" ' - i ^ - V l 4 , ^ l . . i
CRANFIELD
INSTITUTE OF TECHNOLOGY
S Y N T H E S I S O F T H E A I R C R A F T N A V I G A T I O N A L A C R O S S - T R A C K E R R O R by R. N . L O R D A N D D . A . O V E R T O NCRANFIELD INSTITUTE OF TECHNOLOGY
SYNTHESIS OF THE AIRCRAFT NAVIGATIONAL ACROSS-TRACK ERROR
by
-R.N. Lord, M.A., M.I.E.E., M.I.E.R.E., A.F.R.Ae.S.
and
D.A. Overton, M.Sc.
Electrical and Control Engineering Department
S U M M A R Y
Aircraft across-track errors are synthesised by means of a mathematical model using statistical parameters. A convolution program is described in which the sensitivity of the overall error to variation in accuracy of
navigational fix and heading can be determined. Numerical examples relating to present and future North Atlantic navaids are presented for both subsonic and supersonic aircraft, particular attention being paid to the Gaussian/negative exponential error distribution controversy.
Results show that the importance of a good fixing aid diminishes as airspeed increases when the heading becomes of prime importance. The value of synthesis as a design tool for ATC as against the uncertainties of observational methods is demonstrated. Future extensions of synthesis to along-track and inhomogeneous traffic samples is discussed.
INTRODUCTION 1.1 1.2 1.3 1.4 Background
Reason for forming the overall navigational error distribution
1.2.1 Necessity for prediction 1.2.2 Lack of adequate reference
1.2.3 Overall error distribution by observation Forming the overall error distribution by synthesis Conclusions
1.4.1 On synthesis as opposed to analysis of observation NAVIGATIONAL ACCURACY
2.1 Representation of fixing accuracy 2.2 Heading accuracy
2.3 Meteorological error 2.4 Types of error
NORTH ATLANTIC NAVAIDS 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Loran 'A' Loran 'C' Dectra
Astro or celestial nevigation Compass types and accuracy Other forms of heading reference Doppler sensor and computer THE NORTH ATLANTIC ENVIRONMENT 4.1 Trend in wind direction 4.2 Errors in wind forecasting THE ACROSS-TRACK MODEL
5.1 The error due to fixing inaccuracy 5.2 The heading error
5.3 The meteorological or wind error 5.4 Variation of function shapes 5.5 The model structure
COMPUTER PROGRAM CONSTRUCTION
6.1 Estimation of truncation points 6.2 Preliminary studies
6.3 Use of the aircraft models DISCUSSION OF RESULTS
7.1 The subsonic aircraft 7.2 The supersonic aircraft
2 2 2 2 3 4 4 4 4 5 6 6 6 7 7 8 8 9 9 10 10 10 11 11 11 11 11 12 12 12 13 13 14 14 15
16 16 16 17 17 18 18 18 Al
8.1 On the methods employed 8.2 On the results
8.3 Applications FUTURE WORK
9.1 Numerical inputs
9.2 Inhomogeneous samples of aircraft ACKNOWLEDGEMENT S
REFERENCES
Appendix: Determination of functions in terms of their standard deviations.
TABLES
Standard deviations of across-track wind error. Aircraft data.
The 'standard' aircraft used in Figs. 4 to 13. FIGURES
The across-track model.
Across-track errors in forecast winds.
Comparison of Gaussian and exponenential distributions.
The 'all-Gaussian' and 'all-exponential' subsonic aircraft with alternative wind distributions.
The exponential fix, Gaussian heading and wind subsonic model. The exponential fix, Gaussian heading and wind subsonic model showing effect of fixing accuracy.
The effect of adding Doppler to the subsonic aircraft,
The effect of fixing rate on the exponential fix, Gaussian heading and wind subsonic aircraft.
The 'all-Gaussian' Mach 2 aircraft and the effect of the exponential fix.
The effect of heading error distribution on the Mach 2 aircraft. The effect of heading error distribution on the Mach 3 aircraft. The 'all-Gaussian' Mach 3 aircraft and the effect of the exponential fix.
Across-track errors of the three 'standard' aircraft with exponential fixing, Gaussian heading and wind errors.
N O T A T I O N e o v e r a l l e r r o r
e
e w i n d e r r o r V e fixing e r r o r y e heading error 6ft foot linear measure k a Durst parameter nib m i l l i b a r p r e s s u r e nm nautical mile
o.e.d. overall error distribution
r(f) vector correlation coefficient of forecast wind rms root mean square
s.d. standard deviation
t elapsed time after a fix V airspeed
ATC air traffic control CRT cathode ray tube CW continuous wave
HF high frequency (3-30 M H E ) LF low frequency (30-300 kHz) MF' medium frequency (300 kHz-3 MHz) M moment generating function SST supersonic transport aircraft
UHF ultra high frequency (300 MHz - 3 GHz) VHF very high frequency (30-300 MHz)
]i m e a n v a l u e V» m i c r o s e c o n d (1 x 10 s e c o n d ) a s t a n d a r d d e v i a t i o n o v a r i a n c e o standard v e c t o r d e v i a t i o n o f t h e w i n d . w
1. INTRODUCTION
This work provides a second step follov/ing the earlier published work by Lord (Refs. 1,2 & 3) towards a more efficient optimisation of airspace in the North Atlantic region. It could lead to a full synthesis of an aircraft's navigational capability before it even flies. The whole North Atlantic region may then be the subject of a fast-time simulation to achieve the optimum in Air Traffic Control for any particular traffic period.
1.1 Background
The same model for across-track errors (Fig. 1) is used by the present authors as they consider it sufficient for the purpose of this second stage of the project.
In the earlier work Gaussian distribution functions were assumed for the components of the model as this made the early calculations simpler. Because of the reasons explained later the Gaussian distribution is now viewed with some suspicion, and so the authors have based the present model on the option of using alternative error distributions. If mixed distributions are employed however, numerical methods are required to convolute the functions and produce the overall error distribution.
Although the concept of separation standards is mentioned, it is not relevant to this work which is solely concerned with the distribution of the across-track error, and any proposal for changes in the existing arrangement arising out of this report is best left in the hands of those more experienced of the art of international negotiation.
1.2 Reason for forming the overall navigational error distribution 1.2.1 Necessity for prediction
Aircraft need to be separated in space because of their inability to follow a predetermined path accurately, the degree of separation depending on the level of collision risk which may be tolerated. To enable this separation to be achieved at all times, the future position of aircraft must be predicted.
Consequently the overall error distribution (o.e.d.) must be ascertained to indicate the probable future position of the aircraft and the likelihood of any particular deviation occuring.
1.2.2 Lack of adequate reference
When the route lies over densely populated areas the errors in position are small because of the many external radio references. Many of these transmit in the UHF or VHF band and at these frequencies the transmission path is almost rectilinear. Consequently their range is limited by the horizon (and the altitude of the aircraft), although trans-horizon propagation is possible under certain weather conditions, and the aircraft track is subject to more constraints when it lies within range of these beacons and fan markers, the transmissions of which are not liable to considerable atmospheric variations because of the short distances involved.
Additionally when flying over populated areas, ground radar usually provides a further check on the position of the aircraft.
During long flights over uninhabited areas, however, the aircraft cannot be detected by an external radar and therefore the aircraft must report by radio its estimated position to ATC.
However, this long distance radio communication and navigation uses transmission in the LF, HF and MF bands and because of ionospheric variation the integrity of the system becomes suspect. Thus over long transoceanic flights the accuracy of prediction is reduced and the overall error distribution becomes wider. The accuracy of long range fixing aids is discussed later.
1.2.3 Overall error distribution by observation
Hitherto the distribution of overall errors has been based on observed data and considerable effort has been expended in studying the behaviour of aircraft during long transoceanic flights. Examples of this
are:-i) Operation Accordian.
ii) Radar observation (post-Accordian) from Gander, Goose and Killiard. iii) M.O.A./B.O.A.C. Doppler/Loran A trials
iv) North Atlantic systems Planning Group Data Gathering Exercises.
The theory of observational errors is based on a large number of measurements resulting in divergent values for the same quantity. The differences of
individual data from the average constituting a measure of the accuracy of the system under inspection.
It is usual for the observer to fit a standard distribution (a known shape) to the acquired data for ease of manipulation. The Gaussian distribution was often employed because it represents the behaviour of many random events in a manner which often occurs in practice. It has been observed, however, that large deviations occur more often than Gaussian theory would predict, i.e., the tails of the distribution obtained in practice are considerably higher than the tails of the Gaussian distribution (factors of 100 or more have been quoted).
It is important to appreciate that the tails (or skirts) of the error distribution are of prime interest to the ATC organisation as it is on the frequency of occurance of large errors that the safety separation standards are determined.
There is very little observational data available which describes the shape of the tails (or skirts) of the o.e.d. and many hypotheses have been introduced on the basis of very few events.
References 4, 5, 6, & 7 are taken as examples from the Journal of the Institute of Navigation and are concerned with various functions designed to have a shape similar to the Gaussian near the mean but with higher tails.
References 8, 9 & 10 indicate how difficult the problem can become. Acquisition of a pure homogeneous sample is impossible as each aircraft is an entity and not a member of a large population. Different procedures may be undertaken by different airlines, when taking a fix, and each will apply different up-dating rates.
1.3 Forming the overall error distribution by synthesis
The proposed alternative method of forming the o.e.d, is by synthesis and it is with this method that this report is primarily concerned,
A model is constructed having components representing the practical situation. If the model is sufficiently accurate then there is no reason why prediction on the basis of this model should not be as accurate as observation; even more accurate at the tails.
This method is well known in the field of reliability and provides the basis of equipment design. The power of this method does not lie so much in its
integrity (which must rely to a certain extent on observation) but its versatility. Each facet of the model may be examined independently and changed if required so that the effect of this change on the overall picture may be seen.
It should be noted that in the numerical examples the same standard
deviations are used for both Gaussian and exponential distributions, but if these standard hypothetical distributions were fitted to similar sets of observed data the standard deviation would be different. More emphasis is placed on the methods involved rather than on the absolute accuracy of the numbers used. But any numbers chosen are considered representative of operational practice. 1.4 Conclusions
1.4.1 On synthesis as opposed to analysis of observations
The recent work at Cranfield Institute of Technology has shown the versatility of this method of synthesis. To obtain observational results in a quantity
sufficient to establish the shapes of the overall error distributions considered, would have required an enormous amount of flying. The cost would have been prohibitive.
The method of synthesis which is described could provide the data for a fast-time simulation of the North Atlantic region prior to heavy traffic periods. 2. NAVIGATIONAL ACCURACY
The across-track error of an aircraft can be considered to be due to three distinct error components. The across-track component of the initial position fixing error with time causing a further change in the deviation from the
desired track as will also any error in the across-track component of the forecast wind.
2.1 Representation of fixing accuracy
To obtain a fix a navigator selects two (or more) lines of position. Each of these lines will be subject to errors, the distributions of which will most probably have different standard deviations. At the point of intersection of the lines of position the errors will form a 'probability heap' and become three dimensional. If both distributions are considered to be Gaussian the base of the 'heap' will form an ellipse. The size of an ellipse within which 95% of all observations lie is called the 95% ellipse.
When a radio fixing system with hyperbolic lines of position is employed, the ratio of major to minor axes becomes a function of bearing, i.e. the angle of intersection of the hyperbolae. This is fully explained in mathematical
Thus the shape of the ellipse changes with position and consequently this is a difficult contour to handle. However, if a circle is used with its centre lying on the intersection of the two hyperbolic lines the problem becomes easier. The circle represents the root mean square error and the radius becomes the standard deviation of vector error. The coordinate system in which the error is measured has been changed from cartesian to polar. The r.m.s. is taken of the sum of the squares of the diagonals of a parallelogram formed about the true position. In Ref. 12, G. Ulbricht et al. consider this problem in greater detail than is required here.
It is mentioned, however, by Lord and Saunders (1) that the choice of a circular contour may lead to inefficient use of airspace as well as difficult statistical mathematics. The concept of aircraft confined to blocks into which another aircraft must not proceed, makes separation control much easier. This system is inherently orthogonal and for this purpose the circular fix contour is referred to the wrong coordinate system and therefore, a rectangular contour is preferred.
One is then presented with two uni-dimensional problems. As regards computation this report is primarily concerned with the distribution of across-track errors and therefore, only this dimension is here considered.
2.2 Heading accuracy,
It can be seen that the model (Fig. 1) assumes that the aircraft heading error is not affected by wind. The effect of meteorological conditions being considered separately.
The heading error is a function of aircraft velocity and fixing rate and also the inherent random error in the instrument and in reading it. The compass error distribution is naturally truncated at + 180 degrees and - 180 degrees. Another truncation parameter will be provided by instrument inspection. If the maximum compass error is outisde that laid down in the design (or airline) specification the instrument will be rejected. The navigator also provides a truncated distribution when he reads the instrument, the upper and lower limits being successive graduations. If an observation is made which is outside
these limits it becomes a navigational blunder.
When the compass forms part of an integrated system such as with a Doppler sensor and computer, the navigator error is removed. The standard deviation of compass error can then be reduced. There will, however, still be errors in
the angle between the aircraft datum and the heading and these may be of comparable size to that of navigator observation.
A 'Doppler sensor only' will indicate errors in wind prediction and because of this the authors have used reduced figures for meteorological errors, when compared to those calculated from Dursts' formula (13).
The two cases:
(a) Doppler sensor and computer (b) Doppler sensor only,
have been grouped into one, but if it can be shown that (a) has a smaller standard deviation than (b) then alternative figures may be used in the computer programs.
It is also assumed that the time between fixes is not subject to error and consequently is not distributed about its nominal value.
2.3 Meteorological error
Although there will be inherent correlation between aircraft groundspeed and across-track wind error (i.e. correlation between the two wind components) complete independence is assumed as this will not detract from the purpose of the model. Actual meteorological forecast error has been examined by Durst and equations published by him are used as a basis for the computer input data as regards wind errors.
With the advent of the SST meteorological forecast errors, especially with regard to temperature, become increasingly important because of airborne
limitations. A Working Group was formed in 1965 to examine this problem and the results relevant to this work are published in references 14 and 15, although the latter is more concerned with effects beyond the scope of this present paper. 2.4 Types of error
The composite error distribution of any equipment may be broken down into three
components:-(a) Systematic errors which are represented by a deviating of the mean of the error distribution. These errors generally consist
of:-(i) equipment errors which may be dependent on angle, time and frequency, (ii) errors depending upon inertia presented to the instrument during flight, (iii) polarization errors,
(iv) propagation errors (e.g. dual path transmission effect), and
(v) ionospheric errors which may be grouped into (iv) but may be considered to change with diurnal variation.
(b) Random errors which cannot be corrected and are unexpected. These errors fluctuate about the systematic error, about which it is assumed they are symmetrical.
(c) Faults in the equipment or in the reading of it which show themselves in the form of a step in the error distribution. Those are known as blunders and will be discussed in a later report.
It is assumed throughout this work that a systematic error would be detected by the navigator and suitable compensation made. Thus the mean of all the standard
distributions is zero i.e. the largest probability of position lies on the track. 3. NORTH ATLANTIC NAVAIDS
In order to get a better understanding of the problem of navigating transport aircraft across the North Atlantic it is necessary to appreciate both present and proposed navaids in that region.
3.1 Loran 'A'
This is a pulse hyperbolic fixing system which is very widely used where long range navigation aids are required. Loran is a medium frequency (2 MHz) system, the frequency being chosen to take advantage of the E layer, since this ionospheric layer is relatively stable with respect to height.
slaves which comprise a chain. Each master and slave pair transmit on a separate pulse repetition frequency so that the timebase may perform selective action between different signals when pulses are displayed on a cathode ray tube in the aircraft. Changing the sweep speed enables the time difference between the master and slave transmissions to be measured.
Slow, medium and fast sweep speeds are used. Calibration pips are provided so that on the fast sweep 10 ys intervals can be measured and on the medium and slow sweeps 50 Us and 100 ys respectively may be gauged,
From the time difference the line of position may be identified as a lattice line on a Loran chart.
The appearance of the blips on the CRT will change between night and day, as the reception of pulses reflected from the E layer grows stronger as night falls. The main pulses used are from the ground wave and the first hop E wave but other pulses from various layers and also 2nd hop E waves may appear.
The accuracy of Loran is determined by the geometry of the system, the position of the observer, and the accuracy of time measurement. The latter depends upon the synchronisation of the ground stations and also on atmospheric propagation and this can be assumed to be 1 - 5 ys standard deviation (12) when the ground wave is used. If the sky wave is used, the standard deviation is much larger and depends upon the time of day, season etc. Discussion with British Overseas Airways Corporation indicated that 7 ys would be a realistic
figure for the overall standard deviation.
In this work 7.5 nm. s.d. is assumed for a good coverage area and 15 nm taken for a bad coverage area. This will of course differ with bearing and range from the base line, but the authors consider that the above figures are representative of the two situations. In some regions differentiation between ground and sky wave reception is difficult and becomes largely a function of experience.
The average range (coverage) of the ground wave assuming 100 kW peak power and low interference fields (12) amounts to 550 nm across the sea although the coverage at night will be greater.
3.2 Loran C
This equipment is still to be evaluated for area coverage, engineering and pilot compatibility. It is a LF form of standard Loran and has been described as halfway between a pulse and a CW system.
A continuous wave is transmitted within the pulse shape so that signals may be matched not only on the pulse but also on the CW inside it. Automatic counters can be used because of the long time between pulses and also the time for a pulse to build up (up to 80 ys). As a result of this the navigational error is
considerably reduced and in this work a s.d. of 3 nm is used.
This equipment may be used to update the Inertial Navigation system provided in the S.S.T.
3.3 Dectra
This is a long range form of Decca. It operates in the LF band and is a CW system providing hyperbolic lines of position (lanes). The accuracy is similar to that of Loran C but over a longer range (up to 1500 nm). A full description of flight trials undertaken in 1958 will be found in reference 16.
3.4 Astro or celestial navigation
Astro navigation is a huge subject and because of this only a brief description followed by those aspects directly related to the purpose of this paper will be discussed. A fuller description will be found in reference 17,
The stars are assumed to lie on a spherical shell enclosing the surface of the earth i.e. they form concentric spheres. When the line from the fixing star is perpendicular to the surface of the earth, the point at which it joins the surface is known as the sub-point. The distance from the aircraft to the sub-point may be estimated from the altitude of the star. If two such distances are found from different stars a fix will result.
One of the main problems with this form of fixing is the lack of a suitable vertical reference. Even if the artificial horizon or stable platform are used, at high speeds the affect of Coriolis may tilt this reference.
It is most likely that a bubble sextant will be used to 'shoot' a star. This apparatus is extremely dependent on a true vertical and accelerations may tilt this but if sufficient shots are taken in a short space of time its effect may be reduced. In violent weather however, the error will increase and may become considerable.
British Overseas Airways Corporation suggest that a Gaussian distribution be assumed for astro fixing with a standard deviation of 10 nm.
3.5 Compass types and accuracy
The compass types in common use by aircraft crossing the North Atlantic
are gyro-magnetic types, mainly the Bendix Polar Path System and the Sperry Compass System.
These systems combine the functions of a directional gyro with those of a magnetic compass to provide heading information which may then be fed to any other apparatus requiring this reference. Usually magnetic variation can be applied to the systems to enable true heading information to be obtained. The gyros are low drift directional types and these may be corrected for latitude. The systems may be used either in the purely directional mode using the low drift properties of the gyro to maintain heading for short term operation or else in the slave mode. In this mode the earth's magnetic field provides a reference for the gyro in the form of a compass transmitter (flux valve) and can thus be used for long term operation.
The purpose of compass swinging is to eliminate the systematic error and compensate for the magnetic influence of the compass situation: stray magnetic fields and polarisation of material within the vicinity of the compass provides a residual compass reading. The procedure involves rotating the aircraft through 360O and making suitable adjustments to the compass transmitter.
Many airline operators consider that the most probable errors are caused by environmental influences rather than system anomalies. However, magnetic
deviations tend to give rise to relatively long term systematic errors. The lines of magnetic deviation are charted in the form of isogenic lines and corrections can be applied to the compass.
Observation and semi-systematic (very short term systematic) errors will be the only random error. For present generation Mach 0.8 aircraft the authors assume a figure of 1.5° s.d. to be typical.
3.6 Other forms of heading reference (a) Inertial navigator
The accuracy of an inertial navigator will depend on the drift rate of the stabilizing gyro. (17). It is this random drift rate which provides the major portion of the overall system error. It is very difficult to assess the accuracy of such a system and the procedure involved is discussed in references 18, 19 and 20.
If the platform on which the gyros are mounted is Schuier tuned the vertical reference will be extremely accurate and the error in heading will be low, providing
the accelerometers and integrators are stable and noise free. The authors have assumed that if this system is fitted on a Mach 2.0 aircraft the heading error distribution will be Gaussian of 1° standard deviation.* This may be on the high side as the state of the art has improved with the advent of the Boeing 747 which relies on such systems for much of the flight deck apparatus, and may be improved further before the SST is in regular service.
(b) Star trackers
Star trackets are used for space probes to provide the high accuracy required. Telescopes are employed to shoot stars and the system then is fed in the closed loop condition to a digital computer. The cost of these automatic star trackers used in spacecraft may be reduced and possibly fitted in later SST's. If a vertical reference is provided by a Schuier tuned inertial platform the errors may be of the same order as the inertial navigator but the advantage of this system is
that it will not require updating from any other form of aid. Reference 21
indicates the causes of error in such a system and more advanced star-trackers are discussed in reference 14.
3.7 Doppler sensor and computer
The accuracy of a Doppler system depends to a large extent on the surface texture over which the aircraft is flying. If the surface is very smooth the Doppler returns will be very low in signal strength and when this happens the system is switched to 'memory'. In this mode the last indicated airspeed and heading continues to be shown until a signal of such strength is received to cause the system to resume normal operation. British Overseas Airways Corporation experience has shown that the number of times the equipment goes on to memory is extremely small, so much so that they do not now use a ground/sea bias switch. Ocean currents do not provide a bias despite the fact they they may have some influence on the wave motion.
Another form of error peculiar to a Doppler system is that caused by altitude holes although this is minimised by various techniques including fan shaped beams and spectrum wobbulation,
future work will consider IN in detail immediately reliable data on the distribution of errors is forthcoming.
As explained earlier the authors have assumed that the involvement of a Doppler system will serve to reduce the meteorological error.
4. THE NORTH ATLANTIC ENVIROTOIENT
In this report the aircraft are considered to be in the oceanic phase remote from both entry and exit points and it can be shown that the majority of tracks lie between latitudes 40° North and 60° North so only the behaviour of winds in these latitudes will be discussed.
It should however be emphasised that the mathematical model developed is quite general and only the numerical input data to the computer limit the environment to this particular region.
4.1 Trends in wind direction
The sun over the equator heats the air, which rises and flows north and south creating a belt of low pressure at the equator. Some of this air sinks at latitude 30° North forming a high pressure region, and some flows further north to create a polar high pressure belt. At the poles the air is cold and sinks to the surface and begins to drift southward until at about 60° North it becomes warm enough to rise and form another low pressure belt. As winds flow from high pressure to low, the predominant direction in the latitudes of interest is from south to north.
The earth's spin gives rise to a Coriolis component which deflects these winds to the east giving rise to the westerlies over the North Atlantic. Consequently the general trend of winds in this region is west-south-westerly. 4.2 Error in wind forecasting
Wind forecast errors are usually expressed in terms of wind vector deviation. This error is, to some extent, proportional to the updating time. If this
becomes infinite the error approaches the seasonal average for the region. Durst has published a formula (13) for estimating the wind forecast
error:-2 error:-2
2 k 0 ri "^ t c\ ^
In this formula Oy represents the standard vector deviation of the wind and r(f) is the vector correlation coefficient between the forecast and actual wind. Durst tables for estimating k^a^ and \{\-r^{t)} for the North Atlantic region have been extended for Mach 2 and Mach 3 aircraft and are the basis for Fig. 2 (showing the relationship between the across-track wind error and updating rate) and also Table I.
The jet stream effect is only evident between 25,000 ft. and 40,000 ft. and thus only affects subsonic aircraft and not the supersonic aircraft (during the en-route of the flight). It is estimated that the Mach 2 and Mach 3
aircraft will fly at around 50,000 ft. (100 mb) and 70,000 ft, (50 mb) respect-ively. At these altitudes the figures quoted by Durst may be reduced because they take jetstreams into account.
The figures in Table I are for a flight within 6 hours of a meteorological forecast during winter. The wind error in winter is greater than that of the other seasons and consequently the graphs produced from the computations may be assumed to be 'worst case'.
5. THE ACROSS-TRACK MODEL
The model used for across-track errors is shown in Fig. 1, and is based on the earlier work carried out at the College of Aeronautics. The following paragraphs give a brief explanation of the model and outline the various distribution curves used in the present computer calculations.
5.1 The error due to fixing inaccuracy (e )
This is represented by e^ in Fig. 1. Although in practice the shape and size of ey will be a function of flight time (because the cover from long range radio aids changes with position), it is assumed to be independent for the sake of
simplicity. The authors have, however, used two cases for the purpose of forming a representative traffic distribution. A standard deviation of 7.5 nm depicts a good coverage area and 15 nm a bad one. In practice these conditions could exist at the same point on the chart but at different times of day depending on the propagation conditions.
If Astro navigation is employed e-y is assumed to be Gaussian with a standard deviation of 10 nm although for radio fixing both Gaussian and exponential
distributions are used.
5.2 The heading error (vte.)
This is described on Fig. 1 by the product vteg where v is the airspeed of the aircraft, and t is the elapsed time after a fix. The shape of eg, the distribution of heading errors, is presumed to be both Gaussian and negative exponential for the three types of aircraft considered.
It has been suggested (3.6) that future aircraft will have an increased navigational capability and that heading errors will be reduced. Consequently
the authors suggest a 1.5° standard deviation for present aircraft (Mach 0,8) and a 1° standard deviation for future Mach 2 and 3 types as being representative of proposed practice,
5.3 The meteorological or wind error (e t)
The error in meteorological forecasting is formed by the product e^^t where e^^ is the distribution of both wind and temperature inaccuracies. It is assumed
here that there is unity correlation between the two and therefore a common standard deviation may be used.
The distribution of meteorological errors is assumed to be both Gaussian and negative exponential.
5,4 Variation of function shapes
The function shapes used throughout this work are predominantly Gaussian and negative exponential but there is no reason why other functions should not be used if they describe the operational conditions more precisely,*
A
An extension of this synthesis technique is aimed at developing a computer program capable of accepting general purpose input information in any form.
5.5 The model structure
It can be seen from Fig. 1 that if e. is small, say less than 16°
sin e„= e„ e 6
and the total across-track error is the sum of the individual errors, namely e = e + vte„ + e t
o y 9 w
The overall error distribution CQ can therefore be synthesised by
statistically 'summing' the component errors, the summing process being achieved by means of the convolution integral which is formed by adding the products of pairs of probabilities of independent functions.
A detailed analysis of the mathematical processes used in the numerical computation of the o.e.d. will be the subject of a separate report.
6. COMPUTER PROGRAM CONSTRUCTION
The philosophy behind many of the methods used in the programming is to achieve maximum accuracy but also to use the minimum computing time so as to minimise the cost.
The programs are written in FORTRAN IV code and only points of specific interest are mentioned here.
6.1 Estimation of truncation points
In computing the overall error distribution (o.e.d.) for across-track errors with a digital computer, finite limits have to be chosen for the functions. Bearing in mind that each process involves numerical methods, the limits of the functions have to be such that the degree of truncation does not invalidate the hypothetical distribution. If the limits are very great, however, the time (and cost) of the computation would be excessive if the error in the numerical method of integration was to be comparable with that due to truncation.
The frequency functions created by a digital method are in step form and the authors considered it convenient to use 0,5 nautical mile as the step length. This will yield low errors in integration if reasonable limits of integration
(about ± 100 n. miles) are used and will also simplify the determination of the lane width to give a predetermined probability of excess.
Convolution can be described as the sum of products of the probabilities of a point being greater than one datum and less than another. Thus for the final probability distribution to be valid the datum lines must be the same for the three distributions from which it is created. A convenient method of doing this is to assign the same limits to all the distributions and to take an equal number of step intervals in each case.
It is considered (1)(22) that for a collision risk of 1 in 10 , the chance of an error greater than half the separation of the two aircraft intended tracks should be less than 1 in 10-^. This 0.1% data is unlikely to be collected in a data gathering exercise in an amount sufficient to establish the shape of the distribution function at these low levels of probability.
An investigation of typical data on a 'worst case' concept showed that if ± 100 nm was taken as the truncation limit, the probability of excess (i.e. being outside the limit) would be of the order of 3.7 x 10~6. Since this is adequately lower than the figures suggested in the references it was decided to use this limit as the 401 points (assuming 0.5 nm steps) would also be conveniently small and so minimise the overall computational time.
6.2 Preliminary studies
It was decided initially to consider five functions. These were: (a) a Gaussian (or normal) distribution function,
(b) a double sided negative exponential (or first Laplacean) distribution function.
(c) a double sided Gamma distribution function. (d) a rectangular distribution function, and (e) a double sided linear function.
In the event (d) was only used for the blunder distribution (to be reported elsewhere) and (e) was not used at all. For convenience all these functions are best expressed in terms of their standard deviations. It was found (Appendix) that all the functions involved may be defined purely by their standard deviations, so making the computer input requirements simpler.
At first sight the Gamma distribution looks very useful, being nearly normally distributed about the mean yet having higher tails than the Gaussian distribution, but unfortunately it changes radically with its standard deviation; for some values it may be multi-modal but for others it becomes uni-modal and the value of the ordinate at the origin may be very large. Protection against this latter case must be incorporated in the computer program to prevent an overflow execution error from occuring.
6.3 Use of the aircraft models
The numerical examples to be described are based upon 3 aircraft models, one subsonic and two supersonic. Each model is assigned several possible error magnitudes as shown in Tables 1 and 2. Table 1 provides a value for the standard deviation of meteorological forecast wind error appropriate to the required updating period between navigational fixes and Table 2 lists the various standard deviations of navigational fix and heading used in the examples,
The computer input data for any particular calculation will be as follows: (a) Aircraft model (A, B or C ) .
(b) SD of across-track navigational fix error. (c) Distribution of fix (Gaussian, Exponential). (d) SD of heading error.
(e) Distribution of heading (Gaussian, Exponential). (f) Updating period.
(g) SD of across-track wind error appropriate to (f). (h) Distribution of wind (Gaussian, exponential).
The convolution program manipulates this data and produces a graphical output of the frequency of the overall across-track error at the end of the updating period, i.e. immediately prior to a new fix being taken.
The phase of the study reported upon here is concerned mainly with examining the Gaussian/exponential error distribution controversy. The anti-Gaussian protagonists maintaining that at low levels the Gaussian distribution is too low and that the exponential distribution better represents the actual frequency of occurrence of large errors. A graphical illustration of this contention is shown in Fig. 3 where the frequency of occurrence f(x) is plotted on a logarithmic scale against error magnitude x normalised to the standard deviation a.
For comparison purposes composite graphs have been constructed from the individual computer output plots and representative examples are shown in Figs. 4 to 13. These graphs show the frequency of occurrence of the error as a logarithmic plot against the across-track error in nautical miles. It is
emphasised that these graphs show frequency functions and should not be confused with probability distribution graphs which show the total number of errors greater
than a certain displacement, these latter graphs being commonly used when assessing separation standards.
7. DISCUSSION OF RESULTS
Unless specifically stated to the contrary the aircraft models used in the results presented in the composite figures are 'standard' aircraft having the standard deviations listed in Table 3.
It is convenient to begin by examining the 'all-Gaussian' mode and to proceed by using different functions for the components of the model.
7.1 The subsonic aircraft
The usual period between taking fixes in present-day transoceanic aircraft is 20 minutes so this period was generally chosen as giving a representative overall error distribution when comparing alternative distributions.
A typical 'all-Gaussian' mode for aircraft A is shown in Fig. 4. As is to be expected this yields a Gaussian frequency distribution of the overall error which is the one claimed to give too low skirts or tails as compared with real life situations. The 'all-exponential' mode is shown in the same figure from which it can be seen that this has the much higher tail claimed and beyond some
25 nm is practically linear as compared with the increasing negative slope of the 'all-Gaussian' mode.
Opportunity is taken on Fig. 4 to examine the effect of changing the wind distributions. The effect on the 'all-exponential' mode of expressing the wind error as Gaussian is trivial and even with the 'all-Gaussian' mode the effect of taking an exponential wind is negligible near the 1 x 10~3 level which is usually concerned in the determination of separation standards. In view of this, and because the distribution of wind errors is more likely to be Gaussian (normal) than exponential, it was decided to use only Gaussian winds in the representative diagrams given in this report.
The navigational fix is the usual component for which observed data would indicate that the exponential would be a more realistic distribution than the Gaussian so Fig, 5 was prepared, the thick curve representing an exponential fix with both Gaussian and exponential heading and Gaussian wind. (The small
separation of the curves of Gaussian and exponential heading respectively cannot be determined on this scale of graph). The thin curve shows the little effect of a perfect heading condition on the subsonic aircraft.
Enlarging on the concept of exponential fix with Gaussian heading and wind as being reasonably representative of a practical situation the effect of the accuracy of fix in good (7.5 nm) and bad (15.0 nm) Loran situations, and astro fixing (10.0 nm) is shown in Fig, 6, This quite strikingly illustrates the degradation in track-keeping accuracy over different areas of the North Atlantic when using Loran A. It definitely indicates the improvement to be obtained if
astro is used in areas of known poor Loran coverage,
It was earlier postulated that the addition of a Doppler sensor to the aircraft navigational equipment could be considered as improving the forecast wind in that drift was quickly apparent; the magnitude of the improvement to be
expected being shown in Fig. 2 and the improved s.d. values of computer input in Table 1. The effect of this improvement on the overall error is clearly
demonstrated in Fig. 7, where the thick curves represent the 'all-Gaussian' case and the thin curves the exponential fix case.
The final Mach 0.8 composite diagram (Fig. 8) demonstrates the effect of the interval between taking fixes. Once again the exponential fix model is used in a region of good Loran cover. The three solid lines show the variation from a very rapid Loran fixing rate (10 minutes) through the usual 20 minute rate to a more leisurely 30 minute rate. The 60 minute fixing rate is included for the benefit of those who feel the navigation table is overworked during a North Atlantic flight. An interesting comparison with the 'all-Gaussian' cases of 20 minute and 60 minute rate (broken lines) shows how the Gaussian/exponential fixing argument becomes less definite as the fixing rate is reduced.
7.2 The supersonic aircraft
A similar series of computations was carried out for the Mach 2 and Mach 3 aircraft conditions and four graphs are presented as being typical of the results.
The reader is reminded that the authors consider that in the SST era
improved instrumentation will be standard equipment, so the supersonic aircraft models B and C have superior navigational performance figures to aircraft A, as is shown in Table 2. The examples given (Figs. 9 to 12) all relate to Gaussian wind error distributions and 20 minutes updating intervals as being reasonably
typical of a real-life situation.
The Mach 2 comparison of the choice of Gaussian or exponential fixing errors is illustrated in Fig. 9 where the difference can be seen to be considerably
reduced compared with the subsonic aircraft (Fig. 7); this difference being further reduced for the Mach 3 case (Fig. 12). The dynamic error generated by heading becomes more dominant than the fixing error as the airspeed is increased and reference to Fig, 10 clearly illustrates the greater effect of the choice of heading error distribution on the Mach 2 aircraft compared with the minute effect on the subsonic aircraft of Fig. 5.
The deviation between Gaussian and exponential heading error is further
increased for the Mach 3 aircraft as shown in Fig. 11, The considerable improvement in track-keeping to be gained by improvements in the heading error of both Mach 2 and Mach 3 aircraft is shown by the 'perfect' case in thin line in Figs. 10 and 11,
Finally Fig, 13 brings together a comparison of the across-track errors of the subsonic and supersonic aircraft under typical operating conditions, the subsonic aircraft being Doppler equipped and operating in a good Loran coverage area, the SST's carrying improved navigation equipment. In all three cases it is considered that exponential fix with Gaussian heading and wind errors is a fair representation of current thinking on error distributions. It should perhaps be
stated that due to the higher airspeeds of the supersonic aircraft the overall across-track error is little affected by wind error.
Examination of Fig, 13 would appear to indicate that having taken account of the improved navigation of the SST their across-track error distribution will in general be superior to a well-equipped present-day subsonic aircraft operating in the North Atlantic region, even after assuming a somewhat leisurely fixing interval of 20 minutes,
8. CONCLUSIONS
The preceding sections have shown the versatility of synthesis. To obtain observational results in a quantity sufficient to establish the shapes of the overall error distributions considered would have required an enormous amount of flying. The cost would have been prohibitive.
8.1 On the methods employed
The methods used in the basic program show that statistical convolution is both feasible and accurate when performed by a digital computer. The authors, by mixing analytic and step integration, have achieved high accuracy without undue
increase in computing time and costs.
The model used as a basis for these programs is extremely versatile and this versatility has been preserved throughout the work. The programs have been written in such a way as to make it a simple matter to incorporate other distri-bution functions if they describe the practical situation more accurately. Also the functions can be scaled by using different units to those used by the authors or the same units multiplied by some factor to increase the range of the functions. 8.2 On the results
In the case of the subsonic aircraft, the fixing aid error distribution dominates those of the other component errors. As the speed increases, however, so does the leg length (for a given updating rate) and because of this the
heading error distribution becomes more important. Thus for high Mach number aircraft the main emphasis as regards their navigational capability must be put on obtaining a heading error with a very small standard deviation.
Doppler will only improve the navigational accuracy of supersonic transports if it is customary for the navigator to use long updating intervals. The fixing aid does not have a predominant effect on this type of aircraft.
The wind error on a SST are not very great and do not have a significant effect on the overall error distribution. This is due to its high speed and because this type of aircraft usually flies above the region in which jet streams are found.
8.3 Applications
If the error distributions of navigation aids are examined more closely than they have been in the past and particular cases studied, a single aircraft's overall navigational capability may be simulated quite closely. Indeed, if the model is correct and the distribution shapes are correct, there is no reason why
the overall error distribution should not represent the aircraft completely. The program which has been developed can deal with any error distribution which is found in practice, providing that this can be described by a mathematical function.
The collection of a homogeneous set of data for a particular aid is very difficult, since even if the same aircraft and crew are always considered the latter will operate differently under varying conditions of stress. Increasing electronic participation will, however, make navigational performance more predictable and more representative sets of data will be forthcoming,
It has been suggested above that a single aircraft may be simulated quite accurately. If each of the navigation aids are examined in turn, then their effect on the overall error distribution can be shown to an operator and he will then be able to take a decision on whether to install a particular aid. This model will indicate the compatibility of a new aid with his existing system before the aircraft leaves the ground.
Systems are being proposed which use geostationary satellites positioned over the North Atlantic. These may be used to provide an independent measure of aircraft position. The incorporation of such a reference into the navigation system in the aircraft will provide added constraints. The model may be extended to take account of this and its effects on tne capability of the aircraft could then be examined. Since it has been shown that a fixing aid need not be extremely accurate in the case of a supersonic transport the question arises: does the
slight increase in navigational capability justify the cost?
From the synthesis of supersonic aircraft it will be seen that a heading reference of high accuracy is important and this may well become the primary aid of a high Mach number aircraft. Different systems providing this information
(i.e. star trackets, inertial navigators) may be examined under the conditions in which they will be required to operate.
9. FUTURE WORK
This report outlines the concept of simulating an aircraft's navigational performance by considering its capability in the environment into which it is projected. Only the across-track model has been considered here and the along-track must be examined if an overall picture is to be formed. Altitude holding is not a great problem because this is continually monitored and consequently the error distribution will be quite narrow.
By using along and across-track error synthesis of each individual aircraft using an airspace a better optimisation of the traffic density and collision
risk may be made. This is because separation is then based on individual aircraft and not (as it is at present) on 'worst case'.
If the integrity is sufficiently high a tactical system of Air Traffic Control may be used. This could be created by using a computer in the aircraft to estimate probability of excess contours using models such as these described, and to provide the pilot and a master computer at ATC with a position prediction. From a number of aircraft the master computer could estimate the optimum
separation (strategic) or else alarm when a conflct is imminent. The data link between aircraft and ATC could use a satellite as a repeater to provide the required integrity of communication.
9.1 Numerical inputs
The work described here was predominantly concerned with examining the Gaussian/exponential controversy. Current work has extended the program to take in general data in numeric form in order that combinations of analytic and numerical data can be accommodated.
9.2 Inhomogeneous samples of aircraft
The adaption to the along-track error program is simple and straightforward as is also the representation of navigational blunders, so once this phase is completed one will have the capability to express the aircraft total navigational performance under closely controlled restraints. From these overall error distributions an inhomogeneous sample of aircraft can be constructed and the overall navigational error of mixed samples of aircraft groups can be synthesised and applied to the determination of airway widths, etc. A pilot program on these lines has already produced some interesting results.
10, ACKNOWLEDGEMENTS
The authors have pleasure in expressing their indebtedness to Mr. H.E. Smith and his colleagues at British Overseas Airways Corporation for providing factual information on the operational situation, and to their many friends in the
Department of Trade and Industry for helpful encouragement.
The continuing work on ATC carried out at Cranfield is published as a contribution to the advancement of air transportation,
REFERENCES 4. 5, Lord, R.N. and Saunders, D.A. Lord, R.N. Lord, R.N. Anderson, E.W. Crossley, A.F. 6. Parker, J.B. Lloyd, D.A.
8. North Atlantic Systems Planning Group
9. North Atlantic Systems Planning Group
A mathematical derivation of Air Traffic Control separation standards.
College of Aero. Rep. E & C No. 5, 1964. Proposals for the mathematical formulation of separation standards.
lATA 16th Techn. Conf., Aircraft Navigation. April 1965.
Separation standards and aircraft wander. J. Inst. Nav., 19_» l^^^. p.198
Is the Gaussian Distribution normal? J. Inst. Nav. ^ 8 , 1965, p.65.
On the frequency distribution of large errors.
J. Inst. Nav., 19^. 1966, p.33. The exponential integral frequency distribution.
J. Inst. Nav., 19_, 1966. p.526.
A probability distribution for a time varying quantity.
J. Inst. Nav., 19, 1966, p,119,
Summary of the 3rd meeting of the NATSPG, April 1968,
Summary of the 4th meeting of the NATSPG, June 1968.
10. North Atlantic Systems Planning Group
Summary of the 5th meeting of the NATSPG. December 1968.
Marchand, N,
Bauss, W, (Ed)
Durst, C S .
Scherhag, Warnecke & Wehry
Duverge, P.
Hare, E.W.
Anderson, E.W.
Carr, J.G. and Scott, D.
Amacker, L.Z.
Reynolds, P.R.J.
van Duyne, T.A.
Int. Civil Aviation Org. (ICAO)
Error distribution of best estimate of position from multiple time difference hyperbolic networks.
IEEE Trans. Aero. & Nav. Electronics, June 1964.
Radio navigation systems for aviation and maritime use.
AGARDograph 196.
Aircraft separation over the Atlantic, J. Inst. Nav. 10, 1957, p.262.
Meteorological parameters affecting SST operations.
J. Inst. Nav., 20, 1967, p.53.
Meteorological factors in SST operations. J. Inst. Nav., 20, 1967, p.64.
The evaluation and use of the DECTRA navigation system. Read before the
Auschuss fur Funkorting, Berlin, May 1958.
Also J. Inst. Nav., U, 1958, p.377. The principles of navigation.
Hollis and Carter, 1966.
The testing of airborne inertial navigation systems.
J. Inst. Nav., 20, 1967, p.405.
Inertial navigation system testing 1967. J. Inst. Nav., 20, 1967, p.432.
An accuracy evaluation of a civil inertial navigation system.
J. Inst. Nav., 20, 1967, p.449.
Design considerations for a gimballed star tracker.
Bendix techn. J., 1_, 1968, p.70. U.K. Working.Paper No. 14.
APPENDIX
Determination of functions in terms of their standard deviations
All the moments of a continuous frequency distribution can be represented by the moment generating function M (<()) which is defined as
+<» .
M ((t>) = ƒ e''"'f(x)dx A
— OO
Provided M^(<()) has a finite value we can expand e as a power series, so obtaining an expression of the form
M (<|>) = ƒ f(x)dx + ((> ƒ xf(x)dx + •£- ƒ X f(x)dx + ....
—OO —CO 2 ! —^'^
In the distribution functions used here we know that the total is normalised to unity so
ƒ f(x)dx = 1
The first moment about the origin is the mean (y), and this is presented by the coefficient of (|) so
+00
y = ƒ xf(x)dx
— 0 0
More generally we express the moment generating function in terms of displacement from the mean (x-y) as follows:
+00 +00 2 +«> _
M/ _ NC*) = ƒ f(x)dx + ((> ƒ (x-y)f(x)dx + fr ƒ (x-p) f(x)dx + .
^ ^ ' — G O — 0 0 * — 0 0
2
The second moment about the mean is the variance (a ) which is, of course, the square of the standard deviation (a), and is given by the coefficient of
fj-so I
a ^ j (x-y) f(x)dx— 0 0
In the particular frequency distributions here considered we know that they are all symmetrical about zero so the mean must be zero, that is to say
+00
y = ƒ xf(x)dx = 0
- 0 0 I
Using these relationships we can describe frequency distributions purely by their standard deviation (o) and this
yields:-(i) Gaussian distribution ' ~ 1
x2
(ii) Double sided negative exponential (1st Laplacian) distribution
f (x) = — — e - < » ^ x ^ + «> a/2
^"~~" . ^ ^ D A T I N G TIME „•r^^r^.r-T"" « ( M l n u t e s ) AIRCRAFT -—>.i._:^;_;_^^^^^ A A(D) B C MACH 0-8 MACH 0 8 MACH 2 MACH 3 A75 kt (250 mb) A7 5 kt ( 2 5 0 m b ) w i t h doppler sensor 1200 kt (70 mb) 1720 kt (44mb) 10 148 • 7 2 102 8 6 20 138 6 0 8 6 7-1 30 130 5-2 7-6 6 2 40 12 4 4 7 6 8 5-5 50 12 0 42 6-3 5 0 60 11 6 4 0 5 8 4 7
NOTE All the above values are for winter flights and are within 6hrs of a forecast.
TABLE 2. Aircraft Data
AIRCRAFT A B C AIRSPEED 475 kt 1200 kt 1720 kt STANDARD DEVIATION NAVIGATIONAU FIX 7 5 nm 10-0 nm 15 0 nm 3 0 nm 7' 5 nm 3 0 nm HEADING I S ' 1 0 ' 1 0 '
TAB(-E 3. The ' s t a n d a r d ' aircraft used in Figs. 4 to 13
AIRCRAFT A B C AIRSPEED SUBSONIC (MACH 0 8) SUPERSONIC (MACH 2) SUPERSONIC (MACH 3) STANDARD DEVIATION NAVIGATIONAL FIX 7 5 nm 3 0 n m 3 0 n m HEADING 1-5* 10* 10* UPDATING PERIOD 20min 20 min 20min
time t r i ( t o + t ) True Position at Time to s Required Track Overall Error True Position at Time ( t g +-t)
FIG. 1 THE ACROSS-TRACK MODEL
• • o c UJ c u at I i/i (/) O k-u < a 10 20 30 40 50 60 Elapsed Time (Minutes)
1x10' ° 1x10-2 g. 1x10 1x10" 1x10 1x10 ,-6
X,
X
\ ~ <AN
\ ^ * /•^^v
'X,
\ 10 20 30 ÜO 50 6 0 70 80 9 0 100 Acres'-. - t r a c k D e v i a t i o n ( n m )FIG. 4 THE 'ALL-GAUSSIAN' AND 'ALL-EXPONENTIAL' SUBSONIC AIRCRAFT WITH ALTERNATIVE WIND DISTRIBUTIONS
1 0 ti) 1x10" o c ° lxlO"2 ^ 1 x 1 0 1x10 1x10 1x10 -A
px
V
% ^ ^ "i?^ \N
N
^ 10 20 30 ^0 50 6 0 70 80 90 100 A c r o s ? - t r a c k D e v i a t i o n ( n m )FIG. 5 THE EXPONENTIAL F I X . GAUSSIAN HEADING AND WIND SUBSONIC MODEL. (THE EXPONENTIAL HEADING CASE IS INDISTINGUISHABLE ON THIS SCALE).
(» I x l O ' l o c 01 ° 1xl0"2 g - l x l O 1x10 1x10 -4 1x10 1-6
^ v
^ . ^ ^N
^ ^ ^ ^ '—., > .X
^ o „
^^^0^)
\ \ ^ ^ \ >^ ^ 10 20 30 ÜO 50 60 70 8 0 90 A c r o s s - t r a c k D e v i a t i o n ( n m ) 100FIG. 6 THE EXPONENTIAL F I X , GAUSSIAN HEADING AND WIND SUBSONIC MODEL SHOWING EFFECT OF FIXING ACCURACY
1 0 <s 1x10" u c en 3 O ° lx10"2 c g-lxlO 1x10 1x10 1x10 -4 = ^
V
\ ^ ^ ^ \ \ \ \X
X
10 20 30 40 50 60 70 PO 90 Across-track Deviation (nm) 100FIG. 7 THE EFFECT OF ADDING DOPPLER TO T>1E SUBSONIC AIRCRAFT ('ALL-GAUSSIAN' IN HEAVY LINE, EXPONENTIAL FIX IN FEINT)
dl 1x10"' u c 01 O ° 1x10-2 1x10 1x10" 1x10 1x10 " ^ fe ^ V % \ \ \ V \ ^ \ \ \ \ \ % \
X
v_ \ ^X
^ N
10 20 30 40 50 60 70 80 90 100 A c r o s s - t r a c k D e v i a t i o n ( n m )FIG. 8 THE EFFECT OF FIXING RATE ON THE EXPONENTIAL FIX. GAUSSIAN HEADING AND WIND SUBSONIC AIRCRAFT.
1-0 1x10" ° 1x10"2 1x10 1 x 1 0 ' 1x10 1x10 \ \
V
^ ^ \ 10 20 30 40 50 60 70 80 90 100 A c r o s s - t r a c k D e v i a t i o n ( n n i )1x10 '1 •i ° 1x10-2 1x10 1x10 -4 1x10 -5 1x10 ,-6
K
\V
\
^
\
^\
.
\
>\
^ > ^
VN
\
10 20 30 40 50 60 70 80 90 100 A c r o s s - t r a c k Deviation ( n m )FIG. 10 THE EFFECT OF HEADING ERROR DISTRIBUTION ON THE MACH 2 AIRCRAFT
1 0 u 1x10"! u c 01 ° 1x10-2 c
I
1x10
-3 •1x10 1x10 -4 1x10 ,-6 k<K
\^
- ^
\
N
\\
N ^\
\\
,
S
W
V
^k'
\
10 20 30 40 50 60 70 80 90 100 A c r o s s - t r a c k Deviation ( n m )1x10" ° 1xlO"2 1x10 1x10 1x10 -4 1x10 ,-5 = - -^ ,
x«
' \ \ 10 20 30 40 50 60 70 80 9 0 100 A c r o s s - t r a c k D e v i a t i o n ( n m )FIG. 12 THE 'ALL-GAUSSIAN' MACH 3 AIRCRAFT AND THE EFFECT OF THE EXPONENTIAL FIX
1-0 1x10" ° 1xlO"2 1x10 1x10 1x10 1x10 ^ ^
V
N
^ j > ^V
K
V
\ ^ \ ^o 1 \ 10 20 30 40 50 60 70 8 0 90 100 A c r o s s - t r a c k D e v i a t i o n ( n n i )FIG. 13 ACROSS-TRACK ERRORS OF THE THREE 'STANDARD' AIRCRAFT WITH EXPONENTIAL FIXING, GAUSSIAN HEADING AND WIND ERRORS
