# Jump Particle Filtering Framework for Joint Target Tracking and Intent
Recognition

Jiaming Liang, Bashar I. Ahmad and Simon Godsill, , J. Liang, B. I. Ahmad and
S. Godsill are with the Department of Engineering, University of Cambridge,
Cambridge, CB2 1PZ, UK, e-mail: {jl809, bia23, sjg30} @cam.ac.uk.B. I. Ahmad
was also with Land and Air Systems, Thales, UK.This work was supported by a
Defence and Security Accelerator DASA grant (DSTLX1000144447) and the SIGNetS
project grant (W911NF17S0003-US-UK) by the US Army and UK Ministry of Defence.

###### Abstract

This paper presents a Bayesian framework for inferring the posterior of the
extended state of a target, incorporating its underlying goal or intent, such
as any intermediate waypoints and/or final destination. The methodology is
thus for joint tracking and intent recognition. Several novel latent intent
models are proposed here within a virtual leader formulation. They capture the
influence of the target’s hidden goal on its instantaneous behaviour. In this
context, various motion models, including for highly maneuvering objects, are
also considered. The a priori unknown target intent (e.g. destination) can
dynamically change over time and take any value within the state space (e.g. a
location or spatial region). A sequential Monte Carlo (particle filtering)
approach is introduced for the simultaneous estimation of the target’s
(kinematic) state and its intent. Rao-Blackwellisation is employed to enhance
the statistical performance of the inference routine. Simulated data and real
radar measurements are used to demonstrate the efficacy of the proposed
techniques.

###### Index Terms:

Bayesian inference, particle filter, Kalman filter, intent prediction, radar,
drone surveillance

##  I Introduction

In recent years, there has been a growing interest in meta-level tracking for
achieving a higher-level understanding of complex scenes and its evolution
over time by inferring and leveraging latent meta information on the observed
targets, such as their intent and/or any social interactions within a group
[1, 2, 3, 4, 5, 6, 7]. These can drive an object(s) long-term behaviour and
future actions. The aim may be to circumvent conflict or identify
opportunities, automate decision making, detect anomalies and optimize
resources allocation (e.g. more effective operation of sensors and assets).
Meta-level tracking has numerous applications in surveillance, smart
navigation, intelligent vehicles, cybernetics, to name a few. In this paper,
we tackle the problem of revealing, as early as possible, the target intent
(namely destination and any trajectory waypoints) and accurately estimating
its state $\mathbf{x}_{t}$ (e.g. position, velocity, higher order kinematics,
etc.) from the available noisy sensory measurements. Conversely, a plethora of
well-established sensor-level tracking algorithms focus solely on determining
$\mathbf{x}_{t}$, the target’s kinematic state.

###  I-A Problem Statement

The objective here is to develop suitable stochastic models in conjunction
with sequential inference methods to estimate the target’s intent and
kinematic state such that one or more of the following can apply:

  * •

The intent (e.g. destination) can dynamically change over time and take any
value within the state space (e.g. a location or region within a geographical
area constituting a sensor field of coverage).

  * •

The object trajectory can contain a set of waypoints, each can be treated as
an intermediate (or quasi) destination. This is a common behaviour of
(semi-)autonomous systems such as drones [8] and even in manned aviation or
maritime navigation, for example an airliner or a vessel follows a prescribed
route with waypoints [9, 10, 11].

  * •

The target can be agile (e.g. a rotary wing small unmanned air system UAS or
drone) and capable of undertaking sharp sudden as well as frequent manoeuvres
such as swift turns or abrupt accelerations.

###  I-B Related Work

Exploiting known constraints on an object’s state (e.g. future values) is
widely recognised to enhance the quality of the target state estimation. This
can be attributed to the original work by Castanon et. al. [12]. Several
relevant modelling approaches have been introduced since, for example with
reciprocal processes [13, 14, 15, 16, 17], conditionally Markov (CM) processes
[17, 11], embedded stochastic syntactic processes [5], context-free grammars
[16, 1, 18], pseudo-measurements with traditional Markov models [19, 20, 21,
22], Ornstein–Uhlenbeck-type processes [10, 23, 20, 7], intrinsic coordinate
model [24] and more recently embedding the endpoint information into an
(unconstrained) motion dynamics [25]. Contrary to the majority of the
aforementioned destination-constrained models, inclusive of those specifically
devised for trajectories with waypoints [9, 11], the target intent in this
paper is unknown in advance. Instead, a Bayesian modelling framework is
presented to simultaneously track a target and infer its destination.

The proposed formulation here is motivated by the virtual leader [26, 27] and
bulk [28] models, which have proved effective in tracking groups of objects
[27]. The idea is to model the target intent (e.g. final destination and/or
waypoints) as a virtual leader (VL) that has its own dynamics. This serves the
purpose of connecting the object motion to its intent through carefully
designed stochastic equations. The influence of the target’s medium to long
term goal on its instantaneous behaviour is thus captured. This is
fundamentally distinct from the related work on intent prediction in [2, 20,
24, 29, 30, 31, 7], where the target destination belongs to a predefined
finite set of nominal endpoints at fixed locations/values; see [32, 33, 7] for
an overview. Owing to the VL modelling, the target intent here can take any
value within the state space and it can dynamically change over time,
independently from the sensory observations.

In this paper, we build on our preliminary work in [4] and [33], which only
discusses the rudimentary VL model of the target final destination in Section
II-A. For example, in [4] only linear Gaussian models are treated. Below, we
introduce various latent intent models, including for waypoints-driven
trajectories with a piecewise constant model with jumps. This can be combined
with jump diffusion motion models [34, 31] for fast maneuvering targets. The
overall proposed modelling here leads to a jump particle filter [35, 36] for
estimating the sought joint posterior of the target kinematic state and
intent.

Context-free grammars (CFG) and other natural language processing models, with
a discrete state space, are utilised in [16, 1, 18] for meta-level tracking,
namely to detect the target intent and any anomalous behaviour. For example,
CFGs stipulate that an object should pass through a finite number of specified
grid cells to reach an endpoint. A pseudo-measurement technique for estimating
the target state and destination is also described in [22]. It is based on a
linear equality constraint such that the object is assumed to follow a simple
path (e.g. a straight line) to its intended endpoint. On the other hand, the
method introduced below employs continuous state space models which permit
treating asynchronous sensory measurements and does not restrict the target
motion behaviour unlike [1, 18, 22].

There are several data-driven techniques, e.g. using deep neural networks or
Gaussian processes, for intent modelling and prediction [37, 38, 39, 40, 41].
They typically rely on previously observed data to learn the target motion
model and predict its future locations. This is often applied when the
considered tracking scenario is simple and/or object movements are confined to
within small easily observed areas (e.g. a communal space or road junction or
pedestrian crossing with limited set of exits or routes for pedestrians and
cars). Here, we propose a model-based approach within an object tracking
setting. It relies on appropriately designed stochastic models of the object
latent state and intent. As is common in object tracking, these models have a
few unknown parameters which are physically meaningful and can be set manually
or learnt from data (e.g. based on the maximum likelihood criterion).

###  I-C Contributions

The main contributions of this paper are:

  * •

Novel latent intent models based on the virtual-leader formulation such that
the target kinematic state and its unknown intent are intrinsically linked to
capture, in continuous-time, the influence of the latter on the former. In
particular, the continuous and piecewise-constant model represents slow and/or
sudden changes in the destination or waypoint. Fast manoeuvring objects can be
additionally treated within the same framework.

  * •

A Rao-Blackwellised variable rate particle filtering approach to sequentially
infer the joint posterior of the target latent kinematic state and intent from
noisy and potentially asynchronous sensory observations. It offers additional
flexibility in terms of permitting spontaneously examining the likelihood that
any specific spatial location or extended area is the destination of the
tracked object.

Results from synthetic data and real drone surveillance radar measurements are
shown to demonstrate the performance of the proposed techniques.

###  I-D Paper Outline

The remainder of the paper is organised as follows. Several models are
introduced in Section II and a particle-filtering-based inference algorithm is
presented in Section III. Results from synthetic and real data are shown in
IV. Conclusions are drawn in Section V.

##  II Latent Intent Models

In this section, we introduce a range of novel models, each of which represent
different target and intent dynamics. This is preceded by describing the
basic, baseline, model in [4] for completeness. Without loss of generality,
the unobserved target intent, denoted as $\mathbf{r}$, will be considered to
be the Cartesian coordinates of the intended endpoint of a target in the
remainder of the paper. Nonetheless, it should be emphasised that the latent
intent variable can be straightforwardly extended to model higher order
kinematics (e.g. velocity and acceleration).

###  II-A Baseline Model

Here, both the target motion and the destination are driven by Brownian
motions. Specifically, consider first a 1-dimensional motion in Cartesian
coordinate and denote the overall (i.e. extended) state vector at time $t$ as
$\mathbf{s}_{t}=[\mathbf{x}_{t},\mathbf{r}_{t}]^{T}$ with
$\mathbf{r}_{t}=r^{x}_{t}$ being the endpoint location along the $x$-axis and
$\mathbf{x}_{t}=[x_{t},\dot{x}_{t}]^{T}$ the target position and velocity. The
virtual-leader based latent intent/destination model can now be described by
the following stochastic differential equations (SDEs) [42],

$\displaystyle d\dot{x}_{t}$
$\displaystyle=\eta_{x}(r^{x}_{t}-x_{t})dt-\rho_{x}\dot{x}_{t}dt+\sigma_{x}dB_{t},$
(1) $\displaystyle dr^{x}_{t}$ $\displaystyle=\sigma_{r}dB_{t},$ (2)

with $\eta_{x}$, $\rho_{x}$ being the mean reversion constants and
$\sigma_{x}$, $\sigma_{r}$ the diffusion constants. $B_{t}$ is a 1-dimensional
standard Brownian motion. Equation (1) depicts a destination-reverting
behaviour of the target position, hence leading to a destination-driven target
dynamics. Its second component serves to prevent the velocity from being
excessively high.

The target intent is allowed to change dynamically with $\sigma_{r}\neq 0$ in
(2) or it can be fixed with $\sigma_{r}=0$ (i.e. in the sense of having the
minimum uncertainty about the destination). This model can be formalised in
the vector-matrix form

$\displaystyle d\mathbf{s}_{t}=A\mathbf{s}_{t}dt+LdB^{s}_{t},$ (3)

with

$\displaystyle A=\begin{bmatrix}0&1&0\\\ -\eta_{x}&-\rho_{x}&\eta_{x}\\\
0&0&0\end{bmatrix},L=\begin{bmatrix}0&0\\\ \sigma_{x}&0\\\
0&\sigma_{r}\end{bmatrix},B^{s}_{t}=[B_{t},B_{t}]^{T}.$ (4)

A system model in a $k$-dimensional Cartesian coordinate system can be
obtained readily by cascading several such 1-dimensional models. It can be
shown that the solution to the linear time-invariant SDE (3) is given by:

$\mathbf{s}_{n}=e^{A\tau}\mathbf{s}_{n-1}+\int^{\tau}_{0}e^{A(\tau-u)}LdB^{s}_{u},$
(5)

with $\tau=t_{n}-t_{n-1}$ being the time step between two successive time
instants and $e^{(\cdot)}$ is the matrix exponential operator.
Correspondingly, the state transition density is given by

$p(\mathbf{s}_{n}|\mathbf{s}_{n-1})=\mathcal{N}(\mathbf{s}_{n}|F_{\tau}\mathbf{s}_{n-1},Q_{\tau}),$
(6)

where

$\displaystyle F_{\tau}$ $\displaystyle=e^{A\tau},~{}~{}~{}Q_{\tau}$
$\displaystyle=\int^{\tau}_{0}e^{A(\tau-u)}LL^{T}{e^{A(\tau-u)}}^{T}du,$ (7)

and $\mathcal{N}(\cdot|\cdot)$ represents the normal distribution. The state
process covariance $Q_{\tau}$ can be obtained in a closed form or via using
the matrix fraction decomposition [43] to simplify the calculation (especially
in high-dimensional cases). Note that higher-order kinematics may be augmented
to the latent intent variable, e.g. velocity at endpoint $\dot{r}^{x}_{t}$ and
$\mathbf{r}_{t}=[r^{x}_{t},\dot{r}^{x}_{t}]^{T}$. In this case, it resembles
the group virtual leader model in [27], albeit with the number of objects
equal to one.

This rudimentary intent-driven model of linear and Gaussian structure, for
example see [4] and [33], serves here as a baseline method for performance
benchmarking in Section IV.

###  II-B Piecewise-constant Destination Model

The above basic model implicitly stipulates that the target destination
evolves over time in synchronisation with the sensor(s) measurements. This
limits its applicability since in many practical situations the intent is
expected to change at a significantly slower rate compared with the sampling
frequency of the sensor measurements. For instance, a (semi-)autonomous
system, such as a drone, following a way-point trajectory set during the
mission planning, will typically have one or more “intermediate“ destinations
(i.e. waypoints) between which the intent remains unchanged. This is despite
potentially many sensor observations (e.g. from radar or optical sensors) made
during this period. Hence, we propose the following continuous-time model that
can better characterise a target motion with a priori unknown waypoints or
sojourns

$\displaystyle d\dot{x}_{t}$
$\displaystyle=\eta_{x}(r^{x}_{k}-x_{t})dt-\rho_{x}\dot{x}_{t}dt+\sigma_{x}dB_{t},$
(8)

where $t\in(\tau_{k},\tau_{k+1}]$. The latent destination
$\mathbf{r}_{k}=r_{k}^{x}$ is assumed to remain unchanged within the time
interval $(\tau_{k},\tau_{k+1}]$ and it has a distinctive subscript (i.e. $k$)
compared with that in (1). Consequently, the target position is driven
stochastically towards some deterministic destination $r^{x}_{k}$ before the
arrival of an updated destination $r^{x}_{k+1}$ right after $\tau_{k+1}$.

Consequently, the latent destination follows a continuous-time piecewise
deterministic process (PDP) [44] and its value only changes randomly at a
collection of countable jumping times $\\{\tau_{k}\\}$ . In principle, the
intend can be drawn from any suitably chosen density. Here, we model it as a
jump process $W_{t}$ with independent and normally distributed jump sizes
{$J_{k}$} since this can lead to a conditionally linear and Gaussian structure
for the dynamical model. We show in Section III that this particular structure
facilitates the use of the Rao-Blackwellised estimation scheme that can
improve the efficiency of the standard particle filter when the measurement
model is linear-Gaussian. Specifically, we have

$\displaystyle dr^{x}_{t}=dW_{t},$ (9)

where

$\displaystyle dW_{t}$ $\displaystyle=\begin{cases}J_{k},&t=\tau_{k}\\\
0,&\text{elsewhere}\end{cases}$ $\displaystyle J_{k}$
$\displaystyle\sim\mathcal{N}(\mu_{J},\sigma_{J}^{2}).$

The SDE for the 1-dimensional state vector
$\mathbf{s}_{t}=[x_{t},\dot{x}_{t},r_{t}^{x}]^{T}$ can then be written as

$d\mathbf{s}_{t}=A\mathbf{s}_{t}dt+\mathbf{h}_{J}dW_{t}+\mathbf{h}_{B}dB_{t},$
(10)

with

$\displaystyle\mathbf{h}_{J}=\begin{bmatrix}0&0&1\end{bmatrix}^{T},\mathbf{h}_{B}=\begin{bmatrix}0&\sigma_{x}&0\end{bmatrix}^{T}.$

The resulting model can be seen as a variable rate model [35, 36] since the
changepoint pairs $\\{J_{k},\tau_{k}\\}$ are decoupled from the observation
process. For more details regarding variable rate models, see [35, 45, 36]. If
the random jump times $\\{\tau_{k}\\}_{k:\tau_{k}\in[t_{n-1},t_{n})}$ are
assumed known, the following conditionally linear-Gaussian state transition
density can be obtained by solving (10) as in [46],

$\displaystyle
p(\mathbf{s}_{n}|\mathbf{s}_{n-1},\\{\tau_{k}\\}_{k:\tau_{k}\in(t_{n-1},t_{n}]})=\mathcal{N}(\mathbf{\mu}_{n:n-1},\Sigma_{n:n-1}),$
(11)

where

$\displaystyle\mathbf{\mu}_{n:n-1}=e^{A(t_{n}-t_{n-1})}\textbf{s}_{n-1}+\sum_{k:\tau_{k}\in[t_{n-1},t_{n})}\mu_{J}e^{A(t_{n}-\tau_{k})}\mathbf{h}_{J},$
$\displaystyle\Sigma_{n:n-1}=\sigma_{J}^{2}\sum_{k:\tau_{k}\in[t_{n-1},t_{n})}e^{A(t_{n}-\tau_{k})}\mathbf{h}_{J}\mathbf{h}_{J}^{T}{e^{A(t_{n}-\tau_{k})}}^{T}$
$\displaystyle\;\;+\int^{t_{n}-t_{n-1}}_{0}e^{A(t_{n}-t_{n-1}-u)}\mathbf{h}_{B}\mathbf{h}_{B}^{T}{e^{A(t_{n}-t_{n-1}-u)}}^{T}du.$

The inter-arrival times between jumps are assumed to follow an appropriately
chosen distribution. An example we utilise in this paper is the Gamma
distribution as per

$\tau_{k}-\tau_{k-1}\sim\mathrm{Gam}(\alpha_{\tau},\beta_{\tau}),$ (12)

where $\alpha_{\tau}$, $\beta_{\tau}$ are the shape and scale parameters
respectively. Nonetheless, any other suitable distributions can be adopted,
e.g. the exponential distribution in [34].

This model can maintain a consistent intent for a period of time determined by
two consecutive jump times. Beyond way-point driven trajectories, it can be
applied with the velocity-reverting models in [10, 7] for tracking and anomaly
detection in maritime surveillance where prescribed navigation routes followed
by vessels can be represented by a set of piecewise constant typical
velocities. Additionally, we present in Appendix A the multi-hypothesis intent
model, which is based on this piecewise constant destination one. It is
specifically for examining a finite set of predefined $N_{\mathcal{D}}$
destinations. However, unlike prior related work, e.g. [2, 32, 20, 24, 30, 31,
32, 33], it explicitly incorporates the null-hypothesis, i.e. none of the
prescribed endpoints are sought by the target.

###  II-C Jump-diffusion Destination Model

Following from the previous piecewise constant formulation and as a
generalisation, the target latent destination can evolve according to the
following jump-diffusion model

$\displaystyle dr^{x}_{t}=dW_{t}+\sigma_{r}dB_{t},$ (13)

to accommodate both minor and major changes in the underlying target intent.
The resultant SDE for $\mathbf{s}_{t}$ is

$d\mathbf{s}_{t}=A\mathbf{s}_{t}dt+\mathbf{h}_{J}dW_{t}+LdB^{s}_{t},$ (14)

which follows from (10) with $\mathbf{h}_{B}dB_{t}$ replaced by $LdB^{s}_{t}$
as in (3). The piecewise constant model in (10) can thus be obtained from (13)
by setting $\sigma_{r}=0$.

The destination dynamics here can be viewed as a combination of the two
behaviours:

  1. a)

exploration: controlled by $W_{t}$, with which large step sizes can be taken
within the state space when there is a drastic change in intent, i.e. macro
intent-driven motion; and

  2. b)

exploitation: governed by $B_{t}$, with a relatively small diffusion constant
$\sigma_{r}$, and gives the model the ability to make smaller adjustments
(e.g. finer corrections) in the vicinity of the endpoint, i.e. micro intent-
driven motion.

This model is suitable for scenarios where intermediary waypoints and final
destinations are briskly chosen and refined during the target journey, in lieu
of being set in advance. For instance, a manually controlled UAV (e.g. with a
first-person view FPV camera) can suddenly identify and home in on a location
of interest during the flight.

###  II-D Fast Manoeuvring Target with Latent Destination

Here, we extend the modelling to represent fast manoeuvring behaviour that can
entail the target undertaking swift turns and/or accelerations, for instance
due to sudden change in intent, obstacle avoidance, and being subjected to
perturbations (e.g. induced by external environmental factors such as wind,
vibrations, turbulence, etc.). The object motion model along $x$ axis in this
case is described by SDE

$\displaystyle d\dot{x}_{t}$
$\displaystyle=\eta_{x}(r^{x}_{k}-x_{t})dt-\rho_{x}\dot{x}_{t}dt+\sigma_{x}dB_{t}+dW_{t},$
(15)

with the introduction of the jump component $dW_{t}$ into (8). For simplicity,
this is combined with the (baseline) Brownian motion-driven latent destination
model in (2). The SDE for the state vector
$\mathbf{s}_{t}=[x_{t},\dot{x}_{t},r^{x}_{t}]^{T}$ in this case is given by

$d\mathbf{s}_{t}=A\mathbf{s}_{t}dt+\mathbf{h}_{J}dW_{t}+LdB^{s}_{t},$ (16)

with $\mathbf{h}_{J}$, $A$ and $L$ defined in (4).

This model can be solved in exactly the same manner as the piecewise constant
and the jump-diffusion destination models because they share the same
parametric form. The distinction is that the destination variable will neither
jump nor remain piecewise deterministic as it is now driven only by Brownian
motion. Nonetheless, target motion and intent models each containing a
separate jump process, i.e. (13) and (15), with de-coupled jump times, can
also be considered within the presented modelling framework; however this is
outside the scope of this paper. Furthermore, it is worth noting that the
target dynamical model is not limited to the jump-diffusion model, any other
models suitable for fast manoeuvring target may be employed to in place of
that in (15)-(16), e.g. the stable Lévy state-space model as in [33].

##  III Inference Methodology

###  III-A System Model and Inference Task

In addition to the dynamical models, we have the observation model

$\mathbf{m}_{n}\sim p(\mathbf{m}_{n}|\mathbf{s}_{n}),$

where $p(\cdot)$ is some known distribution characterising the observation
process and the noisy measurements
$\mathbf{m}_{0:n}=\\{\mathbf{m}_{0},\mathbf{m}_{1},\ldots,\mathbf{m}_{n}\\}$
made at time instants $t_{0:n}=\\{t_{0},t_{1},\ldots,t_{n}\\}$; $t_{0:n}$ need
not be regularly spaced and asynchronously sampled data can be naturally
treated under the adopted continuous-time state-space modelling. For
simplicity, we assume below the linear Gaussian observation model

$\mathbf{m}_{n}\sim
p(\mathbf{m}_{n}|\mathbf{s}_{n})=\mathcal{N}(\mathbf{m}_{n}|H_{n}\mathbf{s}_{n},R_{n}),$
(17)

where $R_{n}$ is the noise covariance matrix encoding the sensory data
uncertainty. The mapping matrix $H_{n}$ extracts the entries of the overall
state vector $\textbf{s}_{n}$ (e.g. Cartesian coordinates of the object
location), except those associated with the unobservable latent intent
$\mathbf{r}_{n}$. Fixed matrices are assumed here, i.e. $H_{n}=H$ and
$R_{n}=R$. Nevertheless, the models in Section II can be used in conjunction
with any other measurement models, including non-linear and/or non-Gaussian
ones [47].

We recall that the objective of the tackled joint intent and kinematic state
estimation task is to sequentially estimate the posterior distribution
$p(\mathbf{s}_{n}|\mathbf{m}_{0:n})$ or
$p(\mathbf{s}_{n},\bm{\theta}_{t_{0}:t_{n}})$ at each time step $t_{n}$ from
all of the available data $\mathbf{m}_{0:n}$. Whilst the baseline model in
Section II-A with (17) implies that the standard linear Kalman filter is the
optimal estimator, the proposed variable rate intent-driven models in this
paper do not lead to an analytically tractable solution for inferring the
sought posterior distribution. This is due to the inclusion of a change-point
sequence. We hence present in this paper a jump particle filtering approach,
termed variable rate particle filter (VRPF) [35, 36]. For convenience, we use
$\bm{\theta}_{k}=\tau_{k}$ to denote the variable rate state with
$\bm{\theta}_{t_{0}:t_{n}}=\\{\bm{\theta}_{k}\\}_{k:t_{0}\leq\bm{\theta}_{k}<t_{n}}$
as the collection of variable rate state points between $t_{0}$ and $t_{n}$.
Hence we first specify the variable rate state priors before giving a detailed
description of the state inference algorithm.

###  III-B Variable Rate State Priors

From [48, 45, 36], the prior for $\bm{\theta}$ can be expressed as

$\displaystyle
p(\bm{\theta}_{t_{0}:t_{n}})=S(t_{n},\tau_{K_{n}})\prod_{k=1}^{K_{n}}p(\tau_{k}|\tau_{k-1}),$
(18)

where $S(t_{n},\tau_{k})=\int^{\infty}_{t_{n}}p(u|\tau_{k})du$ is the
conditional probability of no additional jump occurring before time instant
$t_{n}$, $K_{n}=\max\\{k:\tau_{k}<t_{n}\\}$ and by convention
$\tau_{0}=t_{0}=0$; refer to Appendix A for the corresponding definition of
$\mathbf{\theta}_{n}$ and $p(\bm{\theta}_{t_{0}:t_{n}})$ of the multi-
hypothesis model.

Similar to standard filtering methods where the inference is normally
performed at the observation times, we formulate the VRPF in terms of
$\mathbf{t}_{0:n}$ albeit the jumps time are not necessarily synchronised with
those of measurements. To this end, it is necessary to obtain the variable
rate state transition density from $t_{n-1}$ to $t_{n}$ using the joint
priors,

$\displaystyle
p(\bm{\theta}_{t_{n-1}:t_{n}}|\bm{\theta}_{t_{0}:t_{n-1}})=\frac{p(\bm{\theta}_{t_{0}:t_{n}})}{p(\bm{\theta}_{t_{0}:t_{n-1}})}$
$\displaystyle\qquad\qquad=\frac{S(t_{n},\tau_{K_{n}})}{S(t_{n-1},\tau_{K_{n-1}})}\prod_{k:t_{n-1}\leq\tau_{k}<t_{n}}p(\bm{\theta}_{k}|\bm{\theta}_{k-1}).$
(19)

A pragmatic way to propose state sequence $\bm{\theta}_{t_{n-1}:t_{n}}$ from
(III-B) is to draw $\bm{\theta}$’s until the most recently simulated jump time
falls beyond $t_{n}$ and then keep the samples falling within
$[t_{n-1},t_{n})$.

###  III-C Rao-Blackwellised Variable Rate Particle Filter

The target posterior distribution at $t_{n}$ can be factorised as

$\displaystyle
p(\mathbf{s}_{n},\bm{\theta}_{t_{0}:t_{n}}|\mathbf{m}_{0:n})=p(\mathbf{s}_{n}|\bm{\theta}_{t_{0}:t_{n}},\mathbf{m}_{0:n})p(\bm{\theta}_{t_{0}:t_{n}}|\mathbf{m}_{0:n}).$
(20)

From Section II the state transition densities are linear and Gaussian
conditional upon the jump time sequence (as well as the indicator sequences
$\\{c\\}$ for the multi-hypothesis model) and the measurement model is linear-
Gaussian. This implies that tractable filtering for the (extended) system
state, namely the estimation of the multivariate Gaussian
$p(\mathbf{s}_{n}|\bm{\theta}_{t_{0}:t_{n}},\mathbf{m}_{0:n})$, can be
performed in an optimal manner via the Kalman filter. The variable rate
particle filter then needs only to operate on the marginal distribution
$p(\bm{\theta}_{t_{0}:t_{n}}|\mathbf{m}_{0:n})$. This inference procedure can
lead to a reduction in the variance of the estimates and it is commonly known
as Rao-Blackwellisation [49, 50]. The VRPF in this case is termed Rao-
Blackwellised VRPF (RBVRPF) [46, 51, 34], which is analogous to the standard
Rao-Blackwellised particle filter [52, 47, 53].

We now detail the particle filtering approach which estimates the posterior
distributions of the variable rate state sequence $\bm{\theta}_{t_{0}:t_{n}}$
and the linear state $\mathbf{s}_{n}$ at time instant $t_{n}$. Similar to the
standard Rao-Blackwellised particle filter, at $t_{n}$ the RBVRPF maintains a
collection of $N_{p}$ weighted particles

$\displaystyle\\{\omega^{(i)}_{n},\bm{\theta}_{t_{0}:t_{n}}^{(i)},\mu_{n|n}^{(i)},\Sigma_{n|n}^{(i)}\\}_{1\leq
i\leq N_{p}},$

where $\omega_{n}^{(i)}$ is the normalised particle weight associated with the
$i$-th particle and $\sum^{N_{p}}_{i=1}\omega_{n}^{(i)}=1$. Given the particle
set, the posterior density for $\theta_{t_{0}:t_{n}}$ can be approximated as

$\displaystyle{p}(\bm{\theta}_{t_{0}:t_{n}}|\mathbf{m}_{0:n})\approx\sum^{N_{p}}_{i=1}\omega^{(i)}_{n}\delta_{\bm{\theta}_{t_{0}:t_{n}}^{(i)}}(\bm{\theta}_{t_{0}:t_{n}}).$
(21)

Each particle also stores the mean and the covariance parameters of the
following Gaussian distribution

$\displaystyle
p(\mathbf{s}_{n}|\bm{\theta}_{t_{0}:t_{n}}^{(i)},\mathbf{m}_{0:n})=\mathcal{N}(\mathbf{s}_{n}|\mu_{n|n}^{(i)},\Sigma_{n|n}^{(i)}),$
(22)

which is computed by a Kalman filter running on the $i$-th particle.
Substituting (22) and (21) into (20) and marginalising over the variable rate
state sequence, the posterior density for the linear state is approximated by
a mixture of Gaussians

$\displaystyle p(\mathbf{s}_{n}|\mathbf{m}_{0:n})$ $\displaystyle=\int
p(\mathbf{s}_{n}|\bm{\theta}_{t_{0}:t_{n}},\mathbf{m}_{0:n})p(\bm{\theta}_{t_{0}:t_{n}}|\mathbf{m}_{0:n})d\bm{\theta}_{t_{0}:t_{n}}$
$\displaystyle\approx\sum^{N_{p}}_{i=1}\omega^{(i)}_{n}\mathcal{N}(\mathbf{s}_{n}|\mu_{n|n}^{(i)},\Sigma_{n|n}^{(i)}).$
(23)

Calculating (21) and (III-C) entails the following two operations.

####  III-C1 Compute particle weights

Assuming that a weighted set of particles has been obtained from the last time
instant $t_{n-1}$, the target posterior distribution of the particle filter at
$t_{n}$ is

$\displaystyle p(\bm{\theta}_{t_{0}:t_{n}}|\mathbf{m}_{0:n})$
$\displaystyle\propto
p(\mathbf{m}_{n}|\bm{\theta}_{t_{0}:t_{n}},\mathbf{m}_{0:n-1})p(\bm{\theta}_{t_{n-1}:t_{n}}|\bm{\theta}_{t_{0}:t_{n-1}})$
$\displaystyle\qquad\times p(\bm{\theta}_{t_{0}:t_{n-1}}|\mathbf{m}_{0:n-1}),$
(24)

with
${p}(\bm{\theta}_{t_{0}:t_{n-1}}|\mathbf{m}_{0:n-1})\approx\sum^{N_{p}}_{i=1}\omega^{(i)}_{n-1}\delta_{\bm{\theta}_{t_{0}:t_{n-1}}^{(i)}}(\bm{\theta}_{t_{0}:t_{n-1}})$.
Sampling from (III-C1) relies on the use of an importance distribution
$q(\bm{\theta}_{t_{0}:t_{n}}|\mathbf{m}_{0:n})$ that has the general form

$\displaystyle q(\bm{\theta}_{t_{0}:t_{n}}|\mathbf{m}_{0:n})$
$\displaystyle=q(\bm{\theta}_{t_{n-1}:t_{n}}|\bm{\theta}_{t_{0}:t_{n-1}},\mathbf{m}_{0:n})$
$\displaystyle\qquad\qquad\qquad\times
q(\bm{\theta}_{t_{0}:t_{n-1}}|\mathbf{m}_{0:n}).$ (25)

The past state trajectory $\bm{\theta}^{(i)}_{t_{0}:t_{n-1}}$ is drawn from a
discrete distribution as per

$\displaystyle
q(\bm{\theta}_{t_{0}:t_{n-1}}|\mathbf{m}_{0:n})=\sum^{N_{p}}_{i=1}v^{(i)}_{n-1}\delta_{\bm{\theta}_{t_{0}:t_{n-1}}^{(i)}}(\bm{\theta}_{t_{0}:t_{n-1}}),$

where the non-negative selection weights $\\{v^{(i)}_{n-1}\\}_{1\leq i\leq
N_{p}}$ sums to one and the state sequence $\bm{\theta}^{(i)}_{t_{n-1}:t_{n}}$
arriving between $t_{n-1}$ and $t_{n}$ is sampled from
$q(\bm{\theta}_{t_{n-1}:t_{n}}|\bm{\theta}_{t_{0}:t_{n-1}},\mathbf{m}_{0:n})$.
Denoting by $\\{\bm{\theta}_{t_{0}:t_{n}}^{(i)}\\}_{1\leq i\leq N_{p}}$ the
$N_{p}$ samples drawn from the joint proposal in (III-C1) at $t_{n}$, the VRPF
weights update is

$\displaystyle\widetilde{\omega}^{(i)}_{n}$
$\displaystyle=\frac{p(\bm{\theta}^{(i)}_{t_{0}:t_{n}}|\mathbf{m}_{0:n})}{q(\bm{\theta}^{(i)}_{t_{0}:t_{n}}|\mathbf{m}_{0:n})}$
$\displaystyle\propto\frac{\omega_{n-1}^{(i)}}{v_{n-1}^{(i)}}\times\frac{p(\mathbf{m}_{n}|\bm{\theta}^{(i)}_{t_{0}:t_{n}},\mathbf{m}_{0:n-1})p(\bm{\theta}^{(i)}_{t_{n-1}:t_{n}}|\bm{\theta}^{(i)}_{t_{0}:t_{n-1}})}{q(\bm{\theta}^{(i)}_{t_{n-1}:t_{n}}|\bm{\theta}^{(i)}_{t_{0}:t_{n-1}},\mathbf{m}_{0:n})},$

where $\widetilde{\omega}^{(i)}_{n}$ is the unnormalised particle weight for
the $i$-th particle. In this paper, we chose to use the prior distribution
(III-B) as the importance distribution for
$\\{\bm{\theta}_{t_{n-1}:t_{n}}^{(i)}\\}_{1\leq i\leq N_{p}}$, that is,
$q(\bm{\theta}_{t_{n-1}:t_{n}}|\bm{\theta}_{t_{0}:t_{n-1}},\mathbf{m}_{0:n})=p(\bm{\theta}_{t_{n-1}:t_{n}}|\bm{\theta}_{t_{0}:t_{n-1}})$.
This leads to a simplified weighting function

$\displaystyle\widetilde{\omega}^{(i)}_{n}\propto\frac{\omega_{n-1}^{(i)}}{v_{n-1}^{(i)}}\times
p(\mathbf{m}_{n}|\bm{\theta}^{(i)}_{t_{0}:t_{n}},\mathbf{m}_{0:n-1}).$ (26)

In order to mitigate the particle degeneracy problem, we apply a popular
adaptive selection (or re-sampling) strategy where the particles are re-
sampled according to the selection weights if the effective sample size (ESS)
is lower than a pre-determined threshold or the particle set remains unaltered
if the ESS is higher than the threshold. In the former case we have
$v^{(i)}_{n-1}=\omega^{(i)}_{n-1}$ and $v^{(i)}_{n-1}=1/N_{p}$ for the latter.
Although not examined here, it is possible to use the incoming and/or future
measurements to better guide the selection process. These more elaborated
“lookahead” strategies have been successfully implemented in the context of
particle filtering, see [54, 35].

To calculate the particle weight from (26), we need to know the predictive
likelihood term
$p(\mathbf{m}_{n}|\bm{\theta}^{(i)}_{t_{0}:t_{n}},\mathbf{m}_{0:n-1})$. Next,
we show that this quantity can be obtained as a byproduct when updating the
Kalman filter associated with the $i$-th particle.

####  III-C2 Kalman filtering

The Gaussian mixture approximation of the filtering distribution of the linear
component requires the mean and covariance for each Gaussian density
$p(\mathbf{s}_{n}|\bm{\theta}_{t_{0}:t_{n}}^{(i)},\mathbf{m}_{0:n})$ as in
(III-C). These two quantities can be computed using a bank of Kalman filters
[55]. Here we detail the calculation steps with a focus on the state
transition density of the piecewise constant model. The resultant is also
applicable to the jump-diffusion model and fast manoeuvring target models.
Similar operations can be readily applied to the multi-hypothesis model with
the transition density (35) (extended to the multiple jumps scenario).
Provided the posterior mean and covariance obtained from the last time step,
the predictive distribution of the $i$-th Kalman filter can be computed as per

$\displaystyle
p(\mathbf{s}_{n}|\bm{\theta}_{t_{0}:t_{n}}^{(i)},\mathbf{m}_{0:n-1})=\mathcal{N}(\mathbf{s}_{n}|\mu^{(i)}_{n|n-1},\Sigma^{(i)}_{n|n-1}),$
(27)

with

$\displaystyle\mu_{n|n-1}^{(i)}$ $\displaystyle=e^{A\delta
t}\mu_{n-1|n-1}^{(i)}+\sum_{k:\tau_{k}\in(t_{n-1},t_{n}]}\mu_{J}e^{A(t_{n}-\tau_{k})}\mathbf{h}_{J},$
$\displaystyle\Sigma_{n|n-1}^{(i)}$ $\displaystyle=e^{A\delta
t}\Sigma^{(i)}_{n-1|n-1}(e^{A\delta t})^{T}+\Sigma_{n:n-1}.$

Note $\delta t=t_{n}-t_{n-1}$ and $\Sigma_{n:n-1}$ is defined in (11). When
the new measurement $\mathbf{m}_{n}$ becomes available the posterior mean and
covariance can be updated via the Kalman correction step,

$\displaystyle\begin{split}\mu_{n|n}^{(i)}&=\mu_{n|n-1}^{(i)}+K_{n}(\mathbf{m}_{n}-H\mu_{n|n-1}^{(i)}),\\\
\Sigma_{n|n}^{(i)}&=(I-K_{n}H)\Sigma_{n|n-1}^{(i)},\\\
K_{n}&=\Sigma_{n|n-1}^{(i)}H^{T}(H\Sigma_{n|n-1}^{(i)}H^{T}+R)^{-1}.\end{split}$
(28)

The predictive likelihood term needed for the computation of the importance
weight can now be calculated using the predictive error decomposition [56]

$\displaystyle
p(\mathbf{m}_{n}|\bm{\theta}^{(i)}_{t_{0}:t_{n}},\mathbf{m}_{0:n-1})$
$\displaystyle=\int
p(\mathbf{m}_{n}|\mathbf{s}_{n})p(\mathbf{s}_{n}|\bm{\theta}_{t_{0}:t_{n}}^{(i)},\mathbf{m}_{0:n-1})d\mathbf{s}_{n}$
$\displaystyle=\mathcal{N}(\mathbf{m}_{n}|\mu_{\mathbf{m}_{n}}^{(i)},\Sigma_{\mathbf{m}_{n}}^{(i)}),$
(29)

with

$\displaystyle\mu_{\mathbf{m}_{n}}^{(i)}=H\mu^{(i)}_{n|n-1},\;\;\;\Sigma_{\mathbf{m}_{n}}^{(i)}=H\Sigma^{(i)}_{n|n-1}H^{T}+R.$

###  III-D Intentionality Estimation

We now illustrate how the target latent intent may be determined from the
inferred posterior using (III-C)-(III-C2) for all introduced models in this
paper, except the basic one in Section II-A. The intermediate (i.e. waypoint)
or final destination can be spatial point or extended regions, which can
chosen impromptu (e.g. by the non-cooperative surveillance system operator)
anytime and anywhere within the state space (e.g. in the analysed scene).

At time $t_{n}$ destination $\mathbf{r}_{n}$ is a sub-state of
$\mathbf{s}_{n}=[\mathbf{x}_{n},\mathbf{r}_{n}]^{T}$ and the intent marginal
posterior can be readily approximated as the following Gaussian mixture

$\displaystyle
p(\mathbf{r}_{n}|\mathbf{m}_{0:n})\approx\sum^{N_{p}}_{i=1}\omega^{(i)}_{n}\int\mathcal{N}(\mathbf{s}_{n}|\mu_{n|n}^{(i)},\Sigma_{n|n}^{(i)})d\mathbf{x}_{n}$
(30)

We can obtain an estimate of the probability of some destination located at
the spatial point $\mathbf{p}_{\mathrm{Des}}$ by evaluating
$\hat{p}(\mathbf{r}_{n}=\mathbf{p}_{\mathrm{Des}}|\mathbf{m}_{0:n})$. When
several positions are concurrently investigated, the relative or normalised
values of the produced probabilities can be compared to make a decision on the
most likely intent.

For an extended spatial region $\mathcal{A}$ of interest, we capitalise on the
fact that the introduced formulation provides the destination posterior
density $p(\mathbf{r}_{n}|\mathbf{m}_{0:n})$ and calculate the integral

$\mathrm{Pr}(\mathrm{Des}=\mathcal{A}|\mathbf{m}_{0:n})=\int_{\mathcal{A}}p(\mathbf{r}_{n}|\mathbf{m}_{0:n})d\mathbf{r}_{n},$
(31)

which can be easily evaluated for a Gaussian (or a mixture of Gaussians)
posterior density; regions of rectangular (or cuboid) shape can also simplify
the calculations (e.g. using the multivariate cumulative distribution
functions). For other distributions and geometric shapes, approximations can
be applied. This will be treated in more detail in future work and it is not
expected to lead to drastically different outcomes to approximating the area
of interest with $k$ simpler shapes (e.g. rectangles, ellipsoids, etc.) via
$\sum_{k}\int_{\mathcal{A}_{k}}p(\mathbf{r}_{n}|\mathbf{m}_{0:n})d\mathbf{r}_{n}$.
Similarly, a number of spatial areas can be routinely treated by computing
(31) for each one independently.

##  IV Results

The performance of the proposed jump particle filtering for the joint
estimation of a target’s state and intent is assessed in this section. First,
we use synthetically generated data to evaluate all models in Section II prior
to evaluating the performance of selected suitable ones on real radar
measurements of a small drone.

###  IV-A Simulated Data

The accuracy of the estimation of the target location and its
destination/waypoint position (i.e. mean root mean square error RMSE of
positional estimates across a trajectory) are listed in Table I for 50
simulated target tracks with a CV model (its dynamical noise has standard
deviation $\sigma_{d}=14$ and $d=x,y,z$) and Table II for 50 fast maneuvering
targets generated with the jump diffusion model ($\mu_{J}=0$ and
$\sigma_{J}=50$). Each of the trajectories has $N=3$ to $N=5$ waypoints
randomly placed within the field of coverage. A Gaussian noise with
$\sigma^{R}_{d}=15$m is added to the 3-D positional data to produce the
measurements (i.e. noisy target locations in Cartesian coordinates) for the
meta-level tracker. Results from the following methods are reported:

  * •

VL-D-KF: baseline model in Section II-A with a Kalman Filter (KF) as in [4].

  * •

VL-PC-RBVRPF for piecewise constant destination model in Section II-B with
Rao-Blackwellised variable rate particle filter.

  * •

VL-PC (known $\tau$) for model in Section II-B and KF since jump/switch times
are assumed known.

  * •

VL-PC (known $\tau$ and $\mathbf{r}$) for model in Section II-B and KF with
known jump/switch times and waypoints/destination.

  * •

VL-MultHyp-RBVRPF for the multiple hypothesis model in Appendix A and RBVRPF;
the $N_{\mathcal{D}}$ nominal destinations are assigned to the true trajectory
waypoints.

  * •

VL-JD-RBVRPF for the jump diffusion model in Section II-C and RBVRPF.

  * •

VL-FMT-RBVRPF for the fast maneuvering target model in Section II-D and
RBVRPF.

Table I shows that the basic VL-D-KF model achieves the poorest estimation
performance compared to the other proposed approaches. Without known
destination switch times, the multi-hypothesis model delivers the best
performance followed by the more general VL piece-wise constant model.
However, in numerous scenarios (e.g. surveillance of non-cooperative targets
such as drones) it is not possible in practice to have information on the
target possible waypoints or nominal destinations as with the multi-hypothesis
model. Given the underlying CV motion model of the simulated data, using more
sophisticated movement models aimed at targets undertaking fast maneuvers can
degrade the overall tracking performance (e.g. compared to models using CV) as
seen in Table I. Conversely, the benefits of such advanced modelling is
visible in the results in Table II where the fast maneuvering target model has
the best target position estimates, whereas the VL-PC has the most accurate
destination/waypoints inference results. This can be attributed to the ability
of the VL-PC to effectively capture dynamic changes in the object intent. On
the other hand, the VL-D-KF for Gaussian and linear settings, as expected,
depicts the least accurate estimates of both target and destination/waypoints
locations in such scenarios.

TABLE I: Position and destination estimation performance across 50 realisations of trajectories containing waypoints for (non-fast-manoeuvring) targets. Methods |  Position est. $\downarrow$ |  Destination est. $\downarrow$
---|---|---
VL-D-KF | 2.213 | 23.773
VL-JD-RBVRPF | 1.905 | 18.539
VL-PC-RBVRPF | 1.907 | 18.515
VL-FMT-RBVRPF | 2.259 | 23.320
VL-MultHyp-RBVRPF | 1.650 | 15.599
VL-PC (known $\tau$) | 1.728 | 16.031
VL-PC (known $\tau$ and $\mathbf{r}$) | 1.561 | -
TABLE II: Position and destination estimation performance across 50 realisations of trajectories containing waypoints for fast-manoeuvring targets. Methods |  Position est. $\downarrow$ |  Destination est. $\downarrow$
---|---|---
VL-D-KF | 2.720 | 28.747
VL-JD-RBVRPF | 2.649 | 21.730
VL-PC-RBVRPF | 2.649 | 21.719
VL-FMT-RBVRPF | 2.367 | 26.547

###  IV-B Real Radar Data for Intent and Threat Detection

We now employ radar observations from Thales Gamekeeper 16U staring radar
system. This sensor has been specifically designed for high performance
detection and automatic recognition of small drones, which can be slow and fly
at low altitudes, within a $7.5$km range in its current configuration. The
data was collected during the SESAR SAFIR live demonstrations near Port of
Antwerp, Belgium, see [57]. It contains the 3-D Cartesian coordinates111Due to
the sensitive commercial nature of the radar raw detections, they could not be
discussed in this paper. Instead, an experimental processing chain, including
clustering, nearest neighbour data association and basic extended Kalman
filter tracker, is applied and its output is used for evaluating the proposed
algorithms. of targets of interest within the radar field of coverage,
specifically: a) a DJI Inspire II quadcopter drone (diameter $\approx 0.5$m
and weight $\approx 3.5$ kg) undertaking a site surveying task within an
authorised flying zone $\mathcal{A}$ whilst following waypoints-driven
trajectories as in Figure 1; and b) unidentified opportune airborne objects
(e.g. microlights, etc.).

Here, the baseline VL-D-KF and the introduced VL-PC-RBVRPF (unknown $\tau$ and
$\mathbf{r}$) techniques are selected for testing where four different
performance aspects are examined, namely accuracy of target tracking with and
without intent estimation, ability to predict the target next
destination/waypoint, revealing malicious intent and early threat detection.
Models for fast maneuvering targets are not considered since the drone
waypoints-guided tracks are relatively smooth, see Figure 1.

Figure 1: Radar observations of a drone surveying an area with predefined
waypoints; truth trajectory from onboard GPS is shown.

####  IV-B1 Intent-aware tracking

The kinematic state estimation results are displayed in Figure 2 for the
measurements of Track 2 in Figure 2; the outcome from a standard Kalman filter
(i.e. without intent modelling) is also shown. It can be noticed that both VL-
D-KF and VL-PC-RBVRPF intent modelling lead to more certain estimates with
smaller ellipsoids compared to the standard KF, especially near the waypoints.
Their tracks are also closer to the ground-truth, albeit marginally. The mean
RMSE of the standard KF, VL-D-KF and VL-PC-RBVRPF for observations of four
trajectories in Figure 1 are $13.4$m, $12.9$m and $12.6$m, respectively. We
recall that the utilised real data has already been processed/filtered, which
limits the ability to achieve bigger improvements in the tracking performance.

Figure 2: Drone track estimation results (green crosses are radar
observations). The dotted solid lines are posterior mean filtering estimates
for VL-D-KF, VL-PC-RBVRPF and standard Kalman filter (i.e. no intent
modelling); posterior confidence ellipses are shown.

####  IV-B2 Waypoints learning

Figure 3 shows waypoints estimation results from VL-PC-RBVRPF and VL-D-KF on
the same drone trajectory in Figure 2. It depicts the inferred posterior for
the $r^{x}_{n}$ and $r^{y}_{n}$, denoting the latent destination X and Y
position, respectively. Both are members of the inferred extended system state
$\mathbf{s}_{n}$. The figure demonstrates the ability of the introduced VL-D-
KF and VL-PC-RBVRPF to predict the waypoints well in advance. The former
exhibits higher variability in predictions (i.e. more spread posteriors)
compared with VL-PC-RBVRPF which delivers higher response to change in
(intermediate) destinations and more certain estimates of the waypoint
location. Figure 4 displays the VL-PC-RBVRPF destination switching time
detection with more jumps visible right after the target aims for a new
waypoint (i.e. vertical dash red lines marking time instant when the drone
reached a waypoint), indicating the ability of the VL-PC-RBVRPF algorithm to
quickly identify new intents. It is noted that beyond improved tracking
results, early prediction of the target next waypoint can enable more
effective deployment of countermeasures and proportionate mitigation
strategies (e.g. against malicious drone activities).

(a) VL-D-KF

(b) VL-PC-RBVRPF

Figure 3: Waypoint estimates showing filtering posterior of the destination
location along the X and Y axes. Vertical red dashed lines indicate time
instants when the drone reached a waypoint. Red solid horizontal lines are
true waypoint positions in the corresponding axis. Figure 4: Destination
switching time $\tau$ with smoothing; vertical red dashed lines indicate time
instant drone reached a waypoint.

####  IV-B3 Drone to leave/enter an authorised/prohibited region $\mathcal{A}$

This uses observations of a track from Figure 1 as in [4] and the results are
depicted in Figure 5. They illustrate the ability of the piece-wise constant
destination model to respond quicker to change in intent compared with the
basic model. This is most visible when the sUAS re-enters $\mathcal{A}$
between waypoints 2 and 3. The VL-PC-RBVRPF predicted re-entry early (i.e.
after the target departed waypoint 2) and reacts to the UAS reaching waypoint
6 near the boarder of $\mathcal{A}$.

Figure 5: Threat detection with the drone exiting and re-entering a permitted
zone $\mathcal{A}$. Left: target trajectory, waypoints and probability
$\mathrm{Pr}(\mathrm{Des}=\mathcal{A}|\mathbf{m}_{0:n})$ as provided by VL-PC-
RBVRPF; $t_{0}$ is the flight start time. Right: probability of destination
being within the zone given by the two the proposed algorithms VL-D-KF and VL-
PC-RBVRPF; vertical dashed lines are time instant the drone reach each of the
waypoints.

####  IV-B4 Malicious intent to reach a restricted region $\mathcal{A}$

The aim is to reveal, as early as possible, if either of the two tracked
targets in Figure 6 intend to enter a protected spatial area; $\mathcal{A}$ is
(artificially) set near the radar. It can be seen both algorithms correctly
determined that object 2 was not posing a threat. Whereas the probability that
$\mathcal{A}$ is the destination Object 1 increased dramatically as the target
approaches and/or heads toward the protected area. The VL-PC-RBVRPF responded
quicker to a potential change of intent (including dip at time instant
$t_{n}\approx 35$sec compared to VL-D-KF where its
$\mathrm{Pr}(\mathrm{Des}=\mathcal{A}|\mathbf{m}_{0:n})$ exceeds 50% up to
20-30 seconds before Object 1 enters $\mathcal{A}$. This demonstrates the
ability of the proposed intent inference approach to provide tactical
advantage in terms of early detection of threats and provide threat
prioritisation capability (e.g. system operator not distracted by Object 2).

Figure 6: Threat detection example with results of estimating the probability
that the protected region is the intended destination of two targets of
interest. The trajectories color reflects the probabilities produced by the
VL-PC-RBVRPF method. Subplots show the calculated probabilities over-time for
VL-PC-RBVRPF and VL-D-KF.

Finally, the presented Bayesian approach provides the means to continuously
estimate the potentially dynamically changing intent (i.e. waypoints and final
destination) of targets and leverage this information to deliver improved
tracking results (i.e. state estimation). It seamlessly and automatically
incorporates salient behavioral attributes (such as distance to destination,
heading, velocity or higher order kinematics) indicative of the object’s
future actions as per the target goal-driven motion model as seen in Figures 5
and 6. On the other hand, there are numerous threat or intent assessment
schemes in the surveillance domain that rely on specific, hand-crafted,
features from the estimated target motion, such as the shape or frequency of
the followed trajectory, acceleration profile, proximity to an asset, heading
angle and other; see [58] for a recent overview. Whilst results from both
methodologies can be combined at the command and control (C2) system level,
performance comparisons require specialised data collection campaigns advised
by end-user operational requirements (e.g. for counter drone applications for
fixed-site protecting such as airports). This is outside the scope of this
work.

##  V Conclusion

In this paper, we propose and illustrate the potential of a novel Bayesian
meta-level tracker for the joint estimation of an object latent kinematic
state and destination/waypoints from noisy sensory observations. It: a) can
handle dynamically changing intents; b) is in principle agnostic to both the
sensing technology and target type; c) is based on a particle filtering
algorithm for sequential inference with minimal training data requirements
(similar to classical well-understood object trackers); and d) utilises
continuous-time models that naturally permit treating irregular and
asynchronous sensory measurements, including from multiple sensors. Future
work will extend the introduced framework to address multi-target scenarios,
such as of coordinated groups of objects, and the data association aspect.

##  Appendix A Multi-Hypothesis Latent Intent Model

For $N_{\mathcal{D}}$ predefined destinations (e.g. spatial points or
regions), we have $N_{\mathcal{D}}+1$ hypotheses inclusive of the null
hypothesis $\mathcal{H}_{0}$. We recall that $\mathcal{H}_{0}$ signifies that
the target intent is none of these nominal endpoints. This multi-hypothesis
intent model follow from that in Section II-B with an additional indicator
parameter $c_{k}$. Assume for simplicity that there is only one jump that
arrives at $\tau_{k}$ within the time interval $[t_{n-1},t_{n}]$;
$\tau^{+}_{k}$ and $\tau^{-}_{k}$ denote the timings right after and right
before the jump time $\tau_{k}$, respectively. The transition process of
$\mathbf{s}=[x,\dot{x},r^{x}]^{T}$ from $t_{n-1}$ to $t_{n}$ can be divided
into the three phases,

  1. 1.

From $t_{n-1}$ to $\tau_{k}^{-}$:

$p(\mathbf{s}(\tau_{k}^{-})|\mathbf{s}_{n-1})=\mathcal{N}(\mathbf{s}(\tau_{k}^{-})|F_{\tau_{k}-t_{n-1}}\mathbf{s}_{n-1},\widetilde{Q}_{\tau_{k}-t_{n-1}}),$
(32)

where $\widetilde{Q}_{\Delta t}=\int^{\Delta t}_{0}e^{A(\Delta
t-u)}\mathbf{h}_{B}\mathbf{h}_{B}^{T}{e^{A(\Delta t-u)}}^{T}du$ and
$F_{(\cdot)}$ is defined in (7).

  2. 2.

From $\tau_{k}^{-}$ to $\tau_{k}^{+}$:

$\displaystyle p(\mathbf{s}(\tau_{k}^{+})|\mathbf{s}(\tau_{k}^{-}))=$ (33)
$\displaystyle\qquad\qquad\mathcal{N}(\mathbf{s}(\tau_{k}^{+})|F^{\pm}(c^{+}_{k})\mathbf{s}(\tau_{k}^{-})+M^{\pm}(c^{+}_{k}),Q^{\pm}(c^{+}_{k})),$

where $c^{+}_{k}$ is the indicator value right after $\tau_{k}$. Note here
that $c^{+}_{k}=c_{k}$ and $c^{-}_{k}=c_{k-1}$.

  3. 3.

From $\tau_{k}^{+}$ to $t_{n}$:

$p(\mathbf{s}_{n}|\mathbf{s}(\tau_{k}^{+}))=\mathcal{N}(\mathbf{s}_{n}|F_{t_{n}-\tau_{k}}\mathbf{s}(\tau_{k}^{+}),\widetilde{Q}_{t_{n}-\tau_{k}}).$
(34)

In the first and the last phases, the transition and covariance matrices take
the same functional forms since during these two periods the latent
destination remains constant.

At $\tau_{k}$, there is an instant state transition taking place under the
guidance of a new indicator value $c^{+}_{k}$. The exact forms of the matrices
in the Gaussian density function in (33) depend on the type of the
corresponding hypothesis. Define $c_{k}\in\\{0,1,\ldots,N_{\mathcal{D}}\\}$
and then we have

$\displaystyle c_{k}=\begin{cases}j,&j\in\\{1,\ldots,N_{\mathcal{D}}\\}\\\
0,&\text{null hypothesis}\end{cases}$

In the case of the $j^{\textrm{th}}$ alternative hypothesis (i.e. the
$j^{\textrm{th}}$ predefined destination), the matrices are given by

$\displaystyle F^{\pm}(c^{+}_{k}=j)=\begin{bmatrix}I_{2\times 2}&0_{2\times
1}\\\ 0_{1\times 2}&0\end{bmatrix},M^{\pm}(c^{+}_{k}=j)=[0,0,p^{x}_{j}]^{T}$

and $Q^{\pm}(c^{+}_{k}=j)=\textrm{diag}([0,0,(\sigma^{x}_{j})^{2}])$;
$I_{a\times b}$ denotes a $a$-by-$b$ identity matrix and similarly $0_{a\times
b}$ is a $a$-by-$b$ zero matrix. Here, $p_{j}^{x}$ is the position of the
$j^{\textrm{th}}$ destination along $x$ axis while $\sigma_{j}^{x}$ describes
the extent-orientation of this endpoint (e.g. the shape and orientation of an
ellipse in a 2/3-D Cartesian coordinate system). For a spatial point location,
$\sigma_{j}^{x}=0$ and the Gaussian in (33) collapses to a Dirac delta
function. On the other hand, the matrices for the null hypothesis ($j=0$) are

$\displaystyle F^{\pm}(c_{k}^{+}=0)=I_{3\times
3},\;\;M^{\pm}(c_{k}^{+}=0)=\mu_{J}e^{t_{n}-\tau_{k}}\mathbf{h}_{J}$

and
$Q^{\pm}(c^{+}_{k}=0)=\sigma_{J}^{2}e^{A(t_{n}-\tau_{k})}\mathbf{h}_{J}\mathbf{h}_{J}^{T}{e^{A(t_{n}-\tau_{k})}}^{T}$.

By marginalising out $\mathbf{s}(\tau_{k}^{-})$ and $\mathbf{s}(\tau_{k}^{+})$
in (32) through (34), it can be shown that the target state transition density
conditioned on the indicator variable and the jump time is

$\displaystyle
p(\mathbf{s}_{n}|\mathbf{s}_{n-1},c_{k},\tau_{k})=\mathcal{N}(\mathbf{s}_{n}|\breve{\mu}_{n:n-1},\breve{\Sigma}_{n:n-1})$
(35)

with

$\displaystyle\breve{\mu}_{n:n-1}$
$\displaystyle=F_{t_{n}-\tau_{k}}\big{(}F^{\pm}(c^{+}_{k})F_{\tau_{k}-t_{n}}\mathbf{s}_{n-1}+M^{\pm}(c^{+}_{k})\big{)}$
$\displaystyle\breve{\Sigma}_{n:n-1}$
$\displaystyle=F_{t_{n}-\tau_{k}}\big{(}F^{\pm}(c^{+}_{k})\widetilde{Q}_{\tau_{k}-t_{n-1}}F^{\pm}(c^{+}_{k})^{T}$
$\displaystyle\qquad+Q^{\pm}(c^{+}_{k})\big{)}F_{t_{n}-\tau_{k}}^{T}+\widetilde{Q}_{t_{n}-\tau_{k}}$

While the density given above is with regard to a single jump occuring between
$t_{n-1}$ to $t_{n}$, it is straightforward to generalise the result to
multiple jumps. In particular, (35) will be equivalent to (11) if the null
hypothesis is in effect at all jump times within $(t_{n-1},t_{n}]$.

There are various possible choices for the prior distribution of the
hypothesis indicator $\\{c\\}$ and the switching times $\\{\tau\\}$. For
example, we adopt the following joint distribution

$\displaystyle
p(c_{k},\tau_{k}|c_{k-1},\tau_{k-1})=p(c_{k}|c_{k-1},\tau_{k})p(\tau_{k}|\tau_{k-1})$
(36)

where the times between jumps are Gamma distributed as in Section II-B and the
indicator variable evolves according to a fixed Markov transition probability
matrix. More sophisticated priors may be devised, e.g. with the switching
probabilities of the indicator dependant on the jump times
$\\{\tau_{k},\tau_{k-1}\\}$.

Let $\bm{\theta}_{k}=\\{\tau_{k},c_{k}\\}$ is for the intent switch time and
the indicator; their joint evolution is described by (36). Compared to (18),
the prior for $\bm{\theta}$ in the multi-hypothesis model is

$\displaystyle
p(\bm{\theta}_{t_{0}:t_{n}})=S(t_{n},\tau_{K_{n}})p(c_{0})\prod_{k=1}^{K_{n}}p(c_{k}|c_{k-1},\tau_{k})p(\tau_{k}|\tau_{k-1}),$

and the inference approach in Section III can be applied with this definition
of $\bm{\theta}_{k}$.

Finally, for the $j^{\text{th}}$ destination located at $p_{j}^{d}$ ($d=x,y,z$
for 3-D space), which can either a point or an extended region as per
$\sigma_{j}^{d}$, and given the available sensory observations
$\mathbf{m}_{0:n}$, e.g. from model in (17), the probability of each of the
$N_{\mathcal{D}}+1$ hypotheses can be obtained via

$\displaystyle\mathrm{Pr}(\mathcal{H}_{j}|\mathbf{m}_{0:n})$
$\displaystyle=p(c_{K_{n}}=j|\mathbf{m}_{0:n}),~{}~{}~{}j=0,1,...,N_{\mathcal{D}}.$
(37)

Since the multi-hypothesis model explicitly incorporates $\mathcal{H}_{0}$ and
its probability can be calculated taking into account the $N_{\mathcal{D}}$
destinations, a reasonable threshold on the intent probability can used to
admit one of the $N_{\mathcal{D}}+1$ hypotheses.

## Acknowledgment

This work was funded by the Defence Science and Technology Laboratory (DSTL)
via the Defence and Security Accelerator (DASA) C-UAS Phase 1 grant
DSTLX1000144447 and the US Army and UK Ministry of Defence via the SIGNetS
project (Cooperative Agreement Number W911NF-20-2-0225). The views and
conclusions contained in this paper are of the authors and should not be
interpreted as representing the official policies, either expressed or
implied, of the US Army Research Laboratory, the MOD, the US Government or the
UK Government.
Authors also thank Thales for providing the real radar measurements and
supporting this work.

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