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Deterministic Relative Attitude Determination of
Formation Flying Spacecraft
Michael S. Andrle∗, Baro Hyun†, and John L. Crassidis‡
University at Buffalo, State University of New York, Amherst, NY, 14260-4400
This paper studies deterministic relative attitude determination of formation flyingspacecraft. The results are intended to allow one to have knowledge of how accuratethe deterministic estimate is given uncertainty information for all the measurements. Aformation of three spacecraft is considered in which each vehicle is equipped with lasercommunication devices. We wish to utilize information obtained by maintaining the align-ment of the lasers to estimate the relative attitude between the spacecraft. This is possiblebecause there are enough measurements to determine the unknown parameters. The pri-mary goal of this work is to provide a covariance analysis aimed at providing insight on thestochastic properties of the estimate error.
I. Introduction
Attitude determination is the calculation of the relative orientation between two reference frames, twoobjects, or a reference frame and an object. The amount of research conducted for this task as well as thequantity of related publications is quite extensive. In the most basic problem, a star tracker is used onboarda space vehicle to observe line of site vectors to stars which are compared with known inertial line of sitevectors to estimate the inertial attitude of the space vehicle. It is obvious why this topic has acquired so muchattention as nearly every spacecraft and satellite ever launched into space required at least the knowledge ofits orientation if not additionally the specification. A survey of attitude estimation algorithms is presentedin Ref. 1.
Manned missions and unmanned missions each have requirements that demand some attitude knowledge.Many manned missions require that critical communications devices be oriented in a specific direction inorder to maintain functionality. Also, because of the space environment, a manned vehicle must alwaysbe changing its orientation with respect to the sun so that it does not attain undesirable temperatures.Unmanned missions often have requirements in common with manned missions. For example, the orientationof solar arrays is critical in maintaining the life of a satellite. Additionally, many well-known space vehicles,such as the Hubble Space Telescope, are scientifically gathering data and must be pointing in a specific,known direction with high accuracy.
Formation flying employs multiple (space) vehicles to maintain a specific relative position, either a staticor a dynamic closed trajectory. Of interest is the dynamics of the relative position of two or more referenceframes, space vehicles, or reference frame(s) and space vehicle(s). An earlier development in this areawas done by Clohessy and Wiltshire2 in which the relative dynamics between two Earth-orbiting craftwere developed. Important applications include spacecraft docking and rendezvous, space station re-supplyoperations, and now satellite formations.
With more and more spacecraft applications employing a formation of multiple vehicles, there has beenan increase in opportunity. For the specific case under consideration, a group of spacecraft performing asingle task require a means to communicate within the formation. To accomplish this, they are equippedwith laser communication devices. The lasers must remain aligned through the use of a gimballed apparatus
∗Graduate Student, Department of Mechanical & Aerospace Engineering. Email: [email protected].†Graduate Student, Department of Mechanical & Aerospace Engineering. Email: [email protected].‡Professor, Department of Mechanical & Aerospace Engineering. Email: [email protected], Associate Fellow AIAA, Member
AAS.
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so that communication is not lost. It is formation flying spacecraft with such a means of communicationthat this work seeks to exploit.
To maintain communication, information characterizing the incident laser is measured to ensure align-ment. This work will show that this information can be used as line-of-site (LOS) vectors between thespacecraft that is capable of being employed to determine the vehicle relative orientations. This is usefulfor three primary reasons. First, information that is being gather by sensors for another purpose can becombined with the onboard algorithms used in attitude estimation to provide a better estimate. Secondly,in the event that any onboard equipment were to fail, a second methodology can be used to assist in stateestimation. Finally, for lower-precision applications or for cost-driven projects, not all of the spacecraft inthe formation are required to fly with all of the sensors required in a mission for a single spacecraft.
II. Problem Definition
The geometry of the problem is well described in figure. 1. There are three spacecraft flying in formation,each vehicle equipped with optical-type sensors, such as VISNAV3 or laser communication system that usesfocal-plane detector (FPD). Through the sensor a vehicle measures the LOS vector to the two other vehicles,and this applies to each vehicle, making three pairs of LOS measurements.
1
1 2/d
d db Deputy 2
SpacecraftDeputy 1
Spacecraft2
2 1/d
d db
D2D1
2
2 /d
d cb 1
1 /d
d cb
2/C
c db
1/c
c db
Chief
SpacecraftC
Figure 1. Space Vehicle Formation.
Both theoretical research and supporting simulations will require LOS vectors that describe with respectto one object or frame, what direction another given object is along. Because these will exist between variousobjects as well as represented using different reference frames, a structured notation is required here as well.A subscript will describe the object the LOS is taken both from and to while a superscript will denote towhich reference frame the LOS is both represented by and measured in. For example, rx
x/y is a LOS vectorbeginning at x and ending at y while it is both expressed in and observed from reference frame X . Now theLOS vectors in Fig. 1 are properly defined.
There are two relative attitudes to be determined since using the characteristic of the attitude matrixthe third attitude is easily obtainable if two relative attitudes are given. For example, knowing Ad1
c andAd2
c gives Ad2d1
= Ad2c (Ad1
c )T . Using the configuration of the LOS vector measurements between vehicles, weare capable of deterministically solving the relative attitude. More detailed literature about deterministicattitude determination can be found in Ref. 4.
Although the unknown relative attitude is deterministically solvable, LOS vector measurements are usu-ally associated with measurement errors. Therefore, it is critical to investigate the confidence of the attitudesolution that was given deterministically with respect to the amount of error that is involved in the LOSmeasurement. A covariance analysis gives an analytical interpretation regarding to this issue. Before showingthis analysis though, we begin with the sensor model that was used.
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III. Sensor Model
The deputy sensor LOS is denoted by s and observes the sensor located in chief coordinates by the vectorb. The relative position between the two spacecraft is related to the sensor position and LOS vector:
scc/d = xcx + ycy + zcz (1a)
bc = b1cx + b2cy + b3cz (1b)rc = bc − sc
c/d = (b1 − x) cx + (b2 − y) cy + (b3 − z) cz (1c)
where cx,y,z denotes the basis of the chief body-fixed frame. This LOS vector in the chief frame is relatedto its observation in the deputy frame by a now familiar attitude mapping
rd = Adcr
c =
A11 A12 A13
A21 A22 A23
A31 A32 A33
(b1 − x)(b2 − y)(b3 − z)
(2)
Defining the focal plane axes as fx, fy, we can write the planar observation as
m = αfx + βfy (3)
where α and β are given by geometric ratios scaled by the focal length, f :
α = fA11 (b1 − x) + A12 (b2 − y) + A13 (b3 − z)A31 (b1 − x) + A32 (b2 − y) + A33 (b3 − z)
(4a)
β = fA21 (b1 − x) + A22 (b2 − y) + A23 (b3 − z)A31 (b1 − x) + A32 (b2 − y) + A33 (b3 − z)
(4b)
A typical noise model used to describe the uncertainty in the focal plane coordinate observations is givenas3
m = m + υm (5)
with
υm∼N (0, Ri) (6a)
Ri =σ2
1 + d (α2i + β2
i )
[(1 + dα2
i
)2 (dαiβi)2
(dαiβi)2 (
1 + dβ2i
)2
](6b)
where σ2 is the variance of the measurement errors associated with α and β, and d is on the order of 1. Thismodel accounts for an increased measurement standard deviation as distance from the camera bore-sightincreases.
Assuming a focal length of unity, the sensor LOS observations can be expressed in unit vector form (herewith unit length) given by
bdi =
1√1 + α2
i + β2i
αi
βi
1
(7a)
bci =
1√(b1 − x)2 + (b2 − y)2 + (b3 − z)2
b1 − x
b2 − y
b3 − z
(7b)
The general attitude determination problem is here observed. One seeks the attitude mapping, Adc , that
satisfies Eq. (2). The measurement vector is defined as
bdi = bd
i + υi (8)
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withυi∼N
(0, RQ
i
)(9)
To characterize the LOS noise process resulting from the focal plane model, Shuster suggests the followingapproximation:5
RQi = E
υiυ
Ti
= σ2
(I3×3 − rirT
i
)(10)
known as the QUEST Measurement Model (QMM). A geometric interpretation of the covariance given bythe QMM can be obtained by first considering the outer-product. The operator formed by the outer productof a vector, r with itself is a projection operator whose image is the component of the domain spanned byr. Similarly, the operator
(I3×3 − rrT
), is also a projection this time yielding an image perpendicular to r.
What this means for the covariance given in Eq. (10) is that the error in the vector, r, is assumed to liein a plane tangent to the focal sphere. It is clear that this is only valid for a small FOV in which a tangentplane closely approximates the surface of a unit sphere. For wide FOV sensors, a more accurate measurementcovariance is shown in Ref. 6. This formulation employs a first-order Taylor series approximation about thefocal plane axes. The partial derivative operator is used to linearly expand the focal plane covariance inEq. (6), given by
Ji =∂ri
∂mi=
1√1 + α2
i + β2i
1 00 10 0
− 1
1 + α2i + β2
i
rimTi (11)
Then the large FOV covariance model is given by
R∗i = JiRiJTi (12)
IV. Covariance Analysis
A. Vector LOS Covariance
A relationship for the true LOS vectors between the two deputy spacecraft can be expressed in terms of theappropriate attitude transformation:
bd2d2/d1
= Ad2c Ac
d1bd1
d2/d1= Ad2
d1bd1
d2/d1(13)
Under the assumption of perfect LOS knowledge, this expression indicates that bd2d2/d1
is the image of bd1d2/d1
through the transformation Ad2d1
. Note that the model in Eq. (13) assumes parallel beams, which must bemaintained through hardware calibrations.
In practice, both bd2d2/d1
and bd1d2/d1
are not exactly known and Eq. (13) is an assumed relationship betweenavailable measurements. The measured LOS vectors are available from the onboard sensors and are relatedto the truth by
bd2d2/d1
= bd2d2/d1
+ υd2d2/d1
(14a)
bd1d2/d1
= bd1d2/d1
+ υd1d2/d1
(14b)
υd2d2/d1
∼N(0, Ωd2
d2/d1
)(15a)
υd1d2/d1
∼N(0, Ωd1
d2/d1
)(15b)
where υd2d2/d1
and υd1d2/d1
are random noise processes that are assumed to be Gaussian with their mean andcovariance given in Eq. (15). It is important to clarify what is implied by this assumption.
Recall from Sec. III that a vector LOS can be expressed in terms of the sensor focal plane coordinates.The sensor measures these coordinates, [α β]T , which are related to the vector LOS in Eq. 7. Under theassumption that the focal plane measurements are normally distributed with known mean and covariance, itis further assumed that under the focal plane transformation, the resulting LOS uncertainty is approximatelyGaussian. Also recall that because a LOS vector is of unit length it must lie on the unit sphere which leads
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to a rank deficient covariance matrix in R3. The covariances in Eq. (15) can be obtained using either of thetwo sensor models discussed in Section III.
After replacing the true LOS vectors in Eq. (13) with their available measurements, we arbitrarily treatbd2
d2/d1as the only random variable in which the uncertainty in the relationship in Eq. (13) is entirely
associated with this vector:bd2
d2/d1= Ad2
d1bd1
d2/d1+ Ad2
d1υd1
d2/d1− υd2
d2/d1(16)
Equation (16) is linear in the noise terms, υ, and as a result bd2d2/d1
has Gaussian distributed uncertainty
that can be described by two parameters: the mean and covariance, µd2d2/d1
and Rd2d2/d1
respectively:
µd2d2/d1
= Ebd2
d2/d1
= Ad2
d1bd1
d2/d1(17)
Rd2d2/d1
= E(
bd2d2/d1
− µd2d2/d1
)(bd2
d2/d1− µd2
d2/d1
)T (18)
Substituting Eqs. (16) and (17) into Eq. (18) and expanding leads to the following expression:
Rd2d2/d1
= E
Ad2d1
υd1d2/d1
(υd1
d2/d1
)T
Ad2d1
T −Ad2d1
υd1d2/d1
(υd2
d2/d1
)T
− υd2d2/d1
(υd1
d2/d1
)T
Ad2d1
T+ υd2
d2/d1υd2
d2/d1
T (19)
Completing the term-by-term expectation in Eq. (19) leads immediately to the covariance expression for thevector LOS bd2
d2/d1:
Rd2d2/d1
= Ad2d1
Ωd1d2/d1
Ad2d1
T+ Ωd2
d2/d1(20)
The covariance of bd2d2/d1
is a function of the true (and not known) attitude matrix as well as the assumedknown noise process characteristics of the space vehicle sensors. The two-term solution in Eq. (20) isindicative of the fact that both the measured LOS as well as the “reference” vector contain uncertainty. Thecovariance associated with bd1
d2/d1needs to be transformed to the D2 coordinate space before being summed
with the covariance of bd2d2/d1
.There are two primary approaches to address the fact that the covariance is a function of the unknown
true relative attitude matrix. First, the true attitude matrix can be approximated by the estimate attitudematrix. This simply requires that the true attitude matrix be replaced by its estimate in all covarianceexpressions. This method is a good approximation that produces second-order error effects which are chosento be ignored. Second, because each pair of LOS vectors are parallel, the focal planes for each of thetwo involved sensors are aligned. Under the logical assumption that both sensors have the same noisecharacteristics, we have Ad2
d1Ωd1
d2/d1Ad2
d1
T= Ωd2
d2/d1. Making this substitution into Eq. (20) leads to the
attitude independent expression for the covariance, namely
Rd2d2/d1
= 2Ωd2d2/d1
(21)
B. Angle Cosine Variance
The angle cosine measurement between D1 and D2 is determined using the inner product between theLOS vector measurements as viewed from the chief spacecraft. Similar to the vector LOS between the twodeputies, a relationship can be expressed in terms of the attitude mapping, Ad2
d1:
bcc/d2
T bcc/d1
=(Ac
d2bd2
c/d2
)T (Ac
d1bd1
c/d1
)= bd2
c/d2
TAd2
d1bd1
c/d1(22)
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Again, the LOS vectors are not perfectly known and the relationship in Eq. (22) is assumed true betweenthe measurements. The remaining LOS vector measurement uncertainty parameters are given by
υcc/d2
∼N(0, Ωc
c/d2
)(23a)
υcc/d1
∼N(0, Ωc
c/d1
)(23b)
υd2c/d2
∼N(0, Ωd2
c/d2
)(23c)
υd1c/d1
∼N(0, Ωd1
c/d1
)(23d)
After replacing the LOS vectors in Eq. (22) with their respective measurements, the resulting expressionrelates true LOS vectors with their associated noise viewed from the chief spacecraft with those viewed formthe deputies with an attitude transformation.
(bc
c/d2+ υc
c/d2
)T (bc
c/d1+ υc
c/d1
)=
(bd2
c/d2+ υd2
c/d2
)T
Ad2d1
(bd1
c/d1+ υd1
c/d1
)(24)
Similar to the vector LOS analysis, Eq. (24) can be expanded and solved for the measured angle. The resultof this is given by
bcc/d2
T bcc/d1
=bd2c/d2
TAd2
d1bd1
c/d1+ bd2
c/d2
TAd2
d1υd1
c/d1+ υd2
c/d2
TAd2
d1bd1
c/d1+ υd2
c/d2
TAd2
d1υd1
c/d1
− bcc/d2
T υcc/d1
− υcc/d2
T bcc/d1
− υcc/d2
T υcc/d1
(25)
The expression in Eq. (25) can be used to determine the mean and covariance of the angle cosine whichdescribes entirely the probabilistic distribution of this relationship. This methodology is again permitted bythe properties assumed of the measurement noise and the linearity of the expression:
µθ = Ebc
c/d2
T bcc/d1
= bd2
c/d2
TAd2
d1bd1
c/d1(26)
Rθd2/d1= E
(bc
c/d2
T bcc/d1
− µθ
)(bc
c/d2
T bcc/d1
− µθ
)T (27)
To complete the expression for the angle variance, Eqs. (25) and (26) are substituted into Eq. (27) and
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then expanded to yield the lofty expression given by
Rθd2/d1= E
bd2
c/d2
TAd2
d1υd1
c/d1υd1
c/d1
TAd2
d1
Tbd2
c/d2+ bd2
c/d2
TAd2
d1υd1
c/d1bd1
c/d1
TAd2
d1
Tυd2
c/d2
+ bd2c/d2
TAd2
d1υd1
c/d1υd1
c/d1
TAd2
d1
Tυd2
c/d2− bd2
c/d2
TAd2
d1υd1
c/d1υc
c/d1
T bcc/d2
− bd2c/d2
TAd2
d1υd1
c/d1bc
c/d1
T υcc/d2
− bd2c/d2
TAd2
d1υd1
c/d1υc
c/d1
T υcc/d2
+ υd2c/d2
TAd2
d1bd1
c/d1υd1
c/d1
TAd2
d1
Tbd2
c/d2+ υd2
c/d2
TAd2
d1bd1
c/d1bd1
c/d1
TAd2
d1
Tυd2
c/d2
+ υd2c/d2
TAd2
d1bd1
c/d1υd1
c/d1
TAd2
d1
Tυd2
c/d2− υd2
c/d2
TAd2
d1bd1
c/d1υc
c/d1
T bcc/d2
− υd2c/d2
TAd2
d1bd1
c/d1bc
c/d1
T υcc/d2
− υd2c/d2
TAd2
d1bd1
c/d1υc
c/d1
T υcc/d2
+ υd2c/d2
TAd2
d1υd1
c/d1υd1
c/d1
TAd2
d1
Tbd2
c/d2+ υd2
c/d2
TAd2
d1υd1
c/d1bd1
c/d1
TAd2
d1
Tυd2
c/d2
+ υd2c/d2
TAd2
d1υd1
c/d1υd1
c/d1
TAd2
d1
Tυd2
c/d2− υd2
c/d2
TAd2
d1υd1
c/d1υc
c/d1
T bcc/d2
− υd2c/d2
TAd2
d1υd1
c/d1bc
c/d1
T υcc/d2
− υd2c/d2
TAd2
d1υd1
c/d1υc
c/d1
T υcc/d2
− bcc/d2
T υcc/d1
υd1c/d1
TAd2
d1
Tbd1
c/d1− bc
c/d2
T υcc/d1
bd1c/d1
TAd2
d1
Tυd2
c/d2
− bcc/d2
T υcc/d1
υd1c/d1
TAd2
d1
Tυd2
c/d2+ bc
c/d2
T υcc/d1
υcc/d1
T bcc/d2
+ bcc/d2
T υcc/d1
bcc/d1
T υcc/d2
+ bcc/d2
T υcc/d1
υcc/d1
T υcc/d2
− υcc/d2
T bcc/d1
υd1c/d1
TAd2
d1
Tbd2
c/d2
− υcc/d2
T bcc/d1
bd1c/d1
TAd2
d1
Tυd2
c/d2− υc
c/d2
T bcc/d1
υd1c/d1
TAd2
d1
Tυd2
c/d2
+ υcc/d2
T bcc/d1
υcc/d1
T bcc/d2
+ υcc/d2
T bcc/d1
bcc/d1
T υcc/d2
+ υcc/d2
T bcc/d1
υcc/d1
T υcc/d2
− υcc/d2
T υcc/d1
υd1c/d1
TAd2
d1
Tbd2
c/d2
− υcc/d2
T υcc/d1
bd1c/d1
TAd2
d1
Tυd2
c/d2− υc
c/d2
T υcc/d1
υd1c/d1
TAd2
d1
Tυd2
c/d2
+ υcc/d2
T υcc/d1
υcc/d1
T bcc/d2
+ υcc/d2
T υcc/d1
bcc/d1
T υcc/d2
+ υcc/d2
T υcc/d1
υcc/d1
T υcc/d2
(28)
Fortunately, some valid simplifications exist that decrease the complexity of the variance. Firstly, as withthe LOS analysis, it has been assumed that all the noise processes defined in Eq. (23) are uncorrelated. Asa result, all the terms with products of unlike noise components are zero. Additionally, third moments ofthe given noise processes are also zero due to parity (in C [−∞,∞], this would parallel the integral of an oddfunction over an even interval). After the cancellation of these terms, the variance is given by
Rθd2/d1= E
bd2
c/d2
TAd2
d1υd1
c/d1υd1
c/d1
TAd2
d1
Tbd2
c/d2+ υd2
c/d2
TAd2
d1bd1
c/d1bd1
c/d1
TAd2
d1
Tυd2
c/d2
+ υcc/d2
T υcc/d1
υcc/d1
T υcc/d2
+ υd2c/d2
TAd2
d1υd1
c/d1υd1
c/d1
TAd2
d1
Tυd2
c/d2
+ bcc/d2
T υcc/d1
υcc/d1
T bcc/d2
+ υcc/d2
T bcc/d1
bcc/d1
T υcc/d2
(29)
Evaluation of the expectation in Eq. (29) requires that most of the terms be examined individually (aspermitted by the linearity of the expectation over summation). The resulting components from the first andfifth terms are immediately acquired by factoring out the deterministic quantities from the expectation:
Ebd2
c/d2
TAd2
d1υd1
c/d1υd1
c/d1
TAd2
d1
Tbd2
c/d2
= bd2
c/d2
TAd2
d1Ωd1
c/d1Ad2
d1
Tbd2
c/d2(30a)
Ebc
c/d2
T υcc/d1
υcc/d1
T bcc/d2
= bc
c/d2
T Ωd2c/d2
bcc/d2
(30b)
It is also helpful to note the property that the inner product is equal to the trace of its outer productcounterpart, given mathematically as aT b = Tr
(baT
). This can be applied to the second and sixth terms
of the variance. By grouping vector quantities, we note that the following equalities are true:
υd2c/d2
TAd2
d1bd1
c/d1bd1
c/d1
TAd2
d1
Tυd2
c/d2=
(υd2
c/d2
TAd2
d1bd1
c/d1bd1
c/d1
T)(
Ad2d1
Tυd2
c/d2
)=
Tr(
Ad2d1
Tυd2
c/d2
)(υd2
c/d2
TAd2
d1bd1
c/d1bd1
c/d1
T) (31)
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υcc/d2
T bcc/d1
bcc/d1
T υcc/d2
= Tr(
bcc/d1
T υcc/d2
)(υc
c/d2
T bcc/d1
)(32)
The expectation operator can be carried inside the trace functional in Eqs. (31) and (32). Terms 3 and 4 ofEq. (29) require an additional step. These terms are first factored into their trace counterparts:
υcc/d2
T υcc/d1
υcc/d1
T υcc/d2
= Tr(
υcc/d1
υcc/d1
T υcc/d2
)(υc
c/d2
T)
(33)
υd2c/d2
TAd2
d1υd1
c/d1υd1
c/d1
TAd2
d1
Tυd2
c/d2= Tr
(υd1
c/d1υd1
c/d1
TAd2
d1
Tυd2
c/d2
) (υd2
c/d2
TAd2
d1
)(34)
Equations (33) and (34) are different than the previous terms dealt with because they involve second momentsof two different random variables.
We recall that given two random variables, x1 and x2, under the assumption that they are zero meanand mutually independent, the expectation of their product squares is given as Ex2
1x22 = Ex2
1Ex22.
Applying this property to the remaining terms and collecting the previous results leads to the angle scalarvariance:
Rθd2/d1=Tr
(Ad2
d1
Tbd2
c/d2bd2
c/d2
TAd2
d1Ωd1
c/d1
)+ Tr
(Ad2
d1bd1
c/d1bd1
c/d1
TAd2
d1
TΩd2
c/d2
)
+ Tr(Ωd1
c/d1Ad2
d1
TΩd2
c/d2Ad2
d1
)+ Tr
(bc
c/d2bc
c/d2
T Ωcc/d1
)
+ Tr(bc
c/d1bc
c/d1
T Ωcc/d2
)+ Tr
(Ωc
c/d1Ωc
c/d2
)(35)
C. Attitude Estimate Covariance
With the uncertainty of all the LOS measurements characterized within Rθ and Rd2d2/d1
, a theoretical boundcan be found for the relative attitude estimate error. As described earlier, a Gaussian distribution requiresonly the mean and (co)variance to describe it. In the current case, the mean is zero and expressions for the(co)variances have been determined. We now seek a characterization of R, the covariance for the attitudeangle errors, δα.
1. Cramer-Rao Inequality
Characterization of the attitude angle error covariance is accomplished using the Cramer-Rao inequality.A theoretical lower bound for the covariance can be found using the Fisher information matrix, F . Theestimate covariance, P , is bounded by the following relationship:
P = E
(x− x) (x− x)T≥F−1 (36)
where the Fisher information matrix is given by
F = −E
∂2
∂x∂xTlnf (y;x)
(37)
Clearly, to bound the estimate covariance, all that is needed is the second derivative of the negative loglikelihood function constructed using vector measurements with their theoretical covariance expressionspreviously calculated.
2. Negative Log Likelihood Function
Though there are no truth quantities available in the system, the uncertainty of all the measurements hasbeen captured in the following measurements:
y =
bd2
d2/d1(bc
c/d2
)T
bcc/d1
(38)
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The remaining measurements can thus be treated as deterministic quantities in the covariance analysis. Thecovariance and variance for the LOS and angle measurement, given by Rd2
d2/d1and Rθ, have been determined
in Eqs. (20) and (35) and are restated as follows
bd2d2/d1
∼ N(bd2
d2/d1, Rd2
d2/d1
)(39a)
(bc
c/d2
)T
bcc/d1
∼ N((
bcc/d2
)T
bcc/d1
, Rθd2/d1
)(39b)
Using both the measurements from Eq. (38), those taken as deterministic quantities, and the known proba-bility density functions described by Eq. (39), a negative-log likelihood function can be constructed:
J(Ad2
d1
)=
12
((bc
c/d2
)T
bcc/d1
−(bd2
c/d2
)T
Ad2d1
bd1c/d1
)T
R−1θd2/d1
×((
bcc/d2
)T
bcc/d1
−(bd2
c/d2
)T
Ad2d1
bd1c/d1
)
+12
(bd2
d2/d1− Ad2
d1bd1
d2/d1
)T
Rd2d2/d1
−1(bd2
d2/d1− Ad2
d1bd1
d2/d1
)(40)
Because the attitude error is not expected to be large, a small error angle assumption is made in Eq. (40). Theattitude estimate can be expressed in terms of the true attitude and the angle errors, δα, here understoodto map D1 to D2:
A = (I + [δα×]) A (41)
This substitution yields a likelihood function that is now a function of the roll, pitch, and yaw angle errors,given by
J(Ad2
d1
)=
12
((bc
c/d2
)T
bcc/d1
−(bd2
c/d2
)T
(I + [δα×]) Ad2d1
bd1c/d1
)T
R−1θd2/d1
×((
bcc/d2
)T
bcc/d1
−(bd2
c/d2
)T
(I + [δα×]) Ad2d1
bd1c/d1
)
+12
(bd2
d1/d1− (I + [δα×]) Ad2
d1bd1
d2/d1
)T
Rd2d2/d1
−1(bd2
d2/d1− (I + [δα×]) Ad2
d1bd1
d2/d1
)(42)
Recalling that [a×]b = −[b×]a, the expression in Eq. (42) can be written as:
J =12
((bc
c/d2
)T
bcc/d1
−(bd2
c/d2
)T
Ad2d1
bd1c/d1
+(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]δα
)T
R−1θd2/d1
×((
bcc/d2
)T
bcc/d1
−(bd2
c/d2
)T
Ad2d1
bd1c/d1
+(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]δα
)
+12
(bd2
d2/d1−Ad2
d1bd1
d2/d1+
[Ad2
d1bd1
d2/d1×
]δα
)T
Rd2d2/d1
−1
×(bd2
d2/d1−Ad2
d1bd1
d2/d1+
[Ad2
d1bd1
d2/d1×
]δα
)
(43)
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Finally, this is entirely expanded for convenience to yield the final expression for the likelihood function:
J =12
(bc
c/d1
)T
bcc/d2
R−1θd2/d1
(bc
c/d2
)T
bcc/d1
−(bc
c/d1
)T
bcc/d2
R−1θd2/d1
(bd2
c/d2
)T
Ad2d1
bd1c/d1
+(bc
c/d1
)T
bcc/d2
R−1θd2/d1
(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]δα
+12
(bd1
c/d1
)T
Ad2d1
Tbd2
c/d2R−1
θd2/d1
(bd2
c/d2
)T
Ad2d1
bd1c/d1
−(bd1
c/d1
)T
Ad2d1
Tbd2
c/d2R−1
θd2/d1
(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]δα
− 12δαT
[Ad2
d1bd1
c/d1×
]bd2
c/d2R−1
θd2/d1
(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]δα +
12
(bd2
d2/d1
)T
Rd2d2/d1
−1bd2
d2/d1
−(bd2
d2/d1
)T
Rd2d2/d1
−1Ad2
d1bd1
d2/d1+
(bd2
d2/d1
)T
Rd2d2/d1
−1[Ad2
d1bd1
d2/d1×
]δα
+12
(bd1
d2/d1
)T
Ad2d1
TRd2
d2/d1
−1Ad2
d1bd1
d2/d1−
(bd1
d2/d1
)T
Ad2d1
TRd2
d2/d1
−1[Ad2
d1bd1
d2/d1×
]δα
− 12δαT
[Ad2
d1bd1
d2/d1×
]Rd2
d2/d1
−1[Ad2
d1bd1
d2/d1×
]δα
(44)
3. Relative Attitude Minimal Covariance
Now we are equipped to determine the Fisher information matrix in Eq. (37) which will allow a boundingof the attitude estimate error covariance. This will be accomplished using the Cramer-Rao inequality. Thesecond derivative of the likelihood function is taken with respect to the error angles, δα. Taking the firstderivative gives
Jδα =(bc
c/d1
)T
bcc/d2
R−1θd2/d1
(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]
−(bd1
c/d1
)T
Ad2d1
Tbd2
c/d2R−1
θd2/d1
(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]
−[Ad2
d1bd1
c/d1×
]bd2
c/d2R−1
θd2/d1
(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]δα +
(bd2
d2/d1
)T
Rd2d2/d1
−1[Ad2
d1bd1
d2/d1×
]
−(bd1
d2/d1
)T
Ad2d2/d1
TRd2
d2/d1
−1[Ad2
d1bd1
d2/d1×
]−
[Ad2
d1bd1
d2/d1×
]Rd2
d2/d1
−1[Ad2
d1bd1
d2/d1×
]δα
(45)
and then taking the second gives
JδαδαT =−[Ad2
d1bd1
c/d1×
]bd2
c/d2R−1
θd2/d1
(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]
−[Ad2
d1bd1
d2/d1×
]Rd2
d2/d1
−1[Ad2
d1bd1
d2/d1×
] (46)
where we now define Rd2d1
as a lower bound for the attitude error covariance. Namely,
EδαδαT
≥ Rd2d1
(47)
with the Cramer-Rao lower bound being:
Rd2d1
−1=−
[Ad2
d1bd1
c/d1×
]bd2
c/d2R−1
θd2/d1
(bd2
c/d2
)T [Ad2
d1bd1
c/d1×
]
−[Ad2
d1bd1
d2/d1×
]Rd2
d2/d1
−1[Ad2
d1bd1
d2/d1×
] (48)
D. Chief to Deputy Mappings
Because the analysis for the relative attitude mappings from C to D1 and from C to D2 follows similarly tothe previous analysis, only the results will be given.
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1. Relative Attitude Mapping C to D1
The measurements for the C to D1 mapping are stated below.
bd1d1/c = Ad1
c bcd1/c (49a)
(bd2
d2/d1
)T
bd2d2/c =
(Ad2
d1bd1
d2/d1
)T (Ad2
c bcd2/c
)=
(bd1
d2/d1
)T
Ad1c bc
d2/c (49b)
The LOS covariance for bd1d1/c can be showed to be
Rd1d1/c = Ad1
c Ωcd1/cA
d1c
T+ Ωd1
d1/c (50)
The variance for the angle cosine,(bd2
d2/d1
)T
bd2d2/c, is similarly given by
Rθd1/c=Tr
(Ad1
c
Tbd1
d2/d1bd1
d2/d1
TAd1
c Ωcd2/c
)+ Tr
(Ad1
c bcd2/cb
cd2/c
T Ad1c
TΩd1
d2/d1
)
+ Tr(Ωc
d2/cAd1c
TΩd1
d2/d1Ad1
c
)+ Tr
(bd2
d2/d1bd2
d2/d1
TΩd2
d2/c
)
+ Tr(bd2
d2/cbd2d2/c
TΩd2
d2/d1
)+ Tr
(Ωd2
d2/cΩd2d2/d1
)(51)
Constructing the appropriate negative-log likelihood function counterpart to Eq. (40), and following throughthe same progression of simplifications will yield an analog to Eq. (44). Applying the Cramer-Rao inequality,the theoretical lower bound for the estimate error covariance for the relative attitude between the chief andthe first deputy is given by Rd1
c :
Rd1c
−1=−
[Ad1
c bcd2/c×
]bd1
d2/d1R−1
θd1/c
(bd1
d2/d1
)T [Ad1
c bcd2/c×
]−
[Ad1
c bcd1/c×
]Rd1
d1/c
−1[Ad1
c bcd1/c×
] (52)
2. Relative Attitude Mapping C to D2
The measurements for the C to D2 mapping are stated below.
bd2d2/c = Ad2
c bcd2/c (53a)
(bd1
d1/d2
)T
bd1d1/c =
(Ad1
d2bd2
d1/d2
)T (Ad1
c bcd1/c
)=
(bd2
d1/d2
)T
Ad2c bc
d1/c (53b)
The LOS covariance for bd2d2/c is given by
Rd2d2/c = Ad2
c Ωcd2/cA
d2c
T+ Ωd2
d2/c (54)
The variance for the angle cosine,(bd1
d1/d2
)T
bd1d1/c, can be shown to be
Rθd2/c=Tr
(Ad2
c
Tbd2
d1/d2bd2
d1/d2
TAd2
c Ωcd1/c
)+ Tr
(Ad2
c bcd1/cb
cd1/c
T Ad2c
TΩd2
d1/d2
)
+ Tr(Ωc
d1/cAd2c
TΩd2
d1/d2Ad2
c
)+ Tr
(bd1
d1/d2bd1
d1/d2
TΩd1
d1/c
)
+ Tr(bd1
d1/cbd1d1/c
TΩd1
d1/d2
)+ Tr
(Ωc
d1/cΩd1d1/d2
)(55)
Constructing the corresponding negative-log likelihood function, counterpart to Eq. (40), and followingthrough the same progression of simplifications yet a third time will yield an analog to Eq. (44). Applyingthe Cramer-Rao inequality, the theoretical lower bound for the estimate error covariance for the relativeattitude between the chief and the first deputy is given by Rd2
c :
Rd2c
−1=−
[Ad2
c bcd1/c×
]bd2
d1/d2R−1
θd2/c
(bd2
d1/d2
)T [Ad2
c bcd1/c×
]−
[Ad2
c bcd2/c×
]Rd2
d2/c
−1[Ad2
c bcd2/c×
] (56)
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V. Simulations
This section considers two different space vehicle configurations and displays simulation results for each.For both configurations, both the QMM and the wide FOV model are used to determine vector LOS stochasticparameters. These results are paired accordingly for comparison. Each configuration is considered for 1000trials. Measurement noise is simulated using Eqs. (7) in Sec. III. Relative attitude estimates are calculatedby a method outlined by Shuster employing the vector LOS and angle cosine measurements analyzed inSec. IV.4 The estimate error covariance for each mapping is determined using either of Eqs. (48), (52), or(56), respectively.
A. First Configuration
The first relative attitude configuration uses the following true LOS vectors:
bcc/d1
=1√5
1√3
−1
, bc
c/d2=
√72
√3
1212
, bd1
d1/d2=
1√6
−1−1−2
(57)
The remaining three LOS truth vectors are determined from those listed in Eq. (57) using the appropriateattitude transformation. For this configuration, the true relative attitudes are given by
qd1c =
1√2
[0 1 0 1
]T
(58a)
qd2c =
1√2
[1 0 0 1
]T
(58b)
100 200 300 400 500 600 700 800 900 1000
−0.02
0
0.02
Rol
l Err
or (
deg)
100 200 300 400 500 600 700 800 900 1000−0.02
0
0.02
Pitc
h E
rror
(de
g)
100 200 300 400 500 600 700 800 900 1000−0.02
0
0.02
Yaw
Err
or (
deg)
Trial Number
(a) Using the QMM.
100 200 300 400 500 600 700 800 900 1000−0.01
0
0.01
Rol
l Err
or (
deg)
100 200 300 400 500 600 700 800 900 1000−5
0
5x 10
−3
Pitc
h E
rror
(de
g)
100 200 300 400 500 600 700 800 900 1000−0.01
0
0.01
Yaw
Err
or (
deg)
Trial Number
(b) With the wide FOV model.
Figure 2. Relative attitude estimate errors and theoretical 3-σ bounds for the D1 → D2 map.
Relative attitude angle errors are displayed with their theoretical 3-σ bounds determined from the resultsof Sec. IV. Those for the D1 to D2 mapping are shown in Fig. 2 with the QMM employed on the left andresults using the wide FOV model on the right. Figure 3 contains the results for the C to D1 mapping andFig. 4 contains those results for the final relative attitude mapping C to D2.
The results for the first configuration using the wide FOV measurement error covariance model are quitesupportive. Theoretically, for a given set of 1000 samples, the attitude error should be outside the 3-σbounds for 3 of those samples. Those results employing this model agree well with this. For the QMM whichassumes that the vector LOS observation angles are small (by approximating the focal plane by a tangentplane), the attitude error covariance poorly approximates the location of the 3-σ bounds. This is a sensibleresult of the large focal plane values that result from the assumed LOS vectors in Eq. (57).
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100 200 300 400 500 600 700 800 900 1000−0.02
0
0.02
Rol
l Err
or (
deg)
100 200 300 400 500 600 700 800 900 1000−0.02
0
0.02
Pitc
h E
rror
(de
g)
100 200 300 400 500 600 700 800 900 1000−0.02
0
0.02
Yaw
Err
or (
deg)
Trial Number
(a) Using the QMM.
100 200 300 400 500 600 700 800 900 1000
−5
0
5x 10
−3
Rol
l Err
or (
deg)
100 200 300 400 500 600 700 800 900 1000−0.01
0
0.01
Pitc
h E
rror
(de
g)
100 200 300 400 500 600 700 800 900 1000
−5
0
5x 10
−3
Yaw
Err
or (
deg)
Trial Number
(b) With the wide FOV model.
Figure 3. Relative attitude estimate errors and theoretical 3-σ bounds for the C → D1 map.
100 200 300 400 500 600 700 800 900 1000
−0.02
0
0.02
Rol
l Err
or (
deg)
100 200 300 400 500 600 700 800 900 1000−0.01
0
0.01
Pitc
h E
rror
(de
g)
100 200 300 400 500 600 700 800 900 1000−0.01
0
0.01
Yaw
Err
or (
deg)
Trial Number
(a) Using the QMM.
100 200 300 400 500 600 700 800 900 1000−0.01
0
0.01
Rol
l Err
or (
deg)
100 200 300 400 500 600 700 800 900 1000
−2
0
2
x 10−3
Pitc
h E
rror
(de
g)
100 200 300 400 500 600 700 800 900 1000
−2
0
2
x 10−3
Yaw
Err
or (
deg)
Trial Number
(b) With the wide FOV model.
Figure 4. Relative attitude estimate errors and theoretical 3-σ bounds for the C → D2 map.
B. Second Configuration
To corroborate the discussion of the results using the first assumed configuration, a second configuration isconsidered in which the 3-σ bounds for the attitude error are expected to better be approximated by theQMM. Thus, it would be expected to see better correlation between the simulation results generated usingthe QMM and the wide FOV model. The LOS vectors for this configuration are given by
bcc/d1
=1√1.02
0.10.11
, bc
c/d2=
1√1.02
0.1−0.1
1
, bd1
d1/d2=
1√1.02
−0.10.11
(59)
Again, the proper attitude transformation can be used to generate the remaining of the six true LOS vectors.The true relative attitudes are defined by
qd1c =
1√2
[1 0 0 1
]T
(60a)
qd2c =
[sin 5π/180√
20 sin 5π/180√
2cos 5π/180
]T
(60b)
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To further emphasize the expectations of this simulation, the true focal plane values are provided in thefollowing table:
Table 1. True Focal Plane Values
bcc/d1
bcc/d2
bd1c/d1
bd1d2/d1
bd2c/d2
bd2d2/d1
α 0.1 0.1 −1 −0.1 0.094 −1β 0.1 −0.1 −10 0.1 0.012 −4.34
As is clear from Table 1, the majority of the LOS vectors do not generate focal plane measurements thatviolate a small angle assumption quite as grossly as in the first configuration. Better results are expected,though not all the theoretical 3-σ bounds will improve on the account of primarily the bd1
c/d1vector.
0 100 200 300 400 500 600 700 800 900 1000−0.1
0
0.1
Rol
l Err
or (
deg)
0 100 200 300 400 500 600 700 800 900 1000−0.5
0
0.5
Pitc
h E
rror
(de
g)
0 100 200 300 400 500 600 700 800 900 1000−0.1
0
0.1
Yaw
Err
or (
deg)
Trial Number
(a) Using the QMM.
0 100 200 300 400 500 600 700 800 900 1000−0.02
0
0.02
Rol
l Err
or (
deg)
0 100 200 300 400 500 600 700 800 900 1000−0.1
0
0.1
Pitc
h E
rror
(de
g)
0 100 200 300 400 500 600 700 800 900 1000−0.02
0
0.02Y
aw E
rror
(de
g)
Trial Number
(b) With the wide FOV model.
Figure 5. Relative attitude estimate errors and theoretical 3-σ bounds for the D1 → D2 map.
0 100 200 300 400 500 600 700 800 900 1000−0.05
0
0.05
Rol
l Err
or (
deg)
0 100 200 300 400 500 600 700 800 900 1000−0.5
0
0.5
Pitc
h E
rror
(de
g)
0 100 200 300 400 500 600 700 800 900 1000−0.05
0
0.05
Yaw
Err
or (
deg)
Trial Number
(a) Using the QMM.
0 100 200 300 400 500 600 700 800 900 1000−0.05
0
0.05
Rol
l Err
or (
deg)
0 100 200 300 400 500 600 700 800 900 1000−0.5
0
0.5
Pitc
h E
rror
(de
g)
0 100 200 300 400 500 600 700 800 900 1000−0.05
0
0.05
Yaw
Err
or (
deg)
Trial Number
(b) With the wide FOV model.
Figure 6. Relative attitude estimate errors and theoretical 3-σ bounds for the C → D1 map.
Relative attitude angle errors are displayed with their theoretical 3-σ bounds determined from the resultsof Sec. IV. Those for the D1 to D2 mapping are shown in Fig. 5 with the QMM employed on the left andresults using the wide FOV model on the right. Figure 6 contains the results for the C to D1 mapping andFig. 7 contains those results for the final relative attitude mapping C to D2.
The results for the two relative attitude mappings including the chief frame, C, in this configuration appearto have much better agreement between the use of the QMM and the wide FOV model. For smaller angles,
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the QMM becomes an increasingly better approximation to the LOS vector covariance better approximatedby the first-order Taylor expansion in the wide FOV model. More specific to this configuration, it can beshown that the effect of Ωd1
c/d1, the covariance for the LOS vector that most greatly violates the small FOV
assumption in the QMM, is greater when determining the relative attitude error covariance for the D1 to D2
mapping.
0 100 200 300 400 500 600 700 800 900 1000−0.05
0
0.05
Rol
l Err
or (
deg)
0 100 200 300 400 500 600 700 800 900 1000−0.01
0
0.01
Pitc
h E
rror
(de
g)
0 100 200 300 400 500 600 700 800 900 1000−0.5
0
0.5
Yaw
Err
or (
deg)
Trial Number
(a) Using the QMM.
0 100 200 300 400 500 600 700 800 900 1000−0.05
0
0.05
Rol
l Err
or (
deg)
0 100 200 300 400 500 600 700 800 900 1000−0.01
0
0.01
Pitc
h E
rror
(de
g)
0 100 200 300 400 500 600 700 800 900 1000−0.5
0
0.5
Yaw
Err
or (
deg)
Trial Number
(b) With the wide FOV model.
Figure 7. Relative attitude estimate errors and theoretical 3-σ bounds for the C → D2 map.
VI. Conclusions
In this paper it is shown that with a three-spacecraft configuration, deterministic relative attitude solu-tions are possible assuming parallel beam information between each spacecraft pair. Covariance expressionswere derived to determine the relative attitude-error 3-σ bounds, which closely matched with Monte Carlosimulations. Comparisons between the small-FOV and large-FOV measurement models indicated that thesmall-FOV measurement model is conservative in its associated 3-σ attitude bound calculations. The finalpaper will expand the three spacecraft case to the N spacecraft case.
References
1Markley, F. L. and Mortari, D., “Quaternion Attitude Estimation Using Vector Observations,” Journal of the AstronauticalSciences, Vol. 48, No. 2/3, 2000, pp. 359–380.
2Schaub, H. and Junkins, J. L., Analytical Mechanics of Space Systems, American Institute of Aeronautics and Astronautics,Inc., 2003.
3Kim, S.-G., Crassidis, J. L., Cheng, Y., Fosbury, A. M., and Junkins, J. L., “Kalman Filtering for Relative SpacecraftAttitude and Position Estimation,” The Journal of Guidance, Control, and Dynamics, Vol. 30, No. 1, 2007, pp. 133–143.
4Shuster, M. D., “Deterministic Three-Axis Attitude Determination,” The Journal of the Astronautical Sciences, Vol. 52,No. 3, 2004, pp. 405–419.
5Shuster, M. D., “Kalman Filtering of Spacecraft Attitude and the QUEST Model,” The Journal of the AstronauticalSciences, Vol. 38, No. 3, 1990, pp. 377–393.
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