Ground target modelling, tracking and prediction with road networks David Salmond
Martin Clark and Richard Vinter
Simon Godsill
Defence Science & Technology Laboratory Farnborough, Hants, U.K. Email:
[email protected]
Department of Electrical and Electronic Engineering Imperial College, London, U.K Email: {j.m.c.clark, r.vinter }@imperial.ac.uk
Department of Engineering University of Cambridge Cambridge, U.K Email:
[email protected]
Abstract— A model for vehicle motion on a road network is developed using an enumeration of feasible routes. Combined with a generic stochastic model of distance travelled, a predicted pdf of vehicle position is derived as a mixture. This approach allows prior information on vehicle intent and behaviour to be included via the mixture weights. Illustrative examples are given using a second-order linear-Gaussian model for vehicle road speed. The value of road map data is shown via a tracking example with poor quality measurements and a substantial period prior to sensor activation. The tracking algorithm is implemented using a standard particle filter. In particular, the scheme has potential for revealing the likely paths taken by the vehicle.
Keywords: Vehicle modelling, road networks, tracking, particle filters.
I. I NTRODUCTION A. Aims This paper is concerned with modelling the growth of uncertainty in the location of a ground vehicle, making use of road map information. More formally, given a vehicle state x0 , the requirement is to construct the probability density function (pdf) p(x(t)|x0 ) of the state x(t) after a time period t without observations. We are particularly interested in the case where the period t is long (of the order of minutes) compared with the usual interval between measurements for tracking problems. Over such relatively long periods, without exploiting information on geographical constraints or preferences, the predicted pdf would spread into a rather bland pool of uncertainty. However, at least in some situations, such as urban environments, it should be possible to infer a much more informative and structured pdf. In this development, we exploit a graphical representation of road networks and a continuous model of vehicle dynamics to efficiently generate a particle representation of the predicted vehicle pdf. The model is also applied to a tracking problem. B. Brief review of related work In recent years there has been a substantial amount of work on map-aided target tracking, particularly using ground moving target indication (GMTI) radar. Kirubarajan et al [1] proposed a variable structure interacting multiple model (VSIMM) filter for exploiting road network information. Each model in this IMM corresponds to a particular road segment, and the network connectivity in the vicinity of the target is used to construct a Markov transition matrix. Road constraint information is used to shape and rotate the dynamics process noise (for a near constant velocity model) so that across-road uncertainty is small compared with along-road uncertainty. This scheme also includes an off-road model - the target is allowed to leave the network if the adjoining terrain is suitable. In Chapter 10 of Ristic et al [2] and Arulampalam [3], a multiple model particle filter is developed along similar lines - i.e. with a Markov transition matrix governing movement between roads and transitions to and from off-road modes. A similar road constrained dynamics model is employed but a truncated Gaussian is used for the process noise to ensure that the target remains on the road. Ulmke and Koch [4] have used a graph theory representation for tracking road vehicles. In particular, they model target dynamics in road co-ordinates and use road-map information to transform the target state into ground co-ordinates. Our development employs this type of model (also see [5], [6]). C. Approach Consider a vehicle that starts at a known position and velocity on a road network and remains on the network. Also suppose that the vehicle may travel a maximum road distance `max in some time period t. Given information on the road connectivity, it is fairly straightforward to find all possible routes (i.e. sequences of road segments) of length less than `max that might be taken by the target. Suppose that there
are N such routes and denote the j th route by Rj (routes are defined precisely in Section II-A below). If a probability Pr{Rj |x0 } can be assigned to each route (from, perhaps, some prior information on target intent), then the pdf of the vehicle state after a period t may be written p(x(t)|x0 ) =
N X j=1
Pr{Rj |x0 }p(x(t)|x0 , Rj ) .
(1)
Here p(x(t)|x0 , Rj ) is the pdf of the vehicle state given the initial state x0 and given that it travels on route Rj . Also PN j=1 Pr{Rj } = 1 since the vehicle must take exactly one of the feasible routes. Note this approach of assigning a probability to a route differs from [1], [3] which treat each junction encountered as an independent decision process. If the road network is modelled as a set of specified continuous curves, for a given route Rj , the conditional pdf p(x(t)|x0 , Rj ) depends only on how far the vehicle has travelled along the route. In principle, one could model the progress of the vehicle by taking account of the detailed gradients, curvature, junctions etc of the route. However, this representation is too cumbersome for our purposes, and instead we propose to use a generic continuous stochastic model of vehicle speed and integrate it to obtain the distribution of (road) distance travelled. The speed model is determined by a few parameters, such as average speed, that depend on the nature of the roads and the type of vehicle. The following sections provide more details of this approach. In Section II, the representation of the road network is described together with the definition of possible vehicle routes. In Section III, a continuous time linear-Gaussian model for vehicle speed is proposed and integrated to give the pdf of road distance travelled. The fusing of network information with the vehicle speed model for target tracking is addressed on Section IV. Illustrative simulation examples are presented within these sections and concluding remarks are given in Section V. II. R EPRESENTATION OF ROAD
NETWORKS
A. Specification of road networks It is convenient to represent a road network as a directed graph [7]. Each junction and road end corresponds to a vertex of the graph and each connecting road is an edge or arc of the graph. An example of a road network in a typical urban area is shown in fig 1. Each vertex is plotted at its geographical position - in this example there are 86 vertices. Those corresponding to junctions are shown as dots, “dead-ends” are shown as asterisks and roads that would continue off the map are terminated in circles (open nodes) at the map boundary. So vehicles that remain on roads cannot pass a dead-end, but they can pass through an open node. The connecting roads are represented by the arcs of the directed graph, where the direction will be associated with the
Vertices and connecting roads (principal directions)
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direction of motion of the vehicle (assuming the vehicle does not change its direction between vertices). We first specify an arbitrary principal direction for each road, so defining one directed arc for each road. The principal directions chosen for the road network example are shown via pointers in fig 1. A second set of arcs is then added which are identical to the first set but with directions reversed. (If some of the roads were “one-way”, and the vehicles of interest could be trusted to obey the convention, then the arcs corresponding to the wrong direction should be deleted from the graph.) The connectivity of the network can be specified by an incidence matrix.
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For this development, each connecting road is modelled as a continuous curve between two points in two, or possibly three, dimensional (geographical) space (also see [4]). So let {ri (u) : 0 6 u 6 1} denote the path of the centre of the road corresponding to arc i. Here u parameterizes the distance along the road in the direction of the arc, where u is proportional to the physical distance along the road segment. Hence ri (0) is the position corresponding to the starting vertex of arc i and ri (1) is the location corresponding to the end vertex of arc i. Also denote the length of the road corresponding to arc i by `(i). So if a vehicle has travelled a distance u`(i) along the road segment (0 6 u 6 1), then its position is given by ri (u) up to an uncertainty in the width of the road. (If the vehicle could be expected to travel on a preferential side of the road, this could also be taken into account.) Furthermore, to a good approximation, the tangent of the curve at u defines the direction of the vehicle’s velocity vector. Thus, given knowledge of the road functions ri (.) for all i, the location and direction of motion of a vehicle on the road network is well approximated by (i, u). In practice, the continuous functions ri (.) could be obtained by fitting a spline curve to a few knot points - the curved roads in fig 1 are generated in this way. B. Possible vehicle routes Suppose that at some instant a vehicle is known to be at (i0 , u0 ) - i.e. on arc i0 travelling in the direction of that arc and with position parameter u0 . To explore possible future behaviour of the vehicle, it is useful to determine all possible routes the vehicle may take which encompass some maximum (road) distance travelled `max . Each possible route (or “walk”) is a sequence of connected arcs Rj = (ij 0 , ij 1 , ij 2 , . . . , ij Nj ) such that ij 0 = i0 , ij k → ij k+1 (j ∈ {0, . . . , Nj − 1}) is a valid transition and Nj −1
X
`(ij k ) < `max + u0 `(i0 )
k=0
i.e. the road distance to the penultimate vertex of the route is less than `max . If there is a valid transition beyond the final vertex of the route (i.e. if there is any arc that is incident from this vertex), it is also required that Nj −1
X
k=0
`(ij k ) < `max + u0 `(i0 ) 6
Nj X k=0
`(ij k ) .
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Junctions Open nodes "Dead−ends"
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Fig. 1. Example road network: the pointers indicate the directions of the principal arcs.
This ensures that the routes do not include any arcs that start at road distances of more than `max from the initial vehicle position (taking account of direction of motion). Note that if a route ends in a vertex from which there is no valid transition (a dead-end), the length of the walk may be less than `max . The set of all possible routes that meet the above conditions may be easily constructed as a “rooted tree”. The stem of the tree is the arc i0 . All arcs incident from the terminal vertex of this stem are used to form branches of the tree (using the incidence matrix of the network). The lengths from the start of the stem to the ends of the branches are recorded. This process is continued from the end of each branch until the route length exceeds `max or a dead-end is encountered. As an example, fig 2 shows all possible vehicle paths of length 0.75 km or less, starting from a position and direction in the upper right region of the network (fig 1). In this case, there are 70 possible routes, and the maximum number of road segments in any of these routes is 12. In constructing this set, we have imposed the extra restriction that the vehicle may not “double back” on itself at vertices and directly travel back along the incident edge (and, as already stated, it may not change direction between nodes). Due to the connectivity of roads in the vicinity of the vehicle, there are a number of the valid routes that include closed walks (loops) and for some of these the vehicle may traverse the same road in different directions. Also in several instances, the closed routes allow the vehicle to return to and pass its starting point. This set of routes also defines a subgraph of the full network which essentially describes (an upper bound on) the “reach” of the vehicle within a maximum road distance of `max . The subgraph for this example is shown in fig 3: it is the union of arcs
Subgraph of arcs from all possible vehicle routes of length 0.75 or less
All possible vehicle journeys of length 0.75 or less −0.1
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Fig. 2. All journeys of length `max = 0.75 or less from indicated initial vehicle position and direction (each route is slightly off-set to show the full set).
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Fig. 3. Subgraph formed by union of arcs from the 70 feasible walks from indicated initial vehicle position and direction for `max = 0.75.
A. Second order linear-Gaussian model A possible second order model for the target speed v is:
traversed by all possible vehicle paths starting from the initial vehicle position with length less than or equal to `max (the arc directions are indicated by pointers in fig 3 - note that in some cases both directions of a connecting road are included). The terminal vertex of each of these routes is shown by a red square (enclosing an asterisk if the vertex is a dead-end). If one has prior knowledge of the mission, intent or behaviour of the vehicle, it may be possible to assign some prior probabilities to the feasible routes within the horizon of `max . For example, it may be unlikely that the vehicle path includes a loop, or that its route terminates in a dead-end. Furthermore, a set of likely destinations for the vehicle may be known so that routes that are in some sense directed towards these destinations should be favoured. As another possibility, the vehicle may be expected to head for the nearest major road. There is considerable scope for including such considerations in the prior probabilities Pr{Rj |x0 }. III. M ODELLING VEHICLE
SPEED
As indicated in the Introduction, we propose to use a generic continuous stochastic model of vehicle speed v and integrate it to obtain the distribution of (road) distance travelled s [4]. So rather than attempting to model the detailed road conditions and vehicle response, we attempt to represent the gross characteristics of vehicle speed such as expected value and bandwidth. A straightforward and convenient approach is to use a continuous linear-Gaussian model. As is well known, this has a Gaussian transition density with mean and covariance that can be expressed in closed form.
v¨ + 2ζω v˙ + ω 2 (v − vR ) = cw
(2)
where w is a zero-mean white Gaussian noise process of unit intensity (power spectral density). This is the familiar second order response model (see, for example, [8] page 335) where ω is the natural frequency, ζ is known as the damping ratio and tc = 2ζ/ω is the time constant (it being assumed 1 > ζ > 0 and ω > 0). The constant c determines the magnitude of the driving noise. If v is the vehicle speed, then v˙ may be interpreted as along-road or tangential target acceleration. vR is the expected value of the vehicle speed in the steady state. The system may be augmented by s˙ = v, so that in terms of the general linear-Gaussian model x˙ = Ax + Bu + Cw we have s 0 1 0 0 0 1 ,B = 0 , x = v ,A = 0 v˙ 0 −ω 2 −2ζω ω2 0 u = vR and C = 0 . c
Given an initial state x0 = (s0 , v0 , v˙ 0 )T , the mean and covariance of the Gaussian transition density (p(x|x0 ) = ¯ , M )) are given by standard results [8], [9]. In parN (x; x ticular, the transition matrix for the model is (augmenting the result quoted by [8]): Φ(τ ) = eAτ =
1
(1/β)(1 − 2ζ 2 )e−ζωτ sin βτ +(2ζ/ω)(1 − e−ζωτ cos βτ )
0
e−ζωτ (cos βτ +
0
−ω2 β
ζω β
sin βτ )
e−ζωτ sin βτ
−(ζ/(ωβ))e−ζωτ sin βτ +(1/ω 2 )(1 − e−ζωτ cos βτ ) 1 −ζωτ βe
sin βτ
e−ζωτ (cos βτ −
ζω β
sin βτ )
p 1 − ζ 2 . So the mean of the state after a period s0 vR t ¯ = Φ(t) v0 − vR + vR x (3) v˙ 0 0
where β = ω t is
In particular, the steady state variance of the target speed is σv2 = c2 /(4ζω 3 ), which can be used to set c to obtain the required variation in the speed. Also the autocorrelation for the steady state speed is R(τ ) = σv2 e−ζωτ (cos βτ + ζω β sin βτ ).
c
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and as t becomes large, the mean tends to vR t + s0 + (2ζ/ω)(v0 − vR ) + v˙ 0 /ω 2 . (4) ¯→ vR x 0 Rt The covariance is given by M = τ =0 Φ(τ )CC T ΦT (τ )dτ (for unit intensity driving noise). The resulting expression for M is given in the Appendix, and for large t, this covariance tends to 2 2 2 1 ω t + 4ω + 3ζ − ωβ 2 + ζ4βω2 + 34 − 4ζ ω 2 2 2 c 2 ω M → 4 − ωβ 2 + ζ4βω2 + 34 0 . 4ζ ω ω ω3 − 4ζ 0 4ζ
Second order speed model (ζ=0.7, t =20 s, v =10m/s, σ =2.5m/s )
0
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Time (sec)
Fig 4 shows a realisation of this model for ζ = 0.7, tc = 20 s, vR = 10 m/s and σv = 2.5 m/s for an initial vehicle state of s = 0 m, v0 = 5 m/s and v˙ 0 = 0 m/s2 . These parameters might be a suitable representation for a somewhat “lumbering” vehicle. As well as the simulated distance, speed and acceleration, the figure shows the evolution of the means and standard deviations from the transition density evaluated above. B. Distribution of predicted vehicle state The pdf of distance travelled describes how far the vehicle has moved along its chosen route Rj . Given the sequence of road segments corresponding to Rj , this pdf of road distance can be mapped onto geographical position (via the road functions ri (.)) to obtain p(x0 (t)|x0 , Rj ), where x0 (t) is denotes vehicle position at time t. If the probability that the vehicle takes route Rj is Pr{Rj |x0 }, then, as in equation (1), the posterior pdf of the PNvehicle location is given by the mixture p(x0 (t)|x0 ) = j=1 Pr{Rj |x0 }p(x0 (t)|x0 , Rj ). Note that different routes will almost certainly share some common arcs, so the positional density on a road segment may be due to several contributing routes. Also if a route contains a closed walk, the positional pdf may accumulate due to a wrap around effect. If a route Rj terminates in a dead-end, then the total probability mass corresponding to the vehicle exceeding the route length accumulates at the dead-end vertex. An example of a positional pdf of a vehicle using the second order speed model (2) is shown in fig 5. The initial position and possible routes are those shown in fig 2 and the prior probabilities Pr{Rj |x0 } are chosen to favour routes that do not include loops or dead-ends. The model parameters (and
Fig. 4. Evolution of transition density for a second order speed model with a 20 sec time constant
initial speed and along-road acceleration) are the same as those used in fig 4 and the prediction period is 60 seconds. In constructing fig 5, it is assumed that the cross-road uncertainty is uniform over a road width of 3m. Also if the vehicle reaches a dead-end, its location is assumed to be uniformly distributed over a “parking area” of 152 m2 - this determines the height of the spikes seen at the dead-end vertices. IV. I LLUSTRATIVE TRACKING
EXAMPLE
To examine the behaviour of the target model an illustrative tracking example has been simulated. The requirement is to learn about the vehicle state vector as measurements are received. Given perfect initial information (i0 , u0 , v0 , v˙ 0 ) at t = 0, it is convenient to define a state vector x(t) = T (j, s(t), v(t), v(t)) ˙ , where j is the label of the route Rj taken, and s(t), v(t), v(t) ˙ are the distance travelled, the vehicle speed and the tangential acceleration. The dynamics of the state vector are governed by the speed model (2). The vehicle cannot change its route, so j is fixed. Note, that given the route Rj = (ij 0 , ij 1 , ij 2 , . . . , ij Nj ) and distance travelled s(t), it can be deduced that the vehicle is traversing arc ij k∗ with position parameter u∗ , where k ∗ satisfies: ∗
k X k=0
`(ij k ) 6 s(t) + u0 `(i0 )
k=0 `(ij k ) ) in which case, k ∗ = Nj and u∗ = 1. The tracking requirement is to construct p(x(t)|Z(t), x0 ), where Z(t) denotes all measurements received up to time t: p(x(t)|Z(t), x0 ) = N X Pr{Rj |Z(t), x0 }p (s(t), v(t), v(t)|Z(t), ˙ x 0 , Rj ) . j=1
A particle filter has been developed for this problem, each particle being a realization of the four element vector x. A standard “bootstrap” or SIR particle filter [2] has been employed for this illustration (however, the structure of the model is amenable to a more subtle particle filter implementation or a multiple hypothesis EKF (also see [4]) the dynamics are linear-Gaussian excluding dead-ends). The particles are projected forward between measurement times by sampling from the predicted Gaussian pdf (using the closed ¯ and covariance form expressions (3) and (7) for the mean x M ). The initial number of particles assigned to each route is determined from the prior probabilities Pr{Rj |x0 }. At each measurement time the particles are resampled according to the measurement likelihood p(Z 0 |x(t)). We assume that a (fictitious) sensor located at (0,0) takes measurements of range only (no range rate or bearing data are available). (This non-standard sensor was employed to show that the scheme can exploit highly nonlinear measurement data.) At each measurement time a frame of measurements
k∗
The simulation example is based on the set of possible routes of fig 2. The actual path taken by the vehicle is shown in fig 6. It is assumed that following initial information on the vehicle state, there is a gap of 60 seconds after which 30 measurement frames are received with an interval of ∆t = 1 second between each frame. The simulated measurements are given in fig 7 (σr = 0.015 km and ρr = 10 km−1 ). Using the same parameter values as in the previous section, the pdf of vehicle position after 60 seconds (before incorporating the first frame of measurements) is the pdf of fig 5. After applying the particle filter (with NS = 4000 samples), an estimate of the posterior pdf of vehicle location at 90 seconds is shown in fig 8. This is simply a normalised histogram of the X,Y components of {rij(n) k(n) (u(n) ) : n = 1, . . . , NS } where j (n) , k (n) and u(n) are the route label, segment number and position parameter corresponding to the nth posterior particle x(n) (t = 90). Most of the particles are clustered around the actual target location, with a few scattered along other arcs in the network. Obviously, the road map information is crucial to the construction of the posterior density; the cluttered and incomplete range measurements are of little value on their own. The posterior probabilities of the routes followed are also of interest. Estimates of these probabilities are obtained by simply counting the number of posterior samples P corresponding to S each route: i.e. Pr{Rj |Z(t), x0 } ' (1/NS ) N n=1 δj ( n) j . Fig 9 shows that two possible routes dominate, one of which is the actual route taken (and examination of the data shows the other route is feasible). The figure also shows the assumed prior probabilities of the routes (where routes involving loops or dead-ends are assigned lower probabilities). This shows that the approach has potential for revealing likely paths of a vehicle from incomplete and cluttered measurements (admittedly for a perfectly matched example). V. C ONCLUDING
REMARKS
A route-based model has been proposed for modelling vehicle motion on a road network. This allows target intent information to be incorporated. A tracking example using a simple particle filter has shown how the model can be used
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Fig. 8.
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to reveal the most likely paths taken by the target vehicle. The scheme may also offer a means of reconstructing tracks that have become fragmented (due, for example, to extended periods of obscuration). The approach could easily be extended to other (perhaps more realistic) vehicle speed models. In particular, we intend to pursue independent random impulse models (resulting in compound Poisson processes) and particle filters using continuous-time jump models [11]. Also, as already indicated, there is much scope for exploiting the structure of the model with more efficient particle filters and in particular hybrid deterministic / random filters (to reduce the Monte Carlo variance of the estimate). An extension to allow off-road vehicle motion will also be included - this will also exploit jump models.
10
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A PPENDIX This Appendix gives a closed form expression for the predicted covariance of the second order speed model. Z t + M = Φ(t − τ )CQ(τ )C T ΦT (t − τ )dτ τ =0 + + m11 m+ 12 m13 + = m+ m+ (7) 12 22 m23 + + m13 m23 m+ 33 where m+ 11
= +
2 ω ζ ω c2 −2ζωt t+ (1 − e )− + 12 ω4 4ζβ 2 4ω β 2 2βe−ζωt ζω ω2 2 cos βt + − 2 sin βt ω2 β β2
+ +
m+ 12
= − +
m+ 13
=
m+ 22
=
m+ 23
=
m+ 33
=
" # 2 βe−2ζωt ζω ζω − 3 cos 2βt 4ω 2 β β " !) 2 # ζω β 1−3 sin 2βt β
! 2 ω2 1 ζω −2ζωt − 2 (1 − e )+ +3 4β 4 β ζω −ζωt cos βt + e sin βt β " #) 2 ! e−2ζωt ζω ζω 1− cos 2βt + 2 sin 2βt 4 β β c2 ω 2 β ω 2 ωζ β −2ζωt −ζωt e 2− 2 cos 2βt − sin 2βt + 4e sin βt − ω 4 4β ωζ β β ωζ e−2ζωt 2 c2 ω 1− ω − ζ 2 ω 2 cos 2βt + ζωβ sin 2βt ω 4 4ζ β2 c2 ω2 e−2ζωt sin2 βt ω 4 2(1 − ζ 2 ) c2 ω 3 e−2ζωt 2 2 2 1− ω − ζ ω cos 2βt − ζωβ sin 2βt . ω 4 4ζ β2 c2 ω4
(
[10] Y. Bar-Shalom and T. Fortmann, Tracking and data association. Academic Press, 1988. [11] S. Godsill, “Particle filters for continuous-time jump models in tracking applications,” ESAIM, to be published.
+ + m+ 22 , m23 and m33 are available from [8] page 336 (although a minor printing error in m+ 33 has been corrected), while + + m+ , m and m have been derived here. 13 12 13
Acknowledgement This research was sponsored by the UK MOD Corporate Research Programme WPE and the Data & Information Fusion Defence Technology Centre. c
Crown copyright 2007. Published with the permission of the Defence Science and Technology Laboratory on behalf of the Controller of HMSO. R EFERENCES [1] T. Kirubarajan, Y. Bar-Shalom, K. R. Pattipati, and I. Kadar, “Ground target tracking with variable structure IMM estimator,” IEEE Transactions on Aerospace and Electronic Systems, vol. 36, pp. 26–45, January 2000. [2] B. Ristic, S. Arulampalam, and N. Gordon, Beyond the Kalman filter. Artech House, 2004. [3] S. Arulampalam, N. Gordon, M. Orton, and B. Ristic, “A variable structure multiple model particle filter for GMTI tracking,” in Fusion 2002: Proceedings of the 5th international conference on Information Fusion, vol. 2, pp. 927–934, July 2002. [4] M. Ulmke and W. Koch, “Road-map assisted ground moving target tracking,” IEEE Transactions on Aerospace and Electronic Systems, vol. 42, pp. 1264–1274, October 2006. [5] C. Agate and K. Sullivan, “Road-constraint target tracking and identification using a particle filter,” SPIE: Signal and Data Processing of Small Targets, vol. 5204, 2003. [6] Y. Zheng and U. Ozguner, “Modelling of grouped vehicles within a cellular spatial structure,” (Minneapolis), American Control Conference, June 2006. [7] J. M. Aldous and R. J. Wilson, Graphs and Applications. Springer, 2000. [8] A. Bryson and Y.-C. Ho, Applied Optimal Control. John Wiley, 1975. [9] Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation with applications to tracking and navigation. Wiley Interscience, 2001.
