2.1

Data storage and management

The main goal of the data storage and management module is the appropriate acquisition of measurement data (plots) and the management of the tracking chain work. Plots are stored in previously prepared containers. Their configuration is dependent on the radar working mode and the algorithm control commands values. The proposed solution collects plots from N scans. It is assumed that the data from one scan (e.g. one revolution of antenna) corresponds to one layer of the container where the lowest layer number contains the oldest data. For the visualization of the whole process this layer is marked at the bottom, as showed in the Figure 3, where N=4.

Figure 3.

The visualization of the data storage container with distinguished layers for N=4.

Beside the distribution of the data to layers accordingly to its time of arrival, there is a spatial separation of the data in range and azimuth. The angular distribution is the consequence of the data acquisition process from detector and the work of the TBD algorithm, EKF tracker and deletion of used or expired data. This also enables the independent access to the data by particular processes that can work in parallel. In order take a full advantage of the capabilities of multicore processors, the Intel Threading Building Blocks (TBB) technology was used in proposed solution. The division of the range domain into sectors gives the possibility of further parallelization of computations and in the same time it limits the amount of data processed in one process.

Figure 4.

The spatial separation of the TBM tracking chain processes.

A side effect of the assumed spatial separation of data is the possibility of initiating multiple tracks with the same plots. Therefore an algorithm of finding and eliminating such tracks had to be implemented.

The data storage and management module is also responsible for storing the TBM trajectories templates and the areas of data processing. The templates database is range dependent, which is the effect of the following facts:

This phenomenon are explained in Figure 5.

Figure 5.

An object trajectory visibility for different launch points and course angles.

The processing areas database includes earlier prepared range and azimuth sectors. Before the start of the TBD algorithm the virtual connection between all sectors and chosen templates is made.

2.2

TBD algorithm module

From the point of view of the whole tracking chain the type of TBD algorithm is insignificant. Depending on the chosen solution data structures of plots or tracks may be extended with additional information. In this article the M/N method called also the binary integrator was used due to its low computational burden and its simple principle of operation. The plots collected from N consecutive scans are associated with earlier prepared templates. If the plots fit into the correlation gate of the template, then a flag variable is set to “one” and otherwise it is set to 0. In a such created sequence of flags there should be at least M “ones” to give a positive decision about object detection.

The templates database contains the correlation gates sizes that has been calculated in a separate application. The type of rocket and its trajectory (lofted, max-range, depressed) along with the earlier described range-dependent phenomenon has been taken into account, as a consequence, the templates are also range- and course-dependent. During the design of the templates the antenna time of revolution has been also considered and the lack of correlation between the launch time and antenna rotations has been assumed. Due to this fact a certain amount of dislocation between the first observable detection of the object and its theoretically assumed position had to be admissible.

Similar assumptions were made in [3], where the correlation gates dimensions are the referred to the radar resolution cell size and the templates include the maximal value of the two. The position of the middle point of the gate is tied to the last detected plot of the candidate track. However this solution has serious consequences. In the case of a stationary or a slowly moving target a false TBM detection could be formed. Moreover, a false candidate plot can dramatically relocate the correlation volume and as a result the object may not be detected. To avoid these side effects, in the presented solution the correlation gate position is dependent on the first analyzed plot and the gates dimensions are described by the minimum and maximum admissible relocation in every coordinate. An example of such prepared template is represented by the black squares in figure 6, where the first plot is marked by the yellow point, the red ones are consecutively detected plots of the object’s trajectory drawn with the blue line.

Figure 6.

An example of template for N = 5.

Considering every coordinate separately allows to narrow the correlation gates dimensions, because in every direction the displacement will vary depending on the object course angle and the range. This idea is presented in figure 7, where by the yellow areas the allowable object’s displacement in reference to the radar in chosen direction is schematically presented. If the object is moving towards the radar, the possible displacement in range is larger than in azimuth. Similarly, if the object course angle is perpendicular to the radar, the displacement in azimuth is dominant. Additional advantages of such solution are the increased separation between templates and the decreased probability of false alarm.

Figure 7.

The maximum displacement of object in reference to its course angle.

The decision about a TBM object detection should be formed as soon as possible. Thus, the TBD algorithm starts working from layer 0, 1… up to N-M, so the first track will be initiated after M scans. Plots used in this processed are marked by appropriate flag and will not be taken into account in further processes. In this way the number of formed tracks is minimized.

New plots stored in the container’s zero layer are firstly used in the updating process of a already formed candidate tracks by their association with the template. Plots uncorrelated neither with older nor newly formed tracks are send to the tracking procedure in the extended Kalman filter.

2.3

Extended Kalmna filter tracking

Basing on the candidate tracks formed by the TBD algorithm the tracking by the extended Kalman filter is initiated. The EKF works with the Newton’s model and the 9-elements long state vector including the position, velocity and acceleration in Cartesian coordinate system. The prediction can be described as [6]:

where:

X_k-1 – state vector at scan k-1,

F – state transition matrix,

X_kp – predicted state vector at scan k,

Q – covariance matrix of discrete process disturbances,

P_k-1 – covariance matrix of prediction errors at scan k-1,

P_kp – predicted covariance matrix of filtration errors at scan k.

The filtration can be described as [6]:

where:

u_kp – predicted measurement error,

H – observation model,

S – matrix of innovation,

R – covariance matrix of measurement errors,

K_k – Kalman gain matrix,

u_k – measurement error,

I – identity matrix

and index xx,n means the n-th element of vector.

Dimensions of correlation gates are calculated based on the matrix of innovation with certain coefficients and the middle point of the gate is located at predicted plot position. Most of the TBM objects do not make turning maneuvers, therefore a single EKF is used. A problem may appear near the apogee of the trajectory, where the direction of the movement in elevation is drastically changing. In effect, the use of the second filter should be considered in further work. This feature could also allow to narrow the correlation gates.

The EFKT module can work in two ways. The first option is designed to initiate the tracking basing on the results of the TBD algorithm. The plots from the candidate track are filtered by the EKF to settle the values of particular matrixes and to shorten the time of its unstable work. Next, the new gate is calculated for the further scan. The plots unused in previous module will be correlated to track, which is the basic function of the EKFT. If any candidate track should be refreshed with a new plot, then the track corresponding to it will also be updated. At the end of this processing chain the plots in the oldest layer are removed from the container leaving the memory space for new data.

Every track contains a parameter describing the number of consecutive prediction passed without updating by a new plot. In the described tracking chain such situation cannot take place more than three times.