Fundamentals of Strapdown Inertial and GPS-Aided Navigation
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1 Fundamentals of Strapdown Inertial and GPS-Aided Navigation Fundamentals of Strapdown Inertial and GPS-Aided NavigationByProfessor Dominick AndrisaniPurdue University, West Lafayette, INTactical Imagery Geopositioning WorkshopMarch 12, 2002Chantilly, VA
2 Purposes of this talkTo provide a tutorial overview of inertial navigation systems (INS).Illustrate ideas with a 2-D navigator.Discuss inertial sensors (simple rate gyros and linear accelerometers).Discuss characteristic errors in the INS.To demonstrate the need by the INS for altitude aiding.To show how GPS aids the INS and leads to far superior navigation accuracy.
3 Types of Inertial Navigation Systems (INS) Inertial Platform based INSStrapdown INS <<(emphasized here)Aided Navigators of either typeAltitude-aidedAltitude and X-aidedHeading-aidedGPS aided
8 Properties of Platforms AdvantagesSimpler gyros (platform rotates at small rates, lower dynamic range).High accuracy (North and East accelerometers do not see a component of gravity).Self alignment by gyro compassing.Sensor calibration by platform rotations.DisadvantagesComplexity and cost.Gimbal magnetics (torquers must not leak magnetic flux).Reliability (bearings and slip rings tend to wear).
9 Properties of Strapdown Systems AdvantagesSimple structure, low cost.More rugged and lighter.Reliability (no gimbal magnetics, no slip rings, no bearings, electronics more reliable then machinery).DisadvantagesMore difficult to align.More difficult to calibrate.Motion induced errors which can only be partly compensated for.Accelerometer errors (each accelerometer may feel 1 g from gravity).Requires a computer that can perform coordinate rotations in <.01 sec).
10 Simple Example: Two Dimensional Motion Xb, Yb, and Zbare body fixedYb=YNED.XbPitch angleqXNED is northerlyYNED is easterlyZNED is downXNEDNorth PoleZNEDZbSouth Pole
11 Equations of Motion of the Aircraft dQ/dt=vx/(Ro+h) + wydvx/dt =vxvy/(Ro+h) + fxdvz/dt =-vz2/(Ro+h) + fz + g(h)dx/dt =vxdh/dt =-vzwherefx=fxbcos(Q)+fzbsin(Q)fz=-fxbsin(Q) + fzbcos(Q)is pitch anglevx is velocity in northerly directionvz is velocity in down directionx is northerly positionh is altitude (positive up)
12 Inertial SensorsRate gyros measure the components of inertial angular rate of the aircraft in the sensitive direction of the instrument.Linear accelerometers are used to measure the components of aircraft linear acceleration minus the components of gravity in its sensitive direction.Newton’s Law for the aircraft isF=ma=Faero+Fthrust+mgAccelerometer measuresa-g=(Faero+Fthrust)/m=specific forceIn this simple two-dimensional example, two linear accelerometers and one rate gyro are used.
13 A Single Axis Angular Rate Gyro This device measures inertial angular rate about its sensitive direction. Three of these arranged orthogonally measure the components of the angular velocity vector.Ref 3.
14 A Simple “Open Loop” Accelerometer This device measures specific force= a-g=(Faero+Fthrust)/m.They cannot distinguish between acceleration and gravity.gRef 3.
19 INS Free IntegratorThe free integrator will create the following types of errors.For initial condition errors on x, the resulting x-position error will neither decay or grow.For initial condition errors on Vx, the resulting x-position errors will grow linearly with time.
21 INS Unstable Altitude Loop The unstable altitude loop results because errors in altitude means that there will be errors in the determination of the acceleration of gravity.This in turn will propagate into an error in vertical acceleration which will be in the direction to drive the altitude error further from the correct value. This is an unstable mechanism since altitude error leads to greater altitude error.
22 INS Simulation (Schuler Pendulum) Schuler Pendulum Loop
23 Schuler Pendulum LoopThe Schuler pendulum loop creates dynamic errors that oscillate with an 84 minute period.The Schuler pendulum loop, while creating persistent oscillations, does limit the growth of errors in velocity (Vx).
24 The Schuler PendulumImagine we have a pendulum to provide a vertical reference.As we accelerate horizontally, the pendulum tilts, giving a false vertical indication.Schuler showed that this would not occur with a pendulum having a period of 84 minutes (a ball on a string with length equal to the radius of the Earth has this period).Correcting an inertial system so that it does not tilt when accelerated is known as Schuler tuning.
25 Error Analysis Via Linearization Nonlinear navigator equations of motiondX/dt=f(X,U)Error modele=XINS-Xsimulationu=UINS-UsimulationLinear error equations of motionde/dt=Ae+BuwhereA=df/dX evaluated at the a reference state X and input U)B=df/dU evaluated at the a reference state X and input U)
26 Error Analysis Via Linearization, continued Linear error equations of motionde/dt=Ae+Bu with initial condition e(0)System matrix A will have 5 poles (eigenvalues), two complex poles for the Schuler Pendulum, two real poles for the altitude modes (one unstable, one stable, equal magnitude), and one pole at zero (X-pole).The error system provides a useful way to study INS error propagation using linear methods and as the basis for designing Kalman filters to implement the various aiding techniques (e.g. altitude aiding).
27 Nonlinear Simulation of Aircraft and INS Given three inputs, we can find all outputs including errors.
30 Error Analysis Using Nonlinear Simulation Examine navigation errors when the IC on X is in error.
31 Error Due to X Initial Condition Errors remain constant.
32 Simulation of Aircraft and INS Examine navigation errors when the IC on H is in error.
33 Errors Due to H initial Condition Errors are dominated by unstable altitude mode.
34 Simulation of Aircraft and INS Examine navigation errors when IC on Vx is in error.
35 Errors Due to Vx Initial Condition In the flat earth navigator the X-error would go to infinity.The Schuler pendulum mode limits the X-error.Note both Schuler oscillation and unstable altitude mode.
36 INS Aiding Altitude Aiding Velocity and position errors in the vertical channel are not bounded and can quickly become quite large.Barometric altitude provides a measure of height above sea level, typically to an accuracy of 0.1%.Most airborne INS operate with barometric aiding in order to bound the growth of vertical channel errors.Many other types of aiding are typically used
42 GPS Aiding of INSGPS can provide aiding to an INS by providing an independent measurement of x, y, and z (altitude).Furthermore, certain GPS implementation can provide velocity aiding by providing independent measurements of Vx, Vy and Vz.A Kalman filter is often used to help blend the GPS measurements with the INS outputs in an optimal way.
44 Measurements from the GPS Receiver Model PseudorangePseudorange Rate: Satellite Position: Platform Position: Pseudorange equvalentClock Bias (Random Walk): Pseudorange rate equivalentClock Drift (Random Walk): Normally Distributed Random Numbers
45 Benefits of Integrated INS/GPS Systems INS gives accurate estimates of aircraft orientation.GPS provides accurate estimates of aircraft position.INS solutions are generally computed 100 times per second.GPS solutions are computed once per second.GPS in subject to jamming, INS is not.Combining GPS and INS provides accurate and robust determination of both translational and rotational motion of the aircraft.Both translational and rotational motion are required to locate targets on the ground from the aircraft.
46 ConclusionsUnaided INS have troublesome errors that grow with time or oscillate with an 84 minute period.Various aiding schemes are often implemented to stabilize the INS errors.GPS aiding of INS is an effective means to stabilize INS position and velocity errors.Integrated INS and GPS systems are useful for determining both the position and orientation of an aircraft. Such systems are therefore helpful in locating of targets on the ground.
47 Additional Purdue Resources Presented at the The Motion Imagery Geolocation Workshop, SAIC Signal Hill Complex, 10/31/011. Dominick Andrisani, Simultaneous Estimation of Aircraft and Target Position With a Control Point2. Ade Mulyana, Takayuki Hoshizaki, Simulation of Tightly Coupled INS/GPS Navigator3. James Bethel, Error Propagation in Photogrammetric Geopositioning4. Aaron Braun, Estimation Models and Precision of Target DeterminationPresented at the The Motion Imagery Geopositioning Review and Workshop, Purdue University, 24/25 July, 20011. Dominick Andrisani, Simultaneous Estimation of Aircraft and Target Position2. Jim Bethel, Motion Imagery Modeling Study Overview3. Jim Bethel, Data Hiding in Imagery4. Aaron Braun, Estimation and Target Accuracy5. Takayuki Hoshizaki and Dominick Andrisani, Aircraft Simulation Study Including InertialNavigation System (INS) Model with Errors6. Ade Mulyana, Platform Position Accuracy from GPS
48 References1. B.H. Hafskjold, B. Jalving, P.E. Hagen, K. Grade, Integrated Camera-Based Navigation, Journal of Navigation, Volume 53, No. 2, pp2. Daniel J. Biezad, Integrated Navigation and Guidance Systems, AIAA Education Series, 1999.3. D.H. Titterton and J.L. Weston, Strapdown Inertial Navigation Technology, Peter Peregrinus, Ltd., 1997.4. A. Lawrence, Modern Inertial Technology, Springer, 1998.5. B. Stietler and H. Winter, Gyroscopic Instruments and Their Application to Flight Testing, AGARDograph No. 160, Vol. 15,1982.A.K. Brown, High Accuracy Targeting Using a GPS-Aided Inertial Measurement Unit, ION 54th Annual Meeting, June 1998, Denver, CO.
54 Poles of Various systems Poles of H and X aided observerWn52 =0.01100.1430Z52 =0.71611.00000.5000P52 =iiiiPoles of H aided observerWn41 =0.01290.0130Z41 =0.44270.8973P41 =iiiiPoles of various systemsOLpoles =ii