite to receiver is .
# The above is extended to the satellite-navigation positioning equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
, , where is atmospheric delay
The error analysis for the Global Positioning System is important for understanding how GPS works, and for knowing what magnitude of error should be expected. The GPS makes corrections for receiver clock errors and other effects but there are sti ...
(= ionospheric delay
The error analysis for the Global Positioning System is important for understanding how GPS works, and for knowing what magnitude of error should be expected. The GPS makes corrections for receiver clock errors and other effects but there are sti ...
+ tropospheric delay
Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere.
As a form of electromagnetic radiation, like light waves, radio waves are affected ...
) along signal path and is the measurement error.
# The Gauss–Newton method can be used to solve the nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
least-squares problem
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
for the solution: , where . Note that should be regarded as a function of and .
# The posterior distribution
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
of and is proportional to , whose mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
is . Their inference is formalized as maximum a posteriori estimation
In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is an estimate of an unknown quantity, that equals the mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the b ...
.
# The posterior distribution
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
of is proportional to .
The solution illustrated
Image:Light cones.svg, Essentially, the solution, , is the intersection of light cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
s.
Image:Evolution light cones 0.gif, The posterior distribution
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
of the solution is derived from the product of the distribution of propagating spherical surfaces. (Se
animation
)
The GPS case
* For Global Positioning System
The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite sy ...
(GPS), the non-closed-form equations in step 3 result in
:
in which is the orbital eccentric anomaly In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position alo ...
of satellite , is the mean anomaly
In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical ...
, is the eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-center, in geometry
* Eccentricity (graph theory) of a v ...
, and .
* The above can be solved by using the bivariate Newton–Raphson
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
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:
* Tropospheric delay
Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere.
As a form of electromagnetic radiation, like light waves, radio waves are affected ...
should not be ignored, while the Global Positioning System
The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite sy ...
(GPS) specification doesn't provide its detailed description.
The GLONASS case
* The GLONASS
GLONASS (russian: ГЛОНАСС, label=none, ; rus, links=no, Глобальная навигационная спутниковая система, r=Global'naya Navigatsionnaya Sputnikovaya Sistema, t=Global Navigation Satellite System) is ...
ephemerides don't provide clock biases , but .
See also
* Time to first fix
Notes
* In the field of GNSS, is called pseudorange
The pseudorange (from pseudo- and range) is the ''pseudo'' distance between a satellite and a navigation satellite receiver (see GNSS positioning calculation), for instance Global Positioning System (GPS) receivers.
To determine its position, a ...
, where is a provisional reception time of the receiver. is called receiver's clock bias (i.e., clock advance).
* Standard GNSS receivers output and per an observation epoch
In chronology and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured.
The moment of epoch is usually decided by ...
.
* 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
The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...
or ITRF) is used in the right side and is the Earth rotating matrix with the argument of the signal transit time. The matrix can be factorized as .
* The line-of-sight unit vector of satellite observed at is described as: .
* The satellite-navigation positioning equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
may be expressed by using the variables and .
* The nonlinearity
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
of the vertical dependency of tropospheric delay
Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere.
As a form of electromagnetic radiation, like light waves, radio waves are affected ...
degrades the convergence efficiency in the Gauss–Newton iterations in step 7.
* The above notation is different from that in the Wikipedia articles, 'Position calculation introduction' and 'Position calculation advanced', of Global Positioning System
The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite sy ...
(GPS).
See also
* Dilution of precision (navigation)
Dilution of precision (DOP), or geometric dilution of precision (GDOP), is a term used in satellite navigation and geomatics engineering to specify the error propagation as a mathematical effect of navigation satellite geometry on positional measu ...
* Global Positioning System#Navigation equations
* Least squares adjustment Least-squares adjustment is a model for the solution of an overdetermined system of equations based on the principle of least squares of observation residuals. It is used extensively in the disciplines of surveying, geodesy, and photogrammetry—t ...
* Precise Point Positioning
Precise Point Positioning (PPP) is a global navigation satellite system (GNSS) positioning method that calculates very precise positions, with errors as small as a few centimeters under good conditions. PPP is a combination of several relatively s ...
* Real Time Kinematic
Real-time kinematic positioning (RTK) is the application of surveying to correct for common errors in current satellite navigation (GNSS) systems. It uses measurements of the phase of the signal's carrier wave in addition to the information conte ...
References
External links
PVT
(Position, Velocity, Time): Calculation procedure in the open-source
Open source is source code that is made freely available for possible modification and redistribution. Products include permission to use the source code, design documents, or content of the product. The open-source model is a decentralized sof ...
GNSS-SDR and the underlying RTKLIB
{{DEFAULTSORT:Gnss Positioning Calculation
Satellite navigation