Spacecraft Attitude Rate Estimation From Geomagnetic Field Measurements. Mark L. Psiaki. Cornell University, Ithaca, N.Y

Transcription

1 Spacecraft Attitude Rate Estimation From Geomagnetic Field Measurements Mar L. Psiai Cornell University, Ithaca, N.Y Yaaov Oshman # echnion Israel Institute of echnology, Haifa 32, Israel Astract his paper develops methods for estimating the rotation rate of a spacecraft using only measured magnetic field data. he goal is to provide rate information for use in applications such as de-tumling, nutation damping, and momentum management without using gyroscopes. wo algorithms are developed, a deterministic algorithm and an extended Kalman filter. Both algorithms employ the magnetic field direction inematics equation and Euler's equation for attitude motion of a rigid ody with momentum wheels. Neither algorithm requires a model of the Earth's magnetic field. he deterministic algorithm solves a nonlinear least-squares prolem for the unnown angular momentum component along the magnetic field direction. he extended Kalman filter estimates the attitude rate vector, corrections to 5 of the 6 inertia matrix elements, and 2 error states of the measured magnetic field direction. It uses an initial rate estimate from the deterministic algorithm in order to avoid divergence. he algorithms have een tested using data from a spinning sounding rocet. hey achieve initial accuracies in the range 2-7 deg/s when the Associate Professor, Siley School of Mechanical and Aerospace Engineering. Associate Fellow, AIAA. # Associate Professor, Department of Aerospace Engineering. Memer, Asher Space Research Institute. Associate Fellow, AIAA.

2 rocet spins at aout 8 deg/s, and their accuracies improve to 1-2 deg/s after the spin rate decays to 2 deg/s. I. Introduction Many spacecraft need estimates of their rotation rates. he rotation rate may e used to apply a control that stops tumling, to manage the total system angular momentum, to aid a star tracer, or as part of an attitude determination system. he most common method of determining attitude rate is y direct measurement using rate gyros. his paper develops 2 new attitude rate estimation methods that rely on Earth magnetic field measurements rather than on rate gyro measurements. here are several important reasons to avoid the use of rate gyros. Gyros are expensive and failure-prone. hey have significant mass and consume a significant amount of power. he desire to manufacture small, light-weight, inexpensive, and reliale satellites militates against the use of rate gyros. Gyro-less rate estimation schemes can e important even for missions that include rate gyros ecause these schemes provide a ac-up capaility. A numer of methods have een developed to estimate attitude rate without using rate gyro data. Some methods wor directly with vector attitude data and use either deterministic differentiation techniques or Kalman filtering techniques to estimate attitude rate 1-3. Another method uses full 3-axis attitude estimates as inputs to a rate estimation filter 4. Reference 5 considers oth types of measurements and compares filters to derivative-ased estimators. Another class of methods estimates attitude and attitude rate simultaneously using vector and scalar attitude measurements ut not rate gyros. hese include deterministic methods that employ differentiation techniques 6,7, and filter-ased techniques 6-1. A significant aspect of the rate estimation techniques of Ref. 3 is that they are completely 2

3 independent of attitude nowledge. he estimators of Refs. 4 and 6-1 explicitly use attitude information. he estimators of Refs. 1, 2, and 5 use 3-axis attitude nowledge implicitly in order to transform the inertial time derivative of an attitude reference vector into ody coordinates. he techniques of Ref. 3 assume that the measured attitude reference vector is fixed in inertial space; thus, there is no inertial time derivative that needs to e transformed into ody coordinates. he only other nown gyro-less rate estimation algorithm that is independent of attitude nowledge is the high-rate-limit algorithm of Ref. 7. Note that attitude-independent rate estimation techniques are suitale for applications such as de-tumling control, momentum management, or nutation damping, ut their outputs should not e used lie rate gyro outputs in an attitude estimation filter. Reference 3 estimates the attitude rate y using a time series of measurements of the Earth's magnetic field vector. Its methods do not require nowledge of the magnetic field direction in inertial coordinates. his feature is a great advantage ecause it allows the estimators to wor without spacecraft position nowledge and without a complicated spherical harmonic model of the Earth's magnetic field. his attitude reference vector, however, has a non-zero inertial rotation rate as a spacecraft moves along its orit, which violates the paper's assumption. Fortunately, this rotation rate is normally less than.12 deg/s, which implies that the method can wor effectively if the required attitude rate resolution is no finer than this lower ound. he approach of Ref. 3 for dealing with nonlinearities is similar to that of Refs. 6 and 7, ut maredly different from that of Refs. 1, 2, and 4. References 3, 6, and 7 use deterministic atch algorithms in conjunction with extended Kalman filters. he atch algorithms are less accurate, ut they deal directly with nonlinearities and cannot diverge. heir outputs can e used to initialize an extended Kalman filter (EKF) near enough to the true state so as to avoid divergence. he techniques of Refs. 1, 2, and 4, on the other hand, use unusual Kalman filters that try to 3

4 maintain linearity y maing various approximations. hese have the advantage that they are less liely to diverge and therefore do not require special algorithms for initialization. heir estimates, however, are liely to e less accurate than those of a converged EKF. he present paper develops improved versions of Ref. 3's two rate estimation algorithms, the deterministic differentiation-ased algorithm and the Kalman filter. he new deterministic algorithm wors directly with Euler's equation to determine the attitude rate along the magnetic field direction, and it uses an improved solution technique that incorporates a gloal nonlinear least-squares solver. he new Kalman filter improves on the Ref. 3 filter y explicitly accounting for the normalization constraint on the measured magnetic field direction vector. It also augments the filter state to include 5 of the 6 inertia matrix elements. hese improvements enale the approach of Ref. 3 to wor well with actual flight data. his paper's algorithms have een tested using flight data from a spinning spacecraft, whereas Ref. 3 tested its techniques using only simulated data. his paper presents its methods and results in the next 5 sections. Section II presents dynamic models of the attitude rate vector and of the magnetic field direction vector. hese models are used to develop the two attitude rate estimation algorithms. Section III develops the deterministic attitude rate estimator. Section IV analyzes the oservaility of the system and determines two cases in which the attitude rate cannot e estimated ased on magnetic field measurements only. Section V descries the extended Kalman filter attitude rate estimator. Section VI presents the results of tests of the two algorithms on actual flight data, and Section VII presents the paper's conclusions. 4

5 II. Dynamic Models for the Attitude Rate and Magnetic Field Direction Vectors Rigid-Body Attitude Dynamics Model Both of the rate estimators use Euler's equations to model the attitude rate vector's dynamics. he model includes a main rigid ody and momentum wheels. It taes the form I m w & h & w ( Iw h) = w n (1) In this equation, I m is the mass moment of inertia matrix, w is the angular velocity vector, h is the angular momentum vector of the momentum wheels, and w n is the net external torque. All of these quantities are defined in spacecraft-fixed coordinates. wo different models of the net external torque are used for the two different estimation algorithms. he deterministic algorithm assumes that w n =. he Kalman filter models w n (t) as eing a white-noise process disturance. hese assumptions could e relaxed in order to include explicit external torque models, ut such models would complicate the estimation algorithms. Fortunately, external torques can e neglected or modeled as white noise in many circumstances without seriously degrading the attitude rate estimation accuracy. Magnetic Field Kinematic Model he estimation algorithms use a inematic model of the motion of the magnetic field unit direction vector. Its taes the form: ˆ & w ˆ = w ˆ ω (2) In this equation, ˆ is the ody-axes magnetic field direction vector, and the vector w ω is the rotation rate of the magnetic field direction vector as measured with respect to inertial coordinates. All of these vectors are expressed in spacecraft ody coordinates. he two estimation algorithms use two different models for w ω. he deterministic algorithm 5

6 assumes that w ω =. In a typical low-earth orit, this assumption causes a maximum attitude rate error on the order of.12 deg/s ecause the magnetic field rotates with respect to inertial space at a rate of twice per orit if the orit passes over the magnetic poles. At lower magnetic inclinations, the field's inertial rotation rate is smaller, which reduces the error of assuming that w ω =. he Kalman filter models w ω (t) as a white noise process disturance. his model allows the filter to gradually de-weight old magnetic field measurements in order to deal with its uncertainty aout the field's slow inertial rotation rate. he white-noise model proaly causes levels of inaccuracy that are similar to those which are caused y the w ω = model. Constant Projection of the Angular Momentum Along the Magnetic Field Direction he deterministic algorithm effectively assumes that the angular momentum vector and the magnetic field direction vector remain fixed in inertial space. Under this assumption, the projection of the angular momentum along the magnetic field remains constant: ˆ ( I m w h) = L, a constant (3) where the scalar constant L is the angular momentum component along the magnetic field. III. Deterministic Attitude Rate Estimation Algorithm wo Components of Attitude Rate from Magnetic Field Kinematics he magnetic field inematics equation, Eq. (2), can e used to determine the components of the attitude rate ω that are perpendicular to the measured magnetic field. If one assumes that w ω =, then Eq. (2) can e used to derive the following expression for the angular rate: w ˆ ˆ & = α ˆ (4) where α = ˆ w is the unnown angular rate component parallel to the measured magnetic field direction. Equation (4) can e derived from Eq. (2) y taing the cross product of oth sides of 6

7 Eq. (2) with ˆ while recognizing that (w ˆ ) ˆ = ( ˆˆ I) w, where I is the identity matrix. A word is in order aout how the time derivative is computed for the measured magnetic field direction vector. he actual measured data consists of, 1, 2,, a sequence of magnetic field vectors that are measured at the sample times t, t 1, t 2, heses raw measurements get processed using atch curve fitting techniques in order to determine ˆ and ˆ &. he atch process wors with all of the magnetic field vectors that have een measured within ±.5 derv seconds of the sample time t. It fits each element of the measured field vector to a quadratic polynomial in t. hese polynomials and their time derivative are evaluated at time t to produce smoothed estimates of ( t ) and & ( t ). he smoothed estimate of ( t ) gets normalized to produce the ˆ value that is used in this paper's deterministic estimation algorithm. he time derivative of the normalization formula is used along with the smoothed estimate of & ( t ) to compute ˆ & for use in the deterministic rate estimator. Care must e taen to use a reasonale length for the quadratic curve-fitting window, derv. If derv is too large, then a quadratic model of (t) will e poor, ut if derv is too small, then the estimated ˆ & will e very noisy. Parameterization of the Rate in erms of an Angular Momentum Component he unnown angular rate component α will e time-varying in the general case, and one wants to e ale to determine α at any sample time t. It is convenient to express this time-varying unnown in terms of the constant angular momentum component along the magnetic field, L. his can e accomplished y sustituting Eq. (4)'s expression for w into Eq. (3) and solving for α. he resulting formula is 7

8 8 m m ˆ I ˆ t ˆ ˆ I ˆ L h )} ( ] [ { = & α (5) Equation (5) allows one to express the time-varying angular velocity in terms of the constant L. If one sustitutes Eq. (5) into Eq. (4), the result is L t h g w w = = ) ( (6) where the vectors g and h are defined y the formulae = m m ˆ I ˆ t ˆ ˆ I ˆ ˆ ˆ ˆ h )} ( ] [ { & & g (7a) = m ˆ I ˆ ˆ 1 h (7) hus, the goal of estimating the attitude rate reduces to the goal of estimating the constant L. A Least-Squares Prolem from rapezoidal Integration of Euler's Equation he unnown angular momentum component L can e determined y using Euler's equation, Eq. (1). If one multiplies Eq. (1) y (I m ) -1, integrates it trapezoidally from sample time t to sample time t 1, sets w n =, and uses Eq. (6) to eliminate w and w 1 from the result, then the equation taes the following vector quadratic form = 2 L L e d c (8) where the vectors c, d, and e are defined y: [ ] ) ( ) ( 2 1 m 1 m 1 m I I I t h h h h = c (9a) ] [ )] ( [ 2 1 { 1 m m 1 m 1 I t I I t = h g g h h h h d ]} [ )] ( [ m m I t I h g g h h (9)

9 e 1 t [ ( ) - ( )] = g 1 g I m h t 1 h t Im 1 { g 1 [ Img 1 h( t 1)] 2 g [ Img h( t )]} (9c) and where t = t 1 - t is the interval etween the two sample times. he use of trapezoidal integration to derive Eq. (8) implies that t must e small compared to typical spin and nutation periods; otherwise, the trapezoidal approximation will produce poor results. he deterministic algorithm estimates L y simultaneously solving multiple versions of Eq. (8) that arise from multiple sample intervals. he resulting over-determined system of nonlinear equations is solved in an approximate least-squares sense. his is eneficial ecause the increased amount of data decreases the effect of noise on the accuracy of the computed L. Suppose that one wants to determine w y computing L and sustituting it in Eq. (6). Suppose, also, that one wants to use data that falls within ±.5 L seconds of t. One first determines j min = [the minimum j such that (t -.5 L ) t j ] and j max = [the maximum j such that t j1 (t.5 L )]. Next, one sees the least-squares solution of the following over-determined system of quadratic equations: 2 cig L digl eig = (1) where the vectors c ig, d ig, and e ig are defined y: cig = c, c, c, K, c jmin ( jmin 1) ( jmin 2) j (11a) max dig = d, d, d, K, d jmin ( jmin 1) ( jmin 2) j (11) max eig = e, e, e, K, e jmin ( jmin 1) ( jmin 2) j (11c) max he over-determined system of equations in Eq. (1) can e solved in a weighted least- 9

10 squares sense. he weighted least-squares cost function taes the form J ( L ) = ( c igl digl eig) Pig( cigl digl eig) 2 (12) he positive definite weighting matrix -1 P ig has een chosen to equal the inverse covariance of a statistical error model for Eq. (1). In this case, the cost function J(L ) can e viewed as the negative of a log-lielihood function, and the minimizing L as a maximum lielihood estimate. he matrix P ig models the correlations etween the errors in Eq (8) for the sample intervals = j min to = j max. here are correlations among the components of the error vector for a given sample index, and there are correlations with the errors of the neighoring sample intervals -1 and 1. he covariance matrix taes the following loc tri-diagonal form: P ig Pj Q j = M min min P Q Q ( j ( j j min min min M 1) 1) Q P Q ( j ( j ( j min min min M 1) 2) 2) Q P ( j ( j min min M 2) 3) K K K K O K P j M max (13) where P = Ptrue( ) {.1 race[ Ptrue( ) ]} I (14a) 2 Ptrue( ) = σ [ I.5 tc 1][ I D 1][ I D 1] [ I.5 tc 1] Q 2 σ [ I.5 tc ][ I D ][ I D ] [ I.5 tc ] 2 = σ [ I.5 tc 1][ I D 1][ I D 1] [ I.5 t 1C 1] (14) (14c) are 3 3 covariance and cross-covariance matrices. he quantity σ is the standard deviation of the errors in the components of ˆ & that are perpendicular to ˆ. he two 3 3 matrices C and D that 1

11 appear in Eqs. (14) and (14c) are defined as follows -1 C = Im ( [{ Img h( t )} ] [ g ] Im ) (15a) D = ˆ ˆ ˆ Im ˆ Im (15) where the notation [z ] indicates the 3 3 cross-product-equivalent matrix that is associated with the 3-dimensional vector z. he ad hoc trace term in Eq. (14a) enforces positive definiteness so that -1 P ig will exist. Without this modification, P ig fails to e positive definite ecause the underlying Euler's equation has zero error in the direction of the magnetic field ˆ due to the use of Eq. (3). he error covariance model in Eqs. (13-15) assumes that the dominant errors are those of the "measured" magnetic field direction time derivative ˆ &. he error model assumes that 2 σ [ I ˆ ˆ ] is the covariance matrix of the noise in ˆ &. his model gets used to calculate the net error in Eq. (8). he resulting net error includes linear noise, quadratic noise, and noise that multiplies L. he latter two noise terms are difficult to manipulate and have een neglected. Only the linear error term has een used in the covariance model. It is possile to use an alternate weighting matrix in the least-squares cost function of Eq. (12). he aove-defined -1 P ig weighting matrix is the est of several alternative weightings that have een tried for the attitude rate estimation case that is reported in Section VI. Alternate weighting matrices might wor etter in different situations. Gloal Solution of the Least-Squares Prolem he gloal minimum of the least-squares cost function in Eq. (12) can e computed analytically. he cost function reduces to a quartic polynomial in L 11

12 J ( L ) = a4l a3l a2l a1l a (16) whose coefficients are: a = c ig -1 ig P c -1 a 3 = digpigcig 2 ig -1 ig ig ig a = e P c -1 a 1 = eigpigdig 1 2 d ig P -1 ig d ig (17a) (17) (17c) (17d) a = 1 e 2 ig P -1 ig e ig (17e) he gloal minimum of this function can e computed y solving the first-order necessary condition for an optimum: dj 3 2 = = 4a4L 3a3L 2a2L a dl 1 (18) Any cuic polynomial can e solved analytically. It may have only one real solution, or it may have 3 real solutions. One of the real roots is guaranteed to e the gloal minimum of the leastsquares prolem ecause a 4 >. If there is just one real solution to Eq. (18), then it is the optimal L estimate. If there are three real solutions, then one is the gloal minimum of J(L ), one is a local minimum, and one is a local maximum. he local maximum lies etween the local minimum and the gloal minimum. he value of J(L ) at the two extreme real solutions can e used to evaluate which of them is the gloal minimum. he gloal minimum of J(L ) might not yield the est attitude rate estimate if there is a second local minimum of the cost function. his can occur ecause of measurement noise and modeling error effects. Such a situation indeed occurs in some of the experimental data that is 12

13 discussed in Section VI. In this case, it may e wise to use oth solutions to initialize the angular rate estimates of two alternate runs of the Kalman filter that will e descried in Section V. he est solution can e determined y examining the Kalman filter's ehavior for the two cases. IV. Oservaility Analysis of Deterministic Algorithm his analysis determines the conditions under which the deterministic algorithm has a unique solution or, at worst, two possile distinct solutions. his uniqueness is equivalent to oservaility. he oservaility analysis wors with a modified version of Section III's least-squares prolem. his modified prolem uses the differential form of Euler's equation. It assumes that the angular momentum vector of the wheels is constant, i.e., that h & =. It uses Eqs. (6-7) and their time derivatives in order to replace w and w& with expressions involving L, I m, h, ˆ, &ˆ & and &ˆ. he resulting over-determined system of equations for L taes the form cl 2 dl e = (19) where the vectors c and d are defined y: c = ˆ ( I ˆ m ) 2 ( ˆ I ˆ m ) (2a) d ( I ˆ ˆ ) I 2 ˆ ˆ ˆ 2 ˆ ˆ = m I m Im ˆ& I I I m ( ˆ & I ˆ ) h 2 m ˆ I ˆ ( ˆ I ˆ ) ˆ I ˆ ˆ I ˆ m m m m (2) he system is oservale if c or d. Equation (19) has a unique least-squares solution, or at most two distinct local minima, if either of these conditions holds. he system is unoservale only if c = and d = for all time. hese latter conditions can e used to determine the characteristics that lead to unoservaility. he condition c = implies that 13

14 ˆ ( I ) ˆ m =. his, in turn, implies that ˆ is an eigenvector of I m : I ˆ m = λ ˆ for some positive scalar eigenvalue λ. Unoservaility also implies that c& =. his condition can e used to derive the requirement that I &ˆ m = λ &ˆ, which is true if &ˆ = or if &ˆ is also an eigenvector of Im with eigenvalue λ. If &ˆ, then I m must have two identical eigenvalues equal to λ ecause &ˆ is perpendicular to ˆ. he unoservaility conditions that result from the equations c = and c& = can e used to simplify d to the following form ˆ d = ( ˆ & I ˆ ) h m ˆ I ˆ m (21) wo unoservale cases can e deduced y setting d in Eq. (21) equal to zero while using the conditions that result from the equations c = and c& =. In one case, &ˆ = and ˆ is aligned with a principal axis of I m. In order for Eq. (21) to yield d = in this case, the wheel angular momentum h must e aligned with ˆ, or h must e zero. he total angular momentum is parallel to the magnetic field vector in this case. In the other case, I m has two equal principal moments of inertia, which is a condition on the spacecraft configuration, and the vehicle rotates so that ˆ and &ˆ remain in the plane of the two corresponding principal axes. he vector ( ˆ & ˆ ) is perpendicular to this plane as is the vector ( ˆ & I ˆ m ). his latter fact is true ecause ( ˆ & ˆ ) must e an eigenvector of I m. Equation (21) implies that h = - ( ˆ& I ˆ m ) in order for d = to hold while h is a constant. he constancy of h implies that ( ˆ & I ˆ m ) is constant, which implies that ˆ undergoes simple rotation in spacecraft coordinates. his second unoservale case also features 14

15 an angular momentum vector that is parallel to the magnetic field vector ecause I m w h = I m α ˆ = λα ˆ. In summary, the deterministic algorithm reas down in two unoservale cases. One is that of a constant magnetic field vector in spacecraft coordinates that is aligned with a principal axis of the spacecraft and with the wheel angular momentum vector. he other is that of an axisymmetric spacecraft. he magnetic field vector is perpendicular to the axis of symmetry and spins aout that axis at a constant rate. he wheel angular momentum vector is directed along the axis of symmetry and is sized so as to null out the total angular momentum component along that axis. In oth cases, the spacecraft's spin rate aout the magnetic field direction vector is the one unoservale component of the attitude rate. V. Extended Kalman Filter Design he other attitude rate estimation algorithm of this paper is an extended Kalman filter. It has een designed for use in conjunction with the deterministic algorithm. he deterministic algorithm can give a rough initial estimate of the attitude rate that properly accounts for all prolem nonlinearities. his estimate can e used as the initial estimate of the EKF in order to minimize the riss of filter divergence. he EKF can then operate on the data to produce a refined attitude rate estimate that taes etter account of measurement and dynamic model errors and that operates in an efficient recursive manner. Single-Stage Extended Kalman Filter Prolem Model he EKF has een designed using a modified sampled data square root information filter (SRIF). Its asic functioning is similar to the EKF that gets used as part of a filter/smoother calculation in Ref. 1, which is an extension of the linear filter of Ref. 11. One adaptation has een needed in order to properly incorporate the dynamic model of the measured magnetic field 15

16 direction vector, Eq. (2). Another adaptation deals with the field direction measurements' normalization constraint. he filter operates recursively from one measurement sample to the next. he modified SRIF algorithm has een developed as an approximate solution to the following constrained weighted nonlinear least-squares prolem. his prolem includes the propagation from measurement sample time t to sample time t 1 and the measurement update at time t 1 : Find: x, x 1, n, n 1, and w (22a) to minimize: J = 1 2 { w() 2 R w 2 R νν() Rν x() ( n ) ( ) -n Rxx() x -x 2 R νν ( 1) n 1 } (22) suject to: A (n ˆ ˆ 1, 1) 1 = f[ A ( n, ˆ ) ˆ, x, w, t, t 1] (22c) x = f x, w,, t ] 1 x[ t 1 (22d) he vectors in this prolem are defined as follows: x and x 1 are filter state vectors at the times t and t 1, n and n 1 are 2-dimensional magnetic field direction measurement error vectors at times t and t 1, and w is the discrete-time process noise that acts from time t to time t 1. he vectors x and n are a posteriori estimates of x and n at time t. he vectors ˆ and ˆ 1 are the raw measured magnetic field direction vectors in spacecraft coordinates at times t and t 1. he filter state vector x is 8-dimensional. It consists of the 3 1 angular rate vector and corrections to 5 of the 6 inertia matrix elements: x = [w, I mxx, I mxy, I mxz, I myy, I myz ] (23) where the true inertia matrix is a nominal inertia matrix, I m, plus the corrections: 16

17 I m = I m I mxx Imxy Imxz Imxy Imyy I myz Imxz Imyz (24) he two vector functions f and f x in Eqs. (22c) and (22d) constitute the filter's discrete-time dynamics model. he 3-dimensional vector function f[ A ( n, ˆ ) ˆ, x, w, t, t 1] computes a magnetic field direction at time t 1 y numerically integrating the magnetic field differential equation, Eq. (2), starting from the initial condition ˆ ( t ) = A(n ˆ ˆ, ). he first 3 elements of the 8-dimensional vector function f x, w,, t ] propagate the angular rate vector w(t) from x[ t 1 time t to time t 1 via numerical integration of the attitude dynamics differential equation, Eq. (1). his integration starts y initializing w(t ) to equal the first 3 elements of x. It uses the inertia matrix in Eq. (24) with perturations as defined y the last 5 elements of x. he resulting w(t) time history also gets used in the integration of Eq. (2) that computes f. he process noise vector w consists of the magnetic field inertial rate in Eq. (2) and the disturance torque in Eq. (1). hese quantities are defined as follows during the numerical integration interval: w ω ( t) = w for t t < t 1 (25) w n ( t) he last 5 elements of the f x discrete-time dynamics function model the dynamics of the inertia matrix perturations in x. hey are modeled as eing constants. hus, ( f x[ x, w, t, t 1]) j = ) j (x for j = 4, 5, 6, 7, and 8 (26) he least-squares cost function in Eq. (22) can e interpreted as the negative of a loglielihood function, which implies that the filter is a maximum-lielihood estimator. he 17

18 weighting matrices R w(), R νν (), R ν x(), x() x R, and R νν( 1) can e interpreted statistically. If the noise/error terms w, ( n -n ), ( x -x ), and n 1 are each Gaussian, zero-mean random vectors, then the weighting matrices define their covariances and cross-covariances: E{ w w } = R -1 - w() R w() E{( n -n )( n -n ) } = E{( n -n )( x-x ) } = E{( x -x )( x -x ) } = ( Rνν ()) ( I Rν x() [ Rxx() ] [ Rxx() ] [ Rν x() ] )( Rνν() ) [ Rνν ()] Rν x() [ Rxx() ] [ Rxx() ] -1 - [ R xx() ] [ Rxx() ] (27a) (27) (27c) (27d) E { n 1 n 1 } = -1 - [ R νν( 1) ] [ Rνν( 1) ] (27e) where the notation () - indicates the transpose of the inverse of the matrix in question. he only two error vectors with a non-zero cross-correlation are ( n -n ) and ( x -x ). he covariances and correlations of these two vectors represent a posteriori values; i.e., values ased on the measurements up through ˆ, which is why the associated R matrices have the () superscript. he 3 3 matrix function A(n, ˆ ) in Eq. (22c) is an orthogonal coordinate transformation matrix. It uses the two elements of the n error vector to define a rotation quaternion aout a rotation axis that is perpendicular to ˆ : q( n, ˆ ) = V ( ˆ ) n 1 n V ( ˆ ) V ( ˆ ) n (28) where the 3 2 matrix V (ˆ ) is constructed so that its two columns are of unit length, orthogonal to each other, and orthogonal to ˆ. he transformation A(n, ˆ ) is the orthogonal transformation matrix that is associated with q( n, ˆ ) as per the equations in Ref. 12. his matrix gets used to 18

19 rotate the measured magnetic field direction ˆ y the measurement errors in n in order to compute the "true" magnetic field direction A(n, ˆ ) ˆ. Unique Contriutions of the Filter Design he unique contriutions of this filter development are in the use of the A(n, ˆ ) matrix and the magnetic field propagation equation, Eq. (22c). hese provide a rational way for the filter to incorporate Eq. (2) into its dynamics model, which is necessary in order to determine the attitude rate ased on the magnetic field measurements. Reference 3 does this y augmenting the state vector with an estimate of ˆ. he filter of Ref. 3 includes a measurement equation that equates the modeled ˆ plus a measurement error to the measured ˆ. his form of the measurement equation is prolematical; its measurement error has a singular covariance matrix ecause ˆ is constrained to have unit magnitude. he present filter avoids the use of a singular measurement error covariance y defining the 2-dimensional measurement error vector n and the corresponding error rotation A(n, ˆ ). he filter state gets augmented y the measurement error n rather than y the measurement ˆ. Dynamic propagation equation (22c) uses the ˆ differential equation, Eq. (2), to define f, ut Eq. (22c) is effectively a propagation equation for n rather than a propagation equation for ˆ. It is an implicit propagation equation, and the extended Kalman filter needs to deal with this equation sensily. he new filter's measurement equation at sample time t 1 is effectively R ( 1) n 1 νν = n 1, where n 1 is a 2-dimensional Gaussian random noise vector whose mean is zero and whose covariance matrix is the identity matrix. his approach for maintaining the normalization of the measured magnetic field is ain to the multiplicative update approach for maintaining quaternion normalization in an attitude determination extended Kalman filter

20 Extended Kalman Filter Algorithm Modifications he EKF functions lie the filter of Ref. 1, ut with the following modifications: a) he full state vector is [n, x ] in the present EKF, as opposed to just x in Ref. 1. ) he a posteriori square root information matrix, information vector, and state estimate are Rˆ, ẑ, and ˆx in Ref. 1. Here they are, respectively, νν() R R R νx() xx(), νν() R R R νx() xx() n x, and n x (29) c) he vectors z w( ) and ~w of Ref. 1, which are part of the a priori process noise statistics model, are zero in the present formulation. d) Reference 1's a priori state estimate at sample time t 1 is f( ˆx, w ~,). he present EKF uses an a priori state estimate at time t 1 of the form [( n 1 ),( x 1 ) ]. he second vector in this expression is straightforward to calculate: x = f x[ x,, t, t 1]. he first vector is harder to calculate. Suppose that one defines the a priori magnetic field direction vector as 1 ˆ 1 = [ (, ˆ ) ˆ f A n, x,, t, t 1 ] hen n 1 is chosen so that A (n ˆ ˆ 1, 1) 1 = ˆ 1, where ˆ 1 (3) is the measured magnetic field direction vector at sample time t 1. If one forms the attitude error vector and the attitude error angle: e = ˆ 1 ˆ 1 and θ = atan2{ e, [ ( ˆ 1 ) ˆ 1 ]} (31) then n 1 can e determined y solving the following system of linear equations 2

21 V ( ˆ 1) n 1 sin( θ/ 2) = e e (32) Although this system is over-determined it consists of 3 equations in 2 unnowns it is guaranteed to have a solution. Recall that the 3 2 matrix V (ˆ ) is defined in the discussion of A( n, ˆ ). e) he state transition matrix Φ in Ref. 1 is replaced y the matrix: Φ νν Φ ν fx x x (33) where Φ νν = 1 [ A( n ) ] 1,ˆ 1 ˆ ( ˆ 1 V 1) V ( ˆ 1 ) n 1 f [ Aˆ ] [ A( n,ˆ ) ˆ ] n (34a) Φ νx = 1 [ A( n ) ] 1,ˆ 1 ˆ ( ˆ 1 V 1) V ( ˆ 1 ) n 1 f x (34) he notation f / [Aˆ ] refers to differentiation of f with respect to its first argument, and the notation refers to evaluation of the corresponding derivative at the arguments [ A ( n, ˆ ) ˆ ], x, w =, t and t 1, whichever are appropriate. he 2 3 matrix V ( ˆ 1 ) in Eqs. (34a) and (34) projects the corresponding linearized version of Eq. (22c) perpendicular to ˆ 1. f) he disturance effectiveness matrix G in Ref. 1 ecomes: 21

22 1 ˆ [ A( n 1,ˆ 1) 1] ˆ ˆ f V ( 1) V ( 1) n w 1 fx w (35) g) he measurement vector z 1 in Ref. 1 ecomes zero, and the measurement matrix A 1 in Ref. 1 ecomes [ 1), ] R νν (. Given the noted changes, the filter can e implemented y using all of the procedures that are outlined in Ref. 1. hese procedures are extended Kalman filter adaptations of the linear SRIF algorithm that is defined in Ref. 11. he filter gets initialized as follows: n =. he first 3 elements of x equal the rate estimate of the deterministic algorithm, and the remaining 5 elements of x equal zero. R νν () = R νν ( ), and R ν x() =. R xx() is initialized to e the square root of the inverse of the covariance of the uncertainty in the estimate x. his initialization accounts for the measurement at sample time t. he first filter operation is dynamic propagation from sample time t to sample time t 1, and the second operation is the measurement update at time t 1. Linearized Oservaility Analysis A linearized oservaility analysis can e performed on the EKF's system model. Such an analysis checs local oservaility. Local oservaility holds if there is a unique local minimum of the least-squares estimation prolem that discards a priori information aout the initial state and that sets the process noise to zero. Such an analysis is needed in order to verify the simultaneous oservaility of the 3 ody-axis angular rates and the 5 inertia matrix perturations. he linearized oservaility analysis uses many of the same equations and operations as the 22

23 EKF. he following are the two principal differences: First, R xx() is initialized to e. Second, w = is enforced, which eliminates w and the equation Rw() w = w() -ν from the operations that the SRIF uses to compute the a priori information equations for n 1 and x 1 during its dynamic propagation from sample time t to sample time t 1. his modification eliminates the first row and the first column from each loc matrix in Eq. (8a) of Ref. 1. Otherwise, the oservaility analysis performs the same dynamic propagation and measurement update operations that the EKF performs on its square root information matrices. It does not, however, perform any operations on associated information vectors. he oservaility analysis uses a model that is linearized aout the EKF's state estimates; therefore, its Φ matrices are the same as those of the EKF. he oservaility analysis evaluates whether x N can e uniquely determined ased on the magnetic field direction vector measurements ˆ through ˆ N. It does this y propagating and updating its state information matrices starting from sample time t and ending at sample time t N. he system is locally oservale if the ran of R xx(n) is 8, i.e., if R xx(n) is full ran. VI. Results Experimental Data from a Spinning Sounding Rocet he attitude rate estimation algorithms of this paper have een tested using data from the Cleft Accelerated Plasma Experimental Rocet (CAPER). CAPER was a sounding rocet that flew in Jan It was launched from the Andoya Rocet Range in Norway, and its flight lasted over 1,2 s. Its attitude and attitude rate time histories have een estimated ased on magnetometer, sun-sensor, and horizon-crossing-indicator data, which have een processed using a nonlinear smoother 9. CAPER was a minor axis spin-stailized spacecraft. It represents an 23

24 interesting and challenging attitude rate estimation case ecause its nutation mode experienced unstale growth from an initial coning half angle of 2 deg to a coning half angle of more than 75 deg at the end of its flight 9. his paper's new algorithms will e used to estimate CAPER's attitude rate from its magnetometer measurement time history, which was sampled at a nominal frequency of 4 Hz. he estimation results of Ref. 9 will then e used to provide "truth" rates for purposes of evaluating the rate estimates that are otained using the new algorithms. Results for the Deterministic Algorithm he deterministic algorithm has een tried using various input parameters. Good results have een otained using a numerical differentiation atch interval of derv = 1 s to compute ˆ &, a least-squares estimation atch interval of L = 12 s to compute L, and the nominal pre-launch estimate of CAPER's moment of inertia matrix I m = diag(185.4, 185.4, 18.73) g-m 2. CAPER's spin period ranged from 4.6 s at the start of the flight to 16.7 s at the end. he differentiation atch interval is reasonaly small compared to the minimum spin period. he L estimation atch interval ranges from 2.6 spin periods at the eginning of the flight to.7 spin periods at the end, which also seems reasonale for purposes of averaging out noise effects. Estimation results for this case are presented in Fig. 1. he horizontal axis' time datum is set to zero at the apogee of the CAPER trajectory. he solid upper curve on the figure plots the magnitude of the estimated attitude rate w est (t) vs t, and the dotted lower curve plots the magnitude of the estimation error w est (t)-w truth (t) vs t. he figure shows that the attitude rate error is small for most of the data run. It hovers near 5% of the true attitude rate. his indicates excellent performance of the deterministic algorithm. During one short period, however, there is an error spie; the magnitude of the rate error jumps to more than 25% of the actual attitude rate at aout 43 s efore apogee. 24

25 his error spie can e explained in terms of the nonlinear least-square cost function in Eqs. (12) and (16). As previously noted, there can e two local minima of this quartic polynomial. he error spie on Fig. 1 corresponds to a situation in which the nonlinear least squares prolem has a second local minimum that is not the gloal minimum. If L for this alternate local minimum is sustituted into Eq. (6) to compute w, then the error spie disappears from Fig. 1. here are two distinct local minima in fewer than 2% of the cases. In those cases, the non-gloalminimizing L provides a etter attitude rate solution only when the two local minima yield leastsquares costs that are close to each other. hese two facts mae it fairly easy to determine whether or not the gloal minimum is liely to provide the est attitude rate estimate. In the great majority of cases, when the gloal minimum is deemed trustworthy due to the lac of a second local minimum or due to a large cost discrepancy etween the two minima, the deterministic algorithm achieves attitude rate estimation errors that are no greater than aout 5% of the actual spin rate. In the other cases, the EKF can e used to distinguish the est solution, as will e demonstrated later in this section. he length of the L estimation atch interval, L, affects the deterministic algorithm's performance. A numer of runs have een done with L = 6 s instead of 12 s. hese produce poorer results: there are more cases in which the minimizing L does not produce the est attitude rate estimate, and there is even an isolated case in which oth local minima of J(L ) produce attitude rate errors of 3% or more. hus, the longer atch interval improves the deterministic algorithm. here is proaly a practical upper limit to L. Beyond this limit the estimation accuracy may tend to degrade due to errors in the dynamics model. he deterministic algorithm is somewhat insensitive to the inertia matrix estimate that gets used for I m. he algorithm has een tried using oth the pre-launch estimate of I m and the est 25

26 estimate of I m that was produced in the study of Ref. 9. he deterministic algorithm performs similarly for oth I m estimates, although there are slightly fewer error spies for the pre-launch I m, which seems counter intuitive. his result may e caused y Ref. 9's use of a CAPER dynamic model that includes flexiility effects. Extended Kalman Filter Results he EKF of Section V has een tested on the CAPER data. One goal of this part of the study has een to test the filter's aility to converge from initial rate estimates that come from the deterministic algorithm. Another goal has een to test the accuracy of the EKF in comparison to that of the deterministic algorithm. A numer of runs have een tried using various filter tunings and various initialization times during CAPER's flight. matrix he oservaility calculations of Section V have een performed during each EKF run. he R xx(n) has een verified to e full ran in every case. herefore, the attitude rate vector and the 5 inertia matrix perturations are oservale in the local linearized sense. he EKF has een tested in a tough situation, one with significant initialization uncertainty due to the existence of two local minima of nearly equal cost in the deterministic rate estimation prolem. A pair of EKF runs have een initialized at t = s. One run uses the initial w that corresponds to the deterministic algorithm's gloal minimum, and the other run uses the initial w that corresponds to the deterministic algorithm's other local minimum. At this initialization time the gloal minimum yields one of the deterministic algorithm's worst initial w estimates, and the other local minimum yields the est initial w estimate review the error spie on Fig. 1. hese runs use the pre-launch estimate of the inertia matrix as the filter's nominal inertia matrix, I m of Eq. (24). he filter uses the same tuning for oth of these cases. he R νν ( ) measurement error 26

27 weighting matrix equals diag(4, 4), which corresponds to 2.9 deg of ˆ direction error per axis. he t R w() process noise weighting matrix equals diag(621, 621, 621, 4.8, 4.8, 1.5), where the first three entries of the diagonal matrix are in units of s.5 /rad, and the last three entries are in units of 1/(N-m-s.5 ). he upper-left 3 3 loc of the filter's initial state square-root information matrix, R xx(), equals diag(2.9, 2.9, 2.9) s/rad, which corresponds to a 2 deg/s initial rate standard deviation per axis, consistent with the error in the poorer rate estimate from the deterministic algorithm. he lower right 5 5 loc of R xx() equals the inertia matrix perturation covariance that results from the following uniform distriutions in principal inertias and principal axis orientations: ± 2% for the principal inertias perpendicular to the nominal spin axis, ± 2 deg for the two orthogonal orientation perturations of the spin axis direction, and ± 45 deg for the orientation perturation aout the nominal spin axis. he remaining elements of x() R x are set to zero. Results from these Kalman filtering runs are presented in Fig. 2. he figure plots the magnitude time histories of the estimated attitude rate vector and the magnitude time histories of the estimation error for the two EKF runs. hese two plot types are the same as those that appear on Fig. 1 for the deterministic attitude rate estimator. It is ovious from Fig. 2 that the poor initialization yields much poorer initial performance: notice the large initial oscillations of the solid gray curve at the top of the figure and the large initial peas of the dashed gray curve at the ottom of the figure. he curves for the good initial estimates, the two lac curves, exhiit almost no initial transients. Both cases eventually converge, however, and their steady-state performance is similar. Figure 2 shows that the rate error for the good initialization starts at a magnitude of aout 4 27

28 deg/s and immediately drops down to aout 2 deg/s or less. By the end of the flight the error is less than 1 deg/s. his contrasts with the deterministic algorithm's error curve in Fig. 1, which starts out oscillating etween 3 and 7 deg/s, slowly decays to an average value of 2 deg/s at apogee, and has oscillation peas of aout 2 deg/s for the entire flight. hus, the Kalman filter's tracing accuracy is superior to that of the deterministic algorithm. his level of performance is consistent with the filter's computed attitude rate estimation error covariance, which indicates that the filter's tuning is reasonale. On the other hand, the filter's estimated measurement error time history, n,, n N, exhiits component standard deviations that are aout 1 times smaller than the 2.9-deg-per-axis tuning value which the filter uses. hus, there may e room for improvement in the filter's tuning. Both initializations yield convergence in Fig. 2, ut it would e preferale to use the initial rate estimate that yields smaller error transients, i.e., the estimate from the deterministic algorithm's local minimum. he superiority of this initialization can e determined from the EKF's estimated measurement error time history, n,, n N. he est initialization yields initial transients in the n history that have significantly smaller pea values than the transients that are caused y the other initialization. hus, the strategy for initializing the EKF is to try two parallel filters if the deterministic algorithm yields two distinct local minima of nearly equal cost. he attitude rate estimate is then chosen as the output of that filter run which yields the smallest initial transient oscillations in its n history. Figure 3 illustrates the EKF's aility to estimate the inertia matrix corrections of Eq. (24). It shows the time histories of the five corrections for the filtering run that corresponds to the good initialization case of Fig. 2. Some of these correction terms undergo nearly step changes at the 28

29 start of the filtering run, ut consistent with the assumption that they are constants, their rates of change tend to slow down as the filter nears the end of the data atch. he final estimated inertia matrix is closer to Ref. 9's "truth" estimate than is the initial matrix I m, 77% closer as gauged y the induced matrix 2-norm. hus, the EKF does a reasonale jo of estimating its inertia perturation states. he performance of the EKF has een evaluated using a numer of different inputs. It has een initialized nearer to the end of the flight. Different tunings have een tried. Different values of I m have een used to initialize the inertia matrix estimate. he performance of the filter is not affected significantly y such changes. A slowing of the filter produces modest accuracy improvements in the w(t) estimate. his slowing is achieved y an increase in the elements of the R w() matrix. An improved value of I m reduces the magnitudes of the inertia correction transients. Initialization of the filter with angular rate estimates from the deterministic algorithm always produces excellent performance, except for the case noted in Fig. 2, which corresponds to the error spie in Fig. 1. A surprising aspect of these results is that the EKF wors well using a rigid-ody model of the spacecraft. Reference 9's main point is that a rigid-ody model is inadequate for attitude estimation for the CAPER mission ecause of the presence of significant flexile ody effects. One might thin that the present results contradict those of Ref. 9. his seeming discrepancy is easily resolved, however. Reference 9's difficulty with a rigid-ody model stems from the fact that its filter/smoother estimates oth attitude rate and attitude. he current filter only estimates attitude rate. If one integrates the difference etween the two studies' attitude rate estimates, then one quicly uilds up attitude errors of 2 deg and more. his is far more error than was noted for the attitude estimates of Ref. 9. he current filter is ale to achieve reasonale attitude rate 29

30 estimation performance ecause it can tolerate small rate errors that get caused y the inaccuracy of its rigid-ody model. he filter/smoother of Ref. 9 does not have this luxury ecause these small rate errors integrate to ecome excessively large attitude errors. VII. Conclusions wo new attitude rate estimation algorithms have een developed for use on spacecraft, a deterministic algorithm and an extended Kalman filter. Both of them wor with a time series of Earth magnetic field direction measurements from a magnetometer, and oth use Euler's equation for the attitude dynamics of a rigid ody with momentum wheels. he deterministic algorithm uses the inematic equation for the Earth's magnetic field direction in spacecraft coordinates in order to estimate the two components of the spin rate that are perpendicular to the magnetic field. It uses Euler's equation to solve for the remaining component via a gloal nonlinear least-squares atch estimation technique. he extended Kalman filter estimates the attitude rate and perturations to 5 of the spacecraft's 6 inertia matrix elements. It wors with Euler's equation and the magnetic field inematics equation and includes measurement error states. he propagation procedure for these error states uses multiplicative-type operations. hese two algorithms have een evaluated using flight data from a spinning sounding rocet whose spin rate started at 78 deg/s and decayed to 2 deg/s y the end of the flight. he deterministic algorithm achieves accuracies ranging from 7 deg/s at the start of the flight down to 2 deg/s near the end, except in unusual cases when its nonlinear least-squares solver pics a wrong local minimum. Convergence of the extended Kalman filter can e ensured y using the deterministic algorithm's output to initialize the filter's rate estimate. he filter's measurement error estimates can e used to distinguish etween good and ad initializations in cases where the deterministic algorithm yields two distinct local minima. he extended Kalman filter's accuracy 3