UNSCENTED KALMAN FILTERING FOR SPACECRAFT ATTITUDE STATE AND PARAMETER ESTIMATION

Transcription

1 AAS UNSCENTED KALMAN FILTERING FOR SPACECRAFT ATTITUDE STATE AND PARAMETER ESTIMATION Matthew C. VanDyke, Jana L. Schwartz, Christopher D. Hall An Unscented Kalman Filter (UKF) is derived in an attempt to solve the spacecraft dual estimation problem with greater accuracy than is attainable with an Extended Kalman Filter (EKF). The EKF is an extension of the linear Kalman Filter for nonlinear systems. Although the EKF has been used successfully in many nonlinear applications, the performance is limited, due mostly to the truncation of all but first-order terms. The UKF is able to achieve greater estimation performance than the EKF through the use of the unscented transformation (UT). The UT allows the UKF to capture first and second order terms of the nonlinear system. INTRODUCTION The Kalman Filter is an optimal filter for estimating a linear system.[1] The simplicity, recursive structure, and mathematical rigour of the derivation of the Kalman Filter make it well-suited and attractive for use in many practical applications. However, many real-world systems, especially mechanical systems, are nonlinear in nature. The Extended Kalman Filter (EKF) was developed to help account for these nonlinearities. The EKF accounts for nonlinearities by linearizing the system about its last-known best estimate with the assumption that the error incurred by neglecting the higher-order terms is small in comparison to the first-order terms. The Kalman Filter measurement update equations are then applied to the linear system, resulting in a suboptimal solution. Although the EKF has been used successfully in many applications, it has several shortcomings. The EKF operates by approximating the state distribution as a Gaussian random variable (GRV) and then propagating it through the first-order linearization of the nonlinear system.[2] The EKF is a suboptimal nonlinear filter due to the truncation of the higher-order terms when linearizing the system. The loss of the higher-order terms can be avoided in the propagation of the state of the system by using the full nonlinear equations. However, a complete description of the state conditional probability density requires a potentially unbounded number of parameters.[3] Attempts to address the first-order approximation shortcomings of the EKF which can lead to instability of the filter are not new.[4, 5, 6, 7, 8, 9] More recent proposed improvements to the EKF have branched out into two areas of research. The two branches offer improved performance against different sources of error.[10] One technique seeks to improve the convergence of the first-order filter by iterating at the measurement update step. These Iterated Extended Kalman Filters (IEKF) reduce the effective measurement noise. As such, they Virginia Space Grant Fellow, Graduate Student. Department of Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, Virginia mavandyk@vt.edu. Student Member AIAA, Student Member AAS. National Science Foundation Fellow, NASA Graduate Student Researcher Program Fellow. Department of Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, Virginia jana@vt.edu. Student Member AIAA, Student Member AAS. Professor. Department of Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, Virginia cdhall@vt.edu. Associate Fellow AIAA, Member AAS. Copyright c 2004 by the authors. Permission to publish granted to The American Astronautical Society. 1

2 can be more tolerant to process noise and errors in initial conditions. Iterating in the measurement step also provides robustness against the first-order approximations of the derivatives.[11, 10] The second family of modified nonlinear filters improve performance by eliminating the Jacobian representation of the derivatives. These filters can yield drastically improved behavior beyond the convergence of the EKF for the same order of floating point operations (flops). Complete coverage of this class of filters, termed the Linear Regression Kalman Filters (LRKF), is beyond the scope of this paper.[12, 13, 14, 3, 15, 16, 10, 17, 18, 19, 20, 21, 22, 23, 2, 24] Some of these techniques have been further enhanced to capitalize on the improvements obtained through iteration in the IEKF and apply similar techniques to these higher-order filters. The Unscented Kalman Filter (UKF), a member of the LRKF family, attempts to remove some of the short-comings of the EKF in the estimation of nonlinear systems. The UKF is an extension of the traditional Kalman Filter for the estimation of nonlinear systems that implements the unscented transformation. The unscented transformation uses a set of sample, or sigma, points that are determined from the a priori mean and covariance of the state. The sigma points undergo the nonlinear transformation. The the posterior mean and covariance of the state are determined from the transformed sigma points.[14] This approach gives the UKF better convergence characteristics and greater accuracy than the EKF for nonlinear systems.[3] The ability of the UKF to accurately estimate nonlinearities makes it attractive for implementation on spacecraft. The state and observation models for spacecraft are inherently nonlinear. The UKF has recently been documented in its application to spacecraft attitude and orbital dynamics; such applications are unusual in a literature dominated by theory papers and neural network applications.[25, 26] The main contribution of this paper is the development of an UKF for the spacecraft dual estimation problem. The UKF dual estimator is tested through numeric simulation of a fully actuated rigid body with attitude sensors that provide two noisy vector measurements and a noisy angular velocity vector measurement. An EKF dual estimator is also developed and is used to gauge the performance of the UKF dual estimator. THE UNSCENTED KALMAN FILTER The UKF was developed with the underlying assumption that approximating a Gaussian distribution is easier than approximating a nonlinear transformation.[14] The UKF uses deterministic sampling to approximate the state distribution as a GRV. The sigma points are chosen to capture the true mean and covariance of the state distribution. The sigma points are propagated through the nonlinear system. The posterior mean and covariance are then calculated from the propagated sigma points. The UKF determines the mean and covariance accurately to the second order, while the EKF is only able to obtain first-order accuracy.[2] Therefore, the UKF provides better state estimates for nonlinear systems. The proceeding summary of the UKF equations follows the presentation by Wan and van der Merwe[2]. The system and measurement noise are assumed to be zero-mean, or additive. The zero-mean noise implementation of the UKF does not require the augmenting of the state vector with the noise variables, thus decreasing the number of points that are required to be propagated through the nonlinear system from 2(L + q) to 2L. State Estimation The filter is initialized with the predicted mean and covariance of the state. } ˆx(t 0 ) E {ˆx 0 { (x(t0 )( ) } T P x0 E ) ˆx 0 x(t0 ) ˆx 0 (1) (2) 2

3 The sigma points are calculated from the a priori mean and covariance of the state using [ ] χ k 1 ˆx k 1 ˆx k 1 + γ P xk 1 ˆx k 1 γ P xk 1 where, γ L + λ. The sigma points are propagated through the nonlinear system χ k k 1 F ( χ k 1, u k 1 ) (3) (4) The posterior mean, ˆx k, and covariance, P x k, are determined from the statistics of the propagated sigma points as follows The weights, W m i ˆx k P x k Wi c and Wi c, are calculated using W m i χ i,k k 1 (5) ( ) ( ) T χ i,k k 1 ˆx k χ i,k k 1 ˆx k (6) W0 m λ L + λ (7) W0 c λ L + λ + 1 α2 + β (8) Wi c Wi m 1 2 (L + λ) (9) The estimated measurement matrix, Υ k k 1, is calculated by transforming the sigma points using the nonlinear measurement model, ) Υ k k 1 H (χ k k 1 (10) The mean measurement, ŷ k, and the measurement covariance, P y k y k, are calculated based on the statistics of the transformed sigma points. ŷ k P yk y k Wi c The cross-correlation covariance, P xk y k, is calculated using P xk x k W m i Υ i,k k 1 (11) Wi c ( Υi,k k 1 ŷ k ) ( Υi,k k 1 ŷ k ) T + Rxk (12) ( χ i,k k 1 ˆx k ) (Υi,k k 1 ŷ ) T k (13) The Kalman gain matrix is approximated from the cross-correlation and measurement covariances using K xk P xk y k P 1 y k y k (14) The measurement update equations used to determine the mean, ˆx k, and covariance, P xk, of the filtered state are ˆx k ˆx k + K ( k yk ŷ ) k (15) P xk P x k K xk P yk y k K T x k (16) 3

4 Parameter Estimation The parameter estimation equations for the UKF are similar to those for the state estimation. This section expounds upon the differences. The filter is initialized with the predicted mean and covariance of the parameters. } ŵ(t 0 ) E {ŵ 0 { (w(t0 )( ) } T P w0 E ) ŵ 0 w(t0 ) ŵ 0 The time update of the parameter vector and the parameter covariance is performed using (17) (18) ŵ k ŵ k 1 (19) P w k P wk 1 + Q wk (20) where, Q w is the system process noise of the time update. The sigma points are calculated from the a priori mean and covariance of the parameters using [ ] Π k 1 ŵ k 1 ŵ k 1 + γ P wk 1 ŵ k 1 γ P wk 1 (21) where, γ L + λ as in the state filter. The expected measurement matrix, Ψ, is determined using the nonlinear measurement model as follows Ψ k k 1 g (Π k 1, u k 1 ) (22) The mean measurement, ˆd k, and the measurement covariance, P dk d k, are calculated based on the statistics of the expected measurements. ˆd k P dk d k Wi c W m i Ψ i,k k 1 (23) (Ψ i,k k 1 ˆd k ) ( Ψ i,k k 1 ˆd k ) T + Rwk (24) The cross-correlation covariance, P wk d k, is calculated using P wk d k Wi c ( Πi,k k 1 ŵ ) ( k Ψ i,k k 1 ˆd ) T k (25) The Kalman gain matrix is approximated from the cross-correlation and measurement covariances using K wk P wk d k P 1 d k d k (26) The measurement update equations are ŵ + k ŵ k + K w k ( d k ˆd k ) (27) P + w k P w k K wk P dk d k K T w k (28) COUPLED SEQUENTIAL PARAMETER ESTIMATION The simultaneous estimation of states and parameters is not a new problem. Techniques particularly applicable to spacecraft applications capitalize on the work done to couple attitude and parameter 4

5 estimation.[27, 28] However, this work produced time-consuming, batch-estimation techniques not desireable for our application. There are two simple extensions that can be applied to any Kalman Filter. These techniques joint and dual filtering use an analogous filter to estimate the parameters concurrently with the states. The joint method is the simpler of the two to conceptualize: the parameter vector of interest is simply appended onto the true state vector. The time-update for the latter portion of the augmented state vector allows no changes beyond the effects of process noise (i.e., the parameters should be constant) but the entire augmented covariance matrix is propagated as one. [12, 29, 11, 30] The dual filtering technique intertwines a pair of distinct sequential filters, one estimating the true states and the other estimating the parameters. [22, 31, 32, 30, 33, 34, 35, 36, 37] The dual filtering technique is investigated in this work. The primary benefit of dual estimation is the ability to temporarily decouple the parameter filter from the state filter as needed. Decoupling can prevent erratic behavior due to poor measurements or initial estimate of the parameter estimation from causing the state filter to diverge. Spacecraft Attitude Dynamics This section provides a brief review of spacecraft attitude dynamics. The attitude of the spacecraft attitude is represented using quaternions, which are defined q [ â T sin ( ) φ 2 ( ) ] T cos φ 2 (29) where, â is the Euler axis and φ is the Euler angle. A benefit of quaternions is that successive rotations can be represented using quaternion multiplication using q a q b q c (30) R( q a ) R( q b )R( q a ) (31) where, R( q) represents the direction cosine matrix associated with q. The vector operator represents the matrix operation a 4 a 3 a 2 a 1 b 1 a b a 3 a 4 a 1 a 2 b 2 a 2 a 1 a 4 a 3 b 3 (32) a 1 a 2 a 3 a 4 b 4 Another useful quaternion identity is the quaternion inverse, q 1, which represents the equal and opposite rotation represented by q, and is defined as The first time derivative of the quaternion is q 1 [ q 1 q 2 q 3 q 4 ] T (33) q 1 2 ω q (34) where, ω [ ω T 0 ] T is the augmented velocity vector, and ω is the angular velocity vector of the spacecraft. The first time derivative of ω is ω I 1 (g ω Iω) (35) where g is the external torque and I is the moment of inertia matrix. 5

6 Spacecraft Dual Unscented Kalman Filter (SDUKF) In this section the development of the specific equations required for the spacecraft UKF dual estimator is presented. The state and parameter estimators are intertwined by the fact that each requires the information generated by the other. State Estimator The three-element error quaternion is used as the attitude representation for the state UKF. Its use requires the addition of several steps to the UKF equations presented earlier, and those steps are described in this section. The use of the error quaternion is prudent because the UKF determines the time update through a weighted average, which in the case of a full four-element quaternion would not always produce a unit quaternion. The error quaternion does not have this constraint.[26] The state vector used by the dual unscented Kalman Filter is x [ δq T ω T ] T (36) where, δq is the three-element error quaternion, which is defined δ q q ˆ q 1 (37) δq [ ] T δq 1 δq 2 δq 3 (38) The state UKF is initialized with ˆx k 1 [ δˆq k 1 ω T k 1 ] T [ ω T k 1 ] T (39) as in Eq. 1. The sigma points are calculated according to Eq. 3, resulting in χ i,k k 1 [ δq σ (i) T ω σ (i) T ] T for i 1,..., 2n + 1 (40) The δq σ are used to determine the four-element sigma point quaternions, q σ, using q σ i (i) [ δq σ (i) T 1 δqσ (i) T δq σ (i) ] T ˆ q for i 1,..., 2n + 1 (41) The sigma points, q σ and ω σ, are propagated numerically using the state space model, q 1 2 ω q ω I 1 (g ω Iω) The propagated δq σ are calculated using Eqs. 37. The attitude state, q σ and ω σ, are used to calculate the expected measurement of each sigma point as defined by the measurement model, Υ i,k k 1 R( qσ )v1 i R( q σ )v2 i (42) ω σ as in Eq. 10. The direction cosine matrix, R, represents the same rotation as q σ and v i is a vector in the inertial reference frame that can measured by an attitude sensor, such as a sun sensor or magnetometer. The mean expected measurement is calculated using Eq. 12. The measurement update of the state and its covariance follow Eqs. 15 and 16, result in ˆx + k [ δˆq + k ω + ] k. The estimated four-element quaternion can now be determined using ˆ q + k [ δˆq + k 1 [δˆq + k ]T δˆq + k ] ˆ q k 1 (43) 6

7 Parameter Estimator The spacecraft parameter UKF does not require all of the additional intermediate steps required by the state filter. The parameter vector is w [ I 11 I 22 I 33 I 12 I 13 I 23 ] T (44) where I 11, I 22, and I 33 are the second moments of inertia, and I 12, I 13, and I 23 are the products of inertia. The time update is performed using Eqs. 19 and 20. The sigma points are calculated using Eq. 21. The expected measurements are formed by numerically propagating the state space model, Eqs. 34 and 35, using the parameter sigma points, Υ i,k k 1, and the state estimate from the previous iteration provided by the state estimator. The expected measurement is calculated using Eq. 23. The current state estimate from the state estimator is the current measurement, d k ˆx k. The measurement update of the parameters and their covariance is then performed using Eqs. 27 and 28. The updated estimate of the parameters is then feed back into the state estimator for the next iteration of the SDUKF. Figure 1: The percent absolute error of the inertia parameters for the UKF with differing errors in the angular velocity measuremnts. SIMULATION RESULTS The dual UKF developed in the previous section was tested through numeric simulation. The results from a dual EKF are also included to provide a baseline for comparison with the dual UKF. A truth model was created by propagating the state space model for the duration of the simulation. Two vector measurements and three angular rate measurements were calculated based on the truth model. Noise was added to the exact measurements to simulate a real-world sensor. Many simulation 7

8 Figure 2: The percent absolute error of the inertia parameters for the UKF with differing initial inertia errors. runs were performed to determine trends in the performance of the UKF for various initial state and parameter estimates and measurement noise levels. The state estimation results are not presented graphically. However, in all of the simulation runs where the parameters did not diverge the angular velocity and the attitude were determined to within the lowest noise value. The dual UKF did not diverge in any of the simulation runs, while the dual EKF did diverge in several simulation runs. Figure 1 depicts the variation in the performance of the dual UKF in the presence of different angular velocity measurement noise levels. All of the other simulation parameters were held constant. The percent error in the inertia is determine using % Error w estimated w exact w exact 100 (45) As expected, the dual UKF was able to better estimate the parameters when the measurement were less noisy, and the estimates converged in each case. The variation in the performance of the dual UKF using different initial errors in the parameters is shown in Figure 2. The dual UKF was able to converge in each simulation run, which is especially impressive in the case of 69.7% error (the dotted curve). The dual EKF, in stark comparison to the dual UKF, was unable to converge in any case where the initial inertia estimate was more than 11.4% off the actual value. Three families of curves are depicted in Figrue 3. In the bottom two plots, the dual EKF parameter estimate is shown to diverge seriously from the actual value, likely due to the poor initial estimates (32.8% and 69.7%) of the parameters used in those simulation runs. The top plot in the figure shows the simulation runs in which the dual EKF s parameter estimate began to converge to the actual value. The two most successful dual EKF simulation runs are compared to the corresponding dual UKF simulation runs in Figure 4. The solid curves show the performance of the dual UKF and the dotted curves show the performance of the dual EKF. The dual UKF converges to a lower error value than 8

9 Figure 3: The percent absolute error of the inertia parameters for the EKF with differing initial inertia errors. Figure 4: A comparison of the performance of the UKF and the EKF. 9

10 the dual EKF in both cases, although only slightly the simulation using the lowest noise value. The ability of the EKF to keep up with the UKF in that simulation is likely due to the fact that a small initial parameter error and low noise measurements were used. Both conditions would decrease the error caused by truncating the higher order terms as in the EKF. Conclusions A dual unscented Kalman Filter was developed in an attempt to solve the spacecraft dual estimation problem. The theoretical background and motivations of the UKF were briefly discussed. The DUKF was tested through numeric simulation, using simulated noisy measurements. The results from a dual EKF were shown to provide a baseline comparison. The UKF was shown to consistently outperform the EKF. The UKF was able to converge with poor initial estimates of the parameters, while the EKF was shown to have a greater tendency to diverge due to poor initial estimates of the parameters. 10

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