Inertial Navigation - University Of Florida
2002 Florida Conference on Recent Advances in RoboticsInertial NavigationKevin J Walchko1University of Florida, Gainesville, FL 32611-6200Dr. Paul A. C. Mason2NASA Goddard Space Flight Center, Greenbelt, MDThis paper will discuss the design and implementation of an inertial navigation system (INS) using an inertialmeasurement unit (IMU) and GPS. The INS is capable of providing continuous estimates of a vehicle’s positionand orientation. Typically IMU’s are very expensive sensors, however this INS will use a “low cost” versioncosting only 5,000. Unfortunately with low cost also comes low performance and is the main reason for theinclusion of GPS into the system. Thus the IMU will use accelerometers and gyros to interpolate between the1Hz GPS positions. All important equations regarding navigation are presented along with discussion. Resultsare presented to show the merit of the work and highlight various aspects of the INS.I. IntroductionOhlmeyer et al [3] developed a GPS/INS system for a newsmart munitions, the EX-171. Due to the high speed of themissile, update rates of 1 second from a GPS only solutionwere too slow, and could not provide the accuracy needed.Navigation has been present for thousands of years in someform or another. The birds, the bees, and almost everythingelse in nature must be able to navigate from one point inspace to another. For people, navigation had originallyincluded using the sun and stars. Over the years we havebeen able to develop better and more accurate sensors tocompensate for our limited range of senses. This paper willdiscuss work using one of these advanced sensors, aninertial measurement unit (IMU). This sensor, coupled withthe proper mathematical background, is capable ofdetecting accelerations and angular velocities and thentransforming those into the current position and orientationof the system.A. OutlineThe first section of this paper will introduce inertialnavigation. Then the IMU and GPS hardware will becovered. Finally experimental results using this INS will bepresented.II. Inertial NavigationThis section will cover strap-down inertial navigation byfirst describing the methods and equations. Next sources oferror for these systems and how the kalman filter will beutilized to account for these errors.Inertial Navigation Systems (INS) have been developed fora wide range of vehicles. Sukkarieh [1] developed a GPS/INS system for straddle carriers that load and unload cargoships in harbors. When the carriers would move from shipto ship, they would periodically pass under obstructionsthat would obscure the GPS signal. Also, as the carriers gotcloser to the quay cranes, it became more difficult to getaccurate positions due to the GPS signal being reflectedabout the cranes metal structure. This increases the time offlight of the GPS signal and results in jumps in the position.During these times the INS would then take over, and guidethe slow moving carrier until a reliable GPS signal could beacquired.A. Overview of Inertial Navigation SystemsA basic flow chart of how inertial navigation works isshown in Figure 1. However, this is not all that needs to bedone to have an INS that works. There are many problemswith noise and unbounded error that must be handled to getany meaningful result out of the INS.Gimballed INSThe first type of INS developed was a gimballed system.The accelerometers are mounted on a motorized gimballedplatform which was always kept aligned with thenavigation frame. Pickups are located on the outer andinner gimbals which keep track of the attitude of thestabilized platform relative to the vehicle on which the INSis mounted. This setup has several detractors which make itundesirable.Bennamoun et al [2] developed a GPS/INS/SONAR systemfor an autonomous submarine. The SONAR added anothermeasurement to help with accuracy, and provided apositional reference when the GPS antenna got submergedand could not receive a signal.1. Graduate Research Assistant, Mechanical Engineering. Student member AIAA.2. Mechanical Engineer, Flight Dynamics Analysis Branch. Member AIAA.1
2Inertial NavigationFIGURE 1 A flow chart of a strap-down INS which takes acceleration and rotation rates from the IMU and producesposition, velocity, and attitude of the system.Strap-down INS Bearings are not frictionless. Motors are not perfect (i.e. dead zones, etc.). Consumes power to keep the platform aligned with the navigational frame which is not always good onan embedded system.Cost is high due to the need for high quality motors,slip rings, bearings and other mechanical parts.Thus the typical customers for such systems weremilitary uses on planes, ships, and intercontinentalballistic missiles.Recalibration is difficult, and requires regular maintenance by certified personnel which could be difficult on an autonomous vehicle. Plus anymaintenance that must be performed on the system(i.e. replace bearings, motors, etc.) must be done ina clean room and then the system must go through alengthy recertification process.FIGURE 2 The XYZ frame is the inertial frame ECEFand the NWU frame is the local navigational frame,where the axes are north (N), west (W), and up (U).A strap-down system is a major hardware simplification ofthe old gimballed systems. The accelerometers and gyrosare mounted in body coordinates and are not mechanicallymoved. Instead, a software solution is used to keep track ofthe orientation of the IMU (and vehicle) and rotate themeasurements from the body frame to the navigationalframe. This method overcomes the problems encounteredwith the gimballed system, and most importantly reducesthe size, cost, power consumption, and complexity of thesystem.B. Reference Frames and RotationsInertial navigation uses several reference frames, which areshown in Figures 2 and 3. To transition between the variousreference frames, several rotation matrices are needed. Thefirst one takes measurements in the body frame and putsthem into the navigation frame,FIGURE 3 Body frame which is aligned with the axes ofthe IMU. The center of this frame is located at the origionof the navigational frame.
Inertial Navigation3cθcψ sφsθcψ – cφsψ sφsψ cφsθcψR bn cθsψ cφcψ sφsθsψ cφsθsψ – sφcψ– sθsφcθcφcθ0where φ is roll, θ is pitch, and ψ is yaw. This rotation isthe sequence 1-2-3, which is typically used in aerospaceapplications. This is a type 1 sequence which hassingularities when the pitch is /- 90 degrees since at thisangle both the roll and yaw have similar effects. Thus forfighter aircraft which typically encounter this range, othermethods must be included to account for this problem.The next rotation will transform points from the ECEFframe to the navigation frame,– sφcλ – sφsλ cφR en – sλcλ0– cφcλ – cφsλ – sφ(2)(3)The last thing to remember with the above equation is thatthe inverse of any orthogonal rotation matrix is equal to itstranspose. If a rotation matrix is not orthogonal (and this aproblem with using Euler angles in navigation) then theprevious statement is invalid.C. Navigation Equations(4)In inertial navigation, accelerometers detect accelerationsdue to forces exerted on the body. These forces are typicallyreferred to as specific forces (S). Thus reading from theIMU will be referred to as specific forces, which areindependent of the mass. The navigation equations for theEarth Centered Earth Fixed (ECEF) system are shownbelow.eeecV· eR be R be 0 g SHCΩ ie– 2Ω ie– Ω ie0 V·P e 0 0 0SbI00 P·Φ000ω00QΦ(5)0ωz –ωy ωx1 –ωz 0 ωx ωyQ --2 ω –ω 0 ωyxz(7)–ωx –ωy –ωz 0D. Sources of ErrorThis section will provide a quick overview of somedifficulties present in inertial navigation. This will providea better understanding for the difficulties encountered withthe IMU.Bias and DriftThese are the most devastating effectors on accuracy to anIMU. Drift rate for the gyros and accelerometer bias aresmall offsets which the IMU incorrectly reads, that must beproperly accounted for. The bias has a quadratic effect onthe position derived from the IMU.Looking at Newton’s second law of motion, a change inmotion occurs as a force is applied to a body. Now, dividingboth sides of the equation by the mass of the object resultsin the specific force.---f- a Sm0(6)where ω ie is the rotation rate of the earth, R is a rotationmatrix between different coordinate systems, P is theposition and V is the velocity vector in the ECEFcoordinate system as denoted by the superscript e. Also theattitude will be changed from euler’s roll, pitch, and yaw toquaternions. Quaternions will help prevent the body tonavigation rotation matrix, which transforms points frombody frame to the navigational frame and back, frombecoming non-orthogonal.0where φ is latitude and λ is longitude. Now with these tworotations we can get another rotation, the one we reallyneed.R be R ne R bn0 – ω ie 0eΩ ie ωie 0 0(1)1error --- bias t 22(8)TABLE 1. Positional error that results frombiases after a time of 100 seconds and 30 mins.Bias (m/s 2)Error (m)t 100 sec.Error (m)t 30 ng at the Table 1 above it becomes apparent thatdetermining the bias is of critical importance if any accuratemeasurement is expected.The drift rate has a similar, and an equally massive impacton the position of a system. If a drift is not properlyaccounted for, and the IMU thinks it is rotating, then the
4Inertial NavigationFIGURE 4 Overview of the extended kalman filter’s integration with the INS.navigation equations will not properly account for gravityand the system will think it is moving due to a maximumacceleration of 9.8 m s 2 depending on how far the systemhas drifted.TemperatureThe IMU’s accelerometers and gyros are sensitive totemperature as shown by Nebot and Durran-Whyte [4].Thus as the temperature of the IMU changes, the associatedbias and drift will change until the temperature reachessteady state or remains the same. This is not critical in ourapplication, we just wait for the IMU to reach steady statebefore trusting the readings. However if this system wasmounted in an aircraft which changed altitude andtemperatures, this would be a problem.HysteresisThe drift rates and accelerometer biases tend to change eachtime the unit is switched on. This is due to the fact thatmeasurements are noisy. Typically there is a low pass filterused to remove some of this noise before the measurementsare used in the navigation equations (also realistically, theretends to be low pass filtering somewhere in the system dueto hardware limitation because not everything has infinitebandwidth). When random noise is filtered, this produceswhat is called a random walk. The integration of thisrandom walk will result in velocity and positions moving atdifferent rates during different runs even though the IMU(and vehicle) are in the same orientation and experiencingthe same accelerations during each run.To give an idea of the performance of a strap-down system,the following quote is taken from an article [5] written byA. D. King, Chief Engineer of Navigation and Electro-opticSystems Division of Marconi Electronic Systems. Marconiproduces INS for virtually all of the RAF’s combat aircraftas well as many other systems.“Many of these instrument errors vary each timeyou switch the system on - INS have good days andbad days. To characterize the performance of anINS, you have to resort to statistics, and take ther.m.s. total error from an ensemble of manyrepresentative missions. A typical standardexpected from a ‘good’ INS produces an error thatincreases with time (not an entirely linear fashion),and reaches .6 miles after one hour (referred to as.6 nautical miles/hour system).”VibrationsVibration in a strap-down system can cause many problemswith the INS. Generally great care must be taken to isolatethe IMU from any resonance frequencies. In high precisionsystems, various tests must be done to try to identify whatthese frequencies are then design elaborate mounts to holdthe IMU.E. Extended Kalman FilterIn addition to the prefiltering of the IMU data, an extendedkalman filter was developed to estimate the biases anddrifts of the system and then update the navigationalsolution. The full kalman filter equations will not bepresented here due to limited space, but an overview of theprocess is shown in Figure 4 and further information can befound in Brown and Hwang [6].The error model was developed based on derivations byChatfield [7] and Rogers [8]. This filter model is smallcompared to other authors have anywhere between 20 and50 different states, depending on how their navigationalmodels were defined. Note that there is also the inclusion oftwo sets of terms which now makes this an extendedkalman filter model. The terms are the errors in bias on theaccelerometers, and drift of the gyros. Each is modelled asa random walk (or could have modelled them as a markovprocess), where the terms with the subscript N on the farright of the equation are zero mean, random white noisewith the appropriate standard deviation. The purpose of thisis to estimate these new parameters, since they are difficultto determine, and (as in the case of the bias) change greatlydepending on temperature, time, and orientation.δV·δP··δφδS· b·δω bδVδPδS Nb A δφ Bδω NbδS bδω b(9)
Hardware5FIGURE 5 Sensors used in the INS. (left) The Crossbow DMU-HDX which is a solid state vertical gyro capable ofmeasuring angular rates and accelerations on all three axes. It also has the capability of measuring the roll and pitch ofthe device too. (right) Garmin 16LVS OEM GPS which is both a reciever and antenna.FIGURE 6 This is a plot of the biases as the IMU wasrotated around the z-axis (yaw). Rotations around theother axes would also efect the biases, thus this mappingis ont useful since the values are changing nonlinearly.e– 2Ω ieΩ ie Ω ieA SeA. Crossbow IMUR be 0 3x3I 3x30 3x30 3x3 0 3x3 0 3x30 3x30 3x3Ω be 0 3x3 – R be(10)0 6x15B 0 6x6(11)I 6x60 – S ze S yeSe S ze0 – S xe– S ye S xeFIGURE 7 Comparison of the unfiltered data (top)produced by the IMU and the filtered data (bottom) usingthe Chebyshev II filter. Data is from one of theaccelerometers while the IMU is sitting still on a table.(12)0III. HardwareThis section will provide an overview of the two primarysensors, the IMU and GPS shown in Figure 5.The IMU is a solid state vertical gyro (DMU HDX) fromCrossbow Technologies intended for airborne applicationssuch as UAV control, Avionics, and Platform Stabilization.This high reliability, strap-down inertial subsystemprovides attitude measurement with static and dynamicaccuracy comparable to traditional spinning mass verticalgyros. Data will be transmitted by the DMU digitally via aserial connection (RS-232). The gyros on the CrossbowIMU are low cost, low performance MEMS (MechanicalElectrical Micro-Systems) gyros. These gyros are muchless expensive to produce, but performed at least an orderof magnitude worse than another low cost IMU systembeing developed by Dr. Crane here at U.F. That system usesan IMU developed from Honeywell which has ring lasergyros. Unfortunately, the gyro performance is a criticalelement in accounting for gravity in the system.Prefiltering IMU DataThe data produced by the IMU is extremely noisy, thus afilter was designed. Matlab’s signal toolbox was used toaccomplish this task. The toolbox is capable of designing
6Inertial NavigationFIGURE 8 This is a test of the GPS accuracy. The GPS was set in a stationary location for 4 hours. The center of theplot was taken to be the average latitude and longitude reported by the sensor. Then the corresponding distances fromthe average were calculated. This GPS reciever is capable of providing the standard 10 meter accuracy 95% of thetime.all of the classic FIR and IIR filters. A IIR filter wasdecided on since it produces the same results as an FIRfilter but with a much lower order. This lower order resultsin a less computational process. The followingspecifications for the filter were decided on: pass bandvalue of 2Hz, stop band of 3 Hz, and a stop band attenuationof -50 dB.Additionally, the desired filter should not have any ripple inthe pass band range, thus the Equiripple, Elliptic, andChebyshev I filters were eliminated as possible designs.The remaining Butterworth and Cheyshev II filters werelooked at. After much testing with various options, theChebyshev II filter was settled one as the best on for the joband its performance can be seen in Figure 7.B. Garmin GPSThe GPS system used in this work is the Garmin 16LVS.Garmin is a common name in commercial civilian GPSsystems, and this OEM device has performance that is onpar with all other GPS systems available currently (i.e.accuracy of about 10 m 95% of the time) as shown in Figure8. However this GPS was specifically bought because itincluded a WAAS (Wide Area Augmentation System)filter which should increase the accuracy to less than 3 m95% of the time.WAAS [9] utilizes ground stations which detect and sendGPS error information to a Master Control site. The MasterControl site uses this information to compute in order ofimportance or effect:1.Integrity information2.Ionospheric and Tropospheric delays3.Short-term and long term satellite clock errors4.Short-term satellite position error (Ephemeris)5.Long term satellite position error (Almanac)This information is relayed to two WAAS geosynchronousInmarsat satellites (AOR-W and POR) from the MasterControl Stations and is re-broadcast to user receivers as agrid of corrections. From this grid, a GPS receiverinterpolates the proper Ionospheric correction based on itsposition in the grid. The "extrapolation" of this informationoutside the WAAS coverage is less and less precise -to thepoint of INDUCING errors. Other errors are not locationdependant.The WAAS correction information is different than RTCMcorrections (transmitted by the Coast Guard for uses inDGPS) because WAAS decomposes the errors into theirprimary elements (Iono, clock, & ephemeris). RTCM, onthe other hand, broadcasts pseudorange corrections whichare the sum of all error sources as observed by the RTCMreference station. This information is only valid relativelyclose to the reference station. This is why spatialdecorrelation is such a large factor for RTCM, but not forWAAS (thus the reason it is "wide area" augmentation).IV. ResultsThe experiment is broken up into two parts. The first part isthe navigation solution which does not utilize the kalmanfilter or the GPS positional corrections. The second partwill include these so the limitation of the IMU and benefitsof the kalman filter and GPS can be seen.A. Test SetupThe experiments were conducted using a car with the IMUand GPS mount on it. A laptop was connected to bothsensors and recorded the data. The data was then taken andanalyzed in Matlab using the proceeding equations.
Results7FIGURE 9 INS attitude solution with out extended kalman filter. The estimated roll, pitch, and yaw are shown bythe solid line, while the true roll and pitch reported by the IMU is the dashed line.FIGURE 10 INS results without GPS and kalman filterintegrated into the system.B. Navigational Solution OnlyThe first set of results was without the use of the extendedkalman filter, to see if it was really necessary. The resultsof estimating the roll, pitch, and yaw without anycorrections is shown in Figure 9. The estimated anglesappear to track the true angles to an acceptable degree. TheIMU is capable of reporting it’s true roll and pitch, but notyaw. Assuming the performance between estimating theyaw angle and the pitch and roll angles are the same, itshould not be necessary to require a compass to update thetrue yaw angle. The couple degrees of error should noteffect INS results much since the car is traveling on flatroads.The performance changes when we look at figure 10. TheGPS and INS (i.e. using the navigation equations and IMUdata only) differ greatly. Thus the GPS with the kalmanFIGURE 11 INS results with GPS and kalman filterintegrated in to the system.filter must be included into the INS to give any goodresults.C. GPS/INSAfter the inclusion of the GPS and kalman filter, the plotshown in Figure 11 is much better. The GPS and INS lieright on top of each other. Taking a closer look at this plot,Figure 12 and Figure 13 show that the two do not really lieexactly on top, but rather the INS transitions smoothlythrough the GPS points.Looking specifically at Figure 12, it can be seen that theIMU is picking up some of the accelerations in the turn andshifted the position left of the GPS points. But going intothe turn and once the turn is completed, the INS and GPSpositions merge back together.
8Inertial NavigationFIGURE 12This plot shows the interpolatingcapabilities of the INS system in X and Y.Figure 13 is better example showing how the INS is able totake the discrete GPS position and the accelerations fromthe IMU and fit a curve through the two. This level ofcontinuos positioning can not be offered by GPS alone.Finally the distances traveled during the experiment werecalculated and the results were close as shown in Table 2.The car’s odometer was felt to be the most accurate and theGPS and INS distances are on either side of the value.TABLE 2. Distances traveled as reported by thedifferent systems.GPSINSkm4.895.16miles3.043.21Car3.1The extended kalman filter attempts to estimate the biasesand drifts present in the system to increase the accuracy ofthe system. However there appeared to be no differencebetween using the estimated biases and drifts estimatedfrom the filter or using constant ones. This is attributed tothe excessive amount of noise from the low cost IMU.Chatfield’s [7] work was the prime motivator for includingthese terms in the extended kalman filter, but he assumedmeasurement that were much better (i.e. less noisy) than theones being produced by the Crossbow IMU. Thus this partof the kalman filter could be eliminated to reducecomputational expense with no loss of performance.V. ConclusionsThis paper has shown the effective combination of twodifferent sensors (GPS and IMU) each with their ownstrengths and weaknesses. The “low cost” IMU used in thiswork is not capable of running by itself and providing anyFIGURE 13This plot shows the interpolatingcapabilities of the INS system in Z.reasonable positioning information. GPS provides goodresults, but is only capable of determining position everysecond. The two sensors combined has the capability ofproducing good estimates of position in between the onesecond updates.AcknowledgmentsThe research conducted in this paper could not have beenaccomplished with out the help of the Machine IntelligenceLab at University of Florida who donated the use of theIMU.VI. References1 Sukkarieh, S., “Low Cost, High Integrity, Aided Inertial Navigation Systems for Autonomous Land Vehicles,” Ph.D. Thesis, University of Sydney, March 2000.2 Bennamoun, M., Boashash, B., Faruqi, F. and Dunbar,M., “The Development of an Intergrated GPS/INS/Sonar Navigation System for Autonomous UnderwaterVehicle Navigation,” 1990 IEEE Symposium onAutonomous Underwater Vehicle Technology, Washington, DC, pp. 256-261, June 5-6, 1990.3 Ohlmeyer, E., Pepitone, T., and Miller, B., “Assessment of Integrated GPS/INS for the EX-171 ExtendedRange Guided Munition,” Naval Surface Warfare Censter.4 Nebot, E., and Durrant-Whyte, H., “Initial Calibrationand Alignment of Low-Cost Inertial Navigation forLand Vehicle Applications,” Journal of Robotic Systems, Vol. 16, No. 2, February, 1999, pp. 81-92.5 King, A. D., “Inertial Navigation - Forty Years of Evolution,” GEC Review, Vol. 13, No. 3, 1998, pp. 140149.
References6 Brown, Robert and Hwang, Patrick, Introduction toRandom Signals and Applied Kalman Filtering, ThirdEd., John Wiley and Sons, 1997.7 Chatfield, A., Fundamentals of High Accuracy InertialNavigation, AIAA, Inc., 1997.8 Rogers, R. M., “Applied Mathematics in IntegratedNavigation Systems,” Reston, VA : American Instituteof Aeronautics and Astronautics, 2000.9 http://www.gpsinformation.net/waasgps.htm9
2 Inertial Navigation Bearings are not frictionless. Motors are not perfect (i.e. dead zones, etc.). Consumes power to keep the platform aligned with the navigational frame which is not always good on an embedded system. Cost is high due to the need for high quality
Visual Inertial Navigation Short Tutorial Stergios Roumeliotis University of Minnesota. Outline . "Visual-inertial navigation: A concise review," IRA'19. Introduction Visual Inertial Navigation Systems (VINS) combine camera and IMU . Continuous-time System Equations: Quaternion of orientation: Rotation matrix: Position: Velocity
Redundant Inertial Navigation Unit (RINU) The RINU is a redundant inertial navigation system manufactured by Honeywell International, Inc (HI). The RINU is derived from the Fault Tolerant Inertial Navigation Unit (FTINU) INS previously flown on the Atlas V launch vehicle. The RINU features a redundant set of five inertial instruments channels.
Inertial Sensors, Precision Inertial Navigation System (PINS). 1 Introduction Presently Inertial Navigation Systems are compensated for gravitational acceleration using approximate Earth gravitation models. Even with elaborate model based gravitation compensation, the navigation errors approach upto several hundred
only inertial navigation system. Objective of the proposal: The objective of the proposal is a combination of the existing inertial navigation system (INS) with global position system (GPS) for more accurate navigation of the launchers. The project's product will be navigation algorithms software package and hardware units.
2.2 Fundamentals of Inertial Navigation, 19 2.2.1 Basic Concepts, 19 2.2.2 Inertial Navigation Systems, 21 2.2.3 Sensor Signal Processing, 28 2.2.4 Standalone INS Performance, 32 2.3 Satellite Navigation, 34 2.3.1 Satellite Orbits, 34 2.3.2 Navigation Solution (Two-Dimensional Example), 34 2.3.3 Satellite Selection and Dilution of Precision, 39
correct the inertial navigation solution and also to constrain future development of navigation errors by correcting the incoming inertial measurements. While different approaches for navigation-aiding can be found in the literature, arguably the most common approach is based on various variants of the well-known extended Kalman filter (EKF).
Inertial Navigation System (AINS) consists of an inertial navigation system (INS), Doppler velocity log, depth meter and intermittent DGPS fixes. The data acquired are fused by an extended Kalman filter. After preliminary tests, this navigation system will be installed on an Autonomous Underwater Vehicle (AUV) where it will
Inertial sensors used in the mechanization of Newton's laws of motion, hereafter called the inertial navigator or inertial navigation system (INS), are gyroscopes and accelerometers. The gyroscopes sense angular orientation or motions of the vessel. The accelerometers sense the vessel's linear accelerations, which are changes in linear
