Projectile guidance with accelerometers and a GPS receiver

[0001] 1. Field of the Invention

[0002] This invention relates to the field of aircraft or projectile guidance using divert propulsion means; the propulsion means using aerodynamic maneuvering or gas propulsion to adjust the flight path of a projectile in response to a command from a guidance system; the guidance system including an attitude determination and estimation means such as a GPS signal and configured to respond to signals from a first and second linear accelerometer for an estimation of the projectile's roll angle, and information from an additional third accelerometer for a pitch angle estimation.

[0003] 2. Description of the Prior Art

[0004] The general problem to be solved is how to guide a gun-fired projectile onto a target with a known geographic location at lowest cost with reasonable reliability. This application addresses a portion of that problem by providing various divert propulsion means based on aerodynamic maneuvering and/or gas propulsion to adjust the flight path of a projectile in response to a command from a guidance system. This application also provides means for enabling the guidance system to determine appropriate commands to communicate to the propulsion system based on an estimate of the trajectory of the projectile from launch, the initial conditions of the projectile immediately after launch and the distance, direction and altitude to the idealized trajectory calculated for the projectile at the time of launch or firing.

[0005] Prior approaches to the problem to be solved used external aerodynamic surfaces, e.g., canards, controlled by a guidance system that relied upon only a GPS receiver and a turns counter. Unfortunately, however, external surfaces cause difficulties in launching the projectiles, e.g., from within a launch barrel, and the GPS signals may be vulnerable to jamming. Accordingly, where external aerodynamic surfaces, e.g., canards, are used, additional provisions must be made to minimize launch tube (barrel) interference and to accommodate large forces imposed on the canards due to projectile acceleration and aerodynamic drag. Furthermore, additional measures may need to be implemented to ensure reliable reception and processing of an uncorrupted GPS signal, e.g., protection of the GPS signal.

[0006] Additional approaches for controlling the aerodynamic surfaces have relied upon a GPS receiver, a turns counter, and a triax of gyros, which further increases the cost of the system. Yet, this approach remains vulnerable to corruption of the GPS signal because, as in the first approach described above, in the event the GPS signal is lost, no mechanism exists to account for any external forces that may act on the projectile throughout its flight.

[0007] Prior attempts to address the vulnerability to GPS signal jamming have focused on either preventing interference with the GPS signal or enabling operation with limited GPS data. Attempts to enable operation with limited GPS data have typically involved increasing the performance and/or functionality of the inertial instruments on board the projectile. For example, to address the inability of the above-mentioned second approach to account for the projectile's reaction to external forces if/when the GPS signal may be lost, additional accelerometers may be incorporated into the system to enable compensation for the projectile's reaction to external forces, i.e., the sensor package may comprise a complete IMU. Gyroscopic instruments for aircraft use are well known and available in a number of technologies such as iron rotor and tuned rotor gyros, ring laser gyros, multi-oscillator gyros, zero lock gyros (ZLG), fiber optic gyros, resonator gyros such as HRGs or hemispherical or tubular ceramic resonant gyros and the like.

[0008] Unfortunately, however, incorporation of additional gyroscopic instruments and/or accelerometers are considered much too costly because of the gyros already in the package. Moreover, use of such instruments imposes additional constraints on the operational envelopes of the projectiles. For example, gyroscopic instruments are typically subject to failure modes and uncertainties relating to launch accelerations in the range of 15,000-30,000 Gs. Further, use of these technologies usually requires that the vehicle carry at least one gyro in a gimbaled or strap-down arrangement with the attendant disadvantages of cost, weight and power dissipation.

[0009] Accordingly, a need exists for a system and process for guiding a projectile to a target while eliminating reliance on aerodynamic surfaces external to the projectile and while minimizing or eliminating the vulnerability of the system and process to interference with, e.g., jamming of, the GPS signal. Thus, it would be advantageous to have an improved, cost-effective system and process for providing projectile guidance in the presence of GPS signal jamming and/or with limited GPS data.

[0010] In accordance with a first exemplary embodiment of the invention system and process, a projectile without gyros is guided toward a target using divert propulsion means substantially internal to the projectile. The divert propulsion means are controlled by a guidance system relying on a GPS signal and a triax of accelerometers. The guidance system uses a computer-implemented process that responds to GPS position and GPS delta velocity data along with data from the accelerometer triax to determine the projectile's estimated attitude in pitch, roll and yaw. With the projectile's position known from data provided by the GPS signal received by a GPS receiver, the computer-implemented process determines the projectile's attitude in navigational coordinates and creates a time indexed record of the projectile's trajectory after the on-board GPS receiver locks on to the required number of satellites. The data in the time indexed record of the trajectory is filtered and smoothed. As the projectile rolls, the accelerometers are used to measure the forces acting on the projectile and the projectile's rotation rates. In some alternative embodiments of the invention, the GPS is used to provide data for the calculation of initial aiming and velocity errors and for the calibration of the accelerometers when positioned in the gun barrel prior to launch.

[0011] In an exemplary embodiment, the divert propulsion means comprises one or more aerodynamic surfaces located substantially within the projectile. In accordance with this embodiment, an exemplary projectile has one or more supply ports located so as to receive a supply of gas having an elevated total pressure (dynamic head), e.g., located on the leading surface of the projectile. The projectile further comprises one or more exit ports situated so as to enable the gas to exit the projectile at a location where the local static pressure is lower than the pressure realized at the one or more supply ports, e.g., at the side or trailing edge of the projectile. The supply ports are in fluid communication with the exit ports through gas metering means and gas directing means which are both controlled by the guidance system. By manipulating the quantity of gas received through the supply ports and/or by manipulating the direction and/or velocity of the gas exiting the projectile through the exit ports, the guidance system may effectively guide the projectile onto a target.

[0012] In another exemplary embodiment, a projectile similarly comprises one or more exit ports in fluid communication with a gas chamber through a similar gas metering means and gas directing means. In accordance with this embodiment, high pressure gas may be supplied via a combustion, or other chemical, process conducted within the projectile, by pre-charging the projectile before or during launch, or by any other means known in the art for producing a compressed volume of fluid. Exemplary methods of supplying the gas chamber include chemical reaction, e.g., combustion, of monopropellants, bi-propellants, solid propellants, liquid propellants, and solid/liquid propellant hybrids. An exemplary method for pre-charging the projectile before launch involves loading the chamber with compressed fluid and closing the supply port as soon as the chamber is loaded. An exemplary method of pre-charging the projectile during launch involves opening a port in the projectile so as to permit high pressure gas from within the launch tube (e.g., gun barrel) to flow into the chamber and closing the port before, or very shortly after, the projectile exits the launch tube, thereby loading the chamber with compressed gas. Accordingly, the guidance system may manipulate the direction and/or velocity of the gas exiting the projectile through the exit ports to effectively guide the projectile onto a target.

[0013] It should be noted that the cost associated with the guidance system may depend upon the accuracy of the accelerometers that must be used, which depends upon the required delivery accuracy of the projectile. Where cost must be reduced, lower accuracy accelerometers can be used with a greater reliance on GPS signal data after launch. Therefore, to reduce cost while retaining reasonable levels of projectile delivery accuracy, greater reliance must be placed on GPS signal data. Thus, an aspect of the present invention provides for improved reliability of GPS signals.

[0014] In an exemplary embodiment, GPS signal data may be acquired from GPS signals transmitted from one or more satellites and received by one or more antennas on the projectile. Where the potential exists to encounter jamming of one or more of the GPS signals, one or more corresponding GPS signal jamming detector(s) may be included to monitor each GPS signal and to detect whether each GPS signal has been subjected to jamming. If such jamming is detected, an anti-jammer will implement GPS signal protection measures.

[0015] Exemplary GPS signal protection measures may include causing the projectile to engage, or remain engaged, in a periodic motion such as rolling, e.g., via the projectile's divert propulsion means, and using the periodic motion of the projectile to selectively sample GPS signals in such a manner as to detect and omit jammed GPS signals and to enable reliable processing of unjammed signals, e.g., by selectively and periodically avoiding GPS signals that exhibit jamming. It should be noted that as a projectile proceeds through each cycle of its periodic motion, i.e., each rolling revolution, each of the one or more antennas will periodically be positioned to receive a GPS signal from each of the one or more satellites. Thus, so long as at least one GPS signal from a satellite remains capable of being received by an antenna in an unjammed condition, each of the one or more antennas will periodically be free from jamming, e.g., during at least a portion of the roll attitude.

[0016] Alternatively, where the projectile motion permits one or more antennas to remain oriented so as to continuously receive an unjammed GPS signal, such as where the projectile is not spinning, exemplary GPS signal protection measures may include detecting whether a GPS signal received by one or more of the antennas is free from jamming and switching a GPS receiver to use only the unjammed signals. It should be noted that in such an exemplary embodiment, the projectile need not be spinning or rolling. Accordingly, this protection measure may be implemented with projectiles that are not roll-stabilized. It should also be noted that, in accordance with this embodiment, the antenna(s) may be decoupled from the main body of the projectile via a mechanism such as a slip ring. Accordingly, the projectile may be spinning or roll stabilized while the antennas are maintained in a fixed orientation with respect to one or more satellites. Similarly, the projectile may be stationary while the antennas undergo periodic motion, e.g., spinning, with respect to one or more satellites.

[0030] FIG. 1 schematically shows the major components in an exemplary embodiment of a projectile guidance system without gyros 10 distributed in the exploded perspective view of a projectile 12. The projectile 12 is depicted in flight after having been fired from a gun (not shown).

[0031] The projectile is shown having an orthogonal body fixed coordinate system. The rotation rate components in roll φ, with a longitudinal axis X_b a pitch axis Y_b and a yaw axis Z_b. FIG. 4 shows an aircraft with a corresponding orthogonal body coordinate system. The rotation rate components in roll φ, pitch θ and yaw ψ are defined in FIG. 4. A locally level fixed earth navigational coordinate system is also depicted in FIG. 4. When the wander angle is zero, the Y_N axis is typically parallel to a meridian of longitude and points true north. The X_N axis is parallel to a parallel of latitude and points east and the Z_N axis extends from the surface of the earth as a gravity vector. The angle α is used in a navigational system to traverse the poles of the earth without encountering position singularities. In operation, components of time indexed data elements such as projectile acceleration and velocity are transformed from the body coordinates, indicated by a subscript “b” into the earth referenced coordinate data elements indicated by a subscript N using conventional direction cosine matrices.

[0032] Referring again to FIG. 1, the projectile, guided by the projectile guidance system without gyros, has a triax of accelerometers 14. The triax has an x-axis accelerometer for providing x-axis acceleration data measured along the X_b axis, a y-axis-accelerometer for providing y-axis acceleration data measured along the Y_b axis and a z-axis accelerometer for providing z-axis acceleration data measured along the Z_b axis. A GPS antenna represented by band 16 and a GPS receiver represented by module 18, in combination, represent a means for providing onboard GPS position and altitude and GPS velocity data in earth referenced navigational coordinates. Alternative antenna patterns 88, 89 are configured to receive GPS data from one or more satellites 78, 79. While only antenna patterns 88, 89 and antenna band 16 are shown, it should be noted that any number, configuration, and/or orientation of antennas may be included on the projectile, each antenna being configured and oriented to receive GPS signals from a different direction or range of directions (e.g., each antenna pattern having a lobe directed toward a different direction or range of directions). Similarly, although only one GPS receiver 18 is shown, it should be noted that a plurality of GPS receivers may be implemented to accommodate processing of the GPS signals received through the specific number, configuration and orientation of antennas implemented on the projectile.

[0033] Module 20 represents a guidance processor which provides the function of a computer, memory and program means for storing and accessing time indexed GPS position and GPS velocity data, for implementing GPS signal protection measures as described herein, and for transforming the x, y and z axis acceleration data from body to navigation coordinates. The acceleration data is arrayed and has time indexes that are common with the time indexes for the GPS position and velocity data. A turns counter 22 provides an ac signal in flight representing the angular rate of rotation of the projectile in the earth's magnetic field.

[0034] The computer, memory and program means on module 20 receives and responds to corresponding time indexed acceleration data, GPS velocity and position data for calculating and outputting corresponding sets of time indexed estimates of the projectile roll, pitch and yaw angles with respect to the locally level earth referenced navigation coordinates for each time index iteration of the program to provide present position, velocity and acceleration data to a flight control and ballistic data system represented by module 26 for it to use in guiding the projectile to a predetermined target location.

[0035] The flight control and ballistic data system controls the projectile by providing commands to a divert propulsion means, such as external control surfaces, e.g., canards, 28, 30, to keep the projectile on a trajectory to the target location. Prior approaches to the problem to be solved used external aerodynamic surfaces, e.g., canards, controlled by a guidance system that relied upon only a GPS receiver and a turns counter. As mentioned above, it has been noted that externally implemented control surfaces cause difficulties in launching projectiles, e.g., from within a launch barrel. Accordingly, where external aerodynamic surfaces, e.g., canards, are used, additional provisions must be made to minimize launch tube (barrel) interference and to accommodate large forces imposed on the canards due to projectile acceleration and aerodynamic drag.

[0036] FIG. 9 illustrates an exemplary embodiment of a projectile with external aerodynamic surfaces for guiding the projectile in flight. As shown in FIG. 9, aerodynamic surfaces 28, 30 may be positioned in a variety of locations about the exterior surfaces of the projectile and may be actuated so that gas flowing past the exterior of the projectile will act on the control surfaces so as change the attitude, i.e., orientation, of the projectile or to modify the flight path of the projectile and to guide the projectile to a target.

[0037] In addition, this invention provides various internally implemented divert propulsion means for guiding a projectile to a target. For example, as shown in FIGS. 10a, 10b, and 10c, an exemplary projectile 1000 may be configured to respond to commands from the guidance system by admitting a flow of fluid 1001 through the leading portion of the projectile in substantially the free stream direction 1002, allowing the fluid 1001 to pass through the interior of the projectile 1000, and causing the fluid 1001 to exit the projectile 1000 in a modified direction 1003. In an exemplary embodiment, a projectile comprises one or more supply ports 1011 located so as to receive a supply of fluid 1001 having an elevated total pressure (combined static pressure and dynamic head), e.g., located on the leading surface 1014 of the projectile 1000. The projectile 1000 further comprises one or more exit ports 1031 situated so as to enable the fluid 1001 to exit the projectile 1000 at a location where the local static pressure is lower than the pressure realized at the one or more supply ports 1011, e.g., at the side 1032 or trailing edge 1033 of the projectile 1000. The supply ports 1011 are in fluid communication with the exit ports 1031 through gas metering means 1021 and gas directing means 1022 which are both controlled by the guidance system. It should be noted that gas metering means 1021 may comprise any valve known in the art for controlling the rate of flow (e.g., by varying the cross-sectional area of a passage) of the fluid (e.g., air, water) through which the projectile may fly. It should also be noted that gas directing means 1022 may comprise an internal control surface, e.g., an internal canard or a flow tube, that, when actuated, may be oriented at an angle to the free stream direction 1002 so as to direct the flow exiting the projectile 1000 in a direction other than the free stream direction.

[0038] As the momentum of the fluid 1001 flowing within the projectile 1000 is changed from its initial state where it enters the projectile 1000, i.e., at supply port 1011, to its final state where it exits the projectile 1000, i.e., at exit port 1031, a force will be exerted on the projectile 1000, causing the linear momentum of projectile 1000 to change. If the exit port is positioned, and the direction of exit flow controlled, in such a manner that the exit flow vector does not pass through both the exit port 1031 and the center of gravity 1080 of the projectile, then a moment will be exerted on the projectile 1000 causing the angular momentum of projectile 1000 to change, i.e., causing the rotation of the projectile to change. It should also be noted that, in accordance with this embodiment, actuation of gas metering means 1021 and gas directing means 1022 may be timed to coincide with one or more desired orientation of projectile 1000 to achieve a desired change, or combination of changes, in the flight path and/or orientation of projectile 1000. Accordingly, by manipulating the quantity of fluid 1001 received through the supply ports 1011, by manipulating the direction and/or velocity of the fluid 1001 exiting the projectile 1000 through the exit ports 1031, by positioning the exit ports to align, or not align, with the center of gravity of the projectile, and/or by timing the actuation of the divert propulsion means, the guidance system may effectively guide the projectile 1000 onto a target.

[0039] In another exemplary embodiment, as shown in FIGS. 11a and 11b, a projectile similarly comprises one or more exit ports 1131 in fluid communication with a gas chamber through a similar gas metering means 1121 and gas directing means 1122. More particularly, FIGS. 11a and 11b illustrate, during and after launch from a gun barrel, an exemplary projectile 1100 having a supply port 1111 positioned to permit launch gasses, e.g., products of combusted gun powder or other propellant, to pre-charge the projectile 1100 for use in guiding the projectile 1100 during flight.

[0040] As shown in FIG. 11a, a supply port 1111 may be opened while the projectile 1100 is in the barrel 1177. Accordingly, high pressure launch gasses are permitted to enter and pressurize chamber 1118. Then, as shown in FIG. 11a, supply port 1111 is closed either before, or very shortly after, the projectile exits the launch tube, thereby retaining in chamber 1118, a supply of compressed gas. Similar to other exemplary embodiments described herein, projectile 1100 may comprise one or more exit ports 1131 in fluid communication with gas chamber 1118 through a similar gas metering means 1121 and gas directing means 1122. Accordingly, the guidance system may actuate gas metering means 1121 and gas directing means 1122 to manipulate the direction and/or velocity of the gas exiting the projectile through the exit ports 1131 to effectively guide the projectile onto a target. It should be noted that difficulties may be encountered where particulates or other undesirable constituents are permitted to enter chamber 1118 during firing. To alleviate difficulties, e.g., clogging, fouling, associated with such constituents, a filter may be incorporated into gas metering means 1121 and gas directing means 1122.

[0041] In another alternative embodiment, as shown in FIGS. 12a, 12b, 12c, 12d, and 12e, high pressure gas may be supplied to chambers 1218 via a combustion, or other chemical, process conducted within a projectile 1200, by pre-charging a chamber 1218 within the projectile 1200 before or during launch, or by any other means known in the art for producing a compressed volume of fluid within chamber 1218.

[0042] As shown in FIG. 12a, an exemplary method for pre-charging the chamber 1218 before launch involves loading the chamber 1218 with compressed fluid and closing the supply port 1211 as soon as the chamber 1218 is loaded. Accordingly, as described above, the guidance system may manipulate the direction and/or velocity of the gas exiting the projectile through the exit ports 1231 to effectively guide the projectile onto a target. As shown in FIGS. 12b, 12c, 12d, and 12e, in accordance with other exemplary embodiments, high pressure gas may be supplied to chamber 1218 via a combustion, or other chemical, process conducted within the projectile 1200, or by any other means known in the art for producing a compressed volume of fluid. For example, as shown in FIG. 12b, compressed gas is supplied to chamber 1218 through chemical reaction, e.g., combustion, of mono- or bi-propellants. Similarly, FIGS. 12c, 12d, and 12e show exemplary embodiments where compressed gas is supplied to chamber 1218, through chemical reaction, e.g., combustion, of solid propellants, liquid propellants, and solid/liquid propellant hybrids.

[0043] The selection and/or design of a flight control and ballistic data system represented by card or module 26 is believed to be a design choice and is not the subject of this invention disclosure. Nevertheless, if the projectile is engaged in periodic motion, e.g., rolling, the present invention provides that trajectory corrections may be made by waiting for an orientation of a control surface or other divert propulsion means to correspond with a direction in which it is desirable to steer the projectile.

[0044] The present invention also contemplates that if the trajectory corrections are large and the projectile guidance section is roll stabilized, then the projectile can be de-spun and the corrections made while the projectile is in a roll-stabilized mode. Then, after completing the corrections, the projectile can go back to being a rolling airframe. The present invention enables these capabilities by providing a mechanism for determining the roll orientation of the projectile. For example, with reference to FIG. 3c, the roll orientation of the projectile may be determined through: 1) a roll gyro, 2) the roll position of GPS satellites as they turn on/off, 3) GPS attitude determination with multi antennas, and 4) a magnetic turns counter.

[0045] In accordance with an exemplary embodiment, GPS signal data may be acquired from GPS signals transmitted from one or more satellites 78, 79 and received by one or more antennas 88, 89 on the projectile. With further reference to FIG. 3c, in an exemplary embodiment, a plurality of GPS antenna patterns 88, 89 receive GPS signals which are passed through RF processing means 98, 99 to produce RF processed GPS signal 3344. Information from a high speed correlator 3354, and roll reference 3364 are transmitted to processor 3375 along with RF processed GPS signal 3344. Based on signals 3344, 3354, and 3364, processor 3374 produces a position signal 3381, a velocity signal 3382, and an attitude signal 3383.

[0046] As shown FIG. 3d, each GPS signal 3407 may be sent through a low noise amplifier 3417 before being received by radio 3427. Also, in an exemplary embodiment, one or more GPS signal jamming detectors 3418 may be included to monitor each GPS signal 3407 and to detect whether each GPS signal has been subjected to jamming. If such jamming is detected, an antijammer 3419 may implement GPS signal protection measures which may, for example, selectively and periodically avoid GPS signals that exhibit jamming based on the periodic motion of the antenna portion of the projectile. As shown in FIG. 3d, jamming detector 3418 and anti-jammer 3419 transmit signals to system processor 3457, which also receives information from roll reference 3467 and GPS receiver 3437. Based on signals received from system processor 3457, digital filters 3428 receive GPS signals from S/W radio 3427 and selectively transmit unjammed portions of those GPS signals to GPS receiver 3437. GPS receiver 3437 then uses pre-stored data 3447 to process and transmit GPS signals to system processor 3457. Finally, system processor 3457 utilizes roll reference 3467 and GPS signal to transmit signals to autopilot 3477, which causes control surfaces and/or divert propulsion means 3487 to be actuated, thereby guiding the projectile to a target.

[0047] In an exemplary embodiment, a GPS signal protection measure involves causing the projectile to engage, or remain engaged, in a periodic motion (e.g., spinning, rolling, or otherwise oscillating) and using the periodic motion of the projectile to selectively sample GPS signals in such a manner as to detect and omit jammed GPS signals and to enable reliable processing of unjammed signals. It should be noted that as a projectile proceeds through each cycle of its periodic motion (i.e., through each rolling or spinning revolution, oscillation, or other periodic motion) each of the one or more antennas 88, 89 will periodically be positioned to receive a GPS signal from each of the one or more satellites 78, 79. Thus, so long as at least one GPS signal from a satellite remains capable of being received in an unjammed state by one or more antenna, each of the one or more antennas will periodically be free from jamming, e.g., during at least a portion of the roll attitude.

[0048] FIG. 8 illustrates three exemplary time-dependent GPS signals as received by three corresponding GPS antenna and receiver means implemented on a projectile undergoing periodic motion, e.g., spinning. As shown in FIG. 8, a first GPS signal exhibits jamming as the projectile rotates through an orientation from approximately 30 degrees to an orientation of approximately 210 degrees. Similarly, a second GPS signal exhibits jamming as the projectile rotates through an orientation from approximately 120 degrees to an orientation of approximately 360 degrees. Finally, a third GPS signal exhibits jamming as the projectile rotates through an orientation from approximately 360 degrees to an orientation of approximately 30 degrees. Accordingly, a substantially continuous unjammed GPS signal may be received by switching on the first antenna as the projectile rotates between approximately 210 degrees and approximately 30 degrees, by switching on the second antenna as the projectile rotates between approximately 360 degrees and approximately 120 degrees, by switching on the third antenna as the projectile rotates between approximately 30 degrees and approximately 360 degrees, by switching off the first antenna as the projectile rotates between approximately 30 degrees and approximately 210 degrees, by switching off the second antenna as the projectile rotates between approximately 120 degrees and approximately 360 degrees, and by switching off the third antenna as the projectile rotates between approximately 360 degrees and approximately 30 degrees.

[0049] With further reference to FIG. 1, in one embodiment, the flight control and ballistic data system 26 is pre-programmed to use information provided via bus 32 from the projectile guidance system without gyros 12, after launch, to de-spin the projectile and to roughly position the top of the projectile skyward, or to otherwise orient an antenna pattern 88, 89, so as to optimize the reception of GPS signals. In accordance with this embodiment, after the projectile is de-spun, the accelerometers are un-caged and activated. The computer, memory and program means then performs its first estimate and update of the state matrix including an estimate of the vehicle rates. In accordance with this exemplary embodiment, where the projectile motion is such that it may permit one or more antenna 88, 89 to remain oriented so as to continuously receive an unjammed GPS signal, the anti-jammer 3419 may detect that a GPS signal 3407 received by one or more of the antennas is free from jamming and may switch or otherwise cause the one or more GPS receiver 18 to use only the unjammed signal. It should be noted that in such an exemplary embodiment, an unjammed GPS signal may be received without requiring the projectile to be rolled. To accomplish this, a computer-implemented process, i.e., a guidance program, may be configured to cause the projectile to maintain an orientation of the GPS antenna and receiver means so as to permit one or more antenna to substantially continuously receive an unjammed GPS signal.

[0050] It should be noted that in such an exemplary embodiment, the projectile need not be spinning or rolling. Accordingly, this protection measure may be implemented with projectiles that are not roll-stabilized. It should also be noted that, in accordance with this embodiment, the antenna(s) may be decoupled from the main body of the projectile via a mechanism such as a slip ring. Accordingly, the projectile may be spinning or roll stabilized while the antennas are maintained in a fixed orientation with respect to one or more satellites. Similarly, the projectile may be stationary while the antennas undergo periodic motion, e.g., spinning, with respect to one or more satellites.

[0051] Thus, in accordance with the invention, the projectile may either be spinning or roll stabilized, e.g., by decoupling it from the main body of the projectile via a slip ring. In the event no unjammed GPS signal may be substantially continuously received, the anti-jamming means may be configured to cause the projectile to engage in a periodic motion and to selectively and periodically avoid GPS signals that exhibit jamming.

[0052] Other exemplary protection measures may include immediate conversion of the GPS signal from an RF signal to a digital signal as well as implementation of an automatic gain control. In such embodiments, each GPS signal may be converted immediately or very soon after it is received by a high speed analog to digital converter. Then, the digital GPS signal may pass through a digitally implemented automatic gain control circuit. Both the analog to digital converter and the automatic gain control circuit may be implemented in the GPS radio receiver.

[0053] In an exemplary embodiment, the system may be configured to be initialized prior to firing. In accordance with this embodiment, prior to projectile firing, the system may determine and store in memory which satellites will be in the field-of-view (FOV) of each antenna pattern 88, 89 throughout the trajectory of the projectile and which combination of satellites 78, 79 gives the best GDOP for the trajectory. This information is pre-stored in the receiver 18 along with any other correction data that may be available.

[0054] In an exemplary embodiment, as shown in FIG. 3d, the GPS signals 3407 are transmitted through a low noise amplifier 3417 to a radio receiver 3427, which may comprise a high speed analog to digital converter (e.g., a 5 gigabit/second analog to digital converter with 14 bits of resolution). Radio receiver 3427 may also comprise a digitally implemented automatic gain control circuit. The signals are then sent to a set of digital filters 3427 as well as one or more jamming detector 3418. It should be noted that a separate filter 3427 may be implemented for each pre-selected satellite 78, 79 that will be used in the TOA computation. It should also be noted that digital filters 3427 may be dynamically tuned in real time to band pass the frequency of each pre-selected satellite. The center of the band pass will be the GPS carrier frequency adjusted for Doppler motion between the projectile and the satellite. The GPS signals are then sent to the GPS receiver 3437 where they are correlated against the known code of each of the pre-selected satellites via a high speed digital correlator 3447, and position and velocity determined.

[0055] If the vehicle is rolling, then rate information is extracted by separating the accelerometer ac signal from the dc or steady state signal. As the projectile rolls, the pitch and yaw accelerometers output signals will vary as a sine wave function. If the vehicle is not rolling, then data is extracted and transferred from the body coordinate system to the earth referenced navigation coordinate system using DCM (direction cosine matrix) relationships as required.

[0056] FIG. 2 shows a mobile artillery weapon, firing a projectile toward a target. In the equation shown, {right arrow over (X)}(t)_True is a vector quantity representing the computed true position of the projectile as a function of time. {right arrow over (X)}(t)_Nominal is a vector representing the ideal location of the projectile on the ideal trajectory. {right arrow over (X)}(0)_Error is a vector representing the position error of the projectile at time zero before launch. {right arrow over (X)}(t)_Error is a vector representing the error in the projectile's position due to accumulated errors resulting from navigational instrument error and GPS errors. The figure shows the sequence of events and schematically portrays an error budget region in dashed lines that concludes with an ellipse representing the TARGET ERROR that can result from launching the projectile using conventional artillery practice with no on-board guidance. The funnel region in a continuous line schematically shows the position error of an on-board GPS progressively diminishing and asymptotically approaching a limit that is much smaller than the TARGET ERROR limit for a trajectory with no on-board guidance. The invention projectile guidance system without gyros 10 uses a triad of accelerometers to provide real time body referenced data to a computer, memory and program means and real time position data from a GPS to establish and output the attitude of the projectile. The outputs are Kalman filtered and coupled to the flight control and ballistic data system represented by module 26 shown in FIG. 1 along with the aiding GPS position data for use in steering the projectile into the region bounded by the funnel region.

[0057] FIGS. 3a and 3b schematically show an embodiment of a data flow used by the projectile guidance system without gyros 10. Phantom block 300 represents the function provided by a fire control and ballistics computer as it executes an initialization program to process data received from data sources such as an external input of a TARGET LOCATION, the geographic location of the weapon launching the projectile, the PROJECTILE TYPE, the PITCH ALIGNMENT IN BARREL, the pitch angle of the launching tube, the outputs of the triax 304 of three components of acceleration a_x, a_y and a_z in body coordinates, the roll angle ROLL φ, and the roll rate ROLL RATE φ from the spin counter 308. The FIRE CONTROL AND BALLISTICS computer processes the data available and prior to launch transfers the initial state data such the initial orientation angles, body fixed angle rates, present position and initial velocity components. The spin counter 308 and the triax 304 provide their outputs directly to the INITIAL PROJECTILE STATE ESTIMATE 312 via signal paths 314 and 333 respectively. It is most likely that all initial conditions would be input into the projectile from an external source such as the FIRE CONTROL AND BALLISTICS function 302 via an electrical, magnetic or optical link, or perhaps mechanical link. If the gun were to be mounted on a ship, the initial conditions would have to be extended to include data such as the ship's velocity, heading, displacement from the ship's inertial guidance computer, and possibly the coordinates of the gun with respect to the station from which the gun is receiving its initial condition information.

[0058] The FIRE CONTROL AND BALLISTICS function 302 also provides roll indexing information to the spin counter 308 to provide a zero reference. The spin counter 308 provides estimates of roll angle φ_b to an accuracy of a few degrees, as well as roll rate φ_b. The FIRE CONTROL AND BALLISTICS function 302 precomputes an expected ballistic trajectory which is also passed to the projectile. This ballistic information, along with accelerometer information, or data directly from the FIRE CONTROL AND BALLISTICS function 302 is used to determine the initial state vector and attitudes of the projectile at initialization of the estimation process. The algorithm is initialized when both the accelerometer and the GPS data are available for state estimation. The FIRE CONTROL AND BALLISTICS function 302 also receives a set of initial states for orientation and position of the projectile via signal path 320 from the ESTIMATION OF NEW INITIAL TARGET STATE function 322 as a result of a failure of a fit test.

[0059] The FIRE CONTROL AND BALLISTICS function 302 outputs a set of initial STATE ESTIMATES to the INITIAL PROJECTILE STATE ESTIMATE functional block 312 which transfers initial BODY STATE ESTIMATES {right arrow over (V)}₀, {right arrow over (x)}₀ via signal path 324 to SMOOTHING PROCESS function block 344 and a set of ATTITUDE STATE ESTIMATES via signal path 330 to the initial state estimate of the initial attitude angle array in ATTITUDE FILE 332.

[0060] The turns or SPIN COUNTER 308 which provided turn count data to the FIRE CONTROL AND BALLISTICS function 302 also provides turn count data via signal line 333 to the INITIAL PROJECTILE STATE ESTIMATE FUNCTION block 312 and to the time indexed SPIN COUNTER MEASURED DATA memory array represented on FIG. 3b by functional block 350 where the total roll angle φ_(n) of the projectile is accumulated and time indexed and transferred via path 351 to the ATTITUDE ESTIMATION ALGORITHM 372. Spin counter technology is a mature technology. The Alliant Techsystems Company has reportedly developed one small enough to fit into a 20 mm projectile. A turns counter for larger diameter projectiles is therefore a product that can be purchased for use in the invention system and processed without difficulty. The output of the turns counter is a sine wave output as the projectile rotates through the Earth's Magnetic Field.

[0061] The INITIAL PROJECTILE STATE ESTIMATE FUNCTION block 312 provides initial ATTITUDE STATE ESTIMATE values for azimuth ψ₀, pitch θ₀ and roll φ₀ angle to the ATTITUDE FILE 332 in FIG. 3b as initial values.

[0062] The outputs of the triax 304 of three components of acceleration a_x a_y and a_z in body coordinates are processed by the function represented by the block 336 to transfer time indexed body referenced changes in velocity via signal path 338 to the ACCEL DATA FILE memory array 340. The ACCEL DATA FILE memory array 340 couples an array of time indexed acceleration data elements to the SMOOTHING PROCESS function block 344.

[0063] The filtering and smoothing process of block 344 will be discussed in connection with FIGS. 6, 7a and 7b later in this disclosure. The smoothing process uses the initial estimates of attitude, velocity and position in conjunction with the present accelerometer data to determine velocity and position of the projectile in body coordinates. The outputs of the smoothing process are STATE VECTOR ESTIMATES, {right arrow over (X)}_B(n), {right arrow over (V)}_B(n) for position and velocity which are coupled via signal path 346 to the BODY TO INERTIAL TRANSFORMATION functional block 348.

[0064] The GPS MEASURED DATA function block 356 periodically receives time indexed velocity and position data {right arrow over (V)}_(n) and {right arrow over (X)}_(n) and transfers the data via signal path 358 into the time indexed array in the computer, memory and program means memory shown as function block 360. The GPS velocity and position data is transferred via signal path 362 to a Kalman filter represented by function block 364. The Kalman filter transfers filtered values of the projectile's estimated present position and velocity vectors via signal path 368 to the FIT ERROR TEST functional block 370 and to the ATTITUDE ESTIMATION ALGORITHM functional block 372. The ATTITUDE FILE 332 transfers time indexed estimated attitude angles and attitude angular rate data to the ATTITUDE ESTIMATION ALGORITHM functional block 372 via signal path 374 and to the BALLISTICS ESTIMATE (n) functional block 376 via transfer path 378 in earth referenced navigational coordinates. The BALLISTIC ESTIMATES uses the known model characteristics of the projectile with the corrected initial position and initial velocity information via path 375 and the history of all past attitude and attitude rates from path 378 to provide a modeled estimate of the present position and velocity via path 382 to the ATTITUDE ESTIMATION ALGORITHM 372.

[0065] The filtered body STATE VECTOR ESTIMATES {right arrow over({circumflex over (X)})}_B(n), {right arrow over({circumflex over (V)})}_B(n) vectors in the orthogonal body coordinate system, are transferred via signal path 346 to the BODY TO INERTIAL TRANSFORMATION block 348. The BODY TO INERTIAL TRANSFORMATION block 348 receives the estimated attitude angles and attitude angular via signal paths 384 and 386 and processes the STATE VECTOR ESTIMATES from representation in the orthogonal body coordinate system into vectors in the earth referenced navigational coordinate system to position and velocity estimates in earth referenced navigational coordinates using conventional direction cosine transformations such as those explained in the text by George Siouris, titled “Aerospace Avionics Systems, A Modern Synthesis”, published by Academic Press, published in 1993. This estimation process can be simplified and improved using measured turns counter data and by computing a new ballistic trajectory based on new initial state information.

[0066] The BODY TO INERTIAL TRANSFORMATION functional block 348 outputs the estimated position and velocity STATE VECTOR ESTIMATES in earth referenced navigational coordinates via signal path 388 to the FIT ERROR TEST functional block 370 where the STATE VECTOR ESTIMATES are compared to the Kalman filtered GPS position and velocity. It is not possible in a statistical sense to make the estimation process fit better than the GPS position and velocity and the errors associated with the GPS. If the error for the differences between the two state vectors is less than 1 or 2 times the GPS error, the algorithm declares the fit to be “statistically good enough” to quit and wait for a new measured data point to arrive. If the test passes, the estimates of position, velocity, attitude and attitude rates are used to update the n−1 position in the estimated position and attitude time indexed memory arrays.

[0067] If the test fails, the algorithm estimates a new set of initial projectile states via a perturbation algorithm and repeats the estimation process again, until the solution converges to limits within the predetermined fit error criteria. If the test fails, a FAIL signal is coupled via path 390 to the ESTIMATION OF NEW INITIAL TARGET STATE functional block 322 which adjusts the initial position, velocity, attitude and attitude rates and couples a modified set of estimated state data via signal path 320 into the INITIAL PROJECTILE STATE ESTIMATE functional block 312 for an additional cycle. Iteration continues until the initial conditions are adjusted to obtain a PASS signal out of functional block 370 followed by the next data iteration using GPS data one second later.

[0068] As discussed previously, in order to avoid excessive reliance on GPS and protection of the GPS signal, higher performing inertial instruments (e.g., inertial instruments performing at levels better than 100 micro g) may be used. Honeywell's VBA accelerometers and Silicon Accelerometer supplied from the Litton Guidance & Control Systems Div. of Northrop Grumman Inc. at Woodland Hills, Calif. 91637, the assignee, have such a capability.

[0069] A GPS receiver by Alliant Techsystems is available for use in the invention system and process. The Alliant GPS uses a type 509 processor that has sufficient processing power and a separate memory which can be used to store user software and the system processor for this application. The Alliant receiver is configured with multiple RF front ends to accommodate multiple antenna inputs as a means of protecting the GPS signal as the projectile rolls. The GPS 509 processor can also be used to process the data and integrate the inertial data with the GPS and turns counter data. Software can be incorporated into the 509 processor to extract rate information from the accelerometer triax and integrate the data from the sensors.

[0070] Alternatively, where cost must be reduced, much lower accuracy accelerometers can be used with a greater reliance on GPS signal data after launch. As described above, in an exemplary embodiment, GPS signal data may be acquired from GPS signals transmitted from one or more satellites 78, 79 and received by one or more antennas 88, 89 on the projectile. One or more GPS signal jamming detector 3418 may be included to monitor each GPS signal 3407 and to detect whether each GPS signal 3407 has been subjected to jamming. If such jamming is detected, an anti-jammer 3419 may implement GPS signal protection measures present in system processor 3457 as shown in FIG. 3d.

[0071] For example, anti-jammer 3419 may cause system processor 3457 to cause the projectile to engage, or remain engaged, in a periodic motion such as rolling or spinning. As the projectile proceeds through each cycle of the periodic motion, i.e., through each rolling revolution, each of the one or more antennas 88, 89 may periodically be positioned to receive a GPS signal from each of the one or more satellites 78, 79. Thus, so long as at least one GPS signal from a satellite remains capable of being by one or more antennas in an unjammed state, each of the one or more antennas will periodically be free from jamming, e.g., during at least a portion of the roll attitude.

[0072] Alternatively, where the jamming detector 3418 does not detect jamming, or detects the substantial absence of jamming, in one or more GPS signal, antijammer 3419 may cause system processor 3457 to de-spin the projectile and to roughly position the top of the projectile skyward, or to otherwise orient an antenna pattern 88, 89, so as to optimize the reception of GPS signals 3407. In accordance with this embodiment, after the projectile is de-spun, the accelerometers are un-caged and activated. The computer, memory and program means then performs its first estimate and update of the state matrix including an estimate of the vehicle rates. In accordance with this exemplary embodiment, the antijammer 3419 may switch or otherwise cause the one or more GPS receiver 18 or the system processor 3457 to use only the unjammed GPS signal received by the one or more of the antennas that, is free from jamming.

[0073] The roll angle of a guided projectile is typically provided by a gyro or a gyro platform. A roll angle signal characterizing the roll angle of the projectile is necessary to enable a flight control system to drive the flight control surfaces, such as canards, to roll the projectile to the left or right as required, to use lift from its lift surfaces to change the heading of the projectile to one directed to a target. It is an objective of this disclosure to provide a mathematical and therefore, a computer programmable process, for generating the roll angle of the projectile without signals from a gyro and by using the accelerometer outputs of a triax and the position and velocity signals from a GPS.

[0074] This technical description uses the following notation. {overscore (x)} denotes a vector with no specific reference frame of resolution. {overscore (x)}^b denotes a vector resolved in a coordinate frame called the body-frame or projectile-frame. All coordinate frames discussed herein, including the body-frame coordinates, are right-handed orthogonal frames that have x, y, and z axes that extend from a location in the body of the projectile, such as the center of gravity, or origin designated as a reference point “O” to form an orthogonal triad extending in the forward or positive “x_b” axis direction, in the right or positive “y_b” axis direction and down in the positive “z_b” axis direction. At least two accelerometers are fixed to and aligned with the body-frame along the y_b and z_b axes. The principal axes of the b-frame coincide with the input axes of the inertial sensors and intersect at the origin “O”. A second coordinate frame of interest is the geographic or earth referenced navigational coordinate system or frame with principal axes X, Y and Z that coincide with the East, North, and Up directions as shown at the bottom of FIG. 4.

[0075] Subscripts on vectors are used to indicate a particular property or identification of the vector. Matrices are designated with capital letters. C_N^b denotes a direction cosine matrix (DCM) that transforms a vector from the N-frame or Navigation-frame to the body-frame, i.e., {right arrow over (x)}^b=C_N^b{right arrow over (X)}^N. Time dependency of a quantity is indicated with round brackets around a time variable or index. For example, C_N^b (t₁) denotes the value of the DCM for the conversion from navigational to body-frame coordinates at time t₁. The transpose of the C_N^b (t₁) DCM provides C_b^N (t₁), the DCM for the conversion from body-frame to navigational-frame coordinates at time t₁.

[0076] An increment of a variable is indicated with the symbol Δ. For example, Δ{overscore (x)} denotes the increment of the vector {overscore (x)} over a predefined time interval. An error in a variable is indicated with the symbol δ. For example, δ{overscore (x)} denotes the error in the vector {overscore (x)}. The=symbol indicates an approximate or substantially close equality.

[0077] In a two accelerometer arrangement, a first accelerometer is positioned to sense acceleration along the y_b body axis that extends outward from the right side of the body. A second accelerometer is positioned to sense acceleration along the z_b body axis that extends downward from the origin forming the yaw axis. A third accelerometer, not required for the computation of a roll variable, is positioned to sense acceleration along the x_b or forward axis.

[0078] The accelerometer outputs are resolved from body coordinates into navigational coordinate values in accordance with the following equation:
A^N=C_b^NA^b={T}{H}^T{P}^T{R}^TA^b  1a.
where A^b represents the acceleration vector in body coordinates, the bracketed H, P and R operators {H}^T, {P}^T {R}^T individually represent the transpose of the positive rotation DCM transformation steps from body to navigational coordinates for roll, about x_b, pitch, about y_b and heading or yaw, about z_b, of the vehicle or projectile fixed coordinate system, in that sequence, to obtain the acceleration vector A^N as elements of vehicle acceleration in navigation coordinates with components along the fixed navigation or earth referenced East, X, North, Y and Up, Z axis.

[0079] DCM {T} represents a unitary matrix that transforms the body referenced acceleration data from a North, East, Down system into an East, North, Up frame and is defined as: $\begin{array}{cc}\left\{T\right\}\stackrel{\Delta }{=}\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& -1\end{array}& 1b.\end{array}$

[0080] The data elements of acceleration A^b, from the triad of accelerometers on the projectile, are integrated over a predetermined incremental time interval such as one second to obtain incremental changes in velocity in navigational coordinates in accordance with the following equation:
ΔV^N=∫C_b^NA^bdt={T}{H}^T{P}^T{R}^TA^bΔT  2.

[0081] Concurrent with each calculation of the incremental change in velocity based on accelerometer data, GPS data is used to calculate an incremental change in velocity. The GPS data is already in navigational coordinates. The following equation is used:
ΔV_GPS=V_GPS2−V_GPS1=ΔV^N={T}{H}^T{P}^T{R}^TA^bΔT  3.

[0082] A double integration is performed on the body accelerometer data along with a body to navigation frame transformation to obtain the change in position based on transformed accelerometer data from the body frame using the following equation:
ΔP^N=∫∫C_b^NA^bdtdτ={T}{H}^T{P}^T{R}^TA^bΔT²  4.

[0083] With each calculation of the incremental change in position based on accelerometer data, ΔP^N, GPS data is used to calculate a substantially equal GPS incremental change in position, ΔP_GPS. The following equation is used:
ΔP_GPS=P_GPS2−P_GPS1=ΔP^N={T}{H}^T{P}^T{R}^TA^bΔT²  5.

[0084] The product of a DCM matrix and its transpose is equal to the identity matrix. Therefore
{P}^T{P}={I}  6a.
 {H}^T{H}={I}  6b.

[0085] From equation 3, it is possible to relate change in velocity components or accelerations components from the GPS data to acceleration components from accelerometer data as follows:
ΔV_GPS={T}{H}^T{P}^T{R}^TA^bΔT=ΔV^N  7a.

[0086] From equation 5, it is possible to relate change in position components or velocity components from the GPS data to velocity components from accelerometer data as follows:
ΔP_GPS={T}{H}^T{P}^T{R}^TA^bΔT²=ΔP^N  7b.

[0087] If the pitch, heading and unitary transformation DCMs are grouped or multiplied to form a single DCM, {Q}, and the increment in time is chosen for simplicity to be 1 second (however, it should be understood that any time increment can be used in the equations):
{Q}={P}{H}{T}  8a.
ΔT=1 Second and Δ²=1 Second squared=1  8b.

[0088] Substituting 1 for ΔT in equation 7a and 1 for ΔT² in equation 7b and multiplying both sides of equations 7a and 7b by the transpose of the unitary, heading and pitch DCMs, in that order, provides the following equations:
{P}{H}{T}(ΔV_GPS)={R}A^b={Q}(ΔV_GPS)  9a.
{P}{H}{T}(ΔP_GPS)={R}A^b={Q}(ΔP_GPS)  9b.

[0089] Using CP to represent Cos, θ SP to represent Sin θ, CH to represent Cos ψ, SH to represent Sin ψ, CR to represent Cos φ and SR to represent Sin φ, then the six DCMs for the rotation conversions from body to navigation coordinates are: $\begin{array}{cc}\begin{array}{cc}\left\{P\right\}\stackrel{\Delta }{=}\begin{array}{ccc}\mathrm{CP}& 0& -\mathrm{SP}\\ 0& 1& 0\\ \mathrm{SP}& 0& \mathrm{CP}\end{array}& {\left\{P\right\}}^T\stackrel{\Delta }{=}\begin{array}{ccc}\mathrm{CP}& 0& \mathrm{SP}\\ 0& 1& 0\\ -\mathrm{SP}& 0& \mathrm{CP}\end{array}\end{array}& 10a\\ \begin{array}{cc}\left\{H\right\}\stackrel{\Delta }{=}\begin{array}{ccc}\mathrm{CH}& \mathrm{SH}& 0\\ -\mathrm{SH}& \mathrm{CH}& 0\\ 0& 0& 1\end{array}& \left\{H\right\}\stackrel{\Delta }{=}\begin{array}{ccc}\mathrm{CP}& -\mathrm{SH}& 0\\ \mathrm{SH}& \mathrm{CH}& 0\\ 0& 0& 1\end{array}\mathrm{and}\end{array}& 10b.\\ \begin{array}{cc}\left\{R\right\}\stackrel{\Delta }{=}\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{CR}& \mathrm{SR}\\ 0& -\mathrm{SR}& \mathrm{CR}\end{array}& \left\{P\right\}\stackrel{\Delta }{=}\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{CR}& -\mathrm{SR}\\ 0& \mathrm{SR}& \mathrm{CR}\end{array}\end{array}& 10c.\end{array}$

[0090] Where the body is a projectile fired from a gun, the pitch angle is the angle of the barrel at firing and the heading is the angle ψ that the barrel makes with true north at firing. Both are available from initialization from the system and are expected to be close to the normal trajectory with time, and therefore nominally known.

[0091] The elements of {Q} are known from 8a, 10a, and 10b and are believed to be slowly varying, so let {Q} be: $\begin{array}{cc}\left\{Q\right\}=\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}& 11.\end{array}$

[0092] Rewriting equations 9a and 9b and substituting {Q} into the result for {P} {H} {T} with ΔT=1 second and ΔT²=1 second provides,
{Q}(ΔV_GPS)={R}A^b  12a.
 {Q}(ΔP_GPS)={R}A^b  12b.

[0093] Reversing the left and right sides:
{R}A^b={Q}ΔV_GPS)  13a.
{R}A^b={Q}(ΔP_GPS)  13b.

[0094] Since the roll DCM is characterized in 10c and accelerometer data is a vector: $\begin{array}{cc}\left\{R\right\}=\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{CR}& \mathrm{SR}\\ 0& -\mathrm{SR}& \mathrm{CR}\end{array}& 10d.\end{array}$  A^b=a₁{circumflex over (x)}_b+a₂ŷ_b+a₃{circumflex over (z)}_b  14.

[0095] where {circumflex over (x)}_b, ŷ_b and {circumflex over (z)}_b are unit vectors along the body axis.

[0096] Equations 13a and 13b are rewritten as: $\begin{array}{cc}\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{CR}& \mathrm{SR}\\ 0& -\mathrm{SR}& \mathrm{CR}\end{array}\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}=\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}\left(\stackrel{\_}{\Delta V_{\mathrm{GPS}}}\right)& 15a.\\ \begin{array}{ccc}1& 0& 0\\ 0& \mathrm{CR}& \mathrm{SR}\\ 0& -\mathrm{SR}& \mathrm{CR}\end{array}\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}=\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}\left(\stackrel{\_}{\Delta P_{\mathrm{GPS}}}\right)& 15b.\end{array}$

[0097] The incremental accelerometer and velocity data from the GPS is made available in fixed navigation coordinates forming two three-element vectors
{overscore (ΔV_GPS)}=ΔV_x{circumflex over (x)}+ΔV_yŷ+ΔV_z{circumflex over (z)} and  16a.
{overscore (ΔP_GPS)}=ΔP_x{circumflex over (x)}+ΔP_yŷ+ΔP_z{circumflex over (z)}  16b.
where, {circumflex over (x)}, ŷ and {circumflex over (z)} are basic unit vectors of the East, North and Up coordinate frame system.

[0098] The matrix for {Q} in 15a and 15b are multiplied by the vectors of 16a and 16b as: $\begin{array}{cc}\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{CR}& \mathrm{SR}\\ 0& -\mathrm{SR}& \mathrm{CR}\end{array}\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}=\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}\begin{array}{c}\Delta V_x\\ \Delta V_y\\ \Delta V_z\end{array}& 17a.\\ \begin{array}{ccc}1& 0& 0\\ 0& \mathrm{CR}& \mathrm{SR}\\ 0& -\mathrm{SR}& \mathrm{CR}\end{array}\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}=\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}\begin{array}{c}\Delta P_x\\ \Delta P_y\\ \Delta P_z\end{array}& 17b.\end{array}$

[0099] The left sides of 17a and 17b are multiplied out forming a 3×1 matrix on the left: $\begin{array}{cc}[\begin{array}{c}{a}_1\\ a_2\mathrm{CR}+a_3\mathrm{SR}\\ -a_2\mathrm{SR}+a_3\mathrm{CR}\end{array}]=\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}\begin{array}{c}\Delta V_X\\ \Delta V_y\\ \Delta V_z\end{array}& 18a.\\ [\begin{array}{c}{a}_1\\ a_2\mathrm{CR}+a_3\mathrm{SR}\\ -a_2\mathrm{SR}+a_3\mathrm{CR}\end{array}]=\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}\begin{array}{c}\Delta P_X\\ \Delta P_y\\ \Delta P_z\end{array}& 18b.\end{array}$

[0100] The first element a₁ of the vector on the left-hand side of equations 18a and 18b, i.e., has no roll angle component because it represents acceleration down the vehicle's longitudinal axis; therefore it is deleted leaving only the second and third row expressions. From Equations 18a and 18b: $\begin{array}{cc}[\begin{array}{c}{a}_2\mathrm{CR}+a_3\mathrm{SR}\\ -a_2\mathrm{SR}+a_3\mathrm{CR}\end{array}]=[\begin{array}{ccc}{q}_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}]\begin{array}{c}\Delta V_X\\ \Delta V_y\\ \Delta V_z\end{array}& 19a.\\ [\begin{array}{c}{a}_2\mathrm{CR}+a_3\mathrm{SR}\\ -a_2\mathrm{SR}+a_3\mathrm{CR}\end{array}]=[\begin{array}{ccc}{q}_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}]\begin{array}{c}\Delta P_X\\ \Delta P_y\\ \Delta P_z\end{array}& 19a.\end{array}$

[0101] The left side of 19a and 19b can be rewritten to form a 2×2 matrix times a 2×1 matrix as: $\begin{array}{cc}[\begin{array}{c}{a}_2\mathrm{CR}+a_3\mathrm{SR}\\ -a_2\mathrm{SR}+a_3\mathrm{CR}\end{array}]=\left[\begin{array}{c}{a}_2{a}_3\\ a_3-a_2\end{array}\right]\left[\begin{array}{c}\mathrm{CR}\\ \mathrm{CR}\end{array}\right]=[\begin{array}{ccc}{q}_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}]\begin{array}{c}\Delta V_X\\ \Delta V_y\\ \Delta V_z\end{array}& 20a.\\ [\begin{array}{c}{a}_2\mathrm{CR}+a_3\mathrm{SR}\\ -a_2\mathrm{SR}+a_3\mathrm{CR}\end{array}]=\left[\begin{array}{c}{a}_2{a}_3\\ a_3-a_2\end{array}\right]\left[\begin{array}{c}\mathrm{CR}\\ \mathrm{CR}\end{array}\right]=[\begin{array}{ccc}{q}_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}]\begin{array}{c}\Delta P_X\\ \Delta P_y\\ \Delta P_z\end{array}& 20b.\end{array}$

[0102] Combining 20a and 2Db obtains: $\begin{array}{cc}\left[\begin{array}{cc}{a}_2& a_3\\ a_3& -a_2\\ a_2& a_3\\ a_3& -a_2\end{array}\right]\lfloor \begin{array}{c}\mathrm{CR}\\ \mathrm{SR}\end{array}\rfloor =[\begin{array}{cccccc}{q}_{21}& q_{22}& q_{23}& 0& 0& 0\\ q_{31}& q_{32}& q_{33}& 0& 0& 0\\ 0& 0& 0& q_{21}& q_{22}& q_{23}\\ 0& 0& 0& q_{31}& q_{32}& q_{33}\end{array}]\left[\begin{array}{c}\Delta V_X\\ \Delta V_Y\\ \Delta V_Z\\ \Delta P_X\\ \Delta P_Y\\ \Delta P_Z\end{array}\right]& 20c.\end{array}$

[0103] A calculation of the roll angle will use the following definitions: $\begin{array}{cc}{\alpha }_1=\left[\begin{array}{cc}{a}_2& a_3\\ a_3& -a_2\\ a_2& a_3\\ a_3& -a_2\end{array}\right]& 20d.\end{array}$
is the accelerometer coefficient matrix. $\begin{array}{cc}{\beta }_1=\left[\begin{array}{cccccc}{q}_{21}& q_{22}& q_{23}& 0& 0& 0\\ q_{31}& q_{32}& q_{33}& 0& 0& 0\\ 0& 0& 0& q_{21}& q_{22}& q_{23}\\ 0& 0& 0& q_{31}& q_{32}& q_{33}\end{array}\right]& 20e.\end{array}$
is the pitch/heading coefficient matrix. $\begin{array}{cc}\stackrel{\_}{M_1}=\left[\begin{array}{c}\Delta V_X\\ \Delta V_Y\\ \Delta V_Z\\ \Delta P_X\\ \Delta P_Y\\ \Delta P_Z\end{array}\right]& 20f.\end{array}$
is the GPS information vector and $\begin{array}{cc}{\stackrel{\_}{\chi }}_1=\left[\begin{array}{c}\mathrm{CR}\\ \mathrm{SR}\end{array}\right]& 20g.\end{array}$
is the unknown roll quantity. Rewriting Equation 20c provides:
α₁{overscore (χ)}₁=β₁{overscore (M)}₁  21.

[0104] The α₁ matrix is a 4×2 matrix. {overscore (χ)}₁ is a 2×1 vector. Solving Equation 21 for {overscore (χ)}₁, recognizing that Equation 21 represents four equations in two unknowns (CR, SR), proceeds with multiplying both sides of Equation 21 by α₁^T to provide:
α₁^Tα₁{overscore (χ)}₁=α₁^Tβ₁{overscore (M)}₁  22a.

[0105] Pre-multiplying both sides by the inverse of the product of the α₁^T α₁ provides:
{overscore (χ)}₁=[α₁^Tα₁]⁻¹α₁^Tβ₁{overscore (M₁)}  22b.

[0106] Therefore, since $\begin{array}{cc}{\stackrel{\_}{\chi }}_1=\left[\begin{array}{c}\mathrm{CR}\\ \mathrm{SR}\end{array}\right]=\left[\begin{array}{c}\chi \left(1\right)\\ \chi \left(2\right)\end{array}\right]& 23a.\end{array}$
the roll angle is:
Roll(φ=tan⁻¹(χ₁(2)/χ₁(1)))=tan⁻¹(SR/CR)  23b.

[0107] This procedure is followed for additional sample times by again solving the over-determined system as follows. For t=t₂ there will be eight equations with two unknowns (cos R, sin R) and for t=t_n there will be 4n equations with two unknowns (cos R, sin R).

[0108] The iterative solution is found by stacking the equations as follows (i.e., for t=t₂): $\begin{array}{cc}\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right]{\stackrel{\_}{\chi }}_2=\left[\begin{array}{c}{\beta }_1\stackrel{\_}{M_1}\\ {\beta }_2\stackrel{\_}{M_2}\end{array}\right]& 24.\end{array}$
where α₁, β₁ and {overscore (M)}₁ have been defined above. The new variables α₂, β₂ and {overscore (M)}₂ have the same structure as α₁, β₁ and {overscore (M)}₁ except that they represent the values at t=t₂. Also, {overscore (χ)}₂ represents the values of cos R, sin R utilizing the variable over two data samples. The solution at t=t₂ is therefore obtained as before and as follows: $\begin{array}{cc}\left[{\alpha }_1^T{\alpha }_2^T\right]\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right]{\stackrel{\_}{\chi }}_2=\left[{\alpha }_1^T{\alpha }_2^T\right]\left[\begin{array}{c}{\beta }_1\stackrel{\_}{M_1}\\ {\beta }_2\stackrel{\_}{M_2}\end{array}\right]& 25.\\ {\stackrel{\_}{\chi }}_2={\left[\sum _{i=1}^2\left({\alpha }_i^T{\alpha }_i\right)\right]}^{-1}\left[\sum _{i=1}^2{\alpha }_i^T{\beta }_i\stackrel{\_}{M_1}\right]& 26.\end{array}$

[0109] Similarly at t=t_n, for n data samples, $\begin{array}{cc}{\stackrel{\_}{\chi }}_n={\left[\sum _{i=1}^n\left({\alpha }_i^T{\alpha }_i\right)\right]}^{-1}\left[\sum _{i=1}^n{\alpha }_i^T{\beta }_i\stackrel{\_}{M_i}\right]& 27.\\ {\mathrm{Roll}}_n={\tan }^{-1}\left(\frac{{\chi }_n\left(2\right)}{{\chi }_n\left(1\right)}\right)& 28.\end{array}$

[0110] For a generalized projectile or vehicle we also need data for the pitch and heading coefficient matrix. The heading and pitch (no gyro) might be approximated from the ground track (arctan Vn/Ve), and the pitch from the flight path angle (arctan of the vertical velocity or rate of change of altitude and the ground speed as determined from delta P North and delta compass available from Honeywell. If the winds aloft are known, they might be used to correct the heading and if the angle of attack were known, it might be used to correct the pitch angle.

[0111] FIG. 5 is a block diagram FIG. 5 is a block diagram that schematically provides an overview of the process functions necessary for roll angle computation. GPS RECEIVER, block 500 is shown with its internal computer providing POSITION and VELOCITY data via POSITION sampler 502, and VELOCITY SAMPLER 504. Each sampler has a respective sampler delay 506 and 508. The outputs of the samplers are Δ POSITION and Δ VELOCITY and this data is coupled to a ROLL ESTIMATE (Iterative Solution) functional block 510. The ACCELEROMETERS block 512 represents the triax of accelerometers aligned with the BODY AXES described earlier. The component of acceleration a_x is aligned with the longitudinal axis of the projectile as shown in FIG. 1. The component of acceleration a_x changes very slowly during the flight from one second to the next. The component of acceleration a_y is the cross track acceleration and the a_z is the component of acceleration along the up axis both being shown in FIG. 1. The PITCH HEADING COEFFICIENT MATRIX 514 is initialized with a_y and a_z accelerometer data. Processing of the data uses present PITCH ESTIMATE and HEADING ESTIMATE for trigonometric values. The input ΔT will be shown to have the preferred value of one second. The PITCH HEADING COEFFICIENT MATRIX is used with the GPS Δ POSITION and Δ VELOCITY inputs by the ROLL ESTIMATE functional block 510 to provide a roll angle φ output to the FLIGHT CONTROL block 516 which includes block 26 in FIG. 1.

[0112] The smoothing process referred to by block 344 in FIG. 3a will now be explained in connection with a discussion of FIGS. 6, 7a and 7b. The invention projectile guidance system with accelerometers and a GPS receiver outputs a smoothed estimate of the past trajectory of the projectile from launch to its present position, throughout its flight, to the FLIGHT CONTROL block 26, shown on FIG. 1, along with the attitude angles to permit the flight control function 26 to adjust the heading or ground track of the projectile to the known target location.

[0113] FIG. 6 provides an embodiment of a forward smoother. The embodiment shown uses Kalman filter. The Kalman filter of block 364, on FIG. 3b is such a filter in which there have been n measurements {y₁, y₂, y₂, . . . , y_n} at times t₁t₂, . . . , t_n.

[0114] A measurement at time t_k might contain acceleration data from an instrument or a velocity or a piece of position data or a sample of velocity or position data from a GPS input or from accelerometer data as delta V data and these values are arrayed as values of the y vector as {y₁, y₂, y₂, . . . , y_n}. We want an estimate of the system state at each update.

[0115] The process is time indexed or sample indexed with GPS information at a one Hz sample rate. Between GPS samples, the system reprocesses all past samples and states using the measurement index counter or k counter to provide an index as it counts from 1 to n.

[0116] The process has the objective of going back in time to obtain an educated estimate of the system state vector at an earlier sample time t_k. Measurement data, such as GPS position data, is indexed as it is acquired or received as samples 1 through k with the last sample received being measurements indexed as sample n.

[0117] FIG. 6 starts at block 600 with the initialization of the covariance matrix P(0)=P(t₀) at t=t₀ based on a knowledge of the initial state of the process at startup. Block 602 shows that the sample index counter, the k counter, is then set to one and block 604 tests to see if time t is greater than t_n,. If time is less than or equal to t_n, the test advances to block 606 and increments the index counter by one.

[0118] Block 608 represents the step of propagating the system state vector or estimated state vector x from time t_k−1 to time t_k. The symbol φ_k−1 represents the system equation or transfer function. The transfer function φ_k−1 propagates the state vector of the system x from time k−1 to time k. The state vector expressed as a state matrix “x” is a column matrix that is “m” by one in dimension. The covariance matrix P is “m” by “m” in dimension.

[0119] The processes advances to block 610. In this step, the flow chart propagates the covariance matrix P from t_k−1 to t_k, using the equation P_k^{{overscore ( )}}=φ_k−1P_k−1φ_k−1^TQ_k−1 In this equation, the term P_k is the error covariance associated with the filter estimate of the state vector {circumflex over (x)}_k^{{overscore ( )}}. The transfer function φ_k−1 and its transpose φ_k−1^T are introduced above. The term Q_k−1 is the covariance matrix of the process noise.

[0120] In block 612, the process computes the Kalman filter gain for time t=t_k. The symbol G_k represents the Kalman gain. The term H_k is the measurement or observation matrix at time t=t_k, and H_k^T represents the transpose of the measurement matrix at time t=t_k. The measurement matrix will typically be formed from elements such as Vx, Vy. The y vector or measurement vector is formed from the product of the H matrix and the state vector x, the product being added to the measurement noise matrix.

[0121] Within the brackets of the expression in block 612, the error covariance matrix P_k^{{overscore ( )}} is multiplied by the transpose of the measurement matrix H_k^T. The product of the two is then multiplied by the measurement matrix H_k. The result of the product is added to R_k, the covariance matrix of the measurement noise vector.

[0122] The R_k matrix is obtained from information that is gathered by the systems engineers through empirical testing. The systems engineers determine what noise is associated with each sensor by collecting historical noise data on each measurement variable. The H_k matrix is given and is hardware and instrument dependent. The Kalman gain is determined and then used to update the estimate of the state vectors in block 614. As each measurement is taken, it is itself a function of a combination of the states. For example, the velocity of a projectile might be the result of x and y velocity components. The result is a linear combination of the individual measurements.

[0123] The process advances to block 614 and the system state estimate vector is updated for time t_k using the equation: {circumflex over (x)}_k={circumflex over (x)}_k^{{overscore ( )}}+G_k[y_k−H_k{circumflex over (x)}_k^{{overscore ( )}}]. The term H_k multiplies the term {circumflex over (x)}_k^{{overscore ( )}}. The result is subtracted from y_k. The result is an observable difference that is multiplied by the Kalman gain and then added to the previously estimated state vector {circumflex over (x)}_k^{{overscore ( )}}.

[0124] The minus sign in the superscript of {circumflex over (x)}_k^{{overscore ( )}} implies that it represents the state a little before the kth update. The sample index “k” can range in value from 1 to n where n is the most recent update index in time. A state variable without a superscript implies that the state variable value is a value that exists just after an update. The symbol {circumflex over (x)}_k represents the kth estimate of the state vector x. When the y_k measurement is made, the kth update is performed for the y_k measurement. The measurement index “k” is a running index.

[0125] After updating the state vector at time t=t_k by evaluating the equation in block 614, the process proceeds to the equation in block 616:
P_k=[I−G_kH_k]P_k^{{overscore ( )}}
which is used to update the covariance matrix P_k The matrix I is an identity matrix. All of its elements are zero except the main diagonal elements which are ones. The dimension of the identity matrix is matched to the product of G_k and H_k matrices. All of the values necessary for the computation of the covariance matrix are available from previous steps.

[0126] After the computation of the P_k matrix, the process leaves block 616 and transfers back to decision block 604 to once again test to see if t>t_n. As time reaches t_n, all of the measurements have been processed. No additional measurements are to be made. A “yes” result transfers the process to the Scenario Complete block 618 and the mission is completed. If the flight had lasted one hundred seconds and the GPS provided position measurements at one per second, there would have been 100 GPS samples and the sample index k would have been incremented from 1 to 100. The index “k” cannot exceed 100.

[0127] FIGS. 7a and 7b form a fixed interval smoother, FIG. 7a providing a forward pass and FIG. 7b providing a backward pass. FIG. 7a can be seen to be identical in function to that of FIG. 6 with the exception of block 700. The process exits block and advances to block 702 also found at the top of FIG. 7b. The backward sweep begins with block 704 as the process sets the sample index counter, i.e., the k counter to current index value of n. The object of the backward sweep process of FIG. 7b is to compute the smoother estimate x_s(k)={circumflex over (x)}(k|n) and the smoother error covariance P_s(k)=P(k|n), using all n measurements as the measurement index counter “k” is decremented k=n, n−1, n−2, . . . , 1. A subscript “s” in the expression x_s (k) and in the expression P_s(k) indicates a smoothed variable.

[0128] The process advances to block 706 to compute smoother gain:
A(k)=P(k|k)φ_k^TP⁻¹(k+1|k)

[0129] The vertical bar followed by a “k” in a term such as (k|k) means that the matrix will be evaluated for all measurements up to and including the kth measurement. The backward sweep begins only after the conclusion of a forward sweep. All k measurements are available at the conclusion of the forward sweep. All k data points form a fixed interval, and all n measurements from t_k back to t₁ are used during the backward sweep.

[0130] The covariance matrix P(k|k) is multiplied with the transpose of the transfer function φ_k at time t=k.

[0131] The rightmost term in block 706 is the inverse of the covariance matrix P(k+1|k) at time t_k+1 using all k measurements. The vertical bar followed by “k” shows that all of the data for measurements through k is to be used.

[0132] The process then advances to block 708. The object of the equation in block 708 is to compute a smoothed state vector {circumflex over (x)}(k|n) using all n measurements. The equation in block 708 is:
{circumflex over (x)}(k|n)={circumflex over (x)}(k|k)+A(k)[{circumflex over (x)}(k+1|n)−{circumflex over (x)}(k+1|k)]

[0133] The gain matrix A(k) is available from block 706. The first term after the equal sign is the kth state vector estimate using measurement data through measurement k. The first term inside of the bracket, the term {circumflex over (x)}(k+1|n) represents the estimate of the state vector at time k+1 using data for all measurements through measurement n. The second term inside of the bracket {circumflex over (x)} (k+1|k) is an estimate of the state vector at time k+1 using all measurements up to time t_k. The difference is calculated as the second term is subtracted from the first. The result is multiplied by the gain matrix, the product is added to {circumflex over (x)}(k|k).

[0134] The process then advances to block 710 to compute a smoothed covariance matrix from the equation:
P(k|n)=P(k|k)+A(k)[P(k+1|n)−P(k+1|k)]A^T(k)

[0135] The first covariance term after the equal sign P(k|k) uses all data through the kth update. The first covariance term in the bracket P(k+1|n) uses all of the data from measurements through measurement n. The second covariance term in the bracket P(k+1|k) uses only data up through the kth update. A covariance difference is calculated from the covariance terms within the brackets. The covariance difference term is then multiplied by the gain matrix and the result is added to P(k|k).

[0136] After the smoothed covariance estimate is calculated, the process advances to block 712 and decrements the measurement index counter from k to k−1. The process then advances to decision block 714 and a test is made to determine if the measurement index matrix k is greater than one. Looping or cycling continues with each “no” decision followed by a cycle back to block 706 until k=1. When k=1, the test at block 714 results in a “yes” decision and the process advances to block 716, the Backward Sweep Complete block. Block 716 signals the conclusion of the program as the projectile nears its target.

[0137] Those skilled in the art will appreciate that various adaptations and modifications of the preferred embodiments can be configured without departing from the scope and spirit of the invention. Therefore, it is to be understood that the invention may be practiced other than as specifically described herein, within the scope of the appended claims.