Research on Tangential Velocity Solution and Accuracy Analysis of Radar Target

. In this paper, a calculation method for the tangential velocity solution of radar target based on linear vector method is proposed. It makes full use of the advantage of high precision radial velocity measurement of radar, and realizes the accurate computation of tangential velocity of radar target. Based on the actual measurement capability of continuous wave (CW) radar, the accuracy of the tangential velocity is analyzed. The feasibility of calculation method and accuracy analysis method is verified by simulation experiment and measured data of shooting range.


Introduction
Radar equipment has always been playing an important role in the weapon test evaluation of range. Radar can realize real-time observation of important information such as velocity, range, azimuth and pitch angle of the target by transmitting and receiving electromagnetic waves [1]. Among many test information, radar target velocity measurement is realized based on Doppler effect [2], which indicates that the condition for the target velocity to be observed is the relative radial movement between the target and the radar, that is, the velocity measured by the radar represents the change speed of the target is moving away from or near the radar. In other words, the velocity is only a velocity component of the tangential velocity (absolute velocity) of the target in the radial direction of the radar. However, in the range test evaluation, people pay close attention to the change of tangential velocity of the target. Therefore, how to put forward an accurate calculation method of tangential velocity of target on the basis of the original observation information of radar has become a technical problem that needs to be solved urgently. In this context, a method for solving tangential velocity of radar target based on linear vector is proposed in this paper. This method makes full use of the advantage of high precision radial velocity measurement of radar, and realizes the accurate computation of tangential velocity of radar target. Simulation experiment and measured data show that the solving method is feasible.

Calculation method of tangential velocity
To set the projectile trajectory is shown in figure 1, Γ(t) represents the target trajectory, the coordinate system direction is taken as the range coordinate direction, the coordinate origin O is taken as the radar coordinates (x 0 , y 0 , z 0 ), v τ represents the ballistic tangential velocity of the target, vrrepresents the radial velocity of the target, the tangential angle ϕ represents the angle between the ballistic tangential velocity and the radial velocity of the target.

Fig. 1 Projectile Trajectory Diagram
The parametric equation of target motion is: As in (1), (x, y, z) represents the coordinates of the target in the range coordinate system, (f 1 , f 2 , f 3 ) represents the coordinate transformation function, and (R, A, E) represents the target range, azimuth and pitch angle measured by radar respectively, which are all functions of time.
As shown in Fig.1, the direction of radial velocity of target at a certain moment is:(x-x 0 ,y-y 0 ,z-z 0 ), and the direction of tangential velocity is:(dx,dy,dz). According to the inner product formula of linear vectors, the tangential angle ф is calculated as: As in (2), {ψ i },i=1,2,3 denotes {x,y,z}. The same as below.
It should be noted that radar measurement is carried out in terms of the measurement period, which cannot be infinitely shortened due to the influence of system hardware performance and other parameters. In other words, radar output data is discrete data. Therefore, (2) needs to be discretized as follows: After the tangential angle between the tangential velocity and radial velocity of the target trajectory at a certain moment is solved, the tangential velocity of the target trajectory at that time can be solved by (4): As in (4), v r represents the radial velocity of the target. It is the direct measurement of radar.

Accuracy analysis
In order to verify the feasibility of the tangential velocity calculation method proposed in this paper and research on the measurement accuracy of the method, the analysis is made as follows. First, we can take the differential from (4): Eq.(5) is the tangential velocity error formula. In Formula (5), dv τ represents the measurement error of tangential velocity, dv r represents the measurement error of radial velocity (depending on the actual radar capability), and dcosϕ represents the measurement error of tangential angle. In the following, the measurement error of tangential angle is analyzed, and the variable substitution is as follows: Then, we can take the differential from (3): According to (6), the measurement error of tangential angle is related to the coordinate change error (dΔψ i ) and the coordinate error(dψ i ). Then, the analysis is made as follows: In a measurement period of radar, the target trajectoryΓ(t) is continuous function of time and has continuous partial derivative, according to Lagrange's mean value theorem [3]: Take the differential from the last equation: As in (7) {dΔξ j } denotes {dΔR,dΔA,dΔE}, which are the measurement errors of range, azimuth and pitch angle variation of radar. These measurement errors are high-order small quantities of range error and angle measurement error of radar. The measurement errors of radar range variation and angle variation are analyzed as follows: According to Lagrange's mean value theorem, the variation of radar range measurement can be expressed as: In the last equation, represents the radial velocity of the target radial distance differentiated with respect to time. Then, we can take the differential from above equation: As in aboveequation, represents the measurement error of radial velocity of differential method. According to error theory [4]: In the last equation, v r represents the radial velocity measured by radar, which is obtained through frequency measurement. v r has high measurement precision. dv r represents the measurement error of the velocity.
Similarly, the measurement error of radar angle variation can be expressed as: As in aboveequation, and respectively represent the target azimuth and pitch angle velocity obtained by differentiating time. It should be noted that target azimuth and pitch angle velocity are not directly measured by radar, and the measurement errors of azimuth and pitch angle velocity are not given in radar technical index. Therefore, the measurement errors of and cannot be directly calculated by error theory. In order to solve this problem, the minimum mean square error optimization algorithm is adopted to estimate the measurement errors of and . (details below). On the other hand, radar coordinate measurement errors (dx,dy,dz) can be expressed as follows according to the coordinate transformation formula: ( 8 ) Therefore, after the coordinate variationmeasurement error dΔψ i and coordinate measurement error dψ i are defined, the measurement error of tangential angle can be solved by (6). And then the measurement error of tangential velocity of radar target trajectory can be solved by (5).

Test Verification
In order to verify the feasibility of the tangential velocity calculation method and precision analysis presented in this paper; Firstly, the ballistic simulation software of the range is used to simulate the trajectory of the projectile. Based on the projectile firing parameters and artillery firing data in the performance evaluation test of a projectile, the simulation test is carried out. The radial and tangential velocity of the projectile obtained are shown in Fig. 2. The tangential velocity (solid red line in Fig. 2) obtained from the simulation test is taken as the truthvalue to verify the feasibility of the tangential velocity accuracy analysis method proposed in this paper. The measurement errors of and are ignored when verifying the accuracy analysis method of the tangential velocity. The test results are shown in  As shown in Fig.3, the solid red line in the figure represents the truth-value of tangential velocity measurement error, which is the result of the difference between the tangential velocity calculation value obtained by linear vector method and the truth-value of tangential velocity obtained by simulation test (solid red line in Fig.2). The blue dashed line in the figure represents the theoretical value of tangential velocity error, which is the calculation result of the accuracy analysis method proposed in this paper. The root-mean-square value of tangential velocity error was statistically analyzed under the two conditions. The root-mean-square value of tangential velocity error represented by the solid red line is 0.0185. The blue dashed line shows that the root-meansquare value of tangential velocity error is 0.0160. Under these two conditions, the trends of red and blue curves are quite different. This is because of the measurement errors of and are ignored. To solve this problem, the minimum mean square error optimization algorithm is adopted to estimate the measurement errors of and . In Fig.3, the red curve represents the truth-value of tangential velocity measurement error. The objective function can be expressed as follows: (9) As in (9),X i ,Y i ,Z i represent measurement matrices, which can be derived from (6), (7)  The root-mean-square value of the tangential velocity theoretical error under this condition is 0.0165, which is closer to the root-mean-square value (0.0185) of the tangential velocity measurement error under the actual condition. Therefore, it is feasible to adopt the minimum mean square error optimization algorithm to estimate the measurement errors of and .  Table Ⅰ. It can be concluded from Table Ⅰ that during the observation time, for different types of projectiles, the root-mean-square value of tangential velocity measurement error based on the linear vector method does not exceed 0.2m/s, and does not diverge because of different types of projectiles. In addition, it should be noted that the fluctuations of measured data are relatively large compared with the simulation data, so the error analysis results based on the measured data are relatively large compared with the simulation results. On the other hand, due to the different shooting conditions of the projectiles, there is a deviation between the velocity errors of different projectiles, which is in line with the general rule.