AN ADAPTIVE GAUSSIAN SUM FILTER FOR THE SPACECRAFT ATTITUDE ESTIMATION PROBLEM

Transcription

1 AAS AN ADAPTIVE GAUSSIAN SUM FILTER FOR THE SPACECRAFT ATTITUDE ESTIMATION PROBLEM Gabriel Terejanu Jemin George and Puneet Singla The present paper is concerned with improving the attitude estimation accuracy by using a finite Gaussian-mixture approximation for the conditional probability density function (pdf. A novel method is discussed to update the weights corresponding to different components of the Gaussian mixture model by constraining the Gaussian sum approximation to satisfy the Fokker-Planck-Kolmogorov equation. Updating the weights not only provides us with better estimates but also with a more accurate pdf. The numerical results show that updating the weights in the propagation step not only gives better estimates between the observations but also provides superior estimator performance where the measurements are ambiguous. INTRODUCTION Spacecraft attitude estimation problem involves determining the orientation of a spacecraft from on-board observations of line-of-sight vectors to other reference points such as celestial bodies, the direction of the Earth s magnetic field gradient, etc. []. Generally, a redundant set of these observations is used to generate more accurate estimates of the spacecraft attitude. Several attitude sensors are discussed in the literature, including three axis magnetometers, Sun sensors, Earth-horizon sensors, global positioning sensors, rate integrating sensors and star-cameras []. The accuracy of the attitude estimation depends on the quality of the attitude sensor used. Mathematical models for attitude sensors are generally based upon the usefulness rather than the truth and do not provide all the information that one would like to know. So, an estimation algorithm is required to extract the useful information from the available sensor measurements, which are often corrupted by sensor noise, biases and sensor inaccuracies. Generally, the attitude estimation algorithms can be divided into two categories: Batch attitude estimation algorithms such as TRIAD, 2 QUEST, 3 ESOQ 45 etc and 2 Sequential attitude estimation algorithms such as Kalman filter, 6 REQUEST, 7 Un-Scented Kalman Filter, 8 Particle filters, 9 etc. A detailed discussion on various attitude estimation algorithms can be found in Ref. []. During normal operations, the attitude estimation problem needs to be solved recursively; i.e., the attitude filter makes new updates and predictions based on present and prior sensor information. Many recursive attitude estimation algorithms are presented in the literature but the Kalman filter [] is one of the most widely used and powerful tools for real-time estimation problems. It is based on the assumption that the nonlinear attitude dynamics can be accurately modeled by a first-order Taylor series expansion. 6 Since the EKF provides us only with a rough approximation of the a-posteriori pdf and solving for the exact solution of the conditional pdf is very expensive, researchers have been looking for Graduate Student, Department of Computer Science & Engineering, University at Buffalo, Buffalo, NY- 426, terejanu@buffalo.edu. Graduate Student, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-426, jgeorge3@buffalo.edu. Assistant Professor, AIAA, AAS Member, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-426, psingla@buffalo.edu.

2 mathematically convenient approximations. However, the accuracy and efficient implementation of these filters have been an issue. The Bayes filter 2 offers the optimal recursive solution to the nonlinear filtering problem. However, the implementation of the Bayes filter in real time is computationally intractable because of the multi-dimensional integrals involved in the recursive equations. An efficient implementation of the Bayes filter is possible when all the involved random variables are assumed to be Gaussian. 3, 4 Several other approximate techniques such as Sequential Monte Carlo (SMC methods, 5 Gaussian closure 6 (or higher order moment closure, Equivalent Linearization, 7 and Stochastic Averaging 8, 9 can also be used to find a solution to nonlinear filtering problem. Sequential Monte Carlo Methods or Particle filters 2 consists of discretizing the domain of the random variable into a set of finite number of particles and transforming these particle through the nonlinear map to obtain the distribution characteristics of the transformed random variable. Although SMC based methods are suitable for highly nonlinear and non-gaussian models and have been applied successfully for the attitude estimation problem, 9 they require extensive computational resources and effort. 2 Alternative to sequential monte carlo method is the exact nonlinear filters based upon the exact solution of the Fokker-Planck-Kolmogorov equation 22 (FPKE as discussed in Refs. [23, 2]. FPKE provides the exact description for the propagation of the state transition pdf through a nonlinear dynamical system. Analytical solutions exists only for stationary pdf and are restricted to a limited class of dynamical systems. 22, 24 Thus researchers are looking actively at numerical approximations to solve the Fokker-Planck equation, 25, 26, 27, 28 generally using the variational formulation of the problem. However, these methods are severely handicapped for higher dimensions and have not found much use in the development of exact nonlinear filters. In Refs. [29], an approximate solution to FPKE equation is presented for the attitude estimation problem. The main idea of this paper is to develop a nearly exact nonlinear filter for the spacecraft attitude estimation problem by approximating the conditional pdf as a mixture of Gaussian components. It can be shown that as the number of Gaussian components increases the Gaussian sum approximation converges uniformly to any probability density function. 3 For a dynamical system with additive Gaussian white noise, the first two moments of the Gaussian components are propagated using the linearized model, and the weights of the new Gaussian components are set equal to the prior weights. In the case where observations are available both the moments and the weights are accordingly updated 3, 32 using Bayes rule to obtain an approximation of the a posteriori pdf, yielding the so called Gaussian Sum Filter (GSF. Extensive research has been done on Gaussian sum filters, 2 which have become popular in the target tracking community. Such efforts have been materialized in methods like Gaussian Sum Filters, 3 GSF with a more advanced measurement update, 32 Mixture of Kalman Filters 33 and Interactive Multiple-Model. 34 However, in all of these methods the weights of different components of a Gaussian mixture are kept constant while propagating the pdf through a nonlinear system and are updated only in the presence of measurement data. This assumption is valid if the underlying dynamics is linear or the system is marginally nonlinear or measurements are precise and available very frequently. The same is not true for the general nonlinear case and new estimates of weights are required for accurate propagation of the state pdf. In our prior work, 35, 36, 37 an adaptive Gaussian sum filter is developed by adapting the weights of Gaussian mixture model in case of both pure propagation and measurement update. In the case of pure propagation, weights are updated by constraining the Gaussian sum approximation to satisfy the FPKE. The present paper is concerned with using this recently developed adaptive Gaussian sum filter for the attitude estimation problem. The structure of paper is as follows. First, a brief review of the attitude sensors and attitude

3 kinematics is presented. Next, an introduction to the conventional Gaussian sum approximation is presented followed by a novel scheme for updating the weights of different components of Gaussian mixture. Finally, numerical examples are considered to illustrate the efficacy of the proposed method. ATTITUDE SENSORS AND DYNAMIC MODELS In this section, a brief review of the attitude sensors and attitude motion models are presented. Attitude Sensor Model We assume that the spacecraft attitude is determined by processing the digital images of the stars by a star-camera. Star positions are the most accurate source of reference celestial bodies for the attitude determination since their position with respect to an inertial frame is fixed. Pixel formats of the order of or larger are commonly used to provide good resolution pictures. The first stage in attitude determination is to identify a star with reference to an on-board star catalog. After star identification is made, image plane coordinates of the stars are given by using a pinhole camera model for the camera. Photograph image plane coordinates of the j th star are given by the following ideal co-linearity equations: x j = f C r xj + C 2 r yj + C 3 r zj C 3 r xj + C 32 r yj + C 33 r zj + x ( y j = f C 2r xj + C 22 r yj + C 23 r zj C 3 r xj + C 32 r yj + C 33 r zj + y (2 where f is the effective focal length of the star-camera and (x, y are the principal point offset, determined by the ground or on-orbit calibration [38]. C ij are the attitude matrix elements (orienting the sensor frame relative to the inertial axes, and the inertial star vector r j is given by r xj r j = r yj r zj = cos δ j cos α j cos δ j sin α j sin δ j Further, choosing the z-axis of the image coordinate system towards the boresight of the star-camera the measurement unit vector b j is given by the following expression: b j = x 2 j + y2 j + f 2 (x j x (y j y f The relationship between the measured star direction vector b j in image space and its projection r j on the inertial frame is given by b j = Cr j + v j (5 where, v j is a zero mean Gaussian white noise process with covariance matrix R j. The attitude determination problem reduces to the estimation of the elements of the attitude matrix C given the well calibrated image plane star-vectors, r j and corresponding inertial plane vector b j. The spacecraft attitude can be represented by many coordinate choices [39, 4], but in this paper, Modified Rodrigues Parameters (MRP (denoted by p are used to parameterize the attitude matrix, C. (3 (4

4 Gyro Model The spacecraft attitude is normally estimated by a combination of the attitude sensor (e.g. Star Tracker measurements along with a model of spacecraft dynamics. The use of densely available angular rate data can omit the need of the spacecraft dynamic model. Rate gyros are used to measure the angular rates of the spacecraft without regard to the attitude of spacecraft. Generally, rate gyro measurements are modeled by the following expression [, 6]: ω = ω b η (6 where ω is the unknown true angular velocity of the spacecraft, ω is the gyro measured angular rate of the spacecraft in the gyro frame, and b is the gyro bias drift vector, which is further modeled by the following first-order stochastic process: ḃ = η 2 (7 where η and η 2 are assumed to be modeled by two independent Gaussian white noise processes with { [ ] η E (t [η T η 2 (t (t + τ ηt 2 (t + τ]} = Qδ(τ The matrix Q is assumed to be diagonal, i.e., Q = diag[q q 2 q 3 q 4 q 5 q 6 ] Attitude Kinematics The kinematic equations for the spacecraft motion using the MRP as attitude parameters can be written as: ṗ = 4 B(pω, B(p = [ ( p T pi + 2[p ] + 2pp T ] (8a B T (pb(p = B(pB T (p = ( + p T p 2 I (8b where [p ] is a 3 3 skew-symmetric cross product matrix given as: [p ] = p 3 p 2 p 3 p p 2 p (9 Among all known three parameter descriptions of attitude, only the MRPs have the remarkable property that the kinematic coefficient matrix, B(p, has orthogonal rows and columns. Generally, these parameters are well behaved for most large motions exclusive of tumbling motions since a full revolution of the rigid-body about any axis is required to encounter the singularity. Their singularity free state space volume (±2π about any axis is substantially larger that of any Euler angle representation (never farther than π/2 from a singularity and the classical Rodrigues parameter representation (±π about any axis. Furthermore, in Ref. [4] it is shown that the MRPs are the most nearly linear three-parameter representation of attitude.

5 GAUSSIAN SUM FILTER In this section, we discuss the adaptive Gaussian sum filter algorithm to estimate the spacecraft attitude parameterized by MRP along with gyro bias. The state vector of the adaptive Gaussian sum filter consists of 3 components of MRP, p and the gyro bias, b: { } p x = ( b We mention that in this case, Eq. (8a along with Eqs. (6 and (7 constitute the assumed dynamic model given by the following equation: ẋ(t = f(x(t, ω(t + g(x(tγ(t, x(t = x ( where, [ ] [ B(p( ω b B(p 3 3 f(x(t, ω(t =, g(x(t = I 3 3 ], Γ(t = [ ] η (t η 2 (t Further, Eq. (5 constitute the measurement model for the attitude estimation problem such that ỹ k = h(t k, x k + v k (2 where ỹ k is a vector of star directions in image frame and therefore, h(t k, x k = C(p k r j Conventional Gaussian Sum Filter In this section, we briefly discuss the development of the conventional Gaussian sum filter. More details about the development of conventional Gaussian sum filter for generic nonlinear dynamical system can be found in Refs. [3, 3]. Now, we consider the 6-dimensional continuous-time disturbance driven nonlinear dynamic system given in Eq. (s and discrete measurement model, given in Eq. (2. The development of the conventional Gaussian sum filter follows from the assumption that the underlying conditional pdf (forecast pdf if t > t k or a posteriori pdf if t = t k can be approximated by a finite sum of Gaussian pdfs ˆp(t, x(t Y k = N w i p g i i= p gi = N (x(t µ i, Pi = 2πP i /2 exp [ 2 ( T ( ] x µ i P i x µ i (3a where Y k denotes a set of k observations, Y k = {ỹ i i =... k}. µ i and Pi represent the conditional mean and covariance of the i th component of the Gaussian pdf, respectively. w i denotes the amplitude of the i th Gaussian in the mixture. The positivity and normalization constraint on the mixture pdf, ˆp(t, x Y k, leads to the following constraints on the amplitude vector: N w i =, wi, t (4 i=

6 Since all the components of the mixture pdf of Eq. (3a are Gaussian and thus, only estimates of their mean and covariance need to be propagated between t k and t k+ using the conventional Extended Kalman Filter time update equations: 3 µ i = f(µi, ( ω k b i (5 Ṗ i = Ai Pi + Pi (Ai T + g(µ i QgT (µ i (6 A i f(x(t, ω(t = x(t (7 x(t=µ i w i = wi k k for t k t t k+ (8 T If we partition the i th mean vector, µ i {p = i }, bi, then the jacobian matrix A i corresponding to attitude estimation problem is given as: [ F(µ A i i = B(µ i ] ( F = [ ( ] ( ω k b i 2 T p i I 3 3 ( ω k b i T pi + p i ( ω k b i T [( ω k b i ] (2 Similarly, both the state vector and the covariance matrix can be updated using the extended Kalman Filter measurement update equations and further the weights of different Gaussian mixture components can be updated using also Bayes rule but under the assumption that P i k+ k as is shown in Ref. [3]. } µ i k+ k+ = µi k+ k + Ki k {ỹ k h(t k, µ i k+ k (2 P i k+ k+ = { I 6 6 K i k k} Hi P i k+ k (22 H i k = h(t k, x k (23 x k xk =µ i k+ k K i k = Pi k+ k (Hi k T { H i k Pi k+ k (Hi k T + R k } (24 w i k+ k+ = w i k+ k βi k N i= wi k+ k βi k (25 where β i k = N ( ỹ k h(t k, µ i k+ k, Hi k Pi k+ k (Hi k T + R k (26 Finally, an optimal state estimate and corresponding error covariance matrix can be obtained by making use of the following relations: µ = P = N w i µi (27 i= N i= w i [ P i + (µi µ (µ i µ T ] (28

7 Notice that it is assumed that weights w i do not change between measurement updates. This assumption is valid if underlying dynamics is linear or the system is marginally nonlinear. The same is not true for the general nonlinear case and new estimates of weights are required for accurate propagation of the state pdf. It is assumed that the covariance of various Gaussian components are small enough 3 such that the linearizations become representative for the dynamics around the means. This is particularly a problem, if the uncertainty in measurement model is large relative to process noise or measurements are not available frequently. In practice this assumption can be easily violated resulting in a poor approximation of the forecast pdf. Practically, the dynamic system may exhibit strong nonlinearities and the total number of Gaussian components, needed to represent the pdf, may be restricted due to computational requirements. The existing literature provides no means for adaption of the weights of different Gaussian components in the mixture model during the propagation of the state pdf. The lack of adaptive means for updating the weights of Gaussian mixture are felt to be serious disadvantages of existing algorithms and provide the motivation for this paper. Adaptive Gaussian Sum Filter In this section, we summarize a recently developed method 36, 37 to update the weights of different components of the Gaussian mixture of Eq. (3a during propagation. The essence of the main idea is that the mixture pdf ˆp(t, x Y k of Eq. (3a should satisfies the Fokker-Planck-Kolmogorov equation (FPKE between observations. t p(t, x Y k = L FP p(t, x Y k (29 where L FP ( is the so called Fokker-Planck operator, and, L FP = 6 i= D ( i (t, x x i + D ( (t, x = f(x(t, ω(t i= j= (t, x (3 x i x j 2 D (2 ij g(x(t Qg T (x(t (3 x D (2 (t, x = 2 g(x(tqgt (x(t (32 We mention that Eqs. (29-(32 represent the Stratonovich interpretation for the FPKE and is more popular among engineers and physicists. An alternative interpretation (Itō interpretation for the FPKE is also commonly used among mathematicians and leads to the same equation if g(. is constant. In this paper, we use the Itō interpretation for the FPKE to illustrate the main idea of updating the weights corresponding to different component of the Gaussian mixture model during propagation of state uncertainty. In Itō interpretation for the FPKE equation, Eq. (3 is replaced with D ( (t, x = f(x(t, ω(t. Now, the main idea is to use the FPKE error as a feedback to update the weights of different Gaussian components in the mixture pdf. In another words, we seek to minimize the FPKE error under the assumption of Eqs. (3a, (5 and (6. The substitution of Eq. (3a in Eq. (29 leads to e(t, x = ˆp(t, x Y k t L FP (ˆp(t, x Y k (33

8 where, ˆp(t, x Y k t = N i= w i p g i µ i T µ i + n n j= k= p gi P i,jk P i,jk (34 where P i,jk is the jkth element of the i th covariance matrix P i. Further, substitution of Eqs. (3 and (32 along with Eq. (34 in Eq. (33 leads to e(t, x = N w i L i(t, x = L T w (35 where w is a N vector of Gaussian weights, and L i is given by L i (t, x = p T n n g i µ i f(µ i, ( ω k b i + p gi j= k= P i,jk P i,jk ( n + f j (x(t, ω(t p g i f j (x(t, ω(t + p gi n x j x j 2 j= where D(x(t is given as: i= k= 2 D jk (x(tp gi x j x k (36 D(x(t = 2 g(x(tqgt (x(t (37 Further, different derivatives in the above equation can be computed using the following analytical formulas: p ( gi = (P i x µ i p gi µ i p gi P i p gi = p g i 2 (Pi ( x = (Pi x µ i 2 [ p gi xx T = (Pi I 6 6 [ ( ( ] T x µ i x µ i (P i I 6 6 p gi ( x µ i ( ] T x µ i (P i p gi Now, at a given time instant, after propagating the mean, µ i and the covariance, Pi, of individual Gaussian elements using Eqs. (5 and (6, we seek to update weights by minimizing the FPKE error over some volume of interest V : 36, 37 min e 2 (t, xdx + w i 2 V N s.t w i = i= w i, N (w i wi k k 2 (38 i= i =,, N

9 Here, the second term in the cost function is introduced to penalize large variations in the weights between two time steps. Since, FPKE error given by Eq. (35 is linear in Gaussian weights, w i, hence, the aforementioned problem can be written as a quadratic programming problem: min w 2 wt Lw + (w w k k T (w w k k (39 s.t T N w = w N Where N is a vector of ones, N is a vector of zeros and L is given by L = L(xL T (xdx V = L L dx V L L 2 dx V V. L L N dx L 2 L dx L N L dx V V L 2 L 2 dx L N L 2 dx.... L 2 L N dx L N L N dx V V V V (4 In Refs. [36, 37], it shown that the aforementioned problem leads to convex optimization problem and guarantees to have a unique solution. For the attitude estimation problem, the expression for L i is: ( [ { L i =N (µ i, Pi f T (µ i, ( ω k b i f T (x, ω }(P i [x µ i ] + 3 ( ωk b i 2 xt ] where 8 3 k= } q k {x T M k x k= j= { } A i Pi + Pi (Ai T + g(µ i QgT (µ i g(xqgt (x 3 6 { (P i [x µ i ][x µi ]T (P i (P i } kj M = diag[5 5 5 ] M 2 = diag[ ] M 3 = diag[ ] Since f(. is polynomial in nature, thus the various integrals in Eq. (4 can be related to higher order moments of various Gaussian components and can be computed analytically. However, in this paper, we will use Gaussian quadrature method to approximate these integrals. The minimization problem will substitute for the Eq. (8 of conventional Gaussian sum filter whenever an estimate has to be computed, even between measurements. The summary of this adaptive Gaussian sum filter with forecast weight update for the continuoustime nonlinear dynamical systems and discrete measurement model is presented in Table. kj

10 Table Summary of the Adaptive Gaussian Sum Filter Continuous-time nonlinear dynamics: ẋ(t = f(x(t, ω(t + g(x(tγ(t Discrete-time measurement model: ỹ k = h(t k, x k + v k Propagation: µ i = f(µi, ( ω k b i Ṗ i = Ai Pi + Pi (Ai T + g(µ i QgT (µ i A i = f(x(t, ω(t x(t x(t=µ i w = arg min w 2 wt Lw + (w w k k T (w w k k subject to T N w = w N Measurement-update: µ i k+ k+ = µi k+ k + Ki k {ỹ k h(t k, µ i k+ k } P i k+ k+ = { I 6 6 K i k k} Hi P i k+ k H i k = h(t k,x k x k xk =µ i k+ k { } K i k = Pi k+ k (Hi k T H i k Pi k+ k (Hi k T + R k w i k+ k+ = wi k+ k βi k N i= wi k+ k βi k β i k = N ( ỹ k h(t k, µ i k+ k, Hi k Pi k+ k (Hi k T + R k

11 .2 x 3 GS Classic GS2 Proposed RMS Error Figure RMS Error for Test Case NUMERICAL EXAMPLES In this section, we demonstrate the effectiveness of the proposed Gaussian sum filter, developed in this paper, by simulating star-camera images. An 8 8 FOV star-camera is simulated by using the pinhole camera model dictated by Eqs. ( and (2. The effective focal length of the star-camera is assumed to be 65mm for both F OV s. For simulation purposes, the spacecraft is assumed to have the following angular velocity in the body fixed reference frame: ω = [ ω ω ] T, ω = 2π rad/sec ( The gyro data is simulated by assuming gyro data frequency to be Hz and corrupted by gyro bias of.deg/hr and Gaussian white noise according to Eqs. (6 and (7 with the following values for noise properties: q = q 2 = q 3 = σ u =.6 6 rad/sec /2 q 4 = q 5 = q 6 = σ v =.55 rad/sec 3/2 For our preliminary simulations, we assume the perfect knowledge of the gyro bias and will estimate only MRP vector p. To show the effectiveness of the proposed idea, we consider three test-cases with star data update frequencies of Hz,.5Hz and.2hz, respectively. It is assumed that 5 lineof-sight vectors are available at all measurement update times for all three test cases. Also, the true line-of-sight vectors are corrupted by Gaussian white noise of standard deviation µ-rad for all three test cases. The initial state covariance matrix, P, is assumed to be 3 I 3 3 for all three test cases. Fig. shows the plots of RMS attitude error for conventional and proposed adaptive Gaussian sum filter corresponding to Test case, i.e., measurement frequency of Hz. It is clear from the plots that the adaptive Gaussian sum filter performs better than the conventional Gaussian sum filter

12 δp δp 2 δp 3 x x x (a Attitude Error Weights (w i (b Weights Figure 2 Conventional Gaussian Sum Results for Test Case δp δp 2 δp 3 x x x (a Attitude Error Weights (w i (b Weights Figure 3 Adaptive Gaussian Sum Results for Test Case

13 .2 x 3 GS Classic GS2 Proposed RMS Error Figure 4 RMS Error for Test Case 2 δp δp 2 δp 3 x x x (a Attitude Error Weights (w i (b Weights Figure 5 Conventional Gaussian Sum Results for Test Case 2 at least for the first one second, i.e., before the first measurement update. Figs. 2(a and 2(b show the plots of attitude error with corresponding 3-σ bounds and weights of different Gaussian mixture components in the case of the conventional Gaussian sum filter while Figs. 3(a and 3(b show the plots of attitude error with corresponding 3-σ bounds and weights of different Gaussian mixture components, respectively in the case of the proposed adaptive Gaussian sum filter. Notice that in the case of the conventional Gaussian sum filter, the weights of different Gaussian components do not change during the propagation while they change gradually in the case of the proposed adaptive Gaussian sum filter. The relatively better performance of the adaptive Gaussian sum filter can be attributed to the adaptation of weights in Gaussian mixture during the propagation step. Fig. 4 shows the plots of RMS attitude error for conventional and proposed adaptive Gaussian sum filter corresponding to Test case 2, i.e., measurement frequency of.5hz. As expected the adaptive Gaussian sum filter performs better than the conventional Gaussian sum filter at least for the first two seconds, i.e., before the first measurement update. Figs. 5(a and 5(b show the plots of attitude error with corresponding 3 σ bounds and weights of different Gaussian mixture components in the case of the conventional Gaussian sum filter while Figs. 6(a and 6(b show the same plots for the

14 δp δp 2 δp 3 x x x (a Attitude Error Weights (w i (b Weights Figure 6 Adaptive Gaussian Sum Results for Test Case 2 Figure 7 RMS Error for Test Case 3 proposed adaptive Gaussian sum filter. Once again, in the case of the conventional Gaussian sum filter, the weights of different Gaussian components do not change during the propagation while they change gradually in the case of the proposed adaptive Gaussian sum filter. These facts are also confirmed by the simulation results for the Test case 3, i.e., measurement frequency of.2hz, as shown in Figs. 7 and 9. These results confirms that the adaptation of weights during the propagation can enhance the performance of the conventional Gaussian sum filter. CONCLUDING REMARKS In this paper, an adaptive Gaussian sum filter algorithm is developed for the attitude estimation problem. The mean and covariance of each of the Gaussian component is propagated by using the extended Kalman filter while the weights of different Gaussian components are updated by constraining the Gaussian sum approximation to satisfy the Fokker-Planck-Kolmogorov (FPKE equation. During the measurement update, Bayes rule is used to update the weights of different Gaussian components. The performance of the proposed method is compared with the performance

15 δp δp 2 δp 3 x x x (a Attitude Error Weights (w i (b Weights Figure 8 Conventional Gaussian Sum Results for Test Case 3 δp δp 2 δp 3 x x x (a Attitude Error (b Weights Figure 9 Adaptive Gaussian Sum Results for Test Case 3

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