A mathematical modeling was performed to obtain and analyze the results of trajectory filtration using linear and nonlinear Kalman filters. It is established that a nonlinear filter and a filter that takes into account the correlation of the errors of the linearized observations have a close efficiency. However, in the case of limited computing resources, it is preferable to use the linear filter. A program was developed in a cross-platform complete integrated development environment Qt Creator on the C++ programming language. This program can be applied to many systems to solve the problem of trajectory filtration after special adjustment.
A number of methods of trajectory filtration are known, the main of which
are based on modifications of algorithms of the Kalman vector estimation.
Application of the nonlinear Kalman filter (NF) can be close to the optimal
solution because observations are made in the polar coordinate system and
estimation of the parameters of the trajectories to be tracked is carried out
in a rectangular coordinate system. However, this approach requires additional
computational costs and is very difficult to implement [
The aim of the research is a comparative analysis of effectiveness of the proposed modifications of linear and nonlinear Kalman filters for various types of trajectories of radar targets.
Operation of most algorithms of the trajectory tracking is based on the use of
various mathematical models, with which it is possible to accurately approximate
the real motion of the target and the process of its observations by the radar
station, and then, in the process of filtration, refine the measurements obtained
assessing the degree of their suitability for the model. The combination of
optimally selected models of motion and observation underlies most methods of
trajectory tracking. Consider the mathematical models of the motion of objects
and observations with that applied to a three-coordinate radar station. As a
model of motion of the accompanied object, we use the Markov random sequence
given up to the stochastic equation [
B0 1 0 0 ti 0 C i = BBBB00 00 10 01 00 t0iCCCC; ti is the time, for which the object position has changed; i = (0 vyi 0 0 0 2 xi speed during the time of flight through the radar coverage area; xi = yi = qvy20 tit , zi = qvz20 tit are the root mean square deviations (RMS) of the projections of velocity; vx0 ; vy0 ; vz0 are the initial values of the projections of velocity; t is the radar scan time.
The use of this model allows one to simulate motion of an object of various degrees of complexity: from rectilinear uniform to motion with large accelerations and maneuvering. If = 0, then vi = v0, that is, the speed does not change. At = 0:01, the speed will change by 1%, etc. Thus, it is possible to estimate the operation of the filter under different conditions.
As a model for observing an object using a three-coordinate radar, we use the following expression: qvx20 tit , zi = h(xi) + ni; (2) 0pxi2 + yi2 + zi21 arctan xyii
C; R ;
A observations by distance, bearing and elevation, respectively.
; are the RMS deviations of the source of
Estimation of the trajectory of a moving object consists in determining
the numerical values of the parameters of its motion. Consider the NF
algorithm [
The covariance matrix of estimation errors is calculated by formula [
1
HiPэi; where Pэi = iPi 1 iT + V i is the covariance matrix of prediction errors Hi = dh(xэi) dxэi 0 xэi pxэ2i + yэ2i pxэ2i + yэ2i zэi
pxэ2i + yэ2i 0 0 01 yэi xэi
xэ2i + yэ2i yэizэi pxэ2i+yэ2i xэ2i+yэ2i+zэ2i
0 pxэ2i+yэ2i xэ2i+yэ2i+zэ2i
C 0 0 0CC :
C C
A
The described algorithm is the most difficult both in the implementation and in
adjustment [
One of the important simplifications laid down in the Kalman filter is the assumption of the linear character of the equations of motion and observation. To reduce computational costs, it is proposed to use a linear filter, to which input linearized observations arrive. To take into account nonlinear dependencies and to achieve acceptable accuracy, we perform the following transformations: 0zxi 1 0zRi cos z i cos z i 1 zi0 = @zyi A = @zRi sin z i cos z i A ;
zzi zRi sin z i where zxi ; zyi ; zzi are the observations of Cartesian coordinates.
As a result of the transformations, the observation model takes the following form: zi0 = Cxi + n0i; (4) 01 0 0 0 0 01 where C = @0 1 0 0 0 0A is the conversion matrix, n0i = (nxi 0 0 1 0 0 0 where Bxyi ; Bxzi ; Byzi are the covariance of observations;
1 Bxyi = M fnxi nyi g = 2 sin 2z i
R2 cos2 z i + Ri2 2 sin2 z i
Ri2 2 cos2 z i ; 1 Bxzi = M fnxi nzi g = 2
1 Bxzi = M fnxi nzi g = 2 cos z i sin 2z i sin z i sin 2z i In this case, the covariance matrix of estimation errors can be found from the following expression:
Pi = Pэi E + CTVn0i 1CPэi 1 ; where E is the unit matrix.
This approach is easier for implementation and adjustment and requires O N N fewer multiplication operations, where N is the number of observed parameters.
To perform a comparative analysis and research of the effectiveness of the
proposed modifications of linear and nonlinear Kalman filters, a mathematical
model was implemented in the computer algebra system Wolfram Mathematica.
Based on the results obtained in [
This program is implemented in the C++ programming language in a free, cross-platform Qt creator complete integrated development environment using solutions from the Boost class library collection. Thus, owing application of these solutions, the configuration of the proposed computational module will not be required if it will be integrated into any automated radar processing system.
In Figure 1, the initial trajectory of the object’s motion, the observation of the radar located at the origin, and the results of the trajectory processing using each filter are presented. The simulation results are shown for the motion of an object with the same modulo velocities along each axis. It is clear that each of the proposed algorithms allows one to get results that are close to the real trajectory of the target.
Fig 1. Initial trajectory, observations and filtration results
In Figure 1a, the simulation results for = 0:01 are shown. In this case, the efficiency of NF and LF filters practically coincides (Fig. 2a).
a) b)
Fig 2. RMS errors in estimating the coordinate x
In Figure 1b, the simulation results for = 0:3 are shown. In this case, the target maneuvers more intensively and moves with greater acceleration. From the graphs of the RMS coordinate x, shown in Fig. 2b, we can conclude that the LF has greater we can conclude that the LF has greater efficiency, since the steady-state LF value is lower. It should also be noted that with the removal of the target from the radar, RMS of the proposed algorithms increases.
In Figure 3, the motion of an object with a higher velocity along the x axis is shown.
Fig 3. Initial trajectory, observations and filtration results
In Figure 3a, the simulation results for = 0:01 are shown. The algorithms of NF and LF have practically the same efficiency (Fig. 4a). In Figure 3b, the simulation results for = 0:1 are shown. In this case, the LF has a greater efficiency since the steady-state value of the RMS of the LF is lower (Fig. 4b). a) b)
Fig 4. RMS errors in estimating the coordinate x
Modeling of the trajectory estimation processes based on observations of objects with different nature of motion allows us to draw the following conclusions.