Alignment, Location and Sensor Range Bias

Simulation and filter models are shown in Table 4.3 where alignment, location and sensor range biases are present.

4.1.5.1 The Universal Bias Estimator

In Section 3.4 it is concluded that joint estimation of target position, alignment bias and sensor bias in universal coordinates is impossible. However these biases are estimable separately. The effect of the alignment bias and sensor elevation bias on target position is small at close range and large at long range, while the sensor range bias is constant. According to Figure 3.9a the range bias has stronger estimability close to the sensors. Thus it seems to be a good assumption to neglect the range bias at long range, and neglect the alignment bias at short range. This means that targets at long range are assumed to only be affected by the alignment bias, while short range targets are only affected by the range bias. The problem is now estimable. In the scenario shown in Figure 4.1 a target is defined to be at short range when the predicted position of each Cartesian coor- dinatex, y and z is in the interval [_{−25 km, 25 km]. If the target exceeds this} interval in any coordinate it is at long range. This threshold is chosen based on Figure 3.9a, but recall that it is only approximate since the constant velocity as- sumption is not included in the static estimability plots.

84 4 Bias Estimation and Data Fusion for Distributed Radars in 3D

Figure 4.6 shows the estimation performance for the UBE. The EKFU per- forms better than the KFBU in Figure 4.6a, while the EKFS has superior perfor- mance compared to the KFBS in Figure 4.6b. However neither the EKFU nor the EKFS come close to the KFU.

4.1.5.2 The Absolute and Relative Bias Estimator

In this simulation the estimation performance of the ABE and the RBE are sim- ilar to the performance in Figure 4.2 and Figure 4.3 respectively. The EKFS converges close to the KFU in both cases. The performance of the EKFU is poor in both cases, which is expected since the master sensor (Sensor 0) has an alignment bias which is not estimated.

4.1.5.3 The Standard Universal Bias Estimator

In this simulation the estimator performance of the SUBE is similar to the perfor- mance in Figure 4.4. The EKFU performs slightly better than the KFBU, while the EKFS performs worse than the KFBS.

4.1.5.4 The Standard Absolute Bias Estimator

In this simulation the estimator performance of the SABE is similar to the perfo- mance in Figure 4.5. The EKFS and KFBS have similar performance while the EKFU performs worse than the KFBU.

4.1.5.5 Conclusions

Comparing Figure 4.4 with the performance of the UBE in Figure 4.6 it is clear that the UBE has superior performance in both_{{u} and {s}0}. The inferior perfor-

mance of the SUBE is explained by the bias model mismatch. The azimuth bias estimation of the SUBE is only able to account for the yaw alignment bias, but not the roll and pitch biases. Since location biases are present the UBE perfor- mance is degraded since it is not able to estimate the location bias. The objective of this simulation is to find a scenario where location biases are present and the UBE estimates as many biases as possible. This is achieved by removing the sensor elevation bias. Other alternatives are to remove the alignment biases roll or pitch.

The performance of the ABE and RBE is approximately equal to the perfor- mance presented in Section 4.1.4.1, Figure 4.2 and Figure 4.3 respectively. The

4.1 Constant Velocity Flight with Monte Carlo Simulations 85 Table 4.4: Simulation and filter (estimator) measurement models for align- ment and sensor bias estimation. The SUBE should ideally estimate all el- ements of the sensor biasbi, but due to poor performance in estimating the

elevation biasbφi, the estimated biases are reduced to rangebρi and azimuth

bθi.

Measurement model Bias model System model (4.8) Present:bsi

ni, bρi, bφi, i∈ {0, 1, . . . , M}

KFU model (4.14) Known:bsi

ni, bρi, bφi, i∈ {0, 1, . . . , M}

KFBU model (4.15) White noise vector asi

i,k accounts for:

bsi

ni, bρi, bφi, i∈ {0, 1, . . . , M}

KFBS model (4.16) & (4.17) White noise vector asi

i,k accounts for:

bsi

ni, bρi, bφi, i∈ {1, 2, . . . , M}

UBE model (4.18) Estimates: bsi

ni, bρi, bφi, i ∈

{0, 1, . . . , M}

ABE model (4.19) & (4.20) Estimates: bsi

s0, i ∈

{1, 2, . . . , M} , bρi, bφi, i ∈

{0, 1, . . . , M} RBE model (4.21) & (4.22) Estimates: bsi

s0, bρi, bφi, i ∈

{1, 2, . . . , M}

SUBE model (4.23) Estimates:bρi, bθi, i∈ {0, 1, . . . , M}

SABE model (4.24) & (4.25) Estimates:b_i, i _{∈ {0, 1, . . . , M}}

fact that the ABE model is correct, and the RBE is almost correct, explains the superior performance in _{s0} of the ABE and RBE in Figure 4.2b and Figure

4.3b with respect to the UBE in Figure 4.6b. In the absolute and relative case there are fewer biases to estimate, and the mathematical model is correct. The superior performance is obtained in_{s0} which is where tracking is done in the

absolute and relative case. In_{{u} the ABE and RBE perform poorly, as seen} in Figure 4.2a and Figure 4.3a with respect to Figure 4.6a. This is because it assumes that the CTMTu

s0 = I, which is not the case. The performance of the

bias ignorant KFBU is superior, which means that there is no point in using the ABE or the RBE when tracking targets in_{{u}. This highlights the importance of} universal bias estimation if tracking is done in sensor independent coordinates.

86 4 Bias Estimation and Data Fusion for Distributed Radars in 3D 0 10 20 30 40 50 60 70 80 90 100 0 200 400 600 800 1000 KFBU EKFU KFU k RMSE

(a) Position RMSE in{u} (EKFU).

0 10 20 30 40 50 60 70 80 90 100 0 200 400 600 800 1000 KFBS EKFS KFU k RMSE

(b) Position RMSE in{s0} (EKFS).

Figure 4.7: Position RMSE in meters versus time stepsk for the UBE.