ON THE TREATMENT OF PROCESS NOISE IN THE UNSCENTED KALMAN FILTER

TOM42 – 2017

Paul F. Easthope^∗

ON THE TREATMENT OF PROCESS NOISE IN THE UNSCENTED KALMAN FILTER

Keywords:Tracking, Unscented Kalman Filter, process noise

1. INTRODUCTION

The Unscented Kalman Filter (UKF) (see, for example, [7, 9, 8, 11, 6, 10, 4]) offers a systematic means of nonlinearly transforming probability distributions, whether as a conse- quence of a nonlinear kinematic model or from a nonlinear measurement model, or from both together. Numerous experiments have shown the superiority of the approach to the more fa- miliar Extended Kalman Filter, in which the nonlinear relations are approximated by linear ones. However, in order to make optimum use of unscented techniques, the treatment of the kinematic process noise requires some consideration.

In the regular Kalman Filter (whether Extended or not), the process noise covariance matrix Q is formed separately and additively augments the extrapolated track covariance matrix ([2]); this is a direct consequence of the additive nature of the process noise itself.

For the UKF, however, the preferred approach is to add extra state components ([9]), while retaining the full nonlinearity in the kinematic equations.

So far as is known, no study has examined the robustness of this approach in the context of general-purpose tracking. As may be readily appreciated, no kinematic tracking model can adequately describe the full range of motions of (for example) an aircraft or a ballistic missile (which is why multiple model solutions are often proposed; see, for example, [1]).

Nonetheless, in the interests of flexibility, it is expected that any specific kinematic model (and associated tracking filter logic) should be capable of tracking a somewhat wider range of motions than is directly specified in its kinematic model.

This paper shows that such expectations are not necessarily realised with the usual for- mulation of the UKF, in which additional state components are included to encompass the process noise. This is demonstrated specifically in the context of a turning target in two di- mensions, chosen to be representative of a wider class of nonlinear tracking models but still simple enough to expose the potential fragility of the standard implementation of the UKF.

The target kinematic model is described in Section 2 and the UKF equations are defined in Section 3.

∗L-3 Communications ASA Ltd, Rusint House, Harvest Crescent, Fleet, Hampshire, GU51 2QS, UK email: paul.easthope@L3T.com

2. TARGET GENERATION

The basic model used for the generation of the underlying target motion is as follows:

¨

x= −α(t) ˙y, (1)

¨

y= α(t) ˙x, (2)

in which x and y represent the two spatial coordinates, t is time and the superscript dot implies time derivative. These are readily solved numerically.

The (potentially) time-dependent quantity α(t) defines a ‘turn parameter’, and two exam- ples are illustrated below. It is evident, however, from the nature of the equations that the acceleration vector is normal to the velocity, so various forms of turn at constant speed are described.

Figure 1 illustrates a target turning in a spiral at 200 ms⁻¹, over a time span of 1000 s.

0 5 10 15 20 25

-2 0 2 4 6 8 10 12 14 16

Y (km)

X (km)

Fig. 1. Spiral target truth data

Here, the value of α is given by α(t) = 0.01 + 10⁻⁴t, giving a total lateral acceleration at the end of the simulation of 22 ms⁻²(about 2.2 g) — not particularly great but sufficient to stress the tracking filters.

An alternative expression for α, namely

α(t) = 0.02 sin ωt, where ω= 0.0314,

gives a weaving form of motion, illustrated in Figure 2.

0 10 20 30 40 50 60

0 10 20 30 40 50 60 70 80

Y (km)

X (km)

Fig. 2. Weaving target truth data

In this case, the data spans 500 s, although the speed is the same at 200 ms⁻¹.

For tracking purposes, Cartesian measurements are provided at one-second intervals with isotropic, normally distributed random errors with a standard deviation of 30 m.

As an aside, it may be pointed out here that the actual nature of the measurement noise distribution is not of great concern in the present paper. While it is appreciated that certain assumptions are made in the formulation of the UKF equations (specifically, that a proba- bility distribution at any particular time step can be adequately encapsulated in terms of its mean and covariance), a practical tracking filter cannot afford to be overly fussy as to the ac- tual distributions of measurement noise (other than their being zero-mean and approximately describable in covariance terms). Indeed, measurements are in any case likely to stem from several independent sources and be of unknown nature.

3. UNSCENTED KALMAN FILTER MODEL

This section describes a UKF implementation of the tracking model, based upon equations (1) and (2). To incorporate the process noise components fully within the nonlinear equations of motion, a state dimension of eight is needed, namely:

x= [x, y, ˙x, ˙y, α, p1, p2, p3]^T ≡[x1, x2, x3, x4, x5, x6, x7, x8]^T,

where the final three states cater for the process noise and seventeen sigma points are needed overall.

These components are then assumed to obey the following differential equations, to be

solved using numerical integration methods:

˙x1= x3,

˙x2= x4,

˙x3= −x5x4+ σax6, (3)

˙x4= x5x3+ σax7,

˙x5= σαx8,

˙x6= ˙x7= ˙x8= 0.

The quantities σaand σαhere stand for process noise magnitudes, with the associated varia- tions encapsulated in the states x6, x7, x8. Note that the derivative of α (≡ x5) is non-zero, thus allowing for an unknown and variable turn parameter. If, on the other hand, α was known to be constant but of unknown value, its derivative could be set to zero.

It may be pointed out that the above system is observable, an issue examined in Ap- pendix A.

The track is formed from the first two measurements, with an initial α = 0. The initial acceleration uncertainty can be set at 0.5 ms⁻², and with an initial α uncertainty of 0.05, plus x6 = x7 = x8 = 0. The initial uncertainties in x6 to x8 are set to unity and with associated cross-components assigned to zero. Subsequent process noise values were set at 0.2 for acceleration and 0.04 for α, and Appendix B provides the operational sequence of the filter.

For the spiral target trajectory, the resulting track behaviour was found to be satisfactory and as expected; the behaviour of the tracked α state over time is shown in Figure 3.

-0.1 -0.05 0 0.05 0.1 0.15

0 100 200 300 400 500 600 700 800 900 1000

Alpha

Time

UKF

Fig. 3. Spiral target, tracked α values for the UKF at 1 Hz

The results are here shown with one-sigma error bars and at the end of the data, the tracked uncertainty in α, namely σα, falls to ∼4.3 × 10⁻⁶.

Although it may appear that such small values of σα are beneficial, this is actually not the case, since the same UKF filter fails to track the weaving target, breaking track at 77 s regardless of the update rate or the process noise settings. In fact, for either data set, it is found that σα and σp3 ≡ σα˙ proceed inexorably toward zero irrespective of process noise settings or data rate.

In summary, the UKF will give satisfactory and accurate results provided the actual turn rate is constant, or monotonically increasing or decreasing with time, but will break track once α starts to vary in a more complex manner. That is, the model is effective provided α is constant or linear in time — both of which are consistent with ˙α = σαx8 — but eventually breaks track for the weaving target case, where α is more variable.

An examination of the components of the error covariance matrix P over time indicates a trend towards zero magnitude. This decay may be inferred from Figure 4, which shows the normalised eigenvalues of P plotted against time, using a circling target tracked over 14000 s.

1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1

0 2000 4000 6000 8000 10000 12000 14000

Normalised covariance eigenvalues

Time

Fig. 4. Normalised covariance eigenvalues vs time

The eigenvalues have here been normalised by their initial values and a logarithmic scale has been used to accommodate the wide range of magnitudes. All of the eigenvalues exhibit decay characteristics.

For a linear Kalman Filter, such behaviour — P(t) → 0 as t → ∞ — would not be unexpected in the absence of process noise (see [2], page 138), although it is less obviously the case for equation set (3). However, it is evident that the filter becomes over-confident and this over-confidence is not cured by using increasingly elaborate models for the time behaviour of α.

It is not that some state components are unobservable, given the analysis of Appendix A and the evidence in Figure 3; rather, the filter behaviour is overly rigid for generic use. Thus, an alternative approach is considered, as discussed in Section 4.

4. UNSCENTED KALMAN FILTER VARIANT

The basic idea invoked here is to make use of particle filter concepts (see, for example, [13], [12]), in which the kinematic equations for the sigma points are perturbed at each in- tegration time step by zero-mean random values with the appropriate levels of uncertainty.

This gives rise to the following form of the differential equations

˙x1= x3,

˙x2= x4,

˙x3= −x5x4+ nx, (4)

˙x4= x5x3+ ny,

˙x5= nα,

thus reverting to five state components and eleven sigma points. Here, nxand nystand for zero-mean, normally-distributed random numbers with standard deviations characteristic of the expected acceleration process noise. Similarly, nαis a random quantity with an expected level of α variability.

Although the above differential equations in set (4) are usually applied in a particle filter context, in which the distribution is represented by a large number of points or particles, it is of interest to see how the same approach behaves when invoked in a UKF arena, with (here) eleven sigma points.

It is found that the new (variant) UKF is both less accurate and less certain than for the UKF considered in Section 3, which can be appreciated from the plot of the tracked value of α over time (with one-sigma error bars) shown in Figure 5.

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

0 100 200 300 400 500 600 700 800 900 1000

Alpha

Time

VUKF

Fig. 5. Spiral target, tracked α values for variant UKF at 1 Hz

This may be compared with Figure 3. Needless to say, more accurate variant UKF α results can be obtained by reducing the magnitude of nαbut it is more important here to show that

the same model is able to track the weaving target successfully, in contrast to the results found in Section 3. To this end, the corresponding tracked α values are shown in Figure 6.

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

0 50 100 150 200 250 300 350 400 450 500

Alpha

Time

Fig. 6. Weaving target, tracked α values for variant UKF

The RMS (root-mean-square) position and velocity uncertainties for the weaving target are shown in Figure 7, and it can be seen that these stabilise at non-zero values, as expected with the continual injection of process noise. The track position uncertainty is about a factor of two smaller than the measurement uncertainty.

0 5 10 15 20 25 30

0 50 100 150 200 250 300 350 400 450 500

RMS position and velocity uncertainties (m and m/s)

Time

Position Velocity

Fig. 7. Weaving target, tracked RMS position and velocity uncertainties for variant UKF

5. CONCLUSIONS

The standard implementation of the Unscented Kalman Filter has been applied to a simple two-dimensional nonlinear turning target model, including additional track states to accom- modate the process noise terms. It is found that provided the tracking model is a faithful representation of the underlying target dynamics (for example, circular or spiral motion), the track results are satisfactory. However, the track diverges and breaks once the target dynam- ics is more variable (e.g. a weaving motion). Analysis of the track covariance components indicate an asymptotic trend to zero values, reminiscent of a zero-process-noise model.

These findings indicate that the incorporation of process noise into the UKF by means of additional state components can give rise to filter fragility. A more robust filter formulation is obtained by borrowing process noise concepts from the particle filter, in which random noise is deliberately injected at each integration time step. This simple approach provides success- ful tracking for all of the target motions considered here, reduces the state dimensionality and broadens the applicability of the same kinematic model.

The same ‘variant’ UKF method has also proved effective in many other nonlinear track- ing problems, including the familiar atmospheric re-entry situation.

A

This work was funded in entirety by L-3 Communications ASA Ltd.

R

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[11] S. Julier, J. Uhlmann, and H. Durrant-Whyte. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Trans. on Automatic Control, 45(3):477–

482, 2000.

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[13] D. Salmond and N. Gordon. An introduction to particle filters. Technical report, 2005.

A. OBSERVABILITY OF THE BASIC UKF MODEL

The assessment of observability here follows the methods of [3] and [5], in which it is assumed that measurements of x, y, u, v and their successive derivatives are available. The relevant kinematic equations may be written:

˙u = −αv + p, (5)

˙v = αu + q, (6)

˙α = r, (7)

˙p = ˙q = ˙r = 0,

ignoring the x and y components, which are directly observable, and absorbing the constant σa, σαterms into p, q and r.

Clearly, if α can be determined, then p and q are also, from equations (5) and (6), so the initial emphasis needs to be on extracting α.

Firstly differentiate (5) and (6):

¨

u= −α ˙v − ˙αv,

= −α²u − αq − rv,

¨

v= α ˙u + ˙αu,

= −α²u+ αp + ru.

Therefore,

u¨u+ v¨v= −α² u²+ v²
+ α (vp − uq) . (8) Next consider the quantity

v˙u − u ˙v = v (−αv + p) − u (αu + q) ,

= −α u²+ v²
+ (vp − uq) , so that

vp − uq= (v ˙u − u ˙v) + α u²+ v²
. Substituting this into equation (8) and rearranging gives:

α=u¨u+ v¨v

v˙u − u ˙v, (9)

which will have a solution unless the denominator is zero.

As mentioned above, once α is determined, p and q are also. To obtain r, differentiate equation (9) and substitute into equation (7), giving

r= 1

(v ˙u − u ˙v)²

h(v ˙u − u ˙v)(u^···u+ ˙u¨u+ v^···v + ˙v¨v) − (u¨u+ v¨v)(v¨u − u¨v)i . Thus, all of the filter states are observable.

B. SUMMARY OF UKF EQUATIONS

For reference, the sequence of operations defining the filter are outlined here. In view of the fact that the measurement is a linear process (Section 2), only the extrapolation step of the filter need be implemented using the UKF equations, thus giving rise to the following operational sequence; the notation is principally that of Gelb (reference [2]), although vectors are here written in index form rather than with a bold font:

1. Carry out Cholesky decomposition of the error covariance matrix P⁽⁺⁾from the prior time step, giving matrix L in lower-triangular form (so that P = LL^T).

2. Define the UKF weight factors w^µin the usual manner (see reference [7], for example).

With N state components, a total of2N + 1 weights and sigma points are required.

3. Define the sigma points in the usual manner, based on the track state ˆx_i(+)from the prior time and the columns of L.

4. Propagate each sigma point independently forward in time to the time stamp of the next measurement, via the above set of nonlinear differential equations (3). A Runge-Kutta fourth-order scheme proved adequate for this purpose, using time steps of 0.01 s. This gives rise to a set of propagated sigma points ζ^µ_i, for i ∈[1, N ]; the superscript index µ ∈[1, 2N + 1] defines the sigma point.

5. Create the mean extrapolated stateˆx_i(−):

ˆ x_i(−)=

2N +1

X

µ=1

w^µζ_i,

and the extrapolated covariance:

P₍₋₎=X

µ

w^µ



ζ^µ_i −ˆx_i(−)



ζ^µ_j−ˆx_{j (−)}

T

.

6. Define the Kalman gain matrix:

K= P(−)H^TS⁻¹,

where H is the measurement matrix, R is the measurement covariance and S = HP₍₋₎H^T + R. Matrix H is principally zero apart from the top-left 2 × 2 section, which is formed from the unit matrix.

7. Update the track state and covariance:

ˆ

x_i(+)= ˆx_i(−)+ K (z_i−Hxˆ_i(−)) , P(+)= (I − KH) P(−),

where I is the unit matrix and z_iis the measurement.

ABSTRACT

A simple simulation of a turning target in two dimensions is used to show that the standard treatment of process noise in the Unscented Kalman Filter (UKF) can give rise to a potentially fragile tracking filter.

In the example used, the turn rate is tracked as part of the state vector and additional state components are introduced to accommodate the process noise terms. This approach works well only while the underlying turn rate is constant or linear with time. With more complex turn rate dynamics, the filter breaks regardless of the process noise settings and such behaviour is indicative of a filter with zero process noise. It is found that a sequential Monte Carlo implementation of process noise gives rise to a much more robust tracking filter.

SZUM A BEZ ´SLADOWY FILTR KALMANA STRESZCZENIE

Na podstawie wyników prostej symulacji obracaj ˛acego si˛e w dwóch wymiarach celu pokazano, ˙ze standardowe potraktowanie problemu istnienia szumu w bez´sladowym filtrze Kalmana (UKF) mo˙ze prowadzi´c do uzyskania filtra o du˙zym stopniu wra˙zliwo´sci. W wykorzystanym przykładzie, ´sled- zona jest pr˛edko´s´c obrotowa, która wchodzi w skład wektora stanu, a przez rozszerzenie wektora stanu uzyskuje si˛e efekt uwzgl˛ednienia składowych pochodz ˛acych od szumu. Podej´scie takie sprawdza si˛e dobrze jednak wył ˛acznie dla stałej pr˛edko´sci obrotowej lub zmieniaj ˛acej si˛e liniowo wzgl˛edem czasu.

Przy bardziej skomplikowanej dynamice obrotów, filtr UKF przestaje spełnia´c swoj ˛a rol˛e, niezale˙znie od parametrów szumu, a wi˛ec jak w przypadku filtru dla braku szumów oddziałuj ˛acych na obiekt.

Pokazano, ˙ze implementacja filtru za pomoc ˛a sekwencyjnej metody Monte Carlo prowadzi do uzyska- nia filtru o wi˛ekszym stopniu odporno´sci.

Received: 2017-11-03 Accepted: 2017-12-05