Performance Bounds for Angle-Only Filtering with Application to

Performance Bounds for Angle-Only Filtering with Application to Sensor Network Management Paul R. Horridge & Marcel L. Hernandez, The Advanced Processing Centre, QinetiQ Ltd., St Andrews Road, Malvern WR14 3PS, UK. email: [email protected] & [email protected] Abstract – In this paper, we examine the Posterior Cram´er-Rao Lower Bound (PCRLB) for bearings-only tracking. We use a minimum detection range, inside which the target cannot be detected and show that the PCRLB tends to zero as this range tends to zero. Hence, in the absence of a minimum detection range, the bearings-only PCRLB is uninformative and identifies only that performance of a filter can be no better than perfect. It is also a feature of bearings-only tracking that no closed-form solution exists for the PCRLB, and numerical approximation is necessary via Monte Carlo integration. We show that in the absence of a minimum detection range the bearings-only PCRLB tends to zero as the number of Monte Carlo sample points tends to infinity. However, simulation results show the convergence can be slow which may account for this phenomenon previously going unnoticed. In the second half of this paper we introduce an alternative performance measure that resembles the error covariance of the Extended Kalman Filter (EKF) with measurements linearised around the true target state. This adapted performance measure is applied to the problem of managing a sensor network when there is a restriction in the total number of sensors that can be utilised at any one time. This measure is shown to closely match the filter performance and therefore can be used to accurately predict the performance of any combination of sensors. As a result, it is shown to allow more efficient management of the sensor network. Keywords: Sensor management, performance measure, target state estimation, Posterior Cram´er-Rao Lower Bound.

1

Introduction

For a general nonlinear filtering problem, no analytic closed-form solution exists and consequently in all practical applications, nonlinear filtering is performed by some form of approximation (e.g. linearisation [2], Monte Carlo c

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approximation [1]). The PCRLB is defined to be the inverse of the Fisher Information matrix and provides a bound on the optimal achievable accuracy of target state estimation. PCRLBs have recently been used to assess the performance of filtering algorithms, both for single target [13] and multi-target tracking [8]. They have also been proposed for radar system design [9], and used to determine the optimal observer trajectory for both passive ranging [11] and terrain navigation [10]. Until recently, calculating Posterior Cram´er-Rao Bounds has proved notoriously difficult. However, reference [15] proves an elegant Riccati like recursion for the general nonlinear filtering problem. Moreover, recent developments by [6] and [17] show that in cluttered environments, with measurements either target generated or false alarms, the measurement origin uncertainty expresses itself as a constant information reduction matrix (IRM). As a result, the PCRLBs can be calculated quickly and efficiently, allowing them to be utilised in real-time sensor management, such as the deployment of sonobuoys in submarine tracking [5]. The PCRLB gives a bound on the optimal performance of a filtering algorithm. In some cases this bound is achievable; for example, in linear filtering the error covariance of the Kalman Filter (e.g. [2]) is equal to the PCRLB. However, this is not generally true; in both nonlinear filtering and cluttered environments, state-of-the-art filters have been shown to perform poorly when compared to the bound [5] [6] [17]. Moreover, [3] observed that the IRM-PCRLB was overly optimistic in the case where there are missed detections, but no false alarms. They introduced an approach based on enumerating the detection/miss sequences. This was proven in [4] to be less optimistic that the IRMPCRLB. The over optimism of the PCRLB is a concern, particularly when the bound is being used as a basis for sensor management (see [5]). In such cases it is then critical that the over optimism of the bound can either be quantified,

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or that real filter estimation errors exhibit the same asymptotic behaviour as the bound (again see [5]). However, in the current paper we will show that in the case of bearingsonly tracking, if there is no minimum detection range, neither of these requirements can be satisfied. In this case, no closed-form solution exists for the PCRLB, and numerical approximation is necessary via Monte Carlo integration. We show that in the absence of a minimum detection range the bearings-only PCRLB is ill-defined and tends to zero as the number of Monte Carlo sample points tends to infinity. The bound is therefore uninformative and identifies only that performance of a filter can be no better than perfect. This phenomenon has previously gone unnoticed (e.g. [5] [6] [13]) because in the main it requires an unfeasibly large number of Monte Carlo sample points to become apparent. Motivated by this result, in section 3 an alternative measure of performance is introduced. This measure is less optimistic than the PCRLB, and importantly closely matches filter performance. In section 4, this new measure is used to manage a sensor network and shows improved results when compared to using the ill-defined PCRLB as the basis for sensor utilisation.

2

Posterior Cram´er-Rao Bounds

2.1

Dk12 Dk21 Dk33 JZ (k + 1)

Z(k) denotes the measurement vector at time k and ∆ is a second-order partial derivative operator. If ak (Xk ) = Ak Xk and wk is a zero mean Gaussian 12 with covariance matrix Σk , then Dk11 = ATk Σ−1 k Ak , Dk = −1 T −1 33 −Ak Σk and Dk = Σk . Applying the Matrix Inversion Lemma to equation (3) we see that in this case:
−1 Jk+1 = Ak Jk−1 ATk + Σk + JZ (k + 1) (9)

2.3

Multiple Sensors and Clutter

We consider an N sensor system in which measurements are available at discrete time epochs. At each time epoch k ∈ N, target generated measurements (of dimensionality, n) at each sensor, i, are of the form Zi (k) = hik (Xk ) + vi (k)

Jk is defined in [16]. The inequality in (1) means that the difference Ck − Jk−1 is a positive semi-definite matrix.1

PCRLBs for Target State Estimation

JZ (k) =

N X   E Hi (Xk )T QRk−1 Hi (Xk )

where Xk is the target state at time epoch k, ak is a possibly nonlinear function of Xk and {wk } is a white noise sequence. The following elegant recursion, giving the sequence of Fisher Information Matrices Jk for k > 0, is courtesy of [15]: Jk+1 = Dk33 − Dk21 (Jk + Dk11 )−1 Dk12 + JZ (k + 1) (3) 1 Throughout this paper, matrix inequalities are understood to be in the positive semi-definite sense, i.e. A ≥ B if and only if A − B is positive semi-definite.

(11)

i=1

where

[Hi (X)]a,b (2)

(10)

where the measurement error, vi (k), is a zero mean Gaussian random variable with covariance Rk , and each measurement is independent conditional on the target state. If false alarms have a uniform distribution throughout the field of view of each sensor, then it can be shown (see [6] and [17]) that the measurement contribution to the PCRLB takes the form

We consider the following general dynamical system Xk+1 = ak (Xk ) + wk

h i k = E −∆X log p(X |X ) (4) k+1 k Xk h i X = E −∆Xk+1 log p(Xk+1 |Xk ) (5) k
T = Dk12 (6) h i Xk+1 = E −∆Xk+1 log p(Xk+1 |Xk ) (7) h i X = E −∆Xk+1 log p(Z(k + 1)|Xk+1 ) (8) k+1

Dk11

Background

Suppose Xk is the (unknown and random) target state, bk . Let Ck which is estimated by an unbiased estimator X be the error covariance. The PCRLB for this covariance is defined to be the inverse, Jk−1 , of the Fisher Information matrix, Jk , [16] i.e.   T  b b C k , E Xk − Xk Xk − Xk ≥ Jk−1 (1)

2.2

where



  ∂ hik a = ∂Xb

(12)

 hik a is the ath component of the measurement vector (10), Xb is the bth component of the target state, and Q is a constant information reduction matrix (IRM) (or information reduction factor (IRF), q if either n = 1 or Rk is diagonal [7]) that expresses the effect of the measurement uncertainty (by inflating the measurement error covariance).

2.4

The Bearings-Only PCRLB

We consider a bearings-only measurement model. If the N sensors are placed at (xSi , yiS ) for i = 1, . . . , N , then   yk − yiS hik (Xk ) = tan−1 (13) xk − xSi

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The target state is denoted by (14)

XkT = (xk , yk , ·, . . . , ·)T

where xk and yk represent the target position in twodimensional space and the unspecified components may represent velocity, acceleration and so on. This ordering reflects the fact that our interest here is in the estimation error bounds on the position (xk , yk ). The following results will hold in all cases except when q = 0. The IRF, q = 0 only if either no target generated measurements are obtained, or the clutter density is overwhelming. In this case the measurement contribution is zero and the PCRLB diverges according to the target dynamics. For the bearings-only model the matrix JZ (k) is given by   j11 j12 0 · · · 0  j12 j22 0 · · · 0     0 0 ··· 0  JZ (k) =  0 (15)   .. .. .. ..   . . . .  0

0

0

···

0

where j11 j12 j22

" # N X (yk − yiS )2 1 E  = q 2 (16) σ2 (xk − xSi )2 + (yk − yiS )2 i=1 i " # N X 1 −(yk − yiS )(xk − xSi ) = q E  2 (17) σ2 (xk − xSi )2 + (yk − yiS )2 i=1 i # " N X 1 (xk − xSi )2 = q E  2 (18) σ2 (xk − xS )2 + (yk − y S )2 i=1 i i

i

Now, let

 −1 JX (k + 1) , Dk33 − Dk21 (Jk + Dk11 )−1 Dk12

(19)

Then the PCRLB at time k is given by

 −1 Jk−1 = JX (k)−1 + JZ (k)

(20)

−1 Let JeX (k), JeZ (k) and Jg be the upper left 2 × 2 subk matrices of JX (k), JZ (k) and Jk−1 respectively. Since JZ (k) is zero except for the upper left 2 × 2 sub-matrix, −1 it can be shown that Jg is given by k

2.5



−1 −1 e Jg + JeZ (k) k = JX (k)

−1

(21)

Measurement Contribution Divergence

Now, provided the probability density function (PDF), p(x, y), of the target position is bounded above zero within

a ball Bε (xSi , yiS ) of radius ε > 0 centred on (an arbitrary) sensor, i, it can be shown that: Z q y2 j11 ≥ 2 p(x + xSi , y + yiS ) dx dy σi R2 [x2 + y 2 ]2  Z y2 q inf p √ ≥ 2 dx dy 2 2 2 σi Bε (xSi ,yiS ) x2 +y 2 ≤ [x + y ]   Z ε  Z 2π  q 1 = 2 inf p dρ cos2 θ dθ σi Bε (xSi ,yiS ) 0 ρ 0 (where (ρ, θ) are polar coordinates)

= ∞

(22)

Similarly it can be shown that j22 = ∞. The PCRLB is therefore ill-defined for this measurement model. This phenomenon can be prevented by imposing a minimum detection range, r, or by approximating the bound (20) using Monte Carlo sampling. We now show that both of these approaches give bounds that converge to zero in an appropriate limit.

2.6

Minimum Detection Range

Suppose a slight modification is made to the measurement model by insisting that each sensor can only detect the target if the distance to the target is no less than some specified distance r > 0. We introduce the following notation: let Bε (x, y) denote the ball of radius ε around (x, y); Aiε = Bε (xSi , yiS ), for i = 1, . . . , N ; and set i Aε = ∪ N i=1 Aε . Suppose that p(x, y) is the PDF of the target position at time k, and define pr (x, y) to be  p(x, y) for (x, y) 6∈ Ar pr (x, y) = (23) 0 for (x, y) ∈ Ar Let JeZr be the upper 2 × 2 sub-matrix of JZ (k) with a minimum detection range of r. Then the entries of JeZr (k) are given by Z N X y2 q r p (x+xSi , y+yiS )dxdy (24) j11 = 2 2 2 2 r σ 2 [x +y ] R i i=1 Z N X q −xy r j12 = p (x+xSi , y+yiS )dxdy (25) 2 2 2 ]2 r σ [x +y 2 R i i=1 Z N X q x2 r j22 = p (x+xSi , y+yiS )dxdy (26) 2 2 2 ]2 r σ [x +y 2 i=1 i R The minimum detection range now prevents the divergence r r of j11 and j22 . We will now prove the following result.

Proposition 2.1 Suppose the target state PDF p(·) is continuous at each (xSi , yiS ) and non-zero at at least one of them. Then  −1  0 0  lim JeZr = (27) 0 0 r→0

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Proof For each i = 1, . . . , N and R > r > 0, let BR,r = BR (0) \ Br (0). Then Z xy S S p (x + x , y + y )dxdy i i 2 [x2 + y 2 ]2 r R Z xy 1 S S ≤ p(x + x , y + y )dxdy + i i 2 2 2 BR,r [x + y ] 2R2 !   R 1 ≤ sup p − inf p log + (28) 2 i r 2R i A AR R and therefore r |j12 |

Also, r j11

N X q ≤ σ2 i=1 i

≥ =

and r j11

≤

"

N  X

!

#  R 1 sup p −inf p log + r 2R2 AiR AiR 



(M ),r Proof For each r > 0, let JeZ be the Monte Carlo approximation that results from ignoring sample points which lie within a distance r of a sensor, i.e.

=

N X i=1



(30)

q M σi2

X

(Xj ,Yj )6∈Ar

Yj − yiS −(Xj − xsi )



γ(·)2 ×

Yj − yiS −(Xj − xsi )

(36) T

(37)

where

" ! #   N X q R 1 π sup p log + 2 (31) σ2 r R AiR i=1 i

and furthermore, since > 0 for some i ∈ {0, . . . , N } and p is continuous at each sensor position, we may choose R > 0 sufficiently small to ensure that C2 > 0. It therefore follows from (31) that  −1 jr (33) JeZr = r r 11 r 2 22 j11 j22 − (j12 ) p(xSi , yiS )

converges to zero as r → 0. In a similar manner it can be  −1 shown that the other entries of JeZr also converge to zero. 2 In this case, it follows from the Matrix Inversion Lemma and (21) that: =

Proposition 2.2 Under the assumptions of Proposition 2.1:  −1   0 0 (M ) = almost surely (35) lim JeZ 0 0 M →∞

(M ),r JeZ

The inequalities (30) and (31) also hold for j22 . By (29) and (30) we see that, for some constants C0 , C1 and C2 depending only on {σi }N i=1 , p, q and R, we have   2   R R r r r 2 j11 j22 −(j12 ) ≥ C2 log +C1 log +C0 (32) r r

−1 Jg k

Monte Carlo Approximation

Consider now the case where there is no minimum detection range and JeZ (k) is approximated by a number of independent samples, (Xj , Yj ) drawn from the distribution (M ) of the target position. Let JeZ be such an approximation with M sample points. Then

Z

y2 dxdy inf p 2 2 2 AiR BR,r [x + y ] i=1   N X qπ R inf p log 2 i σ r A R i=1 i q σi2 !

(29)

2.7

JeZr (k)−1 − JeZr (k)−1 ×  −1 JeX (k)−1 + JeZr (k)−1 JeZr (k)−1 (34)

Therefore, under the assumptions of Proposition 2.1, and provided JeX (k) is non-singular, the PCRLB converges to zero as the minimum detection range converges to zero.

γ(Xj , Yj , xSi , yiS ) =

(Xj −

xSi )2

1 + (Yj − yiS )2

(38)

Since (x y)T (x y) is positive semi-definite for any x, y ∈ (M ) (M ),r R, it is clear that JeZ ≥ JeZ for each M ∈ N and r ≥ 0, and therefore  −1  −1 (M ) (M ),r JeZ ≤ JeZ . (39) By the Strong Law of Large Numbers, we also see that for each r > 0,  −1  −1 (M ),r JeZ → JeZr as M → ∞ almost surely (40)

Now, the following is true for all r: h i−1 h i−1 h i−1 h i−1 (M ) (M ),r e e + JeZr 0 < JZ ≤ JZ − JeZr (41) ii

ii

ii

ii

Hence, if we let M → ∞ and r → 0 and use Proposition 2.1, we have: h i−1 (M ) lim JeZ =0 almost surely (42) M →∞

ii

 −1 (M ) Showing that the off-diagonal entries of JeZ converge to zero in the limit as M → ∞ is then trivial. 2 In the same manner as (34) it then follows from the Matrix Inversion Lemma and (21) that the Monte Carlo approximation of the PCRLB converges to zero almost surely as M → ∞.

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2.8

Simulation Results

Clearly in the case of bearings-only tracking the PCRLB is a trivial bound that tells us that the filter root mean square errors (RMSEs) cannot be negative. In this subsection we attempt to explain how this phenomenon could have gone unnoticed. The relationship between the estimated PCRLB and the number of Monte Carlo sample points used is investigated. In particular, it is shown that if the distance between the sensor and the posterior target PDF is large compared to the target PDF covariance, then a massive number of sample points would be necessary to show that the PCRLB converges to zero. Clearly this presents an impractical computational overhead. It is assumed that the prior distribution of the target position is Gaussian with mean (0, 0) and variance C0 = I2 . Each estimate is based on using a single sensor whose bearing error has standard deviation, σ = 0.01 radians. We determine the PCRLB for 4 different sensor locations (figure 1) to demonstrate that, when the sensor is close to the mean of the target distribution, the PCRLB estimate is highly dependent on the number of sample points used. Each run is repeated ten times at each number of sample points to gauge the variation in successive approximations (each approximation is shown by a blue dot in figure 1). (a)

(b)

0.03

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target location RMSE bound

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We have presented an example of a widely used sensor measurement model for which the PCRLB is ill-defined and Monte Carlo estimates of it can be made arbitrarily small by using a sufficiently large number of sample points. Caution is therefore recommended in using the PCRLB when bearing measurements are present. In the next section we define a new measure of tracking performance that does not suffer the problems associated with the traditional PCRLB approach. Moreover, this new measure is an accurate predictor of tracking performance, and as a result allows the efficient and effective utilisation of the available sensor resource.

3

The measure of performance that follows is dependent on the system model prescribing the target dynamics being a linear function of the target state, i.e. Xk+1

8

J0 (Si ) Jk+1 (Si )

0.14 0.3

= C0−1 =

0.25 0.1 0.2

+

0.08 0.15 0.06

0

1

2

3

4

5

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0

(44)

N X

−1 i i Hj (Xk+1 )T Rk+1 Hj (Xk+1 ) (45)

j=1

0.1

0.05

0.02


T −1

Σk + Ak Jk (Si )−1 Ak

0.12

0.04

(43)

= A k Xk + w k

where {wk } is again a white noise sequence, with wk ∼ N (0, Σk ). An N sensor system is again considered, with the measurement equation given by (10). We do not consider cluttered environments (i.e. q = 1), although we believe there is no reason why the following recursion (45) cannot be adjusted to include an information reduction matrix (or factor) in such cases. Let S denote the set of realisations of the general dynamical system (43). Let Si ⊂ S denote one such sequence: (X1i , X2i , . . .). Now, for each Si we define Jk (Si ) by the following recursion:

(d)

0.35

Alternative Performance Measure

1

2

3

4

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6

7

8

log10 (number of sample points) Figure 1: Monte Carlo estimate of PCRLB versus number of sample points, with a single sensor at (a): (0, 0), (b): (1, 0), (c): (3, 0) and (d): (5, 0). As can be seen from figure 1(a)–(b), when the sensor is close to the mean of the target distribution, the estimated PCRLB decreases as the number of sample points increases. The effect when the sensor is several standard deviations away from the target distribution mean is much less apparent (figures 1(c)–(d)) since it would take a huge number of sample points for enough to land sufficiently close to the sensor to cause a “blow-up” in the terms of the matrix JeZ (k).

and, again, Hj is defined by (12). Note that if we track a target using an Extended Kalman Filter (EKF) and linearise the measurement function around i the state Xk+1 , then Jk (Si )−1 would give the EKF estimation error covariance. An approximate filter mean square error bound, B(enum; k), is then given by   B(enum; k) = ESi Jk (Si )−1 (46)

where the expectation is taken over target trajectories Si . In general, B(enum; k) is intractable, so a Monte Carlo estimate is created by generating M potential target paths, Si , using the initial target distribution and the known dynamics (43). It is straightforward to estimate B(enum; k), by generating a number of simulated trajectories of the required

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• STEP 3: use the recursion given by equations (44) and (45) to determine the APM at each measurement epoch; • STEP 4: choose the combination of sensors that achieves the necessary control over the APM. We note that here we are basing our sensor management strategy purely on the prior target PDF and the known dynamics. However, an approach that uses feedback (based on filter estimates) to refine the sensor management strategy has been shown to give improved results, especially when the target motion is highly uncertain [7].

4.2

Proposition 3.1 For non-linear measurement models2 , provided certain (unrestrictive) regularity conditions concerning Ak , Σk , and C0 are true, then B(enum; k)

>

B(crlb; k)

for all k ≥ 1 (47)

y-coordinate (metres)

where B(crlb; k) is the standard PCRLB, B(crlb; k) = Jk−1 , with Jk given by equation (3).

4

Sensor Network Management

4.1

Methodology

y-coordinate (metres)

We utilise an array of fixed sensors and assume that there is some restriction (possibly as a result of communication bandwidth limitations) on the number of sensors that can be utilised at any one time. To manage the sensor network at each measurement epoch we determine the alternative performance measure (APM) for each possible sensor combination, and select the combination which predicts the lowest target location RMSE. A summary of the key steps is given as follows: • STEP 1: determine M potential target locations (at time 0) by sampling from the initial distribution of the target state; • STEP 2: use the target dynamics (43) to generate M potential target paths, Si ; 2 If the measurement model in linear then it can easily be verified that B(enum; k) = B(crlb; k) for all k.

Simulations

In our simulations each sensor provides a single bearingonly measurement, and there is no clutter. The target is assumed to move according to a nearly constant velocity model (e.g. [2]), with power spectral density l = 10. Measurements are available at 1 second intervals. Figure 2 provides an illustration of the focal scenario. A square array of 16 sensors is considered, and in figure 2, only (N =)2 sensors can be active at each stage (given in red). y-coordinate (metres)

length, running the recursion (45) for each and taking the mean. However, this method does have the drawback that it requires the sequence Jk (Si ), k = 1, 2, . . . to be calculated for each trajectory, and is therefore much slower than the standard PCRLB approach. In [3], the authors were concerned with the problem of calculating the PCRLBs in the case where there may be missed detections, and observed that the standard approach of computing an information reduction factor gave an overly optimistic bound. As an alternative, [3] conditioned on the detection/miss detection sequences and averaged over these sequences. This has been shown to give a measure of tracking performance that is both less optimistic than the IRM-PCRLB [4] and more closely matches actual tracking performance [3]. The approach here is somewhat similar to that of [3], except that an average is taken over the possible target trajectories rather than detection sequences. It is less clear whether our method actually gives a rigorous lower bound, although in the case of no process noise this is true (see [14]). We can now prove the following result (see appendix A), which shows that our performance measure is less optimistic than the traditional PCRLB approach. Importantly, in the case of bearings-only tracking this performance measure therefore gives an informative non-zero bound.

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Figure 2: Sensor array utilisation at six time steps. The black cross gives the estimated target location and the circle is a representation of the covariance, each based on the prior distribution.

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In figure 3, the results for (N =) 1, 2, 3 and 4 active sensors respectively are shown. It can be seen that the APM closely matches the performance of the EKF in this case. We also observe that as the number of sensors increases, both the APM and the EKF RMSEs become smaller, as one would expect. In each case, results are based on 100 simulated scenarios.

4.3

Comparing Sensor Network Performance Under The Two Measures

In figures 4 and 5 we compare the effect of using our APM to manage the sensor network with using the PCRLB. Using the PCRLB to determine the sensor management strategy tends to activate sensors close to the expected target position and so it performs reasonably well. This is especially true in the case of multiple active sensors, where the network constraints ensure reasonable triangulation of the target. However, if only one sensor is active, using the PCRLB as the basis for sensor management leads to individual sensors in the array being active for several time steps, giving poor triangulation. Using the APM then gives a marked improvement in tracking performance in this case (see figure 5). Utilising the APM reduces the RMSE of the EKF estimates by between 7% and 20% depending on the number of active sensors allowed (see figure 5).

5

Conclusions

We have examined the PCRLB for bearings-only tracking. It is shown that in the absence of a minimum detection range the bearings-only PCRLB is uninformative and identifies only that performance of a filter can be no better than perfect. Moreover, in the absence of a minimum detection range the bearings-only PCRLB tends to zero as the number of samples in the Monte Carlo estimate tends to infinity. However, simulation results showed that convergence can be slow, which may explain why this has previously gone unnoticed. In the second half of this paper, an alternative performance measure was introduced. This performance measure was applied to the problem of managing a sensor network when there was a restriction in the total number of sensors that can be utilised at any one time. The measure was shown to closely match the filter performance and therefore can be used to accurately predict the performance of any combination of sensors. As a result it was shown to allow more efficient management of the sensor network, with filter RMSE between 7% and 20% less than when using the uninformative PCRLB to optimise sensor utilisation.

Acknowledgements This research was sponsored by the United Kingdom Ministry of Defence Corporate Research Programme CISP.

References [1] M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking”, IEEE Transactions on Signal Processing, 50(2), pp. 173–188, 2002. [2] Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, “Estimation with Applications to Tracking and Navigation”, Wiley, 2001. [3] A. Farina, B. Ristic and L. Timmoneri, “Cram´er-Rao bounds for non-linear filtering with Pd < 1 and its application to target tracking”, IEEE Transactions on Signal Processing, 50(8), pp. 1316–1324, 2002. [4] M. L. Hernandez, B. Ristic, A. Farina, and L. Timmoneri, “Comparison of Two Posterior Cram´er-Rao Bounds for Non-Linear Filtering with Applications”, submitted to IEEE Transactions on Signal Processing, 2002. [5] M. L. Hernandez, T. Kirubarajan, and Y. Bar-Shalom, “Efficient Multi-Sensor Resource Management Using Cram´er-Rao Lower Bounds”, Signal and Data Processing of Small Targets, Proceedings of the Society of Photo-Optical Instrument Engineers, vol. 4728, 2002. [6] M. L. Hernandez, A. D. Marrs, N. J. Gordon, S. Maskell, and C. M. Reed, “Cram´er-Rao Bounds For Non-Linear Filtering With Measurement Origin Uncertainty”, Proceedings of the 5th International Conference on Information Fusion, vol. 1, pp. 18–25, 2002. [7] P. R. Horridge, and M. L. Hernandez, “Enhanced Multisensor Resource Management Using Theoretical Performance Bounds”, QinetiQ Technical Report, QINETIQ/S&E/APC/TR021664/1.0, 2002 (available on request only). [8] C. Hue, J.-P. Le Cadre and P. P´erez, “Performance Analysis of Two Sequential Monte Carlo Methods and Posterior Cram´er-Rao Bounds for Multi-Target Tracking”, Proceedings of the 5th International Conference on Information Fusion, Annapolis, MD, pp. 464–473, 2002. [9] N. Nehorai and M. Hawkes, “Performance Bounds for Estimating Vector Systems”, IEEE Transactions on Signal Processing [10] S. Paris, and J.-P. Le Cadre, “Planification for TerrainAided Navigation”, Proceedings of the 5th International Conference on Information Fusion, 2, pp. 1007– 1014, 2002.

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Now, firstly it can be seen from equations (4), (7) and (8) that: Dk11 , Dk33 , and JZ (k) are all positive (semi-)definite (50) The proof is then reliant on the following being true: Jk is positive definite (i.e. Jk > 0) Jk (Si ) is positive definite for all Si (i.e. Jk (Si ) > 0 for all Si )

(52)

and indeed we can prove (by induction) the validity of (51) and (52) in all but specially constructed cases. It follows from lemma A.2 that: h
−1 T −1 i T −1 ESi Σ−1 A J (S )+A Σ A Ak Σk k k i k k k k
−1 T −1 T −1 > Σ−1 Ak Σk (53) k Ak Jk + A k Σk Ak

It should be noted that the strict inequality in (53) is a result of the non-linearity in the measurement function, hj (Xk ), ensuring that Jk (Si ) is not a constant matrix (and so the inequality is strict in the statement of lemma A.2). Moreover,   N X −1 i i E Si  Hj (Xk+1 )T Rk+1 Hj (Xk+1 ) = JZ (k+1) (54) j=1

Using the Matrix Inversion Lemma: Jk+1 (Si ) = Σ−1 k

[17] X. Zhang, and P. K. Willett, “Cram´er-Rao Bounds for Discrete Time Linear Filtering with Measurement Origin Uncertainty”, Proceedings of the Workshop on Estimation, Tracking and Fusion: A Tribute to Yaakov BarShalom, Monterey, CA, May 2001.

A

(51)

T −1 −Σ−1 k Ak Jk (Si ) + Ak Σk Ak

+

N X


−1

ATk Σ−1 k

−1 i i Hj (Xk+1 )T Rk+1 Hj (Xk+1 )

(55)

j=1

It then follows from (53), (54) and (55) that

Proof of Proposition 3.1

The proof relies on the following two lemmas. The first is a standard result of matrix algebra. A proof of the second lemma can be found in (for example) [4].

ESi [Jk+1 (Si )]

B(enum; k + 1)   = ESi Jk+1 (Si )−1 −1

> ESi [Jk+1 (Si )] (by lemma A.2 and statement (52))

Lemma A.2 Let X be a random positive definite matrix. −1 Then E X −1 ≥ (E[X]) with strict inequality unless X is constant.

then

Jk+1

(56)

Hence

Lemma A.1 For invertible, symmetric matrices X and Y such that X ≥ Y ≥ 0, we have Y −1 ≥ X −1 .

We prove Proposition 3.1 by induction. It is sufficient to show that if  B(enum; k) ≥ B(crlb; k) for some k (48) ESi [Jk (Si )] ≤ Jk




(57) (58)

−1 Jk+1 (59) (by inequality (56), lemma A.1 and (51)–(52) )

= B(crlb; k + 1).

(60)

The proof is completed by noting that B(enum; 0) = B(crlb; 0) = C0 ESi [J0 (Si )] = J0 = C0−1

(61) (62)

where C0 is the initial target covariance. B(enum; k + 1) > B(crlb; k + 1) ESi [Jk+1 (Si )] < Jk+1

(49)

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Figure 4: Comparison of tracking performance using the PCRLB and the alternative performance metric (APM) with N = 1 (top image) and N = 2 (bottom image) active sensors. Key: green: target location RMSE based on APM, blue: sample based PCRLB RMSE, black: EKF RMSE when using APM to manage the sensor network, red: EKF RMSE when using the PCRLB to manage the network.

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Figure 5: Comparison of tracking performance using the PCRLB (white bars) and alternative performance metric (black bars) as the basis for sensor management.

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