Advances in aircraft-height estimation using distance-measuring equipment~

1 1 Abstract: The authors address the subject of aircraft geometric-height estimation, based on measurements from three distance-measuring equipment (DME) stations. A new improved height computation algorithm is presented, which is computationally efficient and mathematically tractable. The simple form of the new algorithm leads to easy derivation of analytic expressions for both the standard deviation and inherent systematic errors. Evaluation results are presented for representative conditions. The appropriate form of the Kalman filtering algorithm is proposed for producing bias-free and more accurate height estimates.

D.E. Manolakis A.I. Dounis

Indexing terms: Aircraft height monitoring, Altitude, Distance-measuring equipment

1 Introduction

The subject of accurate geometric-height estimation has drawn considerable research interest in the last fifteen years. The accurate height monitoring, performed inde- pendently from an on-board pressure altimeter, is required for establishing a reduced vertical separation minimum of lOOOft applied to aircraft cruising above 29000ft [l]. One of the candidate systems determines the height from three distance measurements derived via on-board DME equipment. A complicated explicit solution has been presented in [2]. However, the guide- lines of the approach followed to derive this formula have not been given. Also, the algorithm performance has been demonstrated for typical geometric arrange- ments, by presenting curves for the standard deviation of the errors in raw height measurements and smoothed estimates obtained by the usual Kalman fil- ter. However, the expected value of the height measure- ment errors has not been examined. We prove that biased height estimates are obtained, since the form of the Kalman filter proposed does not compensate for the biases inherently caused by the nonlinear measure- ment function. Another simpler closed-form solution to the problem of 3-D position estimation, based on three distance measurements, has been derived in [3], with 0 IEE, 1996 IEE Proceedings online no. 19960323 Paper first received 2nd March and in revised form 20th June 1995 D.E. Manolhs is with the Department of Automation, Technological Education Institute of Thessalonilu, 54101 Salonica, Greece A.J. Dounis is with the Hellenic Airforce Academy, Dekelia, Athens, Greece

respect to a specific coordinate frame. Namely, the sta- tion’s plane is the reference plane, the origin coincides with the location of one station and finally a baseline is used as the x-axis of the coordinate system. Thus, if the position is required in any other coordinate system, transformation must be applied to the position vector and the corresponding measurement-error covariance matrix. It is well knoTwn, however, that transformations require many trigonometric function computations. The performance of the proposed algorithm has not been analysed, but rather the method’s principles have been extended for thLe case of the Global Positioning System (GPS), where four pseudorange measurements are required to estimate the 3-D position and the clock bias.

The eixisting algorithms necessitate the calculation of a number of coefficients through complicated algebraic formu1a.e. The coefficients depend on both the station’s coordinates and the ]position of the aircraft relative to the three DME position locations. Consequently, due to aircraft movement, these coefficients are not con- stant, and must be repeatedly calculated each time a new set of distance measurements is obtained.

In this work, we develop a much simpler and effi- cient a1,gorithm based on two polynomial-type expres- sions. The coefficients of the polynomials are independent of the aircraft position, and are completely determined by the positions where the DMEs are located. Since the latter are fixed, the polynomial coef- ficients have to be calculated only once, and then they can be stored in the computer memory. Subsequently, whenever a new se:t of distance measurements is obtained, the height is computed by a simple algebraic formula, which combines the new distance measure- ments with the stored coefficients. The system-perform- ance analysis is not restricted to the variance analysis but also copes with -the expected value of the height- measurement errors. ’We prove that, owing to the non- linearity of the system and the noise contamination of the distance measurements, the height measurements are biased. The bias is generated by the computation algorithm itself and, hence, is called inherent bias. This systematic error appears in the output of nonlinear sys- tems processing noisy signals, even though the noise has zero mean value. Next, we derive closed form ana- lytic expressions for both the standard deviation and the inherent bias error of the system estimates. This enables the correct system modelling and the compen- sation fiar the inherent bias, via use of the appropriate form of Kalman filter.

IEE Proc-Radar, Sonar Navig., Vol. 143, No. 1, February 1996 41

2 performance analysis

Efficient height computation algorithm and

The geometry of the height-estimation system is depicted in Fig. 1, where the three DMEs, SI, S2 and S3, are separately located at points (xi, yi, z J , i = 1,2,3.

Fig. 1 System configuration

The station’s sites must not be collinear, since in that case the system would be singular. The aircraft distance Y , from the DME Si is derived on-board, via the DME interrogation equipment with which the majority of aeroplanes are equipped. The three DMEs are located under an airway, and the data are collected during the time the aircraft remains inside the data acquisition area of the system.

Let R = [rl, r2, r3IT be a brief array notation for the set of three distances, (x,y,z) be the aircraft actual posi- tion, and h be the height computed by the monitoring system. In the Appendix it is proved that h can be com- puted, whenever a measurement array R is obtained, according to the following formula

h = f ( r i , r z , r Q ) = f ( R ) = ( - L + L W ~ ’ ~ ) / ~ P (1) where L and A4 are the following polynomial-type expressions: L = ZO + Zlr: + Zzrz + ~3r32 (2)

( 3 )

M = moo + molr? f mozr; + m03r,” + mll‘r: + 731227‘24

+ m337-34 + m12rtr; + m13r2r: + m23r2r3 2 2

The analytic expressions for the coefficients p , I , and mii can be found in the Appendix. Notice that they are completely determined by the coordinates (x,, y,, ZJ of the DMEs. Consequently, since the latter are at fixed locations, the coefficients are calculated only once, at the moment the aircraft enters the data acquisition area, and in the sequel they can be stored for direct use at any subsequent moment a new height estimate is required.

Compared with the algorithms proposed in [2, 31, the new algorithm proves to be drastically improved. A form of [2] can be found in the Appendix. The compu- tation burden has been significantly reduced, since the height is simply calculated by raising the ranges to the second and fourth power, multiplying by the stored constant polynomial coefficients, and finally summing the terms and taking a square root. In contrast, both the existing algorithms necessitate many computations, which must repeatedly be performed each time a new set of range measurements is received, owing to the

48

dependencies of their polynomial coefficients on both the station’s and aircraft’s coordinates. Consequently, all the coefficients vary, according to aircraft position, and have to be recalculated.

The benefits gained from the new algorithm are not restricted to minimising the computational load. Another advantage offered by the structure of the algo- rithm is that the performance analysis is greatly simpli- fied. Notice that a thorough mathematical performance analysis was difficult to perform with the previous for- mulae. With the new algorithm it is easy to analyse and evaluate the effects caused by both the system geome- try and the distance-measurement errors. The first, and the most usual, characteristic to investigate is the standard deviation of the height-measurement errors. This subject is always addressed in the evaluation of any new measurement system, since the standard devia- tion expresses the accuracy of the system. Let 02rl, 02r2 and G~~~ denote the variances of the errors of the corre- sponding distance measurements. For uncorrelated ranging errors with small magnitudes relative to ranges, the variance 02h of the height-measurement error is given by the following expression [4]

(4) where the terms gi are partial derivatives of the nonlin- ear height-measurement function with respect to the corresponding ranges

i = 1 , 2 , 3 (5)

The specific expressions for gi are easy to derive due to the simple form of the function f . The performance analysis of measurement systems usually terminates at this stage, i.e. in obtaining an expression for the stand- ard deviation. In [2], for example, the subject of per- formance analysis and evaluation was restricted to assessing the standard deviation for various cases of flight patterns and station’s site arrangements.

However, the analysis must extend and include the mean value of height measurements, since they are expected to be biased, owing to the nonlinearity of the measurement equation and the randomness of distance- measurement errors. The existence of biases in the esti- mates obtained by other position estimation systems has been shown either by computer simulations [5 , 61 or analytically [7-IO]. Biased outputs are expected in systems that process noisy input data through nonlin- ear algorithms, even though the inputs are not biased. The difference between the mean and the actual value of the output induced by the nonlinear processing is called inherent bias. In the remainder of this Section we derive an analytical expression to evaluate the inherent systematic error of the present system.

Let rim denote the noisy measurement of the actual distance ri, i = 1,2,3. The noise in distance measure- ments is considered to be additive with zero mean value, which implies that

E[r,m] = Ti i = 1 , 2 , 3 (6)

where E(-) denotes the expected value. Let h, denote the height measurement based on noisy distance meas- urements

hm = f ( T l m , Tam, T Q m )

The height measurement function is smooth, with par- tial derivatives of all orders, consequently it can be

IEE Proc-Radar, Sonar Navig., Vol. 143, No. 1, February 1996

expanded in a convergent Taylor series around the point (r1, r2, r3)

where the partial derivatives are evaluated at the mean values (rl, r2, r3) and e denotes the higher-order terms. The higher-order terms have negligible magnitude for small ranging errors, thus they can be ignored. The dis- tance-measurement errors are considered as being uncorrelated, consequently

E[(rim - ri)(rjm - r j ) ] = o,",Sij i , j = 1 , 2 , 3 (8) where 6, stands for the Kronecker delta. Summing the expected values of the terms to the right of eqn. 7, and taking into account eqns. 1, 6 and 8, the expected value of height measurement ahrn] equals

The simple form of eqn. 1 makes it easy to produce expressions for the second order derivatives. Obviously, they are not equal to zero, consequently the expected value E[h,] of the computed geometric height differs from the actual height h, by an amount, b, referred to as the inherent bias, specifically

The inherent bias is a joint effect of both the system nonlinearity and the randomness of measurement errors. Notice that a linear system processing zero mean-value stochastic inputs yields zero mean-value outputs. Mathematically, this is supported by the fact that the second-order derivatives are zero. Similarly, unbiased outputs will be produced by a nonlinear sys- tem, if it is fed by ideal noiseless inputs.

3 Performance assessment

From eqns. 4 and 10, it results that the performance is affected by the ranging errors and by the system geom- etry which determines the magnitude of nonlinearity. The effect of ranging errors is quite obvious, since oh is proportional to ori whereas the inherent bias is propor- tional to 02rp To examine the general effect of the vari-

Fig. 2

ous geometric factors, the DMEs are considered as being located at the vertices of an equilateral triangle

Typical DME location topology and default flightpath

IEE Proc.-Radar, Sonar Navig., Vol. 143, No. I , February 1996

(see Fig. 2). A Cartesian coordinate frame is assumed, with its 'origin defined by the centre of gravity of the triangle and its x-y plane being identical to the triangle plane. The curves of Figs. 3-5 and 7-9 were derived by

c 12c'o[ \ \ \ s=4 km

o b c r z 0 10 20 30

distance x, km

Fig.3 Standard deviation if height computation errors for various baselines at an altitude, z = 6km, when the aircraft moves along the x-axis iY = 0)

E 8' .- c 0 > a, U

._

F 0 U 0 UI U

Fig.4

IY = 0) lines at

E C' ._ + 0 > a, U

.-

z U

c s m

0-' -30 -20 -10 0 10 20 30

distance x, km

Standard deviation of height computution errors for various base- an altitude, z = 9km, when the aircraft moves along the x-axis

8001 , s d k m

200

-30 -20 -10 0 10 20 30

distance x, km

Fig.5 Standard deviation of hei ht computation errors for various base- lines at an ultitude, z = 12km1, ,fen the aircraji moves along the x-axb (Y = 0)

considering various aiircraft positions along the x-axis line at three different altitudes (6, 9 and 12km), whereas the curves of Figs. 6 and 10 resulted from assuming positions along the y-axis line at a height equal to 9km. The side s of the triangle is referred to as the systein baseline. Five different baseline lengths were considered, namely s = 2, 4, 8, 16 and 32km. The rang- ing errors are assumed to be independent, identically distributed with or1 I= or2 = or3 = or. The graphs shown in Figs. 3-10 have been produced with or = 90m. It is evident from eqns. 4 and 10 that, for any dif- ferent value of or, the corresponding values of oh and b can be easily derived from the presented values via multiplication by the appropriate scaling factors. For

49

precision DME, for example, it holds that G~ = 20m,

ard deviation and 4/81 for the inherent bias.

s=32 km consequently, the scaling factors are 219 for the stand- 0 ~

-n\\-l s=2 km '=&km

s=Ekm s=16km

s=32 km

400

200

-30 -20 -10 0 10 20 30

distance y, km

Fig.6 lines when the aircyaft moves along the y-axis x = 0, z = 9km

Standard deviation of height computation errors for various base-

-160 -30 -20 -10 0 10 20 30

distance x, km

Fig.7 Inherent bias in height computations for various baselines at an altitude, z = 6km, when the aircraft moves along the x-axis (y = 0)

s=32km7 0

E -40

-80

.r -120

-160' I i , I -30 -20 -10 0 10 20 30

distance x , km

Fig.8 altitude, z = 9km, when the aircraft moves along the x-axis ( y = 0)

Inherent bias in height computations for various baselines at an

-160' ' -30 -20 -10 0 10 20 30

distance x, km

Fig.9 altitude, z = 12lcm, when the aircraji moves along the x-axis ( y = 0)

Inherent bias in height computalions for various baselines at an

distance y, km

Fig. 10 Inherent bias in height computations or various baselines the aircraft moves along the y-axis x = 0, z = 9 Lm

when

It is seen in Figs. 3-5 that oh increases rapidly as the aircraft moves away from the centre of the triangle. The increase is stronger the smaller the triangle is and the lower the aircraft flies. However, the baseline length is a more critical factor than the aircraft flight level. As far as the selection of the best baseline length, it is observed that it depends on the width of the area required for height monitoring. If a wide area is to be monitored, for example, larger than 40km (XE [-20, 20]km), then large beselines must be considered, e.g. s = 32km. However, in this case, further investigation needs to be conducted since large baselines are less favoured when the aircraft is close to the centre, and especially when it flies at low altitudes, as can be seen when comparing the central part of Figs. 3 and 4. It is concluded that for a reasonable width of 20 km (XE [- 10, 101km) the most favouring magnitude is s = 16km. Notice that for distances close to the centre and at low flight levels, the performance with this baseline is slightly worse than that achieved with smaller base- lines. Smaller baseline lengths can be preferable only when the length of the data acquisition area is restricted very close to the centre. For example, with s = 8km, best results are obtained when the height- acquisition area is approximately 6km (XE [-3, 3lkm). The above conclusions are also supported by the stand- ard deviation curves shown in Fig. 6 for the alternative flightpath. By comparing Fig. 6 to its respective Fig. 4, it is concluded that the flightpath direction has no essential effects on the height computation accu- racy.

Figs. 7-9 show that the absolute magnitude of the inherent bias increases when the aircraft distance from the centre becomes larger. The rate of increase is more rapid for smaller baselines and lower flight levels. As far as the effect of the flight level for the area close to the centre, we observe that the bias is almost insensitive to the flight level when large baselines are used, but when small baselines are used it becomes larger at higher flight levels. Actually, with the two largest base- lines, the bias remains constant for the central area, retaining an approximate value of l m at aircraft dis- tances in the range [-5, 5lkm. Fig. 10 shows the inher- ent bias when the aircraft flies along the y axis at an altitude of 9km. The symmetrical curves of this Figure support the above observations. The major conclusion derived from these Figures is that the inherent bias has a non-negligible magnitude, and must be taken into account and compensated for. The analytical expres- sion, derived in the previous Section, enables compen-

50 IEE Proc-Radar, Sonar Navig., Vol. 143, No. 1, February 1996

sation for the inherent Systematic error in any algorithm which utilises and further elaborates the height measurements.

4 Bias-free Kalman filter estimation

The usual processing of the primary position measure- ments is performed via a Kalman filter, in order to produce more accurate estimates. The filter is iterated on-board each time a new height measurement is obtained. The ordinary Kalman algorithm, however, which was proposed for this system in [2], will result in biased estimates, since it does not correctly model the measurement errors. This Section presents the correct form of the Kalman filter, to compensate for the sys- tematic errors and produce bias-free and smoothed height estimates. The system is described by the follow- ing state space model

X k + i = &XI. + B k W k

h?”k+l = H X k + l + %+l (11)

where, the first equation expresses the dynamic behav- iour of the system and the second equation relates the height measurement to the aircraft state. A constant- velocity vertical motion model is assumed, although any other model can be similarly used. Specifically:

T = the time elapsed from the previous height measure- ment, T = tk+l-tk zk,vk = the system states corresponding to the aircraft vertical position and velocity, respectively /t,k+l = the height measurement derived at time tktl wk = the process, or plant, noise expressing the random acceleration disturbances with E[wk] = 0, E[wkwJ] =

qk+l = the measurement error, which is assumed to be a gaussian and white stochastic process, with standard deviation (Thk+l calculated from eqn. 4. Notice that, according to the previous Section, the mean value of q cannot be assumed equal to zero. Its value, bk+, = E[T\k+l], can be analytically computed according to eqn. 10. The measurement and process noise are assumed to be uncorrelated, i.e. E[W&] = 0 for all ij.

The iteration of the appropriate Kalman filter is accomplished in the following steps:

(i) state prediction:

G:6kJ

? k + l / k = $ k k k / k

(ii) prediction-error covariance matrix:

(iii) Kalman gain matrix:

(iv) state-estimate update:

% + l / k + l ? k + l / k + Kk+l (hmh+~ - H x k + l / k - b k + l )

(v) estimation-error covariance matrix:

P k + l / k + l = [I - K k + l H I P k + l / k

Notice that the system modelling takes into account that the measurement error has a non-zero mean value. In the sequel, the Kalman filter compensates for it in

IEE Proc.-Radur, Sonar Nuvig., Vol. 143, No. 1, February 1996

the smoothing equation of step (iv) by subtracting it from the usual form of the innovation. Biased estimates are expected in the output of the smoothing algorithm if this fact is ignored.

5 Conlclusions

A new algorithm has been derived for computing the aircraft geometric height using DME. The structure of the algorithm makes it more efficient for computer implementation than the existing algorithms, and facili- tates the performance ;analysis. The analysis has proved that the height compui,ed on the basis of three distance measurements is biased, owing to the joint effect of both the system nonlinearity and the random ranging errors. Performance evaluation has been conducted for typical conditions, and has shown that the systematic error has a non-negligible magnitude. The perform- ance, in terms of standard deviation and mean value of height-measurement errors, strongly depends on both the systcm baseline and the horizontal distance from the station’s arrangement centre. The flight level has a minor influence. The ;analytical expression derived can be suitably exploited to compensate for the bias in algorithms, which further process the height measure- ments such as the Kalman filter.

6 Achnowledgment

The authors would lilke to thank Mr. Leonidas Asla- nidis of the TEI of Piraeus, Greece, for writing the computer code and performing the runs.

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References

COX, M.E., TEN, HAVE, J.M., and FORRESTER, D.A.: ‘European studies to investigate the feasibility of using l000ft ver- tical separation minima’, J. Nuvig., 1991, 44, (2), pp. 171-183 REKKAS, C.M., LEFAS, C.C., and KRIKELIS, N.J.: ‘Improv- ing the accuracy of aircraft absolute altitude estimation using DME measurements’, htt. J. Syst. Sci., 1990, 21, (7), pp. 1381- 1392 FANG, B.T.: ‘Trilateration and extension to global positioning system navigation’, J. Guid. Control Dyn., 1986, 9, (6), pp. 715- 717 PAPOlULIS, A.: ‘Probability, random variables and stochastic processes’ (McGraw-Hill, 1984, 2nd ed.) NAGAOKA, S.: ‘ Possibility of detecting a non-level-flight air- craft by the navigation accuracy measurement system (NAMS)’. ZCAO RGCSP-WPI180, 7th meeting, Montreal, 1990 MANOLAKIS, D.E., and LEFAS, C.C.: ‘Aircraft geometric height computation using secondary surveillance range differ- ences’, ZEE Proc. - Radar, Sonar Nuvig., 1994, 141, (2), pp. 119- 124 MANOLAKIS, D.E., LEFAS, C.C., STAVRAKAKIS, G.S., and REKKAS, C.M.: ‘(Computation of aircraft geometric height under radar surveillance’, ZEEE Trans., 1992, AES-28, (l), pp. 241-24-8 MANOLAKIS, D.E., and LEFAS, C.C.: ‘Systematic errors in ground referenced geometric height monitoring’, ZEE Proc. F, 1993, 1140, (2), pp. 138-144 HO, KX., and CHAN, Y.T.: ‘Solution and performance analysis of geolocation by TDOA’, ZEEE Trans., 1993, AES-29, (4), pp. 13 1 1-1 322 MANOLAKIS, D.E., LEFAS, C.C., and DOUNIS, AI. : ‘Inher- ent bias assessment in height computation employing mixed-type radar (data’, IEEE Tran.s., 1994, AES-30, (4), pp. 1045-1049

Appendix: Vertical position estimation

The square of distance rl is expressed as T: = (z - x2)2 -I- (y - yz)’ + (X - z2)2

and after some manipulation it becomes x2 + y2 + z2 - 22:c, - 2yy, - 222, - g: = 0

2 = 1 , 2 , 3 (13)

(14) a = 1 , 2 , 3

(J2 1 r2 - U2 z 2. z 7 UP =zP+Yz + z , 2 2

SI

Subtracting the equations with i = 2, and i = 3 from the equation at i = 1, yields

Hp = g-6.25 (15)

D , 1x2 - 51 Dy = ~2 - yl D, ~2 - 21

d, = 2 3 - ~1 dv = y3 - 1 ~ 1 d, = 2 3 - ZI

If the station’s baselines are not collinear, then matrix H will not be singular, consequently the horizontal position vector is expressed in terms of z

Substituting this expression for x and y in eqn. 14 at i = 1, the following quadratic equation is obtained

p = H - l ( g - 6z) (16)

(17) pz2 + q + r = o p = 44 + 4a;(D: + D i + D:)

q = 4alb - 44 (-2y1 D , D , + 221 DE + (91 - 9,”) D ,

r = b2 - a;c

- 2x10,0, + 2z1D;)

Thus the vertical position can be computed as

The two roots are mirror symmetric to the station’s plane, consequently, we accept the positive sign in the rest of this paper. One can prove that the expressions for the quadratic coefficients are identical to that pre- sented in [2]. The expressions for q and r have compli- cated forms and depend on both the aircraft distances and the station’s coordinates. This implies that they must be continuously evaluated each time a new set of distance measurements is received. Notice that the

range measurements are used only in the expressions of g, defined in eqn. 14. Thus, significant computational advantages will be achieved by manipulating further eqn. 18, and expressing the height computation for- mula according to

h = (-L + M 1 J 2 ) / 2 p L = 10 + Zlrf + LZr ,2 + /3r,”

M = m oo + molrf + moar,2 + mo3.i + ml l r? + maar;

(19)

f m33.,” + mlZT,”T,” + m13$T,” + m 2 3 T ; T i

The coefficients 1, and my are determined by the sta- tion’s coordinates, exclusively, hence they may be eval- uated only once. First, define the following quantities:

SI = D; + D i

s3 = ZlD, + YlD,

3 2 = D,d, + Dydy

s 4 = 21D, - YlD, and then evaluate 1, and my through the following for- mulae: lo = 4al(sl(sl(u; - U ! ) - S Z ( U , 2 - U ; ) ) + 2s4a2)

- a;(40,(~; - U ? ) + ~ ( Z I S I - DZs3))

11 = 4 U l S l ( S l - s2) - 444DZ

Z2 = 4alsls2 + 4a$D, Z3 = -4nls1 2

52 IEE ProccRadar, Sonar Navig., Vol. 143, No. 1, February 1996