In this work, we aimed to develop and investigate algorithms for filtering radar information, which could provide the required accuracy of multiple-object tracking procedures. During the research, modifications to algorithms based on Kalman filtering were developed and analysed. A comparative assessment of the accuracy of the developed and existing algorithms was carried out.

Determination of target coordinates by the on-board interceptor radar requires filtering of the received data containing different noise interferences. The majority of filtering algorithms are oriented towards a certain model of target motion and ensure optimum estimation in terms of a minimum (representing a sum of dynamic and fluctuation) tracking error.

For modern manoeuvring aircraft (AC) of class Su-35 with thrust-vectoring module, AC axes angular velocities are 1.5-2.5 rad/s. Dynamic analysis of the processes developed in the course of tests with Su-35 tracking has shown that angular velocities of the line of sight may reach 30°/s. The above entails the need for developing a measurement filtering algorithm ensuring optimum estimation of target coordinates under conditions of manoeuvring operations [1][2][3][4][5]. In this study, it was assumed that the radar carrier was performing a manoeuvre while the target was moving uniformly and linearly. Modelling was carried out using software package MATLAB.

For the study, automatic tracking circuit block diagram [9], shown in Figure 1, was used.

Fig. 1. Block diagram of automatic angle tracking circuit mathematical model

The following designations are adopted in Figures 1 and 2: ACS – antenna coordinate system; TCS – translational coordinate system; FCS – fixed coordinate system; IMS – interceptor motion simulator; TMS – target motion simulator; KL – “interceptor-target” kinematic link; IIS – interceptor inertial system; εг им, εв им, φг им, φв им – true values of angular coordinates received from simulator; εг э, εв э, φг э, φв э – extrapolated values of angular coordinates; DC – algorithm of angular coordinates εг, εв conversion from TCS into angular coordinates φг, φв of ACS; IC – algorithm of angular coordinates εг εв conversion from ACS into angular coordinates εг, εв of TCS; Хц, Уц, Zц – target coordinates in FCS; Хп, Уп, Zn – interceptor coordinates in FCS; ψ, υ, γ – interceptor heading, pitch and roll angles relative to TCS axes.

Fig. 2. Used coordinate systems

The coordinate values are formed during modelling in the range between t = 0 and Ткон with an increment of ∆t. To describe interceptor motion, at each simulator operational increment [7, 8] spatial angles ψ(i), υ(i) are calculated, which represent the horizontal and vertical angles of interceptor velocity vector relative to TCS axes.

Based on the obtained values of angles ψ(i), υ(i), interceptor coordinates are calculated in FCS using simultaneous equations (1), and three-dimensional target coordinates are calculated in a similar way.

Then, kinematic link (KL), shown in Figure 1, determines target position coordinates and calculates distance to the target in TCS in accordance with simultaneous equations (2). Based on the data obtained from (1) and (2), target angular coordinates εв им and εг им are calculated in translational coordinate system using formulas (3) and (4), respectively.

Since the discriminator shown in Figure 1 operates in the antenna coordinate system and the extrapolator – in the translational coordinate system, a task arises of converting target angular coordinates from one system to the other. This conversion is carried out to decrease the dynamics of angles shown in Figure 3, when radar carrier performs a manoeuvre.

Fig. 3. Target azimuth in translational and antenna coordinate systems

Conversion of target angular coordinates εг, and εв into coordinates φг, and φв of antenna coordinate system involves sequential transition from translational coordinate system to the coordinate system related to the aircraft construction axes. Then, a transition relative to the aircraft construction axes is carried out to antenna coordinate system through rotation by adjusting angles.

In Figure 1, linear statistical equivalent of discriminator is used as a discriminator, which models measured angles φг изм and φв изм based on the data obtained from motion simulator.

Additive noise mixture ξ(i) with target angular coordinates φг им and φв им arrives at the discriminator input shown in Figure 4. Mean amplitude of noise impact was selected based on flight tests and it equals to 10 angular minutes. Random number generator with a Gaussian distribution from package MATLAB was used as a noise source. Then, target angular coordinates in the azimuthal plane φг изм at the ideal discriminator characteristic are calculated in accordance with equations (5) and (6). Operations with angular coordinate φв изм, range and radial velocity are carried out in a way similar to (5) and (6). As previously noted, extrapolator operates in TCS, consequently, target coordinates measured in ACS are converted using invert conversion (IC in Fig. 1) into target coordinates in TCS through invert rotation by heading, pitch and roll angles.

Fig. 4. Block diagram of elevation channel discriminator

This paper provides a study of (α – β)-filter [1][2][3][4][5][6] and Kalman filter [1][2][3][4][5][6] algorithms with different modifications [3][6].

Let’s consider the algorithm of data processing in (α – β)-filter. At each filtration increment, tracking error by elevation angle (7) and azimuth (8) is calculated in TCS. Then, considering the specified errors, smoothing of obtained data is carried out and angular coordinates for the next increment are predicted. Filtration algorithm for azimuth angle is described with simultaneous equations (9), where Тф is filter operation period. Target elevation angle is filtered in a similar way.

Multi-dimensional Kalman filter performs smoothing through summing the measured and predicted data with the corresponding weight. Vectors of measured Xизм(i) and predicted Xэ(i) target angular coordinates in translational coordinate system are described with equations (10) and (11), respectively.

where Tф is filter operation period.

Vector of extrapolated target coordinates (11) is formed using system status vector transfer matrix (12).

As a result, coordinate evaluation vector takes the form (13).

where I is unit matrix,

static matrix of status vector recalculation into vector of observable parameters,

weighting matrix.

where R is measurement error covariance matrix.

At each filtration increment, dispersion matrix (14) is calculated using angle coordinates evaluation vector. Combination (15) is called a predicted dispersion matrix. At each filtration stage, elements of Kalman coefficient matrix are calculated based on the minimum mean-square error (minimization of dispersion matrix elements). As a result, elements of Kalman coefficient matrix are calculated by formula (16).

where added matrix is a model noise matrix that has the form (18).

To decrease dynamic tracking error, this paper considers different modifications of Kalman filter.

One of the modifications (17) introduces model noise matrix Q. Parameters a and b represent third derivatives of target azimuth and elevation angles in translational coordinate system: a = and b = and have the form of fixed nonzero values.

Another version of modification is represented by the lower limit of diagonal elements of variance matrix P11(i) > Pmin(i), P44(i) > Pmin(i) at each filtration step, which leads to an increase of Kalman coefficients and, consequently, to a decrease of time of filter response to a manoeuvre.

Numerical calculations were carried out for five filtration algorithms: Alpha-Beta-filter, Kalman filter and its 3 modifications. They all were investigated at different motion paths of the interceptor (radar carrier) and the target, including during aerobatics operations. During the analysis of data obtained in the course of tests at Su-35tracking, the following basic operation parameters were selected:

As a result of numerical modelling, target angular coordinates and tracking errors in transla tional coordinate system were obtained as shown in Figures 5–8. Based on the obtained data, it may be concluded that Kalman filter features a significant dynamic lag and underperforms compared to (α – β)-filter in terms of dynamic tracking errors.

Table 1

Root-mean-square error. Frequency of target addressing 20 Hz

The limiting algorithms proposed for Kalman filter variance matrix elements allow to decrease the tracking error down to the values comparable with those of adaptive (α – β)-filter.

To address the demanding requirements for the radars with respect to the number of tracked targets, there is a need to determine optimum allocation of time between the tracked targets, and, consequently, there is a task to increase the quality of angular coordinates filtration at a decreased frequency of target addressing. This paper gives an overview of the aforementioned case of target tracking at a frequency of target addressing decreased to 5 Hz.

Table 2

Root-mean-square error. Frequency of target addressing 5 Hz

Table 3

Root-mean-square error. Frequency of target addressing 20 Hz. Noise amplitude increased to 30´

In accordance with Figures 9–12, at a decrease of target addressing frequency, Kalman filter modifications tracking errors by angular coordinates reach the level of adaptive (α – β)-filter.

Another quality criterion of filtration algorithm operation is its resistance to external noise impact. Thus, numerical calculation was carried out subject to an increase of noise in angular channels to 30 angular minutes. Based on the data shown in Figures 13–16, it may be concluded that the dynamic characteristics of the described modifications of Kalman filter are similar to those of (α – β)-filter. At the same time, rootmean-square error of target position estimation and angle tracking errors obtained when using thedescribed modifications are significantly lower than when using the described (α – β)-filter.