1. Introduction
Tracking a ground target using an airborne sensor platform is frequently used in various applications, such as surveillance, search, and rescue missions [
1,
2,
3]. Airborne radar sensors are of particular interest in various surveillance missions, because of their ‘day-and-night’ operational capabilities [
4]. Airborne synthetic aperture radars (SAR) are often used to acquire high-resolution images of ground targets [
5,
6]. In [
7,
8], authors presented the application of airborne sensors for magnetic anomaly detection (MAD). MAD is widely used in maritime surveillance, detection of shipwrecks, geophysical studies, etc. [
7].
Estimating the state of a ground target using the measurements from an angle-only airborne sensor is one of the most practical applications. A number of works have extensively studied the angle-only tracking problem in the 2-D Cartesian Coordinate System (CCS) [
9,
10]. However, as the authors in [
2] pointed out, the number of works on 3-D angle-only tracking problems is relatively low. Some of the earlier works involving tracking ground targets in the 3-D coordinate system using angle-only sensors are reported in [
11,
12]. In [
11], tracking an air target using a ground-based angle-only sensor is considered. Tracking a ground target using an airborne angle-only sensor is a slightly different problem since additional information about the target height will be available. In this work, our focus is to track a ground target using an airborne angle-only sensor platform.
One of the major challenges in angle-only tracking is observability. Considering the additional information of the height of the target from the sea level, such observability issue can be addressed. Since the target is on the ground, the target’s height is the same as the height of the terrain from the sea level, which can be obtained from pre-stored Digital Terrain Elevation Data (DTED) [
13]. Most of the aforementioned works in 3-D angle-only tracking considered the height of the ground target from the sea level is known accurately [
14,
15,
16]. However, in practical setups, such height information is associated with uncertainty due to the errors in the DTED data. Ignoring the height uncertainty (i.e., using a wrong height value) will lead to a bias in the estimated state. To the best of our knowledge, not many analysis is performed on tracking a ground target with terrain uncertainty. This is the motivation for this work to develop algorithms to handle the terrain uncertainty with angle-only sensors.
Apart from the terrain uncertainty, possible biases in the sensor measurements play a key role in determining the quality of the estimates. Possible sources of bias include sensor alignment bias, sensor altitude bias, location bias, etc. [
17]. In this work, we consider only the measurement biases, i.e., the biases in the elevation and bearing angles. Sensor biases and terrain uncertainty should be handled jointly in order to obtain better tracking results.
In this work, we propose a filtering algorithm to track a target with bias and altitude uncertainties. With a larger bias uncertainty, a filter will take a longer time to reduce the target state bias. If we have an option to reduce the sensor bias uncertainty before we start tracking the target of interest, that will help to obtain a better estimate of the target faster. Usually, an airborne platform will fly from a base station to the Region Of Interest (ROI), where we have the target of interest. On the way to the ROI, a bias estimation could be performed by pointing the angle-only sensor toward one more multiple stationary ground object, called as targets of opportunity. In this work, this bias estimation is considered an optional step. Note that the bias uncertainty will not be completely removed even with this optional step. Hence, the filtering algorithm used for tracking the target should still consider the bias uncertainty.
In this work, we explore the possibility of improving the bias estimation using targets of opportunity by changing sensor trajectory. We also study the effect of increasing the number of targets of opportunity and changing their locations with respect to the sensor trajectory on bias estimation. Our proposed approach considers the bias estimation when the x and y coordinates of the target of opportunity are known as well as unknown. A number of bias estimation approaches are proposed in the literature [
18,
19,
20]. However, the challenges in the bias estimation with the terrain uncertainty are not considered in any of the papers.
Predicting the performance of an estimator is essential to decide the optimal sensor trajectory or optimal targets of opportunity locations. The covariance of an unbiased estimator is bounded by the Posterior Cramer–Rao Lower Bound (PCRLB) [
21,
22]. When the estimator is biased, the estimated covariance can not be directly bounded by the PCRLB. In [
23], the authors proposed a performance bound considering the gradients of the bias state. Such a performance bound on the total variance of the estimator is referred to as
biased PCRLB [
24]. The central idea behind using the gradient of the bias state is to have a dependency on the non-constant part of the bias. In other words, the bias can not be removed from the measurements by simple subtraction. In this work, PCRLB and a biased PCRLB are derived for angle-only tracking problems with bias and terrain uncertainties.
In this paper, we consider two possible scenarios: ground target with terrain uncertainty (1) remains stationary; (2) moves with a nearly constant velocity (CV) model [
25]. We assume that the ground target (while moving in nearly CV) moves along the x–y plane, i.e., there is no velocity along the z-axis. Although the dynamic model of the target state is linear, the angle-only measurements are non-linear functions of the target and the sensor state [
11]. For such an estimation problem, several non-linear filtering algorithms are proposed in the literature. Some of the examples include Extended Kalman Filter (EKF) [
26], Cubature Kalman filter (CKF) [
27], Unscented Kalman Filter (UKF) [
28], and Feedback Particle Filter (FPF) [
29].
The computational complexity of the UKF is of the same order as the EKF while providing improved estimation accuracy addressing the approximation issues of the EKF, as shown in [
30,
31]. As a result, we propose a filtering algorithm using UKF. However, UKF can easily be replaced by any other non-linear filter. One of the challenges in the ground target tracking problem is filter initialization with terrain uncertainty. In this work, we use the measurements obtained by the biased angle-only sensor to initialize the target state by incorporating the terrain uncertainty. The estimate errors of the target trajectory are compared with the PCRLB and a conventional approach to evaluate the accuracy and the benefit of the proposed algorithms. The simulation results show that the proposed approach provides better tracking results with all the given uncertainties.
The key contributions of our work can be stated as follows: (1) We derive the PCRLB for this problem to predict/evaluate the performance of the estimator and optimize bias estimation; (2) Bias estimation using a separate target(s) of opportunity is proposed with optimal platform trajectory and optimal target of opportunity location selection; (3) We propose a filtering approach to estimate a target with bias and terrain uncertainties.
This paper is organized as follows. We discuss the problem description in
Section 2. System model detailing the coordinate system, measurement generation, and the system dynamics are introduced in
Section 3. A discussion on performance bounds is presented in
Section 4. Bias estimation approaches and related analysis are detailed in
Section 5. Filter initialization, optional bias compensation, and ground target tracking are discussed in
Section 6. Simulation results are shown in
Section 7 and the paper ends with the concluding remarks in
Section 8.
2. Problem Description
In this paper, our main objective is to track a ground target in 3-D using a biased airborne angle-only sensor. The ground target can either remain stationary or move at a nearly constant velocity. The height of the ground target from the sea level is obtained from DTED; hence it has uncertainty. The two major sources of uncertainty are measurement bias and terrain uncertainty.
To reduce the measurement bias uncertainty, the possible biases could be estimated using separate target(s) of opportunity on the way to the region of interest from the base station. Two possible cases that can happen with the target(s) of opportunity are (1) x and y coordinates of the target(s) are known accurately, but the z coordinate is obtained from the DTED data, which has error; (2) x and y coordinates of the target(s) are unknown, but the z coordinate is obtained from the DTED data as in the first case. Platform trajectory and the location of the target(s) of opportunity can be optimized to obtain a better bias estimate by minimizing the additional time required to reach the destination. That is, we prefer if we do not need to change the trajectory of the platform.
In this work, we make the following assumptions:
Only a single target is considered. However, the proposed algorithm can be used for multiple well-separated targets without any modification;
The height of the ground target from the sea level is fixed, but not known accurately;
The ground target can either remain stationary or move with a nearly constant velocity;
Bias affecting the angle-only measurements are unknown constant and additive. Time-varying bias is not considered in this work. However, the proposed approach can be easily extended for time-varying biases;
Data association issues are not considered, i.e., false alarms are not considered.
In the next section, we introduce the coordinate system, the system dynamics and the measurement model.
5. Bias Estimation Using Targets of Opportunity
Recalling the discussions from
Section 2, an optional two-step bias estimation using targets of opportunity is presented in this section. To bring clarity and avoid confusion with the original target, throughout this paper, we reserve the term ‘target of opportunity’ to indicate the target for which some prior information is known, or we are not interested in estimating that target’s state. Note that this step is usually performed on the way to the target region from the base station. Hence, we need to consider only a reduced bias error when estimating the target state.
The two steps involved in our proposed bias estimation approach are as follows. The first step is to identify one or more stationary targets of opportunity on the sensor’s path to the tracking region. The next step is to estimate the biases in the measurements using the identified targets of opportunity. Such a bias estimation approach has two following benefits:
The uncertainty in the sensor bias is reduced before the original ground target appears in the sensor’s field of view. Hence, the tracking can provide better estimates from the beginning. Otherwise, we may need to make more maneuvers to reduce the biases in the state estimate to obtain a reasonable tracking accuracy;
The sensor can choose multiple targets of opportunity to improve the convergence of the bias estimates. When we use only the interested target to correct the bias, it will take a longer time to converge.
In this section, we analyze the impact on bias estimation caused by the change in sensor trajectory, the number of targets of opportunity used and targets’ proximity to the sensor trajectory. We consider both scenarios where the locations of the targets of opportunity are known as well as unknown, but the terrain heights are not known accurately. PCRLB is used to quantify the estimation quality in order to find the optimal platform trajectory and the locations of the targets of opportunity. The terms ‘target of opportunity’ and ‘target of opportunity with terrain uncertainty’ are used interchangeably in this section.
5.1. Known Location with Terrain Uncertainty
In this section, the bias estimation is performed with a known location of the target of opportunity. To emphasize, the x and y coordinates of the target of opportunity are known, and the height (z) information is obtained from DTED. Hence, there is an uncertainty in the target’s z value. The following factors impact the bias estimation: the number of targets of opportunity used, change in sensor trajectory and the proximity of the target of opportunity with the sensor trajectory. In this section, two possible scenarios are analyzed, and a conclusion on the optimal bias estimation is presented using the PCRLB. In the first scenario, the sensor bias is estimated using one target of opportunity with various sensor trajectories. In the second scenario, the sensor bias is estimated using two targets of opportunity with possibly different terrain heights.
Let us denote the state vector of the stationary target of opportunity as . Here, the superscript ‘’ indicates the target of opportunity. The uncertainty in the height of the target of opportunity is modeled as a zero-mean Gaussian with a standard deviation of . Although the location of the target of opportunity is known, is needed to be estimated because of the presence of the terrain uncertainty. Therefore, the state vector of the bias estimation problem at the time step k is expressed as . Here, the post-fix ‘’ refers to the augmented state for the first scenario, i.e., known location of the target of opportunity.
Our next step is to initialize the filter for bias estimation. Modeling the error associated with the terrain uncertainty by a Gaussian with zero-mean and
standard deviation, we can write
. Hence, we can initialize the height of the target of opportunity with the DTED information
. The bias states are initialized as
. Therefore, the state vector is initialized as
. Moreover, the initial state covariance
is calculated using the FIM evaluated at the initial time step as
. After filter initialization, the non-linear filter (UKF in our work) is used to estimate the augmented state. Note that, for initialization
is expressed as,
The biased measurement covariance matrix, during initialization, of the augmented state is expressed as
. Once
and
are obtained for
, the FIM is evaluated from (
9) as,
After initialization, we obtain the PCRLB for the time steps
. First, we obtain the Hessian matrix
of the bearing and elevation measurement model as,
where the relative state
. The measurement contribution
, where measurement noise covariance is
. With
,
and
, PCRLB is evaluated using (
9). As the sensor bias and the target height are constants, we consider
, i.e., identity matrix of appropriate dimension.
In order to improve the bias estimation further, we consider two different cases below.
5.1.1. Change in Sensor Trajectory
For a given target of opportunity, we restrict our observations to two types of sensor trajectories. The sensor can either follow the CV model and make a fly-by while estimating the bias state or follow a combination of CV and CT models to make a turn around the target to estimate the bias. In this work, the combination of CV and CT models is referred to as CV-CT model. The aforementioned two types of sample sensor trajectories are shown in
Figure 2a,b, respectively.
Let us assume that the target of opportunity remains in the sensor’s field of view for K time steps. The x-axis indicates the sensor heading at the start time step. In other words, when the sensor follows the CV model, .
Let us now consider the case when the sensor follows the CV-CT model. We denote as the total number of time steps the sensor follows the CV model in this CV-CT model. The platform switch to the CT model when the platform reaches the closest distance from the target of opportunity. Considering the sensor velocity to be Vm/s and the sampling rate to be T, we can write the total number of samples obtained while the sensor remains in the CV model to be . Once the sensor starts following the CT model, the total number of measurements obtained by completing one full cycle is denoted by , where is the turn rate. We consider the total number of time steps needed to obtain samples to be .
Now we shift our focus into finding
, for the sensor to follow the CT model. Let us denote the x–y coordinates of the sensor at
as
. From the discussions of the previous paragraph, the maximum change in the y-coordinate occurs at
. In this work, the idea behind finding
is to ensure that the target of opportunity is inside the circle formed by the CT model,
Now, using the CT model (
A1), we can show that
From (
25) and (
24), we can obtain,
However, the platform has a constraint on maximum turn, , that it can make, hence we pick the as to reduce the additional time required to reach the region of original target.
From (
9), it is evident that the Hessian matrix
significantly affects the PCRLB. When considering (
23), the elements corresponding to the differentiation involving
in
are constant. Therefore, the measurement contribution from the bearing bias does not depend on the sensor trajectory. On the other hand, the terms corresponding to the differentiation involving
in
depends on the location of both the sensor and the target of opportunity. As a sensor trajectory formed by the CV-CT model reduces the relative distance between the target of opportunity and the sensor,
and
reduces. As a result, reduction in
(from (
23)) leads to the reduction in PCRLB. Thus, the sensor trajectory formed by the CV-CT model provides a better elevation bias estimation when compared to that of the sensor trajectory formed by the CV model. A comparative analysis between bias estimation performance while the sensor follows both the CV and the CV-CT model is shown in
Section 7.3.1. Note that we need to spend more time on this bias estimation when we use the CV-CT model. With the CV model, no additional time is needed to scan the target of opportunity since there is no change in the platform trajectory.
We now expand our analysis to show the effect of using two targets of opportunity with known locations and additional terrain uncertainty.
5.1.2. Bias Estimation with Multiple Targets of Opportunity
In the previous section, we concluded that the sensor following a CV-CT trajectory improves the estimate
, when compared to the case where the sensor follows only a CV trajectory. In this section, we analyze the significance of adding a second stationary target of opportunity in the sensor’s field of view. The goal here is to analyze whether the presence of the second target of opportunity coupled with the sensor following the CV model provides better
so that we do not need to change the platform trajectory. Note that the second target of opportunity with a different terrain height could be located anywhere in the sensor’s field of view.
Figure 3 shows an example of two targets of opportunity located at a distance of
m and
m from the sensor trajectory.
Based on the assumption that only one target of opportunity is tracked at any given time, the second target of opportunity is picked far away from the first target so that the second target will be in the sensor’s field of view for a reasonable time after completing the tracking of the first target. For instance, let us assume that the total number of time steps taken by the sensor for estimating the bias using one target of opportunity with the CV-CT model is K. Now, let us consider that the bias estimation is performed using two targets of opportunity. Denoting (where ) as the total number of time steps for which the first target of opportunity is used, we obtain the total number of time steps used for the second target of opportunity as . Note that an equal number of time-steps are considered in both cases for a fair comparison. However, in practice, the total number of time steps depends on the sensor’s field of view and the target locations.
A better elevation bias estimate can be obtained while the target of opportunity is located closer to the sensor trajectory. To explain such a result, we analyze the reduction in PCRLB.
To validate the above notion, we provide the following experimental analysis. Denoting the first target of opportunity as
and the second target target of opportunity as
, from
Figure 3, we can write the x-coordinates as
m and
m. Note that, in presence of multiple targets of opportunity, we denote first, second, and third target of opportunity with lower-case letter ‘
’, ‘
’, and ‘
’, respectively. Such notation is used to avoid conflict with the sampling time of sensors, which is denoted by the upper-case letter ‘
T’. The y-coordinates are changed (while keeping
and
unchanged) to locate both the ‘
’ and ‘
’ at various relative distances from the sensor trajectory. A comparison of the PCRLB of
(in degree) at the end of the bias estimation is shown in
Table 1. In this analysis, the performance of the CV model with two targets is compared with a CV-CT model with one target (
), as described in the previous section.
From
Table 1, the following conclusions are drawn:
The addition of the second target of opportunity with different terrain height (i.e., different error in the assumed height) provides additional information to the estimator. Therefore, in most of the above simulation scenarios, estimation of with the CV model for sensor trajectory and two targets of opportunity provides better performance than that of the CT model for sensor trajectory and one target of opportunity;
Bias estimation accuracy diminishes with the distance of the targets from the sensor. If the second target of opportunity is further away from the sensor trajectory, the additional information contributed to the bias estimation is insignificant. In such a scenario, the location of the first target of opportunity plays a significant role in the performance of bias estimation. As shown on the {8 and 9}-th row of
Table 1, when the targets are relatively far away from the sensor trajectory, sensor trajectory with CV-CT model and one target of opportunity outperforms bias estimation obtained by the CV model along with two targets of opportunity;
The bias estimation also depends on the error of the assumed height of the targets of opportunity. For our analysis in
Table 1, different height errors are used for different targets. As shown on the 3-rd and 7-th rows of
Table 1, we obtain different PCRLB estimates even with same
y values ((
,
) and (
,
)).
Additional results involving the Root Mean Square Error (RMSE) plots along with the scenarios of positioning the second target of opportunity on two opposing sides of the sensor trajectory are shown in
Section 7.3.
From this analysis, we can conclude that we can obtain a better estimate of the biases by using multiple targets of opportunity without wasting additional time that we discussed in the previous section for the CV-CT model with a single target.
Although the bias estimation discussed in this section only considers targets of opportunity having known locations, we may need to pick an unknown stationary object as a target of opportunity. In the next section, bias estimation with an unknown location of the target of opportunity is introduced.
5.2. Unknown Location with Terrain Uncertainty
Following the same notations from
Section 5.1, we consider a stationary target of opportunity with unknown
and
to estimate the bias. Let us denote the unknown stationary target and bias state as
and
, respectively. In order to estimate both
and
simultaneously, we form the augmented state vector as
. Our goal here is to estimate the augmented state vector
, even though we are not interested in the target location.
We can write the initial augmented state vector
, where
and
are converted position coordinates and
is the assumed target height (equations are provided later in (
29) and ((
30) of
Section 6.2). The measurement covariance matrix
is used to find the initial covariance. Details about the construction of the Hessian matrix, i.e.,
, is shown in (
A6). Once
and
are obtained, we obtain
from (
22). Initial covariance is evaluated as
. Non-linear filter is used to update the state
.
In order to evaluate the bias estimation performance, PCRLB is evaluated following the formulation of
Section 4. First, we evaluate the Hessian matrix
(details about the matrix construction is shown in (
A5)) and the measurement covariance matrix
. With
and
, the measurement contribution of the PCRLB is evaluated as
. Substituting
,
and
into (
9), PCRLB is evaluated.
In this section also, we study the possibility of improving bias estimation by,
5.2.1. Change in Sensor Trajectory
For a known location of the target of opportunity, in
Section 5.1.1, we concluded that a sensor trajectory comprised of the CV-CT model provides a relatively improved bias estimation when compared to that of the CV model. Now, considering the targets of opportunity with unknown locations are considered, we use PCRLB to explain the performance of bias estimation. Considering
from (
A5) (
Appendix A), the differentiation involving
is dependent on
and
. As a result, a sensor trajectory formed by the CV-CT model provides a reduced estimation error of
by reducing the relative distance between the sensor and the target of opportunity. Note that this was not the case with the known target location. However, for the differentiation involving
, the relative height
is present in the numerator. As the terrain uncertainty is considered, a similar conclusion on a preferred trajectory can not be drawn for the estimation of
, as opposed to the estimation of
.
Similar to
Section 5.1, we now expand our analysis by introducing multiple targets of opportunity with unknown locations and terrain uncertainty.
5.2.2. Bias Estimation with Multiple Targets of Opportunity
In this section, we investigate the possibility of improving the bias estimation by introducing multiple targets of opportunity in the sensor’s field of view. The sensor follows a trajectory formed by the CV model as opposed to the CV-CT model.
As discussed in
Section 5.1, the bias estimation depends on the proximity of the targets of opportunity to the sensor trajectory. To analyze similar dependency for bias estimation with unknown locations, we perform a comparison of PCRLB. Let us consider the ground truth of x-coordinates of the first and second target of opportunity as
m and
m. For 3 different values of
and
, we obtain three different locations of the target of opportunity based on its proximity to the sensor trajectory. Note that by location, we refer to the ground truth needed to evaluate the PCRLB.
Let us analyze the PCRLB evaluations from
Table 2. Firstly, we can draw the following conclusions for
:
When the sensor follows a trajectory formed by the CV-CT model, one target of opportunity is sufficient to estimate the bearing bias. See
Section 5.1.2 for explanations.
Secondly, the following conclusions can be drawn for :
When both the targets of opportunity are relatively far away from the sensor trajectory, a better estimation of is obtained when the sensor follows a trajectory formed by CV-CT model. The reason behind such result can be attributed to the reduction in relative distance and , which, in turn, reduces the PCRLB;
When
is away from the sensor trajectory, we draw the same conclusions as the known locations of the targets of opportunity, discussed in
Section 5.1.
Following the above discussions, following the CV-CT model with one target of opportunity is a better choice than following the CV model with two targets of opportunity. However, adding one more target of opportunity after completing the CV-CT model with the first target of opportunity on the way to the destination with the CV model will help to improve the elevation bias estimate.