Attracting manifolds for attitude estimation in flatland and otherlands

Attracting Manifolds forAttitude Estimation in

Flatland and Otherlands1

Maruthi R. Akella,2 Dongeun Seo,3 and Renato Zanetti4

Abstract

Non-convex and non-affine parameterizations of uncertainty are intrinsic within everyattitude estimation problem given the fact that minimal and/or nonsingular representationsof the attitude matrix are invariably nonlinear functions of the unknown attitude variables.Of course, this fact remains true for rotation matrices both in the 2-D plane (flatland) and inhigher dimensional spaces (otherlands). Therefore, estimation problems involving minimalnonsingular representations of unknown attitude matrices bring significant challenges to theadaptive estimation community. This paper develops a novel algorithm for attitude estima-tion. The proposed algorithm relies upon the design of an adaptive update law for the atti-tude estimate while preserving its inherent orthogonal structure. The underlying approachborrows from the classical Poisson differential equation in rigid-body rotational kinematicsand endows certain manifold attractivity features within the adaptive estimation algorithm.Consequently, we are not only able to efficiently handle the non-affine and non-convex na-ture of the parameter uncertainty, but are also ensured of estimation algorithm stability androbustness under bounded measurement noise. In addition to a rigorous discussion on theoverall methodology, the paper provides example simulations that help demonstrate the ef-fectiveness of the attracting manifolds design.

Introduction

Attitude estimation problems routinely arise in numerous aerospace engineeringand robotics applications. More specifically, relative navigation and attitude deter-mination problems are linearly parameterized by unknown proper orthogo-nal matrices SO(3). Numerous attitude estimation/determination algorithms areavailable in the literature developed by both control and estimation communities.Of immense significance is the fact that every orthogonal matrix is nonlinear in its

3 � 3

The Journal of the Astronautical Sciences, Vol. 54, Nos. 3 & 4, July–December 2006, pp. 635–655

635

1Dedicated to Malcolm D. Shuster for his friendship and inspiring presence. 2Associate Professor, Department of Aerospace Engineering and Engineering Mechanics, University of Texasat Austin. Tel: (512) 471-9493, Fax: (512) 471-3788, E-mail: [email protected]. Member AAS. 3Graduate Research Assistant and Ph.D. Candidate, Department of Aerospace Engineering and EngineeringMechanics, University of Texas at Austin. E-mail: [email protected]. 4Graduate Research Assistant and Ph.D. Candidate, Department of Aerospace Engineering and EngineeringMechanics, University of Texas at Austin. E-mail: [email protected].

degrees of freedom [1, 2]. Virtually every existing attitude estimation method con-verts this nonlinear parameterization into a linear over-parameterization [3]. De-pending on how the underlying attitude estimation algorithm is implemented,existing methods can be broadly categorized into two classes: batch-type andsequential-type estimators.

Batch-type estimators such as QUEST [4] and FOAM [5] utilize more than two ob-servations at each observation instant to determine the attitude matrix in three di-mensions, thereby necessitating the use of two or more independent sensors. Theunderlying estimation is accomplished by minimizing a quadratic cost function origi-nally proposed by Wahba [6]. TRIAD [7] is another variant among existing batch-typeattitude estimation algorithms which requires at most linearly independentobservation vectors for attitude determination in an n-dimensional space [8].

Sequential-type estimators, in contrast with batch-type estimators, need only oneobservation at each time. The most common implementation of sequential estima-tors adopt an extended Kalman filter [9]. Instead of fusing measurements from mul-tiple sensors at each instant, sequential-type estimators utilize an analytic model ofthe system to forward propagate the observation data. Attitude estimates are thengenerated by comparing predicted observations with actual measurements whileupdating the underlying analytic model through minimization of a suitable opti-mality criterion. The standardly adopted optimality criterion for Kalman filtering isthe minimization of variances between estimates from sensor measurement andpredicted values derived from the system analytic model. Recent applications of ex-tended Kalman filter type sequential estimators are documented for missions suchas the Earth Radiation Budget Satellite (ERBS) [10, 11] and the Solar AnomalousMagnetospheric Particle Explorer (SAMPEX) [12, 13].

Even though batch- and sequential-type estimators have been successfully ap-plied to a wide array of actual spacecraft missions, both classes of methods usuallysuffer from a crucial limitation—that of over-parameterization and the resulting non-enforcement of the orthogonality structure on the attitude matrix estimate. In orderto eliminate problems associated with over-parameterization, an adaptive algorithmfor orthogonal matrix estimation was developed by Kinsey and Whitcomb [14]. Theconvergence proof for their estimation algorithm, applicable only for attitude esti-mation in the three-dimensional space, utilizes a matrix logarithmic map definedover the attitude estimation error matrices. Further, this logarithmic map is notinjective for a certain class of orthogonal matrices. To be precise, this restrictionpertains to non-inclusion of SO(3) matrices whose trace is equal to (the corre-sponding Euler principal rotation angle

This paper aims at deriving new classes of adaptive estimation algorithms for un-certain proper orthogonal matrices. The proposed estimation methodologyinvolves introduction of an attracting manifold about the “true” but “unknown” at-titude matrix which helps in automatically enforcing the attitude matrix estimate tobe proper and orthogonal at every time step. Design of the attracting manifold re-sults in significant reduction of computational burden associated with: (a) beingable to fully utilize the available prior information on orthogonality of the unknownattitude and thereby avoiding over-parameterization; and (b) not having to performthe re-orthogonalization process at each time step. Additionally, under standard as-sumptions involving availability of persistence in excitation (PE), we are able toshow that the attitude estimate is guaranteed to converge to the corresponding truevalue. Convergence proof for the estimation algorithm presented in this paper isaccomplished in such a way that not only are the restrictions associated with the

n � n

� � ��.�1

n � 1

636 Akella, Seo, and Zanetti

logarithmic map of Kinsey and Whitcomb [14] completely eliminated but our re-sults also generalize nicely for attitude matrices on all n-dimensional spaces.

The paper is organized as follows. In the next section, we present our main re-sults that establish a new orthogonality preserving attitude matrix estimation algo-rithm valid for the general n-dimensional case. Robustness analysis for thisestimation algorithm under the influence of measurement noise is also presented.Implications of the proposed attitude estimation algorithm for two-dimensions(flatland case) and three-dimensions (otherlands) are then discussed. Numericalsimulations are presented to demonstrate and validate the various technical claimsof this paper. The final section provides a discussion with concluding remarks.

Problem Statement and Main Result

Problem Definition

Succinctly stated, the attitude estimation problem is that of finding a directioncosine matrix describing the orientation of any body fixed frame with respect to theinertial frame. In order to formulate this problem within an analytical framework,we consider estimation of an unknown and time-varying proper orthogonalmatrix (i.e., satisfying and for allt) defined through the following continuous-time input-output mapping

(1)

where unit vectors and respectively correspond to the input andoutput signals, both of which are assumed accessible for all time t. Further, we as-sume the input vector to be a differentiable function of time with a boundedderivative.

From a practical standpoint, in three-dimensions, the measurement model equa-tion (1) is typical for single input-output type unit vector measurement sensors suchas star trackers, Sun sensors and magnetometers [15]. More specific examples of starsensor modeling in attitude determination problems can be found in [16, 17, 18, 19].

The evolution of in equation (1) is governed by the Poisson differentialequation as described by

(2)

where is any prescribed/measured bounded signal for and is a skew-symmetric matrix function such that It isa well known fact that if is a proper orthogonal matrix, then will remaina proper and orthogonal matrix for all t whenever it evolves along equation (2) [20].Now, the estimation objective is to find an appropriate adaptive algorithm that gen-erates the proper orthogonal matrix as an estimate for the “true” butunknown matrix at each instant t. The output model based on can be es-tablished by

(3)

Discrete-time analogs of the aforementioned attitude estimation problem rou-tinely arise within the field of spacecraft attitude determination. In such cases,and are respectively interpreted as the star catalog values (inertial) and thecorresponding star tracker measurements. Estimation problems within this frame-work also occur in space applications of rendezvous and proximity operations thatinvolve computation of the relative navigation solutions.

y�t�r�t�

y�t� � C�t�r�t�

C�t�C�t�C�t� � Rn�n

C�t�C�0�ST � �S.S��m l �n�n

m � n�n � 1��2,��t� � �m

C�t� � �S��t��C�t�

C�t�

r�t�

y�t� � �nr�t� � �n

y�t� � C�t�r�t�

det�C�t�� 1C�t�CT�t� � CT�t�C�t� � In�nC�t�n � n

Attracting Manifolds for Attitude Estimation in Flatland and Otherlands 637

The attitude estimation error matrix defined by

(4)

represents the element-by-element error between the estimate and the “true”attitude Obvious from the definitions in equation (1) and equation (3) is thefact that the attitude error convergence, automatically impliesthe output estimation error, as In the following de-velopments, unless considered necessary, the time argument t is omitted from var-ious signals for the sake of notational simplicity.

Main Result

In this section, we present a new attitude estimation algorithm which preservesorthogonality of the estimated attitude matrix

THEOREM 1. If the attitude estimate is generated according to

any real any proper orthogonal (5)

then, remains a proper orthogonal matrix for all implying that attitudematrix estimation error and the output estimation error remain boundedfor all In addition, the estimation process for is driven along an “at-tracting manifold” according to the following convergence condition

(6)

PROOF. Just the same way that the Poisson differential equation equation (2) enforcesothogonality on for all t, the attitude estimate is also ensured to be properand orthogonal by virtue of the fact that it is generated by another Poisson differ-ential equation given in equation (5). This similarity can be made more explicit byrecognizing that the additional term within equation (5) given by isalso a skew-symmetric matrix. Since is guaranteed to be an orthogonal matrixfor all it follows that and remain bounded for all This is suf-ficient to demonstrate boundedness of the output estimation error

Next, in order to prove the convergence claim along the attracting manifold inequation (6), we consider the Lyapunov candidate function

(7)

where is the matrix trace operator. Then, the time derivative of evaluatedalong equation (2) and equation (5) is given by

� ��

2rT�In�n � CTCCTC � CTCCTC In�n�r

� ��rT�In�n � CTCCTC�r

� � tr��rrT CTCCTCrrT� � � tr��CT � CT� �CrrT � CrrTC TC�� tr�CT�yyT � yyT�C�

V�t� � tr��CTS��C �CT�yyT � yyT�C�

V�t�,tr��

V�t� �1

2 tr�CT�t�C�t��

e�t�.t � 0.C�t�C�t�t � 0,

C�t��yyT � yyT�

C�t�C�t�

limtl�

�In�n � CTCCTC�r�t� � 0

C�t�t � 0.e�t�C�t�

t � 0C�t�

C�0� � �n�n� � 0,

C�t� � ��S�� yyT � yyT�� C�t�;

C�t�

C�t�.

tl �.e�t� � y�t� � y�t�l 0limtl� C�t� � 0,

C�t�.C�t�

C�t� � C�t� � C�t�

C�t�


(8)

which demonstrates and thereby uniform boundedness of Sinceby definition from equation (7) and from equation (8), we have

existence of Further, from the fact that is bounded (seen bydifferentiating both sides of equation (8)), using Barbalat’s lemma, we conclude

REMARK 1. In the case that the attitude estimate converges to the correspon-ding “true” attitude (i.e., using the matrix orthogonalityproperty, it is easy to recognize that as which obviouslysatisfies the convergence condition of Theorem 1 given in equation (6). On theother hand, mere presence of the attracting manifold as governed by equation (6)does not in general guarantee regulation of attitude estimation error to zerounless certain persistence of excitation (PE) conditions are additionally satisfiedby the input signal These PE conditions will be further elaborated upon inthe later sections of the paper.

Robustness to Measurement Noise

We next present robustness properties of the estimation algorithm from Theorem 1to account for possible presence of bounded measurement noise. Introducing abounded noise signal in equation (1) to reflect the presence of error in themeasurement signal, we have

(9)

wherein we assume the noise signal satisfies Even though not re-quired for any of our further developments, from a practical standpoint, it is per-fectly meaningful to assume that While retaining the same estimation rulefor the attitude matrix as given by equation (5), we consider the Lyapunov can-didate function that enables demonstration of the robustness properties

(10)

Since presence of measurement noise in no way changes the fact that estimated according to equation (5), is a proper and orthogonal matrix for all the function is uniformly bounded. Further, using equation (3), equation (5),and equation (9), the time derivative of can we written as

(11) = ��

2 $$�In�n � CTCCTC�$$ �$$�In�n � CTCCTC�r$$ � 2vmax�

� ��rT�I � CTCCTC�r � � vTC�I � CTCCTC�r

� � tr�CT�yyT � yyT�C� Vr�t� � tr�CC�

Vr�t�Vr�t�

t � 0,C�t�,v�t�

Vr�t� �1

2 tr�CTC�

C�t�vmax 3 1.

vmax � supt�v�t��.

y�t� � C�t�r�t� v�t�

v�t�

r�t�.

C�t�

tl �CT�t�C�t�l In�n

limtl� C�t� � 0�,C�t�C�t�

�limtl��In�n � CTCCTC�r�t� � 0.

V�t�V� � limtl�V�t�.V�t� � 0V�t� � 0

V�t�.V�t� � 0

� ��

2 $$�In�n � CTCCTC�r$$2

� ��

2rT�In�n � CTCCTC�T�In�n � CTCCTC�r

� ��

2rT�In�n � CTCCTC � CTCCTC �CTCCTC�T�CTCCTC��r


In the presence of measurement noise, the convergence condition of Theorem 1along the attracting manifold is no longer preserved.However, it is also obvious from equation (11) that whenever

and therefore the size of the residual set gov-erning is ultimately dictated by the upper bound on the noise signal Theforegoing analysis of the proposed attitude estimation algorithm provides confir-mation that while all the computed signals remain bounded in the presence of meas-urement uncertainty, presence of large magnitude noise leads to increasing dilutionof attitude estimation accuracy.

Estimation Within the Certainty-Equivalence Framework

The main result of this paper derived in Theorem 1 will now be compared with analternative estimation mechanism based on the traditional/conventional certainty-equivalence (CE) framework [21]. In order to be consistent with the standard assump-tions of the CE methodology, we restrict the attitude matrix in equation (1) tobe a constant (i.e., in equation (2)), and that there exists no measure-ment noise Important to mention here is the fact that recognition andutilization of prior information on the structure of the unknown parameter leadsus to nonlinear parameterization of the unknown elements in equation (1). On theother hand, if we ignore this prior information on orthogonality of then the un-known parameter appears linearly (affinely) in the governing input-output relation-ship, equation (1), thereby making application of the CE-based adaptive estimationmethod feasible. This convenient simplification comes at a heavy price: over-parameterization and/or permitting the parameter estimation (search) process for

to potentially evolve outside the region where the true parameter lies—ultimately leading to the unacceptably slow/poor estimator performance. On theother hand, any attempt to explicitly utilize the a priori known parameter structurecauses nonlinear parameterization which is not readily amenable to most existingCE-based formulations. For the case when a simple CE-based formula-tion based on standard methods [21] may be obtained as given by

any real any (12)

where just as before, the output estimation error is defined by Stability for the CE-based estimation algorithm given in equation (12)

can be demonstrated by considering the Lyapunov candidate function

(13)

where is defined in equation (4). Then, the time derivative of taken alongsolutions generated by equation (12) is given as

(14)

Since and we have boundedness for the parameter estima-tion error and therefore boundedness for the attitude estimate matrix Also, exists and is finite because and Integrating both sides of equation (14), we can show that

which implies that Furthermore, from the fact thate � L 2 � L�.��0� ��e�t��2 dt

V�ce � Vce�0� �

V ce�t� � 0.Vce�t� � 0V�ce � limtl�Vce�t�

C�t�.C�t�V ce�t� � 0,Vce�t� � 0

� ��$$Cr$$2 � ��$$e$$2 � 0

� �� tr�CTrrT� � ��rTCTCr

V ce�t� � tr�CTC� � �� tr�CTerT�

Vce�t�C�t�

Vce�t� �1

2 tr�CTC�

y�t� � y�t�.e�t� �e�t�

C�0� � �n�n� � 0,C�t� � ��erT;

C�t� � C *,

C *C�t�

C *,

C *�v�t� � 0�.

��t� � 0C *C�t�

v�t�.Vr�t�$$�In�n � CTCCTC�r$$ � 2vmax � 0

Vr�t� � 0�In�n � CTCCTC�r�t� � 0


whose terms are all bounded signals, we have Therefore,from Barbalat’s lemma, we guarantee that the output estimation error as

One very important point we should emphasize here is that doesnot always imply unless the reference signal satisfies certainadditional persistence of excitation (PE) conditions. For example, if (two-dimensional case) and the reference input signal i.e., a constant vector in

then can always be satisfied without equalling zero provided thevector resides within the kernel (null space) of the matrix

A few important remarks are now in order to highlight the limitations of the CE-based attitude estimation scheme.

1. A proof of stability, boundedness, and convergence can be derived for theCE-based method (as outlined in the foregoing) only when the unknown attitudematrix remains constant with time. No such restriction exists on the result pro-posed under Theorem 1.

2. Under the CE-based estimation scheme, even when one selects the initial esti-mate to be proper and orthogonal, there is no assurance whatsoever that estimated through equation (12) remains orthogonal for On the otherhand, by the very fact that the attitude estimation under the proposed approachproceeds along equation (5) (Poisson differential equation), orthogonality of theestimate matrix is rigorously assured for all

3. Further, in the next sections, we show that when compared with CE-based atti-tude estimation, the main result of this paper needs weaker (less-restrictive) PEconditions on the reference input signal so as to ensure that the parameter esti-mation error converges to zero.

Attitude Estimation in Flatland

In this section, we specialize the main result for attitude matrix estimation to thetwo-dimensional (flatland) case, i.e., in equation (1). While doing so, we willalso be able to precisely characterize the persistence of excitation (PE) conditions thatwould regulate the attitude estimation error matrix to zero. Our use of the term“flatland” is inspired and motivated to a great extent by the work of Shuster [22]that addressed attitude analysis in two-dimensions.

Note that for the flatland case, the true/unknown attitude matrix and its es-timate can be parameterized in terms of scalar “angle-like” variables respec-tively designated by and in the fashion

(15)

where J is the matrix generalization of and is given by

(16)

For the flatland case, the following result can be established as a direct conse-quence of Theorem 1.

COROLLARY 1. Consider the input-output system described by equation (1), withIf the initial value of the attitude estimate is such that

where,�0� � >s

C�0� � eJ,�0�n � 2.

J � �0

1

�1

0 ��12 � 2

C�t� � eJ,�t� � �cos ,

sin ,

�sin ,

cos , �C�t� � eJ,�t� � �cos ,

sin ,

�sin ,

cos , �,

,�t�,�t�C�t�

C�t�

C�t�

n � 2

t � 0.C�t�

t � 0.C�t�C�0�

C.r*Ce � Cr* � 0�2,

r�t� � r*,n � 2

r�t�limtl�C�t� � 0limtl�e�t� � 0

tl �.e�t�l 0

e � L�.e � Cr C r


then the attitude estimate matrix generated through equation (5) exponen-tially converges to the unknown true value for all nonzero (unit-vector) ref-erence inputs

PROOF. We begin by specializing the Lyapunov candidate function given inequation (7) for the two-dimensional (flatland) case as which may readily beexpressed in terms of the parameterization defined by equation (15) as

(17)

It is clear from the last step of the preceding development that not only is bounded for all but in fact, we have for all Further, itcan be seen that its maximum value, whenever for all integers k. Thus, if is satisfied. On the other hand,

its minimum value, whenever (estimated variable exactlyequals the true/unknown variable value). Of course, more generally,whenever for integers k. All the aforementioned interesting andimportant characteristics of the Lyapunov candidate function for the flatlandcase are illustrated in Fig. 1.

Additional insights into the characterization of dynamics of can be obtainedby the specializing function in equation (8) with for the flatland case.Since the quantity can be written as in the flatlandcase, making using of equation (17), we may derive a simplified expression for

in the fashion VF�t�

CTCCTC � eJ�2,�2,�CTCCTCn � 2V�t�

VF�t�

VF�t�,�t� � ,�t� � 2k

VF�t� � 0,�t�,�t�,�t� � ,�t�VF�t� � 0,

,�0� � >sVF�0� � 4,�t� � ,�t� � �2k 1�VF�t� � 4,

t � 0.0 � VF�t� � 4t � 0VF�t�

� 2�1 � cos�, � ,�� 4 sin2�, � ,

2 �

1

2 tr�2I2�2 � �CTC CTC��

1

2 tr�2I2�2 � e�J�,�,� � eJ�,�,��

VF�t� �1

2 tr�CT�t�C�t��

1

2 tr��C � C�T�C � C��

VF�t�V�t�

r�t� � �2.C�t�

C�t�

k � 0, �1, �2,� � �>s � � � �� ,�0� � �2k 1�,


FIG. 1. Plot Showing Variation of the Bounded Lyapunov Candidate Function for Flatland Case with Respect to the Estimated Variable ,�t�.

VF �t�

(18)

As stated before, the reference input is a unit vector for all and thus,we have

(19)

which indicates the fact that the dynamics of is governed by an infinite num-ber of equilibrium points whenever is satisfied along the -axis as can beseen in Fig. 1. However, those equilibrium points clearly segment into two distinctcategories. One is an unstable branch in the sense that while i.e., for integer values of the variable k. Thus, if then for all irrespective of the reference input vector The otherset of equilibrium points form a stable branch satisfying the conditions and for all integers k. The equilibriumpoints corresponding to the condition are designated unstable becausewhenever for all due to the fact that for all Clearly, if then for is assured resulting in the fact that

only when (stable and attracting manifold). The convergence properties of starting from any initial value

can be established by recognizing that the first-order ordinary differential equationgoverning in equation (19) admits an analytical solution as given by

for all where (20)

Therefore it immediately follows that any initial condition implyingleads to exponential convergence of to zero. In terms of the attitude

estimate matrix, this result means exponential convergence along for all integers k, and accordingly, we conclude that exponentially as

In light of Corollary 1, the following observations are in order.

1. In flatland, whereas the matrix has four elements that need to be es-timated, the process of updating estimates of can be accomplished by up-dating a single scalar variable thereby eliminating any scope foroverparameterization. More specifically, instead of directly estimating through the matrix differential equation of equation (5), one can efficiently gen-erate the same matrix through the identity given from equation(15). It is straightforward algebra to show that the “angle” variable needed,�t�

C�t� � eJ,�t�C�t�

C�t�

C�t�C�t�2 � 2

�tl �.C�t�l 02k

,�t� � ,�t�lVF�t�VF�0� � 4

,�0� � >s

c �VF�0�

4 � VF�0�t � 0,VF�t� �

4ce�2�t

1 ce�2�t

VF�t�

VF�0� � 0, 4VF�t�VF�t�l 0VF�t�l 0

t � 0VF�t� � 4,�0� � >s,t � 0.VF�t� � 0t � 0VF�0� � 4,

VF�t� � 4VF�t� � 0 ⇔ ,�t� � ,�t� � 2k

VF�t� � 0r�t�.t � 0VF�t� � 4

VF�0� � 4,,�t� � ,�t� � �2k 1�VF�t� � 4,VF�t� � 0

,VF�t� � 0VF�t�

VF�t� � ��

2VF�t� �4 � VF�t��

t � 0r�t�

� ��

2VF�4 � VF� ��r��2 � 0

� �8� sin2�, � ,

2 cos2�, � ,

2 ��r��2

� ��1 � cos�2, � 2,�� r��2

� ��rT�1 � cos�2, � 2,�sin�2, � 2,�

�sin�2, � 2,�1 � cos�2, � 2,��r

VF�t� � ��rT�I2�2 � eJ�2,�2,��


for computation of the attitude estimate matrix is updated through the scalardifferential equation

(21)

where the scalar can be physically interpreted as an “angular velocity” vari-able such that in equation (2), and denotes a unit vector normalto the flatland.

2. Exponential convergence of to zero is guaranteed regardless of (nonzero) values so long as This remarkable feature holds truewithout requiring the unknown matrix to be a constant, i.e., inequation (2). Further, the same exponential convergence conditions hold evenwhen (any nonzero constant vector) and thus every nonzero referenceinput vector is persistently exciting in the flatland case. As already mentioned inthe previous section, similar convergence assurances are impossible from theconventional CE-based estimation framework.

3. If then for all (the unstable equilibriumbranch) and obviously, there is no way to regulate the estimation error tozero irrespective of the reference input However, from a practical stand-point, even the smallest of perturbations and/or numerical drift that causes to deviate from at some would ensure that for all

(the stable and attracting manifold) and consequently we are again assuredof exponentially converging to zero on

4. In the presence of bounded measurement noise the robustness result for thegeneral n-dimensional case from the previous section can be specialized forthe flatland case by rewriting equation (11) as

(22)

Recalling the reference input to be a unit vector, the previous inequality mayfurther be simplified and arranged as

(23)

It is clear from here that whenever Further,equation (23) can also be adopted for the case of relatively low magnitude meas-urement noise (i.e., is small enough for the small angle approximation

to hold), so that as the inequality servesas a useful approximate upper bound on the estimation error. Clearly, if isapplied in this approximation result (the ideal no-noise case), we immediatelyrecover the (exponential) convergence property which wasestablished earlier in this section.

Attitude Estimation in Three-Dimensions

Although the input-output model in equation (1) is for a single measurementsystem, the results from Theorem 1 can be readily extended to the systems havingmultiple measurement devices. Let M denote the total number of measurement de-vices, i.e.

(24)k � 1, 2, . . . , Myk�t� � C�t�rk�t�,

limtl�,�t� � ,�t� � 0

vmax � 0$,�t� � ,�t�$ � vmaxtl �,sin�vmax� � vmax

vmax

$sin�, � ,�$ � vmax � 0.Vr�t� � 0

Vr�t� � �2� $sin�, � ,�$ �$sin�, � ,�$ � vmax�

r�t�

� �2� sin2�, � ,� ��r��2 2� sin�, � ,�vTr

Vr�t� � �2� sin2�, � ,� ��r��2 � ��cos , � cos�2, � ,��vTr

v�t�,t � t*.C�t�

t � t*,�t� � >st � t* � 0VF�0�

VF�t�r�t�.

C�t�t � 0VF�t� � VF�0� � 4,�0� � >s,

r�t� � r*

��t� � 0C�t�,�0� � >s.

r�t�C�t�

ek��t� � ��t�ek

��t�

,�t� � ��t� ��y2�t�y1�t� � y1�t�y2�t��

C�t�


Then, the update law from equation (5) needs to be modified to account for thepresence of more than one input-output pair and is now given by


wherein the kth estimated output is defined through

(26)

From the same definition of Lyapunov candidate equation (7) with a subscript M,we can derive the time derivative of as

(27)

which identically covers equation (8) when The stability proof remains thesame as Theorem 1 except for the summation. One thing which should be noted inequation (27) is that we can obtain the same effect from a single measurement deviceby tuning � in equation (5) instead of adopting M measurement devices. Given thatall technical details remain unaltered with single or multiple measurements, to keep thenotation simple, we retain the single measurement model while discussing all fur-ther implications of our proposed attitude estimation algorithm for three dimensions.

We adopt the four-dimensional unit-norm constrained quaternion vector repre-sentation for a singularity-free parameterization of the attitude matrix whilespecializing the results of Theorem 1 to the case of three dimensions Ac-cordingly, we assume that the true attitude matrix is represented by the quater-nion vector where the subscripts “o” and “v” respectivelydesignate the scalar and vector parts of the quaternion representation. This quater-nion parameterization is mathematically realized through the relationship

(28)

where the skew-symmetric matrix operator designates the vector crossproduct operation such that for all three-dimensional vectors a andb. Similarly, the estimate matrix is represented through the quaternion param-eterization so that we have

(29)

Further, if we designate the quaternion to parameterize theproper orthogonal matrix it follows that

(30)CT�t�C�t� � I3�3 � 2zo�t�S�zv�t�� 2S 2�zv�t��

CT�t�C�t�,z�t� � �zo�t�, zv�t��T

C�t� � I3�3 � 2qo�t�S�qv�t�� 2S 2�qv�t��

q�t� � �qo�t�, qv�t��TC�t�

S�a�b � a � bS��3 � 3

C�t� � I3�3 � 2qo�t�S�qv�t�� 2S 2�qv�t��

q�t� � �qo�t�, qv�t��TC�t�

�n � 3�.

M � 1.

� ��

2 �Mk�1

$$�In�n � CTCCTC�rk$$2

� ��Mk�1

�rkT�In�n � CTCCTC�rk

� � tr��Mk�1

��rkrkT CTCCTC rkrk

T� VM�t� � tr��CTS��C � �M

k�1CT�ykyk

T � ykykT�C

VM�t�

k � 1, 2, . . . , Myk�t� � C�t�rk�t�,

yk�t�

C�0� � �n�n� � 0,

C�t� � ��S�� Mk�1

�ykykT � ykyk

T��C�t�;


where, by virtue of the quaternion multiplication property [23], the following iden-tity holds

(31)

In other words, if the vectors q and are aligned in the same direction (i.e., ifor in other words, we have which from the unit vector

constraint on the quaternion implies that Recalling the definition of theattitude estimation error matrix in equation (4), it is obvious that

and accordingly, we have whenever Thus, the vector has the interpretation of the error quaternion such that

if and only if matrix Note that the cascade matrix is encountered in the convergence result

of Theorem 1. Accordingly, we adopt yet another quaternion representation for theproper orthogonal matrix given by such that

(32)

Through the quaternion multiplication property, it is easy to establish the identities

(33)

We are now ready to state the following result applicable to the three-dimensional case.

COROLLARY 2. For the given input-output system of equation (1) in the three-dimensional case, suppose the true/unknown attitude matrix evolving ac-cording to equation (2) is parameterized by the quaternion vector throughequation (28). If the attitude estimate matrix is parameterized by the unitquaternion according to equation (29) and is updated according to equation (5)subject to the condition that where

then for all nonzero (unit-vector) reference inputs the following con-vergence condition holds asymptotically

(34)

where is a quaternion representation for the cascaded properorthogonal matrix

PROOF. Consider the same Lyapunov candidate function from equation (7) spe-cialized to the three-dimensional case as

(35)

Making use of the quaternion parameterization for the proper orthogonalmatrix given in equation (30), we may rewrite as

Vo�t� �1

2 tr��2S2�zv� � 2S2�zv�� 2 tr�S2�zv��

Vo�t�CT�t�C�t�z�t�

Vo�t� �1

2 tr�CTC� �

1

2 tr�2I3�3 � �CTC CTC��

�n � 3�

CT�t�C�t�.z�t� � �zo�t�, zv�t��T

limtl�

��r�t� � zv�t�� 0

r�t� � �3

&u � � � �4�� 1; �Tq�0� � 0

q�0� � &u

q�t�C�t�

q�t�C�t�

wv � 2zozvwo � 2zo2 � 1,

CT�t�C�t�CT�t�C�t� � I3�3 � 2wo�t�S�wv�t�� 2S 2�wv�t��

w�t� � �wo�t�, wv�t��TCTCCTC

CTCCTCCT�t�C�t� � I3�3 ⇔ C�t� � 0.zo�t� � �1 ⇔ zv�t� � 0

z�t�CT�t�C�t� � I3�3.C�t� � 0C�CTC � I3�3�,

C �C�t��zv�� 0.

zo � �1C � C�,q � �qq

zo�t� � qT�t�q�t�


(36)

where Now, it is clear that is a uniformly bounded functionfor all More specifically, and there exists a very interestinggeometric interpretation for in equation (36) which can be explained throughthe quantities and q using the quaternion multiplication identity of equation (31).If is on the hyperplane normal to q (i.e., then its maxi-mum value. On the other hand, if is aligned along q (i.e., or then

its minimum value. Physically, corresponds to an error in the Eulerprincipal rotation angle (for the matrix given by and therefore, the prop-erties of in the three-dimensional case are very much analogous to the func-tion of the flatland case.

Starting from equation (8), and making use of the quaternion representationgiven in equation (32) for the matrix together with the identities listed inequation (33), the time derivative of function can be written as

(37)

From the foregoing analysis, it is clear that the dynamics of have an equi-librium manifold corresponding to which corresponds to the set of allpossible initial conditions In this case, and forall resulting in the fact that remains fixed at its initial value. Instabilityof this equilibrium manifold can be demonstrated by the argument that if

with � being arbitrarily small, then due to from equation (37), weare guaranteed that for all and accordingly, there is no way forconvergence to happen as We geometrically depict this interpre-tation of the unstable manifold in Fig. 2.

On the other hand, assuming we know that thus preclud-ing the possibility Using the facts and we areassured the limit as of exists and is finite. As a consequence, the inte-gral also exists as Further, from boundedness of (seen by dif-ferentiating both sides of equation (37) and recognizing that each term therein isbounded), we have uniform continuity of Using Barbalat’s lemma, we aretherefore led to conclude that From equation (37), since we al-ready ruled out the possibility of from happening, the only otherremaining way for to be satisfied is when thereby completing the proof.

For the case of attitude matrix estimation in three dimensions, the following ob-servations are now in order.

�

limtl��r�t� � zv�t�� 0limtl�Vo�t� � 0limtl�zo�t� � 0

limtl�Vo�t� � 0.Vo�t�.

Vo�t�tl �.�0t Vo�t� dt

Vo�t�tl �Vo�t� � 0,Vo�t� � 0limtl�zo�t� � 0.

Vo�0� � 4q�0� � &u,

tl �.zo�t�l 0t � 0�zo�t��

Vo�t� � 0� � 0,zo�0� � &u

Vo�t�t � 0Vo�t� � 0Vo�0� � 4q�0� � &u.

zo�0� � 0Vo�t�

� �8�z02��zv � r��2 � 0

� �8�z02rTST�zv�S�zv�r

� ��rT��1 � 2z02� �2z0�S�zv� � 8z0

2S 2�zv��r

Vo�t� � ��

2 $$�I3�3 � CTCCTC�r$$2

Vo�t�CTCCTC

VF�t�Vo�t�

�CTC �zo � 0Vo � 0,

�1�,z0 � 1qVo � 4,z0 � qTq � 0�,q

qVo�t�

0 � Vo�t� � 4t � 0.Vo�t�zv � �z1, z2, z3�T.

� 4�1 � z02�

� 4�z12 z2

2 z32� � 4zv

Tzv

� �2 tr��z32 � z2

2

z1z2

z1z3

z1z2

�z12 � z3

2

z2z3

z1z3

z2z3

�z22 � z1

2 �


1. An important consequence of Corollary 2 is the fact that whenever not only are we assured of convergence but in fact, theoutput estimation error also converges to zero as This result can beeasily established as follows: first, from equation (30) we note that

implies subsequently, starting with the definition of theoutput estimation error in the limit we obtain

which proves the stated assertion. 2. In three dimensions, the attitude estimate is a matrix and hence has

nine entries that need to be updated if we are to adopt the matrix differentialequation described by equation (2). However, a computationally efficient methodfor estimating the matrix would be to update the four-dimensional quater-nion vector satisfying the parameterization of equation (29). Such an updatedifferential equation for can indeed be derived. To do so, we first recognizethat for three-dimensional vectors and the skew-symmetric matrix

listed in equation (2) can be expressed in terms of the vector cross-product as

(38)

Accordingly, an update law for the unit-quaternion may beexpressed by

(39)

3. The error quaternion vector parameterizing the proper or-thogonal matrix according to equation (30) has a time-evolution de-scribed by

CT�t�C�t�z�t� � �zo�t�, zv�t��T

qv�t� �1

2 �qo I3�3 S�qv�� y � y�qo�t� � �

1

2 qv

T�� y � y�;

q�t� � �qo�t�, qv�t��T

��yyT � yyT� � �S��y � y�

yyT � yyTy�t�,y�t�

q�t�q�t�,

C�t�

3 � 3C�t�

e�t� � y�t� � y�t� � C�t�r�t� � C�t� �CT�t�C�t� � I3�3�r�t� � 0

tl �.e�t�,CTCr � r;S�zv�r � 0

zv � r �tl �.e�t�

limtl��r�t� � zv�t�� 0q�0� � &u,


FIG. 2. Illustration of the error quaternion vector representing the proper orthogonal matrixThe hyperplane represents an unstable equilibrium manifold.zo � 0C TC.

z � �zo, zv�T

(40)

The rotational invariance property of the cross product

can be substituted in equation (40) to establish

(41)

4. From Corollary 2, it is possible to show that when together with(i.e., the reference input not a constant vector), then the attitude estima-

tion error asymptotically converges to zero. In order to prove this statement,we start from Corollary 2 where we already proved that 0. This result can be applied in equation (30) to infer that as which may further be substituted in equation (41) leading us to

Further, from uniform continuity of we have

which implies that or 0. So far, we have proved that as we have quantities and

i.e., the vector is simultaneously parallel to both and This can be possible only if since every unit vector satis-fies and therefore vectors and remain normal to one otherfor all t. Now that we have from equation (30), it is possible toconclude that as and accordingly, the asymptotic con-vergence result for the attitude estimation error matrix.

5. We note that our proposed attitude estimation algorithm is developed under theassumption that the angular velocity vector is perfectly measured/determined.This assumption is easily violated in practical applications due to the fact thatgyros that measure angular velocity exhibit drift over time. Typically, such gyrodrift is slow and small (at least for high-grade gyros) and hence the bias in an-gular velocity measurements may be considered to remain constant. When angu-lar velocity biases are included, the stability of the proposed estimator may bepresented in a robustness context. Suppose a small constant bias b exists in themeasurement of the angular velocity i.e.

and the attitude estimate update law from equation (5) is accordingly modified as


Then, the dynamics of the vector part of the error quaternion vector is mod-ified from equation (41) and can be determined as

zv�t�

C�0� � �n�n� � 0,

C�t� � ��S�� yyT � yyT��C�t�;

��t� � ��t� b

��t�,

��t�

limtl� C�t� � 0tl �CT�t�C�t�l I3�3

limtl�zv�t� � 0,r�t�r�t�rT�t�r�t� � 0,

r�t�zv�t� � 0r�t�.r�t�zv�t�zv�t� � r�t�l 0,

zv�t� � r�t�l 0tl �,limtl��zv�t� � r�t�� limtl��zv�t� � r�t� zv�t� � r�t�� 0

limtl�

�z v�t� � r�t�� 0 ⇒ limtl�

d

dt�zv�t� � r�t�� 0

zv�t� � r�t�,limtl� zv�t� � 0.tl �

CT�t�C�t�r�t� � r�t�limtl��zv�t� � r�t��

C�t�r�t� � 0

q�0� � &u

zv�t� ��

2�zoI3�3 S�zv�� r � CT�t�C�t�r�zo�t� � �

�

2 zv

T�r � CT�t�C�t�r�;

CT�t� ��y � y� � �CT�t�y � CTy � �r � CT�t�C�t�r

zv�t� �1

2�zoI3�3 S�zv��CT�t� ��y � y�zo�t� � �

1

2 zv

TCT�t��y � y�;


(43)

When compared to the no-bias case of equation (41), we note that the presenceof gyro bias introduces the bounded perturbation term seen from the last term ofequation (43). This perturbation vanishes when but is otherwise nonvan-ishing when

From a dynamic stability standpoint, obviously, we are now looking at an as-ymptotically stable system equation (41) being perturbed by a bounded pertur-bation. Linearization of the unperturbed system equation (41) and evaluating aboutthe equilibrium results in a time-varying Jacobian matrix whose eigenvalues are evaluated as The fact that the eigenvaluesdon’t all have strictly negative real parts prevents us from using any of thestability/robustness results available through the linearized approximations. Onthe other hand, the unit norm constraint on obviously holds even in the pres-ence of the bias error, and hence boundedness of trajectories for equation (43) inthe presence of perturbations is not an issue. However, an interesting theoreticalquestion is whether a meaningful upper bound can be derived on the differencebetween the solutions of the unperturbed system of equation (41) and the per-turbed system in equation (43). It is reasonable to hypothesize that small biaserrors (small perturbations) would maintain solutions of the perturbed system toremain “close” to those of the asymptotically stable unperturbed system. How-ever, such conclusions are meaningful and may only be drawn on compact timeintervals since the difference between the unperturbed and perturbed solutions isgoverned through exponential terms involving the Lipschitz constants of the un-perturbed system which grow to infinity as A great technical obstacle fordrawing bounds valid for all time t is the fact that we only have proof of asymp-totic stability for the states in the unperturbed case. In other words, a lack ofproof for exponential stability for the nonlinear unperturbed system prevents usfrom making any authoritative comments on how far the trajectories of the per-turbed system deviate from those of the unperturbed (no gyro bias case) nominalsystem. Of course, the fact that we cannot show that the difference between theunperturbed and perturbed trajectories remains small (at least when the bias erroris small) doesn’t mean it is not true.5

6. It is possible to derive further insights into the convergence properties of the at-titude estimation algorithm for the case of constant reference input, i.e.,for all To enable this analysis, using equation (40), we derive the projec-tion of along as given by

(44) � ��r*Tzv� ��zv � r*��

2

� �r*T�zoI3�3 S�zv�� zo�zv � r*� � r* � �r*

Tzv� �zv � r*��

r*T zv�t� �

�

2r*

T�zoI3�3 S�zv��CT�t� �y � y�

r*z v�t�t � 0.

r�t� � r*

zv�t�

tl �.

zv�t�

�1, �1, 0.�rrT � I3�3�zv � 0

zv�t�l 0.b � 0

zv�t� ��

2�zoI3�3 S�zv�� r � CT�t�C�t�r�

1

2�zoI3�3 S�zv��CT�t�b


5We acknowledge the insight provided to us by an anonymous reviewer who performed numerical simu-lations including a small gyro bias term and discovered satisfactory performance by the proposed attitudeestimator.

The preceding expression for has two important consequences. The firstis that when the component of along is zero, will always remainbounded to a plane that is normal to the vector. This implies that the vector partof the quaternion estimation error vector as whenever

i.e., because of the fact that asprovided by Corollary 2. The other consequence of equation (44) is that anynonzero component of along the direction of has the same sign as that ofthe component of along Therefore, the absolute value of increasesmonotonically so that the vector part of the error quaternion never convergesto zero.

The foregoing discussion can be formalized through an elegant geometric in-terpretation about a unit sphere in three dimensions. Note that the vector part ofthe quaternion estimation error evolves on or inside a unit three-dimensionalsphere. Now, consider the constant reference input vector as the axis of thissphere so that its intercepts with the sphere are respectively the North and SouthPoles. The plane normal to the direction may be designated the equatorialplane. With these definitions, now, depending on various possibilities for the ini-tial conditions on and we have the following convergence conditions:(a) If is on the equator, i.e., and then will al-

ways remain on the equator. (b) If is on the equatorial plane but not on the equator, i.e., and

the vector part of the quaternion estimation error asymp-totically converges to zero (the center of the sphere). Accordingly, the atti-tude estimation matrix also converges to zero.

(c) If is inside the sphere but not on the equatorial plane, then willconverge to a point on the axis towards the closest pole.

(d) If is on the surface of the sphere but not on the equator, then willconverge to one of the poles. More precisely, if is in the Northern Hemi-sphere, then converges to the North Pole. Likewise for in theSouthern Hemisphere, the vector part of the quaternion estimation error converges to the South Pole.

Numerical Simulations

In this section, we present results from numerical simulation of the attitude esti-mation algorithm. We restrict attention to the three-dimensional case so as to mimicthe problem of estimating the attitude of a satellite stationed in a geosynchronousorbit. Accordingly, the reference input signal and angular velocity of the satel-lite � expressed in body frame are taken as

(45)

For all our simulations, the initial value of the quaternion representing thetrue/unknown attitude matrix according to equation (28) is fixed as

Three different simulations are performed with By changing the � value,we can adjust the convergence speed of the attitude estimator. For the first case, theinitial condition for the quaternion estimate parameterizing the attitude esti-mate matrix is selected to be C�0�

q�0�

� � 1.q�0� � �0.5, 0.5, 0.5, 0.5�T.

C�0�

� � �0, 0, 1�Tr�t� � �sin t, cos t, 0�T,

r�t�,

zv�t�zv�0�zv�t�

zv�0�zv�t�zv�0�

zv�t�zv�0�C�t�

zv�t�zv�0�Tr* � 0,zo�0� � 0zv�0�

zv�t�zv�0�Tr* � 0,zo�0� � 0zv�0�zo�0�,zv�0�

r*

r*

zv�t�

zv�t�r*

Tzv�t�r*.zv�t�r*zv�t�

limtl��zv�t� � r*� � 0zo�0� � 0,q�0� � &u,tl �zv�t�l 0

r*

zv�t�r*zv�t�r*

T zv�t�


(46)

For this choice, it is easy to verify that and therefore, Giventhat the reference input satisfies and based on the discussion from theprevious section, we know that the attitude estimation error asymptoticallyconverges to zero. Simulation results for this case are presented in Fig. 3 where itcan be seen, quite expectedly, that

As a second case for our simulation studies, we select a different value for theinitial condition for the estimate quaternion This time the estimation starts from

Obviously, for this choice, we have and therefore theoretically speaking, we should have for all

However, from the fact that is an unstable equilibrium manifold, wesee in Fig. 4 that the vector part of the error quaternion converges to nearlyzero after about 40 seconds of simulation time.

Finally, the measurement noise is introduced to illustrate the robustness proper-ties of the attitude estimation algorithm. The bounded measurement noise hasuniform random distribution on the interval The initial value of theestimate quaternion is again taken according to equation (46) so that

Simulation results including the effects of measurement noise areshown in Fig. 5. As discussed earlier, we no longer have the asymptotic conver-gence of error quaternion to zero. Instead, each component of the estimationerror quaternion remains bounded so that ultimately the magnitude of is dic-tated by the magnitude of the measurement noise.

��zv�t��zv�t�

q�0� � &u.q�0�

��0.05, 0.05�.v�t�

zv�t�zo � 0t � 0.

zo�t� � 0, ��zv�t�� 1zo�0� � 0q�0� � �0.5�2, 0, �0.5�2, 0�T � &u.

q�0�.

limtl�zv�t� � �z1�t�, z2�t�, z3�t�� 0.

C�t�r�t� � 0r�t�

zo�0� � 0.q�0� � &u

q�0� � ��0.9, �0.1�3, �0.1�3, �0.1�3 �T


FIG. 3. The trajectory of the error quaternion as a function of time t. The initial condition so that zo�0� � 0.

q�0� � &uz�t�


FIG. 4. The trajectory of the error quaternion as a function of time t. The initial condition (the unstable equilibrium manifold) so that Accumulation of numerical round-off errors

ultimately results in regulation of the attitude estimation error to zero.zo�0� � 0.

q�0� � &uz�t�

FIG. 5. The trajectory of the error quaternion as a function of time t. The initial condition Measurement noise is introduced as a signal having a uniform distribution between and 0.05.�0.05

q�0� � &uz�t�

Concluding Remarks

This paper presents an adaptive estimation algorithm, together with a conver-gence proof that holds for all proper orthogonal matrices. The importance ofthis problem, particularly in three dimensions, lies in the fact that no attitude rep-resentation is ever directly measured. Instead, such information always needs to bereconstructed through techniques presented in this paper while using availableinput-output observations. In practice, measurements from sensors that generateinput-output type vectors may be used in conjunction with the proposed methodol-ogy to reliably reconstruct the attitude information.

The results obtained here have broader implications, particularly for purposes ofgenerating feedback control signals on the group of rigid body motions whereinsignals corresponding to attitude representations (Euler angles, Gibbs vector,and/or quaternions) are estimated on the basis of measurement models.

Acknowledgments

This work was supported in part by the National Science Foundation under Grant CMS-0409283.

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[22] SHUSTER, M. D. “Attitude Analysis in Flatland: The Plane Truth,” The Journal of the Astro-nautical Sciences, Vol. 52, Nos. 1&2, 2004, pp. 195–209.

[23] SCHAUB, H. and JUNKINS, J. L. Analytical Mechanics of Space Systems, AIAA EducationSeries, 2003, Chapter 3.

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Attracting manifolds for attitude estimation in flatland and otherlands

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