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Technical ReportUCAM-CL-TR-696ISSN 1476-2986Number 696Computer LaboratoryAn introduction to inertial navigationOliver J. WoodmanAugust 200715 JJ Thomson AvenueCambridge CB3 0FDUnited Kingdomphone 44 1223 763500http://www.cl.cam.ac.uk/

c 2007 Oliver J. WoodmanTechnical reports published by the University of CambridgeComputer Laboratory are freely available via the Internet:http://www.cl.cam.ac.uk/techreports/ISSN 1476-2986

An introduction to inertial navigationOliver J. WoodmanAbstractUntil recently the weight and size of inertial sensors has prohibited their use in domains such ashuman motion capture. Recent improvements in the performance of small and lightweight micromachined electromechanical systems (MEMS) inertial sensors have made the application of inertialtechniques to such problems possible. This has resulted in an increased interest in the topic ofinertial navigation, however current introductions to the subject fail to sufficiently describe the errorcharacteristics of inertial systems.We introduce inertial navigation, focusing on strapdown systems based on MEMS devices. A combination of measurement and simulation is used to explore the error characteristics of such systems.For a simple inertial navigation system (INS) based on the Xsens Mtx inertial measurement unit(IMU), we show that the average error in position grows to over 150 m after 60 seconds of operation.The propagation of orientation errors caused by noise perturbing gyroscope signals is identified as thecritical cause of such drift. By simulation we examine the significance of individual noise processesperturbing the gyroscope signals, identifying white noise as the process which contributes most tothe overall drift of the system.Sensor fusion and domain specific constraints can be used to reduce drift in INSs. For an exampleINS we show that sensor fusion using magnetometers can reduce the average error in position obtainedby the system after 60 seconds from over 150 m to around 5 m. We conclude that whilst MEMSIMU technology is rapidly improving, it is not yet possible to build a MEMS based INS which givessub-meter position accuracy for more than one minute of operation.1IntroductionThis report provides an introduction to inertial navigation and the error characteristics of inertial navigation systems. Its purpose is to address the lack of a readable introduction into the subject which doesnot oversimplify or ignore the error properties of inertial navigation systems. The report also aims toexplain the meaning and significance of performance specification measurements such as ‘noise density’and ‘bias stability’ which are often stated by manufacturers.The reader should note that whilst this report aims to provide a broad introduction to the subjectof inertial navigation, the latter chapters focus mainly on strapdown type inertial navigation systemsusing micro-machined electromechanical systems (MEMS) devices. MEMS technology is of particularinterest at the current time since it offers rugged, low cost, small and lightweight inertial sensors relativeto the other available technologies. The performance of MEMS inertial devices is also improving rapidly.Throughout the report a simple inertial navigation system (INS) is developed based on an Xsens Mtxdevice. The report is structured as follows: Section 2 introduces the reader to inertial navigation, its uses, and the two main varieties of inertialnavigation system. Sections 3 and 4 describe gyroscopes and accelerometers in detail. Both sections contain an overviewof the different types of sensors available, as well a description of error sources. Section 5 introduces Allan Variance, a technique which can be used to detect and measure the noisecharacteristics of gyroscope and accelerometer signals.3

Section 6 describes strapdown inertial navigation in more detail and explains how errors in individual gyroscopes and accelerometers propagate through the navigation system as a whole. Theperformance of a simple INS is analysed in order to illustrate the relative importance of noiseperturbing the gyroscope and accelerometer signals. Section 7 describes how simulation can be used to analyse the relative importance of different noisesources. A simple simulator is constructed and verified against the real system developed in Section6. Section 8 introduces several methods for reducing drift in inertial systems.4

Figure 1: The body and global frames of reference.2Inertial NavigationInertial navigation is a self-contained navigation technique in which measurements provided by accelerometers and gyroscopes are used to track the position and orientation of an object relative to a known startingpoint, orientation and velocity. Inertial measurement units (IMUs) typically contain three orthogonalrate-gyroscopes and three orthogonal accelerometers, measuring angular velocity and linear accelerationrespectively. By processing signals from these devices it is possible to track the position and orientationof a device, as described in Section 2.1.Inertial navigation is used in a wide range of applications including the navigation of aircraft, tacticaland strategic missiles, spacecraft, submarines and ships. Recent advances in the construction of MEMSdevices have made it possible to manufacture small and light inertial navigation systems. These advanceshave widened the range of possible applications to include areas such as human and animal motioncapture.2.1Inertial System ConfigurationsNearly all IMUs fall into one of the two categories outlined below. The difference between the twocatagories is the frame of reference in which the rate-gyroscopes and accelerometers operate. Throughoutthis report we will refer to the navigation system’s frame of reference as the body frame and to the frameof reference in which we are navigating as the global frame, as shown in Figure 1.2.1.1Stable Platform SystemsIn stable platform type systems the inertial sensors are mounted on a platform which is isolated fromany external rotational motion. In other words the platform is held in alignment with the global frame.This is achieved by mounting the platform using gimbals (frames) which allow the platform freedom inall three axes, as shown in Figure 2. The platform mounted gyroscopes detect any platform rotations.These signals are fed back to torque motors which rotate the gimbals in order to cancel out such rotations,hence keeping the platform aligned with the global frame.To track the orientation of the device the angles between adjacent gimbals can be read using anglepick-offs. To calculate the position of the device the signals from the platform mounted accelerometersare double integrated. Note that it is necessary to subtract acceleration due to gravity from the verticalchannel before performing the integration. The stable platform inertial navigation algorithm is shown inFigure 3.5

Figure 2: A stable platform IMU.Figure 3: Stable platform inertial navigation algorithm.Figure 4: Strapdown inertial navigation algorithm.6

2.1.2Strapdown SystemsIn strapdown systems the inertial sensors are mounted rigidly onto the device, and therefore outputquantities measured in the body frame rather than the global frame. To keep track of orientation thesignals from the rate gyroscopes are ‘integrated’, as described in Section 6. To track position the threeaccelerometer signals are resolved into global coordinates using the known orientation, as determined bythe integration of the gyro signals. The global acceleration signals are then integrated as in the stableplatform algorithm. This procedure is shown in Figure 4.Stable platform and strapdown systems are both based on the same underlying principles. Strapdownsystems have reduced mechanical complexity and tend to be physically smaller than stable platformsystems. These benefits are achieved at the cost of increased computational complexity. As the cost ofcomputation has decreased strapdown systems have become the dominant type of INS.7

Figure 5: A conventional mechanical gyroscope (source: [1]).33.1GyroscopesTypes of GyroscopeIn this section the main types of gyroscope are presented. Note that this is far from an exhaustive list.In particular there are many different varieties of mechanical gyroscope which are not described. A morecomprehensive survey can be found in [1].3.1.1MechanicalA conventional gyroscope consists of a spinning wheel mounted on two gimbals which allow it to rotatein all three axes, as show in Figure 5. An effect of the conservation of angular momentum is that thespinning wheel will resist changes in orientation. Hence when a mechanical gyroscope is subjected to arotation the wheel will remain at a constant global orientation and the angles between adjacent gimbalswill change. To measure the orientation of the device the angles between adjacent gimbals can be readusing angle pick-offs. Note that a conventional gyroscope measures orientation. In contrast nearly allmodern gyroscopes (including the optical and MEMS types outlined in Sections 3.1.2 and 3.1.3) arerate-gyros, which measure angular velocity.The main disadvantage of mechanical gyroscopes is that they contain moving parts. Moving partscause friction, which in turn causes the output to drift over time. To minimise friction high-precisionbearings and special lubricants are used, adding to the cost of the device. Mechanical gyroscopes alsorequire a few minutes to warm up, which is not ideal in many situations.3.1.2OpticalA fibre optic gyroscope (FOG) uses the interference of light to measure angular velocity. A FOG consistsof a large coil of optical fibre. To measure rotation two light beams are fired into the coil in oppositedirections. If the sensor is undergoing a rotation then the beam travelling in the direction of rotationwill experience a longer path to the other end of the fibre than the beam travelling against the rotation,as illustrated in Figure 6. This is known as the Sagnac effect. When the beams exit the fibre they arecombined. The phase shift introduced due to the Sagnac effect causes the beams to interfere, resultingin a combined beam whose intensity depends on the angular velocity. It is therefore possible to measurethe angular velocity by measuring the intensity of the combined beam.Ring laser gyroscopes (RLGs) are also based on the Sagnac effect. The difference between a FOG andRLG is that in a RLG laser beams are directed around a closed path using mirrors rather than optical8

Figure 6: The Sagnac effect. The dashed line is the path taken by the beam travelling in the directionof rotation. The solid line is the beam travelling against the rotation. θ is the angle through which thegyro turns whilst the beams are in flight.fibre.Unlike mechanical gyroscopes, optical gyros contain no moving parts and require only a few secondsto start-up. The accuracy of an optical gyro is largely dependent on the length of the light transmissionpath (larger is better), which is constrained by the size of the device.3.1.3MEMS GyroscopesDespite years of development, mechanical and optical gyroscopes still have high part counts and a requirement for parts with high-precision tolerances and intricate assembly techniques. As a result they remainexpensive. In contrast MEMS sensors built using silicon micro-machining techniques have low part counts(a MEMS gyroscope can consist of as few as three parts) and are relatively cheap to manufacture.MEMS gyroscopes make use of the Coriolis effect, which states that in a frame of reference rotatingat angular velocity ω, a mass m moving with velocity v experiences a force:Fc 2m(ω v)(1)MEMS gyroscopes contain vibrating elements to measure the Coriolis effect. Many vibrating elementgeometries exist, such as vibrating wheel and tuning fork gyroscopes. The simplest geometry consists ofa single mass which is driven to vibrate along a drive axis, as shown in Figure 7. When the gyroscopeis rotated a secondary vibration is induced along the perpendicular sense axis due to the Coriolis force.The angular velocity can be calculated by measuring this secondary rotation.At present MEMS sensors cannot match the accuracy of optical devices, however they are expectedto do so in the future. Below is a list of the advantageous properties of MEMS sensors, taken from [1]. small size; low weight; rugged construction; low power consumption; short start-up time; inexpensive to produce (in high volume);9

Figure 7: A vibrating mass gyroscope (source: [2]).SizeWeightStart-Up TimePowerOperating Temperature RangeAngular Random WalkBias StabilityGG1320AN (Laser Gyro)88 mm 88 mm 45 mm454 g 4s15 Vdc, 1.6 watts nominal5 Vdc, 0.375 watts nominal 54 C to 85 C 0.0035 / h0.0035 /hGG5300 (MEMS 3xGyro)50 mm 50 mm 30 mm136 g 1s5 Vdc, 800 mA 45 C to 85 C0.2 / h 70 /hTable 1: Specifications for the Honeywell GG1320AN and GG5300 gyroscopes. high reliability; low maintenance; compatible with operations in hostile environments;As previously mentioned, the major disadvantage of MEMS gyroscopes is that they are currently far lessaccurate than optical devices. Table 1 illustrates the advantages and disadvantages of MEMS technologyby comparing the specifications of two gyroscopes manufactured by Honeywell1 . The GG1320AN is asingle axis digital laser gyro. The GG5300 is a three axis MEMS rate gyro. Note that the MEMSpackage not only contains three gyros as opposed to one, but also has a lower power consumption, ashorter start-up time, and is both smaller and lighter than the optical device. The main disavantage ofthe MEMS device is that it is far less accurate, as indicated by the bias stability and angular randomwalk measurements. These measurement types are explained in Section 3.2.3.2MEMS Gyro Error CharacteristicsIn this section we examine the errors which arise in MEMS gyros, and their effect on the integrated(orientation) signal.1 http://www.honeywell.com10

3.2.1Constant BiasThe bias of a rate gyro is the average output from the gyroscope when it is not undergoing any rotation(i.e: the offset of the output from the true value), in /h. A constant bias error of ǫ, when integrated,causes an angular error which grows linearly with time θ(t) ǫ · t.The constant bias error of a rate gyro can be estimated by taking a long term average of the gyro’soutput whilst it is not undergoing any rotation. Once the bias is known it is trivial to compensate for itby simply subtracting the bias from the output.3.2.2Thermo-Mechanical White Noise / Angle Random WalkThe output of a MEMS gyro will be perturbed by some thermo-mechanical noise which fluctuates at arate much greater than the sampling rate of the sensor. As a result the samples obtained from the sensorare perturbed by a white noise sequence, which is simply a sequence of zero-mean uncorrelated randomvariables. In this case each random variable is identically distributed and has a finite variance σ 2 .To see what effect this noise has on the integrated signal we can do a simple analysis in which it isassumed that the rectangular rule is used to perform the integration. Let Ni be the ith random variablein the white noise sequence. Each Ni is identically distributed with mean E(Ni ) E(N ) 0 and finitevariance Var(Ni ) Var(N ) σ 2 . By the definition of a white sequence Cov(Ni , Nj ) 0 for all i 6 j.The result of using the rectangular rule to integrate the white noise signal ǫ(t) over a timespan t n · δtisZ tnXNi(2)ǫ(τ ) dτ δt0i 1where n is the number of samples received from the device during the period and δt is the time betweensuccessive samples. Using the standard formulae E(aX bY ) aE(X) bE(y) and Var(aX bY ) a2 Var(X) b2 Var(Y ) 2abCov(X, Y ) (where a and b are constants and X and Y are random variables)it follows that Z t Eǫ(τ ) dτ δt · n · E(N ) 0(3)0 Z t Varǫ(τ ) dτ δt2 · n · Var(N ) δt · t · σ 2 .(4)0Hence the noise introduces a zero-mean random walk2 error into the integrated signal, whose standarddeviation σθ (t) σ · δt · t(5)grows proportionally to the square root of time.Since we are usually interested in how the noise affects the integrated signal it is common for manufacturers to specify noise using an angle random walk (ARW) measurementARW σθ (1)(6) with units / h. For example the Honeywell GG5300 has an ARW measurement of 0.2 / h. This means that after 1 hour the standard deviation of the orientation error will be 0.2 , after 2 hours it willbe 2 · 0.2 0.28 and so on. Other measurements used to specify noise are power spectral density(units ( /h)2 /Hz) and FFT noise density (units /h/ Hz). It is possible to convert between the variousdifferent noise specifications using the equations: ARW ( / h) ARW ( / h) 1 p· PSD (( /h)2 /Hz)60 1· FFT ( /h/ Hz)60(7)(8)For more information about angle random walk and noise specifications see [3].2 Here a random walk is defined as a process consisting of a series of steps, in which the direction and size of each stepis randomly determined.11

3.2.3Flicker Noise / Bias StabilityThe bias of a MEMS gyroscope wanders over time due to flicker noise in the electronics and in othercomponents susceptible to random flickering. Flicker noise is noise with a 1/f spectrum, the effects ofwhich are usually observed at low frequencies in electronic components. At high frequencies flicker noisetends to be overshadowed by white noise. Bias fluctuations which arise due to flicker noise are usuallymodelled as a random walk.A bias stability measurement describes how the bias of a device may change over a specified periodof time, typically around 100 seconds, in fixed conditions (usually including constant temperature). Biasstability is usually specified as a 1σ value with units /h, or /s for less accurate devices. Under therandom walk model bias stability can be interpreted as follows; If Bt is the known bias at time t, then a1σ bias stability of 0.01 /h over 100 seconds means that the bias at time (t 100) seconds is a randomvariable with expected value Bt and standard deviation 0.01 /h. Over time this property creates arandom walk in the gyro bias, whose standard deviation grows proportionally to the square root of time.For this reason bias stability is occasionally specified by a bias random walk measurement BS ( /h)BRW ( / h) pt (h)(9)where t is the timespan for which the bias stability is defined.As usual we are interested in how this error affects the orientation obtained from integrating the rategyro signal. If we assume the bias random walk model, then the result of integrating the bias fluctuationsis a second-order3 random walk in angle. In reality bias fluctuations do not really behave as a randomwalk. If they did then the uncertainty in the bias of a device would grow without bound as the timespanincreased. In practice the bias is constrained to be within some range, and therefore the random walkmodel is only a good approximation to the true process for short periods of time.3.2.4Temperature EffectsTemperature fluctuations due to changes in the environment and sensor self heating induce movementin the bias. Note that such movements are not included in bias stability measurements which are takenunder fixed conditions.Any residual bias introduced due to a change in temperature will cause an error in orientation whichgrows linearly with time, as described in Section 3.2.1. The relationship between bias and temperatureis often highly nonlinear for MEMs sensors. Most inertial measurement units (IMUs) contain internaltemperature sensors which make it possible to correct for temperature induced bias effects. Some IMUssuch as the Xsens4 Mtx perform such corrections internally.3.2.5Calibration ErrorsThe term ‘calibration errors’ refers collectively to errors in the scale factors, alignments, and linearitiesof the gyros. Such errors tend to produce bias errors that are only observed whilst the dev

2.1.1 Stable Platform Systems In stable platform type systems the inertial sensors are mounted on a platform which is isolated from any external rotational motion. In other words the platform is held in alignment with the global frame. This is achieved by mounting the platform using gimbals (frames) which allow the platform freedom in

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