Spacecraft Orbit Anomaly Representation Using Thrust-Fourier-Coefficients with Orbit Determination Toolbox Hyun Chul Ko and Daniel J. Scheeres University of Colorado - Boulder, Boulder, CO, USA ABSTRACT A new way of describing a spacecraft orbit anomaly using an event representation technique with Thrust-Fourier-Coefficients (TFCs) is introduced and its application in the satellite orbit determination (OD) is studied. Given unconnected two orbit states across a spacecraft anomaly, TFC representation of unknown events allows us to obtain a unique control law that can generate the desired secular behavior for a given orbit anomaly. This event representation technique with TFC is able to capture the fundamental element of unknown dynamics for a given anomaly and to connect two separate orbit states across an unknown event. The represented orbit dynamics using TFCs can be used to improve the quality of orbit fits through an anomaly period and therefore helps to obtain a good OD solution after the anomaly. By directly appending TFCs and the rigorously represented anomaly dynamics to the pre-built least square filter in Orbit Determination Toolbox (ODTBX), the modified filter with TFCs enables us to blend postanomaly tracking data to improve the post-event orbit solution in the absence of observation data during the unknown event period. Satellite OD solutions using ODTBX with TFCs for different anomaly cases are illustrated to demonstrate the efficiency and versatility of the TFC event representation technique. Also, the estimated unknown thrust accelerations from ODTBX with TFCs are compared with the true perturbing acceleration.. INTRODUCTION An essential part of safe space operation is related to tracking satellites and maintaining orbit determination (OD) of them by estimating the satellite state and its uncertainty throughout their operations. However, spacecrafts often deviate from their nominal trajectories when they undergo space events such as encountering external forces or sudden changes in the space environment. If such an event is unpredicted, it may cause a lost track of satellite due to abrupt changes of orbit states, which is referred as spacecraft orbit anomaly in this paper. Generally, recovering of a lost space object after an unknown space event using post-event tracking data is a manual job and requires competent skill as well as properly trained experts on the matter []. Therefore, developing an efficient way to represent an unknown space event could play an essential role in analyzing an orbit anomaly or estimating a post-event trajectory of a space object. Maintaining OD across an orbit anomaly period requires to model unknown perturbing accelerations related to a space event. A conventional method used at the time of encountering an unknown event is to use the least square filter over a span following the event []. However, OD across any space event is generally not performed in the least square method due to its complexities introduced in a computation of the inverse of the information matrix [3]. One of the common methods to perform OD across an unknown event is to add process noises to the sequential estimation filter in order to compensate unknown perturbations (e.g., State Noise Compensation [3], Dynamic Model Compensation [3], optimal sequential filter with smoother in Orbit Determination Tool Kit (ODTK) []). This approach is usually a timeconsuming process to find valid parameters to compensate for unknown dynamics for an orbit anomaly. Even if the OD filter with process noise can estimate an unknown acceleration, this approach does not consider any fundamental element of the unknown orbit dynamics for a given anomaly. Moreover, this method requires observation throughout the orbit anomaly period, which often cannot be achieved in a data-starved space surveillance environment. To deal with an orbit anomaly in data sparse environments, new approaches have been introduced to detect and reconstruct unknown space events by using minimum fuel cost functions [], or applying optimal control performance metrics [6]
[7]. However, the perturbing acceleration related to a space event may not be always optimal, specially for an orbit anomaly caused by thrust malfunctions, or debris hittings. In this paper, we present an alternative way to maintain OD across an unknown space event using Thrust-Fourier- Coefficients (TFCs) in data sparse environments. Hudson and Scheeres used Fourier series representation of the thrust components in order to explain perturbing motion of orbital dynamics using averaging technique [8] [9]. Ko and Scheeres adapted this approach to efficiently represent the secular effect of any perturbation related to a space event with a selected minimum set of 6 TFCs [], and have shown that the chosen TFCs can be rigorously estimated as a part of augmented state in the batch filter in order to tie together two disparate states []. This paper uses this event representation technique with 6 TFCs and adapts it into the least square filter available in Orbit Determination Toolbox (ODTBX) to show the versatility of our approach. We chose to use the ODTK least square filter because it is a pre-built MATLAB based software package that can be modified easily. The modified least square filter with TFCs is able to provide valid post-event OD solutions as well as perturbation estimations for orbit anomalies. Section summarizes the event representation approach with TFCs and Section 3 outlines the modified least square OD algorithm in ODTBX. Section presents several orbit anomaly simulations in which the modified filter is implemented to provide OD solutions, and Section concludes with a summarization.. EVENT REPRESENTATION WITH THRUST-FOURIER-COEFFICIENTS This section is intended to be a quick overview of how to represent an unknown space event with TFCs. The event representation with TFCs is a mathematical model whose purpose is to account for perturbing motions of a satellite. Using the perturbing acceleration represented as a Fourier series expansion in eccentric anomaly, the perturbation in each direction (radial, R; along-track, S; cross-track, W) can be expressed in terms of TFCs (α k, β k ) [] : U R,S,W = k= [ ] α R,S,W k cos ke + β R,S,W k sin ke () Substituting the Fourier series representation of perturbing acceleration components into the Gauss equations, the averaged dynamics equations were found to be a function of only TFCs [8]. To achieve a given orbital transfer in the 6-dimensional orbit space, the most efficient set of 6 TFCs, which is referred as the essential TFC set, is selected to make a fast assessment of any perturbing acceleration []: c ess = [α R, α S, α S, β S, α W, β W ] The selected essential set consists of six TFCs, which are zero- or first-order terms: one from the radial direction, two from the cross-track direction, and three from the along-track direction of perturbation as described in Fig.. Any U = S S α + S α cos E + S β sin E U = W W α cos E + W β sin E α U = R R Figure : Essential TFC set spacecraft orbit anomaly transitioning between two arbitrary orbital states can be represented as an equivalent orbital Available at: http://opensource.gsfc.nasa.gov/projects/odtbx/.
transfer and its required controlling acceleration can be obtained with this TFC event representation which provides an analytical solution of control profile for an unknown event. The advantage of representing a space event with this essential TFC set is able to obtain a fundamental basis of unknown perturbing accelerations for a given spacecraft anomaly with less computational requirements. Also, any orbital anomaly can be easily represented by this essential TFC set that rigorously represents a unique control profile to connect two separated states across an unknown event without requiring observation data during the event. A priori information and post-event measurement data are only requirements to estimate the essential TFCs. Depends on the control profile reconstructed from the estimated TFC values, the perturbing acceleration can be analyzed to determine the nature of the orbit anomaly and possible characterization of it. 3. MODIFYING THE LEAST SQUARE FILTER IN ODTBX WITH TFC SET This section highlights how the least square filter in ODTBX can be modified with TFCs to maintain OD solution across unknown space events. It can be easily done by appending TFCs and the represented dynamics to the least square formulation in ODTBX to estimate unknown perturbing accelerations as well as the orbit state. The modified state vector X becomes -dimensional vector (time-dependent satellite position and velocity, 6 constant TFCs) that can completely characterize the system. By representing an orbit anomaly with the essential TFCs, the governing equations of motion can be rewritten as follows : d dt #» X(t) = F #» ( X, #» t) + B U( X, #» t) () #» #»ẋ x #»ẋ 3 3 c ess = µ #» x #» x 3 6 + I 3 3 6 3 U( #» X, t) (3) where #» F ( X, t) are the known dynamics and U( #» X, t) is the represented unknown perturbing acceleration which is as follows : U = α R ˆr + (αs + α S cos E + β S sin E)ŝ + (α W cos E + β W sin E)ŵ () where ˆr = r r, ŵ = r v r v, ŝ = ŵ ˆr. Along with the augmented state and represented dynamics, the state Jacobian as well as the measurement Jacobian matrices have to be modified to complete the least square algorithm : dφ(t, t ) dt = #» X [ F ( #» X, t) + B U( #» X, t) ] = F ( #» X, t) #» X = [ F ( #» X, t) #» X #» X(t, X #» ) X #» + B U( #» X #» + B U( #» X, t) X #» () #» X(t, X #» ) X #» (6) ] X, t) Φ(t, t ) = A Φ(t, t ) (7) H = H Φ(t, t ) = G(#» X, t) #» X Φ(t, t ) (8) in which Φ(t, t ) is the State Transition Matrix (STM) and G( #» X, t) is the measurement equation. All the modifications associated with the TFC event representation are shown at the flow chart in Fig., which is based on the general batch processing algorithm [3]. This modified least square filter simply takes a priori information from the pre-event epoch, integrates reference trajectory and STM across the period of the unknown event according to the anomaly representation with TFCs. Then, the filter takes in a post-event observation, compares it with a computed measurement using the propagated state with the represented dynamics, and adjusts the TFCs and the initial state until the event representation connects all the post-event measurements with a priori state. It provides the least square solution that minimizes the sum of the squares of errors and the best estimate is determined by repeating this iteration
Step. Initialization for iteration a. Set i =, t i = t, X(t #» i ) = X #», Φ(t i, t ) = Φ(t, t ) = I b. Set Λ = P and N = P #» x where P is an initial covariance matrix and #» x = Step. Read next observation and perform Integration & Accumulation(until t f ) a. Integrate to find #» X(t i ) : #» X(t) = F ( X, #» t) + B U( X, #» t) b. Integrate to find Φ(t i, t ) : Φ(t, t ) = A Φ(t, t ), A = c. Accumulate current observation Λ = Λ + Hi T R i H i, N = N + Hi T R i H i = H i Φ(t i, t ), [ F ( #» X,t) #» y i = Y #» i G( X #» i, t i ), Hi = G( X,t #» i) X #» where Y #» is observation and G( X #» i, t i ) is computed measurement y i #» X + B U( #» X,t) X #» ] Step 3. Solve normal equation and check the observation residual RMS a. Solve the normal equation : Λˆx = N b. RMS= ( l k= ˆɛT i R i ˆɛ i /m) /, ˆɛ i = #» y i H iˆx Has process converged? No Go to Step a. Update #» X with #» X + ˆx b. Shift #» x with #» x ˆx c. Use the original value of P Yes Step. Stop and illustrate the results #» a. Obtain OD solution across a space event : X = X #» + ˆx, P = Λ b. Reconstructing unknown perturbation using the estimated TFCs : U = α R ˆr + (α S + α S cos E + β S sin E)ŝ + (α W cos E + β W sin E)ŵ Figure : Flow chart for the modified least square algorithm (blue : modified part) until the estimation process has converged. To check the convergence, the root mean square (RMS) of observation residuals is computed [3]: ( l / RMS = ˆɛ T i R i ˆɛ i /m) (9) R i k= m = l p : the total number of observation = W : weighting matrix ˆɛ i = #» y i H iˆx () where l, p are the number of observation data and the dimension of error vector respectively. When the RMS no longer changes, the best estimate of ˆx is assumed to be converged. Once the process has converged, the ODTBX illustrates the results using various graphical presentations including reconstruction of a control profile.
. SIMULATIONS AND RESULTS In order to validate the modified least square filter with the TFC event representation, several simulated orbit anomaly cases are analyzed with the filter. All the simulations are performed on a low-earth orbiting (LEO) satellite and a typical initial condition is chosen as Table. To check the performance of the modified filter in a simple way, the Table : Initial state of LEO satellite h e i(deg) Ω(deg) ω(deg) ν(deg) 3. 8 3 h : altitude at perigee; Earth Radius : 6378.37km [3] two-body dynamics under the uniform gravity field is assumed to be the only known dynamics that exist between a spacecraft and the Earth. There are two possible kinds of perturbing acceleration for an orbit anomaly []. One is an impulsive perturbation like any collision with other space object, structural break up, or an explosion. The other is a relatively small continuous perturbation such as a low thrust malfunction or significant environmental change. Whether it is an impulsive or a continuous perturbation, the unknown acceleration has to be modeled to be compensated in the OD process. In this paper, 6 different cases of orbit anomaly are tested with the modified filter, which are shown in Table. Case -3 are for the continuous low thrust malfunction cases on each direction. Each malfunction starts at Table : Perturbations for different orbit anomaly cases case type V (m/s) Direction Continuous Along-track Continuous Radial 3 Continuous Cross-track Impulsive Along-track Impulsive Radial 6 Impulsive Cross-track minutes after the last pre-event measurement and ends at minutes before the first post-event measurement. Case -6 are for the orbit anomalies related to an impulsive V burn on each direction. Suppose all ground stations lose tracking of a satellite as soon as this impulsive event happens. After one orbital period (.88 hour) following the event, they are able to take measurements every minute for next 3 hours. To start the algorithm, the filter needs an initial state estimate, an associated initial state covariance, and an initial TFC estimate. Preliminary OD was performed to obtain the initial data by filtering a pre-event satellite tracking data (range and range-rate for 3 orbital periods). As a result, a relatively accurate a priori information (σ position = m, σ velocity = cm/s) was obtained and used as a priori information in our simulations. For the initial guess for TFCs, the modified filter can simply start with zeros without tuning those coefficients. With a priori information and the true event dynamics, the measurement generating function built in ODTBX produces post-event measurements. For the post-event observation, 3 ground tracking stations are selected from the ground station list in ODTBX and are used to generate range and range-rate data from true position and velocity vectors at every minute. The sensor measurement error (m, m/sec) is imitated by adding white Gaussian noises to the truth and it is assumed that there is no observation available during the unknown space event. To compare how an unmodified least square filter perform over an orbit anomaly, the original least square filter in ODTBX is processed through the unknown event of case without representing the event and the result is shown in Fig. 3. Without modeling the orbit anomaly, the filter delivers large state estimation errors that stay outside of the 3 sigma uncertainty boundaries which are relatively too small. It shows that the original least square filter fails to maintain OD across an orbit anomaly without compensating the perturbing effect. By modifying the least square filter with the TFC event representation, OD was performed on the same case and the result is shown in Fig.. The figure describes the error of the state estimate from the true trajectory as a red line and 3 sigma covariance boundary of the estimate as a green line. The modified filter successfully process all the measurements with a priori information, and the state errors as well as the measurement errors are within uncertainty boundaries. All the results on various cases are consistent with the expected behavior that the estimated state values are within confidence intervals and the filter
x x x 3 3 x x x 3 3 Figure 3: OD result (state errors) with the unmodified filter on case x x 3 x... x 3. 3 range. Measurement Error x x x 6 x... 3 3.. x... 3 3.. x 3. 3 rangerate x 3 Figure : OD result (state, measurement errors) with modified filter on case
provides a converged solution with the TFC event representation. For the impulsive perturbation case, Fig. illustrates the OD solution with the modified filter on case 6. x x 3 x... x 3. 3 range. Measurement Error x x x 6 x... 3 3.. x... 3 3.. x 3. 3 rangerate x 3 Figure : OD result (state, measurement errors) with modified filter on case 6 The performance of the modified least square filter is also validated through use of 3 Monte Carlo simulations. The linearly propagated covariance results are compared to confidence intervals associated with the Monte Carlo analysis available in ODTBX. Each Monte Carlo run in ODTBX generates random deviations from the reference as initial conditions, and then it integrates each deviated case and uses this as truth for measurement generation and estimation in OD simulation []. Each run delivers the time series of estimation errors and residuals for each case. Figure. 6 shows the results of 3 Monte Carlo runs on case, where red dots are individual actual errors while blue line indicates the mean error of ensembles from 3 Monte Carlo runs. In each subfigure, cyan bands show -3 sigma confidence intervals of the ensemble standard deviations and the green line displays mean value of 3 sigma uncertainty boundaries from 3 OD solutions. These uncertainty boundaries from the Monte Carlo runs with the modified filter enclose state and measurement errors. Thus, it confirms that the OD solutions from the modified filter are consistent and the modified least square filter is able to maintain OD across an unknown space event. Utilizing a priori information and an event representation of an anomaly, the modified filter is able to provide valid orbit states and uncertainty information without measurement data during the event. This OD solution can be used to improve the accuracy of orbit state right after an anomaly by making further observation after the event. The modified filter also provides a representation of an unknown acceleration by constructing a control profile
. x x x 3 x state error e ensemble mean of e ensemble to 3 STD of e mean of 3 sigmas from filter.. 3 3.. 3 x x. 3 3..... 3 3..... 3 3..... 3 3... x x6 x. dy.. 3 3.. measurement error dy ensemble mean of dy ensemble to 3 STD of dy mean of 3 sigmas from filter.. dy x3. 3 x x. x... 3 3. Figure 6: OD results from 3 Monte Carlo runs on case
with Eq. (). Figure. 7 compares the reconstructed perturbing acceleration components to the true ones for the case, 3 and. For those anomalies associated with continuous perturbations, the reconstructed accelerations are found to a R (m/s ). True Represented... a R (m/s ). True Represented... a R (m/s ) x 3 True Represented.. a S (m/s ).... a S (m/s ).... a S (m/s ) x 3.. a W (m/s )... time(hr). a W (m/s )... time(hr). a W (m/s ) x 3.. time(hr) (a) Case (b) Case 3 (c) Case Figure 7: Represented vs True acceleration components in body frame be the same order of magnitude as the true perturbations. For the impulsive cases, the modeled thrust acceleration is spread over the whole period of an anomaly, therefore the shape of the recovered control profile is different from the true one. Although this filter does not recover the actual perturbing acceleration precisely for an orbit anomaly, yet it provides insight into the control effort for the unknown event. The control effort ( V ) can be estimated by integrating the thrust acceleration profile over the time interval, which still can be used to quantify the magnitude of an actual perturbation. The represented perturbing acceleration components for different cases are shown in Table. 3. Even Table 3: Estimated V from the filter Continuous Impulsive Direction case case case 3 case case case 6 Along-track 99..3.3. Radial...9.9 Cross-track.. * True V : m/s for continuous, m/s for impulsive though the computed V s do not exactly match with true values, it still provides an estimate of controlling effort, which can be used to bound the control effort necessary for linking two separate states across an unknown event.. CONCLUSION Our TFC event representation is a very simple and effective way to represent an unknown space event. This paper applied this simple approach to the least square filter in Orbit Determination Toolbox (ODTBX) using the essential Thrust-Fourier-Coefficient (TFC) set. Case study with simulated tracking data shows that the modified least square filter with TFCs allows one to make use of previous orbit information in order to maintain orbit determination across an unknown event as well as to provide insight into the unknown accelerations leading to a given orbit anomaly. This is particularly valuable for the case with no measurement available during an orbit anomaly period and could allow ground station operators to obtain a valid orbit trajectory right after an unknown event. Also, with the estimated thrust
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