IDENTIFICATION OF A TETHERED SATELLITE USING AN EXTENDED KALMAN FILTER Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration with my advisory committee. This thesis does not include proprietary or classified information. _________________________________________ Elizabeth Jo Volovecky Hayes Certificate of Approval: _________________________ _________________________ John E. Cochran, Jr. David A. Cicci, Chair Professor Professor Aerospace Engineering Aerospace Engineering _________________________ _________________________ Robert S. Gross Joe F. Pittman Associate Professor Interim Dean Aerospace Engineering Graduate School IDENTIFICATION OF A TETHERED SATELLITE USING AN EXTENDED KALMAN FILTER Elizabeth Jo Volovecky Hayes A Thesis Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the Degree of Master of Science Auburn, Alabama December 17, 2007 iii IDENTIFICATION OF A TETHERED SATELLITE USING AN EXTENDED KALMAN FILTER Elizabeth Jo Volovecky Hayes Permission is granted to Auburn University to make copies of this thesis at its discretion, upon the request of individuals or institutions and at their expense. The author reserves all publication rights. ______________________________ Signature of Author ______________________________ Date of Graduation iv VITA Elizabeth Jo Volovecky Hayes was born on August 30, 1980 in Fairhope, AL. Her parents are Joe and Debbie (Probst) Volovecky of Daphne, Alabama. Elizabeth graduated from Daphne High School in May of 1998. She enrolled at Auburn University in the fall of 1998, where she began her aerospace studies. Upon receiving her Bachelor of Aerospace Engineering degree in May of 2002, she began her graduate studies in the Aerospace Engineering Department at Auburn the following fall. She married her high school sweetheart, Andy Hayes, in November 2004 and she currently works as an Intel Analyst for SAIC in Huntsville, Alabama. v THESIS ABSTRACT IDENTIFICATION OF A TETHERED SATELLITE USING AN EXTENDED KALMAN FILTER Elizabeth Jo Volovecky Hayes Master of Science, December 17, 2007 (B.A.E., Auburn University, 2002) 62 Typed Pages Directed by David A. Cicci Recent studies involving a tethered satellite system(s) (TSS) have increased due to the importance of accurately identifying and analyzing the motion of a TSS. If the motion of a tethered satellite is not accurately identified, the satellite could be mistaken as a ballistic threat. Standard orbit determination methods used today are unable to identify a tracked satellite as part of a TSS, due to the non-Keplerian nature of its motion. Accurate identification of a TSS becomes more complicated with the need to perform this process quickly using a small set of observational data. Once this ?quick- look? identification process is performed, it is necessary to calculate the critical orbit determination parameters used for future TSS tracking and prediction. An extended Kalman filter (EKF) has been developed to perform both the state estimation and quick-look identification processes for a tethered satellite not known vi a priori as being part of a TSS. In the application of the EKF to a TSS, both, manual tuning and adaptive tuning methods were used. The adaptive tuning method used is based upon ridge-type filtering techniques involving the computation of a biasing parameter that is used as input into the process noise matrix, which is required in tuning the EKF. The overall performance of the EKF is presented for varying tether lengths, tether orientation, and observation noise levels. The results obtained from the adaptively-tuned EKF are presented in this thesis and are compared to those obtained from a batch filter and manually-tuned EKF presented in recent studies. vii ACKNOWLEDGMENTS I would like to thank my family and friends for being so patient with me throughout my life, but especially through my college journey. You all mean the world to me. I would not be where I am right now without the guidance from the Aerospace Engineering faculty at Auburn University. They have all made a lasting impression in my life. I especially want to thank Dr. David Cicci for all of his patience and understanding through my graduate studies. He never once turned me away even when I thought I asked a ?stupid? question and for that, I am most thankful. viii Style manual or journal used: The Journal of the Astronautical Sciences ________ ________________________________________________________________________ Computer software used: FORTRAN PowerStation 4.0, Microsoft Excel, Microsoft Office Word 2003, and Microsoft Equation 3.0 ix TABLE OF CONTENTS LIST OF FIGURES.........................................................................................................x LIST OF TABLES .........................................................................................................xi 1. INTRODUCTION.......................................................................................................1 2. QUICK-LOOK ORBIT DETERMINATION METHOD DESCRIPTION ...................7 2.1 Preliminary Orbit Determination Method (POD, 1 st Stage).............................7 2.2 Identification Using an Extended Kalman Filter (2 nd Stage)..........................10 2.3 Adaptive Tuning Method Using a Biasing Parameter ...................................14 3. PROCEDURE DESCRIPTION .................................................................................17 4. TEST CASES............................................................................................................23 5. RESULTS .................................................................................................................25 6. CONCLUSIONS .......................................................................................................32 REFERENCES..............................................................................................................35 APPENDIX A: TEST CASE DATA..............................................................................37 x LIST OF FIGURES FIG. 1. Tethered Satellite System (TSS) Model ..............................................................1 FIG. 2. TSS Model with Force Components and Libration Angle ....................................8 FIG. 3. TSS Center-of-Mass and Tether Length Measurements .......................................9 xi LIST OF TABLES TABLE 1. Parameter Variations for Scenarios..............................................................23 TABLE 2. Baseline Orbit for Data Generation...............................................................23 TABLE 3. Varying Combinations of the Biasing Function ...........................................24 TABLE 4. Comparisons for No Tether cases, ? cm = 0 m................................................27 TABLE 5. Comparisons for 1 km cases, ? cm = 909 m, LOW noise................................28 TABLE 6. Comparisons for 1 km cases, ? cm = 909 m, MEDIUM noise.........................28 TABLE 7. Comparisons for 1 km cases, ? cm = 909 m, HIGH noise ...............................28 TABLE 8. Comparisons for 10 km cases, ? cm = 9091 m, LOW noise ............................29 TABLE 9. Comparisons for 10 km cases, ? cm = 9091 m, MEDIUM noise.....................29 TABLE 10. Comparisons for 10 km cases, ? cm = 9091 m, HIGH noise .........................29 TABLE 11. Comparisons for 50 km cases, ? cm = 45455 m, LOW noise ........................30 TABLE 12. Comparisons for 50 km cases, ? cm = 45455 m, MEDIUM noise.................30 TABLE 13. Comparisons for 50 km cases, ? cm = 45455 m, HIGH noise .......................30 TABLE 14. Comparisons for 1 km UP cases, ? cm = -909 m, LOW noise.......................31 TABLE 15. Comparisons for 1 km UP cases, ? cm = -909 m, MEDIUM noise................31 TABLE 16. Comparisons for 1 km UP cases, ? cm = -909 m, HIGH noise......................31 TABLE 17. Test Case Data: No Tether (? cm = 0) .........................................................38 TABLE 18. Test Case Data: ? = 1 km (? cm = 909 m), LOW noise ...............................39 xii TABLE 19. Test Case Data: ? = 1 km (? cm = 909 m), MEDIUM noise ........................40 TABLE 20. Test Case Data: ? = 1 km (? cm = 909 m), HIGH noise ..............................41 TABLE 21. Test Case Data: ? = 10 km (? cm = 9091 m), LOW noise............................42 TABLE 22. Test Case Data: ? = 10 km (? cm = 9091 m), MEDIUM noise ....................43 TABLE 23. Test Case Data: ? = 10 km (? cm = 9091 m), HIGH noise...........................44 TABLE 24. Test Case Data: ? = 50 km (? cm = 45455 m), LOW noise..........................45 TABLE 25. Test Case Data: ? = 50 km (? cm = 45455 m), MEDIUM noise ..................46 TABLE 26. Test Case Data: ? = 50 km (? cm = 45455 m), HIGH noise......... ...............47 TABLE 27. Test Case Data: ? = 1 km up (? cm = -909 m), LOW noise ......... ...............48 TABLE 28. Test Case Data: ? = 1 km up (? cm = -909 m), MEDIUM noise .. ...............49 TABLE 29. Test Case Data: ? = 1 km up (? cm = -909 m), HIGH noise ........................50 1 1. INTRODUCTION A tethered satellite system (TSS) refers to a system that includes two or more satellites, or space bodies, which are connected by a tether, or cord. One idea behind tethered satellites is to control the motion of one satellite by attaching it to another satellite. In a two-bodied TSS, the primary satellite is generally the larger of the two and is referred to as the ?parent? satellite. The second satellite, usually smaller than the parent satellite, is referred to as the ?daughter? satellite. Either satellite may be located in a higher or a lower orbit with respect to the system?s center-of-mass or they can be in the same orbit. A two-bodied TSS model is shown in Fig. 1 below. FIG. 1. Tethered Satellite System (TSS) Model The study of tethered satellites originated in the late 1800?s and has continued on through today with actual ?in space? applications. In general, tethered satellites can offer many important uses, such as: providing power between satellites or other space vehicles for transfer of energy or momentum purposes, providing support to astronauts during an Earth Surface Parent Daughter Tether 2 Extra-Vehicular Activity (EVA) maneuver, which connects them to the spacecraft, or providing aid in the control of a space vehicle?s motion. It has only been recently that studies have addressed the orbit determination problem of tethered satellites. In one study, a satellite, which was a member of a TSS, was incorrectly identified to be a re- entry object?s trajectory [1]. This is due to the tether force perturbing the motion of the tracked satellite and causing it to behave differently than the motion of an untethered satellite [2]. The orbit determination of any space object involves the ability to accurately track, identify, and predict the motion of the object of interest. The need to perform a proper orbit determination analysis of a TSS has increased, since its future use has become important to both the private and public industries. If a TSS is inaccurately analyzed, its motion could result in a tethered satellite being incorrectly identified as a possible threat, resulting in many unnecessary and costly measures being taken to counteract the threat. It is possible for this type of scenario to occur since tethered satellites behave differently than untethered ones. This is due to the force present in the tether, which acts on all satellites connected to the tether. Each satellite in the TSS is perturbed by the tether force, causing it to vary from the Keplerian-type motion, typically found in untethered satellites [1,2]. When a tethered satellite is stationed in a higher orbit than the center-of-mass of the TSS, its velocity will be larger in magnitude than the results predicted by classical Keplerian motion due to the presence of the tether force. Likewise, if a tethered satellite is in a lower orbit than the center-of-mass of the TSS, its velocity will be smaller than that indicated from classical Keplerian motion due to the tether force. 3 Classical orbit determination methods are not capable of distinguishing between a tracked tethered satellite and an untethered satellite; therefore the satellite may be incorrectly identified as an untethered one if the classical methods are used. Incorrect identification will also result in an inaccurate prediction of the satellite?s future motion. In order to prevent a TSS from being incorrectly identified, the need to quickly perform the identification process is very important. This process of identification, referred to as ?quick-look? identification, is performed by processing measurements of the satellite?s motion made from tracking stations. These measurements are acquired by the use of radar, infrared, radio and/or optical techniques and include parameters of the observed satellite, such as: range, range-rate, azimuth, elevation, azimuth-rate and/or elevation-rate. The measurements are then used in the orbit determination process to estimate the satellite?s orbit. The first few measurements obtained are used in the preliminary orbit determination (POD) procedure in order to establish a set of initial conditions, which will then be used in the filtering process. In order to more accurately determine the set of initial conditions, a larger set of observational measurements, accumulated from one or more tracking stations, must then be processed. The need to accurately estimate the state of a tethered satellite, which can include parameters, such as: position, velocity, dynamical constants, etc., involves the differential correction process. Once the POD provides the set of initial conditions, the differential correction process improves the accuracy of the solution acquired by the POD procedure. These improvements are made by processing all of the observational data available in the filtering process. There are several filtering techniques available to use in the differential correction process. The three most commonly used are batch-type filters, Kalman (or 4 sequential) filters, and extended Kalman (or extended sequential) filters. Batch-type filters process an entire set of observational data at once in order to estimate the state for a specified epoch. Kalman and extended Kalman filters process observations as they are received. A Kalman filter provides the estimate of the satellite?s state at each observation time where an extended Kalman filter updates the reference trajectory at each observation to reflect the best estimate of the true trajectory. Each filter has its distinct advantages and disadvantages; however this study will utilize the extended Kalman Filter (EKF) and compare its results to previous studies using batch-type and sequential estimators. A POD method was recently developed [3,4] for the use of both tethered and untethered satellites. Following this method, several different batch-type filters for the estimation of the state of a tethered satellite were presented and their performances were compared [5]. In this study a two-dimensional dynamical model of a TSS was considered. The model maintained a vertical orientation and as a result did not possess the capabilities to include out-of-plane motion of the system, or any apparent oscillatory motion of the TSS. There have been additional studies where out-of-plane libration of a TSS was modeled, in an enhanced batch filter [6]. The accuracy of quick-look TSS identification was improved by the use of ridge- type estimation methods [7]. Once a satellite is identified as being a tethered satellite, more sophisticated models of a TSS [8-12] can then be used to predict its long-term motion. When long arcs of observational data are available, these enhanced dynamical models and filtering techniques are more useful. A more recent presentation involved a method that combined all of the desired characteristics needed in both the quick identification and the prediction of long-term 5 motion of a TSS. The three-stage TSS identification and orbit determination methodology [13] included: a 1 st Stage POD procedure, a 2 nd Stage ridge-type filter, and a 3 rd Stage long-term prediction filter. The performance of this methodology was demonstrated using a series of simulated cases with varying TSS geometry and observation noise levels, as well as on real TSS data obtained from the Tether Physics and Survivability Experiment (TiPS) [14]. An extended Kalman filter (EKF) was recently developed in order to address the TSS identification problem using extended sequential processing of the observational data [15]. The primary emphasis is on the quick-look aspects of tethered satellite identification rather than on the long-term orbit prediction aspects referred to in [13]. The manually-tuned EKF utilizes the POD results from [13] as initial conditions, and short arcs of observations are processed in order to determine the best estimate of the satellite?s state. The performance of the EKF is evaluated through the analysis of simulated data using differing tether lengths, tether orientations, observational error levels, and observation arcs. An adaptive or automated tuning methodology, used in angles-only tracking and intercept problems [16], was applied to the EKF to improve overall filter performance and to make the entire filtering process less tedious. The tuning method involves the use of a biasing parameter that is computed within the EKF. The biasing parameter is an integral part of ridge-type estimation techniques, which have shown improved accuracy in batch and sequential solutions of ill-conditioned orbit determination problems. The biasing parameter provides a measure of the overall solution error and is input into the process noise matrix for tuning the EKF. The performance of the filter depends on 6 correct propagation of the covariance matrix added with the a priori covariance, the observational covariance, and the process noise [16]. A ?tuned? extended Kalman filter means that the best possible filter performance is achieved. The elements of the process noise covariance and measurement noise covariance matrices need to be properly determined through the tuning process in order to improve the filter?s ability to provide accurate state estimates. The results from the adaptively- tuned EKF are presented in this study and are compared to the results obtained using the batch filter from the 2 nd Stage of the TSS methodology presented in [13] along with those results from the manually-tuned EKF presented in [15]. Conclusions for their use are also provided. 7 2. QUICK-LOOK ORBIT DETERMINATION METHOD DESCRIPTION The suggested orbit determination method for TSS involves two different dynamic models. The first stage of the methodology is a POD strategy that uses a simple dynamic model, while an enhanced dynamic model is used in the EKF of the second stage quick-look identification process. The related TSS models are specifically designed for those stages in order to yield the most accurate results for a given span of observations. The 1 st Stage uses only a few observations in its process where the results are used as input into the 2 nd Stage. The 2 nd Stage performs an adaptive, sequential analysis using up to 15 minutes of the observational data. Each stage, along with their models, is described in more detail in the following paragraphs. 2.1 Preliminary Orbit Determination (POD, 1 st Stage) The TSS model used in the POD stage consists of a ?daughter? satellite, m, and a ?parent? satellite, m p , where both are considered to be point masses. These satellites are connected by a massless tether, as illustrated in Fig. 2. Also shown in Fig. 2, is the effective tether force that acts upon the daughter satellite. The radial and tangential force components, F r and F t , respectively, which make up the tether force, are shown acting on the daughter satellite, since its motion is the one being observed. Depending on where the satellites are located in their orbit and in relation to each other, their velocities will be affected due to the change in acceleration imposed by the force components. For 8 example, the radial force component will create a radial acceleration, a r , which acts on the daughter satellite, and will cause its velocity to decrease if it lies in a lower orbit than the parent satellite, but will cause its velocity to increase if it lies in a higher orbit than the parent satellite. Likewise, the tangential force component will create a tangential acceleration, a t , on the daughter, causing its velocity to decrease if it precedes the parent, but causing its velocity to increase if it trails the parent. In the case where the parent satellite is being observed, the opposite of these dynamical characteristics will be true. FIG. 2. TSS Model with Force Components and Libration Angle One satellite preceding or ?leading? another satellite depends on the in-plane libration angle, ?, as shown in Fig. 2. The value of ? will also determine the directions of the tether force components, where the positive radial direction is defined from the center of Earth toward the daughter satellite and the positive tangential direction is that of the direction of orbital motion. The radial acceleration of a satellite?s motion can be obtained using a POD method along with other information, including the gravitational parameter, ?. Earth ? F r F t orbit direction m m p 9 Specifically, a modified gravitational parameter, ?*, can be calculated during the POD process by a relationship between the parameters ? and a r , which is presented in [3,4] as: 2 * ra r ?= ?? (2.1) Where, r, is the distance from the center of the Earth to the daughter satellite. Upon obtaining ?*, it can be used to find an approximate value of the distance from the daughter satellite to the center-of-mass of the TSS, which is designated by, ? cm , and illustrated in Fig. 3.. FIG. 3. TSS Center-of-Mass and Tether Length Measurements This distance is measured along the tether length and can illustrate whether the daughter satellite is above or below the parent satellite. When libration is present in the TSS, ? cm will represent the projection of the tether length (to the center-of-mass) in the radial direction and will be denoted by the parameter, ? cm *, and is presented in [3,4] as: () r cm ? ? ? ? ? ? + ? = *2 * * ?? ?? ? (2.2) Earth CM m p m ? cm ? cm * ? 10 If ? cm * is positive, then the daughter satellite is below the parent and ?* < ?. If it is negative, then the daughter is above the parent satellite where ?* > ?, which indicates that the satellite being observed is above the parent satellite. In addition, the value of ? cm will approach the actual tether length as the ratio of the point masses, m/m p , approaches zero. Several classical POD methods were modified to include the capabilities to determine ?* for a TSS and were presented in [3,4]. Due to superior convergence characteristics, the 9 th order f and g series method proved to perform best and those results were used in this study. The calculated value of ?* can be used quickly to determine whether the observed satellite is part of a TSS due to the few observations that are used in the POD process. The output from the 1 st Stage POD method, in the form of position, velocity, and a r , is used as input into the 2 nd Stage Extended Kalman Filter. The POD results provide no information regarding a t , therefore a t is initially assumed to be zero. 2.2 Identification Using an Extended Kalman Filter (2 nd Stage) The enhanced dynamical TSS model used in the ?quick-look? identification process, presented in [6], is similar to the model presented in the 1 st Stage, but includes additional dynamical effects. The model for the 2 nd Stage considers the tether to be inextensible and allows for oblate Earth effects, as well as, in-plane libration in the TSS dynamics. To determine the dynamical TSS characteristics, including: acceleration components, libration angle, etc., a batch-type filter has been used [5] to generate an 11 estimation of the daughter satellite?s state vector. This state vector includes the satellite?s position and velocity components and the acceleration components due to the tether force. The acceleration components, a r and a t , are both assumed to be constant over short observation arcs. This implies that the libration angle will remain constant through the observation arc as well. Since these parameters are tether-specific, including them in the filter allows for the satellite to be identified as tethered or untethered. A satellite is found to be untethered when its acceleration components are determined to be zero. This means that there is no tether force perturbing the acceleration components. Likewise, nonzero acceleration terms indicate that the satellite being observed is tethered. No other forces, such as: thrust, drag, which creates orbit decay, etc. are considered in this study. Once the acceleration terms are obtained, they can be used to calculate the libration angle of the TSS. In order to perform the 2 nd Stage process in a sequential manner, an extended Kalman filter (EKF) is used to quickly identify the observed satellite with a short arc of observational data. Classic EKF equations are used in this study and are explained in more detail below. To better describe the EKF, it is appropriate to summarize the process of the Kalman (or sequential) filter first and then provide a comparison. Swerling originally developed the sequential algorithm in 1958 [17], yet Kalman and Bucy have been recognized more for their work with the algorithm since 1961 [18]. The most important difference between the two filters is that the sequential algorithm processes observational data as it is received, while the EKF does the same and in addition, updates the reference trajectory after each observation is processed. The disadvantage of the sequential algorithm is the significant amount of errors due to neglecting higher order 12 terms in the linearization procedure. The EKF is used to decrease the effects of those errors, which allows for more rapid convergence. When using the sequential algorithm, if the true trajectory and the reference trajectory are too far apart, which is often the case at the beginning of a simulation, the estimation process may diverge due to the errors from the linearization process, previously described. The benefit in using the EKF is that the best estimate of the state will be reached more quickly. Below is a description of the EKF algorithm: 1. Given the following: - 1? ? kX , the estimate of the state vector at tk-1, (n x 1); - P k-1 , covariance matrix at time t k-1 , (n x n); - Y k , p-vector of observations taken at time t k , (p x 1); - R k , observation covariance at time t k , (p x p); 2. Integrate from t k-1 to t k , ()ttXFX ),(= & , () 1 1 ? ? ? = k k XtX (2.3) () () )( ),( tX ttXF tA ? ? = (2.4) )()()( 11 tQtAPPtAP T kkk ++= ?? & (2.5) where )(tX is the updated state vector (n x 1), F( )(tX , t) is the system dynamics function (n x 1), A(t) is the state sensitivity matrix (n x n), and where Q(t) is the process noise matrix (n x n). 3. Compute, ( ) )( ),( k kk k tX ttXG H ? ? = (2.6) () 1? += k T k k k T k k k RHPHHPK (2.7) 13 ()k kkk PHKIP ?= (2.8) ( ) kkkk ttXGYy ),(?= (2.9) kkk yKx = ? (2.10) kkk xXX ?? += (2.1) where H k is the measurement sensitivity matrix (p x n), ( ) kk ttXG ),( is the observation-state relationship vector (p x 1), K k is the Kalman gain (n x p), P k is the covariance matrix (n x n) associated with the best estimate of the (n x 1) state vector kX ? , y k is the observation residual vector (p x 1), Y k is the current observation, and k x ? is the state correction vector (n x 1). All variables are computed at time t k . 4. Replace k with k-1, return to step 2 and substitute. Repeat until all observations have been read. The algorithm above assumes that the process noise matrix, Q(t), is known. This will be discussed in more detail in the following section. The state vector, kX ? , used in the 2 nd Stage process will include the observed satellite?s position, velocity, and tether acceleration components. Upon obtaining the initial estimate of the state vector at each observation time, several determinations can be made relative to the tether acceleration components [13], such as: 1. If the values of a r and a t are found to be zero, it can be assumed that the observed satellite is an untethered one and standard techniques can be used to analyze its motion. 2. If the values of a r and a t are nonzero, then the following applies: a. The libration angle, ?, can be calculated from, ? ? ? ? ? ? ? ? = ? r t a a 1 tan? (2.12) 14 where the signs of a r and a t will determine the appropriate quadrant for ?. b. The magnitude of the acceleration due to the tether force can be calculated from, () 22 tr T aa m F += (2.13) c. The value of ? cm * , the radial projection of the tether length to the center- of-mass of the TSS, can be approximated through the use of a r and Equations 2.1 and 2.2. 3. In the case where one of the acceleration components is equal to zero and the other is not, then the following applies: a. For a t = 0 and a r ? 0, the satellite will be tethered and the system?s orientation will be vertical. This will be the case when ? = 0? and a r > 0 or when ? = 180? and a r < 0. b. For a r = 0 and a t ? 0, the satellite will be tethered and the system?s orientation will be horizontal. This will be the case when ? = 90? and a t > 0 or when ? = 270? and a t < 0. The methodology from the 2 nd Stage process combined with the implementation of the tuning method described in the following section will complete the quick look identification method. 2.3 Adaptive Tuning Method using a Biasing Parameter Before describing the adaptive tuning algorithm it is appropriate to stress the importance of ?tuning? an EKF in general. An EKF must be properly tuned in order to prevent filter divergence. This is achieved by adding adequate values of the a priori covariance and the observational covariance, if known, to the propagation of the covariance to properly generate the filter?s performance envelope; any ?process noise? is also added. A major disadvantage to tuning an EKF is that the results are usually 15 achieved manually, which is very tedious and allows for significant error. These problems present the need for an aid which ?automatically? tunes an EKF. In this presentation, an adaptive tuning method presented by Cicci [16], typically used in angles- only tracking and intercept problems, was applied to the EKF described in the previous section; yielding an adaptively-tuned EKF. The adaptive tuning method used in this study is based upon ridge-type estimation methods and requires the calculation of a biasing parameter, which represents the overall error in the solution and is used as input into the process noise matrix. The process noise matrix, Q(t), defined above in the 2 nd Stage description, is necessary to successfully tune the EKF and can be expressed as a function of the biasing parameter and written as Q(k), as described by Cicci [16]. The biasing parameter provides the following advantages to the EKF: 1. The diagonal terms of the process noise matrix are updated after each observation time as opposed to remaining constant for all of the observations. 2. Adaptive tuning allows the user to define how the biasing parameter is implemented at the beginning of the simulation, eliminating manual changes to the process noise. The form of the biasing parameter, k, is determined through converting the batch solution into a sequential solution, described in [16]. The sequential form of the biasing parameter, k, is presented below: () ()()[] ()[]xHIRDDRDDDHx xHIRDDRDDDHxRDDtr k mmmmmm TT mmmmmm TT mm ?? ?2? 22 222 + ++ = (2.14) where D m is a normalizing diagonal matrix (p x p) and its ith diagonal term is defined as: 16 T m HPH D 1 = (2.15) As stated in [16], the inspection of equation 2.14 above shows that k will always be greater than one. As the covariance matrix, P , decreases with each processed observation, the biasing parameter, k, also decreases in value and approaches one. As described before, k represents the overall error in the solution and Q(t) represents the process noise, therefore k is used to compute the Q(k) matrix, which includes the effects of k-biasing. With each observation k is computed and then used to update the appropriate Q(k) terms. Only those Q(k) terms that will effect the acceleration terms of the propagated covariance matrix [16] will be updated with the biasing parameter, since that is where most of the error exists. In summary, the description of the biasing parameter and its use within the EKF concludes the quick-look orbit determination process. Implementation of the process, test cases and results are described in the following sections. 17 3. PROCEDURE DESCRIPTION At this point it is necessary to expand on the methods previously explained by summarizing the overall procedure that was implemented in this thesis study. Beginning with the POD (1 st Stage) process, only the output data obtained from the POD presentations referenced in [3,4] was used as input into the adaptive EKF (2 nd Stage); no additional POD processes were considered. The output data from the POD process provides a reference trajectory for the tracked satellite with results in the form of the position vector, R , velocity vector, V , and ?*, which includes the standard gravitational parameter, ?, and the radial acceleration tether force component acting on the satellite. Using equation 2.1, this radial acceleration component, a r , can be calculated since the remaining variables are known. With the zero tangential acceleration component, a t , these parameters form the starting state vector for the daughter satellite which is used as input into the EKF (2 nd Stage). To implement the adaptive EKF in this study, the FORTRAN program developed for [15] was modified to include the adaptive parameters. The entire estimation process used in this study will be described in the paragraphs below, however the form that the state vector assumes for this study, must be explained first. The state vector includes the parameters from the POD process, satellite position, velocity, and acting tether 18 acceleration force. The state variables related to the dynamics of the satellite?s motion are defined in equation 2.16, below: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? == 8 7 6 5 4 3 2 1 )( X X X X X X X X a a z y x z y x tXX r t & & & (2.16) The equation of motion, including perturbation effects acting on the satellite, used in this study and those being compared to this study, is provided in equation 2.17. Definitions of the perturbation forces, p , which represents the effects of oblateness, and, t F , which represents the tether acceleration effects on the satellite, are provided in equations 2.18 and 2.19, respectively. t Fp R R R ++ ? = 3 ? && (2.17) ? ? ? ? ? ? ? ? ?= 2 2 3 2 2 3 1 2 R z R RJ p e ? (2.18) V Va R Ra F tr t += (2.19) In equation 2.18, 2 J is the oblateness constant coefficient, e R is the radius of the earth, and z is the z-component of the satellite?s position. In equation 2.19, r a and t a represents the satellite?s radial and tangential tether force acceleration terms, respectively. R and V are the magnitudes of the satellite?s position and velocity, respectively. 19 Take the time derivative of the state vector dynamics to find X & and substitute equations 2.17-2.19; yielding equation 2.20, where X & is a function of the state variables. () ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ++ ? ? ? ? ? ? ? ? +??? ? ? ? ? ? ++ ? ? ? ? ? ? ? ? +??? ? ? ? ? ? ++ ? ? ? ? ? ? ? ? +??? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? == 0 0 2 15 2 9 1 2 15 2 3 1 2 15 2 3 1 ),( 6738 4 2 3 2 2 2 2 2 3 3 5728 4 2 3 2 2 2 2 2 3 2 4718 4 2 3 2 2 2 2 2 3 1 6 5 4 8 7 6 5 4 3 2 1 V XX R XX R XRJ R RJ R X V XX R XX R XRJ R RJ R X V XX R XX R XRJ R RJ R X X X X X X X X X X X X ttXFX EE EE EE ? ? ? & & & & & & & & & (2.20) The observation vector, y, as defined in equation 2.9 requires the use of an observation- state relationship vector, ()()ttXG , , defined here in equation 2.21. ()( )() ( () () () () )() ? ? ? ? ? ? ? ? ?+?+?+?+ ?+?+?+?+ = ? ? ? ? ? ? ? ? ???+?+?+ ?+??+? = ?+?+?+?+= 2 3 2 21 2 21 21321 3 32121 2121 2 2 3 2 21 2 211 cossinsincos sincoscossinsincoscossincos sin cossinsincossincossinsincos sinsincoscoscossin tan cossinsincos SSS SSS SSS SS SSS ZXYXXXXX YXXZXXXX aG ZXYXXXXX XXXYXX aG ZXYXXXXXG ???? ????????? ????????? ?????? ???? (2.21) In equation 2.9, Y, represents the current observation?s range, azimuth, and elevation values of the satellite as measured relative to the tracking station. In equation 2.21, ? is equal to the ? E at the current observation time. Xs, Ys, and Zs are the known position parameters (in Earth-Centered Earth Fixed, ECEF) for the tracking station location. The latitude and longitude of the tracking station, also known, are ? and ?, respectively. The observation vector equations include the transformations between the tracking station?s local coordinate frame and the ECEF coordinate frame. 20 The equations provided for X & and the observation vector, y, were also used in presentations [3,4,15]. They are used in this study to provide consistency for comparing results from test cases. After explaining the forms of the state and observation vectors, the simulation description proceeds. Initially, the diagonal values for the a priori covariance and process noise matrices are given to begin the quick-look EKF analysis. The values for the a priori covariance matrix used in this thesis will be explained in more detail in the Test Cases section. 1. First, the state estimate found from the POD results are read into the program. Processing one observation at a time, in 5 second intervals and up to 15 minutes of data, the range, azimuth, and elevation defining the satellite?s location relative to the tracking station is recorded. Also processed is the observation covariance. 2. Compute EKF equations 2.5 through 2.10 and calculate the predicted state vector and covariance matrix (with process noise and the k-biasing function). 3. Update the state estimate and a priori covariance matrix. 4. Calculate the Root Mean Square (RMS). 5. Calculate ? cm * using equation 2.2. 6. Begin processing the next observation with the updated state estimate and process noise matrix and continue until all observations are processed. Once the k-biasing parameter is calculated using equation 2.14 it is used in a biasing function to update the appropriate terms of the process noise matrix for use in the next observation. The Root Mean Square (RMS) is also calculated at each observation to measure any process error, which aids in identifying the convergence of the best solution. The projection of the tether length (to the center-of-mass) in the radial direction, ? cm *, is 21 then calculated. The next observation is processed using the updated process noise matrix (with biasing included) and the best estimate of the state vector. This procedure continues until all observations are processed. In this study, those terms in the process noise matrix that effect the tether acceleration components will be the only terms considered. These terms are functions of the k-biasing parameter used to automatically tune, or adjust, the propagation of the state covariance matrix. As mentioned before, the process noise matrix, Q(k), is updated with functions of the k-biasing parameter. Only the acceleration terms of the matrix will be updated as shown in the following form. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = )(0000000 0)(000000 00000000 00000000 00000000 00000000 00000000 00000000 )( 2 1 kf kf kQ (2.2) The functions used for f 1 (k) and f 2 (k) are dependent upon the types of scenarios that are being analyzed. There are no limitations to the types of function that can be used, but some effort may need to be put forth at the beginning of the analysis to make sure the appropriate functions are chosen to use. For instance, in angles-only tracking intercept problems, the functions should be chosen so that larger values of the process noise are provided to the components that contain high levels of error [16]. Other cases may not include levels of error that are significantly high; requiring a biasing function to yield smaller values. 22 Using the biasing function in the process noise matrix, which is updated at each observation, should provide faster tuning results to the EKF. This allows the EKF to perform more powerfully and with high confidence of the best converging solution. 23 4. TEST CASES Many different scenarios were considered in this thesis study to illustrate the adaptive EKF?s performance when identifying whether the observed satellite is tethered or untethered. In the end, over 2,000 scenarios were generated to test the EKF. This involved scenarios, limited to the data provided from simulated data acquired from a previous study [3], with varying tether lengths, orientations, observation noise levels, and observation arcs; specific values and levels are summarized in Table 1 below. TABLE 1. Parameter Variations for Scenarios Tether Lengths (?) 0 km, 1 km, 10 km, 50 km, and 1 km UP Orientation: In-Plane Libration Angle (?) 0?, 5?, 10? (N/A for 0 km cases) Observation Noise Levels LOW (5 m/0.002?), MEDIUM (25 m/0.01?), and HIGH (50 m/0.02?) Observation Arcs 5, 10, and 15 minutes The simulated data was generated using a baseline circular orbit [3]. The orbital elements for this circular orbit are provided in Table 2 below. TABLE 2. Baseline Orbit for Data Generation Orbital Elements a 6621 km e 0.00 i 5.73 deg ? 5.73 deg 24 Each test case scenario used as input the a priori covariance matrix found with the best manually-tuned results from the EKF presented in [15]. The idea behind this is to provide commonality among both filters for comparison purposes and to ultimately determine which filter yielded faster and more accurate results. As mentioned when describing the procedure for implementing all presented methodology in this study, many combinations for the k-biasing function, used in the process noise matrix, Q(k), were also considered. Some of the functions used were taken from [16], in order to provide a common foundation for comparing results to the adaptive EKF. A description of the combinations used in the k-biasing functions is presented in Table 3 below. TABLE 3. Varying Combinations of the Biasing Function Functions used for both, f 1 (k) & f 2 (k) k and k 2 , from [16] n k 1 , where n = 1, 2, 3, 4, ?21 (k-1) and (k-1) 2 , from [16] n k )1( 1 ? , where n = 1, 2, 3, 4, ?9 (k+1) and (k+1) 2 n k )1( 1 + , where n = 1, 2, 3, 4, ?21 The results from the two best biasing functions from [16] and two from the remaining combinations from Table 3 are presented for all test case scenarios in Appendix A. The same function is not ideal for all test cases due to observation, process, and user errors. Out of the four cases presented for each test case scenario, the case yielding the best solution represents optimum results found from the adaptive EKF for that test case. 25 5. RESULTS From the results presented for each test case in Appendix A, a variety of different combinations of biasing functions were used to acquire the end results for this study. The best solutions from these test cases were taken and compared to the best solutions from the batch filter presented in [13] and the manually-tuned EKF presented in [16]. Tables 4 through 16 organize this information from all three studies so that appropriate conclusions may be drawn. Overall the adaptive EKF produced the most accurate results within 15 minutes of observed data for all tether lengths and k-biasing functions presented. The tables of data provided observe the adaptive EKF?s performance at three different time spans (5, 10, and 15 minutes); this provides the opportunity to monitor whether or not a test case converges sooner than the anticipated 15 minute time span. Where this occurred, which was in very few cases, it was determined that the early convergence was due to the proper selection of the biasing function. Additionally, the rapid divergence that was found in the following time span, for these cases, demonstrated that the best solution had already occurred for the test case. The adaptive EKF was able to produce the best results for the no tether cases within 10 minutes of data and to match those of the manually-tuned EKF. The cause for 26 this result illustrates the ability to choose the biasing function to achieve the desired results. In these cases, the process noise matrix values were zero for the manually-tuned EKF, therefore the biasing function used in the adaptive EKF needed to provide small values for the process noise so that the solutions would be similar when compared. This means that the value of k needed to be larger when using a function similar in form to () n k ? +1 , where n is larger; the output from this biasing function is driven to the expected limit, or floor, more quickly. The adaptively-tuned EKF was more sensitive to observation error and libration levels than those results found from the manually-tuned EKF. Results from the adaptive- tuned EKF for test cases with high noise levels performed better than the low and medium noise level cases for some tether lengths. In the cases where low levels of noise were present the results varied depending on the amount of libration present. Higher amounts of libration provided better results (i.e. RMS closest to 1 and ? cm * closest to desired value) for these cases. Results from the adaptively-tuned EKF became more accurate with higher tether lengths, which did not seem to have an affect in either the batch filter or manually-tuned EKF processes. In all three filter cases, there was commonality among the no tether cases diverging as observation noise increased. The amount of error the input conditions and process noise are expected to contain, aids in selecting the ?best? biasing function to use. Varying the k-biasing function provided a unique look into the overall convergence to the best solution for each case. Because many biasing functions were considered for each scenario in this study, selecting the biasing function that provided the best solution to each test case was simple. 27 As verification that the best solution had been achieved, the very next variation of the biasing function used in the scenario, and any thereafter, provided an immediate and significant divergence. This pattern provided confidence throughout the entire adaptively-tuned EKF analysis that the best solution for each test case had been determined. The RMS values calculated at each observation and recorded for all three time spans provided good results in most of the cases. Where the RMS values were significantly higher than the ideal value of one, the conclusion is that the adaptively- tuned EKF procedure required more observations, than either the batch or manually- tuned EKF processes needed, to find the best converged solution. TABLE 4. Comparisons for No Tether cases, ? cm = 0 m Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) low 5 min: 1.021 -116 0.243 -2416 0.2431 -2416 10 min: 1.025 310 0.443 -39 0.4433 -39 15 min: 1.006 152 0.686 92 0.6862 92 med 5 min: 1.042 4246 0.233 -6559 0.2329 -6559 10 min: 1.021 624 0.363 320 0.3631 320 15 min: 1.011 708 0.709 685 0.7086 685 high 5 min: 1.033 4613 0.188 -9622 0.1879 -9622 10 min: 1.021 1400 0.362 551 0.3622 551 15 min: 1.009 1493 0.706 1297 0.7059 1297 28 TABLE 5. Comparisons for 1 km cases, ? cm = 909 m, LOW noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.022 277 13.211 -233381 16.03 30885 10 min: 1.022 991 3.218 9275 7.760 30051 15 min: 1.007 1077 0.570 1227 4.507 1082 5? 5 min: 1.022 230 21.458 -343299 19.65 29076 10 min: 1.022 943 4.320 10599 7.835 16121 15 min: 1.008 1028 0.599 810 4.837 -492 10? 5 min: 1.021 260 0.620 9537 1.884 -13500 10 min: 1.021 956 0.487 2668 0.842 697 15 min: 1.010 979 0.669 1006 0.785 1024 TABLE 6. Comparisons for 1 km cases, ? cm = 909 m, MEDIUM noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.020 -1318 6.831 693696 8.238 613209 10 min: 1.021 1001 4.017 -39847 11.36 104116 15 min: 1.008 1521 0.952 1423 3.638 566 5? 5 min: 1.020 -1364 13.007 910863 6.816 203867 10 min: 1.021 950 3.101 58191 3.533 56500 15 min: 1.008 1472 0.800 970 2.489 6855 10? 5 min: 1.021 3381 13.075 918297 6.842 204010 10 min: 1.021 1236 3.115 58121 3.541 56510 15 min: 1.010 1416 0.799 840 2.491 6734 TABLE 7. Comparisons for 1 km cases, ? cm = 909 m, HIGH noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.015 -892 0.473 20009 2.342 -8226 10 min: 1.023 840 0.677 7081 1.273 2355 15 min: 1.006 1003 0.594 1502 0.9346 963 5? 5 min: 1.020 -1432 8.636 1743440 4.763 -157130 10 min: 1.021 1096 2.381 90619 4.373 119816 15 min: 1.009 1950 0.680 1533 3.238 930 10? 5 min: 1.021 -8401 8.756 1784948 4.787 -161501 10 min: 1.022 1509 2.386 91278 4.380 119851 15 min: 1.011 2196 0.679 1458 3.238 744 29 TABLE 8. Comparisons for 10 km cases, ? cm = 9091 m, LOW noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.022 8423 17.910 -591062 6.060 -217109 10 min: 1.025 9192 1.382 22633 0.4890 8391 15 min: 1.010 9339 0.637 10154 0.6134 9632 5? 5 min: 1.021 7948 11.251 26732 11.77 53852 10 min: 1.025 8714 5.229 14630 5.429 13736 15 min: 1.065 8845 1.755 9770 2.720 12807 10? 5 min: 1.021 7444 35.889 -1443476 12.54 551718 10 min: 1.195 8406 0.370 27736 1.721 13200 15 min: 1.028 8267 0.604 9959 1.747 8343 TABLE 9. Comparisons for 10 km cases, ? cm = 9091 m, MEDIUM noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.020 6801 0.544 -2396 0.6496 9091 10 min: 1.022 9179 0.472 19235 0.5880 15966 15 min: 1.008 9787 0.657 12817 1.050 22614 5? 5 min: 1.020 6341 4.986 -360337 14.66 1506834 10 min: 1.021 8708 2.010 21618 7.127 68631 15 min: 1.010 9294 0.780 10351 2.398 6134 10? 5 min: 1.020 5842 4.986 -360842 14.24 1383031 10 min: 1.022 8274 2.010 20967 9.143 115028 15 min: 1.016 8855 0.779 9671 2.926 4684 TABLE 10. Comparisons for 10 km cases, ? cm = 9091 m, HIGH noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.015 7211 0.380 25179 0.2834 10359 10 min: 1.023 8992 0.478 14181 0.3606 10416 15 min: 1.006 9225 0.529 10105 0.4563 9186 5? 5 min: 1.020 6301 18.581 1603925 6.635 -1426637 10 min: 1.021 8888 0.997 148312 0.8132 71679 15 min: 1.010 9791 0.758 10630 0.6790 10518 10? 5 min: 1.020 5811 18.580 1603099 6.634 -1427007 10 min: 1.021 8450 0.996 147601 0.8133 71021 15 min: 1.011 9355 0.758 10014 0.6795 9873 30 TABLE 11. Comparisons for 50 km cases, ? cm = 45455 m, LOW noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.022 44523 7.125 -255455 3.986 -64684 10 min: 1.027 45385 1.066 51432 0.4422 45428 15 min: 1.023 45659 0.890 46422 0.6646 44962 5? 5 min: 1.017 42148 76.441 1804829 2.504 59790 10 min: 1.035 42994 12.660 -20485 1.207 43754 15 min: 1.844 43146 4.399 42729 1.220 45149 10? 5 min: 1.016 39625 270.15 -5901658 3.873 46189 10 min: 1.105 40795 14.464 33030 1.976 42401 15 min: 3.202 40910 0.503 46430 1.725 40072 TABLE 12. Comparisons for 50 km cases, ? cm = 45455 m, MEDIUM noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.021 42789 5.219 -302465 13.13 648210 10 min: 1.022 45482 0.619 43857 9.663 194554 15 min: 1.008 46154 0.653 49841 4.583 44944 5? 5 min: 1.019 40517 0.534 -138322 12.87 1202841 10 min: 1.021 43138 0.696 62693 8.772 144106 15 min: 1.054 43703 0.540 48204 3.384 43042 10? 5 min: 1.019 38056 0.345 -217184 1.432 -691193 10 min: 1.022 40931 0.693 58296 0.5688 45995 15 min: 1.178 41358 0.540 44817 0.7569 44486 TABLE 13. Comparisons for 50 km cases, ? cm = 45455 m, HIGH noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.020 42834 2.775 1758469 6.549 -1435114 10 min: 1.022 45584 0.732 142976 0.8103 107429 15 min: 1.009 46644 0.518 50734 0.6709 48181 5? 5 min: 1.019 40629 3.369 2836331 19.21 3636757 10 min: 1.021 43221 0.771 169977 0.9604 77643 15 min: 1.022 44137 0.519 48170 0.8773 45485 10? 5 min: 1.019 38232 5.070 5227532 153.7 7650025 10 min: 1.022 41013 0.938 203335 39.36 -1267781 15 min: 1.058 41908 0.528 46766 31.57 30505 31 TABLE 14. Comparisons for 1 km UP cases, ? cm = -909 m, LOW noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.022 -1532 65.567 1254150 15.31 18527 10 min: 1.022 -833 5.161 -50243 7.075 26939 15 min: 1.006 -761 0.903 -1122 4.110 118 5? 5 min: 1.022 -1484 21.456 -344736 19.65 27320 10 min: 1.022 -785 4.320 8959 7.835 14476 15 min: 1.006 -712 0.600 -804 4.835 -2144 10? 5 min: 1.022 -1532 21.456 -344685 19.65 27378 10 min: 1.022 -740 4.320 9024 7.835 14542 15 min: 1.007 -688 0.600 -736 4.836 -2069 TABLE 15. Comparisons for 1 km UP cases, ? cm = -909 m, MEDIUM noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.020 -3121 3.225 654906 0.8382 108676 10 min: 1.021 -809 4.278 -40633 8.219 138718 15 min: 1.008 -318 0.943 -1744 8.659 -20563 5? 5 min: 1.020 -3074 8.911 638134 10.39 763366 10 min: 1.021 -757 3.134 33660 9.381 94157 15 min: 1.008 -269 0.833 -624 3.203 -606 10? 5 min: 1.020 -3025 8.911 638192 10.39 763419 10 min: 1.021 -709 3.134 33726 9.381 94216 15 min: 1.008 225 0.833 -624 3.203 -540 TABLE 16. Comparisons for 1 km UP cases, ? cm = -909 m, HIGH noise Batch Filter [13] Manually-tuned EKF [16] Adaptively-tuned EKF with k-biasing ? ?t RMS ? cm * (m) RMS ? cm * (m) RMS ? cm * (m) 0? 5 min: 1.020 -3194 11.369 277325 2.556 -2158 10 min: 1.021 -683 2.465 124955 2.705 204906 15 min: 1.009 -155 0.764 -1779 3.004 112275 5? 5 min: 1.020 -3149 7.018 2823610 4.762 -158748 10 min: 1.021 -637 0.728 67245 4.373 118088 15 min: 1.009 203 0.612 -1209 3.237 -1239 10? 5 min: 1.020 -3100 7.073 1424384 5.194 198245 10 min: 1.021 -593 2.486 71591 4.528 144550 15 min: 1.008 246 0.679 -901 3.265 -836 32 6. CONCLUSIONS In summary, this study generated interesting and unique results for the quick-look identification methodology presented for identifying a satellite as being a member of a TSS using an EKF. The 2 nd Stage TSS identification method used for both the manually- tuned EKF [15] and the adaptive EKF provided correct and quick identification of the tracked satellite. Both EKFs were sufficient in producing accurate results within the 15 minute time span. Where at least 15 minutes of data is processed, identification of a tethered satellite as part of a TSS can almost certainly be made. One major disadvantage to using either EKF is that the results are very sensitive to the filter tuning parameters. Specifically, the choice of the a priori covariance and process noise parameters greatly affect the accuracy of the outcome of the filter. For the manually-tuned EKF cases, both parameters were manually-tuned which contributed to the difficulty of achieving acceptable filter performance. The adaptive EKF?s filter tuning was slightly more automated with the addition of the biasing function and took advantage of using the a priori covariance found in the best manually-tuned cases. This required that only the process noise matrix be determined for the adaptively-tuned cases. Finding the appropriate biasing function to use for tuning adaptive EKF cases depends on the level of error expected during process noise; and in this study, how accurately tuned the a priori covariance was from the manually-tuned cases. Selecting the appropriate biasing function form and variation of the form can be tedious, but 33 through experience with filter tuning the appropriate function can more easily be determined. Once the proper biasing function is selected, it is more advantageous when compared to the manually-tuned EKF because the process noise is updated each time an observation is processed in the adaptive EKF, rather than remaining constant throughout the analysis when using a manually-tuned filter. Comparing the results of both EKFs to those of the batch filter, shows that a batch filter can be used to identify a tethered satellite more quickly than the EKFs can. Although the EKFs provided more accurate results than those found in the batch filter, the EKFs required more observations to do so. Several recommendations can be made for future use of EKFs in identifying a satellite as a member of a TSS. In addition to using the most accurate data available to achieve optimum results, it is highly advised that at least 15 minutes of observation data be used in the 2 nd Stage identification process. This observation data should be recorded as often as possible, since observations are processed by the adaptive EKF at each time step. With the application of the adaptive EKF in this study, satisfying results were achieved; however additional analysis can be performed to streamline the entire tuning process. An iterative tuning process can be established when an EKF is required or desired for use in any study. This can be achieved by establishing an iteration procedure for any parameters needing to be tuned, such as the a priori covariance, the biasing function form and variation or any other parameter requiring tuning. Critical output values can be monitored within defined tolerances so that the desired filter performance can be achieved. 34 An iterative tuning procedure would benefit in the use of EKFs since it would alleviate the need for a trial and error process, which has traditionally been the means for tuning an EKF. However, extensive analysis must be performed early in the study when defining the iteration details. More effort may be required in the setup of an iterative process of this nature, but the caliber of the results achieved along with the ease of the tuning process would produce a useful and effective software solution where the application of EKFs are needed. 35 REFERENCES 1. Asher, T. A., D. G. Boden, and R. J. Tegtmeyer, ?Tethered satellites: The orbit determination problem and Missile Early Warning Systems,? AIAA Paper 88-4284, AIAA/AAS Astrodynamics Conference, Minneapolis, MN, August 15-17, 1988. 2. Hoots, F. R., Roehrich, R. L., and Szebehely, V. G., ?Space Shuttle Tethered Satellite Analysis,? Directorate of Astrodynamics, Peterson AFB, CO, August 1983. 3. Qualls, C., and Cicci, D. A., ?Preliminary Orbit Determination of a Tethered Satellite,? Paper AAS 00-191, presented at the AAS/AIAA Astrodynamics Specialist Conference, Clearwater, FL, January 23-26, 2000. 4. Qualls, C., ?Preliminary Orbit Determination of a Tethered Satellite,? AU MS Thesis, 2000. 5. Kessler, S. A., and Cicci, D. A., ?Filtering Methods for the Orbit Determination of a Tethered Satellite,? The Journal of the Astronautical Sciences, Vol. 45, No. 3, July- September 1997, pp. 263-278. 6. Cicci, D. A., Lovell, T. A., and Qualls, C., ?A Filtering Method for the Identification of a Tethered Satellite,? The Journal of the Astronautical Sciences, Vol. 49, No. 2., April-June 2001, pp. 309-326. 7. Cicci, D. A., Qualls, C., and Lovell, T. A., ?A Look at Tethered Satellite Identification Using Ridge-Type Estimation Methods,? Paper AAS 99-415, presented at the AAS/AIAA Astrodynamics Specialist Conference, Girdwood, AK, August 16- 19, 1999. 8. Cochran, J. E., Jr., Cho, S., Cheng, Y-M, and Cicci, D. A., ?Dynamics and Orbit Determination of Tethered Satellite Systems,? The Journal of the Astronautical Sciences, Vol. 46, No. 2, April-June 1998, pp. 177-194. 9. Cho, S., Cochran, J. E., Jr., and Cicci, D. A., ?Modeling Tethered Satellite Systems for Detection and Orbit Determination,? The Journal of the Astronautical Sciences, Vol. 48, No. 1, January-March 2000, pp. 89-108 36 10. Cochran, J. E., Jr., Cho, S., Lovell, T. A., and Cicci, D. A., ?Evaluation of the Information Contained in the Motion of One Satellite of a Two-Satellite Tethered System,? The Journal of the Astronautical Sciences, Vol. 48, No.4, October- December 2000, pp. 477-493. 11. Cho, S., Cochran, J. E., Jr., and Cicci, D. A., ?Approximation Solutions for Tethered Satellite Motion,? AIAA Journal of Guidance, Control, and Dynamics, Vol. 24, No. 4, August 2001, pp. 746-754. 12. Cho, S., Cochran, J. E., Jr., and Cicci, D. A., ?Identification and Orbit Determination of Tethered Satellite Systems,? Applied Mathematics and Computation, Vol. 117, 2001, pp. 301-312. 13. Cicci, D. A., Cochran, J. E., Jr., Qualls, C., Lovell, T. A., ?Quick-Look Identification and Orbit Determination of a Tethered Satellite,? The Journal of the Astronautical Sciences, Vol. 50, No. 3, July-September 2002, pp. 339-353. 14. ?TiPS: Tether Physics and Survivability Satellite Experiment? web pages, Naval Center for Space Technology, http://hyperspace.nrl.navy.mil/TiPS/data.html. 15. Cicci, D. A., Volovecky, E. J., Qualls, C., ?Identification of a Tethered Satellite Using a Kalman Filter,? AIAA Paper 04-165, AIAA/AAS Astrodynamics Conference, Maui, HI, March 2004. 16. Cicci, D. A., ?An Adaptive Extended Kalman Filter for Angles-Only Tracking/Intercept Problems,? The Journal of the Astronautical Sciences, Vol. 41, No. 3, July-September 1993, pp. 411-435. 17. Swerling, P. ?First Order Propagation in a Stagewise Smoothing Procedure for Satellite Observations,? The Journal of the Astronautical Sciences, Vol. 6, 1959, pp. 46-62. 18. Kalman, R. E., and Bucy, R. S., ?New Results in Linear Filtering and Prediction Theory,? ASME Journal of Basic Engineering, Vol. 83, 1961, pp.95-108. 37 APPENDIX A: TEST CASE DATA 38 TABLE 17. Test Case Data: No Tether (? cm = 0) Noise Level f(k) ?t RMS ? cm * (m) LOW (k-1) 2 5 min. 1.948E+7 -7408104 10 min. 26720 -1.358E+9 15 min. 15512 -1.227E+8 k 5 min. 9508 -4.909E+10 10 min. 536754 -3.489E+7 15 min. 776503 -1.381E+7 1/(k+1) 20 5 min. 0.2431 -2416 10 min. 0.4431 -39 15 min. 0.6860 93 1/(k+1) 21 5 min. 0.2431 -2416 10 min. 0.4433 -39 15 min. 0.6862 92 MEDIUM (k-1) 5 min. 1.010E+8 -3.033E+9 10 min. 304318 -4.504E+9 15 min. 20177 -1.558E+9 k 5 min. 145829 -8.204E+8 10 min. 7.976E+7 -1.397E+9 15 min. 5595 -1.013E+9 1/(k+1) 20 5 min. 0.2329 -6559 10 min. 0.3631 320 15 min. 0.7086 686 1/(k+1) 21 5 min. 0.2329 -6559 10 min. 0.3631 320 15 min. 0.7086 685 HIGH (k-1) 5 min. 0.1572 258513 10 min. 0.2502 556325 15 min. 0.4262 613046 k 5 min. 0.1531 31660 10 min. 0.2779 1264131 15 min. 0.8096 473790 1/(k+1) 19 5 min. 0.1879 -9622 10 min. 0.3622 551 15 min. 0.7058 1297 1/(k+1) 20 5 min. 0.1879 -9622 10 min. 0.3622 551 15 min. 0.7059 1297 39 TABLE 18. Test Case Data: ? = 1 km (? cm = 909 m), LOW noise ? f(k) ?t RMS ? cm * (m) 0? k 2 5 min. 79246 -2.629E+7 10 min. 54037 -6.210E+9 15 min. 15664 -1.004E+9 (k-1) 5 min. 9.770 185433 10 min. 18598 -8.038E+9 15 min. 90578 -8320845 1/(k+1) 8 5 min. 11.18 63636 10 min. 5.368 12295 15 min. 3.782 1397 1/k 8 5 min. 16.03 30885 10 min. 7.760 30051 15 min. 4.507 1082 5? k 5 min. 1.582E+7 -3.289E+10 10 min. 844865 -4796600 15 min. 34161 -4.853E+8 k 2 5 min. 208974 -1.162E+8 10 min. 1.429E+7 -4.033E+8 15 min. 5381568 -2.482E+7 1/(k+1) 6 5 min. 16.25 31850 10 min. 6.801 24515 15 min. 4.047 -766 1/(k+1) 8 5 min. 19.65 29076 10 min. 7.835 16121 15 min. 4.837 -492 10? (k-1) 5 min. 41970 -1.039E+10 10 min. 3316391 -8.432E+9 15 min. 1190777 -1.641E+8 k 5 min. 0.9509 -3578792 10 min. 9710738 -1.894E+8 15 min. 860739 -3.679E+7 1/(k+1) 19 5 min. 2.055 -14873 10 min. 2.096 -662 15 min. 0.697 1027 1/(k+1) 18 5 min. 1.884 -13500 10 min. 0.842 697 15 min. 0.785 1024 40 TABLE 19. Test Case Data: ? = 1 km (? cm = 909 m), MEDIUM noise ? f(k) ?t RMS ? cm * (m) 0? k 5 min. 103.1 1818062 10 min. 0.4188 46684 15 min. 105657 -4.427E+8 (k-1) 5 min. 22.26 -2.336E+7 10 min. 2.181 -84327 15 min. 5230 -2.208E+8 1/k 6 5 min. 4.723 161874 10 min. 4.876 74596 15 min. 3.059 506 1/(k+1) 13 5 min. 8.238 613209 10 min. 11.36 104116 15 min. 3.638 566 5? k 5 min. 5.621 250947 10 min. 3.045 325449 15 min. 28066 -6080792 (k-1) 5 min. 5.628 358120 10 min. 2.998 449543 15 min. 1.092 486845 1/(k+1) 5 5 min. 7.538 -225738 10 min. 3.755 44240 15 min. 2.625 7176 1/(k+1) 4 5 min. 6.816 203867 10 min. 3.533 56500 15 min. 2.489 6855 10? (k-1) 5 min. 5.629 250800 10 min. 3.046 324929 15 min. 6938 -7375386 k 5 min. 5.638 358068 10 min. 3.001 448444 15 min. 1.139 486206 1/k 7 5 min. 6.662 121095 10 min. 3.468 44970 15 min. 2.491 7993 1/(k+1) 4 5 min. 6.842 204010 10 min. 3.541 56510 15 min. 2.491 6734 41 TABLE 20. Test Case Data: ? = 1 km (? cm = 909 m), HIGH noise ? f(k) ?t RMS ? cm * (m) 0? k 2 5 min. 215727 -1.252E+8 10 min. 60705 -3.087E+8 15 min. 47348 -1.666E+8 (k-1) 5 min. 2.286E+7 -3.890E+8 10 min. 40120 -4.000E+8 15 min. 17875 -1.082E+8 1/k 16 5 min. 1.789 -3662 10 min. 1.131 2921 15 min. 0.7322 891 1/k 20 5 min. 2.342 -8226 10 min. 1.273 2355 15 min. 0.9346 963 5? k 5 min. 4.319 418352 10 min. 4.249 196265 15 min. 78.05 -6102592 (k-1) 5 min. 4.358 384777 10 min. 4.206 202737 15 min. 30.01 -724931 1/k 5 5 min. 4.748 -259218 10 min. 4.354 113148 15 min. 3.245 3201 1/(k+1) 3 5 min. 4.763 -157130 10 min. 4.373 119816 15 min. 3.238 930 10? k 5 min. 4.331 417880 10 min. 4.253 196247 15 min. 71.67 -6049962 (k-1) 5 min. 4.370 376580 10 min. 4.213 202907 15 min. 29.36 -654351 1/k 5 5 min. 4.772 -263485 10 min. 4.362 113210 15 min. 3.245 3014 1/(k+1) 3 5 min. 4.787 -161501 10 min. 4.380 119851 15 min. 3.238 744 42 TABLE 21. Test Case Data: ? = 10 km (? cm = 9091 m), LOW noise ? f(k) ?t RMS ? cm * (m) 0? k 2 5 min. 9333 -1.945E+9 10 min. 16554 -4.025E+7 15 min. 16904 -5.611E+9 (k-1) 2 5 min. 10.79 187180 10 min. 4.014E+7 -6.773E+7 15 min. 9369 -3.998E+8 1/k 14 5 min. 34.75 -393739 10 min. 5.419 17539 15 min. 4.153 9784 1/k 9 5 min. 6.060 -217109 10 min. 0.4890 8391 15 min. 0.6134 9632 5? k 2 5 min. 1384498 -1.951E+7 10 min. 205806 -7.384E+8 15 min. 29305 -4.252E+9 (k-1) 2 5 min. 9.251 174365 10 min. 27515 -7.132E+7 15 min. 38093 -2.057E+9 1/k 6 5 min. 13.29 42465 10 min. 6.268 15103 15 min. 3.298 4459 1/k 5 5 min. 11.77 53852 10 min. 5.429 13736 15 min. 2.720 12807 10? k 5 min. 17918 -7.760E+8 10 min. 25090 -1.508E+7 15 min. 211137 -2.656E+9 (k-1) 5 min. 46356 -1.644E+8 10 min. 556165 -3.057E+8 15 min. 24391 -1.139E+7 1/(k+1) 9 5 min. 45.71 -1787109 10 min. 1.496 15401 15 min. 1.254 10057 1/(k+1) 8 5 min. 12.54 551718 10 min. 1.721 13200 15 min. 1.747 8343 43 TABLE 22. Test Case Data: ? = 10 km (? cm = 9091 m), MEDIUM noise ? f(k) ?t RMS ? cm * (m) 0? k 5 min. 17853 -2.590E+11 10 min. 184594 -1.598E+8 15 min. 10008 -4.272E+8 k 2 5 min. 1.557E+7 -3.012E+9 10 min. 157668 -1.627E+8 15 min. 107549 -4732578 1/(k+1) 12 5 min. 1.006 65842 10 min. 0.6383 15992 15 min. 0.8980 10545 1/(k+1) 6 5 min. 0.6496 9091 10 min. 0.5880 15966 15 min. 1.050 22614 5? (k-1) 2 5 min. 3.197 256493 10 min. 17718 -8.348E+8 15 min. 5936 -3.960E+9 (k-1) 5 min. 3.191 254015 10 min. 1.979 802715 15 min. 44311 -3.185E+8 1/(k+1) 13 5 min. 14.24 1383702 10 min. 9.139 115611 15 min. 2.928 5340 1/k 21 5 min. 14.66 1506834 10 min. 7.127 68631 15 min. 2.398 6134 10? k 5 min. 3.158 362342 10 min. 2.540 836567 15 min. 449341 -2.630E+9 (k-1) 5 min. 3.191 253351 10 min. 2.109 1030993 15 min. 9.151 -1709413 1/k 20 5 min. 15.07 1567105 10 min. 4.800 6979 15 min. 2.038 14877 1/(k+1) 13 5 min. 14.24 1383031 10 min. 9.143 115028 15 min. 2.926 4684 44 TABLE 23. Test Case Data: ? = 10 km (? cm = 9091 m), HIGH noise ? f(k) ?t RMS ? cm * (m) 0? (k-1) 2 5 min. 86174 -2.786E+11 10 min. 222068 -3.655E+8 15 min. 520551 -1.107E+9 k 2 5 min. 74891 -2.003E+8 10 min. 46188 -7.888E+7 15 min. 21370 -1.053E+8 1/(k+1) 21 5 min. 0.2406 11202 10 min. 0.4266 9307 15 min. 0.7085 8814 1/k 15 5 min. 0.2834 10359 10 min. 0.3606 10416 15 min. 0.4563 9186 5? k 5 min. 4.735 513295 10 min. 2.552 250591 15 min. 3.387 545368 (k-1) 5 min. 4.336 79536 10 min. 2.047 375226 15 min. 1.050 343727 1/(k+1) 6 5 min. 9.441 -800925 10 min. 0.8150 54226 15 min. 0.7375 11919 1/k 10 5 min. 6.635 -1426637 10 min. 0.8132 71679 15 min. 0.6790 10518 10? k 5 min. 4.735 512591 10 min. 2.553 249809 15 min. 3.558 564122 (k-1) 5 min. 4.336 78918 10 min. 2.048 374348 15 min. 1.074 340145 1/(k+1) 6 5 min. 9.441 -801358 10 min. 0.8149 53564 15 min. 0.7352 11274 1/k 10 5 min. 6.634 -1427007 10 min. 0.8133 71021 15 min. 0.6795 9873 45 TABLE 24. Test Case Data: ? = 50 km (? cm = 45455 m), LOW noise ? f(k) ?t RMS ? cm * (m) 0? (k-1) 2 5 min. 5.395 231609 10 min. 801420 -4.4474E+11 15 min. 1.693E+7 -2.602E+8 k 2 5 min. 10994 -5502491 10 min. 16017 -1.009E+9 15 min. 264641 -3.167E+7 1/k 15 5 min. 2.604 57065 10 min. 0.4719 44662 15 min. 0.8280 45715 1/k 16 5 min. 3.986 -64684 10 min. 0.4422 45428 15 min. 0.6646 44962 5? k 2 5 min. 1.446E+8 -2.454E+11 10 min. 193956 -3.943E+9 15 min. 2604 -7301015 (k-1) 2 5 min. 5.396 228756 10 min. 42646 -3.877E+8 15 min. 51900 -5525326 1/k 7 5 min. 2.438 64344 10 min. 1.186 45427 15 min. 1.186 49021 1/k 8 5 min. 2.504 59790 10 min. 1.207 43754 15 min. 1.220 45149 10? k 2 5 min. 3.548 447265 10 min. 8726927 -2.807E+8 15 min. 3381018 -2.204E+8 (k-1) 2 5 min. 864520 -1.155E+7 10 min. 4.396E+7 -3.929E+8 15 min. 76714 -4.701E+7 1/k 6 5 min. 0.8651 50501 10 min. 0.7711 47243 15 min. 0.5045 47104 1/k 10 5 min. 3.873 46189 10 min. 1.976 42401 15 min. 1.725 40072 46 TABLE 25. Test Case Data: ? = 50 km (? cm = 45455 m), MEDIUM noise ? f(k) ?t RMS ? cm * (m) 0? k 5 min. 3.175 404529 10 min. 1.878 225780 15 min. 22242 -1.102E+7 k 2 5 min. 2.675 675655 10 min. 1315 -7.271E+7 15 min. 2152 -9912064 1/(k+1) 13 5 min. 10.448 753723 10 min. 8.351 147276 15 min. 3.649 49440 1/(k+1) 14 5 min. 13.13 648210 10 min. 9.663 194554 15 min. 4.583 44944 5? (k-1) 5 min. 3.192 291176 10 min. 1.655 798882 15 min. 2.886 892367 k 5 min. 3.175 401593 10 min. 1.569 690194 15 min. 6.579 469003 1/k 20 5 min. 12.86 1324856 10 min. 5.037 49065 15 min. 2.499 52389 1/(k+1) 13 5 min. 12.87 1202841 10 min. 8.772 144106 15 min. 3.384 43042 10? (k-1) 2 5 min. 3.184 284810 10 min. 1.354E+7 -1.763E+9 15 min. 10182 -7.710E+7 k 2 5 min. 2.455 726058 10 min. 515473 -7.197E+7 15 min. 6302 -1.803E+7 1/(k+1) 11 5 min. 44.32 3140658 10 min. 2.368 -77223 15 min. 1.104 46131 1/(k+1) 7 5 min. 1.432 -691193 10 min. 0.5688 45995 15 min. 0.7569 44486 47 TABLE 26. Test Case Data: ? = 50 km (? cm = 45455 m), HIGH noise ? f(k) ?t RMS ? cm * (m) 0? (k-1) 5 min. 4.350 117136 10 min. 2.048 439609 15 min. 51874 -3881839 k 5 min. 4.750 556965 10 min. 2.553 300542 15 min. 2.055 665919 1/k 14 5 min. 41.33 3.437E+7 10 min. 4.218 -115323 15 min. 1.140 48648 1/k 10 5 min. 6.549 -1435114 10 min. 0.8103 107429 15 min. 0.6709 48181 5? k 5 min. 3.583 400650 10 min. 665978 -2.094E+8 15 min. 3679 -9128196 (k-1) 5 min. 4.119 291700 10 min. 1.945 529531 15 min. 1.295 356237 1/k 13 5 min. 15.90 2601556 10 min. 0.7851 95354 15 min. 0.9108 46854 1/(k+1) 8 5 min. 19.21 3636757 10 min. 0.9604 77643 15 min. 0.8773 45485 10? (k-1) 2 5 min. 19.54 -2344955 10 min. 3.616 231902 15 min. 26125 -4.056E+8 (k-1) 5 min. 8.483 -1325513 10 min. 34868 -1.097E+9 15 min. 2647 -8848180 1/k 20 5 min. 152.9 1.221E+7 10 min. 40.52 -1325599 15 min. 33.98 20924 1/(k+1) 12 5 min. 153.7 7650025 10 min. 39.36 -1267781 15 min. 31.57 30505 48 TABLE 27. Test Case Data: ? = 1 km up (? cm = -909 m), LOW noise ? f(k) ?t RMS ? cm * (m) 0? k 5 min. 19052 -2.120E+8 10 min. 34387 -3.174E+8 15 min. 21728 -9.479E+8 (k-1) 5 min. 12.19 186597 10 min. 1.384E+7 -2.584E+8 15 min. 50100 -5.576E+8 1/k 15 5 min. 61.00 -255588 10 min. 24.29 33578 15 min. 14.07 -2049 1/(k+1) 5 5 min. 15.31 18527 10 min. 7.075 26939 15 min. 4.110 118 5? (k-1) 5 min. 12.74 187621 10 min. 8.131E+7 -1.886E+9 15 min. 38714 -2.556E+8 (k-1) 2 5 min. 11.79 177763 10 min. 36909 -4.148E+7 15 min. 1849706 -5546073 1/(k+1) 6 5 min. 16.25 30146 10 min. 6.801 22862 15 min. 4.044 -2528 1/(k+1) 8 5 min. 19.65 27320 10 min. 7.835 14476 15 min. 4.835 -2144 10? k 2 5 min. 174303 -5196165 10 min. 10543 -3.563E+11 15 min. 43525 -1.850E+9 (k-1) 2 5 min. 11.79 177828 10 min. 82339 -7.497E+8 15 min. 407830 -2.402E+8 1/(k+1) 6 5 min. 16.25 30206 10 min. 6.801 22928 15 min. 4.044 -2450 1/(k+1) 8 5 min. 19.65 27378 10 min. 7.835 14542 15 min. 4.836 -2069 49 TABLE 28. Test Case Data: ? = 1 km up (? cm = -909 m), MEDIUM noise ? f(k) ?t RMS ? cm * (m) 0? k 5 min. 4.040 1711062 10 min. 332479 -5.537E+8 15 min. 9223 -2.146E+8 (k-1) 2 5 min. 9463021 -3.204E+10 10 min. 4138 -1.782E+9 15 min. 762 -5482947 1/k 12 5 min. 4.414 426439 10 min. 15.24 106843 15 min. 53.03 -54076 1/(k-1) 3 5 min. 0.8382 108676 10 min. 8.219 138718 15 min. 8.659 -20563 5? (k-1) 5 min. 6.645 252247 10 min. 3.381 232577 15 min. 59352 -1.100E+7 k 5 min. 6.614 358337 10 min. 3.326 337699 15 min. 1.071 492506 1/(k+1) 14 5 min. 14.62 699033 10 min. 12.01 164880 15 min. 4.301 -9869 1/(k+1) 13 5 min. 10.39 763366 10 min. 9.381 94157 15 min. 3.203 -606 10? k 2 5 min. 6.614 493022 10 min. 766.6 -1.042E+8 15 min. 153437 -6506473 k 5 min. 6.614 358405 10 min. 3.326 337777 15 min. 1.065 492748 1/(k+1) 4 5 min. 6.934 119266 10 min. 3.643 56365 15 min. 2.568 4557 1/(k+1) 13 5 min. 10.39 763419 10 min. 9.381 94216 15 min. 3.203 -540 50 TABLE 29. Test Case Data: ? = 1 km up (? cm = -909 m), HIGH noise ? f(k) ?t RMS ? cm * (m) 0? (k-1) 5 min. 3.266 1274895 10 min. 2.674 1734528 15 min. 3874 -9.433E+7 k 5 min. 2.351 633480 10 min. 5228083 -2.215E+9 15 min. 6232 -5.042E+7 1/(k-1) 4 5 min. 2.042 -8681 10 min. 2.923 191447 15 min. 3.063 101456 1/(k-1) 2 5 min. 2.556 -2158 10 min. 2.705 204906 15 min. 3.004 112275 5? k 5 min. 4.319 416463 10 min. 4.250 194422 15 min. 208.4 -6786659 (k-1) 5 min. 4.358 382826 10 min. 4.207 200965 15 min. 31.04 -846237 1/k 4 5 min. 4.599 342338 10 min. 4.329 140397 15 min. 3.207 991 1/(k+1) 3 5 min. 4.762 -158748 10 min. 4.373 118088 15 min. 3.237 -1239 10? k 5 min. 4.501 445663 10 min. 4.313 195575 15 min. 44.87 -5428452 (k-1) 5 min. 4.601 386415 10 min. 4.246 202680 15 min. 29.96 -702344 1/k 5 5 min. 5.374 -307660 10 min. 4.571 115868 15 min. 3.322 -796 1/k 4 5 min. 5.194 198245 10 min. 4.528 144550 15 min. 3.265 -836