1. Introduction
Ground tracking radars mounted on airborne platforms play a key role in many applications, especially those for military purposes; surveillance, airstrike, and escort missions done by aircraft commonly require precise tracking of ground targets. In several modern military campaigns, ground moving target indicator (GMTI) radar on-board the Joint Surveillance Target Attack Radar System (STARS) has been proven strategically and tactically significant [
1]. Accordingly, algorithms that track ground targets running on radars are becoming more important. Although there have been great advances in target tracking, tracking ground targets is still a challenging problem. The reason is that the characteristics of ground target tracking are different from those of tracking other types of targets. (e.g., high clutter, terrain obscuration, etc.) [
2].
Because exploiting appropriate assumptions other than the state-space model can help to improve the statistical inferences of the system [
3], many studies have tried to introduce useful assumptions to ground target tracking. They can be classified based on two criteria of what or how assumptions were applied.
Based on the first criteria, existing studies can be classified into two further categories. The first category considers the behavior characteristics of the ground target that are distinguished from those of airborne targets. Fosbury [
4] and Kastella [
5] each created the terms ‘trafficability’ and ‘hospitability of maneuver’, which represent how easily a vehicle can go through a particular area. These notions adaptively modify the target dynamics so that the dynamics can reflect the tendency of the target to prefer directions with small gradients. The second category involves empirical constraints. For instance, in the work of Streller [
6], ground targets are assumed to move along the infrastructure such as roads, bridges, etc. More specifically, the assumption encourages the prior probability density which is propagated by the system model to align with the road network. The same assumption is utilized in the works of Pannetier [
7,
8]. The works of Mallick [
9] and Kim [
10] are also classified into the same category and share the same motivation as ours. To compensate for inaccurate GMTI measurements, both utilized the assumption that the position of a ground target is restricted to the terrain surface. This idea can be extended even further by adding another assumption that the velocity of the target is tangent to the terrain surface, which allowing the system to estimate the velocity more precisely [
11,
12].
From an other perspective, existing studies can be classified based on the second criteria, namely, how assumptions are applied. The first category involves modifying the target dynamics so that it can reflect the tendency of the target. Similar to the aforementioned works [
4,
5,
6,
13], the system dynamics of the filter are adaptively modified. In other words, external knowledge is embedded in the state-space model. Thus, we have the freedom to control only the tendency of a target. The second category involves transforming the assumption into a state constraint. This type of approach explicitly limits the state of a target to a specific subspace ([
7,
8] for example).
Extensive studies have attempted to deal with such constrained state estimation problems [
11], including the methods that do not rely on the state-space model [
14,
15]. In the case of linear system dynamics and linear constraints, the following methods are applicable: model reduction [
16], perfect measurement [
17,
18,
19], estimate [
20]/system [
21]/gain [
22] projection, pdf truncation [
22], etc. If either system dynamics or constraint is nonlinear, the combination of linearization and linear methods is an available option. Other possible choices are variants of the Unscented Kalman Filter (PUKF [
12,
23], ECUKF [
12], 2UKF [
24], etc.), variants of the Particle Filter [
25,
26,
27,
28] (CLIP, COMP [
29]), and the Smoothly Constrained Kalman Filter (SCKF) [
30]. Moreover, many works in the literature have paid attention to state estimation problems with soft constraints [
18,
19,
31,
32,
33,
34]. Soft constraints, conditions that the state approximately satisfies, are utilized in most practical engineering applications [
11,
33] because uncertainty may appear during the transformation of external knowledge into the constraint. For example, in the case of ground target tracking constrained to a road, the roadmap may be inaccurate. Among promising methods dealing with soft constraints, some regard the degree of constraint satisfaction as measurement and extend the likelihood function [
18,
19,
31,
32,
35]. Especially, this approach can be intuitively extended to a nonlinear soft constraint; scPF (soft-constrained Particle Filter) [
35] is a good example. scPF has the advantage of preserving the nonlinearity of the constraint because it is based on an SIR (Sequential Importance Resampling) particle filter. However, scPF is not sample-efficient because the constraint is reflected by the generalized likelihood. More specifically, while particles are propagated through the system dynamics, they can be scattered in a direction that does not satisfy the constraint. Therefore, the propagated particles that do not satisfy the constraint would be given a low likelihood and eventually vanish, which makes the whole algorithm inefficient.
Thus, in this paper, we propose a particle filter that considers the stochastic terrain constraint. The term ‘terrain constraint’ not only represents the assumption that the position of a ground target should be located on the terrain surface but also that the velocity vector of the target should be tangent to the terrain surface. Contributions are the following:
We propose a sample-efficient particle filter to which the terrain constraint can be applied. The proposed algorithm is named Soft Terrain Constrained Particle Filter (STC-PF). Given the assumption of target motion, STC-PF performs sampling in a direction for which the state satisfies the constraint during the propagation step. As a result, STC-PF is more sample-efficient than scPF. Furthermore, in the numerical simulations, STC-PF using soft terrain constraint outperforms Smoothly Constrained Kalman Filter (SCKF) [
30] using hard constraint in terms of tracking performance.
Using a Gaussian process, terrain constraint is formulated as a soft position constraint along with a soft velocity constraint. Because kinematics states that position and velocity is not independent, a constraint on the position of a target implies that the velocity of the target will be constrained as well. Therefore, terrain constraint includes both position constraint and velocity constraint. Furthermore, terrain constraint requires exact terrain elevation and its gradient at an arbitrary position, but DTED (Digital Terrain Elevation Data) [
36] cannot provide them. To overcome this issue, we model the ground-truth terrain elevation with a Gaussian process (GP) and treat DTED as a noisy observation [
37] of it.
Technically, we used SRTM (Shuttle Radar Topography Mission). However, we will use the term DTED and SRTM interchangeably as they both are data that map terrain elevation of the entire globe.
The structure of this paper is as follows: In
Section 2, tracking of a ground target with a terrain constraint is formulated.
Section 3 presents the proposed algorithm, STC-PF.
Section 4 provides detailed explanations, the results, and a discussion of the numerical simulation. Finally, in
Section 5, we conclude.
2. Problem Formulation
In this section, tracking of a ground target with terrain constraint is formulated as a constrained state estimation problem.
Consider a system described by the following state-space model:
where
is the system state vector at time
k,
the measurement vector,
the system function,
the observation function,
the process noise vector, and
the measurement noise vector. The system state vector
consists of the position (
,
,
) and the velocity (
,
,
) in local Cartesian coordinates at time
k. The system function is a possibly nonlinear function but is assumed to be a constant velocity model in this paper.
is the measurement, which consists of range, azimuth angle, and elevation angle measured from the radar.
is white Gaussian process noise, and
is white Gaussian measurement noise. Subsequently, Equations (
1) and (
2) are realized as follows:
The final goal of the state estimation problem is to infer the state sequence of the dynamical system from the series of observations .
Now, the terrain constraint can come into play to incorporate the additional information that the state-space model cannot reflect. The terrain constraint not only represents the assumption that the position of a ground target should be located on the terrain surface but also that the velocity vector of the target should be tangent to the terrain surface. Both assumptions can be transformed into state constraints as follows:
where
,
, and
are the latitude, longitude, and altitude (LLA) of the target at time
k.
is ground-truth terrain elevation at latitude
and longitude
. Note that we do not have direct access to
, but only noisy observations,
such that
3. Soft Terrain Constrained Particle Filter
In this section, the newly proposed algorithm, Soft Terrain Constrained Particle Filter (STC-PF) is derived. In
Section 3.1, mathematical modeling of ground-truth terrain elevation is presented. Then, we propose a technique for the transformation of velocity between the LLA coordinates and the local Cartesian coordinates in
Section 3.2. Necessary assumptions required for algorithm derivation are described in
Section 3.3. After the algorithm derivation in
Section 3.4, we show the similarity between STC-PF and scPF [
35] in
Section 3.5.
3.1. Modeling of Ground-Truth Terrain Elevation
Although the terrain constraint (Equation (
5)) requires the ground-truth elevation, it is almost impossible in practice to retrieve it at an arbitrary position. The reason is that DTED provides neither accurate ground-truth terrain elevation (Equation (
7)) nor terrain elevation at arbitrary positions. (Equation (
6)) This challenge can be met by reconstructing the ground-truth terrain elevation with a Gaussian process (GP) and treating the DTED as independent observations:
where the observation noise
can be estimated from the work of Rodriguez [
37]. (see
Appendix B) Because GP assigns a probability for each possible terrain, the terrain constraint becomes stochastic. An advantage of this approach is that it enables us to compute the gradient of
analytically, which is required to apply the velocity constraint. (Equation (
5)) More strictly, joint predictive distribution for ground-truth terrain elevation and its gradient can be expressed in a closed-form, (detailed description is in
Appendix A)
provided that the kernel function is differentiable.
Figure 1 shows an example of prediction results when zero mean function and squared exponential kernel are utilized.
3.2. Velocity Transformation
Another major challenge when applying the terrain constraint to the filter is that the conversion of velocity between the local Cartesian coordinates and the LLA coordinates is not straightforward. More specifically, the terrain constraint (Equation (
5)) requires the velocity in LLA coordinates.
This challenge can be met by multiplying the Jacobian, which is obtained by numerical differentiation. Additionally, because velocity in local Cartesian coordinates is relative while that in LLA coordinates is absolute, the velocity of the radar
should be added (or subtracted) after (or before) multiplying by the Jacobian.
Conversion from LLA to local Cartesian can be done in a converse way.
where
.
3.3. Assumptions
Regarding the motion of the target, we assume the followings:
The vertical position (h) can be determined provided that the horizontal position (, ) is fixed.
Then, the vertical velocity () can be also determined when the horizontal velocity (, ) is fixed.
In
Figure 2, assumptions 1 and 2 correspond to the red arrows that inbound to
h and
, respectively. They comprise the ’elevation model’.
Due to the recursive Markovian structure, it is possible to infer the current latent state from the previously inferred latent state and the current measurement. Mathematically, by Bayes’ rule, the joint distribution of
given
can be expressed as
The dynamic model
and the likelihood model
are found in the above equation. Respectively, they correspond to the blue arrows and the green arrows in
Figure 2. Note that the measurement
is only affected by the position of the target (
,
,
), as stated in Equation (
4).
3.4. Algorithm
The proposed algorithm is based on the SIR(Sequential Importance Resampling) particle filter. In the SIR algorithm, which forms the basis of most sequential Monte Carlo (MC) filters [
38], the posterior probability density function
is characterized by the set of support points
(or particles) and the corresponding weights
, where
is the number of particles [
39]. The posterior density at time
k is approximated as
such that
where
represents the Dirac delta function. We assume that the particles are sampled from a well-known proposal distribution,
Then, by the principle of importance sampling, the corresponding weight is calculated as
Because we have freedom to choose the proposal distribution, we consider a proposal distribution that has a form of
In other words, one can draw new support points by augmenting each of the previous support points with the new state .
Starting from Equation (
17),
Together with Equation (
18) and the recursive relation (Equation (
13)), the weight update equation can be simplified.
A further assumption regarding the proposal distribution,
yields
This means that the weight of each particle is updated proportionally to its corresponding likelihood. Note that the above weight update equation implicitly includes the normalization given by Equation (
15).
The proposed algorithm, STC-PF, is summarized in Algorithm 1. In a vanilla SIR PF, the next state is propagated through the dynamic model only. In contrast, in STC-PF, the next state is propagated through the dynamic model first and then propagated through the elevation model (line numbers 1 and 1 in Algorithm 1).
In
Figure 3, a detailed implementation of the elevation model propagation is shown. It is worth mentioning that elevation model propagation can be accelerated by two techniques: parallelization and use of local data during the GP inference. Because the propagation process for each particle does not require information on other particles, it can be parallelized. Furthermore, during the GP inference, only the neighborhood data of DTED are utilized. The range of the neighborhood is defined by the spatial window size
L.
Algorithm 1: Soft Terrain Constrained Particle Filter (STC-PF). |
|
3.5. Remark on an Existing Work
As mentioned in
Section 1, from a mathematical perspective, the proposed algorithm (STC-PF) is similar to scPF (soft-constrained Particle Filter) [
35]. Similar to STC-PF, scPF is based on the SIR particle filter; however, the two differ in the sense that scPF utilizes generalized likelihood.
where
is a pseudo-measurement that represents how much the given state
satisfies the constraint. If Equation (
21) is replaced by
then the weight update rule is also changed.
Thus, the generalized likelihood function can be identified by equating the elevation model with the pseudo-measurement. As a result, scPF can be reduced to STC-PF as long as the assumption for target motion holds.
5. Conclusions
To sum up, we have proposed a particle filter to improve the performance of ground target tracking. To estimate the velocity more accurately, not only a position constraint but also a velocity constraint has been introduced in the terrain constraint. Although DTED provides terrain elevation of the entire globe, it provides inaccurate values at discrete positions. Thus, the ground-truth terrain elevation included in the terrain constraint has been modeled with a Gaussian process, and DTED has been regarded as noisy observations of it. As a result, terrain constraint has become a soft constraint that can reflect the uncertainty of DTED. Finally, we have proposed a particle filter, STC-PF, given the assumption of the motion of the target. STC-PF is based on SIR PF, but the major difference is that STC-PF uses the elevation model. Due to the elevation model, knowledge of the horizontal position and velocity of a target enables us to infer the vertical position and velocity more precisely. In the numerical simulation, STC-PF has been compared with SCKF which can incorporate hard constraints only. Furthermore, to reflect the uncertainty in DTED, filters have made use of DTED contaminated by noise, whereas the ground-truth trajectory of the target is generated by the original DTED. The simulation results showed that STC-PF outperforms SCKF in terms of RMS error, for two possible reasons. The first is that particle filters are more expressive than simple Gaussian filters. The second is that the state estimation with soft constraint is less sensitive to uncertainty of the constraint than that with hard constraint.