US 3535789 A Abstract available in Claims available in Description (OCR text may contain errors) Oct. 27, 1970 e. E. ROBERTS 3,535,789 I CHART POSITION INDICATING APPARATUS Filed April 12, 1968 5 Sheets-Sheet 1 Oct. 27, 1970 e. E. ROBERTS ,5 CHART POSITION INDICATING APPARATUS Filed April 12, 1968 5 Sheets-Sheet z Oct. 27, 1970 ROBERTS CHART POSITION INDICATING APPARATUS 3 Sheets-Sheet 5 Filed April 12. 1968 3,535,789 CHART POSITION INDICATING APPARATUS Graham Estyn Roberts, London, England, assignor to Decca Limited, London, England, a British company Filed Apr. 12, 1968, Ser. No. 720,948 Claims priority, application Great Britain, July 11, 1967, 31,899/ 67 Int. Cl. B431 /00 U.S. Cl. 33-1 5 Claims ABSTRACT OF THE DISCLOSURE A chart position indicating apparatus in which input signals, each representing one out of a family of hyperbolic position lines in a radio navigation system, are used to drive an index relative to a chart in two cartesian coordinate directions. The chart is distorted so that each family of position lines is represented on the chart by a family of straight lines inclined to both the cartesian coordinate directions and the apparatus is specially constructed to effect an associated coordinate transformation of the signals. This invention relates to chart position indicating apparatus. Chart position indicating apparatus is commonly used in conjunction with a radio navigation system in order to indicate on a chart a geographical position using signals obtained at a receiver which may be, for example, arranged to compare the phases of signals radiated by various pairs of stations in the radio navigation system to obtain outputs respectively representing one of two or more position lines whose intersection defines the geographical position of the receiver. In order to display the aforementioned geographical position, the chart position indicating apparatus commonly includes an index which is relatively movable in two coordinate directions with respect to a chart, the relative movement of the index and chart being controlled in each direction from an output or outputs of the receiver (that is, using at least one signal or set of signals representing a determined position line). In general, however, the coordinate system of the movement of the chart and index (usually cartesian) is different from that of the radio navigation system, whose position lines are commonly hyperbolic but may be elliptical or circular. It is necessary therefore to effect a co-ordinate transformation of the receivers outputs before they can be used to control the relative movements of the index and chart. Although it is possible to do this with a suitably programmed or constructed computer, it is in many ways simpler and cheaper to draw the chart with a projection that effects the required co-ordinate transformation and to use a sim plified computer. This introduces some distortion of the geographical features of the chart and in the past the distortion has often been excessive. It is one of the objects of the present invention to provide chart position indicating apparatus for which the charts can be less distorted than hitherto and be at the same time easily prepared. According to the present invention, apparatus for indicating on a chart the geographical location represented by the intersection of two position lines which are respectively members of two families of position lines of a radio navigation system adapted to provide outputs identifying a position line in each family, comprises a chart of .a geographical area embracing said location, the chart being distorted such that the two families of position lines are each representable on the chart by a set of lines such that the rates of change of position, with respect to the respective output of the radio navigation system, of the intersections of a line in a set with all the lines in the other set 3,535,789 Patented Oct. 27, 1970 are all constant, an index relatively movable in two orthogonal co-ordinate directions over the surface of the chart, the lines in at least one set being inclined to both the two orthogonal directions; and computing means adapted to obtain the orthogonal co-ordinates of the geographical location with respect to the chart by solving two equations which have the aforementioned outputs as variable quantities using a programme that relies on the constancy of the aforementioned rates of change, means being provided for displacing said index relative to the chart in the two orthogonal co-ordinate directions in accordance with the values of said orthogonal co-ordinates determined by said computing means. Although the chart is used with an index movable in an orthogonal co-ordinate system, the particular manner of drawing the chart makes it possible, in general, to produce a chart with very little distortion from a geographical chart, using conventional navigational charts which are usually drawn with polyconical or mercator projections. This may be seen from the following considerations. If the lattice lines of the radio navigation system are drawn on a conventional navigational chart, they are usually curved, but for a small region it is possible to draw straight lines approximating very closely to these lattice lines. With chart position indicating apparatus heretofore it has been necessary to distort the chart so that the lattice lines are co-ordinate lines in a cartesian co-ordinate system: with the apparatus of the present invention on the other hand, the charts are arranged so that although at least the lines in one set are normally straight, the lattice lines need not be co-ordinate lines in a cartesian co-ordinate system and in practice they may be arranged so that distortion only arises by assuming the constant nature of the aforementioned rates of change. It is found that the approximation can for most practical purposes be very good. Thus, one of the usual features of the invention is that the position lines in both families are when represented on the chart inclined to both the orthogonal directions of relative movement between the index and chart. The present invention may be better appreciated if the most usual form of it is discussed in some detail. Preferably both families of lines are representable on the chart by a set of straight lines and preferably the aforementioned orthogonal co-ordinate directions are cartesian. With this form of the invention, the conversion of the navigation systems outputs into the cartesian co-ordinate system of the chart and index may proceed as follows, reference being made to the accompanying FIG. 1, in which M and N represent the values of two intersecting position lines (and for convenience denote the lines themselves). . It is possible to define, in each set of lines, two reference lines, M M and N N whose intersections a a etc. (where a is the intersection of M and N and the parts of M M N and N between the intersections form a quadrilateral when drawn in the said cartesian co-ordinate system. If, for example, 0 and are the points at which the line M intersects respectively the lines joining a a and 1 1 then, by simple ratio, the x co-ordinate of c divides the x co-ordinates of a and a in the same ratio as M divides M and M and in the same ratio as the x co-ordinate of c divides the x co-ordinates of @1 and a A similar equality is true for N and thus it is possible to express X and Y the cartesian co-ordinates of Z, the intersection of the lines M and N, by equations in which only the cartesian co-ordinates of the a (r, 5:1 or 2), the navigation system outputs corresponding to position lines through the a (all known constants) and M and N appear. Thus the computing means need only be fed with a few constants and M, N and the values of X, and Y may easily be computed to control the displacing means for the index. It should be made clear, and will be explained in more detail hereinafter, that the resultant expressions for X and Y can be more conveniently stated in the form k MN+k M+k N+k and k' MN+k M+k' N-|-k' where all the k are constant for any point on the chart and each are derived from the cartesian co-ordinates of the a This leads to a simplification in the computation of X and Y According to this aspect of the invention, chart position indicating apparatus for use with a radio navigation system adapted to provide, at a mobile receiver, two outputs, M and N, each identifying a position line in a respective one of two families of intersecting position lines, comprising a chart drawn by assuming the position lines all to be straight computing, in cartesian co-ordinates with respect to the chart, the co-ordinates of the four intersections of a pair of position lines in one family with a pair in the other family and plotting the chart such that for any point on the chart, either position line through the point divides either selected position line in the other family in the same ratio as the cartesian co-ordinates of said point divide the respective cartesian co-ordinates of the intersections lying on said selected position line in the other family; an index relatively movable in the two cartesian co-ordinate directions over the surface of the chart, means for displacing said index relative to the chart to indicate the chart position (X Y of said location automatically in accordance with two outputs, defining X and Y respectively, of a computing means responsive to the navigation system outputs, said computing means being arranged to compute X, and Y as the values of expressions of the form k MN+k M+k N+k and k' MN +k M+k' N+k' or expressions which can be stated in that form, wherein k k k and k., and k k k;., and k' are constant for any point on the chart and are derived from the cartesian co-ordinates of the intersections. The aforementioned intersections would normally lie outside the area covered by the chart; although it is possible to use directly the values of the co-ordinate defining the positions of the intersections to obtain the values of the cartesian co-ordinates of any other point, it is preferable (as already mentioned) to define various constants comprising combinations of the co-ordinates of the intersections to obtain simple expressions for calculating X and Y These various constants are peculiar to a chart and signals defining them would usually be automatically fed to the computing means when a particular chart was in use. They are usually called chart constants. The present invention is particularly applicable but not restricted to radio navigation systems of the hyperbolic phase comparison kind. In such a system, the position lines are families of confocal hyperbolae each having two transmitting stations in the navigation system at their foci. It is well known that if one of the two stations is common to two pairs of stations it is possible to define intersecting families of hyperbolae and to define the position of a receiver as the intersection of two members of each family. For convenience, the present invention will be particularly described with reference to such a system 'but it will be understood that the present invention is also applicable to other navigation systems. For example, it may be used with a two range system which would provide outputs defining one each in two families of concentric circular position lines the intersection of the defined position lines determining the location of the receiver. Furthermore, the invention might also be used with time displacement radio navigation systems which, like phase comparison radio navigation systems, possess position lines in the form of families of confocal hyperbolae. A preferred, but not essential feature of the invention is accordingly its use in conjunction with a radio navigation system of the kind having at least three radio transmitting stations arranged to radiate phase-locked radio frequency signals of different but harmonically related frequencies and in which a mobile receiver is arranged to compare the phases of signals received from two pairs of stations, one station being common to the two pairs, so as to obtain two outputs each representing a different position line through the location of the receiver. The invention includes within its scope a chart for use in chart position indicating apparatus of the kind which is controlled by two outputs of a radio navigation system, each output representing a position line in one of two families of intersecting position lines and which cooperates with the chart to provide relative movement of an index and the chart in two orthogonal co-ordinate directions so that the index and chart together indicate the geographical location defined by the intersection of the position lines defined in turn by said outputs, the chart comprising a chart member bearing markings representing a geographical area that includes said location, the chart being distorted so that the two families of position lines are each representable on the chart by a set of straight, parallel lines, at least the lines in one set being inclined to both said co-ordinate directions. It is convenient to arrange (as previously indicated), that the chart provides the constants necessary for the programme used by the computing means; accordingly, the member may carry sensible indications of at least some of a plurality of constant terms which are normally required by the apparatus for providing a conversion of the said outputs into signals appropriate for effecting relative movement of the index and chart, whereby the chart in use can co-operate with sensing means to provide signals defining the constant terms. In the following, reference will be made to the rest of the accompanying drawings in which: FIGS. 1 and 2 illustrate in part transmitting stations and position lines in a hyperbolic phase comparison navigation system; FIG. 3 illustrates the arrangement of a chart in accordance with the present invention; and FIGS. 4 and 5 illustrate in simplified block diagram form a receiver and chart position indicating apparatus, constructed in accord with the invention and for use with a radio navigation system of the kind shown in FIG. 2. Before explaining the various approximations and mathematical computations contemplated by the present invention it is appropriate to describe by way of example a typical radio navigation system that is suitable for the present invention in order that the features of the present invention may be fully appreciated. In FIG. 2 is shown the configuration of four transmitting stations in a typical hyperbolic phase comparison radio navigation system. The four transmitting stations are spaced apart and are arranged in a star-shaped pattern with the master station M near the centre of the pattern, the red, green and purple slave stations R, G and P being disposed around the master station. For the purposes of the present invention it may be assumed that these stations transmit continuous radio frequency signals on different but harmonically related frequencies. The transmissions from the slave station are locked in phase to the signals from the master station further, for example, by measurement at a mobile receiver 10 of the phase relationship between signals from the master M and the slave station R a position line 12 through the geographical location of the receiver is determined. If as is usual, the frequencies radiated by the two stations are different it is necessary to bring them to a common comparison frequency before measuring their phase relationship. The position line 12 is one of a family of confocal hyperbolae having the master and slave stations M and R as foci. Some others of these hyperbolae are shown at 11, 13 and 14. By using a further phase comparison between the signals radiated from the same master station M and the slave station P a further hyperbolic position line 22 is obtained, the intersection of lines 12 and 22 defining the location of the receiver 10. The position line 22 is one of a family of confocal hyperbolae having the master and purple slave stations as foci, other position lines in this family being 21, 23 and 24. The other halves of the position lines in the master-red set have been omitted and the position lines for the master-green set have been entirely omitted for convenience. It is common practice to refer to the results of the phase comparison between the master-red signals as the red co-ordinate and to that between the master and purple signals as the purple co-ordinate and thus they will be called hereinafter. Also shown in FIG. 2 is an area 17 which is to be covered by a chart and which embraces the geographical location of the receiver 10. The area shown is approximately rectangular but in practice it may deviate slightly from rectangular. It will be understood, however, that the chart itself is in normal practice always rectangular. It will be noted from FIG. 2 that the two families of position lines are divergent over the area covered by the chart; this will not always be so, the lines in a family being parallel if the chosen area is on a base line of the navigation system. It will also be noted that at least over the area covered by the chart the various position lines are approximately straight. In FIG. 3 is shown a chart 18 having some of the position lines of FIG. 2 projected on it in accordance with the present invention. In FIG. 3, the position lines 11, 12, 13'etc. are shown as the straight lines 11a, 12a, 13a etc. The latter lines form sets of lines which are not, of course, families of position lines, but which represent the families of the respective position lines. It will be appreciated that it is not necessary to draw lines corresponding to the aforementioned position lines actually on the chart but it is necessary to be able to calculate them so that the chart is suitable for use with the computing apparatus to be described later and so that the positions of the various geographical features on the chart may be properly located. For each chart, it is convenient to choose from each family of position lines two reference position lines which are for the red set, the lines 11 and 13 and for the purple set are the lines 21 and 23. The present invention places no particular restriction on the choice of the two lines except that the four points A, B, C and D which are defined as the intersections of the two pairs of reference position lines (or their corresponding lines in the charts co-ordinate system), are preferably outside, but enclose, the area represented by the chart. It is at this point convenient to recall that in a hyperbolic phase comparison system it is usual to make a phase comparison at a high frequency in order to define a number of possible position lines (which differ by integral multiples of a complete cycle of phase change at the appropriate comparison frequency) and then to make a further, coarse comparison at a much lower frequency in order to define which of the possible position lines is the one passing through the position of the receiver. It is usual to call a complete cycle of phase change at the lower frequency a zone whereas it is usual to call a complete cycle of phase change at the higher comparison frequency a lane. It will be seen that a zone comprises a number of lanes, the number being equal to the ratio between the upper and lower comparison frequencies. Since a chart position indicating apparatus and modern-day computing means are digital in nature it is desirable that constant terms defining the reference position lines or the intersections thereof are chosen to have as few digits as possible and it is with the present invention quite possible to chose the various points A, B, C, D so that their coordinates (or, as will be seen, constants derived therefrom) associated with them are integral powers of two or numbers having only a few digital places. A chart would be associated with an index (not shown in FIG. 3), the chart and the index being relatively movable in two orthogonal directions which will be called the X and Y (cartesian) co-ordinate directions hereinafter. These directions are for convenience chosen to be parallel to a respective edge of the chart but this restriction is not essential to the present invention. It is normally only essential when one of the movements in a co-ordinate direction is effected by movement of the chart whereas the movement in the other co-ordinate direction is effected only by movement of an indicator carrying a marker to define the index. However, such considerations will be discussed in detail hereinafter. Referring again to FIG. 3: for calculating the position of the point which is the intersection of the two determined position lines it is necessary to make use of the special manner in which the charts are drawn. As has been mentioned, the chart is drawn so that (for example) equal changes in the red hyperbolic co-ordinate produce equal changes in the positions of the intersections of the line relating to that co-ordinate with all the lines representing position lines in the purple family and vice versa. With this assumption it will be seen (for example) that the ratio EC/DC is equal to the ratio ZJ/H] and also equal to the ratio FB/AB. Similar relationships hold for the purple position lines. The assumption can only be exactly true and produce no distortion if the lines 11 to 13 (in FIG. 2) were parallel and evenly spaced, and the lines 21 to 23 were evenly spaced and parallel, with lines defining intermediate co-ordinates also parallel to the other members in the respective family. In the following, for any point Q, R,,, P.,, X and Y define respectively the red co-ordinate, the purple coordinate, the X cartesian (chart) co-ordinate and the Y cartesian (chart) co-ordinate of point Q. The red and purple co-ordinaes relate to the radio navigation system whereas the X and Y co-ordinates relate to the co-ordinate system of the chart. It will be understood that R and P are examples of the M and N lines mentioned hitherto. For the time being, it will be assumed that it is possible to consider that the red, purple, X and Y co-ordinates of the points A, B, C and D are defined and that the red and purple co-ordinates of the point Z are known. Although this would appear to require 16 fixed quantities and two variable quantities, there are in fact, only 12 known quantities since (for example) R =R etc. The X co-ordinate of point B, i.e., X divides the interval X -X in the same proportions as the co-ordinate interval P,,P is divided by point E. Thus, it may be assumed that and and 7 and z d r f d(- b n)+ a From Equation 2 z d r r s d (A e)+ o Substituting Equations 3 and 4 in Equation 2, there is obtained z d a d) X. X..+X. X. X.) z d) a d]+ Td e d)+ d (5) It will be seen that Equation 5 defines the unknown X co-ordinate of the point Z in terms of the X co-ordinates of the points A, B, C and D, the red and purple coordinates of the same points (all known quantities) and the respective inputs, R, and P,,, from the radio navigation system; a similar equation holds true for the Y coordinate for the point Z but for convenience only the manipulation and solving of Equation 5 will be discussed in detail. One way of handling Equation 5 is to make provision (as hereinafter explained) for the storage of all the constant terms appearing therein and when it is required to compute the required value of X to perform the various additions, subtractions and multiplications defined in Equation 5. However, it is appropriate to simplify the computation by, instead of storing the values of the coordinates X X etc., to store various combinations and derivations thereof as will now be explained. It is convenient to recognise a particular set of constants in Equation 5 which are as follows: C =P P =2 where p: any integer between +5 and 0 C =R,,R :2 where r=any integer between +5 and 0 C =P =any integer (the units being Mi zone) C =R =any integer (the units being Mt zone) C and C are in the same units as O; and C Equation 5 then reduces to: X (2 (r+p)) z 8) z 1) 4 Since the multiplication by an integral power of 2, where digital computing apparatus is being used, only amounts to a rightor left-shift by a number of digital places, it is convenient to define this in one of the constant terms. This is equivalent to using new constants defined as follows: When the corresponding equation for Y is written, the constants C and C reappear in the same role; in fact, R is never used by itself: only R -C is used. Similarly, P is never used by itself. New values are required for the other constants and the total list of constants required is When the corresponding equation for Y is written, the constant C and C reappear in the same role; in fact, R is never used by itself: only R C is used. Similarly, P is never used by itself. New values are required for the 8 other constants and the total list of constants required is K Kz, XK3, C K 1(2, K3, C C7 and C8 is only ten constants. Because C7 and C are common to X and Y, it may be convenient to form R C and call it R. This can be stored and used for both X and Y determinations. The subtraction of C consists mainly of zeros when expressed as a binary number. if G, and C are quoted in the negative form, no subtractions are needed, i.e., -R becomes R +(C and it will be seen that the solution of Equations 6 and 7 require only eight multiplications and eight additions. It will be seen that although the constants in Equations 6 and 7 are related to and are defined by the various coordinates of the points A, B, C and D, the connection between the two is complex. It might be possible to simplify Equation 5 in other ways but essentially the computation of the values X and Y relies on the fact that the programme for solving X and Y must always rely on the constancy of the aforementioned rates of change. This is the basis of the Expressions 1 and 2 from which the Equations 6 and 7 are derived. It is by the same token a measure of the distortion present in the chart. From the foregoing explanation of FIG. 3 it will be appreciated that in order to move an index and a chart relatively in the X and Y co-ordinate directions it is necessary to provide computing means for solving, using the appropriate outputs from the navigation systems and constants, Equations 6 and 7. How the outputs are derived and the constants may be stored for use in a computation and how the chart and index may be driven will now be explained. For convenience of explanation it will be assumed that the stations P, M, G and R in FIG. 2 radiate continuous radio frequency signals of frequencies 5 6f, 9] and 8f respectively, 1 being (for example) in the region of 14 kHz. In a mobile receiver as shown in FIG. 4, to determine the hyperbolic co-ordinates the signals are picked up by an aerial 40 and are amplified by amplifiers 41 and and - 44 tuned respectively to the frequencies 5 6 9 and 8). The outputs of the amplifiers are fed to frequency multipliers 45 to 48. The frequency multiplier 46 associated with the 6 signal from the master station separately multiplies the received 6] signals by factors of 3, 4 and 5 to provide outputs of 18], 24] and 30]. The 30 output from the multiplier 46 is fed to one input of a phase discriminator 50 to other input of which is fed a 30 output from the multiplier 45 which multiplies the received purple 5f signal by a factor of 6. The discriminator 50 provides two outputs representing the sine and cosine of the phase angle beween the master and purple slave station signals at the receiver. The outputs are fed to a digitiser 51 which provides a digital output representative of the phase angle, that is, it defines the master-purple slave hyperbolic co-ordinate of the receiver. The 18 output from the multiplier 46 is fed to one input of a discriminator 52 to the other input of which is fed an 18 output from the multiplier 47 which multiplies the received 9 green slave signal by a factor of 2. The outputs from the discriminator 52 are fed to a digitizer 53 to yield an output in digital form representative of the phase difference between the l8 signals from the multipliers 46 and 47, i.e. the master-green slave hyperbolic co-ordinate. A 24) output from the multiplier 48, which multiplies the frequency of the received red slave signals by a factor of 3, is fed to one input of a discriminator 54 to the other input of which is fed the 24f output from the multiplier 46. The outputs from the discriminator 54 are fed to a digitiser 55 to provide an output in digital form representative of the phase difference between the 24 signals from the multipliers 46 and 48, i.e. the masterred slave hyperbolic co-ordinate. A hyperbolic phase comparison navigation system of the above kind is with the exception of the provision of digitisers more fully described in the specification of British Pat. No. 620,479 and reference may be made to that specification for further explanation of the construction and manner of operation of such a system. The digitisers 51, 53 and 55, which might readily be replaced by a single time-shared digitiser or other digitisers capable of being driven by the sine and cosine signals from the discriminators, accordingly provide indications with respect to three sets of confocal hyperbolae, each set having a master station and the appropriate slave station as its two foci. The positional information in binary digital form from the digitisers 51, 53 and 55 is fed through gates 56, 57 and 58 which are controlled as required by a computer 59 to feed the positional information into the computer. The computer 59, which is not illustrated in detail since its manner of construction will be readily apparent to those skilled in the art, essentially comprises a multiplier and adder 60 to which are fed at appropriate times two hyperbolic co-ordinates obtained from the phase comparisons and the constants hereinbefore mentioned, binary to Gray code converter 66 receiving the determined values of X and Y and providing two digital outputs in Gray code defining X and Y The computer is programmed to feed to the multiplier and adder the signals at appropriate times so that the multiplier and adder yields X and Y by solving Equations 6 and 7. The description of the invention so far has been concerned with the drawing of a chart to conform with this invention and with the obtaining of signals which define with respect to the chart the position of the receiver as determined by the phase comparisons between signals received from various stations in the navigation system. The X and Y signals, fed out from the computer on lines 74 and 75 are to be used to control the displacement of an index with respect to a chart automatically so that a chart display can automatically indicate the geographical location of the receiver. A number of ways of controlling the displacement of the index are possible. It is possible to provide means for storing determined values of X and Y and comparing these stored values with further values of X and Y obtained from a succeeding determination, the displacement of the index being made in accordance with each change in X and Y from the stored to the further values. Such a chart display is known as an integrating display since after an initial setting the index and chart are relatively displaced by the integral of the changes in the X and Y cartesian co-ordinates of the chart and index system. However, with such a system it is necessary to set the display initially to indicate the correct position of the receiver at the instant of setting, which for obvious reasons may not always be possible. It is preferable to provide, for each co-ordinate direction of displacement of the index a servo system which responds to the respective (X or Y) co-ordinate determined from the phase comparisons and to the actual position indicated by the index relative to the chart for that co-ordinate direction and to effect relative displacement of the chart or index in accordance with any difference between the actual and indicated co-ordinates. For this purpose, the chart may be movable to and fro relative to the index to indicate one co-ordinate direction and the index may be movable to and fro in the transverse direction to indicate the other co-ordinate, there being associated with both the chart and the member bearing the index a separate set of digitally coded tracks arranged to co-operate with a respective sensing means to provide coded signals defining the coordinate indicated by the chart or member, a respective servo mechanism being arranged to compare the respective coded signals with a respective value of X or Y from the navigation system and to drive the chart or member to the co-ordinate position defined by said value of X or Y. Such a system is more particularly described and claimed in the specification of British Pat. No. 914,185 or the corresponding US. Pat. No. 3,113,313 and reference may be made to the specification of that patent for further details of the construction and manner of operation of such a system; it will be understood, however, that what follows, which is given by way of completeness, is by no means essential to the present inven tion and considerable variations, which will be pointed out as they occur, may be made. Referring to FIG. 5, the signal on line 74 is fed to a servo system 76 together with another input on a line 95 obtained in a manner to be described. The servo system 76 is arranged to produce an error signal, which is fed to a motor 78 to drive a lead screw 79 on which is mounted a carriage 80. Rotation of the motor moves the carriage 80 transversely with respect to a chart 83 so that one co-ordinate of the position of the receiver (for the sake of example, the Y co-ordinate) is determined by the position relative to the chart of an index line 81 on a transparent cursor 82 which is carried by the carriage 80. On the line 81 there is an index cross or circle 81a which serves as a reference for the other co-ordinate and hence is to indicate when used with a chart the position of the receiver. The servo 76 drives the carriage 80 until the value of the co-ordinate indicated by the line 81 is the same as that determined for the receiver by the computer 59. This type of drive and automatic drive display is fully described in the aforementioned British patent specification No. 914,185. A similar servo system 77 is responsive both to an input on a lead 75 from the computer and to an input on a lead 94 which indicates the position of the chart with respect to the index 81a. The chart is one of a number on a chart roll 72 mounted for movement between the two spools and 91 which may be driven to move the chart in its longitudinal direction by a motor 86. The output from the servo 77 is fed to the motor 86 via a shaft 87 and links 88 and 89 (which have been illustrated diagrammatically for convenience) and thus the index 81a is movable relative to the chart in the two orthogonal X and Y directions. It is not essential to use a movable chart; in particular two cursors carried on respective carriages may be movable in orthogonal directions across the chart, which for this purpose would be fixed in position. Also, as is disclosed in the specification of British Pat. No. 1,090,- 153 it is possible to use an index which is formed as the intersection of two orthogonal lines drawn on respective flexible members which are movable in the longitudinal direction with respect to the chart. One line would be drawn in an oblique direction on one transparent flexible member and the other drawn on the other member but at right angles to that on the first member. Reference may be made to the specification of that patent for further details. For controlling the servo mechanism, the chart and the carriage are each associated with digital tracks which cooperate with sensing means to define the actual co-ordinate indicated by the chart and carriage. Along the sides of the chart member 72 parallel to and along one side of the chart 83 is a plurality of tracks of opaque digital markings 93 performing two functions. Firstly, a number of tracks are arranged in a repetitive cyclic code such as Gray code so that a corresponding number of photocell sensing means 92 may provide an output indicating the actual position of the chart. The digits sensed by the individual photocells 91 are fed on line 94 to the servo system 77 to enable the necessary comparison between the actual and required chart positions to be made. However, further markings 93a are provided to supply the aforementioned constants which are used by the computer to define the X and Y co-ordinates. It will be appreciated that the constants defining the various coordinates and constants are peculiar to each chart and when a chart is inserted in the apparatus, the photocells read the respective constants and feed them to a temporary store from which they are fed out at appropriate times by a gating system to the multiplier and adder. It will be apparent that the photocells are indicated only diagrammatically and typically there may be several rows of them since each constant in general comprises a plurality of digits. As will be made clear, however, the extreme latitude in the choice of the reference position lines that is provided by the present invention enables the number of digits in each constant to be kept to a minimum. For the other co-ordinate direction, the carriage 80 is associated with plurality of Wipers 84 co-operating with digitally coded tracks 85 to provide on the line 95 Gray code signals representing the position of the carriage and therefore the index 81a in the Y direction. The servo 76 forms the difference between the signals on the lines 74 and 95 and drives the motor 78 to move the carriage 80 to the correct position in the known manner. I claim: 1. Apparatus for indicating on a chart the geographical location represented by the intersection of two position lines which are respectively members of two families of position lines of a radio navigation system adapted to provide outputs identifying a position line in each family, comprising a chart of a geographical area embracing said location, the chart being distorted such that the two families of position lines are each representable on a display surface of the chart by a set of parallel lines, the rates of change of position, with respect to the respective output of the radio navigation system, of the intersections of a line in a set with all the lines in the other set being all constant; an index and the chart being movable relative to each other in two orthogonal co-ordinate directions parallel to the display surface of the chart, the lines in at least one set being inclined to both the two orthogonal directions whereby each orthogonal co-ordinate of the geographical location with respect to the chart is determined by a respective one of two equations which each have the aforementioned outputs as variable quantities; and computing means for forming said two equations and thereby determining each said orthogonal coordinate; drive means being provided for displacing said index and chart relative to each other in the two orthogonal co-ordinate directions in accordance with the values of said orthogonal co-ordinates determined by said computing means. 2. Apparatus as claimed in claim 1 in which the chart is distorted such that both families of position lines are representable on the chart by a set of straight, parallel lines. 3. Apparatus as claimed in claim 2 in which the two orthogonal co-ordinates are cartesian. 4. Chart position indicating apparatus for use with a radio navigation system adapted to provide, at a mobile receiver, two outputs, M and N, each identifying a position line in a respective one of two families of intersecting position lines, said outputs defining thereby two position lines through the location of the mobile receiver, said apparatus comprising a chart drawn by assuming the position lines all to be straight, computing, in cartesian coordinates with respect to the chart, the co-ordinates of the four intersections of :a pair of position lines in one family with a pair in the other family and plotting the chart such that for any point on a display surface of the chart, either position line through the point divides either selected position line in the other family in the same ratio as the cartesian co-ordinates of said point divide the respective cartesian co-ordinates of the intersections lying on said selected position line in the other family; an index and said chart being movable relative to each other in the two cartesian co-ordinate directions which directions are paral lel to the display surface of the chart, means for displacing said index and chart relatively to each other to indicate the chart position (X Y of said location automatically in accordance with two outputs, defining X and Y respectively, of a computing means responsive to the navigation system outputs, said computing means being arranged to determine X and Y by computing the values of expression of the form k MN+k M+k N+k and k' MN+k' M+k N+k' wherein k k k and k and k k k and k',; are constant for any point on the chart and are derived from the cartesian co-ordinates of the intersections and a representation on the chart of a position line in either family of position lines is inclined to both the orthogonal co-ordinate directions of relative movement between the index and chart. 5. Apparatus as claimed in claim 4 in which the radio navigation system is of the kind having at least three radio transmitting stations arranged to radiate phaselocked radio frequency signals of different but harmonically related frequencies and in which a mobile receiver is arranged to compare the phases of signals received from two pairs of stations, one station being common to the two pairs, so as to obtain two outputs each representing a different position line through the location of the receiver. References Cited UNITED STATES PATENTS 6/1952 Copping. Patent Citations
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