Active tracking

Array radar resource management Alexander Charlish* and Fotios Katsilieris*

3.2 Task management

3.2.3.1 Active tracking

whether to track using active tracking or TWS. If the target is actively tracked, the track manager must also decide the revisit interval time as well as the transmit waveform to use. For tracking, the revisit interval time is the time between mea- surements of a target. When an active track update is executed, the radar beam is directed towards the estimated position of the target in angle space. A beam posi- tioning power loss occurs when the true target angle is offset from the estimated target angle.

Van Keuk and Blackman [26] assume a Singer target motion model, whereby the motion is driven by independent Gaussian Markov acceleration processesq_iin each spatial dimension with specified correlation timeQand standard deviationS. Accordingly, the noise autocorrelation is given by:

RðtÞ ¼E q _iðtÞq_jðtþtÞ

¼d_ijS²exp t

Q (3.14)

whered_ij¼1 for i¼j and 0 otherwise. Speed and position then follow by inte- gration with respect to time. More on the Singer model, including the resulting discrete time state transition matrix, can be found in standard texts [9,28]. In addition, it is assumed that the multiple targets are well-separated point targets, each occupying a single range-Doppler resolution cell. The power loss is modelled by a Gaussian loss function that is matched to the antenna beamwidth. Coherent integration followed by a square law detector is modelled, and the target amplitude is assumed to fluctuate according to a Swerling 1 model.

0 20 40 60 80 100 120

0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31

Time (s)

Sharpness

Track initiation

Non- manoeuvre

manoeuvre Non-

manoeuvre

V₀B

Figure 3.9 Adaptive tracking process. The revisit interval time is selected for when the filter predicted estimation error standard deviation in angle is equal to a specified fraction of the beamwidth. Increased process noise in track initiation and manoeuvre stages results in faster increases in the angular track uncertainty, resulting in more frequent measurements

Based on these assumptions, the track revisit interval timet_r which achieves a specified track sharpnessn₀ can be calculated according to [26]:

t_r¼0:4 rs ffiffiffiffi pQ S 0:4

U^2:4

1þð1=2ÞU² (3.15)

wherer is the target range. This is an approximation to the revisit interval time when the tracking filter is in steady state, given the specified target parameters. The variance reduction ratioU is the ratio of the track estimation error to the mea- surement errors:

U ¼q_Bn₀

s (3.16)

whereqBis the antenna half beamwidth, i.e. half of the antenna 3 dB beamwidth.

The measurement noise standard deviation s is calculated assuming that unbiased measurements of the angular position of a target are corrupted by additive Gaussian noise that in high SNR has a standard deviation according to (8.29) in [29, Section 8.2.4]:

s¼ 2qB

k_m ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2SNR

p (3.17)

wherek_mis the slope of the normalized monopulse error curve, which is taken by Van Keuk and Blackman ask_m¼ ffiffiffi

p2

. SNR is the SNR encountered in the range- Doppler cell.

Due to a non-unity probability of detection, it may be necessary to have numerous looks for the target on a single track update. Van Keuk and Blackman describe a search strategy that minimizes the number of looks required at each update. They present a formula for the expected number of looks, which is a function of track sharpness.

Based on these formulas, the relative track loading for a target can be plotted for varying choices of track sharpness parameter and received SNR, as illustrated in Figure 3.10. It can be seen that the minimum track loading can be found by selecting the probability of false alarmP_FA, the track sharpnessn₀ and the desired SNR without beam positioning loss, which is denoted SN₀. Van Keuk and Black- man recommend selecting the probability of false alarm between 10⁴ and 10⁵, the track sharpness to n₀0:3q_B and a coherent dwell length which achieves SN₀16 dB.

The Van Keuk and Blackman formulas are a set of rules for selecting active track control parameters with low computational complexity; however, the approach is limited. The formulas only describe the asymptotic performance and therefore it may take many measurements before the actual angular error achieves the angular error predicted by the model. Moreover, the formulas are specific to the use of a Singer target motion model. Finally, they enable the track to be maintained with the minimum resource; however, they cannot be easily adapted for other cri- teria, such as a Cartesian error requirement or operational requirements.

Covariance analysis

Instead of using the Van Keuk and Blackman rules of thumb, the current track can be predicted forwards in time to find when the track sharpness threshold is reached, which gives the track revisit interval time [31]. This approach has the benefit that the actual track at the current time is used, and it is therefore not based on the asymptotic performance. In addition, this approach is more flexible, as it can be applied with any target motion model or set of models.

Let x_tk be the target state vector at time tk, commonly comprising position and velocity in two or three dimensions, e.g.x_tk ¼ ½xtk;ytk;x⁰_tk;y⁰_tk^T. Then, the track at time tk based on all the measurements up to timetk is comprised of the state estimate x_tkjtk ¼ ½^x_tkjtk;^y_tkjtk;^x⁰_tkjtk;^y⁰_tkjtk^T and the filter posterior error covariance matrixP_tkjtk, which represents the uncertainty in the state estimatex_tkjtk, e.g.:

P_tkjtk ¼

E x_tk ^x_tkjtk

x_tk ^x_tkjtk

E x_tk ^x_tkjtk

y⁰_tk ^y⁰_tkjtk

h i

... ... ...

E y⁰_tk ^y⁰_tkjtk

xtk ^xtkjtk

h i

E y⁰_tk ^y⁰_tkjtk

y⁰_tk ^y⁰_tkjtk

h i

2 66 66 4

3 77 77 5 (3.18)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

10 20 30 40 50 60 70 80 90 100 110

Track sharpness (beamwidth)

Relative load

Relative load for varying track sharpness for Van Keuk model

10 dB 13 dB 16 dB 19 dB 22 dB

Figure 3.10 Relative tracking loading for varying choice of track sharpness parameter and SNR. Image taken from [30]

The error covariance matrixP_tkjtk can be predicted to time t_kþt_r, using the standard Kalman filter prediction equation:

P_tkþtrjtk ¼FðtrÞPtkjtkF^TðtrÞ þQðtrÞ (3.19) whereFðtrÞandQðtrÞare the target state transition and process noise covariance matrices, respectively, for the revisit interval tr. The state transition matrix describes the assumed change to the target state during the revisit intervaltrand is therefore derived based on an assumed motion model [9,28,32]. Deviations to the assumed model are accounted for by assuming that the state transition also includes a noise sequence that has an associated covarianceQðtrÞ.

The predicted track at timetkþtr can be converted into spherical coordinates (angles, range and Doppler) to give the predicted error covarianceP_tkþtr. Lets^b_t

kþtr

ands^e_t

kþtr be the filter predicted error standard deviation in bearing and elevation, respectively. For a set of possible revisit intervals tr2Tr, for example Tr¼f0:5;1:0;1:5;2:0;2:5;3:0gs, then the following procedure can be applied [31]:

1. Start withtr as the greatest revisit interval inTr

2. Predict track covarianceP_tkþtrjtk

3. Convert covariance into spherical coordinates and extracts^b_t

kþtr ands^e_t

kþtr

4. If eithers^b_tkþtr ors^e_tkþtr exceedsv0, then decreasetr, else use the currenttr

By following this procedure, the largest revisit interval is found that does not exceed the specified track sharpness. This procedure can easily be adapted to be based upon alternative track criteria that may be more operationally relevant, such as the Cartesian error. Other popular measures are also based on the covariance of the track estimate, e.g. the trace of the covariance matrix, see the analysis in [33].

An important aspect of this adaptive tracking procedure is the use of inter- acting multiple model (IMM) filtering [34]. IMM filtering maintains a number of parallel filters each with a different motion model. Based on the observed mea- surement innovations, the probability of each model being currently active can be evaluated. The estimates and covariances for the all models are then mixed at each time step. The use of multiple models ensures that the filter-predicted covariance is consistent given varied target manoeuvres. When using IMM, the track revisit interval selection procedure is the same; however, step 2 is based on IMM prediction [34].

Benchmark problems

The benchmark problems [35,36] were a set of common scenarios, radar model and performance assessment criteria, that enabled the comparative assessment of adaptive tracking approaches.

As discussed above, adaptive tracking matches the revisit interval time to the estimated manoeuvre state of the target under track. Consequently, the quality of the estimator has a strong influence on the selection of the track control parameters and the subsequent resource allocation. This was clearly demonstrated through the numerous solutions [31,37,38] to the benchmark problems. It was shown that IMM filters are crucial for adaptive tracking, as they ensure that the target dynamic

model in the filter is matched to the current target manoeuvre. Consequently, the revisit interval time decreases as the target manoeuvres and increases as the target follows predictable motion. From the benchmark problems, the combination of IMM and multiple hypothesis tracking [9, Section 14.8] emerged as the tracking methodology that resulted in lowest radar loading for each target track [37,38].