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NASA NTRS2001

Controlling Malaria and Other Diseases Using Remote Sensing

Remote sensing offers the vantage of monitoring a vast area of the Earth continuously. Once developed and launched, a satellite gives years of service in collecting data from the land, the oceans, and the atmosphere. Since the 1980s, attempts have been made to relate disease occurrence with remotely sensed environmental and geophysical parameters, using data from Landsat, SPOT, AVHRR, and other satellites. With higher spatial resolution, the recent satellite sensors provide a new outlook for disease control. At sub-meter to I 10m resolution, surface types associated with disease carriers can be identified more accurately. The Ikonos panchromatic sensor with I m resolution, and the Advanced Land Imager with 1 Om resolution on the newly launched Earth Observing-1, both have displayed remarkable mapping capabilities. In addition, an entire array of geophysical parameters can now be measured or inferred from various satellites. Airborne remote sensing, with less concerns on instrument weight, size, and power consumption, also offers a low-cost alternative for regional applications. NASA/GSFC began to collaborate with the Mahidol University on malaria and filariasis control using remote sensing in late 2000. The objectives are: (1) To map the breeding sites for the major vector species; (2) To identify the potential sites for larvicide and insecticide applications; (3) To explore the linkage of vector population and transmission intensity to environmental variables; (4) To monitor the impact of climate change and human activities on vector population and transmission; and (5) To develop a predictive model for disease distribution. Field studies are being conducted in several provinces in Thailand. Data analyses will soon begin. Malaria data in South Korea are being used as surrogates for developing classification techniques. GIS has been shown to be invaluable in making the voluminous remote sensing data more readily understandable. It will be used throughout this study to clearly demonstrate the spatial relationship between the disease intensities, geophysical variables, and socioeconomic parameters. Asides from malaria and filariasis, application of remote sensing to the control of other diseases have been vigorously pursued by NASA's Environment and Health Initiative. The current program includes projects on Rift Valley fever, St. Louis encephalitis, dengue fever, ebola, African dust and diseases, meningitis, asthma, bartonellosis, cholera, and urban health concerns. Results from these projects indicate that remote sensing will play an increasingly important role in disease control in the future.

Kiang, Richard K.↗

NASA NTRS2001

A Comprehensive C++ Controller for a Magnetically Supported Vertical Rotor

This manual describes the new FATMaCC (Five-Axis, Three-Magnetic-Bearing Control Code). The FATMaCC (pronounced "fat mak") is a versatile control code that possesses many desirable features that were not available in previous in-house controllers. The ultimate goal in designing this code was to achieve full rotor levitation and control at a loop time of 50 microsec. Using a 1-GHz processor, the code will control a five-axis system in either a decentralized or a more elegant centralized (modal control) mode at a loop time of 56 microsec. In addition, it will levitate and control (with only minor modification to the input/output wiring) a two-axis and/or a four-axis system. Stable rotor levitation and control of any of the systems mentioned above are accomplished through appropriate key presses to modify parameters, such as stiffness, damping, and bias. A signal generation block provides 11 excitation signals. An excitation signal is then superimposed on the radial bearing x- and y-control signals, thus producing a resultant force vector. By modulating the signals on the bearing x- and y-axes with a cosine and a sine function, respectively, a radial excitation force vector is made to rotate 360 deg. about the bearing geometric center. The rotation of the force vector is achieved manually by using key press or automatically by engaging the "one-per-revolution" feature. Rotor rigid body modes can be excited by using the excitation module. Depending on the polarities of the excitation signal in each radial bearing, the bounce or tilt mode will be excited.

Morrison, Carlos R.↗

NASA NTRS2001

Numerical Study of Buoyancy and Different Diffusion Effects on the Structure and Dynamics of Triple Flames

Numerical simulations of 2-D triple flames under gravity force have been implemented to identify the effects of gravity on triple flame structure and propagation properties and to understand the mechanisms of instabilities resulting from both heat release and buoyancy effects. A wide range of gravity conditions, heat release, and mixing widths for a scalar mixing layer are computed for downward-propagating (in the same direction with the gravity vector) and upward-propagating (in the opposite direction of the gravity vector) triple flames. Results of numerical simulations show that gravity strongly affects the triple flame speed through its contribution to the overall flow field. A simple analytical model for the triple flame speed, which accounts for both buoyancy and heat release, is developed. Comparisons of the proposed model with the numerical results for a wide range of gravity, heat release and mixing width conditions, yield very good agreement. The analysis shows that under neutral diffusion, downward propagation reduces the triple flame speed, while upward propagation enhances it. For the former condition, a critical Froude number may be evaluated, which corresponds to a vanishing triple flame speed. Downward-propagating triple flames at relatively strong gravity effects have exhibited instabilities. These instabilities are generated without any artificial forcing of the flow. Instead disturbances are initiated by minute round-off errors in the numerical simulations, and subsequently amplified by instabilities. A linear stability analysis on mean profiles of stable triple flame configurations have been performed to identify the most amplified frequency in spatially developed flows. The eigenfunction equations obtained from the linearized disturbance equations are solved using the shooting method. The linear stability analysis yields reasonably good agreements with the observed frequencies of the unstable triple flames. The frequencies and amplitudes of disturbances increase with the magnitude of the gravity vector. Moreover, disturbances appear to be most amplified just downstream of the premixed branches. The effects of mixing width and differential diffusion are investigated and their roles on the flame stability are studied.

Chen, Jyh-Yuan↗

NASA NTRS1994

Numerical Simulations of Self-Focused Pulses Using the Nonlinear Maxwell Equations

This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations. Abstract of a proposed paper for presentation at the meeting NONLINEAR OPTICS: Materials, Fundamentals, and Applications, Hyatt Regency Waikaloa, Waikaloa, Hawaii, July 24-29, 1994, Cosponsored by IEEE/Lasers and Electro-Optics Society and Optical Society of America

Goorjian, Peter M.↗

NASA NTRS2001

Large Aperture Scanning Lidar Based on Holographic Optical Elements

Lidar remote sensing instruments can make a significant contribution to satisfying many of the required measurements of atmospheric and surface parameters for future spaceborne platforms, including topographic altimeters, atmospheric profiles of, wind, humidity, temperature, trace molecules, aerosols, and clouds. It is highly desirable to have wide measurement swaths for rapid coverage rather than just the narrow ribbon of data that is obtained with a nadir only observation. For most applications global coverage is required, and for wind measurements scanning or pointing is required in order to retrieve the full 3-D wind vector from multiple line-of-sight Doppler measurements. Conventional lidar receivers make up a substantial portion of the instrument's size and weight. Wide angle scanning typically requires a large scanning mirror in front of the receiver telescope, or pointing the entire telescope and aft optics assembly, Either of these methods entails the use of large bearings, motors, gearing and their associated electronics. Spaceborne instruments also need reaction wheels to counter the torque applied to the spacecraft by these motions. NASA has developed simplified conical scanning telescopes using Holographic Optical Elements (HOEs) to reduce the size, mass, angular momentum, and cost of scanning lidar systems. NASA has developed two operating lidar systems based on 40 cm diameter HOEs. The first such system, named Prototype Holographic Atmospheric Scanner for Environmental Remote Sensing (PHASERS) was a joint development between NASA Goddard Space Flight Center (GSFC) and the University of Maryland College Park. PHASERS is based on a reflection HOE for use at the doubled Nd:YAG laser wavelength of 532 nm and has recently undergone a number of design changes in a collaborative effort between GSFC and Saint Anselm College in New Hampshire. The next step was to develop IR transmission HOEs for use with the Nd:YAG fundamental in the Holographic Airborne Rotating Lidar Instrument Experiment (HARLIE). The HOE spins like a compact disk in a large ring ball bearing. In an aircraft the HOE faces down, looking out through a window at an angle of 45 degrees off-nadir. The HOE diffracts 85% of the incident 532 nm light into a 160 micron spot at a focal length of 1 meter. HARLIE is a field deployable lidar measuring aerosol, cloud, and boundary layer backscatter for atmospheric research. It has flown several times and is also used from a ground-based trailer in an upward-looking mode. The HOE generates a 45 degree conical scan pattern by rotating at speeds up to 30 rpm. Like PHASERS, the HOE in HARLIE serves both as the laser collimating lens as well as the receiver telescope primary optic. The telescope is coupled to the receiver package via fiber optic. The transmitter is a diode pumped Nd:YAG laser operating at 1064 nm, delivering 1 mJ pulses at a 5 KHz rep-rate. The receiver has a 200 microradian field-of-view and a 0.5 nm optical bandpass. The photon counting data system utilizes a single Geiger-mode silicon avalanche photodiode detector, This new technology has also presented us with new data visualization challenges as well as new measurement techniques. The backscatter data obtained from a stationary (i.e. ground-based) scanning HOE lidar is on the surface of a cone, which when viewed over many consecutive scans can reveal atmospheric motions on this surface over time as the atmosphere advects over the site. In a moving platform such as an airplane or satellite, the data from consecutive scans cover different areas under the flight path, revealing atmospheric structure in 3-dimensions. An example of a visualization of HARLIE ground-based data is presented, showing aerosol backscatter on a 90 degree conical surface generated from one 360 degree scan of the lidar during the HOLO-1 field campaign on the afternoon of 10 March 1999. Higher backscatter levels are rendered as lighter signal against a dark background. Breaking Kelvin-Helmholtz waves are evident on the north side of the scan at an altitude of 10-11 km. Time series of successive scans made at regular intervals render unique views of atmospheric motions, from which vertical profiles of atmospheric wind vectors can be obtained using a unique data analysis approach. Wind vectors obtained from the lidar were compared with co-located radiosonde wind profiles during an intensive operating period in September-October 2000 at the Atmospheric Radiation Measurement Program's Southern Great Plains Central Facility.

Schwemmer, Geary K.↗

NASA NTRS1999

Coupling of Carbon Nanotubes to Metallic Contacts

The modeling of carbon nanotube-metal contacts is important from both basic and applied view points. For many applications, it is important to design contacts such that the transmission is dictated by intrinsic properties of the nanotube rather than by details of the contact. In this paper, we calculate the electron transmission probability from a nanotube to a free electron metal, which is side-contacted. If the metal-nanotube interface is sufficiently ordered, we find that k-vector conservation plays an important role in determining the coupling, with the physics depending on the area of contact, tube diameter, and chirality. The main results of this paper are: (1) conductance scales with contact length, a phenomena that has been observed in experiments and (2) in the case of uniform coupling between metal and nanotube, the threshold value of the metal Fermi wave vector (below which coupling is insignificant) depends on chirality. Disorder and small phase coherence length relax the need for k-vector conservation, thereby making the coupling stronger.

Anantram, M. P.↗

NASA NTRS1995

The Performance of the NAS HSPs in 1st Half of 1994

During the first six months of 1994, the NAS (National Airspace System) 16-CPU Y-MP C90 Von Neumann (VN) delivered an average throughput of 4.045 GFLOPS while the ACSF (Aeronautics Consolidated Supercomputer Facility) 8-CPU Y-MP C90 Eagle averaged 1.658 GFLOPS. The VN rate represents a machine efficiency of 26.3% whereas the Eagle rate corresponds to a machine efficiency of 21.6%. VN displayed a greater efficiency than Eagle primarily because the stronger workload demand for its CPU cycles allowed it to devote more time to user programs and less time to idle. An additional factor increasing VN efficiency was the ability of the UNICOS 8.0 Operating System to deliver a larger fraction of CPU time to user programs. Although measurements indicate increasing vector length for both workloads, insufficient vector lengths continue to hinder HSP (High Speed Processor) performance. To improve HSP performance, NAS should continue to encourage the HSP users to modify their codes to increase program vector length.

Bergeron, Robert J.↗

NASA NTRS2002

The Flare Genesis Experiment

Using the Flare Genesis Experiment (FGE), a balloon-borne observatory with an 80-cm solar telescope we observed the active region NOAA 8844 on January 25, 2000 for several hours. FGE was equipped with a vector polarimeter and a tunable Fabry-Perot narrow-band filter. It recorded time series of filtergrams, vector magnetograms, and Dopplergrams at the Ca(I) 6122.2 angstrom line, and H-alpha filtergrams with a cadence between 2.5 and 7.5 minutes. At the time of the observations, NOAA 8844 was located at approximately 5 N 30 W. The region was rapidly growing during the observations; new magnetic flux was constantly emerging in three supergranules near its center. We describe in detail how the FGE data were analyzed and report on the structure and behavior of peculiar moving dipolar features (MDFs) observed in the active region. In longitudinal magnetograms, the MDFs appeared to be small dipoles in the emerging fields. The east-west orientation of their polarities was opposite that of the sunspots. The dipoles were oriented parallel to their direction of motion, which was in most cases towards the sunspots. Previously, dipolar moving magnetic features have only been observed flowing out from sunspots. Vector magnetograms show that the magnetic field of each MDF negative part was less inclined to the local horizontal than the ones of the positive part. We identify the MDFs as undulations, or stitches, where the emerging flux ropes are still tied to the photosphere. We present a U-loop model that can account for their unusual structure and behavior, and it shows how emerging flux can shed its entrained mass.

Rust, D. M.↗

NASA NTRS1998

Achieving High Performance on the i860 Microprocessor

The i860 is a high performance microprocessor used in the Intel Touchstone project. This paper proposes a paradigm for programming the i860 that is modelled on the vector instructions of the Cray computers. Fortran callable assembler subroutines were written that mimic the concurrent vector instructions of the Cray. Cache takes the place of vector registers. Using this paradigm we have achieved twice the performance of compiled code on a traditional solve.

Lee, King↗

NASA NTRS2002

GPS-Based Navigation and Orbit Determination for the AMSAT Phase 3D Satellite

This paper summarizes the results of processing GPS data from the AMSAT Phase 3D (AP3) satellite for real-time navigation and post-processed orbit determination experiments. AP3 was launched into a geostationary transfer orbit (GTO) on November 16, 2000 from Kourou, French Guiana, and then was maneuvered into its HEO over the next several months. It carries two Trimble TANS Vector GPS receivers for signal reception at apogee and at perigee. Its spin stabilization mode currently makes it favorable to track GPS satellites from the backside of the constellation while at perigee, and to track GPS satellites from below while at perigee. To date, the experiment has demonstrated that it is feasible to use GPS for navigation and orbit determination in HEO, which will be of great benefit to planned and proposed missions that will utilize such orbits for science observations. It has also shown that there are many important operational considerations to take into account. For example, GPS signals can be tracked above the constellation at altitudes as high as 58000 km, but sufficient amplification of those weak signals is needed. Moreover, GPS receivers can track up to 4 GPS satellites at perigee while moving as fast as 9.8 km/sec, but unless the receiver can maintain lock on the signals long enough, point solutions will be difficult to generate. The spin stabilization of AP3, for example, appears to cause signal levels to fluctuate as other antennas on the satellite block the signals. As a result, its TANS Vectors have been unable to lock on to the GPS signals long enough to down load the broadcast ephemeris and then generate position and velocity solutions. AP3 is currently in its eclipse season, and thus most of the spacecraft subsystems have been powered off. In Spring 2002, they will again be powered up and AP3 will be placed into a three-axis stabilization mode. This will significantly enhance the likelihood that point solutions can be generated, and perhaps more important, that the receiver clock can be synchronized to GPS time. This is extremely important for real-time and post-processed orbit determination, where removal of receiver clock bias from the data time tags is needed, for time-tagging of science observations. Current analysis suggests that the inability to generate point solutions has allowed the TANS Vector clock bias to drift freely, being perhaps as large as 5-7 seconds by October, 2001, thus causing up to 50 km of along-track orbit error. The data collected in May, 2002 while in three-axis stabilized mode should provide a significant improvement in the orbit determination results.

Davis, George↗

NASA NTRS2002

An Introduction to Tensors for Students of Physics and Engineering

Tensor analysis is the type of subject that can make even the best of students shudder. My own post-graduate instructor in the subject took away much of the fear by speaking of an implicit rhythm in the peculiar notation traditionally used, and helped us to see how this rhythm plays its way throughout the various formalisms. Prior to taking that class, I had spent many years "playing" on my own with tensors. I found the going to be tremendously difficult but was able, over time, to back out some physical and geometrical considerations that helped to make the subject a little more transparent. Today, it is sometimes hard not to think in terms of tensors and their associated concepts. This article, prompted and greatly enhanced by Marlos Jacob, whom I've met only by e-mail, is an attempt to record those early notions concerning tensors. It is intended to serve as a bridge from the point where most undergraduate students "leave off" in their studies of mathematics to the place where most texts on tensor analysis begin. A basic knowledge of vectors, matrices, and physics is assumed. A semi-intuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and higher vector products. The reader must be prepared to do some mathematics and to think. For those students who wish to go beyond this humble start, I can only recommend my professor's wisdom: find the rhythm in the mathematics and you will fare pretty well.

Kolecki, Joseph C.↗

NASA NTRS2002

Autonomous Reconfigurable Control Allocation (ARCA) for Reusable Launch Vehicles

The role of control allocation (CA) in modern aerospace vehicles is to compute a command vector delta(sub c) is a member of IR(sup n(sub a)) that corresponding to commanded or desired body-frame torques (moments) tou(sub c) = [L M N](sup T) to the vehicle, compensating for and/or responding to inaccuracies in off-line nominal control allocation calculations, actuator failures and/or degradations (reduced effectiveness), or actuator limitations (rate/position saturation). The command vector delta(sub c) may govern the behavior of, e.g., acrosurfaces, reaction thrusters, engine gimbals and/or thrust vectoring. Typically, the individual moments generated in response to each of the n(sub a) commands does not lie strictly in the roll, pitch, or yaw axes, and so a common practice is to group or gang actuators so that a one-to-one mapping from torque commands tau(sub c) actuator commands delta(sub c) may be achieved in an off-line computed CA function.

Hodel, A. S.↗

NASA NTRS2003

An Efficient and Robust Singular Value Method for Star Pattern Recognition and Attitude Determination

A new star pattern recognition method is developed using singular value decomposition of a measured unit column vector matrix in a measurement frame and the corresponding cataloged vector matrix in a reference frame. It is shown that singular values and right singular vectors are invariant with respect to coordinate transformation and robust under uncertainty. One advantage of singular value comparison is that a pairing process for individual measured and cataloged stars is not necessary, and the attitude estimation and pattern recognition process are not separated. An associated method for mission catalog design is introduced and simulation results are presented.

Juang, Jer-Nan↗

NASA NTRS2003

Observational Investigation of Solar Interior and Atmosphere

The Imaging Vector Magnetograph (IVM) has been modified to make it easier to observe at more than one spectral line. The cell holding the blocking filter has been replaced by a four-position filter wheel, so that changing to a different line is a matter of a few minutes rather than the several hours it used to take to disassemble the cell and install a new filter. Three new filters have been obtained, for Na 1589.6 nm, Fe 1630.25 nm, and H 1656.3 nm. The new filters have better bandpass profiles than the ones they replaced: somewhat wider, with flatter tops and steeper wings. This results in a reduction of parasitic light coming from adjacent Fabry-Perot orders, from seven percent to about two percent, and flattens the apparent continuum. The Mees CCD Imaging Spectrograph (MCCD) was upgraded under this grant, with a new control computer and data system. The camera was replaced with a faster, larger-format frame-transfer camera. Final integration of the upgrades is not yet complete, but tests indicate that the system cadence will be improved by a factor of five to ten, while increasing the spatial coverage by a factor of two (depending on observation options). Synoptic observations with the IVM and MCCD continue to be conducted daily, to the extent permitted by the fact that we have a single observer responsible for the observations. The older Haleakala Stokes Polarimeter is also used to make a daily vector magnetogram, normally of the region selected by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) duty scientists. This instrument, however, is showing its age to the extent that its maintenance is becoming something of a challenge. We also run a white light full-disk imager and a video H alpha prominence camera, continuously during times of observations. Of particular interest, we obtained rapid-cadence observations of the 2003 July 15 white light flare with both the IVM and MCCD. The vector magnetograms show no obvious difference between the preflare and postflare configurations. The photospheric magnetogram sequence and H alpha spectra we obtained can be combined with spacecraft observations to construct a rather detailed picture of the events of this flare. Several papers are in preparation.

Kuhn, Jeffrey R.↗

NASA NTRS1990

Space Vehicle Deployment from Space Station Orbit

When launching a spacecraft from Earth parking orbit to deep space, it is highly desirable to have the hyperbolic excess velocity vector (v-infinity) contained in the parking orbit plane. Ground launches can force the parking orbit plane to contain the v-infinity vector by using launch azimuth and lift-off time as independent variables. When launching from the Space Station, a new set of variables comes into play. The Station orbit is of fixed inclination but precessing due to the Earth's oblateness. Its plane will seldom (and may never) contain the desired v-infinity vector. Consequently, the departure strategy will usually require multiple burns and include a plane change. Also, the concept of "launch period" will be somewhat different from Earth surface launches. An analysis of the deployment of interplanetary spacecraft from Space Station is described, with emphasis on the effect of the trajectory characteristics on station operations. Several planetary mission types are analyzed for manned Mars missions. In addition, high declination departures of spacecraft on unmanned missions to an asteroid are examined. The constraint of Station orbit nodal position is quantified and the operational implications for station reboost strategy are examined.

Henry, Paul K.↗

NASA NTRS2003

CME Prediction from Line-of-Sight Magnetogram

We have previously shown for bipolar active regions that measures of active-region nonpotentiality from vector magnetograms are correlated with active-region CME productivity. We have now obtained a measure from line-of-sight magnetograms that is well correlated both with our measures of active-region nonpotentiality from vector magnetograms and with active-region CME productivity. The measure is the length of strong-gradient main neutral line (L(sub G)). This is the length of the bipolar region's main neutral line on which the potential transverse field is greater than 150G, and the gradient in the line-of-sight field is greater than 50G/Mm. From the sample of 17 MSFC magnetograms of 12 basically bipolar active regions used in our previous paper, we find that L(sub G) is strongly correlated with one of our vector-magnetogram measures of nonpotentiality, the length of strong-gradient main neutral line L(sub SS) (99.7%). We also find that L(sub G) is as strongly correlated with CME productivity (99.7%) as is L(sub SS). Being obtainable from line-of-sight magnetograms, L(sub G) makes the much larger data set of line-of-sight magnetograms (i.e. from SOHO/MDI and Kitt Peak) available for CME prediction study. This is especially important for evolutionary studies, with SOHO/MDI having no daylight, cloudy weather, or atmospheric seeing problems. This work was supported by funding from NSF's division of Atmospheric Sciences (Space Weather and Shine Programs) and by NASA's office of Space Science (Living with a Star program Solar and Heliospheric Physics Supporting Research and Technology program).

Falconer, D. A.↗

NASA NTRS2003

A Performance Evaluation of the Cray X1 for Scientific Applications

The last decade has witnessed a rapid proliferation of superscalar cache-based microprocessors to build high-end capability and capacity computers because of their generality, scalability, and cost effectiveness. However, the recent development of massively parallel vector systems is having a significant effect on the supercomputing landscape. In this paper, we compare the performance of the recently-released Cray X1 vector system with that of the cacheless NEC SX-6 vector machine, and the superscalar cache-based IBM Power3 and Power4 architectures for scientific applications. Overall results demonstrate that the X1 is quite promising, but performance improvements are expected as the hardware, systems software, and numerical libraries mature. Code reengineering to effectively utilize the complex architecture may also lead to significant efficiency enhancements.

Oliker, Leonid↗

NASA NTRS2004

Pseudo Linear Attitude Determination of Spinning Spacecraft

This paper presents the overall mathematical model and results from pseudo linear recursive estimators of attitude and rate for a spinning spacecraft. The measurements considered are vector measurements obtained by sun-sensors, fixed head star trackers, horizon sensors, and three axis magnetometers. Two filters are proposed for estimating the attitude as well as the angular rate vector. One filter, called the q-Filter, yields the attitude estimate as a quaternion estimate, and the other filter, called the D-Filter, yields the estimated direction cosine matrix. Because the spacecraft is gyro-less, Euler's equation of angular motion of rigid bodies is used to enable the estimation of the angular velocity. A simpler Markov model is suggested as a replacement for Euler's equation in the case where the vector measurements are obtained at high rates relative to the spacecraft angular rate.

Harman, Richard R.↗