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Using the Kalman Filter to Estimate the State of aManeuvering Aircraft

K. Meier and A. Desai

Abstract—Using sensors that only measure the bearing angleand range of an aircraft, a Kalman filter is implemented to trackthe range, range rate, bearing, and bearing rate of a maneuveringaircraft with unknown varying accelerations. Simulations willdemonstrate the tracking performance of the Kalman filter withsingle and multiple prediction steps between the measurementstep. Then, a Kalman filter will be implemented with assumedcorrelated measurement and process noise.

Keywords: Kalman filter, Correlated noise, Estimation, Track-ing

I. INTRODUCTION

In applications such as target tracking, channel tracking incommunications etc. It is required to measure and estimateunknown quantities. Within the significant toolbox of mathe-matical tools that can be used for stochastic estimation fromnoisy sensor measurements, one of the most often used tools isthe Kalman filter. Typically, physical systems are described bya system model equation and a measurement equation. Whenthe physical system equations are both linear, the Kalmanfilter can be used to estimate the state vector recursively.When the system equations are nonlinear, the physical systemcan be approximated by an extended Kalman filter (EKF).This paper presents a solution to a linear system using aKalman filter. The physical system is an aircraft and the sensorused to measure its state is a radar. The remainder of thispaper proceeds as follows: section 2 describes the Kalmanfilter when the measurement and process noise are correlated.Section 3 implements a Kalman filter to estimate the state of anaircraft when the received sensor measurements are range andbearing. Section 4 presents the simulated data from section 3.The simulated data demonstrates the accuracy of the Kalmanfilter in tracking performance. To enhance filter performance,extra prediction steps were run between measurement updates.These results given are given in section 4.

II. KALMAN FILTER AND COVARIANCE MATRICES

The Kalman filter is a set of mathematical equations thatprovides an efficient computational (recursive) means to esti-mate the state of a process in a way that minimizes the mean ofthe squared error[7]. For a derivation of the Kalman filter see[1]. The discrete time Kalman filter equations are as follows,

System dynamic model:

xk = Φk−1xk−1 + wk−1 (1)

Measurement model:

zk = Hkxk + vk (2)

State estimation:

xk(−) = Φk−1xk−1(+) (3)

Error covariance (a priori):

Pk(−) = Φk−1Pk−1(+)ΦTk−1 +Qk−1 (4)

Kalman Gain:

Kk = Pk(−)HTk [HkPk(−)HT

k +Rk]−1 (5)

Error covariance update (a posteriori):

Pk(+) = [I −KkHk]Pk(−) (6)

State estimate update:

xk(+) = xk(−) +Kk[zk −Hkxk(−)] (7)

Where xk denotes the state vector at time k, and zkdenotes the corresponding measurement. Φk represents thestate transition matrix and Hk represents the measurementmatrix. wk is the Gaussian zero-mean white process noiseandvk is the Gaussian zero-mean white measurement noise.The covariance of w(k) and v(k) is given by,

E[w(k)w(j)T ] = Qkδkj

E[v(k)v(j)T ] = Rkδkj

Where δ denotes the Kronecker-delta function. Figure 1shows each state before (a priori state estimate) and after (aposteriori state estimate) a measurement is received. From thecumulative knowledge of each past a priori and a posterioristate estimates, a Kalman filter can be implemented to estimatethe future states.

Figure 1. Each state with knowledge of measurement and estimation.

Figure 2 shows a typical time sequence of values assumedby the ith component of the estimated state vector (plottedwith solid circles) and its corresponding variance of estimationuncertainty (plotted with open circles). The arrows show thesuccessive value assumed by the variables, with the annotation

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(in parentheses) on the arrow indicating which input variablesdefine the indicated transitions. Note that each variable as-sumes two distinct values at each discrete time: it’s a priorivalue corresponding to the value before the information in themeasurement is used, and the a posteriori value correspondingto the value after information is used.[1]

Figure 2. General diagram to show how each step is calculated

Thus in general, a recursive algorithm is formed. If you havexk(−) you can calculate the a priori error covariance matrixbefore the sensor update. With the a priori error covariancematrix, the Kalman gain can be computed. from the Kalmangain, the a posteriori error covariance matrix, (Pk(+)), can becalculated. Once Pk(+) is calculated it is easy to calculate aposteriori estimate of xk, and repeat the process (see figure3).

Figure 3. The Kalman filter recursive equation

The previous discussion only involved uncorrelated processand measurement noise. If the process and measurement noiseare correlated the Kalman filter equations are modified asfollows. Let the correlation between the measurement noiseand process noise be given by,

E[w(k)v(j)T ] = Ckδkj (8)

Then the discrete-time estimate equations have the sameinitial conditions and state estimate extrapolation and error

covariance extrapolation equations. However, the measurementequations are modified as,

Kk = [Pk(−)HTk +Ck][HkPk(−)HT

k +Rk+HkCk+CTk HT ]−1

Pk(+) = [Pk(−) −Kk[HkPk(−) + CTk ]

xk(+) = xk(−) +Kk[zk −Hkxk(−)]

III. BEARING AND RANGE

Radars are used to track and control aircraft. As a radarrotates, it continuously sends out pulses of electromagneticradiation. Pulses intercepted by objects are reflected back andintercepted by the radar. The time delay from when the pulseis transmitted to when the pulse is received by the radar isused to calculate the range from the radar to the aircraft. Tocalculate the bearing angle, a straight line is drawn from theaircraft to the radar. The angle between the straight line andtrue north is the bearing angle. Figure 4 shows the range andbearing angle.

Figure 4. The range and bearing angle of an aircraft

For control purposes, it is desired to track the range andbearing angle of an object as it navigates. In tracking, itis desired to have an improved sensor measurement of thebearing angle and range at regular sample intervals and tohave a predicted estimate of the objects navigation route.Furthermore, if it assumed the only measurements receivedfrom the radar are range and bearing information, it is desiredto accurately reconstruct the range rate and bearing rate even ifthe aircraft acceleration is unknown. By assuming reasonableflight characteristics, the appropriate Kalman filter can beconstructed to handle these conditions. Let r(k) denote therange, r(k) denote the range rate, u1(k) denote the rangeacceleration, θ(k) denote the bearing angle, θ(k) denote thebearing angle rate, u2(k) denote the bearing acceleration, andw1(k), and w2(k) denote the zero-mean white process noisewith variance σ2

w at time sample k. Then, using a generalmodel of the singer equations[6], The state equation for timek + 1 can be written as,

x(k + 1) = Φx(k) + w(k)

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where

x(k) =

r(k)r(k)u1(k)θ(k)

θ(k)u2(k)

w(k) =

00

w1(k)00

w2(2)

Φ =

1 T 0 0 0 00 1 1 0 0 00 0 ρ 0 0 00 0 0 1 T 00 0 0 0 1 10 0 0 0 0 ρ

The variable ρ is defined in [2]. It a value to account for the

correlated acceleration between measurements and is given by,

ρ =

{1 − λT

0

T ≤ 1/λ

T ≥ 1/λ

where λ is the inverse of the average maneuver durationand T is the time between radar measurements.

To calculate the sensor measurements, let z1(k) denote therange sensor measurement, z2(k) denote the bearing measure-ment, and v1(k) and v2(k) denote the zero-mean white sensornoise measurements with variance σ2

r and σ2θ respectively at

time k, then the sensor measurements are given by,

z(k) = Hx(k) + v(k)

where

z(k) =

[z1(k)z2(k)

]H =

[1 0 0 0 0 00 0 0 1 0 0

]v(k) =

[v1(k)v2(k)

]By assuming the sensor noise is uncorrelated, The noise

covariance matrix, Rk, is given by,

Rk = E[v(k)v(k)T ] =

[σ2r 0

0 σ2θ

]By assuming the process noise is uncorrelated, The process-

noise covariance matrix is given by,

Uk = E[w(k)w(k)T ] =

0 0 0 0 0 00 0 0 0 0 00 0 σ2

1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 σ2

2

The derivation used to calculate σ2

1 and σ22 is given in [2].

Using that model, the acceleration of the aircraft is assumedto be a uniformly distributed Random Variable with maximumacceleration, A, and variance, σ2

u, given by,

σ2u =

A2

3(1 + 4P1 + P2)

where P1is the probability it will accelerate at the maximumacceleration, and P2 is the probability it will undergo zeroacceleration.

From σ2u, the values of σ2

1 and σ22 are given by,

σ21 = σ2

uT2 = A2T 2

3 (1 + 4P1 − P2)

σ22 = σ2

uT 2

R2 = A2T 2

3R2 (1 + 4P1 − P2)

IV. SIMULATIONS

The figures below demonstrate the tracking performanceof the Kalman filter. To produce these results, the followingvalues were assumed: A = 2.1, P1 = 0, P2 = 0, T = 3,R = 1.6e5,σr = 1000, σθ = 1, and 1

λ = 30. From these values,σ21 = 13.23, σ2

2 = 5.17e − 10, and ρ = .9. The left side offigure 4 and the left side of figure 5 show the error covarianceof the range and bearing before and after a measurement isreceived. The right half of figure 4 shows the Kalman gainfor the range and bearing. The remaining figures demonstratethe ability of the Kalman filter to track a maneuvering aircraftwith unknown varying accelerations. The figures display thetrue, estimated, and measured range and bearing of the aircraft;the range and bearing error of the aircraft; and, the true andestimated bearing and range rate.

Figure 5. The error covariance of the range and the Kalman gain for theBearing and Range as a function of time

Figure 6. The error covariance of the bearing and the true, measured, andestimated bearing

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Figure 7. The range error and bearing error of the Kalman filter estimates

Figure 8. The true bearing rate and estimated bearing rate and the true rangerate and estimated range rate of the Kalman filter

Figure 9. The measured, estimated, and true range of the aircraft

In performing these simulations, it was assumed that oneprediction of the aircraft state was given for each receivedmeasurement. To improve the filter performance, a simulationwas run that predicted the state of the aircraft multiple timesbetween sensor measurements. The only noticeable improve-ment from this change was in the range rate and the range asseen in figure 10 and 11.

Figure 10. The range rate of the aircraft using single and multiple predictionestimates between sensor measurements. The figure on the left is the singleprediction estimate.

Figure 11. The Range Error of the aircraft using single and multipleprediction estimates between sensor measurements. The figure on the leftis the single prediction estimate

The final simulation demonstrates when the process noiseand measurement noise are Gaussian-correlated. The resultsare given in figure 12.

Figure 12. The range error and range rate using a Kalman filter withcorrelated process and measurement noise

V. CONCLUSION

For a linear system, the Kalman filter is the best estimatorof a state in the presence of zero-mean white Gaussian processand measurement noise. From the data, it is seen that theKalman filter closely tracks the true state of the aircraft in thepresence of unknown accelerations. If the filter is modified sothat multiple prediction steps between measurements are made,then the Kalman filter performance is improved. Finally, ifthere are correlations between process noise and measurementnoise, the Kalman filter becomes suboptimal, but it still tracksthe aircraft with excellent accuracy.

REFERENCES

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[3] S. Julier, J. Uhlmann, H.F. Durrant-Whyte. “A new method forthe nonlinear transformation of means and covariance in filtersand estimators’’. IEEE Trans on Automatic Control, vol.45, pp.477-482, March 2000.

[4] M.S. Arulampalam, Maskell, S.N. Gordon. and T. Clapp, “Atutorial on particle filters for online nonlinear/non-GuassianBayesian tracking”. IEEE Trans on Signal Processing,vol. 50, pp.174-188, Feb 2002.

[5] O. Cappe, S.J. Godsill, and E.Moulines. “An overview ofexisting methods and recent advances in sequential MonteCarlo”. Proceedings of the IEEE, vol. 95, pp. 899-924,May 2007.

[6] R. A. Singer. “Estimating Optimal Tracking Filter Perfor-mance for Manned Maneuvering Targets”. IEEE Trans onaerospace and electronic systems, vol. 6, pp. 473-483, July1970.

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[7] G. Welch, G. Bishop. “An Introduc-tion to the Kalman Filter.” Internet:http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf[11/29/2011].