Attitude Error EstimatorDaniele Mortari

Universita degli Studi di Roma, Via Salaria 851, 00138 Roma (Italy).

Abstract

This paper describes a geometrical method to predict the attitude error, that is, thedistance between the estimated and the unknown true attitude matrices. The method,which is based on vector observations and on the sensors’ noise knowledge only, isdeveloped according to the attitude error, whose definition and statistical parameters,are here given. The theory of the Attitude Error Estimator is developed forn �2 observations. Finally, numerical tests to compare the proposed method with thestandard trace of covariance matrix, are provided. Tests show that the Attitude ErrorEstimator describes the reliability of an attitude estimation faster and more accurately.

Introduction

Several different approaches exists to evaluate spacecraft attitude based on vectorobservations. During the last two decades, the improvement in this field have beenachieved in term of developing faster algorithms satisfying the Wahba optimality def-inition. Once the attitude is evaluated, the problem to establish how much reliable theestimation is, still remains. The reliability is worldwide accepted to be represented bythe trace of the covariance matrix, which is usually time consuming. Therefore, theattitude reliability evaluation usually slows down the attitude estimators. In this papera geometrical method, which is based on the attitude error definition, is proposed asan alternative method to quantify the attitude reliability.

The idea of an “Attitude Error Estimator” (AEE) was firstly presented in a previouswork (Ref. [1]), and then applied in Ref. [3] for multiple fields of view star trackers.However, the solution presented was found under linearization, and no comparisonswere performed versus the trace of the covariance matrix, tr(P). This paper, presentsthe complete, non approximated AEE solution, and compares the results by meansof numerical tests which validates the AEE as a better and faster tool to evaluate theattitude reliability with respect to the tr(P).

However, prior to enter in the mathematical details of the algorithm, the attitude error(how far an estimated attitude matrix is with respect to the unknown true attitudematrix), should be clearly defined.

The Attitude Error

The position error is easily described by the distance between two vectors, whichidentify, for instance, the estimated and the true positions. Unfortunately, the error

on rotation (the attitude is a rotation) cannot be so easily represented. The problemconsists to define thedistance between two orientations, that is, between two differentattitude matrices. To this end, letT andA be the true and the estimated attitudes,respectively. MatrixA can even be randomly chosen, provided that the conditionsATA �AAT � I, and det�A� ��1, which assureA to be a rotation matrix, are satisfied.LetC � TAT. SinceC is a product of rotation matrices,C is a rotation matrix too. Thismatrix represents the correttive rotation that, applied to our estimated attitude, movesthe estimationA to the true attitudeT . In fact,

CA � TATA � T (1)

SinceC is a rotation matrix, it has it own principal axise and principal angleΦ. Sincee is the principal axis of the correttive matrixC, its direction does not change duringthe correction and, therefore, its direction is error-less (Ce � e) independently of thechoice of theA matrix.

Let us consider the spherical triangle ofvertexe, b andCb, shown in Fig. 1, andwhereb is an arbitrary direction, displacedfrom e by the angleϑ. Due to theC rota-tion, the direction ofb will be corrected toCb, which is displaced frome by the sameangleϑ.This implies thatb is affected by the er-ror ε, where cosε � bTCb. Now, the sinelaw allows us to write sin�Φ�2� sinϑ �sin�ε�2�, that yields to the relationship

Fig 1. Direction Error Geometry

cosε � �1�cosΦ�cos2ϑ �cosΦ (2)

where cosϑ � eTb � eTCb � eTTATb, and cosε � bTCb � bTTATb. Equation (2) pro-vides the attitude error distribution (shown in Fig. 2), and establishes that it existsa maximum value for the errorεmax � Φ associated with directions orthogonal toe(displaced frome by ϑ � π�2), whileε � 0 for Φ � 0, for any value ofϑ (all direc-tions are error free). WhenΦ �� 0, then it resultsε � 0 for all the directions alignedwith e, that is, for those havingϑ � 0 or ϑ � π, regardless the value ofA! This im-plies that, for instance, having a spacecraft with orientation described by the matrixA(with A any rotation matrix), whileT is an attitude which allows a correct pointing ofan on-board telescope, then in the lucky case that the mounting angle of our telescopecoincide with the principal axis ofC, then the telescope would point correctly!

Usually, the practical values of the maximum attitude error are so small that cosΦ ��1�Φ2�2 is a well accepted approximation. Sinceε � Φ, then cosε �� 1� ε2�2 iscertainly satisfied. By this substitution, Eq. (2) provides the following simple ap-proximated expression of the error associated with a direction displaced frome bythe angleϑ

ε�ϑ� �� Φsinϑ � εmaxsinϑ (3)

Figure 2: Rotation Error Geometry

that holds for smallΦ. This relationship allows us to provide simple expressions forthe statistical parameters of the direction errorε.

In fact, since the distribution ofε, provided by Eq. (2), is axial symmetric withrespect toe (see Fig. 2), then it does not depend on the azimuthϕ. Therefore, thepartial derivative with respect toϕ of the probability density functionP�ϑ�ϕ�, mustbe zero, that is,∂P�ϑ�ϕ��∂ϕ � 0, which impliesP�ϑ�ϕ� � P�ϑ�. Now, any directionmust have equal probability (otherwisee would have preferential directions), thenthe infinitesimal probabilityP�ϑ�dϕ dϑ to have directions in the infinitesimal areadS � �sinϑ dϕ�dϑ must be proportional todS. This impliesP�ϑ� � k sinϑ, and theconstraint

� π0 P�ϑ�dϑ � 1, providesk � 1�2. This allows us to evaluate the expected

value

E�ε�� ε �� π

0ε�ϑ�P�ϑ�dϑ ��

� π

0Φ sinϑ

sinϑ2

dϑ �π4

Φ (4)

the mean square error

E �ε2�� ε2 ��

� π

0Φ2 sin2ϑ

sinϑ2

dϑ �23

Φ2 (5)

the variance

V�ε�� E��ε� ε�2�� ε2� �ε�2 ���2�3� �π�4�2�Φ2 (6)

and the standard deviation

D���

V�ε� �� Φ�

2�3� �π�4�2 (7)

These equations demonstrates that the statistical parameters of the errorε (becauseAdiffers fromT ), all depend on the maximum valueεmax� Φ. Sinceε is constrainedto an assigned shape by Eq. (2), what is characterizing it is its maximum value only(principal angleΦ of matrixC). The angleΦ, or the expected valueE�ε� � πΦ�4,is the parameter identifying thedistance between two orientations.

Summarizing: associated with the translation we have an isotrope distribution for theerror (there is no preferential direction) which constitutes an open set (no upper limitfor the error) while, associated with rotation the error distribution is anisotrope orpolarized along one direction (error always zero alonge and maximum along orthog-onal directions) which constitute a closed set (maximum error limited and always lessor equal toπ).

The Attitude Error Estimator

The AEE algorithm consists of estimating the attitude error directly from the knowl-edge of the observed directions, and the associated relative precision. The knowledgeof the spacecraft attitude is not required and the attitude reliability can be quantifieda priori, before the attitude estimation process.

To this end, let us consider (see Fig. 3) the case ofn� 2 observationsb 1 andb2, whichhave a cone of uncertainty ofβ1 andβ2, respectively. For a Gaussian error distributionwe can considerβ1 � 3σ1 andβ2 � 3σ2, whereσ1 andσ2 are the Gaussian standarddeviations. Now, Ref. [2] has demonstrated that, forn � 2 observations Wahba’soptimality definition implies a co-planarity among the directionsb 1, b2, Ar1, andAr2,wherer1 andr2 are the reference directions associated withb1 andb2, andA is theoptimal attitude satisfying Wahba’s cost function. It is clear from Fig. 3 that theproblem to find the attitude error can be seen as made of two components: 1) an in-plane error, and 2) an out-of-plane error. The in-plane error occurs whenT differsfrom A in such a way that the plane identified byb1 andb2 coincides with the planeidentified byTr1 andTr2, while the out-of-plane error increases as the inclination ofthese two planes increases. This means that the in-plane error cannot exceed the valueof min�β1�β2�, which is usually small, and which does not vary with the observationinterstar angleϑb. In the contrary, the out-of-plane error is represented by the angleδ, and this angle increases as the interstar angleϑ b decreases. In particular, when theuncertainty cones start to touch each other, then the angleδ is maximized. The angleδis strictly related to the maximum attitude errorΦ. In particular, this angle representsthe upper limit of all the maximum attitude errors associated to all the possible trueattitude matricesT . Numerical tests have demonstrated that the relationshipΦ � k δ,holds, where the value ofk � 1 depends on the number of observationn.

Therefore, the maximum out-of-plane error (δ) is achieved forT such thatTr 1 andTr2 both lie on the cones of uncertainty with axesb1 andb2 and aperturesβ1 andβ2,respectively. Reference [1] has provided the approximated expression sin2δ� �β2

1�β2

2�2β1β2 cosϑ��sin2ϑ, which was based on some linearizations, and on theϑ �ϑr � ϑb approximation. In this paper, the complete (non approximated) analytical

Figure 3: Attitude Error forn � 2 Observations

solution forδ, who works better than the approximated solution as the values ofϑbecome critical, is given and compared versus tr[P] to quantify the attitude estimationreliability.

Based on Fig. 3 (where cosα � eTAr1� cosϑ1 � bT1Ar1, and cosϑ2 � bT

2Ar2), thespherical triangles [e, b1, Tr1] and [e, b2, Tr2], allows us to write [C��� � cos���, andS��� � sin���]

�CδSαS�ϑ1�α��CαC�ϑ1�α� �Cβ1

CδS�ϑr�α�S�ϑr�ϑ2�α��C�ϑr�α�C�ϑr�ϑ2�α� �Cβ2

(8)

whereϑ1�β1�ϑb�ϑr���β1�β2�, andϑ2�β2�ϑb�ϑr���β1�β2� are demonstratedin Ref. [2], and where cosϑ r � rT

1r2, and cosϑb � bT1b2. Now, settingϑ � ϑ r �ϑb,

and rearranging Eq. (8), we obtain�������

cos�2α �ϑ1� �2Cβ1

�Cϑ1�1�Cδ�

�1�Cδ�

sin�2α �ϑ1� �2Cβ2

�Cϑ2�1�Cδ��Cϑ �2Cβ1�Cϑ1�1�Cδ��

�1�Cδ�Sϑ

The condition cos2�2α�ϑ1��sin2�2α�ϑ1� � 1, allows us to obtain a solving equa-tion in term of cosδ

A cosδ�2B cosδ�C � 0 (9)

whereA � S2ϑ�C

2ϑ1�1���Cϑ1Cϑ�Cϑ2�

2, B � S2ϑ�ξ1Cϑ1�1���ξ1Cϑ�ξ2��Cϑ1Cϑ�

Cϑ2�, andC � S2ϑ�ξ

21� 1� � �ξ1Cϑ � ξ2�

2, and whereξ1 � 2Cβ1�Cϑ1, andξ2 �

2Cβ2�Cϑ2. The second order Eq. (9) provides the two solutions

cosδ� �B�

B2�AC��A (10)

These solutions are associated with the directionsTr1 andTr2 placed on the sameside or on the opposite side with respect to the [b1, b2] plane, respectively. Thus,the searched solution is the greatest one. Now, at any possible true attitude matrixT ���, there is an associated maximum attitude errorΦ��� Therefore,δ represents themaximum attitude error associated with theT ��� who maximizesΦ���. Hence,δ �max�Φ����. Numerical tests have demonstrated thatδ� hE�εmax� whereh � h�n� isa proportional constant.

As already described in Refs. [1, 3], when the observations aren � 2, then thesearched solutionδ is simply represented byδ � min�δi j�βi�β j��, whereδi j�βi�β j�is associated with thebi andb j observations.

Covariance Matrix

The covariance matrixP is often derived from the Fisher information matrix and itsinverse, the Cramer-Rao Lower Bound, which quantifies the error variance of theestimator. In particular, in the attitude estimation problem, letB � Σ i wi bi rT

i be the

attitude profile matrix, thenPb �κ I�BBT

λκ �det�B�, andPr �

κ I�BTBλκ �det�B�

are the co-

variance matrices in the body and in the reference frames, respectively, and where2κ � λ2� tr�BBT�. At λ � λmax, we haveP � �λmaxI�Aopt BT��1. It is clear thatPprovides more information thanE�εmax�. However, in order to compare tr[P] againstE�εmax�, let us follow Ref. [4], which has shown thatAoptBT � Σi wi bi bT

i , and whichevaluates the covariance matrix as

P � �λ I�Aopt BT��1 � �Σi wi �I�bi bTi ��

�1 (11)

Numerical Tests

Some numerical tests have been performed to compare AEE against the tr[P] pro-vided by Eq. 11. The first plot of Fig. 4 shows the simulated errors as a function ofthe interstar angleϑ b for n � 2 observations together with the linear best fitting lineand the value provided by AEE and tr[P]. In the second plot the ratio with respect thelinear best fitting is given. From Fig. 4 it becomes evident that the AEE results betterdescribe the slope of the line fitting the data. As for the speed tests (not included),AEE is much faster than tr[P], especially when the approximated solution is used.

Conclusions

This paper presents an alternative method to quantify the reliability of an attitudeestimation. The proposed Attitude Error Estimator is derived from the geometricalnature of the attitude error, here shown, as well as from the spatial displacement of theobserved vectors and they precision. Compared to the trace of the covariance matrix,

100

101

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10−1

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103

104

Angle between the two observed directions (deg)

Max

imum

Atti

tude

Err

or (

deg)

and

trac

e[co

varia

nce]

AEE Estimation

trace[covariance]

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103

104

Angle between the two observed directions (deg)

Rat

io w

ith th

e Li

near

Bes

t−F

ittin

gAEE Estimation

trace[covariance]

Figure 4: Numerical Test Results

worldwide adopted as the standard mathematical tool to quantify the attitude esti-mation reliability, the resulting algorithm presents two advantages: 1) the consumedtime is dramatically shorter, and 2) the accuracy to describe the attitude reliability isbetter.

References

[1] Mortari, D. Range Limits of Attitude Determination Accuracy,Advances inthe Astronautical Sciences, Vol. 97, Pt. I, pp. 167-178. Paper 97-611 of theAAS/AIAA Astrodynamics Conference, Sun Valley, ID, August 4-7, 1997.

[2] Mortari, D. EULER-2 and EULER-n Algorithms for Attitude Determinationfrom Vector Observations,Space Technology, Vol. 16, Nos. 5/6, 1996, pp. 317-321.

[3] Mortari, D., Pollock, T.C., and Junkins, J.L. Towards the Most Accurate Atti-tude Determination System Using Star Trackers,Advances in the AstronauticalSciences, Vol. 99, Pt. II, pp. 839-850. Paper AAS 98-159 of the 8th AnnualAIAA/AAS Space Flight Mechanics Meeting, Monterey, CA, Feb. 9-11, 1998.

[4] Markley, F.L. Attitude Determination using Vector Observations: a Fast OptimalMatrix Algorithm, Journal of the Astronautical Sciences, Vol. 41, No. 2, April-June 1993, pp. 261-280.