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<ul><li><p>Effective Use of Magnetometer Feedback forSmart Projectile Applications</p><p>JONATHAN ROGERS and MARK COSTELLOGeorgia Institute of Technology, Atlanta, GA, 30332</p><p>THOMAS HARKINSUS Army Research Laboratory, Aberdeen, MD, 21001</p><p>MOSHE HAMAOUIData Matrix Solutions, Inc., Aberdeen, MD, 21001</p><p>Received September 2010; Revised March 2011</p><p>ABSTRACT: The use of magnetometers for orientation estimation on rapidly-spinning flight bodies is analyzed.Specifically, the effect of spin-induced magnetic field distortion is discussed, with particular attention to its impli-cations for magnetometer-based orientation estimation. First, the nature of spin-induced field distortion isdescribed and it is shown that, if not properly accounted for, distortion can lead to significant estimation errorsin artillery projectiles. Then, an orientation estimator is constructed driven by magnetometer, gyroscope, andGPS feedback. A novel feature of this algorithm is its compensation for spin-induced distortion of the Earthsfield. The algorithm also incorporates in-flight magnetometer calibration performed simultaneously with projec-tile orientation estimation. The comprehensive algorithm is built as a coupled set of Extended Kalman filters.Observability of the estimation problem is discussed and unobservable modes are identified. Finally, exampleresults and Monte Carlo simulations compare estimation performance to algorithms which neglect spin-induceddistortion effects or do not perform in-flight calibration. These results demonstrate that magnetometer-based sys-tems on-board spinning projectiles should incorporate corrections for field distortion, and that overall accuracy isgreatly enhanced by performing in-flight calibration.</p><p>INTRODUCTION</p><p>The availability of low-cost, lightweight sensorsand digital microprocessors has enabled weaponsdesigners to equip artillery projectiles with fullguidance and control capability. However, in com-parison to electronics packages on-board missiles,projectile sensor suites and guidance units must beable to withstand large acceleration loads at launchand high spin rates while maintaining low cost.One challenge routinely faced by designers hasbeen the development of attitude estimators forthese vehicles. While accurate attitude informationis critical to control system performance, it isoften impossible to obtain by integrating outputsfrom low-cost rate sensors. One common solutionis to use magnetometer measurements to provideperiodic updates to integrated rate sensor data.Magnetometers are attractive due to their lowpower requirements, rugged construction, and lowcost.</p><p>The use of magnetometers to obtain orientationinformation is not a new idea and has been used formany types of flight vehicles, from satellites toUAVs. Typically, an algorithm is employed to deter-mine a solution for the direction cosines matrix(DCM), Euler angles, or quaternions of the vehiclebased on a set of magnetometer measurements.Wahba [1, 2] initially proposed a batch least squarestechnique for determining the DCM of a satellitebased on vector measurements. Other researchersproposed purely deterministic methods [3] and re-cursive algorithms [4] for computing the DCM. Bar-Itzhak and Oshman [5] extended this work by devel-oping a method for recursively estimating quatern-ions from a set of vector measurements. Similaralgorithms [6] were developed for Euler angle deter-mination. Most recently, Mortari [7] has shown thatwhen using more than two vector measurements,the optimal Euler axis and the principle Euler anglecan be obtained separately without iterative proce-dures. Psiaki [8] developed a magnetometer-onlyattitude and angular rate estimator for low-costspacecraft and showed reasonable performanceusing flight data.</p><p>NAVIGATION: Journal of The Institute of NavigationVol. 58, No. 3, Fall 2011Printed in the U.S.A.</p><p>203</p></li><li><p>While much of the above research has beendirected toward spacecraft applications, several in-vestigators in the projectile community have re-cently developed algorithms specifically tailored tosmart projectiles. Ohlmeyer, Fraysse, and Pepitone[9] incorporated magnetometers in a low-cost INSalong with accelerometers and GPS. Magnetometerbias was estimated in-flight to improve perform-ance. Wilson [10] also proposed the use of magneto-meters as the primary attitude sensor on-boardsmart projectiles. He showed that complete attitudesolutions could be obtained only by combining mag-netometers with additional sensors such as acceler-ometers or solar sensors. Most recently, Lee et al.[11] developed a roll attitude estimator for smartmunitions using magnetometers based on anunscented Kalman filter.</p><p>Two recurring problems have consistently hin-dered efforts to implement magnetometer-basedestimators on-board guided artillery shells. The firstis that distortion and attenuation of the Earths mag-netic field can be significant inside the body of a spin-ning projectile due to the formation of eddy currentswithin the conducting metallic body. Harkins [12]has explored this phenomenon experimentally, con-cluding that these effects can lead to significanterror in magnetometer-based estimators if propercompensation is not used. The second, and perhapsless application-specific, is that bias, scale factor,and misalignment errors can often have significantdetrimental impact on overall attitude estimationperformance. Calibration values, especially bias, canchange during launch or in flight. Several techni-ques have previously been developed to mitigate thisproblem by performing on-line estimation of thesenuisance parameters (autocalibration). Lerner andShuster [13] first developed a method to estimatemagnetometer nuisance parameters on-board space-craft given a priori knowledge of attitude. Alonsoand Shuster [1416] have proposed so-calledattitude independent autocalibration methods forspacecraft that rely on changes in the Earths mag-netic field magnitude over one orbit cycle. Crassidiset al. [17] expanded on this work, developing threealgorithms to perform real-time magnetometer cali-bration based on observed differences in field magni-tude. Most recently, Gebre-Egziabher [18] developedan autocalibration algorithm for UAVs by fitting anellipsoid to measured magnetic field data. Magne-tometer data used to define this ellipsoid is gener-ated by rotating the vehicle through prescribedturns during the calibration process.</p><p>Previously-developed autocalibration algorithmsare for the most part unsuitable for projectile appli-cations for several reasons. First, the Earths mag-netic field does not change enough throughoutflight to employ attitude-independent solutions.Second, prescribed calibration maneuvers such as</p><p>those outlined in [18] are typically not an optionduring projectile flight. Standard pre-flight sensorcalibration procedures using reference sensors aretoo expensive given the low-cost nature of gun-launched munitions, and cannot account for cali-bration changes after launch due to large shocks.Although it may be possible to develop sensors thatmaintain calibration through launch and do notrequire extensive calibration, cost considerationsfor projectile applications make this solution lessattractive. Thus, new techniques are required toperform in-flight estimation of as many nuisanceparameters as possible.</p><p>The contributions of this article are twofold.First, an in-depth analysis of the problem of mag-netic field distortion inside projectile bodies is pre-sented. An analytical model for this field distortionis built and compared to experimental results.A new magnetometer model is built which incorpo-rates this distortion, and is used to generate anExtended Kalman filter that estimates field distor-tion parameters. The second main contribution isthe development of a coupled set of Extended Kal-man filters that estimate projectile orientation, nui-sance parameters, and distortion effects simultane-ously. Using this filter, a direct analysis of the bene-fits of in-flight nuisance parameter and fielddistortion estimation is performed. This is accom-plished through example and Monte Carlo simula-tions in which performance of estimators with andwithout the capability to perform in-flight nuisanceand distortion parameter identification is directlycompared. Results show a significant benefit to in-corporating distortion effects and performing nui-sance parameter estimation in flight. The articlebegins by describing magnetic field distortion withinrapidly-spinning projectile bodies. Then, algorithmsare developed to estimate distortion effects, magne-tometer nuisance parameters, and projectile orien-tation simultaneously. An observability analysis ofthe entire estimation problem is then performed andunobservable parameters are identified. Finally,simulation results show that attitude estimationperformance is almost always improved when dis-tortion effects are incorporated and nuisance param-eter estimation is performed.</p><p>SPIN-INDUCED MAGNETIC FIELD DISTORTION</p><p>When a conductive body is subjected to a chang-ing magnetic field, eddy current effects occur insidethe body. Equivalently, a spinning cylinder im7-mersed in a transverse uniform magnetic field isessentially subject to two radial oscillating fieldswhich are 90 degrees apart in space and time. ByFaradays law, these oscillating magnetic fields willinduce electric fields according to</p><p>204 Navigation Fall 2011</p></li><li><p>r 3 ~E 2 @~B</p><p>@t(1)</p><p>If the cylinder is conductive, eddy currents willbe induced to flow along the length of the cylinder.These currents in turn give rise to a secondarymagnetic field which, in superposition with theexternal field, creates a distorted magnetic field inand around the cylinder [19]. Because on-boardelectronics packages used for smart weapon guid-ance and control are located within a rapidly spin-ning projectile body, magnetometer-based orienta-tion estimators must compensate for this field dis-tortion to avoid significant error.</p><p>Modeling Spin-Induced Distortion:Infinite Cylinder</p><p>Maxwells equations lead to a diffusion equationfor the magnetic vector potential describing thetime evolution of the magnetic field, given by</p><p>r2~A lr @~A</p><p>@t(2)</p><p>where ~A is the magnetic vector potential, l is themagnetic permeability, and r is the electrical con-ductivity. Note that in the above equation the sim-plifying assumption that the displacement currentin Maxwells equations is negligible is made, sinceprojectile spin rates correspond to wavelengths thatare orders of magnitude larger than typical projec-tile body dimensions. Recently, Ziolkowski and Gra-bowski [20] imposed appropriate boundary condi-tions on Equation (2) to obtain an analytic solutionfor the case of an infinite, hollow cylinder spinningin a uniform, transverse magnetic field. Figure 1shows magnetic field lines for an infinite, non-magnetic, conducting cylinder of inner radius 50mm and outer radius 75 mm rotating counterclock-wise at a rate of 20 rad/s placed in a constant trans-verse external field (computed using the expressionsfrom reference [20]). Note that, for an infinitely-longconducting cylinder, the field inside the body is uni-form (i.e., constant distortion angle throughout) andattenuated with respect to the external transversefield. Any axial component of the external field isunaffected for cylinders of infinite length.</p><p>Modeling Spin-Induced Distortion: Finite Cylinder</p><p>For cylinders of finite length, no closed-form solu-tion for the magnetic field is possible. Thus, finiteelement methods must be employed to model mag-netic field distortion inside rotating bodies of finitelength. To analyze distortion effects for finitebodies, a finite element (FE) model was constructedusing ANSYS software. First, for validation pur-poses, the FE model was used to generate the spin-</p><p>distorted field for an infinite cylinder, and resultswere compared to those generated with the closed-form solution. The magnetic fields generated bythese two models matched to within 1%. Then, theFE model considered a nonmagnetic hollow cylindercomposed of 6061 Al, this time with inner and outerradii of approximately 50 mm and 57 mm respec-tively and length of 175 mm. This finite-length cyl-inder was immersed in a uniform transverse mag-netic field of 1 Gs and spun at selected frequencies.Figure 2 shows example magnetic field results gen-erated with this FE model for this finite cylinderspinning at a rate of 80 Hz.</p><p>Fig. 1Magnetic field lines for infinite cylinder rotating counter-clockwise in an external magnetic field</p><p>Fig. 2Magnetic field vectors near a cylinder spinning at 80 Hz,generated by FEA. The cylinder spins about~IP</p><p>Vol. 58, No. 3 Rogers et al.: Effective Use of Magnetometer Feedback for Smart Projectile Applications 205</p></li><li><p>Figure 2 highlights some key differences betweendistortion effects from infinite- and finite-lengthcylinders. First, in contrast to the infinite-lengthcase, the field inside finite-length cylinders is notuniform. Second, in the finite-length case, the dis-torted field inside the cylinder has a noticeableaxial component. This axial component is zeroalong the spin axis and in the transverse plane ofthe centroid, but grows considerably near the edgesof the cylinder.</p><p>Experiments were conducted to validate resultsfrom the FE model. A Helmholtz coil was used togenerate a near-uniform rotating magnetic fieldnear a cylinder composed of 6061 Al with dimen-sions approximately equal to those described above.The experiment was performed at selected frequen-cies between 0 and 250 Hz, and the magnetic fieldwas measured approximately at the centroid of thecylinder.</p><p>In order to quantitatively compare results be-tween the analytical model, FE model, and experi-ment, three variables are defined that describespin-induced distortion: the attenuation factor (AF),the transverse distortion angle, cD, and the inducedaxial component, fD. Attenuation factor (AF) is avalue representing the attenuation of the compo-nents of the Earths field transverse to the projec-tile spin. Distortion angle (cD) represents thechange in direction of the transverse components ofthe field (as shown in Figure 1). The induced axialcomponent (fD) is a value between 0 and 1 thatintroduces distortion in the axial direction as a per-centage of the overall transverse field strength. Letthe components of the Earths magnetic field vectorinside the projectile body expressed in the body ref-erence frame be given by ~mxD; ~myD; ~mzD. Therefore,</p><p>~mxD~myD~mzD</p><p>8=&gt;; (3)</p><p>where ~mx; ~my; ~mz are components of the Earthsmagnetic field expressed in the body referenceframe. Note that in Equation (3), and in the re-mainder of this article, ca denotes cos(a), sa denotessin(a), and ta denotes tan(a). Given both the nomi-nal and distorted sets of magnetic field components,it is also possible to solve Equation (3) for AF, cD,and fD such that</p><p>AF </p><p>ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~m2y ~m2z</p><p>q</p><p>ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~m2yD m2zD</p><p>qffiffiffiffiffiffiffi...</p></li></ul>