PARTICLE FILTERS FOR CONTINUOUS-TIME JUMP MODELS IN TRACKING APPLICATIONS

In this article we summarise recent work in modelling and estimation of continuous-time jump models for application in tracking scenarios. The models are constructed such that random jumps occur in the driving function (typically the applied force on an object being tracked) at random times, and in general asynchronously with the observation times. The sojourn times between jumps are modelled as general distributions (here gamma or shifted gamma), and hence we are in the class of semi-Markov models, since the arrival times do not form a Poisson point process (see [21] for an overview of such models in the tracking setting). In contrast with other such models in the tracking literature we allow a fully continuous random set of manoeuvre parameters, rather than a discrete set of switching models, and deterministic paths during the sojourns, obeying a set of nonlinear kinematic equations for point mass motion, thus modelling the path of the object in a smooth and parsimonious fashion. These models are aimed at capturing the highly random manoeuvres of real objects in a simple way. Estimation is carried out using a Variable Rate Particle Filter (VRPF) that parameterises the model explicitly in terms of the jump times and their parameters [12, 13]. Extensions to the models and algorithms are also presented that allow for a diffusion component of the model, which captures continuous random disturbances to the object in addition to jumps. These are illustrated in a 3-dimensional linear Gaussian setting where the entire path, except for jump times, may be marginalised, hence making a more efficient and effective particle filter. Introduction In this paper we give a brief overview of recent work in tracking of highly manoeuvrable objects by the use of nonlinear jump processes and particle filters. The models developed are expressed in a natural intrinsic coordinate system which, it is postulated, represent realistic conditions more accurately (see [21] for evidence to back up this claim). New models and algorithms are presented for linear jump processes that also contain a Brownian driving component, and a fully Rao-Blackwellised particle filtering implementation is demonstrated for this setup. Initial proof-of-principle results are given for simulated trajectories. ∗ The work of the author is partially funded by the EPSRC project Bayesian Inference for Continuous Time Diffusions, and by the UK’s Data and Information Fusion Defence Technology Centre 1 Signal Processing and Communications Group University of Cambridge c © EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.edpsciences. org/proc or http://dx.doi. org/10.1051/proc:071907 40 ESAIM: PROCEEDINGS 1. Models General Nonlinear Case 1.1. Dynamical models The class of models we consider initially comprises continuous-time trajectories having piecewise deterministic sections. The type of models proposed fits closely with the general class of piecewise deterministic processes, see [5], although we do not need to restrict to the Markovian case. In our models the kth section, k = 0, 1, . , is parameterised by a time τk at which the section commences (τk > τk−1 > .), and the parameters of the manoeuvre θk. A Markovian update on the k scale is defined for the combined kth state xk = [θk, τk]: xk ∼ f(xk|xk−1) = fθ(θk|θk−1, τk, τk−1)fτ (τk|θk−1, τk−1). (1) The distribution fτ (τk|θk−1, τk−1) is quite general, subject to the constraint τk > τk−1, although we do constrain ourselves to finite activity processes, i.e. those having a finite number of manoeuvre points τk in any finite time interval. We have typically used gamma, or shifted gamma, distributions for the inter-arrival times. Note that the parameters of the time distributions can explicitly depend on the manoeuvre parameters θk, and could themselves be made non-Markovian if required. Having defined a skeleton for the process in this way in terms of a countable collection of states {xk}k=0, the continuous-time path of the process is defined as a deterministic function of these states. In general for models of this type the path could be a function of several neighbouring state variables, see [12, 13]. Here however we restrict ourselves to causal constructions in which the path at any time t ≥ τ0 depends only upon the two closest state points, Nt = {k, k − 1; τk−1 < t ≤ τk}. Note that the underlying model here is a type of Marked point process , and hence the inference schemes proposed are for filtering of marked point process dynamical models, see [5, 6, 19] for example. 1.1.1. Example: Intrinsic Coordinates Motion Model [13] To give a specific example, we construct the state variable θk for an object undergoing planar motion, subject to applied forces TT,k (tangentially to motion) and TP,k (perpendicular to the motion), as θk = [TT,k, TP,k, v(τk), ψ(τk), z(τk)] where v(τk) is the speed at time τk, ψ(τk) is the heading angle relative to a fixed coordinate axis, and z(τk) is the Cartesian position of the object in the fixed Cartesian coordinate frame. This model is distinct from others in the area in that forces are applied relative to the body frame of the object, which is regarded as a more realistic model for motion of objects under self-forcing [21]. These models are however avoided in standard settings owing to the high degree of nonlinearity involved. In the particle filter implementation this is not an issue and we may go for greater realism. The kinematic equations for the object between times τk and τk+1 are then obtained from standard considerations as TT,k = λ ds dt +m ds dt2 , TP,k = m ds dt dψ dt , τk ≤ t < τk+1 where s(t) is the total distance moved along the path from time 0. The parameters m and λ denote the mass of the object and the linear coefficient of resistance, respectively. Note that a linear resistance model is unrealistic for many scenarios, but has been found to be a reasonable approximation for the trajectories that we have so far encountered. Current research is investigating extensions that model a more realistic square law effect in the resistance, for which closed form solutions to the differential equation also exist. The model as posed above falls within the general class of curvilinear models [21], which are some of the most powerful generic models of target motion currently available. 1By ‘generic’ we mean that the kinematics are not obtained from a direct modelling of the dynamics of a particular aircraft or boat, for example. ESAIM: PROCEEDINGS 41 In order to obtain the interpolated value of the path z(t), τk−1 ≤ t < τk, the kinematic equations are solved as standard coupled differential equations. The tangential equation is readily integrated from time τk to τk + τ to give the speed v(t) along the path at time t = τk + ∆τ : v(τk + ∆τ) = 1 λ ( TT,k − (TT,k − λv(τk))e−∆τλ/m ) (2) and the distance moved along along the path: s(τk + ∆τ) = s(τk) + ∆τ λ TT,k + m λ2 (TT,k − λv(τk))(e−∆τλ/m − 1) (3) Then using the above expression for the speed v(t), the perpendicular equation is rearranged and integrated to give: dψ dt = TP,k mv(t) , τk ≤ tτk+1, v(t) = 0 ψ(τk + ∆τ) = ψ(τk) + ∫ τk+∆τ