Comparison between least squares method and Kalman filter (Kalman)
Single-point positioning is based on the distance observation between the GPS satellite and the user's receiver antenna, and determines the coordinates of the observation point corresponding to the receiver antenna based on the satellite coordinates. Because of its advantages such as simple model calculation and fast positioning, it is widely used in fields such as vehicles, ship navigation, and geological exploration.
At present, the classic method for solving pseudorange positioning is to expand the observation equation according to Taylor series, linearize it with a linear term, and then solve it according to the least squares principle.
1 Least squares method
The basic principle of finding the optimal solution using the least squares iteration method is to minimize the sum of squares of the observation errors .
The least squares solution simply considers that each observation epoch is independent of each other, and the observation results at the previous moment will not have any impact on the observation results at the next moment.
The least squares method can only use the current observation quantity and cannot perform error analysis on the observation quantity. Therefore, the positioning result is greatly affected by the error of the observation quantity and the accuracy is not high. However, the least squares method converges very quickly in the initial iteration and is subject to acceptance. The impact of the initial rough coordinates of the machine is small.
2 Kalman filter
The principle of Kalman filtering method to find the optimal solution is to minimize the variance of the state error , and then obtain a better estimate of the state.
Kalman filter is a set of recursive filtering algorithms based on state space. Its model includes state space model and observation model . The former describes the state transition changes at adjacent moments through state equations, while the latter reflects the relationship between actual observations and state variables. The filtering problem is to obtain the optimal estimate of the system state by combining observation information and state transition rules.
Kalman filter believes that each observation epoch is correlated in time, that is, the observation result at the previous moment will have an impact on the observation at the next moment. The difference is only the degree of influence, which must be considered when solving the results at the next moment. The effect of the last moment's results on the present, so Kalman filtering has good smoothness.
Kalman filter does not need to store a large amount of data and can easily process dynamic data in real time. However, when using pseudorange as an observation quantity to perform Kalman filtering, the approximate coordinates of the initial position of the receiver are required. If the approximate coordinate deviation is large, the measurement equation will be inaccurate, it will easily cause filter divergence, and the real-time performance will be poor.
3 The connection between the two
If the time factor is fixed in the application of Kalman filter, Kalman filter degenerates into the classic least squares method. Under normal circumstances, iterative calculations must be performed, and the calculation process is more complicated than the classical least squares method, but the accuracy of the solution will not be improved in any way.
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