The Global Positioning System (GPS) is short for the acronym for the English Global Positioning System. It means using navigation satellites for time measurement and ranging to form a global positioning system. It is a new generation of satellite navigation and positioning system developed by the US Department of Defense with a full range of real-time 3D navigation and positioning capabilities in sea, land and air. The core of the GPS user part is the GPS receiver. It is mainly composed of two parts: baseband signal processing and navigation solution. The baseband signal processing part mainly includes two-dimensional search, acquisition, tracking, pseudorange calculation and navigation data decoding of GPS satellite signals. The navigation solution part mainly includes real-time calculation of each visible satellite position according to ephemeris parameters in the navigation data; star clock error, relativistic effect error, earth rotation effect, signal transmission error (mainly including ionization) according to various error parameters in the navigation data Calculation of various real-time errors such as layer real-time transmission error and tropospheric real-time transmission error, and eliminate it from pseudorange; perform calculation of receiver PVT (position, velocity, time) according to the above results; DOP) performs real-time calculations and monitoring to determine the accuracy of the positioning solution. This article focuses on the navigation solution of the GPS receiver. The baseband signal processing section can refer to the relevant information. The assumptions discussed in this paper are that the GPS receiver has effectively captured and tracked the GPS satellite signals, calculated the pseudorange, and decoded the navigation data.
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1 Earth coordinate system
To describe the position of an object, there must be an associated coordinate system. The position of the GPS receiver on the surface of the Earth is relative to the Earth. Therefore, to describe the position of the GPS receiver, it is necessary to use a coordinate system that is fixed to the Earth along with the Earth, that is, the Earth coordinate system as a reference system.
The Earth coordinate system has two geometric expressions, the Earth Cartesian coordinate system and the Earth's Earth coordinate system. The Earth's Cartesian coordinate system is defined as: the origin O coincides with the Earth's centroid, the Z axis points to the Earth's North Pole, the X axis points to the intersection of the Earth's equatorial plane and the Greenwich meridian (ie 0 longitude direction), and the Y axis is in the equatorial plane with XOZ Forms the right-handed coordinate system (ie, pointing 90 degrees east longitude). The Earth's geodetic coordinate system is defined as: the center of the Earth's ellipsoid coincides with the Earth's centroid, and the short axis of the ellipsoid coincides with the Earth's axis of rotation. The geodetic latitude of any point on the surface of the earth is the angle φ between the normal of the ellipsoid and the equatorial plane of the ellipsoid. The longitude is the angle λ between the meridian plane of the ellipsoid and the meridian plane of the Greenwich. The height h of the point is the distance of the point along the ellipsoid normal to the ellipsoid. It is assumed that any point P on the surface of the earth is expressed as P(x, y, z) in the Earth's Cartesian coordinate system, and is expressed as P (φ, λ, h) in the Earth's geodetic coordinate system. Then the relationship between the two is: the geodetic coordinate system becomes a Cartesian coordinate system: (1) where: n is the radius of curvature of the ellipsoid, and e is the first eccentricity of the ellipsoid. If the long radius of the ellipsoid is a and the short radius is b, then (2) the Cartesian coordinate system becomes the geodetic coordinate system, which can be obtained by the following method: φ is obtained by the iterative method for the eccentricity of the earth, and ep is the ellipticity.
The initial value φ=φc can be set to be iterated until |φi=1-φi| is less than a certain threshold.
These two coordinate systems are often used interchangeably in positioning systems and must be familiar with the conversion relationship between the two coordinate systems.
2 Main error and elimination algorithm in GPS positioning
The main errors in GPS positioning are: star clock error, relativistic error, earth rotation error, ionosphere and tropospheric error. 1) Star clock error The star clock error is formed by the error between the star clock and the GPS standard. The GPS measurement is based on the precision time measurement. The star clock error time can reach 1ms, and the resulting distance deviation can reach 300Km. Eliminate it. The binomial error is generally expressed by a binomial formula. (3) In the GPS ephemeris, the coefficient of the binomial is transmitted to achieve the purpose of correction. After this correction, the error between the star clock and the GPS standard time can be controlled within 20 ns. 2) Relativistic error From the theory of relativity, after a clock with frequency on the ground is installed on a satellite operating at speed, the clock frequency will change, and the amount of change is:
That is, the clock on the satellite is slower than on the ground. To correct this error, a coefficient improvement method can be used. This coefficient is broadcast in the GPS ephemeris to eliminate the relativistic error, and the relativistic error can be controlled within 70 ns. 3) Earth rotation error GPS positioning uses a protocol earth coordinate system fixed to the Earth, and rotates around the z axis with the earth. The position (coordinate value) of the satellite relative to the agreed Earth system is relative to the epoch. If the satellite is in a certain position in the protocol coordinate system at a certain moment when the signal is transmitted, when the local receiver receives the satellite signal, the satellite is no longer at the instantaneous position (coordinate value) due to the rotation of the earth. That is to say, in order to solve the position of the receiver receiving the satellite signal at the moment in the protocol coordinate system, the coordinate system at that moment must be used as the reference coordinate system for solving. The time used to solve the satellite position is the time at which the satellite transmits the signal. In this way, the satellite position solved at that moment must be converted to a position in the reference coordinate system. Let the angular velocity of the Earth's rotation be we, and the signal propagation delay of the transmitted signal instantaneously to the instantaneous value of the received signal is Δt, then the longitude of the ascending node is adjusted to the three-dimensional coordinate during this time. (4) The positioning error caused by the Earth's rotation is in the meter. Level, precision positioning must be considered to eliminate. 4) Ionosphere and tropospheric error The ionosphere refers to the atmosphere above the earth at a height of 50-1000 km from the ground. The gas molecules in the ionosphere are strongly ionized by various radiations of celestial bodies such as the sun, forming a large amount of free electrons and positive ions. The ionospheric errors are mainly composed of ionospheric refractive error and ionospheric delay error. The error caused by the vertical direction can reach about 50 meters, and the horizontal direction can reach about 150 meters. At present, it is not possible to use a strict mathematical model to describe the magnitude and variation of electron density. Therefore, the elimination of ionospheric errors is corrected by ionospheric correction models or dual-frequency observations. The troposphere refers to the atmospheric bottom layer within about 40 km from the ground, accounting for 99% of the total mass of the atmosphere. Its atmospheric density is larger than that of the ionosphere and its atmospheric state is more complex. The troposphere is in contact with the ground, and radiant heat is obtained from the ground, and the temperature decreases as the height rises. The tropospheric refraction consists of two parts: one is due to the propagation speed of the electromagnetic wave or the slowing of the light in the atmosphere, which is the main part; the second is because the GPS satellite signal passes through the troposphere, and the propagation path is also bent, thus making the measurement The distance is deviated. It can reach 2.5 meters in the vertical direction and 20 meters in the horizontal direction. Tropospheric errors are also corrected by empirical models. The ionospheric and tropospheric errors are eliminated in the GPS ephemeris by given the ionospheric tropospheric model and model parameters. The experimental data show that the effectiveness of the model is improved by 75% for the ionospheric error and 95% for the tropospheric error.
3 GPS ephemeris structure and solving process
To get the position of the receiver, in the case where the receiver clock and the GPS standard are strictly synchronized, the position to be solved is three unknown variables, and three independent equations are needed to solve. However, in actual situations, it is difficult to strictly synchronize the receiver clock and the GPS standard. Therefore, we take the receiver time and the GPS standard time deviation as an unknown variable. Thus, the solution requires four independent equations, that is, it needs to be 4 observation satellites. Figure 1 GPS positioning diagram (without considering the time deviation) Assuming that the receiver position is (xu, yu, zu) and the receiver time deviation is tu, the distance deviation due to the time deviation is the obtained pseudo-range observation value. We can get the simultaneous equation (5) to linearize the above equation, that is, the Taylor series expansion in the real position (xu, yu, zu), ignore the high-order term, and get (6) where, equation (6) is the actual The calculated iterative formula, the iterative termination condition is that the change of the true position (xu, yu, zu) is less than a certain threshold, and finally can be used as the basis for adjusting the receiver time deviation. The calculation is generally solved by matrix. To solve the equation, we also need to know the position of the four satellites (xj, yj, zj) in advance, and the satellite position can be obtained from the satellite's ephemeris. The GPS satellite ephemeris gives the ephemeris of the star, and the real-time position of the satellite can be calculated according to the ephemeris. The parameters of the satellite star clock error, relativistic error, earth rotation error, ionosphere and tropospheric error are given in the ephemeris. According to the calculated satellite position of these parameters, the above error can be substantially eliminated. The basic steps for solving the satellite position are: calculating the average angular velocity of the satellite operation 1 to calculate the naturalization time; 2 calculating the near-point angle of the observation time; 3 calculating the near-point angle; 4 calculating the satellite vector; 5 calculating the true near-point angle of the satellite; 6 Calculate the angular distance of the ascending node; 7 Calculate the perturbation correction term; 8 Calculate the ascending angle, satellite diameter, or orbital inclination after perturbation correction; 9 Calculate the longitude of the ascending node at the observation time; 10 Calculate the satellite in the geocentric coordinate system The location in . It is particularly worth pointing out that in calculating the true near-point angle Vk of the satellite, the formula (7) should be used, where e is the eccentricity and Ek is the satellite near-point angle. Some reference books have incorrect formulas for calculating the true near-point angle of the satellite, which will cause the quadrant of the satellite to be near-point angle blurred, so that the correct position of the satellite cannot be obtained. After performing the above calculation, the errors are further eliminated according to the error parameters broadcasted in the ephemeris. In this way, we get a complete process of using GPS ephemeris for navigation and positioning.
4 Conclusion
We describe in detail the navigation and positioning principle of GPS satellites and the algorithm of positioning and solving, and analyze the main sources of error and elimination methods. Of course, the algorithm for the number of satellites with more than 4 stars and the differential GPS algorithm can be studied in depth based on this algorithm.
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