Abstract:
An improved navigation and display system especially suitable for small aircraft rapidly determines the attitude (roll and pitch) of an aircraft from measurements made by an inexpensive, single receiver and antenna responsive to signals obtained from a satellite positioning system, e.g., the Global Positioning System. In contrast to conventional attitude navigation systems, which determine and display conventional (body-axis referenced) pitch angle, the present system determines and displays the “pseudo-attitude” or stability axis roll and pitch (flight path) angle, so that the pilot has an instantaneous comprehension of the actual flight path angle of the aircraft without needing to correct for angle of attack as in conventional attitude information systems. A Kalman filter with a short (˜0.5 second) time constant provides data at a sufficient rate (e.g., ˜10 Hz) to enable real-time flight with the system. With the additional availability of measured or estimated angle of attack, this system can also display conventional roll and pitch attitude angles. The pseudo-attitude data is incorporated into an integrated flight information system which derives all its data from the single GPS sensor and which presents all the information needed for navigation and control of the aircraft in which it is mounted.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application is a continuation in part of U.S. patent application Ser. No. 08/890,268, filed Jul. 9, 1997, now abandoned and entitled “Aircraft Attitude Indicator”. 
    
    
     BACKGROUND OF THE INVENTION 
     A. Field of the Invention 
     The invention relates to aircraft instrumentation and control and, more particularly, comprises a method and apparatus for improved attitude determination and navigation, and an improved integrated flight information and control system in connection therewith. 
     B. Prior Art 
     Safe control and navigation of an aircraft requires a continuous stream of relatively accurate information concerning the dynamics of the aircraft as it moves along its flight path from origin to destination. Information such as the airplane&#39;s altitude, velocity, heading, attitude (including at least pitch and roll information), among other data, are important for piloting the aircraft. Extensive, highly accurate, and correspondingly expensive, instrumentation has been developed for commercial and military aircraft to fill the required need. Such instrumentation is far too expensive for smaller, private aircraft, and the latter therefore must make do with simpler, cruder navigation systems providing more limited, and commonly less accurate, navigation data. 
     In larger commercial and military aircraft, inertial navigation systems have commonly been the instrumentation of choice. Such systems rely on the ability of gyroscopes to maintain their orientation in space once initialized, and to provide relatively sensitive indications of accelerations tending to disturb that orientation. These systems typically have significant mass and bulk, are expensive to acquire, and require continued, frequently costly, calibration, maintenance and repair to ensure continued acceptable performance. Their use has therefore been confined largely to larger commercial and military aircraft. 
     The advent of satellite positioning systems, such as the Global Positioning System (GPS) established and maintained by the United States, or the GLONASS system established and maintained by Russia, offers the possibility of significantly reducing the mass and bulk of many present navigation systems, and possibly their cost as well. Navigation systems using these facilities rely on the measurement of phase differences in received radio signals from a number of satellites in order to determine the position in space of the receiver, and therefore the platform on which the receiver is carried, with respect to the satellites. Because the position and velocity of the satellites relative to earth at any given time is known, the position and velocity of the receiver with respect to an arbitrary earth-based reference system can be determined from measurements with respect to the satellites. 
     In addition to navigation functions, such systems can also be used to determine the attitude of the vehicle in which the system is mounted. Numerous attitude determination systems based on GPS measurements have been proposed and such systems take many forms. For example, U.S. Pat. No. 5,548,293 issued Aug. 20, 1996 to Clark E. Cohen and entitled “System and Method for Generating Attitude Determinations Using GPS” proposes the use of a multiplicity of antennas on a vehicle whose orientation with respect to a reference frame is to be determined. The antennas provide a multiplicity of baselines from which the orientation may be found. Multiple baselines are used in order to resolve the position ambiguity inherent in measurements from antennas typically separated by many meters resulting from the short wavelengths used in GPS signaling (on the order of 0.2 meters). The use of a number of antennas, of course, increases the cost of the system, as well as the cost of installation, and inhibits the application of such a system to small aircraft in particular. Further, because of the very short wavelength, significant errors are introduced in the measurement whenever the distance between the antennas changes, as it is susceptible to do in response to stresses imposed on the aircraft during flight. 
     Some systems, such as that described in U.S. Pat. No. 5,534,875 issued Jul. 9, 1996 to Debra Diefes et al., entitled “Attitude Determining System for Use with Global Positioning System”, utilize a single GPS receiver and antenna on board the vehicle, but use standard inclinometers to determine the pitch and roll of the vehicle platform. Such hybrid systems fail to make use of the capabilities of GPS for attitude determination. 
     Still other systems, such as that described in U.S. Pat. No. 5,451,963, issued Sep. 19, 1995 to Thomas A. Lempicke, entitled “Method and Apparatus for Determining Aircraft Bank Angle Based on Satellite Navigational Signals”, utilize a single on-board GPS system that determines certain attitude information, such as bank angle, only under conditions of level flight, thus precluding effective use of the system in arbitrary maneuvers such as climbing or descending turns in which accurate attitude information is often most essential, particularly in connection with takeoff and landing. Further, the system posits a mode of operation (determining bank angle as inversely proportional to aircraft speed) which is not explained and not achievable by anything described in the patent. 
     Still another GPS-based system is described in U.S. Pat. No. 5,406,489, issued Apr. 11, 1995 to LaMar K. Timothy et al., entitled “Instrument for Measuring an Aircraft&#39;s Roll, Pitch and Heading by Matching Position Changes Along Two Sets of Axes”. This patent uses both a GPS receiver and a multiplicity of accelerometers oriented along the three aircraft body axes, respectively, to determine attitude and other navigation information. Again, the hybrid nature of the system increases its cost, complexity, and maintenance requirements. 
     SUMMARY OF THE INVENTION 
     A. Objects of the Invention 
     Accordingly, it is an object of the present invention to provide an inexpensive but relatively accurate and reliable attitude determination method and apparatus for aircraft navigation. 
     Another object of the invention is to provide a simple, low-cost navigation system for determining attitude (roll, pitch) information for small aircraft. 
     Further, it is an object of the invention to provide an inexpensive attitude determination system that is useful as a backup for more elaborate navigation instrumentation systems. 
     Still a further object of the invention is to provide a simple, relatively inexpensive attitude indicator. 
     Yet another object of the invention is to provide an improved, economical integrated flight information and control system for control and navigation of aircraft. 
     B. Brief Summary of the Invention 
     In accordance with the present invention, we provide a method and apparatus for readily and inexpensively determining and displaying flight path angle and roll angle of an aircraft despite its engagement in arbitrary, but balanced, maneuvers such as ascent or descent accompanied by banked turns, conditions which present problems for many navigation systems. Because of its simplicity, and its reliance on a single source of measurement data for the requisite input information, the system is extremely simple and inexpensive to construct, install, and maintain. It is particularly suited for installation and use in small aircraft, where the cost of more elaborate and more expensive systems considered essential for navigation on larger aircraft effectively preclude their acquisition and use. Further, the system is sufficiently accurate and reliable to be used as a supplemental system on larger aircraft for use in integrity checking, as well as for backup in the event of failure of the primary system. The present invention obviates the use of a multiplicity of receivers or antennas or supplemental orientation indicators, and is useful throughout the entire range of flight dynamics commonly encountered in air navigation. 
     In particular, in accordance with the preferred embodiment of the invention, we determine the flight path angle γ and roll angle φ s  of an aircraft and display these as parameters to the pilot as a principal measure of the aircraft attitude at a given moment. For convenience of reference, the stability axis roll angle φ s  and flight path angle γ will on occasion be referred to hereinafter as the “pseudo-attitude” parameters or, more simply, as “the pseudo-attitude”, in contrast to the conventional attitude parameters based on a body axis, namely, roll φ and pitch θ. The roll angle φ s  (“pseudo roll”) is advantageously determined with respect to the stability axis of the aircraft, as opposed to the body axis, since determination of the stability axis roll angle requires no knowledge of the angle of attack, a parameter which is frequently not known or readily determinable with accuracy in small aircraft. Similarly, the flight path angle γ or “pseudo pitch” is determined about the stability axis of the aircraft in order to present a direct indication of the path of the vehicle through space. This is in contrast to conventional attitude navigation systems which determine and display roll angle and pitch about the aircraft&#39;s body axis. Although the body-axis and stability-axis roll angle are typically nearly equal to each other, this is not the case with pitch angle, which differs from flight path angle by the angle of attack. Thus, in conventional systems, when a pilot wishes to maintain the aircraft on a particular flight path angle, he/she can not navigate by the pitch angle alone, but must correct it by the current angle of attack, a value which is frequently known only imprecisely at best in small aircraft, and which can change from moment to moment, dependent on the flying situation. With the aid of the present invention, the pilot is presented directly with the flight path angle and can thus navigate and control the aircraft more readily. 
     Further in accordance with the present invention, we have developed a simple and reliable method and apparatus for determining the desired fight path and roll angles under arbitrary conditions as long as the flight dynamics are balanced, i.e., the forces required for any centripetal acceleration associated with a maneuver are balanced by the lift and gravitational forces associated with that maneuver, as described more fully below. Specifically, from the data obtained from a measurement system such as a GPS system which provides at least periodic measurements of the position of the aircraft, we obtain velocity and acceleration data from which the desired attitude (flight path angle, roll) information is determined. In accordance with the preferred embodiment, a Kalman filter receives the measurement data and provides the required information from which the desired attitude is determined. In our filter, the aircraft is modeled as a triple integrator system, the filter then providing estimates of the jerk (impulse), velocity, and acceleration of the aircraft during its maneuvers. The latter (velocity and acceleration) estimates provide the requisite information for determination of the desired attitude as described more fully below. 
     In particular, in accordance with the preferred embodiment of the invention, we determine the flight path angle γ and roll angle φ s  of an aircraft and display these as parameters to the pilot as a principal measure of the aircraft attitude at a given moment. The roll angle is advantageously determined about the stability axis of the aircraft, as opposed to the body axis, since determination of the stability axis roll angle requires no knowledge of the angle of attack, a parameter which is frequently not known or readily determinable with accuracy in small aircraft. This is in contrast to conventional aircraft navigation systems which determine and display roll angle and pitch about the aircraft&#39;s body axis. Although the body-axis and stability-axis roll angle are typically nearly equal to each other, this is not the case with pitch angle, which differs from flight path angle by the angle of attack. Thus, when a pilot wishes to maintain the aircraft on a particular flight path angle, he/she can not navigate by the pitch angle alone, but must correct it by the current angle of attack, a value which is frequently known only imprecisely at best in small aircraft, and which can change from moment to moment dependent on the flying situation. 
     The present invention enables construction of an integrated flight information and control system that presents to the pilot all the information necessary to safely pilot an aircraft but which eliminates costly equipment such as attitude and heading reference systems (AHRS) and other sensors heretofore required to generate the necessary data. In particular, all the essential parameters commonly measured for use in controlling and navigating the aircraft can now be obtained from a single sensor, i.e., a single GPS receiver that provides the requisite outputs from which the position, altitude, vertical speed, ground speed, ground heading, and now, pseudo attitude, of the aircraft can be determined. Thus, an integrated flight information and control system of the type commonly found only in large, commercial aircraft such as a Boeing 747 can now feasibly be provided in small aircraft such as a Piper Arrow. The result is expected to revolutionize aircraft flight information and control systems on such aircraft. 
     In addition to utilization in small aircraft, the attitude indicator and flight and control system of the present invention is adapted to larger, commercial and military aircraft as well, where it may serve either as the principal navigation and control system or as a backup system or a reference system for integrity checking. Because of the dramatically reduced cost of the system, a significantly greater degree of redundancy may be provided at a reasonable cost, thus further enhancing flight safety overall. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The foregoing and other and further objects and features of the invention will be more readily apparent in consideration of the following detailed description of the invention, when taken in conjunction with the accompanying drawings in which: 
     FIG. 1 is a block diagram of a navigation system in accordance with the present invention; 
     FIG. 2 is a sketch of an aircraft showing the orientation of the body, stability and wind axes with respect to the aircraft in relation to a local, ground-fixed reference frame; 
     FIG. 3 is a stylized sketch of an aircraft, as seen from the rear, during an arbitrary balanced maneuver and showing the orientation of the forces and accelerations established by the dynamics of the aircraft; 
     FIG. 4 is a chart showing the track of an aircraft carrying the navigation system of the present invention and illustrating various maneuvers performed to test the system of the present invention; 
     FIG. 5 is an enlarged view of sections of the track of FIG. 4; 
     FIG. 6 is a plot of the roll angle as determined by the system of the present invention, superimposed on a plot of the roll angle as measured by a standard MIGITS (“Miniature Integrated GPS/INS System”) system for the flight path of FIG. 4; 
     FIG. 7 is a schematic of the plant model of the motion of the aircraft; 
     FIG. 8 is a flow diagram showing the determination of the state estimates (velocity and acceleration); and 
     FIG. 9 is an illustration of a flight instrumentation system of a type commonly used in commercial aircraft but modified in accordance with the present invention, the system encompassing a primary flight display (FIG. 9A) and a navigation display (FIG.  9 B). 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Turning, now, specifically to FIG. 1, an antenna  10  provides GPS signals to a receiver  12  for determination of the instantaneous position of the receiver with respect to a multiplicity of GPS satellite transmitters (not shown) and thus, since the position of the satellites themselves is known, with respect to a ground-based reference frame shown in FIG. 2 as N, E, D (for “north”, “east” and “down”, respectively). In implementation of the present invention, we have used a NovAtel 3151 R receiver board which provides as output not only the instantaneous position in the ground-based frame, but also the velocity components V′ N , V′ E , and V′ D  of the receiver in that frame. Preferably, the receiver  12  provides its outputs at a sufficiently fast rate that the path of the receiver (and thus, of the aircraft to which it is attached) through space can be very closely tracked. In our implementation, the receiver is configured to provide data at a rate of  10  sets of data per second. 
     The output of the receiver  12  is applied to a filter  14 . Preferably, this filter is of the Kalman type, a filter that is widely used in control systems for smoothing data measurements and for providing further data based on the input data. In the present case, the filter is configured to provide an estimate of the receiver (and thus aircraft) velocity as it moves along the flight path and, additionally, to provide an estimate of the acceleration of the aircraft along the flight path. Of course, where the receiver itself provides the desired acceleration data, as some do, the receiver acceleration data is also provided to the filter  14  (as indicated by the dotted line  13  in FIG. 1) which then need only filter it. 
     A navigation processor  16  receives the velocity and acceleration outputs from the filter  14  and, based on this data and on the flight dynamics of the aircraft, determines the roll and pitch of the aircraft in the manner described more fully below. In accordance with the present invention, the (stability axis) roll angle φ s  and flight path angle γ are presented to a display  18  for utilization in navigation. 
     The conventional roll angle φ and pitch angle θ commonly used in attitude display and navigation are referred to the body axis of the aircraft as will be described more fully below in connection with FIG. 2, while the stability roll angle φ s  and pitch angle (flight path angle γ) are referred to the stability axes of the aircraft. The stability axes define the path of the center of gravity of the aircraft as it moves along its flight path. The stability pitch angle (flight path angle) is an important parameter, not commonly provided to the pilot, for use in navigation; it provides a more direct indication of the rate of descent or ascent of the aircraft at a given time than the conventional pitch angle θ and is not dependent on the angle of attack. 
     As is the case with conventional attitude indicators, the display  18  has a horizon line  20  separating the sky  22  from the ground  24 . An aircraft, stylistically indicated by wings  26  and tail  28 , indicates the stability roll angle φ s  by means of the angular displacement of the tail marker  28  with respect to a roll angle scale  30 . In FIG. 1, the aircraft is illustrated as banked to the right by 20 degrees. The flight path angle γ is indicated by a marker  32  with respect to a scale  34 . As illustrated, the marker  32  is positioned on the horizon line, and thus the aircraft as shown in FIG. 1 is neither climbing nor descending but is instead engaged in level flight. When the aircraft climbs, the marker  32  will move to the upper area  36  of the scale  34 , thus clearly indicating a climb and its magnitude. Conversely, when the aircraft descends, the marker will move to the lower area  38  of scale  34 , thus indicating the descent and its magnitude. Thus, flight path angle and roll angle are directly presented to the pilot, without need for approximation or estimation of the flight path angle by the pilot. 
     The presence of a wind (referred herein as a “ground wind” to distinguish it from the apparent wind seen by the aircraft as it moves along its flight path) will typically not have a significant effect on measurements discussed herein, provided that it is of no more than moderate strength, i.e., less than about 20% of the aircraft speed. However, where it is desired to obtain enhanced accuracy, particularly in high wind conditions, a wind indicator  40  may be used to correct the measured velocity V′ prior to the application of the latter data to the filter  14 . In particular, the N, E and D components of the wind as measured by the indicator  40  are subtracted, in a summing junction  42 , with the corresponding components of the measured velocity V′, and the resultant velocity is then applied to the filter  14 . 
     Turning now to FIG. 2, an aircraft  50  is shown in motion relative to a ground-based N, B, D reference frame  52 . A body axis system  54  (x b , y b , z b ) is defined with respect to the airplane as follows: The x b  axis points longitudinally along the fuselage of the aircraft towards its nose; the y b  axis is orthogonal to the x b  axis and points along the right wing  55  of the aircraft; and the z b  axis is orthogonal to the other two axes and points downwardly with respect to them. 
     A “stability” axis system  56  (x s , y s , z s ) is defined with respect to the motion of the center of gravity of the aircraft as it moves along its fight path. The y s  axis is coincident with the y b  axis; the x s  and z s  axes are rotated about the y b  axis with respect to the x b  and z b  axes, respectively, by the angle of attack α. Finally, a wind axis system  58  (x w , y w , z w ) is defined with respect to the apparent wind (i.e., the apparent direction of the air seen by the aircraft as it moves along its flight path), with z w  coincident with z s  and x w  and y w  rotated about z w , z s  by the angle β with respect to the x s  and y s  axes, respectively. 
     The conventional pitch θ and roll φ angles are defined by the relation of the body x and y axes (x b , y b ) respectively with respect to a plane parallel to the local horizontal ground reference plane established by the N and E axes, respectively. The conventional roll angle is defined by the relation of the y b  axis to the local horizontal ground reference plane as measured about the x b  axis. Similarly, the stability axis roll angle φ s  is defined by the relation between the y s  axis and the local horizontal ground reference plane as measured about the x s  axis, and the flight path angle γ is defined by the angle between the x s  axis and its projection onto that horizontal ground reference plane. 
     In light of FIG. 2, the determination of the pitch and roll angles in accordance with the present invention may now be more readily understood. For the moment, it will be assumed that the aircraft is flown along its flight path with negligible side slip, that is, β is zero and the wind axis systems and stability axis systems coincide. Further, in order to simplify the discussion, it will be assumed in the following description that the wind is zero, although the presence of wind in practice is accounted for by modifying the velocity components as measured by the GPS system. FIG. 3 then shows the relevant dynamics (forces and accelerations) of the aircraft as it undergoes an arbitrary but balanced flight path maneuver, in this instance a balanced turn (i.e., the lift is perpendicular to the wings and the components of the lift, centripetal acceleration, and gravity normal to the flight path balance each other) in which the aircraft may be climbing, stationary, or descending with respect to the ground reference plane. 
     FIG. 3 shows the aircraft as seen from the rear, with its right wing  55  pointing downwardly with respect to a horizontal reference vector H by an amount φ s . The gravitational acceleration g is divided into two components, the first, g n , being taken normal to the flight path followed by the plane, and a tangential component g t  being taken along this flight path. It will be understood that if the aircraft is following any path other than a level path, the normal component g n  will not coincide with the local gravitational acceleration. 
     The normal and tangential components of the gravitational acceleration vector g are related by: 
     
       
         g=g n +g i   (1) 
       
     
     where the components in this and the following equations are vector quantities. The tangential component is given by:                  g   t     =         g   ·   V            V        ·        V               V       ,           (   2   )                                
     where V (Vx,Vy,Vz) is the velocity vector of the aircraft along its flight path, where the dot indicates a dot product and the bars “||” indicate the magnitude of the corresponding vectors. 
     Thus, g n  is given by:                g   n     =       g   -     g   t       =     g   -         g   ·   V            V        ·        V               V                 (   3   )                                
     The gravitational acceleration g is a known constant obtainable for the various positions on earth, and acts vertically downward along the ground axis D. The velocity V is the aircraft velocity as provided to the navigation processor by the Kalman filter  14 . Thus, the normal gravitational acceleration, g n , is determinable from the information provided. 
     For a balanced turn, the acceleration, L, is perpendicular to the wings  55 ,  57  as shown in FIG.  3 . The acceleration of the aircraft has two components, namely, a normal (centripetal) component an perpendicular to the flight path and a tangential component a t  along the flight path, i.e.: 
     
       
         a=a n +a t   (4) 
       
     
     The acceleration a is provided as an output of the Kalman filter  14  based on inputs from the GPS receiver  12 . The tangential component a t  of the aircraft acceleration a, is given by:                a   t     =         a   ·   V            V        ·        V               V             (   5   )                                
     and is thus completely determined from the estimated acceleration and velocity as provided by the filter  14 . Accordingly, the normal component a n  is determined by the processor  16  as:                a   n     =     a   -         a   ·   V            V        ·        V               V               (   6   )                                
     A local horizontal reference, H, is determined as: 
     
       
         H=g n ×V ,  (7) 
       
     
     where the “x” indicates the vector product of the vectors g n  and V. Because the forces are balanced, the following relationships holds: 
     
       
         L=a n −g n   (8) 
       
     
     Since a n  is determined as set forth in equation 6, and g n  is determined as set forth in equation  3 , L is therefore completely determined. From this, the angle between the acceleration vector L and the reference vector H may be determined from the dot product, namely:              δ   =       cos     -   1            {       L   ·   H            L                H            }               (   9   )                                
     Finally, the (stability) roll angle φ s  may be determined as: 
     
       
         φ s =π/ 2 −δ  (10) 
       
     
     and the flight path angle y then obtained as:              γ   =       tan     -   1            {       -     V   z           (       V   x   2     +     V   y   2       )       1   2         }               (   11   )                                
     where the minus sign is used to provide a positive value for the flight path angle for an upward (negative) velocity in accordance with convention. A navigation processor performing the calculations of equations 1-11 has been implemented on an IBM laptop personal computer. As may be seen from the preceding equations, the required computations are not complex and are readily and quickly performed. 
     In order to test the capabilities of the system, it was installed in a Piper Arrow IV aircraft flown on a path between Hanscom Air Force Base in Bedford, Mass. and Plymouth, Mass. Various maneuvers were executed on this flight path, including straight and level flight, climbing flight, and a variety of banked turns, including descending turns of the kind one might make in preparing for landing, as well as level turns. The performance of the instrument was compared with data provided in a MIGITS (Miniature Integrated GPS/INS System) system, and the results are shown in FIGS. 4-6. 
     FIG. 4 shows the flight path, which extended from approximately 42.5° to 41.5° north latitude, and 71.6° to 70.9° west longitude. As may be seen from FIG. 4, a variety of flight conditions were tested, including straight, gently turning, and strongly turning flight segments. 
     FIG. 5 is an enlarged view of a portion of FIG. 4, showing the portion of the flight path extending from approximately 71.3° to 71.5° west longitude. In a first portion of the flight generally indicated by numeral  60 , the plane was flown in an approximately level condition. FIG. 6 shows the roll angle as measured by the MIGITS system and as determined in accordance with the present invention. As there shown, the two determinations are practically identical, that is, so closely they track each other that one cannot reasonably distinguish between the two of them. 
     Comparing FIGS. 5 and 6 again, a shallow level turn indicated by numeral  62  was followed by a steep level turn  64 . Again, the system of the present invention tracked the reference MIGITS system nearly identically. 
     The next maneuver comprised a 500 foot climb  66 , followed by a 500 foot descending turn  68 . Again, the system of the present invention tracked the reference MIGITS system almost identically. 
     The accuracy and response rate of the system were proven to be sufficiently great as to enable the second portion of the flight (i.e., extending from approximately 71.3 to 70.9 degrees west longitude) to be flown solely with the system of the present invention for the required attitude determination, including an instrument landing system (ILS) landing approach involving descending, banked turns. The system performed exceptionally well, and clearly demonstrated its usefulness as a principal attitude determination and display system. 
     As noted previously, the filter  14  is preferably a Kalman filter. In accordance with the present invention, the filter consists of three separate and independent filters, one each to estimate motion in the local north, east and downward directions. For each such direction we define a triple integrator model for the process (plant). In particular, referring to FIG. 7, first, second and third integrator outputs  70 ,  71  and  72  respectively represent jerk (j), acceleration (a) and velocity (v). In addition, there is an input  73  to the first integrator which is a white noise process. The standard form of the process model is the vector matrix equation 
     
       
         {dot over (x)}= A x+ B u 
       
     
     where the dot indicates the first derivative; x is the state vector        x   =     [         V           a           j         ]                            
     u is white noise; A is the 3×3 matrix        A   =     [         0       1       0           0       0       1           0       0       0         ]                            
     and B is the column matrix:        B   =     [         0           0           1         ]                            
     In order to run the Kalman filter on a computer, the continuous time model, described above, is converted to a discrete time model of the form: 
     
       
           Xκ+ 1 =φκXκ+Wκ   
       
     
     where Xκ is the value of the state of vector at time tκ; Xκ+1 is the updated value, i.e., its value at tκ+1; φκ is the state transition matrix given by        φκ   =     [         1         Δ                 t           Δ                 t        2   2               0       1         Δ                 t             0       0       1         ]                            
     and Wκ is a white noise sequence whose covariance is given by:          Q   k     =       [             Δ                   t   5       20             Δ                   t   4       8             Δ                   t   3       6                 Δ                   t   4       8             Δ                   t   3       3             Δ                   t   2       2                 Δ                   t   3       6             Δ                   t   2       2           Δ                 t           ]     ·   q                            
     where q is the process noise strength for the particular process (plant) under consideration. The conversion from continuous to discrete time is well known in the art. 
     For the Piper Arrow IV on which we have tested the system, we have found the process noise in the North and East direction, q NE , to be approximately 0.001 (m 2 /sec 7 ) and the process noise q D  in the down direction to be approximately 0.0005 (m 2 /sec 7 ). The measurement model was defined as: 
     
       
         
           Yκ=HκXκ+Vκ 
         
       
     
     where Yκ is the measurement of the velocity in the North, East, Down directions provided by the GPS receiver at time t κ ; H κ is the measurement connection matrix H κ=[ 1 0 0]; and V κ is the measurement noise, which is assumed to be a wide noise sequence with variance R κ . 
     The determination of the state estimates {circumflex over (X)} κ , {circumflex over (X)} κ+1  is then made in accordance with FIG. 8, which is a flow diagram of the computations that have been implemented on a personal computer running the Windows operating system. In FIG. 8, a carot above a quantity indicates an estimate of the quantity; a minus a sign to the upper right of a quantity indicates the prior value of the quantity; a plus sign to upper right of the quantity indicates the updated value of the quantity; a “T” to the upper right indicates matrix transpose; and a “− 1 ” to the upper right indicates matrix inversion. 
     As shown in FIG. 8, an updated value of the gains K κ are computed at step  90 , followed by the acquisition of new measurements Y κ and the determination of new estimates {circumflex over (X)} κ at step  92 . The error covariance matrix Q κ is then updated in step  94 , and the estimate finally projected ahead to the next step in step  96 . The required computations have been found sufficiently non-demanding as to allow updating of the display at a rate of approximately ten times per second, thereby enabling use of the attitude indicator as a real-time navigation instrument. 
     It will be appreciated that if the aircraft angle of attack (α) is measured or estimated, then conventional pitch angle (θ) and roll angle (φ) (with respect to the body axes) are determined well in accordance with the present invention by a simple rotation about the y s , y b  axis by the measured or estimated angle of attack. Hence θ and φ can thus be readily displayed to the pilot. This transformation is well known in the art and will not be further described. 
     A significant aspect of the present invention is that it enables the provision of a simple, integrated flight information and control system for navigation and control of aircraft of all sizes, small as well as large. Integrated flight information and control systems are known, but heretofore their use has been confined to large, commercial aircraft or to military aircraft, because of the cost and complexity of such systems. In accordance with the present invention, however, the essential flight navigation and control data, i.e., position, altitude, vertical speed, ground speed, ground heading, and now, pseudo attitude, are obtained from a single GPS receiver, thus eliminating expensive and cumbersome sensors such attitude and heading reference systems (AHRS) which have heretofore put integrated flight information and control systems out of the reach of other than commercial and military aircraft. 
     Thus, referring again to the drawings, FIG. 9A shows a primary flight display, and FIG. 9B a navigation display, of the type used to form the electronic flight instrument display of a Boeing 747-400 aircraft, but modified to present directly to the pilot the pseudo attitude (pseudo roll angle φ s ) and flight path angle γ) as determined in accordance with the present invention, in place of the conventional body-axis attitude (roll angle φ and pitch angle θ). The pseudo attitude presents a more direct measure of the actual path of the aircraft with respect to the earth than does the conventional body-axis attitude, and thus is more immediately useful in navigation and control of the aircraft. 
     Thus, in FIG. 9A, a primary flight display unit  100  includes an attitude indicator  102  on which are displayed angular roll indicia  104  indicative of the stability axis roll angle φ s  and linear indicia  106  indicative of the flight path angle γ, both as determined in accordance with the present invention. An aircraft indicator  110  provides a reference for determining the magnitude of these angles at any given moment. A ground speed indicator  112  provides a direct digital indication,  112   a , and analog indication,  112   b , of the ground speed as determined from the measured velocity V (Vx, Vy, Vz). The attitude and ground speed parameters are thus presented to the pilot in a familiar format, but provide data that is more immediately useful for control and navigation. 
     A vertical speed display element  114  shows the vertical speed in both digital,  114   a , and analog,  114   b , form. Similarly, an altitude display  116  shows the altitude in both digital,  116   a , and analog,  116   b , form. An azimuth deviation indicator  118  provides an indication of the horizontal deviation from the desired flight path, while an elevation deviation indicator  120  provides an indication of the vertical deviation from the desired flight path. Further in accordance with the present invention, a pitch command bar  122  and a roll command bar  124 , which heretofore served as reference indicators for the desired (body-axis) roll φ and pitch θ now provide the desired reference for the pseudo roll angle φ s  and the flight path angle γ, which are used to maintain the aircraft on a desired flight path, and thus form pseudo-attitude command bars. The display of these bars is controlled by a flight computer in accordance with the measured position of the aircraft, the aircraft dynamics, and the desired flight path as established by the pilot. 
     Supplementing the primary flight display  100  (FIG. 9A) is a navigation display  150  (FIG.  9 B). The display comprises an underlying map containing navigation symbols  152  and identifiers  154  of the area that the aircraft is traversing at a given time. These symbols and identifiers are retrieved by a flight computer from a database in accordance with the current position of the aircraft and presented on the display  150 . Typically, they form a standardized set for the area being traversed. The display similarly includes waypoint symbols  156  and identifiers  158  also retrieved from the database by the flight computer. The waypoint symbols and identifiers are selected by the pilot in connection with planning the flight. 
     An aircraft indicator  160  indicates the current position of the aircraft. In accordance with the present invention, a digital,  162 , and analog,  164 , ground track direction display indicates the current ground track direction of the aircraft. The data for the display is derived from the single GPS sensor measurements described above. A digital ground speed indicator  166  indicates the current speed relative to the ground. Completing the display  150  is a digital display  168  and an analog display  170  indicating the distance to the next waypoint, and a time display  172 . With respect to physical implementation, the displays  100  and  150  are advantageously formed as video displays on a CRT (cathode ray tube), an LCD (liquid crystal device), or the like. 
     In accordance with the present invention, all the data for the above displays is derived from a single GPS receiver. In particular, the pseudo-attitude is obtained as described in detail above. The vertical speed, ground speed and ground track direction are obtained directly from the velocity vector V (Vx, Vy, Vz) that is determined as described above. The altitude and position of the aircraft and the current time are determined from the GPS sensor in the conventional manner. Thus, additional, frequently elaborate and expensive sensors such as gyros are completely eliminated and the attitude and other information previously provided by them is now obtained in a reliable manner from the GPS system instead. As a result, the cost, complexity, and weight of the navigation and control system is greatly reduced, and it is now possible for the first time to include a fully integrated flight information and control system in even small, non-commercial, non-military aircraft. 
     Although it is contemplated that the system of the present invention will find most widespread use in piloted aircraft navigation and control in which the flight information display will be used to provide the necessary information to a pilot for navigation and control in accordance with the information displayed, it should be understood that it is not so restricted. Specifically, the data generated by the system as described herein may be used to navigate and control the aircraft in autopilot mode; in that case, the flight display is used simply to monitor the autopilot control. Further, the system may be incorporated into pilotless vehicles to control their flight from one point to another without human intervention. 
     From the foregoing, it will be seen that the system of the present invention has wide application in aircraft navigation, from the smallest of private aircraft to large commercial craft. The system is not restricted to fixed wing aircraft, but is expected to also find application in the navigation of helicopters, where the determination of flight path can otherwise be quite complicated. Thus, the term “aircraft”as used herein should be understood in its broadest sense as applicable to any vehicle moving in three dimensional space. Other applications and modifications will readily suggest themselves to those skilled in the art, and it is intended that the foregoing be taken as illustrative only, the scope of the invention being defined with particularity in the claims appended hereto.