CA1125419A - Method of correcting doppler navigator - Google Patents

Abstract

METHOD OF CORRECTING DOPPLER NAVIGATOR
Abstract of the Disclosure An improved method of processing the data from an airborne monopulse Doppler navigational radar, thereby to determine the aircraft velocity, is disclosed. In such method, the difference channel return data from each of three sequentially formed radar beams are rotated through an angle to insure that the monopulse difference pattern null associated with each of the three beams re-mains parallel to a selected isodop. Following rotation, the data are normalized by the sum channel data to reduce the effect of scintillation errors. After normalization the data undergoes a processing technique to determine the Doppler frequency associated with the monopulse null for each of the three beams. The Doppler frequency data are subsequently converted to velocity data and combined with velocity data obtained from an inertial platform in order to reduce the long term drift errors associated with the inertial platform.

Description

1 1'2 ~ ~9 Background of the Invention This invention relates generally to airborne radar systems utilizing the Doppler effect to derive navigational information and particularly to signal processing techniques for improving the accuracy of systems of such character.
As is known in the art,in any so-called Doppler navigation radar system in an aircraft, components of velocity o~ the air-craft may be derived with respect to orthogonal axes having a known relationship to the surface of the earth. Because there are three orthogonal components of velocity of interest, radar echo signals in a plurality ~at least three) of radar beams illuminating different portions of the underlying terrain must be used. A Doppler navigation system, employing three non-coplanar radar beams, which has found wide acceptance is the so-called Janus system. Such a system has both forward and rearward looking beams, and is usually comprised of two forward looking beams and one rearward looking beam arranged in such a manner as to form the ~reek letter ~. In such a system, the difference between the Doppler shifts received from the rearward looking beam and the forward looking beam on the same side of the aircraft provides a measure of the "along heading" velocity VH; while the difference between Doppler shifts received from the two forward looking beams provides a measure of the "cross-~- heading" or "drift" velocity VD. The vertical velocity component is derived by combining the Doppler shifts obtained from the rear-ward looking beam and the forward looking beam an the opposite side of the aircraft.
~ ecause each Doppler radar beam has a finite beamwidth, the return signal associated with any beam comes from different direc-tions. Thus, return signals for each radar beam contain Doppler 3~'2~

frequencies within a frequency spectrum rather than a singleDoppler frequency. The Doppler frequency of interest, which is proportional to the velocity component along any beam, is the mean frequency of the Doppler frequency spectrum. Because of the appreciable spectrum width, the instantaneous central fre-quency of the Doppler signal is, however, subject to random fluctuations about its mean value, giving rise to an error in tracking the center of the beam and, therefore, in measuring velocity. In addition, signal returns from isolated large targets or the effect known as "sea bias~ can cause measurement errors by apparently changing the center frequency of the Doppler frequency spectrum.
As is known in the art, errors which result from some apparent changes in the return signal spectrum may be reduced by utilizing a monopulse radar having beams with antenna patterns directed toward the earth fore and aft of the aircraft velocity vector, as is described in an article entitled: "Null-Tracking Doppler Navigation Radar" by P. G. Smith, IEEE Transactions on Aerospace and Navigational Electronics, March 1963, Pages 50-55.
In such a system, the Doppler frequencies corresponding to the null plane of the monopulse elevation difference channel are tracked. The null plane is used because antenna gain is apparently zero at the null, and the slope of the apparent antenna gain curve on either side of the null plane is very steep. The accuracy of null tracking depends upon the average slope of the difference pattern in the ViCillity of the null. Thus, any known null tracking system is useful only when the null plane of the elevation differ-ence pattern is substantially orthogonal to the ground track of the aircraft, i.e. when the azimuth difference null is essentially parallel to an isodop, in order to produce a Doppler spectrum in

-2--1~5~i~

the elevation difference channel which has a sharp null. As the null plane deviates from parallelism with an isodop, degradation of the difference pattern null occurs. This means that roll or yaw angles in excess of 15 experienced by the aircraft may de-grade the null to an unacceptable degree to the detriment of system accuracy. In addition, because any known null tracking system is restricted to use with beams which are substantially in the ground track directlon, returns from sideward looking beams are still subject to beam spreading with errors due to apparent changes in the center frequency of the Doppler frequency spectrum as described above.
As is also known in the art, tile degradation of system accuracy as a function of drift angle may be reduced by using an antenna which is azimuth-stabilized along the ground track. In such a system, the difference between the Doppler shifts obtained from two sideward looking beams on opposite sides of the ground track is used to drive a servo which turns the antenna in azimuth ~.
until the difference is forced to zero. This condition occurs -^, when the ground intersections of the two beams lie on the same isodop, thereby aligning the antenna with the aircraft track.
Ground track stabilization by such a method is slow and is imple-mented only by relatively complex, heavy, expensive hardware~
Such a system is, in any event, not suitable to prevent degradation ~; in accuracy resulting from aircraft roll and yaw.
As is also known in the art, an antenna radome will diffract a radar beam passing therethrough. This diffraction, which is dependent on the beam pointing direction, causes an error to be introduced in the determination of the angle associated with the Doppler frequency of the return signal and therefore in the com-putation of the aircraft velocity.

Summary of the_Invention With this background of the invention in mQnd, it is an object of this invention to provide an improved signal processing techni~ue for an air-borne Doppler navigation radar system using a monopulse radar.
Another object of this invention is to provide a method of increas-ing the accuracy of an airborne manopulse radar used in a Doppler navigation system without any need for a complex mechanical drive for the antenna of such a ra~r.
Still another object of this invention is to provide an airborne m~nopulse D~ppler navigation radar ~hose performance is not affecbed by air-craft roll, pitch, or drift angles.
Still another object of this invention is to provide an airborne monopulse D~ppler navigation radar whose performance is not affected by radome induced antenna beam point mg errors.
These and other Qbjects of the invention are generally attained by providing an airborne monopulse radar in a Doppler navigation system and a pro-cessor wherein a selected monopulse null plane may be rotated to make it appear that ~uch selected null plane is parallel to the isodops in the portion of the illuminated terrain, regardless of the orientati of the radar beam.
In a preferred ecbodiment of this invention, such rotation is accomplished by weighting the difference channel signals by, respectively, the sine and cosine of a so-called mon~pulse rotation angle. ~he m~nopulse angle (B) is evaluated for each different orientation of the radar beam in accordance with the attitude and velocity of the aircraft. m e attitude and velocity of the aircraft in turn are determined i~ part by inertial measurements which are continuously processed in an onboard computer.
In accordance with the present invention, there is provided an im-proved method of operating an airborne Doppler navigation radar system utiliz-ing an inertial platform and a monopulse radar having an antenna adapted to produce a plurality of directive bea~s, the centerline of each one of such beams being in a fixed angular relationship with respect to the radiatin~ face J

g of such antenna, for illuminating different portions of the terrain underlying an aircraft carrying such system so that return signals in each one of such beams may be processed in an azimuth difference and an elevation differenoe channel to derive measurements:of the velocity vector of the aircraft with respect to the earth, in particular, the components of the velocity vector along said beams, such improved method comprising the steps of: (al aligning the antenna so that the center of the portion of the terrain illuminated by a first selected one of the directive beams is disp~sed at an angle with respect to the aircraft flight path; (b) energizing the antenna to form, sequentially, each one of the plurality of directive beams; and (c) pro oessing the returns from ea~h sequentially formed beam as received by the antenna to derive the weighting vzlues required to rotate the null of the elevation difference chan-nel for the next group of sequentially formed radar beams by "N" degrees, where N is the angular difference in degrees between the null through the oe nterline of the beam which is tangent to a selected isodop.

.
~.~

- 4a -"

Brief Description of the Drawings The above-mentioned and other features of the invention will become more apparent by reference to the following descript-ion taken in connection with the accompanying drawings in which:
FIG. lA is a sketch, somewhat simplified, of an air-craft employi~g a monopulse Doppler navigation radar to deter-mine its velocity by directing a series of radar beams toward the terrain beneath the aircraft;
FIG. lB is a sketch, greatly simplified, of the portion lQ of the terrain illuminated by radar beam 22 of FIG. lA useful in understanding how the radar on board the aircraft of FIG. lA
processes the return signals from such portion to determine air-craft velocity;
FIG. lC is another sketch of the terrain illuminated by radar beam 22 of F~G. lA illustrating portions of the terrain ~-~: from which returns of constant Doppler shift are obtained;
, ., FIG. lD is a sketch of a Doppler sensitivity curve obtained by processing the returns from the portion of the terrain ; illustrated in FIG. lC;
'1 ~ 20 FIGS. 2A, 2B and 2C are sketches of the portion of the :~ terrain illuminated by radar beam 40 of FIG. lA which correspond, respectively, to FIG6. lB, lC and lD;
FIG. 3 is a block diagram, somewhat simplified, of a radar sy-stem incorporating our invention;
:: FIGS. 4A and 4B are sketches of a uniform sphere surrounding the aixcraft of FIG. lA which are useful in under-standing the contemplated signal processing technique;
FIGS. 5, 5A and 5B are vectox diagrams useful in unde~standing the contemplated signal processing technique; and 3Q FIG 6 is a vector diagram illustrating the effect of monopulse axis rotation according to our in~ention.

- 4a -~125~9 The geometry involved with a Doppler navigation radar system which gives rise to the problem to be solved is shown in FIG. lA.
Referring now to FIG. lA, an aircraft 10, equipped with a mono-pulse radar 12 having an antenna 14 (which conveniently may be a conventional phased array) fixed relative to the longitudinal axis of the aircraft 10, is shown to be flying parallel to an assumed flat terrain 16 at a constant altitude "h~. Three orthogonal axes marked "x", "y", and "z" having an origin at the aircraft 10 are shown. The components of interest of the velocity of the air-craft 10 are the components along such axes. The aircraft 10 here is shown to be moving so that its ground track 19 is parallel to the z-axis. The hyperbolas 20a, 20b, and 20c represent the loci ~e' of points on the terrain 16 from which echo signals having a constant Doppler shift are received. Such loci are commonly referred to in the art as "isodops".
- The antenna 14 is shown to direct a radar beam 22 toward the terrain 16 in such a manner that the centerline 24 of beam 22 forms an angle (A) with the velocity vector 18. In the absence of air-craft yaw and drift, the aircraft heading vector 26 is coincident with the velocity vector 18 and the radar beam 22 i5 symmetrlcal about the ground track vector 19.
Referring now to FIG. lB, which is an enlargement of the area illuminated by radar beam 22, the monopulse elevation null 28 of the monopulse elevation difference pattern 30 is shown to be parallel to isodop 20b while the azimuth null axis 32 (sometimes herein referred to as the elevation error axis 32 for reasons to be clear hereinafter) is orthogonal to the isodop 20b. In order to gain a better understanding of the processing technique involved in determining the desired Doppler shift, reference is made to l~Z~i~19 FIG. lC wherein another representation of the area illuminated by radar beam 22 is given. The vertical sections labeled fO, fl f 1~ ... fn~ f n represent areas of constant Doppler shift, each of which is centered on a corresponding Doppler frequency.
Doppler frequency fO corresponds to the isodop which is tangent to the elevation null, here isodop 20b. The negative subscripts represent negative Doppler shifts or decreased frequency with respect to the center frequency fO, while the positive subscripts represent positive Doppler shifts or increased frequency with respect to center frequency fO. The x's 34 represent unit re-flectors at each of the Doppler frequencies.
Referring back to FIG. lB, it may be seen that unit reflectors - to the left of elevation null 28 are on the negative side of the elevation difference pattern 30, while unit reflectors to the right of elevation null 28 are on the positive side of the eleva-tion difference.pattern 30. As is well known, the sign of the signals out of a monopulse receiver are positive (+) for returns coming from unit reflectors located on one side of the elevation null 28, and negative (-) for returns coming from unit reflectors on the opposite side of elevation null 28.
The Doppier navigation radar 12 processes the return signals along the elevation error axis 32, plotting net signal strength versus Doppler frequency. The result is an S-shaped sensitivity curve 36, as shown in FIG. lD. The Doppler frequency of interest is that which corresponds to the cross-over point 38 of the sensi-tivity curve 36. The slope of the sensitivity curve 36 in the region of the cross-over point 38, which is related to the depth of the elevation difference pattern null, determines how accu-rately the cross-over point may be tracked.
Referring back to FIG. lA, it may be seen that when aircraft ~i25419 10 is subjected to a wind velocity vector 40', the aircraft must be yawed (here at an angle ~) to maintain its flight path along the velocity vector 18. Centerline 24 of radar beam 22 moves with the aircraft heading vector 26 such that it is no longer in the same vertical plane with the velocity vector 18, and there-fore, the elevation null 28 is no longer parallel to isodop 20b.
The net result is that the elevation null is degraded and the slope of the sensitivity curve 36 is diminished with a concomitant loss of tracking accuracy. This effect may be more clearly under-stood by referring to FIG. 2A where isodop 20b is shown crossingthe elevation null 28 at an angle ~; and to FIG. 2B where isodops ' fl~ f 1~ ... f ~ f , are shown crossing the elevation error axis 32 at an angle ~. As the isodops are no longer parallel to the elevation null 28, the unit reflectors 34 on any isodop fall on both the positive and negative sides of the elevation differ-ence pattern 30. The resulting sensitivity curve 42 shown in FIG. 2C,which is obtained by plotting signal strength versus ` Doppler frequency along the elevation error axis 32, has a more .~ ~ shallow slope than the sensitivity curve 36 of FIG. lD.
Referring now back to FIG. lA, it may be seen that a loss in tracking accuracy will also result if the antenna 14 directs a beam to one side of the aircraft 10, i.e. beam 40. Thus, there is a loss in tracking accuracy associated with radar beams whose centerlinés are not coplanar with aircraft velocity vector 18.
;

llZ5~9 .
Referring now to FIG. 3, a radar system having a monopulse receiver for null tracking is shown. Such system includes a microwave front end (not numbered), comprised of four antenna elements, 46a-46d, connected in a conventional manner to form a sum channel 50a,an elevation channel 50b (i.e. a difference channel), and an azimuth channel 50c for receiver section 52.
It is noted that the sum channel is coupled, selectively, to receiver section 52 and transmitter 54, in a conventional manner, here by circulator 56.
Receiver section 52 is a conventional heterodyne receiver and includes RF amplifiers 58a-58c, a local oscillator 60, mixers 62a-62c, and IF amplifiers 64a-64c, which are all of conventional design and arranged to convert RF signals applied to receiver section 52 into IF signals on lines 66a-66c. Such IF signals are passed through range gates 67a-67c, which are o-~ conventional design and are controlled by range gate generator 69, also of conventional design. The azimuth and elevation difference signals, on lines 66c, 66b, respectively, are directed to the monopuIse rotation processor 68 wherein they are subsequently wei~hted by the sine and cosine of the rotation angle B in rotation angle multipliers 70a-70b (i.e. IF sine and cosine phase multipliers which are of conventional design). The rotation angle B is the angle through which the monopulse axes must be rotated to maintain proper alignment to the generated isodops and thereby prevent mono-pulse null fill-in.
The monopulse rotation angle B is calculated in rotation angle computer 72, in a manner to be described hereinafter, from data received from inertial platform 74 and motion compensator 75.
The aircraft attitude is determined from inertial platform 74 which is comprised of gyros and accelerometers. Analog signals l~Z~

from inertial platform 74 are digitized by A/D converters 76a-76n and are processed in rotation angle computer 72 along with velocity information from motion compensator 75 to yield the required rota-tion angle B.
The binary representations of the sine and cosine of the rotation angle B are translated to suitable analog signals in DlA
converters ~8a-78b and are sent to the monopulse rotation processor 68 on lines 80a-80b, respectively. The rotated monopulse data are processed in combiner 84 in such a manner as to form new monopulse 10 axes Xl and yl on lines 86a-86b which may be represented, respec-tively, by the following equations:
xl = Az cos B + El sin B (1) yl =-A sin B + El cos B (2) where:
A = the azimuth difference channel signal El = the elevation difference channel signal For the purpose of clarity, the remainder of the radar system :~ in FIG. 3 is shown to process only the data on line 86a, correspon-ding to the yl axis. The signal processing for the Xl axis data on 20 line 86b is identical to that for the yl axis data described here-inafter.
rlixers 88a-88b, in conjunction with local oscillator 90, ; translate each of the IF signals on lines 86a and 92, corresponding to the yl axis and the sum channel data respectively, to a suitable video frequency so that they may be digitized by A/D converters 94a-94b. Before being fed to mixer 88b, the sum channel data on line 66c is attenuated in ~ariable attenuator 98 in order to com-pensate for the insertion loss experienced by the difference channel data in passing through the rotation processor 68 and the 30 combiner 84. The digital data from A/D converters 94a-94b is then applied to the Doppler filter bank Fast Fourier Transform (FFT) g i9 signal processors 96a-96b which perform Doppler processing, signal to noise enhancement, and thresholding for each Doppler filter. The FFT is performed on a number of range gates for each coherent transmitter frequency to produce corresponding sets of frequency spectra which represent the resolvable target Dopplers obtained during the radar dwell time interval.
The rotated difference channel data and the sum channel data ~` are passed to a processor, here designated normalization processor 100, to normalize the former in a conventional manner. Such pro-cessing is performed independently for each Doppler filter in the filter bank. The result of the normalization is a reduction in the effect of scintillation in radar return signals. The normal-ized data represent a plot of monopulse error angle versus Doppler frequency (i.e. a discriminator curve similar to that shown in FIG. lD). Such data are then passed through processor 102 wherein only data from filters below a preset threshold are accepted for processing. The accepted data in the vicinity of the difference pattern null then undergo a "least squares" curve fitting process from which the zero gain intercept is extracted. The zero gain intercept is the Doppler frequency estimate corresponding to the monopulse null. This process is repeated for each range gate, obtaining independent measurements of the Doppler frequency at each monopulse null. The average of such measurements yields the final estimate of the desired null Doppler frequency. Such average is passed to motion compensator processor 75 wherein it is converted to velocity coordinates and is subsequently used to compute the track and climb angles, in a manner to be described in greater detail hereinafter. The revised estimates of the track and climb angles are sent to the rotation angle computer wherein they are used to update the rotation angle B.

5~

The data from processor 102 represents Doppler frequency data measured in beam coordinates. In motion compensator processor 75 J
such data is first converted to velocity data and is then trans-formed into reference space coordinates by means of multiplying the data by a matrix comprising the direction cosines of the beam pointing angles. This transformation may be expressed as:

V = [DCI V ~3) REFERENCE BEAM
;` ~
The radar derived velocity data is then combined with the velocity data from inertial platform 74 to correct for the long term errors in the velocity data from inertial platform 74. The data is combined as shown in Equation 4.

~Vx = Vx ~radar) - Vx (platform) + Nx ~Vy = Vy ~radar) - Vy (platform) ~ N
~V e V (radar) - V (platform) ~ N

The ~V terms represent the errors between the radar derived velo-city data and the platform derived velocity data, and may be seen to include a noise term (N). The noise term is eliminated and the error data is smoothed by submitting the data to a filtering or smoothing process. Such a process, which is described in an article entitled, "Optimizing the Dynamic Parameters of a Track-While-Scan System" by J Sklansky, RCA Review, June 1957, Pages 163-185, may be represented as shown in Equations 5 and 6. For the sake of clarity only, the x component of the velocity vector li;~S~l~

V is shown in Equations 5 and 6, it being understood that the y and z components of V undergo an identical process.

(ll) = aVx(n) + ~QVx(n) ~ ~x(n)] (5) ~x(n) = ~Vx(n) + ~/T[~Vx(n) - ~Vx(n)] (6) The following definitions apply to the terms of Equations 5 and ~ Vx(n) is the smoothed value of velocity error for the Nth radar dwell .rX(n) is the predicted value (from the N-l radar dwell) of the velocity error for the Nth radar dwell Vx(n) is the rate of change of the smoothed value of velocity error for the Nth radar dwell is a constant 0<~1 ~ is a constant given approximately by ~
; T is the duration of the Nth radar dwell The velocity error predictions for the N+l radar dwell are given by ~x(n~ Vx(n) + T~Vx(n) /~ _ (n+l) = ~Vx(n) ~8) where Vx(n+l) is the predicted rate of change of the velocity error for the N+l radar dwell The velocity error predictions for the N+l ràdar dwell are combined with the platform derived velocity data to calculate new values for the climb (C~ and track (T) angles. Referring now to 11~5~9 FIGS. SA and 5B, it may be seen that the aircraft track angle T
involves rotation of the aircraft ground velocity vector, V
about the Y-axis. The following definitions therefore apply:
cos T = Vz/Vgs sin T = ~x/vgs where Vgs = ~x + Vz ~9) The aircraft climb angle C involves rotation of the aircraft velocity vector V about the X-axis and therefore the following definitions apply:

sin C = -Vx/V cos C = Vg/V (10) where V = ~x + Vy + Vz The updated values for the climb ~C) and track (T) angles are used to calculate the value of the monopulse rotation angle (B) for the N+l radar dwell.
The process for computing the rotation angle B, through which the monopulse difference channel data are rotated to form the new monopulse axes Xl and yl~ will now be described. As prevlously mentioned, in order to maintain the depth of the monopulse differ-ence pattern null so as to identify a frequency or velocity with a given beam direction, it is desirable that the data in the two monopulse difference channels be processed along an axis normal to an isodop. This is accomplished by rotating the two receiver channels through an angle B so as to effectively form two new mono-pulse receiver channels Xl and yl where:
[ 1 ]= (B)~ ] (11) The rotation angle (B) must be found for each of the radar beams and will, in general, be different for each beam because the aircraft attitude and velocity vectors will affect each beam differently, The rotation angle (B) is formed in rotation angle computer 72 from inputs received from inertial platform 74 and motion compensator processor 75. The rotation process involves the transformation of the radar beam coordinates in "velocity space" to corresponding coordinates in "antenna space".
Before proceeding, it should be noted that W. H. Von Aulock, in an article entitled "Properties of Phased Arrays," Proceedings of IRE, Oct. 1960, Pages 171S-1727, has shown that in discussing the scanning properties of phased array antennas. it is advanta-geous to use contour maps of the antenna pattern in so-called "T-space" which is a projection of a unit sphere of constant range on the plane of the array.
Referring now to FIG. 4A, a unit range sphere 200 is shown surrounding an aircraft 202 which is equipped with a Doppler navi-gation radar ~not shown~. The intersection of a Doppler cone 204 (which is herein defined as a conical contour of constant Doppler frequency) with the unit sphere 200 is shown to form a circle 206.
The intersection of the Doppler cone 204 with a ground plane 208 is shown to form a hyperbola 210, and the intersection o the unit sphere 200 with the ground plane 208 is shown to form a circle 212.
Referring now to FIG. 4B, the aircraft 202 is shown to be flying in the XZ plane with its velocity vector V coinciding with the Z axis. The aircraft 202 is shown to direct a Doppler cone 204 in the direction of the Z-axis. The Doppler cone 204 inter-sects the unit sphere 200 at points marked, respectively, 214a, 214b. As previously mentioned, the intersection of the Doppler cone 204 with the unit sphere 200 forms a circle 206 li ~ 5~1 9 which is projected into "T-space" or "antenna space" as a circular isodop 218 The circular isodop 218 in "T-space" or "antenna space" is shown to lie in $he XY plane. The angle G, which is shown to be one-half the Doppler cone angle, is defined as the angle between the velocity vector V, which here is coincident with the Z-axis, and the radius of the unit sphere 216, which here is defined as being equal to unity. The angle G is here also defined as the predetermined beam pointing angle. The vertical distance then, from the Z-axis, to the intersection points 214a, 214b, is seen to be equal to sine G. As the intersection of the Doppler cone 204 with the constant range sphere 200 is projected into "antenna space" or "T-space" as a circular isodop 218, the radius of the circular isodop 218 may also be seen to be equal to the sine G. It may be seen therefore, that specifying a particular beam pointing angle G will also define a particular circular isodop in "antenna space" or "T-space". Any point T on the selected circular isodop may now be defined by means of the following expressions:
X = sin G sin t ~12 Y = sin G cos t ~13~
where t ls defined as the angle between the radius of the isodop (sine G) and the Y-axis.
A vector equation relating a beam in "velocity space" to a beam which has been selected a priori in "antenna space" is:

Yv I = [C] [T] [A]t y ¦ (14) where llZS419 [A] = the transpose of the attitude matrix and is com-prised of the roll, pitch, and heading matrices ~T] = the track matrix ~ C~ = the climb matrix The subscript "v" represents the beam coordinates in velocity space, and the subscript "a" represents the beam coordinates in antenna space.
The transformation process will now be described with refer-ence to PIG. 5. The two-dimensional vector V in FIG. 5 has com-ponents x and y in one coordinate system. The same vector hascomponents xl, yl, in a coordinate system which is rotated from the first coordinate system through an angle ~. The components xl, yl can be expressed in terms of the components x, y by:

xl = x cos ~ ~ y sin ~
y = y cos ~ - x sin ~ (15) or in the matrix notation by:

~ xl 1 ~ cos ~ sin ~ 1 ~x 1 L yl ~ -L -sin ~ cos ~ ~ ly ~ ~16) The two-element column matrices in Equation~l~represent the vector V of FIG. 5 in two different coordinate systems. The second-order matrix represents the rotational transformation from .~ the coordinate system x, y to the coordinate system xl, yl, A vector in three dimensional space is expressed in terms of its three mutually orthogonal components x, y, z. Since this triad has three degrees of rotational freedom, the relationship between it and another orthogonal triad xl, yl, zl can be il~54~9 expressed as an angular rotation about each of the three axes in some particular succession. Thus, as mentioned hereinabove, the attitude matrix is comprised of the roll, pitch, and heading matrices and is formed by successively rotating about the Z, X, and Y axes through the roll, pitch, and heading angles, respec-tively. Such transformation relates a beam in "antenna space" to the inertial platform onboard the aircraft, which is herein defined as reference space. This transformation may be expressed as:

a- cos R sin R O 1 0 0 cos H O sin H XR-a = -sin R cos R O O cos P sin P O 1 0 R ~17) a O O 1 O -sin P cos P -sin H O cos H R

Equation tl7) is inverted to yield:

YR ¦ ~ [A] ¦Ya ~18) where [A] is defined as the transpose of the attitude matrlx lA].
~quation ~18) relates a beam in "antenna space" to a beam in "reference space". Transforming a beam from "reference space" to "velocity space" involves rotating that beam successively through the aircraft track and climb angles. This transformation process is given by:

1 0 0 cos T O sin T ~XR~
v = O cos C sin C O 1 0 R ~19) v O -sin C cos C -sin T O cos T R

112~ 9 Finally, combining equations (18) and ~19) results in:

Y ¦= [C] ~T~ ~A]t lly (1~) Equation ~14)may be further simplified as follows:

IY ¦ = [D]t ¦ (20) v BE~I a BEAM
Where [D]t is a composite matrix given by:

[D]t = [C] ~T] [A] (21) ;

Expanding Equation (20) results in:

X (beam) = dl A (beam) + dl Y ~beam) + dl3 Z ~beam) ~ Y ~beam) = dl X ~beam) + dl22 Y (beam) ~ dl3 Za tbeam) ~22) : where dll, dl2, etc. are elements of the composite transformation ; matrix ¦D] , and Z ~beam) = [1 x2 (beam) y2 (beam)] 1/2 ~23) Inverting vector Equation ~20) yields:

Ya = [Dl ¦YV (24) The procedure now is to map an isodop from velocity space into ~antenna space," obtain the slope of the isodop in 'antenna spacel' and then evaluate this slope at the coordinates of a predetermined beam pointing angle. In "velocity space", the locus of points of equal Doppler (i.e. an isodop) is given by the following expres-sion:

Xv~ sin G sin t Yv = sin G cos t ~25) _Zv cos G

where the angles G and t have been previously defined with reference to FlG. 4b. The isodop of Equation (25) maps into "antenna space'~
X , Y , Z where:

20-xa- i sin G sin t a = [D] sin G cos t t26) LZa cos G

Expanding Equation ~26) results in the following:

Xa dll sin G sin 12 13 ~27) a 21 sin G sin t + d22 sin G cos t + d23 cos G ~28) Z = (1 - Xa a) (29) where dll, dl2, etc. are elements of the transformation matrix L~-Taking the first derivative of Equations ~27) and (28~ yields:

dY
a = d sin G cos t -d sin G sin t ~30 dX
a = dll sin G cos t -dl2 sin G sin t (31) : Dividing Equation ~30) by (31) yields:

dYa d sin G cos t -d sin G sin t dXa dll sin G cos t -dl2 sin G sin t ~32) which further reduces to:

dY d Y -d X
a 21 v 22 v : = (33) dXa dll Yv -dl2 Xv Equation (33) represents the slope of the isodop in antenna space~
This slope should now be evaluated at the coordinates of the beam by means of combining equations (33) and (22~ to yield:
.

~ZS~l9 !

dY ~d dl X + d dl Y + d dl z a 21 21 a21 22 a 21 23 a (34) dX (d dl X + dl Y + d dl z a 11 21 a22 a 11 23 a 22 11 ad22 dl2 Ya + d22 d (d dl X + d dl Y + d dl 12 11 a12 12 a 12 13 a Simplifying Equation (34): yields:

dY (d dl -d dl )X +(d dl -d dl )Y +(d dl -d dl )Z
a 21 21 22 11 a 21 22 Z2 12 a 21 23 22 13 a dX (d ldll-dl2dll)Xa+(dlldl22-dl2dl2)Ya (dll 23 12 13 a where Xa, Ya, Za are the coordinates of the predetermined beam.
Referring now to FIG. 6, the pertinent angles and slopes are shown.
The tangent of the Angle B between the lines labeled respectively X, Xl is given by:
`:
M - M
tan B -2 1 (3~) : 1 + M2 M

where:

M2 is the slope of X
Ml is the slope of X

Since the slope of X - Ml = O and the slope of Xl = M2 = M, then it 30 follows that:

tan B = M = - 1 dYa/dXa evaluated at the beam since Xl is normal to the isodop.
Also, it follows that:

cos B = ~
~2 + 1 sin B = M
~M2 + 1 Thus, the rotated axes Xl, yl of FIG. 6 may be related to the unrotated axes X, Y by means of the following expression:

[~ ] [ cos B sin Bl [X] (38) yl -sin B cos B

or xl = X cos B + Y sin B ~39) yl = -X sin B + Y cos B (40~

Referring back now to FIG. 3, the inputs to the monopulse rotation processor 68 correspond to the sin B, cos B terms in Equations ~39) and (40). The X, Y terms represent the azimuth and elevation signals on lines 66c and 66b, respectively, and the Xl, yl terms are the rotated azimuth and elevation signals shown on lines 86b and 86a.

ll~S~i9 Having described a preferred embodiment of this invcntion, it is now evident that other embodiments incorporating its concepts may be used. It is felt, therefore, that this invention should not be restricted to its disclosed embodiment but rather should be limited only by the spiTit and scope of the appended claims.

Claims (2)

What is claimed is:

1. An improved method of operating an airborne Doppler navigation radar system utilizing an inertial platform and a monopulse radar having an antenna adapted to produce a plurality of directive beams, the centerline of each one of such beams being in a fixed angular relationship with respect to the radiating face of such antenna, for illuminating different portions of the terrain underlying an aircraft carrying such system so that return signals in each one of such beams may be processed in an azimuth dif-ference and an elevation difference channel to derive measurements of the velocity vector of the aircraft with respect to the earth, in particular, the components of the velocity vector along said beams, such improved method comprising the steps of:
(a) aligning the antenna so that the center of the portion of the terrain illuminated by a first selected one of the directive beams is disposed at an angle with respect to the aircraft flight path;
(b) energizing the antenna to form, sequentially, each one of the plurality of directive beams; and (c) processing the returns from each sequentially formed beam as received by the antenna to derive the weighting values required to rotate the null of the elevation difference channel for the next group of sequentially formed radar beams by "N" degrees, where N is the angular difference in degrees between the null through the centerline of the beam which is tangent to a selected isodop.

2. The method recited in claim 1 wherein the processing step comprises the steps of:
(a) computing the components of the velocity vector along each of the sequentially formed beams and resolving these components into reference space coordinates;
(b) subtracting the reference space velocity components from the velocity data derived from the inertial platform to form velocity error signals;
(c) filtering the velocity error signals; and (d) combining the filtered velocity error signals with the velocity data derived from the inertial platform.