AAS-04-115

UNSCENTED KALMAN FILTERING FOR SPACECRAFTATTITUDE 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 thespacecraft dual estimation problem with greater accuracy than is attain-able with an Extended Kalman Filter (EKF). The EKF is an extensionof the linear Kalman Filter for nonlinear systems. Although the EKF hasbeen used successfully in many nonlinear applications, the performance islimited, due mostly to the truncation of all but first-order terms. The UKFis able to achieve greater estimation performance than the EKF throughthe use of the unscented transformation (UT). The UT allows the UKF tocapture 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, recursivestructure, and mathematical rigour of the derivation of the Kalman Filter make it well-suited andattractive for use in many practical applications. However, many real-world systems, especiallymechanical systems, are nonlinear in nature. The Extended Kalman Filter (EKF) was developed tohelp account for these nonlinearities. The EKF accounts for nonlinearities by linearizing the systemabout its last-known best estimate with the assumption that the error incurred by neglecting thehigher-order terms is small in comparison to the first-order terms. The Kalman Filter measurementupdate 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 EKFis a suboptimal nonlinear filter due to the truncation of the higher-order terms when linearizingthe system. The loss of the higher-order terms can be avoided in the propagation of the stateof the system by using the full nonlinear equations. However, a complete description of the stateconditional probability density requires a potentially unbounded number of parameters.[3] Attemptsto address the first-order approximation shortcomings of the EKF — which can lead to instabilityof 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. Thetwo branches offer improved performance against different sources of error.[10] One technique seeks toimprove the convergence of the first-order filter by iterating at the measurement update step. TheseIterated Extended Kalman Filters (IEKF) reduce the effective measurement noise. As such, they

∗Virginia Space Grant Fellow, Graduate Student. Department of Aerospace & Ocean Engineering, VirginiaPolytechnic Institute & State University, Blacksburg, Virginia 24061. mavandyk@vt.edu. Student Member AIAA,Student Member AAS.

†National Science Foundation Fellow, NASA Graduate Student Researcher Program Fellow. Department ofAerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, Virginia 24061.jana@vt.edu. Student Member AIAA, Student Member AAS.

‡Professor. Department of Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University,Blacksburg, Virginia 24061. cdhall@vt.edu. Associate Fellow AIAA, Member AAS.Copyright c©2004 by the authors. Permission to publish granted to The American Astronautical Society.

1

can be more tolerant to process noise and errors in initial conditions. Iterating in the measurementstep 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 Jacobianrepresentation of the derivatives. These filters can yield drastically improved behavior beyond theconvergence of the EKF for the same order of floating point operations (flops). Complete coverageof this class of filters, termed the Linear Regression Kalman Filters (LRKF), is beyond the scope ofthis paper.[12, 13, 14, 3, 15, 16, 10, 17, 18, 19, 20, 21, 22, 23, 2, 24] Some of these techniques havebeen further enhanced to capitalize on the improvements obtained through iteration in the IEKFand apply similar techniques to these higher-order filters.

The Unscented Kalman Filter (UKF), a member of the LRKF family, attempts to remove some ofthe short-comings of the EKF in the estimation of nonlinear systems. The UKF is an extension of thetraditional Kalman Filter for the estimation of nonlinear systems that implements the unscentedtransformation. The unscented transformation uses a set of sample, or sigma, points that aredetermined from the a priori mean and covariance of the state. The sigma points undergo thenonlinear transformation. The the posterior mean and covariance of the state are determined fromthe transformed sigma points.[14] This approach gives the UKF better convergence characteristicsand greater accuracy than the EKF for nonlinear systems.[3]

The ability of the UKF to accurately estimate nonlinearities makes it attractive for implementa-tion on spacecraft. The state and observation models for spacecraft are inherently nonlinear. TheUKF has recently been documented in its application to spacecraft attitude and orbital dynam-ics; such applications are unusual in a literature dominated by theory papers and neural networkapplications.[25, 26] The main contribution of this paper is the development of an UKF for thespacecraft dual estimation problem. The UKF dual estimator is tested through numeric simulationof a fully actuated rigid body with attitude sensors that provide two noisy vector measurements anda noisy angular velocity vector measurement. An EKF dual estimator is also developed and is usedto gauge the performance of the UKF dual estimator.

THE UNSCENTED KALMAN FILTER

The UKF was developed with the underlying assumption that approximating a Gaussian distributionis easier than approximating a nonlinear transformation.[14] The UKF uses deterministic sampling toapproximate the state distribution as a GRV. The sigma points are chosen to capture the true meanand covariance of the state distribution. The sigma points are propagated through the nonlinearsystem. 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 isonly able to obtain first-order accuracy.[2] Therefore, the UKF provides better state estimates fornonlinear systems.

The proceeding summary of the UKF equations follows the presentation by Wan and van derMerwe[2]. The system and measurement noise are assumed to be zero-mean, or additive. Thezero-mean noise implementation of the UKF does not require the augmenting of the state vectorwith the noise variables, thus decreasing the number of points that are required to be propagatedthrough the nonlinear system from 2(L + q) to 2L.

State Estimation

The filter is initialized with the predicted mean and covariance of the state.

x(t0) = E{x0

}(1)

Px0 = E{(

x(t0)− x0

)(x(t0)− x0

)T}

(2)

2

The sigma points are calculated from the a priori mean and covariance of the state using

χk−1 =[xk−1 xk−1 + γ

√Pxk−1

xk−1 − γ√

Pxk−1

](3)

where, γ =√

L + λ. The sigma points are propagated through the nonlinear system

χk|k−1 = F(χk−1,uk−1

)(4)

The posterior mean, x−k , and covariance, P−xk

, are determined from the statistics of the propagatedsigma points as follows

x−k =2L∑i=0

Wmi χi,k|k−1 (5)

P−xk

=2L∑i=0

W ci

(χi,k|k−1 − x−k

) (χi,k|k−1 − x−k

)T

(6)

The weights, Wmi and W c

i , are calculated using

Wm0 =

λ

L + λ(7)

W c0 =

λ

L + λ+ 1− α2 + β (8)

W ci = Wm

i =1

2 (L + λ)(9)

The estimated measurement matrix, Υk|k−1, is calculated by transforming the sigma points usingthe nonlinear measurement model,

Υk|k−1 = H(χk|k−1

)(10)

The mean measurement, y−k , and the measurement covariance, Pykyk, are calculated based on the

statistics of the transformed sigma points.

yk =2L∑i=0

Wmi Υi,k|k−1 (11)

Pykyk=

2L∑i=0

W ci

(Υi,k|k−1 − yk

) (Υi,k|k−1 − yk

)T + Rxk(12)

The cross-correlation covariance, Pxkyk, is calculated using

Pxkxk=

2L∑i=0

W ci

(χi,k|k−1 − x−k

) (Υi,k|k−1 − y−k

)T(13)

The Kalman gain matrix is approximated from the cross-correlation and measurement covariancesusing

Kxk= Pxkyk

P−1ykyk

(14)

The measurement update equations used to determine the mean, xk, and covariance, Pxk, of the

filtered state are

xk = x−k + Kk

(yk − y−k

)(15)

Pxk= P−

xk−Kxk

PykykKT

xk(16)

3

Parameter Estimation

The parameter estimation equations for the UKF are similar to those for the state estimation. Thissection expounds upon the differences.

The filter is initialized with the predicted mean and covariance of the parameters.

w(t0) = E{w0

}(17)

Pw0 = E{(

w(t0)− w0

)(w(t0)− w0

)T}

(18)

The time update of the parameter vector and the parameter covariance is performed using

w−k = wk−1 (19)

P−wk

= Pwk−1+ Qwk

(20)

where, Qw is the system process noise of the time update. The sigma points are calculated fromthe a priori mean and covariance of the parameters using

Πk−1 =[wk−1 wk−1 + γ

√Pwk−1

wk−1 − γ√

Pwk−1

](21)

where, γ =√

L + λ as in the state filter. The expected measurement matrix, Ψ, is determined usingthe nonlinear measurement model as follows

Ψk|k−1 = g (Πk−1,uk−1) (22)

The mean measurement, dk, and the measurement covariance, Pdkdk, are calculated based on the

statistics of the expected measurements.

dk =2L∑i=0

Wmi Ψi,k|k−1 (23)

Pdkdk=

2L∑i=0

W ci

(Ψi,k|k−1 − dk

) (Ψi,k|k−1 − dk

)T

+ Rwk(24)

The cross-correlation covariance, Pwkdk, is calculated using

Pwkdk=

2L∑i=0

W ci

(Πi,k|k−1 − w−

k

) (Ψi,k|k−1 − dk

)T

(25)

The Kalman gain matrix is approximated from the cross-correlation and measurement covariancesusing

Kwk= Pwkdk

P−1dkdk

(26)

The measurement update equations are

w+k = w−

k + Kwk

(dk − dk

)(27)

P+wk

= P−wk−Kwk

PdkdkKT

wk(28)

COUPLED SEQUENTIAL PARAMETER ESTIMATION

The simultaneous estimation of states and parameters is not a new problem. Techniques particularlyapplicable to spacecraft applications capitalize on the work done to couple attitude and parameter

4

estimation.[27, 28] However, this work produced time-consuming, batch-estimation techniques notdesireable for our application.

There are two simple extensions that can be applied to any Kalman Filter. These techniques — jointand 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 simplyappended onto the true state vector. The time-update for the latter portion of the augmented statevector 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 filteringtechnique intertwines a pair of distinct sequential filters, one estimating the true states and theother estimating the parameters. [22, 31, 32, 30, 33, 34, 35, 36, 37] The dual filtering techniqueis investigated in this work. The primary benefit of dual estimation is the ability to temporarilydecouple the parameter filter from the state filter as needed. Decoupling can prevent erratic behaviordue to poor measurements or initial estimate of the parameter estimation from causing the statefilter to diverge.

Spacecraft Attitude Dynamics

This section provides a brief review of spacecraft attitude dynamics. The attitude of the spacecraftattitude is represented using quaternions, which are defined

q =[

aT sin(

φ2

)cos

(φ2

) ]T

(29)

where, a is the Euler axis and φ is the Euler angle. A benefit of quaternions is that successiverotations can be represented using quaternion multiplication using

qa = qb ⊗ qc (30)R(qa) = R(qb)R(qa) (31)

where, R(q) represents the direction cosine matrix associated with q. The ⊗ vector operator repre-sents the matrix operation

a⊗ b =

a4 a3 −a2 a1

−a3 a4 a1 a2

a2 −a1 a4 a3

−a1 −a2 −a3 a4

b1

b2

b3

b4

(32)

Another useful quaternion identity is the quaternion inverse, q−1, which represents the equal andopposite rotation represented by q, and is defined as

q−1 =[

q1 q2 q3 −q4

]T (33)

The first time derivative of the quaternion is

˙q =12

ω ⊗ q (34)

where, ω =[

ωT 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

Spacecraft Dual Unscented Kalman Filter (SDUKF)

In this section the development of the specific equations required for the spacecraft UKF dualestimator is presented. The state and parameter estimators are intertwined by the fact that eachrequires the information generated by the other.

State Estimator

The three-element error quaternion is used as the attitude representation for the state UKF. Its userequires the addition of several steps to the UKF equations presented earlier, and those steps aredescribed in this section. The use of the error quaternion is prudent because the UKF determinesthe time update through a weighted average, which in the case of a full four-element quaternionwould 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 =[

δqT ωT]T

(36)

where, δq is the three-element error quaternion, which is defined

δq = q⊗ ˆq−1 (37)

δq =[

δq1 δq2 δq3

]T (38)

The state UKF is initialized with

xk−1 =[

δqk−1 ωTk−1

]T=

[0 0 0 ωT

k−1

]T(39)

as in Eq. 1. The sigma points are calculated according to Eq. 3, resulting in

χi,k|k−1 =[

δqσ(i)T ωσ(i)T]T for i = 1, ..., 2n + 1 (40)

The δqσ are used to determine the four-element sigma point quaternions, qσ, using

qσi (i) =

[δqσ(i)T

√1− δqσ(i)Tδqσ(i)

]T ⊗ ˆq for i = 1, ..., 2n + 1 (41)

The sigma points, qσ and ωσ, are propagated numerically using the state space model,

˙q =12

ω ⊗ q

ω = I−1 (g − ω × Iω)

The propagated δqσ are calculated using Eqs. 37. The attitude state, qσ and ωσ, are used tocalculate the expected measurement of each sigma point as defined by the measurement model,

Υi,k|k−1 =

R(qσ)vi1

R(qσ)vi2

ωσ

(42)

as in Eq. 10. The direction cosine matrix, R, represents the same rotation as qσ and vi is a vectorin the inertial reference frame that can measured by an attitude sensor, such as a sun sensor ormagnetometer. The mean expected measurement is calculated using Eq. 12. The measurementupdate of the state and its covariance follow Eqs. 15 and 16, result in x+

k =[

δq+k ω+

k

]. The

estimated four-element quaternion can now be determined using

ˆq+k =

[δq+

k

√1− [δq+

k ]Tδq+k

]⊗ ˆqk−1 (43)

6

Parameter Estimator

The spacecraft parameter UKF does not require all of the additional intermediate steps required bythe state filter. The parameter vector is

w =[

I11 I22 I33 I12 I13 I23

]T (44)

where I11, I22, and I33 are the second moments of inertia, and I12, I13, and I23 are the products ofinertia. The time update is performed using Eqs. 19 and 20. The sigma points are calculated usingEq. 21.

The expected measurements are formed by numerically propagating the state space model, Eqs.34 and 35, using the parameter sigma points, Υi,k|k−1, and the state estimate from the previousiteration 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, dk = xk. Themeasurement update of the parameters and their covariance is then performed using Eqs. 27 and28. The updated estimate of the parameters is then feed back into the state estimator for the nextiteration of the SDUKF.

Figure 1: The percent absolute error of the inertia parameters for the UKF with differing errors inthe angular velocity measuremnts.

SIMULATION RESULTS

The dual UKF developed in the previous section was tested through numeric simulation. The resultsfrom a dual EKF are also included to provide a baseline for comparison with the dual UKF. A truthmodel was created by propagating the state space model for the duration of the simulation. Twovector 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

Figure 2: The percent absolute error of the inertia parameters for the UKF with differing initialinertia errors.

runs were performed to determine trends in the performance of the UKF for various initial state andparameter estimates and measurement noise levels. The state estimation results are not presentedgraphically. However, in all of the simulation runs where the parameters did not diverge the angularvelocity and the attitude were determined to within the lowest noise value. The dual UKF did notdiverge 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 angularvelocity measurement noise levels. All of the other simulation parameters were held constant. Thepercent error in the inertia is determine using

% Error =||westimated −wexact||

||wexact||× 100 (45)

As expected, the dual UKF was able to better estimate the parameters when the measurement wereless noisy, and the estimates converged in each case. The variation in the performance of the dualUKF using different initial errors in the parameters is shown in Figure 2. The dual UKF was ableto converge in each simulation run, which is especially impressive in the case of 69.7% error (thedotted curve).

The dual EKF, in stark comparison to the dual UKF, was unable to converge in any case wherethe initial inertia estimate was more than 11.4% off the actual value. Three families of curves aredepicted in Figrue 3. In the bottom two plots, the dual EKF parameter estimate is shown to divergeseriously from the actual value, likely due to the poor initial estimates (32.8% and 69.7%) of theparameters used in those simulation runs. The top plot in the figure shows the simulation runs inwhich 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 UKFsimulation runs in Figure 4. The solid curves show the performance of the dual UKF and the dottedcurves show the performance of the dual EKF. The dual UKF converges to a lower error value than

8

Figure 3: The percent absolute error of the inertia parameters for the EKF with differing initialinertia errors.

Figure 4: A comparison of the performance of the UKF and the EKF.

9

the dual EKF in both cases, although only slightly the simulation using the lowest noise value. Theability of the EKF to keep up with the UKF in that simulation is likely due to the fact that a smallinitial parameter error and low noise measurements were used. Both conditions would decrease theerror 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 estimationproblem. The theoretical background and motivations of the UKF were briefly discussed. TheDUKF was tested through numeric simulation, using simulated noisy measurements. The resultsfrom a dual EKF were shown to provide a baseline comparison. The UKF was shown to consistentlyoutperform 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 theparameters.

10

REFERENCES

[1] R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Transactionsof the ASME–Journal of Basic Engineering, D, vol. 82, pp. 35–45, 1960.

[2] E. A. Wan and R. van der Merwe, Kalman Filtering and Neural Networks, ch. 7, The UnscentedKalman Filter. Wiley, September 2001.

[3] S. J. Julier and J. K. Uhlmann, “A New Extension of the Kalman Filter to Nonlinear Systems,”in Proceedings of the SPIE AeroSense International Symposium on Aerospace/Defense Sensing,Simulation and Controls, (Orlando, Florida), April 20–25, 1997.

[4] R. Kalman and R. Bucy, “New Results in Linear Filtering and Prediction Theory,” Transactionsof the ASME–Journal of Basic Engineering, D, vol. 83, pp. 95–108, March 1961.

[5] R. Bucy, “Nonlinear Filtering Theory,” IEEE Transactions on Automatic Control, vol. AC-10,p. 198, April 1965.

[6] M. Athans, R. P. Wishner, and A. Bertolini, “Suboptimal State Estimation for Continuous-Time Nonlinear Systems from Discrete Noisy Measurements,” IEEE Transactions on AutomaticControl, vol. 13, October 1968.

[7] R. Bass, V. Norum, and L. Schwartz, “Optimal Multichannel Nonlinear Filtering,” Journal ofMathematical Analysis and Applications, vol. 16, pp. 152–164, 1966.

[8] H. Kushner, “Dynamical Equations for Optimal Nonlinear Filtering,” Journal of DifferentialEquations, vol. 3, pp. 179–190, 1967.

[9] H. Kushner, “Approximations to Optimal Non-Linear Filters,” in Proceedings of the IEEE JointAutomatic Control Conference, pp. 613–623, June 1967.

[10] T. Lefebvre, H. Bruyninckx, and J. D. Schutter, “Kalman Filters for Nonlinear Systems: AComparison of Performance,” Internal Report 01R033, Department of Mechanical Engineering,Katholieke Universiteit, Leuven, Belgium, October 2001. Submitted as Regular Paper to IEEETransactions on Automatic Control, October 2001.

[11] A. Gelb, ed., Applied Optimal Estimation. The M.I.T. Press, 1974.

[12] J. L. Crassidis and J. L. Junkins, Optimal Estimation of Dynamic Systems. Boca Raton, Florida:CRC Press, to be published 2004.

[13] S. J. Julier, J. K. Uhlmann, and H. F. Durrant-Whyte, “A New Approach for Filtering NonlinearSystems,” in Proceedings of the American Control Conference, vol. 3, pp. 1628–1632, June 21–231995.

[14] S. J. Julier and J. K. Uhlmann, “A General Method for Approximating Nonlinear Transfor-mations of Probability Distributions,” tech. rep., Robotics Research Group, Department ofEngineering Science, University of Oxford, 1994.

[15] S. J. Julier and J. K. Uhlmann, “A Non-divergent Estimation Algorithm in the Presence of Un-known Correlations,” in Proceedings of the American Control Conference, vol. 4, (Albuquerque,New Mexico), pp. 2369–2373, June 4–6, 1997.

[16] S. J. Julier, “The Scaled Unscented Transformation,” in Proceedings of the American ControlConference, vol. 6, pp. 4555–4559, 2002.

[17] T. Lefebvre, H. Bruyninckx, and J. D. Schutter, “A Non-Minimal State Kalman Filter forNonlinear Parameter Estimation Applied to Autonomous Compliant Motion,” in Proceedingsof the IEEE International Conference on Robotics and Automation, (Taipei, Taiwan), May12–17, 2003.

11

[18] M. Nørgaard, N. K. Poulsen, and O. Ravn, “Advances in Derivative-Free State Estimation forNonlinear Systems,” Tech. Rep. IMM-REP-1998-15, Technical University of Denmark, 2800Lyngby, Denmark, April 7, 2000.

[19] M. Nørgaard, N. K. Poulsen, and O. Ravn, “New Developments in State Estimation for Non-linear Systems,” Automatica, vol. 36, pp. 1627–1638, 2000.

[20] T. S. Schei, “A Finite-Difference Method for Linearization in Nonlinear Estimation Algorithms,”Automatica, vol. 33, no. 11, pp. 2053–2058, 1997.

[21] E. A. Wan and R. van der Merwe, “The Unscented Kalman Filter for Nonlinear Estimation,”in Proceedings of the IEEE Symposium 2000: Adaptive Systems for Signal Processing, Commu-nications, and Control, (Lake Louise, Alberta, Canada), October 1–4, 2000.

[22] E. A. Wan, R. van der Merwe, and A. T. Nelson, Advances in Neural Information ProcessingSystems 12, ch. Dual Estimation and the Unscented Transformation, pp. 666–672. MIT Press,2000.

[23] R. van der Merwe and E. A. Wan, “Efficient Derivative-Free Kalman Filters for Online Learn-ing,” in Proceedings of European Symposium on Artificial Neural Networks, (Bruges, Belgium),April 2001.

[24] R. L. Bellaire, E. W. Kamen, and S. M. Zabin, “A New Nonlinear Iterated Filter with Applica-tions to Target Tracking,” in Proceedings of the International Society for Optical Engineering(SPIE) Conference on Signal and Data Processing of Small Targets, vol. 2561, pp. 240–251,1995.

[25] D.-J. Lee and K. T. Alfriend, “Precise Real-Time Orbit Estimation Using the Unscented KalmanFilter,” in Proceedings of the 13th AAS/AIAA Space Flight Mechanics Winter Meeting, no. 03-230, (Ponce, Puerto Rico), February 9–13, 2003.

[26] J. L. Crassidis and F. L. Markley, “Unscented Filtering for Spacecraft Attitude Estimation,”Journal of Guidance, Control, and Dynamics, vol. 26, July–August 2003.

[27] F. L. Markley, “Attitude Determination and Parameter Estimation Using Vector Observations:Theory,” Journal of the Astronautical Sciences, vol. 37, pp. 41–58, January–March 1989.

[28] F. L. Markley, “Attitude Determination and Parameter Estimation Using Vector Observations:Application,” Journal of the Astronautical Sciences, vol. 39, pp. 367–382, July–September 1991.

[29] R. F. Stengel, Optimal Control and Estimation. Dover Publications, Inc., September 1994.

[30] A. T. Nelson, “Nonlinear Estimation and Modeling of Noisy Time-Series by Dual KalmanFiltering Methods,” Doctor of Philosopy, Oregon Graduate Institute of Science and Technology,September 2000.

[31] E. A. Wan and A. T. Nelson, Kalman Filtering and Neural Networks, ch. 5, Dual ExtendedKalman Filter Methods. Wiley, September 2001.

[32] S. Haykin, ed., Kalman Filtering and Neural Netowrks. John Wiley & Sons, Inc., 2001.

[33] A. T. Nelson and E. A. Wan, “A Two-Observation Kalman Framework for Maximum-LikelihoodModelling of Noisy Time Series,” in Proceedings of the IEEE Interational Joint Conference onNeural Networks, pp. 2489–2494, 1998.

[34] L. W. Nelson and E. Stear, “The Simultaneous On-Line Estimation fo Parameters and Statesin Linear Systems,” IEEE Transactions on Automatic Control, vol. AC-21, no. 2, pp. 94–98,1976.

12

[35] E. A. Wan and A. T. Nelson, “Neural Dual Extended Kalman Filtering: Applications in SpeechEnhancement and Monaural Blind Signal Separation,” in Proceedings of the IEEE Workshopon Neural Networks for Signal Processing VII, (Florida), September 1997.

[36] E. A. Wan and A. T. Nelson, Advances in Neural Information Processing Systems 9, ch. DualKalman Filtering Methods for Nonlinear Prediction, Smoothing, and Estimation, pp. 793–799.MIT Press, 1997.

[37] J. L. Schwartz and C. D. Hall, “Comparison of System Identification Techniques for a SphericalAir-Bearing Spacecraft Simulator,” in Proceedings of the AAS/AIAA Astrodynamics SpecialistConference, no. 03-611, (Big Sky, Montana), August 3–7, 2003.

13