1998 - Draper Laboratory

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Several targeting strategies may be employed to define the target.The first and most primitive strategy sets the target value to thenominal value at the next iteration time. This is equivalent tobringing the difference between the actual and nominal state inthat element to zero at the next iteration time. This strategy canalso be described as "shooting" for the nominal. A second, moresophisticated strategy may be described as "overshooting" thenominal. In this case, the target value is set to the valueapproximately opposite its current value. (The amount secondarytargeted elements are corrected depends on how far from theirnominal value they are when they were chosen for targeting.)In actual applications, it is desirable to take advantage of theconstellation dynamics. For the 8:1 Ellipso Borealis TMconstellation, enhancements to the above strategies will beemployed. With other elements controlled within reasonablebounds, it is noted that the node error is driven by the inclinationrate error (approximately constant), and behaves as a quadraticfunction of time. Similarly, the mean anomaly error growthbehaves as a quadratic function of time, depending on thesemimajor axis rate error (again approximately constant). ASKScontrols the 8:1 Borealis TM constellation using these relationshipsby targeting the semimajor axis to control the mean anomaly andby targeting the inclination to control the ascending node. Theeccentricity will be targeted via the "overshoot" method, and theargument of perigee will be targeted to the next predicted value(i.e., not controlled). Node control via inclination is (at leastpartially) reasonable in this situation because the inclinationdiverges from reference at approximately a constant rate due tothe deep 8:1 resonance. (This is not true for the 81:10 case,which is in shallow resonance and has a much more stableinclination.)Impulsive Maneuver Generation FunctionThe impulsive maneuver generator applies Primer Vector Theoryto calculate a near-optimal n-impulse transfer trajectory to thefuture target state, given the initial and final state over the definedtransfer time. The theoretical basis for primer vector theory wasfirst stated by Lawden in the form of necessary conditions, whichif met, ensure that an impulsive trajectory is optimal (Ref. [19]).In 1968, Lion and Handelsman developed a method by whichone can determine if the addition of an interior impulse to a givenimpulsive trajectory will decrease the total fuel expenditure (Ref.[20]). Soon after, Jezewski and Rozendaal (Ref. [21]) developedan efficient method of calculating optimal two-body n-impulsetrajectories by applying the Lion and Handelsman work in theframework of a computational search scheme. Further work onthe subject was accomplished by Prussing (Ref. [22]) and a seriesof his students. They applied Primer Vector Theory for orbitaltransfers involving path constraints (Ref. [23]), circular orbits(Ref. [24]), cooperative transfers (Ref. [25]), and direct ascentrendezvous (Ref. [26]). Carter and Brient further improved onPrimer Vector Theory by applying it to noncircular Keplerianorbits (Ref. [27]).The main flow of this process is displayed in Figure 5 and fallsinto the "Primer Vector" box of Figure 4. First a two-burn"Lambert" solution is generated. The Primer Vector history isthen examined. If the maximum magnitude of the Primer Vectoris greater than 1, one or more interior impulses are generated. Wehave adopted a direction-set optimization algorithm due toPowell (Ref. [28]) to refine the burn times to produce nearoptimalimpulses. This algorithm finds the minimum of a multidimensionalfunction without using derivatives. It operates byminimizing the function along a series of multidimensionaldirections until global minimization occurs. The function valuein this application is the characteristic velocity or total ∆V. Theindependent variables are the intermediate burn times.The maximum number of impulses allowed in each transfer,n max , and the discretization size of the time grid between theinitial and final state times are both defined by the user in theconstellation input file. The minimum allowable value for n max is2. Increasing n max will result in a lower total ∆V expenditure.Near-optimaln-impulse burnlistInitial state Transfer time Final stateNoCalculate terminal positionand velocity vectorsEstablish state and primer vectortrajectories between initialand final stateDetermine 2-burn burnlistFind maximum value of primervector magnitudeDetermine estimate of timewhere addition of an intermediateburn results in lowest aggregate ∆VIs |λ| max >1 andand n
However, since optimization and recalculation of the PrimerVector and state trajectories are computationally expensive, theminimum value of n max that yields acceptable burn magnitudesshould be used. In practice, increasing n max past 4 or 5 results inonly marginal improvement in the characteristic velocity. Notethat Figure 5 identifies n < n max and |λ| max > 1 as the onlyconditions for exiting the control loop. In actuality, if theimprovement in the cost with the addition of the (n+1) st burn isnot greater than a specific tolerance value, then the extra burn isnot included and the loop is terminated. The output of thetrajectory planner algorithm is an n-element burnlist containingthe impulsive velocity vectors to apply and the times to applythem.Orbit Propagation FunctionThe DSST orbit propagator is employed by ASKS to propagateboth the reference and actual satellite trajectories in theconstellation. It is currently being employed in the missionsupport system for the Canadian Space Agency Radarsat system(Ref. [29]). The DSST propagator models realistic satellitedynamics, including complex perturbations, and propagatesthrough impulsive maneuvers.The DSST propagator includes the following perturbations in themean element dynamics (Ref. [30]):• Central body gravitational spherical harmonics ofarbitrary degree and order.• J 22 second-order effect. An explicit analyticalexpression, truncated to the first power of the satelliteeccentricity, is used for the mean element rates ofchange.• Third-body point mass effects.• Atmospheric drag on a spherical satellite. Both Jacchia-Roberts and Harris-Priester atmospheric density modelsare available.• Solar radiation pressure on a spherical satellite witheclipsing.The following perturbations are included in the short periodicvariations (Ref. [30]):• Central body gravitational zonal harmonics of arbitrarydegree.• Central body gravitational m-daily sectoral and tesseralharmonics of arbitrary degree and order.• Lunar/solar perturbations to the short periodicvariations have been added in a new version of theDSST stand-alone (Ref. [31]).All the perturbation models listed above can be turned on or offby the user. The user must also define the epoch time, epochstate, output request times, and physical properties of thespacecraft. At each request time, the orbit generator calculates thefollowing values:• Mean equinoctial elements.• Osculating equinoctial elements.• Position and velocity vectors.• Mean equinoctial element rates.• J 2 mean element state transition matrix.Station-Keeping Analysis:The Case of EllipsoBorealis TMThe Ellipso Borealis TM constellation will occupy a dynamicallycomplicated environment. Eight Borealis TM orbit configurationshave been analyzed by Sabol et al. (Ref. [11]), who point out that“when the Borealis TM orbit is put into operational use, thecoupling effect between the perturbations at critical inclinationwill require careful monitoring in order to maintain constellationintegrity.” These perturbations will include both zonal andtesseral geopotential harmonics, solar radiation pressure,third-body point mass, and atmospheric drag.Before embarking on an examination the ASKS station-keepingalgorithms, consider the coverage obtained without stationkeepingby the baseline Ellipso Borealis TM /Concordiaconstellation, (employing the 8:1 Node at Noon/Node atMidnight Borealis TM planes). Although coverage is not the onlymetric by which to measure constellation performance, lack ofproper coverage will compromise system performance. As seen inFigures 6 and 7, both one-way and two-way coverage begins todegrade badly in the Northern Hemisphere after the first 200 daysof the mission. Clearly, in this case, station-keeping is needed, ifonly to maintain coverage.% Coverage10110099989796959493One-way Coverage-Ellipso 8:1Northern Hemisphere 0-90N degSouthern Hemisphere 0-55S deg920 100 200 300 400 500 600 700Days Since Epoch (Epoch:1 Jan 97)Figure 6. Ellipso TM one-way coverage (uncontrolled Borealis TM 8:1node at noon/midnight and Concordia).800Automated Station-Keeping for Satellite Constellations8
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Letter from thePresident and CEO,Vi
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Information TechnologyMilton AdamsE
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BiographyMilton Adams has been at D
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Figure 1 represents a functional de
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Programs. In effect, these controll
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Although the terminal area traffic
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Table 2. ATFM performance evaluatio
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In the experiments, a nominal capac
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[3] Wambsganss, Michael C. “Colla
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Guidance, Navigation, and Control A
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A Control Lyapunov FunctionApproach
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x( 0) ∈ X and w(t) ∈Wfor all t
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(b) Select a quadratic RCLF V i (x)
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at each grid point. In the case w 1
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References[1] Ball, J.A. and A.J. v
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Page 35 and 36: Relative and Differential GPSData T
Page 37 and 38: The first term on the right in the
Page 39 and 40: H R# δρ R,GPS -H A# δρ A,GPSThi
Page 41 and 42: selection; and (3) shown that the a
Page 43 and 44: Guidance, Navigation, and Control A
Page 45 and 46: Segmentation of MR ImagesUsing Curv
Page 47 and 48: (3)where ν now represents a contin
Page 49 and 50: Experimental ResultsThe results of
Page 51 and 52: Table 1. A summary of segmentation
Page 53 and 54: Guidance, Navigation,and ControlJim
Page 55 and 56: BiographyGeorge SchmidtGeorge Schmi
Page 57 and 58: clock and ephemeris errors, as well
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Page 61 and 62: Table 5. “Typical” absolute GPS
Page 63 and 64: performed, then the target location
Page 65 and 66: tightly-coupled system, however, ca
Page 67 and 68: Concluding RemarksRecent progress i
Page 69 and 70: As real-time systems evolve into th
Page 71 and 72: Advanced Fault-TolerantComputing fo
Page 73 and 74: The Viking and Voyager were both in
Page 75 and 76: Containment Regions (FCRs). There a
Page 77 and 78: well as reversing the whole process
Page 79 and 80: As real-time systems evolve into th
Page 81 and 82: Automated Station-Keepingfor Satell
Page 83 and 84: Figure 2. Minimum elevation angles
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Page 95 and 96: Draper’s primary goal is to Drape
Page 97 and 98: )Rotordynamic Modelingof an Activel
Page 99 and 100: Eq. (9) becomes:λ[ R ] { Φ } = [
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Page 105 and 106: BiographyJeffrey Borenstein is curr
Page 107 and 108: process step. Process information i
Page 109 and 110: Figure 4. Control chart for boron d
Page 111 and 112: References[1] Barbour, N., J. Conne
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Page 117 and 118: calculates miscellaneous terms, suc
Page 119 and 120: Table 1. Suggested specification sh
Page 121 and 122: User Accuracy as aFunction of Simul
Page 123 and 124: 20-min averaging, this clock lockin
Page 125 and 126: Table 2. Sample high-level summary
Page 127 and 128: AcknowledgmentR.L. Greenspan, J.A.
Page 129 and 130: Systems IntegrationRich MartoranaPe
Page 131 and 132: BiographyAnthony Kourepenis is an A
Page 133 and 134: control is employed to maintain the
Page 135 and 136: Table 1. Summary of automotive yaw
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Resolution (60 Hz) deg/h10000000100
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References[1] Greiff, P., B. Boxenh
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An Integrated Safety AnalysisMethod
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Infrastructure ModelsSystemRequirem
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Figures 6 and 7 illustrate the bloc
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Notice that each flight track descr
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Table 7. Safety statistics at 1700-
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An Optimal Guidance Law forPlanetar
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Note that the states in the three d
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Crossrange (Kft)10090807060504030Cl
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The 1997 Charles StarkDraper PrizeT
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The 1997 Charles StarkDraper Prize1
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“Draper encourages its personnel
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Gimballed Vibrating GyroscopeHaving
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Optical Source Isolator withPolariz
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Hunting Suppressor forPolyphase Ele
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Sensor Having an Off-Frequency Driv
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proof mass from transients and enha
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1997 Published PapersThe following
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monitoring of space structures and
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measured by kinematic degrees of fr
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i.e., what percent of the earth’s
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McConley, M. W.; Dahleh, M. A.; Fer
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unaffordable, or even misguided. Bu
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The Draper DistinguishedPerformance
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Educational Activitiesat Draper Lab
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1998 - Draper Laboratory

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