DETERMINATION ILLUYUNATED AFIT/GA/ENY/89J-3 REPRODUCED *~BSTAVILBL DEPARTMENT UNIVERSITY TECHNOLOGY Wright-Patterson to"nmlmL89 AFIT/GA/ENY/89J-3 ORBIT DETERMINATION OF SUNLIGHT ILLUMUNATED OBJECTS DETECTED OVERHEAD PLATFORMS P. Osedacz Captain, USAF AFIT/GA/ENY/89J-3 Approved for public release; distribution AFIT/GA/ENY/89J-3 Determination of Sunlight Illuminated Objects by Overhead 3 Presented to the Faculty of the School of Engineering Air Force Institute of Technology Air University In Partial Fulfillment Requirements if Master of Science in Astronautical Engineering Unnnnounced - Jutlflcoti0n I Approved for public release; distribution Acknowledgements I am deeply indebted to a number of people for all of the support in performing this First of all like to thank Dr. Wiesel and Capt Bain and patience the members of the Powered Group, Flight Performance of the Foreign Technology I sponsoring me in this task. In particular I would like to Creehan for getting me started project, Larry Lillard who coached me in the intricacies of estimation theory to trajectory reconstruction, Jim Bernier from Geody- Corporation for stimulating my interest in trajectory estimation applications keeping my spirits up during dark and gloomy extremely grateful wife Carol understanding and support those long hours which spent with the computer aole of Contents m Acknowledgements.. I. Introduction ............. II. Orbit Generation Equations and Methods ....6 Coordinate Systems and Transformations 6 Geopotential Determination Orbital Element Generation .......... Orbital Element Sets ........... Transformation Equinoctial order Runge-Kutta-Nystrom (RKN) Numerical Integrator State Transition Matrix .......... 3 IV. Refining the Initial State Unobservability ..... ............ 3 Range Determination Event A and B Analysis Results ...... .. 47 Multiple Collection VI. Summary and Conclusions......... VII. Recommendations Bibliography List of Figures 1. Fastwalker Problem Geometry ........ .......... 2. Ellipsoidal Earth Model 3. Fastwalker Sensor Geometry ......... .......... 10 4. Equinoctial Coordinate Frame .. ......... .. 23 5. Single Data Set Focal Plane Traces ...... .. 48 6. Simulated View Angle Rates for Various Targets 49 7. Actual Angular Rate Ratio vs Range Ratio 8. Event A Azimuth Residuals .... .......... .. 52 9. Event A Elevation Residuals B Azimuth Residuals .... ........... .. 53 11. Event B Elevation Residuals ... .......... 12. oa' vs Slant Range for Single Data Set Events 55 13. Multi-Day Event Focal Plane Traces ...... .. 55 14. Multiple Day Event Azimuth Residuals ..... 15. Multiple Day Event Elevation Residuals ....57 Description Statistics A technique determining sunlight-illuminated (when passing through the sensor's field of view) is devel- determination consisting components sufficiently analyst-supplied indicating determination impossible consecutive algorithm, presented. Determination Sunlight-Illuminated by Overhead Introduction the geostationary field of view. These objects, known as fastwalkers, are a suspicion uncatalogued cross-tagged data base. Technology Performance (FTD/SQDF) to analyze data tracks and determine identifying this thesis feasibility determining from space-based commonality/occurrence The project next occurrence preventative to protect over-saturation. In order to start the analysis, a few underlying tions must be made. First, the sensor ephemerides are known exactly, since this is the best true baseline information available. Also, sensor location is a type of Q-parameter, where Q-parameters are defined by Day as those parameters effect the observations but, for some reason, cannot be estimated (5:3-1). Other examples of Q-parameters are atmo- spheric density. data biases, and the like. P-parameters, the other hand, are those parameters which can be estimated the given data. the objects (targets) are assumed to be non- thrusting bodies within 1000 km from the sensor since the col- lected data tracks can be as long as 30 minutes, implying the object's orbital speed is nearly the same as the sensor's speed. This assumption along with the fact perturbations will have no visible effect during the span of the data track, the two body of motion will be sufficient to estimate the orbital element set. simulated data for this project were generated by the Modularized Vehicle Simulation/Trajectory Reconstruction Program (MVS/TRP), a batch weighted least squares estimator originally developed to validate the guidance equations various space boosters. The data were generated Gaussian white noise at one point every ten Description of Simulated Elevation: fastwalker safe to assume problem has is most prevalent missile early system located at various geostationary locations. in his paper using various and a least that there is a family squares solutions in r which satisfy the collected azimuth and Little was resolve this single possible fastwalker, his conclusion the problem unsolvable. undocumented simulations were performed Surveillance the sensor was boosted a higher orbit circularized 500 km above its original angle a collection other satellites field of view were listed at a particular epoch time. This proved to be futile in singling out one particular the fastwalker. Other than the above mentioned instances, no other work has been performed on fastwalker Since pure angular data is given along with the ephemeris over the data span, the Gauss orbit determination will be used to determine an intial orbit. The initial target (fastwalker) state vector can then be refined batch weighted least squares estimator to further update The ephemeris is reported in latitude, longitude, alti- tude and time since no velocity information is required the data (i.e. the data is also position related. No doppler 3 or other velocity related effects need to be taken into count to correct the data model). However, an accurate veloc- m ity component of the sensor state vector must be determined if a six element state vector will be propagated over a few Assuming the first ephemeris position data point is exact, the remainder of the position ephemeris can be using a least squares algorithm. The first difference is an first guess to the velocity required to "hit" all of the remaining position ephemeris points. Once converged, the sensor state vector will be the best possible estimate given the data at hand. The next step is to refine the derived state vector the Gauss algorithm using a least squares estimator. Assuming the Gauss-derived estimate is reasonable, the target vector improved up to the error of the data. The 1-a figures of merit from the sensor specifications are an input into the program. These inputs form the data matrix required by the least squares algorithm normalize the data entries With the fastwalker orbit now defined, it can be propa- gated forward to determine the next encounte' with another sensor in this or another constellation. This is easily done Air Force Jet Propulsion Laboratory's Long Term Orbit (LOP). The accuracy of this predictor has been with simulations matching probe flight and highly eccentric orbit trajectories. Generation and Methods the motion of an orbiting over short long periods This includes the differential of motion, transformation inertial state to classical orbital elements. Aiso included are the sensor coordinate coordinate frame, which is the computational coordinate transformations squares (also known as differential correction) process the observed measurements are compared to the assumed or estimated measurements. Coordinate Systems and Transformations Many different coordinate systems are quite useful for expressing the orbit of an artificial satellite. Two major coordinate systems, the sensor coordinate frame and the iner- tial coordinate frame will be described here. The computational coordinate system for this algorithm, 3I the frame in which the equations of motion are expressed and integrated, is the Earth Centered Inertial, or ECI coordinate 3I system shown in Figure 1, p. 7. The origin of this frame ic at the center of the ellipsoidal earth model; the X-Y plane is the equatorial plane of the earth and the Z-axis is coinci- Fastwalker Problem axis of rotation. coordinate is fixed in space with X pointing toward the Ares at some arbitrarily chosen time this program, the midpoint of the data set for the of the state and the time of the first point for the WLS estimation Two latitude normally associated earth model geocentric and (see Figure 9). The geocentric defined as angle between the equatorial the vector equatorial perpendicular horizontal). geocentric respectively. Geocentric geocentric. coordinate sensor-fixed is pointing equatorial right-handed coordinate elevation. Local Vertical 6 Eccentricity: 0.08182 V Geocentric 2 Ellipsoidal Earth Model COS A = p Z (R, R, and R, are the inertial sensor vector elements and Projecting the range unit vector upon the sensor coordinate frame produces the direction cosines L,, Lv, and L,, where Fastwalker Sensor .=-sin el s in and scale factor is added to each azimuth and elevation calculation to account for other metric data formats. coordinate rotations. then about Mathematically, frame slant range vector a = -local hour 9,.- Greenwich hour angle at midnight I, -earth's A, = sensor i =-. -,sensor Geonotential and Gravity Determination Assuming a non-thrusting above the earth's atmosphere, significant force acting on the body a data span is gravity. Lillard in his thesis assumes negli- dependence, geopotential becomes (16:18) P,(sin(,p)) gravitation equatorial = associated geocentric potential, sufficient perturbations four expansion acceleration to gravity is derived potential. By setting geopotential gravitational acceleration be written computational coordinate eliminating coordinates. Therefore, the gravita- acceleration (') denotes derivative are integrated a Runge-Kutta- integrator Orbital Element Generation For this specific problem, Escobal states the Gaussian method of orbit determination is "second to none" when the time between the observations is small (6:272). The Laplace was found to be unsuitable since the direction cosine and accelerations were small enough to produce near- singular matrices. The Gauss method hinges on the fact that only two linear- ly independent vectors are needed to define the orbit assuming negligible perturbation effects during the data span. a third vector can be written as a linear of the independent vectors. Thus it is possible to a set of constants a, b, and c not all zero such Arbitrarily choosing r, as the dependent vector and redefining to c, and c, Crossing r, with r, and r, three parallel vectors