Initial Relative Orbit Determination Using Stereoscopic Imaging and Gaussian Mixture Models

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1 SSC3-VIII-6 Initial Relative Orbit Determination Using Stereoscopic Imaging and Gaussian Mixture Models Keith LeGrand Missouri University of Science and Technology 4 W. 3th St., Rolla, MO; (573) kal7cd@mst.edu Faculty Advisors: Henry Pernicka, Kyle DeMars Missouri University of Science and Technology ABSTRACT The unobservability of space-based angles-only orbit determination can be mitigated by the inclusion of angle measurements from a second optical sensor fixed at a known baseline on the observing spacecraft. Previous approaches to the problem have used these stereoscopic angles to triangulate the position of a second satellite at a given time step. However, due to the nonlinearity of stereo triangulation, zero-mean Gaussian noise of these measurements cannot be assumed. This work investigates a modified approach in which the uncertainty of both angle measurements is used to bound a region for all possible positions of the second satellite. A Gaussian mixture that represents uniform uncertainty across the bounded region for the position of the second object is constructed at two initial time steps. Linkage of the Gaussian mixtures is performed using a relative Lambert solver in order to formulate a full state probability density function that can be further refined through processing subsequent measurement data in a Bayesian framework. INTRODUCTION Due to the increasing number of untracked space objects in Earth orbit, both the military and scientific communities have identified the need for improved, persistent Space Situational Awareness (SSA). By definition, advancements in SSA depend on the ability to determine a Resident Space Object s (RSO) current state, and from that, predict its future trajectory. Resident space objects encompass all Earth-orbiting objects, whether active spacecraft or space debris. One approach to RSO orbit determination involves the use of space-based ranging sensors. This approach typically requires the use of RADAR, LIDAR, or optical sensors to measure the range of a nearby RSO with respect to the observing spacecraft. When the nearby object is a non-cooperative spacecraft, the problem becomes much more complicated, as there is likely little or no a priori knowledge of the resident spacecraft. Furthermore, in non-cooperative scenarios, detection of the observing spacecraft by the RSO may be undesirable. In such a situation, angle measurements from a passive optical sensor can be used. Ref. has shown that Cartesian states characterizing a nearby satellite s relative motion are unobservable with angles-only measurements. Ref. 3 showed that full state estimation was achievable if angle measurements from a second camera are included in the measurement model. By utilizing angles from two cameras separated at a known baseline, depth information can be recovered in order to provide a relative position vector direction and magnitude. By utilizing stereoscopic vision, the relative state of a nearby object can be measured with no a priori knowledge of its physical geometry. Traditional stereoscopic measurement schemes rely on the minimization of the Euclidean distance between two the skew line-of-sight (LOS) vectors generated from each camera. The assumption of zero-mean Gaussian noise is consequently invalidated, and thus, a better quantification of relative position uncertainty is sought. In this work, the applications of new ground-based angles-only orbit determination techniques to the relative orbit determination problem are investigated. In Ref. 4, ground-based LOS measurements were bounded by orbital range constraints. By generating possible positions within the bounded LOS segment for multiple measurements, candidate orbits were computed by performing a Lambert s routine solver for each possible point combination. Ref. 5 presents a similar approach in which the orbiting body s admissible region was constructed to determine bounds on range and range-rate LeGrand 7 th Annual AIAA/USU

2 relative to the observer. Gaussian mixture approximations were then applied to the admissible region in order to generate an initial probability density function (pdf) associated with uniform ambiguity within the admissible region. In this paper, a hybrid approach motivated by the work of Refs. 4 and 5 is considered and applied to relative orbit determination. Stereoscopic measurements and their associated geometry are used to bound potential range values along a single LOS. Gaussian mixtures are generated to approximate the the range uncertainty. After Gaussian mixtures at two measurements are constructed, a Lambert s solver connects all Gaussian component combinations to formulate a full state initial pdf. This initial pdf is then further refined through processing subsequent angles-only measurement data in a Bayesian framework. RELATIVE MOTION The motion between two close-orbiting satellites in nearly circular orbits can be approximated using the Hills-Clohessy-Wiltshire (HCW) linearized equations of motion (EOMs), given by ẍ ωẏ 3ω x = f x ÿ + ωẋ = f y z + ω z = f z (a) (b) (c) ω is the angular velocity magnitude of the reference satellite s circular orbit, and f x, f y, and f z are external perturbations to the natural motion. These equations describe the relative motion of a target satellite with respect to the inspector satellite and are expressed in terms of the Local-Vertical, Local-Horizontal (LVLH) frame. Fixed at the center of mass of the inspector satellite, the x axis of the LVLH frame is directed radially outward, the z axis parallels the orbit normal, and the y axis completes the triad. Assuming no perturbations act on either spacecraft, that is f x = f y = f z =, the solution to the HCW EOMs can be expressed in state transition matrix form as r(t) = Φ rr r t + Φ rv v t v(t) = Φ vr r t + Φ vv v t (a) (b) For brevity, the four state transition matrices Φ are not provided here, but can be found in Reference 6. If the position of the target satellite is known at two separate times, the unique arc connecting the positions can be computed. This calculation is analogous to the familiar Lambert s problem for Keplerian motion. Given two positions at t = t and t = t, Eqns. () can be rearranged to find the initial relative velocity as v t = (Φ rv ) (r t Φ rr r t ) (3) STEREOSCOPIC IMAGING Stereoscopic imaging is widely used in terrestrial robotics applications. Only recently, however, has it been proposed as a mechanism for small satellite proximity operations. 3, 7 9 Advantages of stereoscopic imaging for space-based imaging, in comparison to the previously mentioned relative sensing alternatives, include its passive sensing nature, and the availability of inexpensive low-power commercial-off-the-shelf stereo cameras for small satellite applications. Depth Recovery By capturing images of a nearby satellite from two cameras at a known baseline separation, full depth information can be recovered. Due to the difference in camera perspective, the nearby spacecraft s image projection will differ in each camera s image. This difference, commonly known as disparity, is defined as the difference of the x coordinate projections in the left and right image planes, respectively. For a given pair of images captured simultaneously at time t i, the disparity d i is defined as d i ɛ xi l ɛ xi r (4) The disparity, coupled with an image plane projection coordinate set ɛ i, enables the calculation of the object s relative position ρ i with respect to either camera s center of projection (COP). In the simplified case the right camera s COP resides at the origin and the left camera s COP lies on the x axis (Figure ), the relative position is calculated as ρ i = x i y i z i bɛ xi r = d i b bɛ zi r = bɛ x i l d i b d i bɛ z i l d i b (5) baseline b is the physical separation distance between the two cameras COPs and f is the camera focal length. Because the nonlinear triangulation of the two lines of sight results in measurement noise that is not centered about the object s position mean, a different approach to stereoscopic range vector determination is considered. The second camera s LOS direction and its associated projection angle uncertainty can be used to bound a region along the first camera s LOS, as shown in Figure. LeGrand 7 th Annual AIAA/USU

3 COP l b ɛ l i COP r Figure : The stereoscopic imaging geometry. It is then assumed that the object s range is equiprobable between the bounded range values, yielding a uniform distribution on ρ [ρ min ρ max ]. The uniform uncertainty over the bounded region can then be approximated by a Gaussian mixture. ρ max ɛ r i f ρ l i z ρ i x y approximated by a Gaussian mixture (GM). Previous works, have shown that a large class of pdfs, including uniform pdfs, can be modeled by a Gaussian mixture. The uniform distribution, which is given by p(x) = b a, a x b, otherwise (6) can be approximated by a Gaussian mixture pdf of the form q(x) = w l p g (x ; m l, P l ), (7) l= p g (x ; a, A) represents a Gaussian pdf for the random variable x with mean a and covariance A, such that { p g (x ; a, A) = πa / exp } (x a)t A (x a) To quantify the difference between a truly uniform pdf and its Gaussian mixture approximation, the distance between p(x) and q(x) is taken as the L norm via L [p q] = (p(x) q(x)) dx (8) ρ min σ θ Ref. 5 has shown in detail that the L distance can be reduced to L [p q] = b a + w πσ w b a i= j= exp { Mi,j } [erf {B l } erf {A l }], l= θ θ b Camera Camera Figure : The possible ranges ρ bounded by a second camera. GAUSSIAN MIXTURE APPROXIMATION In order to capture the uniform uncertainty associated with a stereoscopic measurement, a uniform pdf can be A l = ( ) ( ) a ml b ml, B l =, σ σ ( and Mi,j mi m j = σ and requires optimization over only the common standard deviation parameter, σ. The aforementioned optimization consists of finding the roots of the L derivative with respect to σ, given as dl [p q] dσ = w πσ w (b a) πσ i= j= ) [ M i,j ] exp { Mi,j} [ A l exp { A l} Bl exp { Bl } ] l= LeGrand 3 7 th Annual AIAA/USU

4 Library of Solutions Ref. 5 demonstrated that by performing the optimization for the case of a = and b =, a generalized library of solutions can be produced, which can then be easily scaled to any arbitrary uniform distribution. For the case of a = and b =, the optimal standard deviation (i.e. the σ value that yields a zero derivative), is denoted as σ. It follows that component weights and means can be calculated as w = L and m l = l L + l {,,..., L} Here, the notation denotes values obtained from the generalized case of a = and b =. The prominent advantage in generating a general library is that it needs to be computed only once, as the obtained values are directly scalable to any approximation. To extend the library to an arbitrary approximation, the generalized parameters are scaled as follows: w = w, m l = a + (b a) m l, σ = (b a) σ It is intuitive that increasing the number of components L used will result in a closer approximation to the true uniform distribution (Figure 3). Because computational cost also increases with L, an appropriate value must be chosen, especially when considering small satellite flight-computer limitations. To control the accuracy/computational cost reciprocity, the acceptable maximum deviation parameter, σ max is specified. The precomputed general library is then searched to find L, such that σ(b a) σ max. This method ensures that no superfluous computations are made for the given mission requirements. GM APPROXIMATION OF STEREOSCOPIC MEASUREMENTS In this work, two dimensional relative motion in the x-y plane is considered. Preliminary investigations suggest that the following algorithm can be scaled to three dimensions, and will be addressed in future works. The relative state of an imaged object with respect to a single camera can be expressed in polar coordinates as the range magnitude and the angle from the image plane center to the object s projection as ρ = ρ (9) θ θ = atan (ɛ xi i, f) () Here atan (a, b) is defined as tan (a/b) < tan (a/b) < π based on which quadrant of an xy-plane contains the point. For a single camera, the range component ρ is unknown. However, the second camera s angle θ and associated uncertainty σ θ provide bounds on the values which the range can be, given by ρ min and ρ max. In this work, it is assumed that the possible range values follow a uniform distribution which is described by p(ρ ) = ρ max ρ min, ρ min ρ ρ max, otherwise Specification of ρ min and ρ max, along with the accuracy parameter, σ max, provides the required information to determine a Gaussian mixture approximation to the uniform range distribution using the previously described method. The position pdf can then be constructed in polar coordinates by combining the range Gaussian mixture with the observed angle measurement, which gives L ρ p ρ (ρ) = w ρ,l p g (ρ ; m ρ,l, P ρ,l ), () l= w ρ,l = L, P ρ,l = ((ρ max ρ min ) σ ρ ) σθ and m ρ,l = ρ min + (ρ max ρ min ) l L + θ l {,,..., L}. The variable σ θ represents the standard deviation of the image projection angle, and is a physical parameter of the chosen hardware. Figure 4 provides a graphical representation of the mixture of Gaussian components over the bounded LOS. LINKAGE Equation () gives the probability of the position variable ρ in polar coordinates. To construct the full state, both the relative position and relative velocity in Cartesian coordinates are needed. In a well-known theorem of orbital mechanics, the velocity of an orbiting body can be computed given two unique times and position vectors by solving the familiar Lambert s problem. Similarly for relative dynamics, the relative velocity can be approximated using the linearized state transition matrix (Eqn. 3). LeGrand 4 7 th Annual AIAA/USU

5 .8.8 p(x).6.4 p(x) x (a) -component approximation x (b) -component approximation.8.8 p(x).6.4 p(x) x (c) -component approximation x (d) -component approximation Figure 3: Gaussian mixture approximation of a uniform pdf for different numbers of components. The uniform pdf is given by the solid line and the GMM pdf is given by the dashed line. Adapted from Ref. 5. Unscented Transform ρ max In order to transform a set of position vectors ρ at t and t to a full Cartesian state x, the nonlinear function g(z) is defined as ρ min y = g(z) () r t g(z) = z = ρ t = v t ρ t ρ t θ t ρ t θ t Camera Figure 4: Gaussian mixture approximating uniform uncertainty [ρ min, ρ max ]. The relative position r describes the position of the nearby spacecraft with respect to the origin of the LVLH frame and requires mapping from polar to Cartesian coordinates and the addition of the camera s position. At times t and t, the positions are computed as r t =ρ t cos θ t + d t sin θ t r t =ρ t cos θ t + d t sin θ t LeGrand 5 7 th Annual AIAA/USU

6 Here, d is the position of camera with respect to the LVLH frame origin. With these two positions, Eqn. (3) can be applied to resolve the velocity at t, that is v t = (Φ rv ) (r t Φ rr r t ) In order to estimate the velocity mean and covariance from a pair of position means at times t and t, an unscented transformation is employed. First, given the n mean vector z, n sigma points Z (i) are chosen as Z (i) = z + z (i) i =,..., n (3) ( ) T z (i) = np i =,..., n i ( ) T z (n+i) = np i =,..., n i np is the matrix square root of np such that ( np ) T np ( np ) = np, and is the ith row of i np. The transformed sigma points are then calculated as ( Y (i) = g Z (i)) i =,..., n (4) Algorithm : Gaussian Mixture Construction Compute p ρ (ρ) for t and θ,t (Eqn. ) Compute p ρ (ρ) for t and θ,t (Eqn. ) for l =,...L t do for j =,..., L t do Construct z l,j from: l th mean of p ρ (ρ) at t j th mean of p ρ (ρ) at t Construct P l,j from: l th covariance of p ρ (ρ) at t j th covariance of p ρ (ρ) at t Compute Z (i) (Eqn. 3) for i =,..., n do Compute Y (i) l,j (Eqn. 4) end for Compute w l,j (Eqn. 8) Compute µ l,j (Eqn. 5) Compute Σ l,j (Eqn. 6) end for end for Compute q t (x) (Eqn. 7) The transformed full Cartesian state mean and covariance can then be calculated as µ = n Σ = n n i= n i= Y (i) (5) ( ) ( T Y (i) µ Y (i) µ) (6) This transformation is repeated for every position combination of the GM components at time t and t, as illustrated in Figure 5. Each linkage generates a new Gaussian component, resulting in L t L t total components. These components are then summed in the same fashion of Eqn. (7) to complete the full initial state pdf approximation at t, that is t t θ,t θ,t L t L t q t (x) = w l,j p g (x ; µ l,j, Σ l,j ) (7) l= j= w l,j = w l w j (8) and µ l,j and Σ l,j are found by the application of Eqns. (5) and (6) for every (l,j) component link. The complete algorithm is summarized in Algorithm. RECURSIVE STATE ESTIMATION To predict the state of the spacecraft at future times, standard Bayesian filtering techniques are employed. The Figure 5: Gaussian mixture linkage between t and t. dynamical system being modeled is linear in the sense that x k = F k x k In this case, F k is simply the STM Φ from Eqns. (), and x k is the state of the system at time t k with an initial LeGrand 6 7 th Annual AIAA/USU

7 condition given by x t. The initial condition is taken to be random with pdf p(x t ), which was constructed using the GM approach and linkage process that has been previously detailed and outlined in Algorithm. This initial pdf is described by q(t ) given in Eq. (7), or through a simple rewriting as p(x t ) = w + l, p g(x t ; m + l,, P + l, ) (9) l= In additional to the linear dynamical system, nonlinear measurements are received in the form of y k = h(x k ) + v k, h(x k ) represents single angle measurements at t k, that is h(x k ) = θ k and v k is the measurement noise, which is assumed to be zero-mean with covariance R k. Starting from the constructed initial condition, a predictor-corrector structure is employed which propagates each Gaussian component of the pdf in parallel and applies a Kalman filter update with angle measurements from camera at each time step to refine the pdf. Predictor In order to propagate the pdf between measurement times, a standard predictor mechanism is used in which the conditional pdf is represented by the Gaussian sum p(x k Y k ) = w l,k p g(x k ; m l,k, P l,k ) () l= The weights, means, and covariances for each of the components are propagated independent of one another. The weights are held constant during propagation, such that and w k,l = w+ k,l, () m l,k = Φm+ l,k P l,k = ΦP + l,k ΦT () Corrector At measurement time t k, a measurement y k is made available. The corrector step of the recursive estimation process applies a Bayesian update to fuse the prior information (given by the predictor output of Eqn. ()) with the newly acquired measurement data; this gives the updated pdf to be p(x k Y k ) = w + l,k p g(x k ; m + l,k, P + l,k ) (3) l= the updated weights, means, and covariances of the GM are w + l,k = β l,kw l,k / L j= β j,kw j,k (4a) m + l,k = m l,k + K l,k(y k ŷ l,k ) P + l,k = P l,k K l,kp l,y K T l,k (4b) (4c) In order to compute the updated parameters, the weight gain and the Kalman gain are computed via β l,k = p g (y k ; ŷ l,k, P l,y) K l,k = P l,xy P l,y (5a) (5b) For each component, ŷ l,k, P l,y, and P l,xy are found to be ( ) ŷ l,k = arctan rk,() (6a) r k,() P l,y = H k P l,k H T k + R k r k = P l,xy = P l,k HT k m l,k,() m l,k,() d k H k = [ r r k k,() r k,() ] (6b) (6c) (7a) (7b) Equations (4) are well-known. A comprehensive derivation of these equations can be found in the referenced literature., 3, 4 The algorithm for Gaussian mixture generation, linkage, and filtering is now complete. A graphical summary of this complete algorithm is provided in Figure 6. SIMULATION RESULTS To evaluate the performance of the stereoscopic GMM algorithm, several Monte Carlo simulations are considered. Each individual simulation includes the GM construction and linkage process described in the previous sections to generate the initial pdf. The pdf is then refined though the processing of subsequent angle measurements. LeGrand 7 7 th Annual AIAA/USU

8 Data y y y k GM GM p(x t ) p(x k Y k ) p(x k Y k ) Linkage Predictor Corrector Figure 6: Block representation of the GM pdf construction and refinement algorithm. Initial Conditions For all cases, the following orbit initial conditions (Table ) are chosen: Table Orbit Initial Conditions. Altitude 4 km Eccentricity. Angular Velocity.3 3 rad/s Linkage Measurements at times t = and t = 36 s are arbitrarily chosen for GMM construction and linkage. Figures 7(a) and 7(b) show the generated initial position and velocity pdfs for a single run. The relative initial conditions of the target spacecraft are specified as r t = [m] v t =.566 [m/s] (a) Position pdf approximation at t. The preceding initial conditions are propagated using the linear HCW equations and are treated as the truth model. The target satellite is assumed to be in free motion with no applied control. Measurement Model Measurements are provided by camera s measurement angle θ every six minutes. The angles are corrupted with zero-mean Gaussian noise, with σ θ = 5 arcseconds, a value typical of a narrow-field high-definition camera. Each simulation is run for 48 hours, over which 5 measurements are taken. (b) Velocity pdf approximation at t. Figure 7: Initial position and velocity pdfs. LeGrand 8 7 th Annual AIAA/USU

9 Test Case The first test case consists of simulations of the same orbit with a camera baseline of m. The covariance of the pdf at each measurement is computed, from which the position root-sum-square (RSS) and velocity RSS are extracted. The average position and velocity RSS over runs are plotted in Figures 8 and 9, respectively. A notable sinusoidal pattern is exhibited in these plots, which incidentally is a permeation of the camera angle θ. In other words, larger θ angles induce lower position resolution a consequence mitigable through active pointing control. The position and velocity tracking error are also plotted in Figures and. To visualize how the pdf evolves over time, the position pdf is plotted at different measurement times in Figure. As shown, the pdf quickly evolves into a Gaussian distribution. Position tracking error [m] Measurement Number x 4 Figure : Position tracking error..6 Avg. Position RSS [m] Velocity tracking error [m/s] Meaurement Number Figure 8: Average position RSS over runs Measurement Number Figure : Velocity tracking error. Test Case Avg. Velocity RSS [m/s] 6 x Meaurement Number Figure 9: Average velocity RSS over runs. In the second test case, the sensitivity of camera baseline is investigated. 5 simulations are ran for baseline distances of, 4, and 8 m. The average position RSS and velocity RSS are plotted for each baseline value in Figures 3 and 4, respectively. As expected, widening the camera baseline results in higher accuracy initial determination. For small satellites, baselines such as 8 m are unlikely for body-mounted camera configurations, but could be potentially obtainable through the use of deployable booms. CONCLUSIONS The applications of Gaussian mixture models and spacebased stereoscopic imaging to small satellite close proximity operations were presented. The limitations of typical stereoscopic measurement schemes were mitigated by using stereoscopic geometry to bound relative position range, for which uniform uncertainty was assumed. LeGrand 9 7 th Annual AIAA/USU

10 (a) Initial position pdf. (b) Position pdf at measurement 5. (c) Position pdf at measurement. (d) Position pdf at measurement 5. Figure : Evolution of the position pdf over subsequent measurements..6.4 b = [m] b = 4 [m] b = 8 [m] 8 x 4 7 b = [m] b = 4 [m] b = 8 [m]. 6 Avg. Position RSS [m] Avg. Velocity RSS [m/s] Measurement Number Figure 3: Sensitivity of camera baseline to average position RSS Measurement Number Figure 4: Sensitivity of camera baseline to average. velocity RSS. LeGrand 7 th Annual AIAA/USU

11 It was shown that the uniform range uncertainty can be approximated with a mixture of Gaussian components. By applying these approximations over two discrete measurements and linking all possible combinations of the Gaussian components with a relative Lambert solver, a full state Cartesian pdf was then composed. Further refinement of the resultant pdf was achieved by processing subsequent angle measurements in a Bayesian framework, which resulted in high accuracy relative orbit determination. FUTURE WORK Future work will investigate the relaxation of several assumptions made in this work. Firstly, the relative motion will be extended to three dimensions. Secondly, the assumption of no perturbative forces will be relaxed to include perturbations due to Earth s oblateness and differential drag. The inclusion of these perturbations will require a higher fidelity relative Lambert s solver routine, such as the methods presented in Ref. 5. In this work, the angles measurements from camera were used only twice (in the construction of the initial pdf). Future work will investigate an intelligent and dynamic selection of the measurement source in the recursive state estimation stage of the algorithm. ACKNOWLEDGMENTS The author would first like to thank his advisors, Dr. Kyle DeMars and Dr. Henry Pernicka, for their persistent support and technical guidance over the duration of this research. The author would also like to thank the past and present members of the Missouri S&T Satellite Research Team. Their research in proximity operations and stereoscopic imaging has provided the foundation which made this research possible. REFERENCES [] Space Policy, DoD Directive 3.,. [] Woffinden, D. and Geller, D., Observability Criteria for Angles-Only Navigation, Proceedings of the AAS/AIAA Astrodynmics Speciliast Conference, Mackinac Island, MI, No. AAS 7-4, 7. [3] Segal, S., Carmi, A., and Gurfil, P., Visionbased Relative State Estimation of Non-cooperative Spacecraft Under Modeling Uncertainty, IEEE Aerospace Conference,, pp. 8. [4] Sabol, C. A., Segerman, A., Hoskins, A., Little, B., Schumacher, P. W. J., and Coffey, S., Search and Determine Integrated Environment (SADIE) for Space Situational Awareness, Tech. rep., Air Force Research Laboratory, Directed Energy Directorate, Kihei, HI,. [5] DeMars, K. J. and Jah, M. K., Initial Orbit Determination via Gaussian Mixture Approximation of the Admissible Region, Proceedings of the nd AAS/AIAA Space Flight Mechanics Conference, No. AAS -6,. [6] Wiesel, W. E., Spacecraft Dynamics, Beavercreek, OH: Aphelion,. [7] Segal, S., Carmi, A., and Gurfil, P., Stereoscopic Vision-based Spacecraft Relative State Estimation, AIAA Guidance, Navigation, and Control Conference and Exhibit, 9. [8] Darling, J., LeGrand, K., Pernicka, H., Lovell, T. A., and Mahajan, B., Close Proximity Spacecraft Operations Using Stereoscopic Imaging, Proceedings of the 3rd AIAA/AAS Space Flight Mechanics Meeting, 3. [9] McCall, P., Torres, G., LeGrand, K., Adjouadi, M., Liu, C., Darling, J., and Pernicka, H., Many-Core Computing for Space-based Stereoscopic Imaging, Proceedings of the IEEE Aerospace Conference, 3. [] Alspach, D. and Sorenson, H., Nonlinear Bayesian Estimation Using Gaussian Sum Approximations, IEEE Transactions on Automatic Control, Vol. 7, No. 4, 97, pp [] van der Merwe, R. and Wan, E., Gaussian Mixture Sigma-point Particle Filters for Sequential Probabilistic Inference in Dynamic State-space Models, IEEE International Conference on Acoustics, Speech, and Signal Processing, Vol. 6, 3, pp. VI 7 4 vol.6. [] Julier, S. and Uhlmann, J., Unscented Filtering and Nonlinear Estimation, Proceedings of the IEEE, Vol. 9, No. 3, 4, pp [3] Simon, D., Optimal State Estimation : Kalman, H and Nonlinear Approaches, Wiley- Interscience, Hoboken, N.J, 6. [4] Crassidis, J., Optimal Estimation of Dynamic Systems, CRC Press, Boca Raton, FL,. [5] Jasper, L., Anthony, W., Lovell, T. A., and Newman, B., Application to Relative Satellite Motion Dynamics to Lambert s Problem, Proceedings of the 3rd AIAA/AAS Space Flight Mechanics Meeting, 3. LeGrand 7 th Annual AIAA/USU