2.1

Kalman filter

Practical solutions of modern radar tracking systems are usually based on the Kalman filter. The Kalman filter is an optimal estimator in the sense of minimizing mean square prediction error for linear objects.⁴ The Kalman algorithm assumes that the object model and measurement model are described by linear equations, and the random processes used during modeling are Gaussian, and are described by normal distributions with known mean values and covariances. The discrete-time Kalman algorithm to describe dynamic and measurement model use following equations:

Where:

1 Update:

2 Prediction

Where:

For this paper the state vector is written as x(k) = [x, y, z, v_x,v_y, v_z, a_x, a_y, a_z]^T where x, y, z describe position,v_x, v_y, v_z velocity and a_x, a_y, a_z acceleration in carthesian coordinate system respectively. Following matrices were used to describe the state space model:

2.2

Gating

In radar tracking system, association gate is defined around a predicted target position, which accumulates uncertainty of prediction, resulting from tracking filter parametrization and past performance of the tracking process. When the following relation of norm of residual error d² is satisfied

measurement is considered for track update, with ỹ being the innovation vector described in (3) and S is the innovation covariance matrix described by (4). In the literature equation (14) is known as statistical distance or Mahalanobis distance, and is a typical metric used in gating algorithms. The parameter T_G is responsible for the effective size of the association gate and can be defined in several ways. In this paper T_G value has been determined using chi-square statistics. This approximation can be used keeping in mind the assumed Gaussian nature of random processes used during modeling in the tracking filter. This means that the value is the sum of squares of M independent Gaussian random variables with zero mean value and uniform standard deviation, where M is the size of the measurement vector.⁵

3.1

Shape and orientation of association gates

First tests examines the shape of the association gate in the Cartesian system depending on the orientation of the detected object relative to the radar. For this examination assumed radar measurement error is 0.25 [deg] in azimuth 0.5 [deg] in the elevation and 15[m] in the distance, and the covariance of the model matrix is:

The results of simulation tests are shown in Figure 1 and Figure 2. Figure 1 shows the change in the angle of inclination in the X-Y plane and the size of the association gate during movement of the object from east to west. Figure 2 shows the relationship of the angle of inclination in the Y-Z plane and the size of the gate depending on the height of tracked object.

Figure 1.

Shape and orientation of association gates depending on detection point in X-Y plane

Figure 2.

Shepe and orientation of associatioin gates depending on hight of detection object

Another test concerns the change in the shape of the association gates depending on the radar parameters. Assumed radar measurement error is 0.25 [deg] in azimuth 0.25 [deg] in the elevation and 15 [m] in distance.

Figure 3.

Shape and orientation of association gates depending on detection point in X-Y plane with various R matrix

Next study concerns the impact of matrix selection on the size of the association gate. For this examination assumed covariance of the model matrix is:

Results are shown in Figure 4. As we expect orientation of the association gate (direction of the axis of the large ellipse - the association gate on the X-Y plane) is closely related to a straight line perpendicular to the so-called line of sight (a straight line connecting the prediction point and radar). The angle formed by the perpendicular line with the X axis approximately describes the orientation of the association gate (Figure 5).

Figure 4.

Shape and orientation of association gates depending on detection point in X-Y plane with various Q matrix.

Figure 5.

Geometric dependencies of the association gate orientation.

Height at which the object was detected also has a great impact on the orientation of the association gate in space. Depending on the height, the association gate is inclined to the X-Y plane, where inclination is increasing with height (Figure 2). Size of the association gate, specified in the cartesian coordinate system, is mostly influenced by the distance of the tracked object from the radar. Farther the tracked object is from the radar, the more the angle measurement errors translate into the absolute error of position measurement, which consequently enlarges the gate (Figure 1). The radar parameters themselves are another important factor affecting the shape of the gate. Comparing Figures 1 and 3, it can be seen that reducing the elevation error two times reduces the height of the association gate twice. Comparison of Figure 1. with Figure 4. allows comparision of the effect of the matrix on the shape of the association gate. Decreasing the value of matrix parameters visibly reduces the size of gate, but much less than the radar measurement error.

3.2

Influence of state vector on the shape of association gate

Shape and orientation of association gates is also influenced by the object state vector. Figure 6 and 7 show impact of track direction on orientation of association gates. Figure 6 show the average value of angle of orientation for 10 simulations for opposing direction of simulated object velocity. The tests assumed that in the first case the object moves from east to west and in the second case from west to east. The object is moved only in the X axis with a constant position value in the Y axis = 6000 [m] and a constant height of 200 [m].

Figure 6.

Shape and orientation of association gate depending on direction of tracked object move.

Figure 7.

Shape and orientation of association gate depending on direction of tracked object maneuver.

Figure 7 compares the average values of orientation angle of association gate from 10 simulations for object maneuver in opposite directions (course maneuver). The starting point for both maneuvers is point 1, the blue bar on the graph represents the average angle of orientation of the association gate relative to the radar for an object moving in the counterclockwise direction, and orange for the object maneuvering in a clockwise direction.

Results presented in Figures 6 and 7 show that the orientation of the association gate is also affected by the state vector. Depending on the direction of movement (Figure 6) and maneuver direction (Figure 7), the orientation of the association gate slightly differs for the same object position relative to the radar.

3.3

Impact of detection loss on association gate

In radar tracking data loss due to lack of detection is a common problem. In this situation, the tracking algorithm extrapolates track performing track update without measurement. Such operation means also extrapolation of the state covariance matrix, which effectively results in a relative increase in value in the future innovation covariance matrix. The impact of that loss on the size of the association gate is presented in Figure 8. Figure 9 shows the reference sizes of subsequent association gates for the case without lack of detection.

Figure 8.

Shape and orientation of association gate during lack of detection of object.

Figure 9.

Shape and orientation of association gate during lack of detection of object, reference datas.

Lack of detection almost doubled size of the gate. However, subsequent update with measurement quickly reduces association gate to regular size.