small-time local controllability and stabilizability of spacecraft attitude dynamics under cmg...

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SIAM J. CONTROL OPTIM. c© 2014 Society for Industrial and Applied MathematicsVol. 52, No. 2, pp. 797–820

SMALL-TIME LOCAL CONTROLLABILITY ANDSTABILIZABILITY OF SPACECRAFT ATTITUDE DYNAMICS

UNDER CMG ACTUATION∗

SANJAY P. BHAT† AND ADITYA A. PARANJAPE‡

Abstract. The purpose of this paper is to examine whether the presence of singular configu-rations poses an obstruction to the small-time local controllability (STLC) and stabilizability of theattitude dynamics of a spacecraft actuated by single-gimbal control moment gyroscopes (CMGs).We identify a class of singular CMG configurations called critically singular configurations, at whichthe linearized dynamics fail to be controllable. Linear controllability, and hence STLC and stabiliz-ability, hold at equilibria in which the CMG array is in a nonsingular or singular, but not criticallysingular, configuration. We give sufficient conditions and necessary conditions on critically singularconfigurations for STLC to hold. In particular, we show that STLC and stabilizability fail to hold atan equilibrium if the magnitude of the total spin angular momentum of the CMG array has a localmaximum at the equilibrium.

Key words. attitude control, local controllability, stabilizability, control moment gyroscopes

AMS subject classifications. 93B05, 93D15, 93B29

DOI. 10.1137/130918903

1. Introduction. Control moment gyroscopes (CMGs) form an important classof torque actuators for spacecraft attitude control. Because of their large torqueproducing capability and smooth operation, CMGs are preferred over other torqueactuators in applications involving precision pointing or momentum management oflarge, long duration spacecraft.

A single-gimbal CMG comprises a gimbal carrying an axisymmetric rotor spin-ning at a constant rate. The orientation of the rotor axis can be changed by applyingtorque to the gimbal, and the resulting reaction torque serves to control the spacecraftattitude. The magnitude of torque produced by the CMG is a function of the con-stant rotor speed and gimbal rotation rate, while, under additional assumptions, thedirection is orthogonal to the rotor and gimbal axes. Clearly, a single CMG cannotproduce torques in all directions. In order to obtain torques along three independentdirections, as well as for actuation redundancy, CMGs are used in arrays consisting ofmultiple CMGs. An array of q single-gimbal CMGs gives rise to a function ν : Tq → R

3

giving the components of the total spin angular momentum of the CMG array, whichwe simply refer to as the CMG angular momentum, as a function of the vector of qgimbal angles, which may be viewed as an element of the q-dimensional torus Tq.

Under certain assumptions that we state later, the actuation torque produced bythe CMG array in a CMG configuration θ ∈ T

q is given by [7]

(1.1) −ν′(θ)θ,∗Received by the editors April 29, 2013; accepted for publication (in revised form) November 20,

2013; published electronically March 4, 2014. This paper greatly extends preliminary results from apaper of a similar title that was presented at the Symposium on Mathematical Theory of Networksand Systems (MTNS08) held in Blacksburg, VA, from June 28 to August 1, 2008.

http://www.siam.org/journals/sicon/52-2/91890.html†TCS Innovation Labs Hyderabad, Tata Consultancy Services, Madhapur, Hyderabad 500081,

India ([email protected]).‡Department of Mechanical Engineering, McGill University, Montreal, QC H3A 0C3, Canada

([email protected]).

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http://www.siam.org/journals/sicon/52-2/91890.html

mailto:[email protected]

mailto:[email protected]


798 SANJAY P. BHAT AND ADITYA A. PARANJAPE

where ν′(θ) represents the differential of ν at θ, and the vector θ of gimbal ratesrepresents a tangent vector to T

q at θ. Equation (1.1) represents a kinematic mapfrom the gimbal rates at the configuration θ ∈ T

q to the actuation torque and isanalogous to the kinematic map relating the end-effector velocity of a revolute-jointedrobotic manipulator to its joint velocities [4]. A CMG array may thus be viewed as aninput-output device whose inputs are gimbal rates and outputs are actuation torquesgiven by (1.1). Accordingly, a common approach to attitude control using CMGs isto first compute the torque profiles needed to elicit the required spacecraft behavior,and then invert the kinematic map (1.1) to obtain the gimbal rates that produce thosetorque profiles.

Unfortunately, every CMG array possesses singular configurations at which thedifferential of ν loses rank and fails to be surjective. At such a singular configuration,there exists a singular direction such that no combination of gimbal rates yields anactuation torque along the singular direction, and the spacecraft dynamics becomemomentarily underactuated. In fact, given any direction, there exist singular config-urations for which the given direction is a singular direction. A brief and accessibleillustration of CMGs and singular configurations may be found in [7], while detailedexpositions are given in [17, 18, 29, 32]. Since the common approach described inthe paragraph above involves inverting the kinematic map (1.1), the approach en-counters difficulties at singular configurations. Consequently, a considerable amountof research related to CMGs has focused on the geometry of singular configurations[4, 17, 18, 32] as well as on steering algorithms to maneuver CMG arrays to producedesired torque profiles while avoiding singular configurations [5, 14, 16, 21, 22, 30, 31].The primary issues in this entire body of work, which parallels the work on the inversekinematics problem for redundant robotic manipulators, relate to the set of torqueprofiles that the CMG array is capable of producing.

Although not theoretically guaranteed to always work, steering algorithms com-bined with the approach described above have been highly successful in practice. Yet,certain fundamental questions remain unanswered. The main problem posed by thepresence of singular configurations is that it makes it more difficult to devise gim-bal motions that result in any given actuation torque profile. While certain controlobjectives such as attitude trajectory tracking clearly require the ability to generatearbitrary torque profiles, other control objectives such as asymptotic stabilization orstate-to-state steering do not. Hence, it is not evident that singular configurationsaffect the ability to perform control tasks such as asymptotic stabilization or state-to-state steering. Indeed, one may view the problem of singular configurations asessentially that of local underactuation. In the wider systems and control literature,underactuation is the norm rather than the exception, as the degrees of actuationin a system of interest are often fewer than the mechanical degrees of freedom, andalmost always fewer than the number of state variables. Yet, underactuation by itselfis never seen as an obstruction to stabilizability or controllability. These observationslead us to ask exactly which system-theoretic properties are adversely affected by thepresence of singular CMG configurations.

The question stated above cannot be answered by only considering the CMGarray in isolation, and instead requires the application of system-theoretic tools tothe combined dynamics of the spacecraft and the CMG array. The central issues insuch an approach would concern the spacecraft behaviors, rather than torque profiles,that the CMG array is capable of producing. The number of research papers that takethis alternative approach is very small. In fact, even the properties of the Jacobian

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STLC AND STABILIZABILITY OF ATTITUDE DYNAMICS 799

linearization of the combined dynamics at an equilibrium do not appear to have beenreported. Reference [14] applied feedback linearization to the combined dynamicsof the spacecraft and the CMG array, and gave a steering law to implement thecorresponding input transformation. More recently, [7] used nonlinear controllabilityresults to show that the combined spacecraft-CMG system can be steered betweenany two states possessing the same total angular momentum despite the presence ofsingular CMG configurations. In other words, the presence of singular configurationsdoes not act as an obstruction to global controllability. In this paper, we use thesetting provided by [7] to investigate whether the presence of singular configurationsposes an obstruction to the small-time local controllability (STLC) and stabilizabilityof the spacecraft attitude dynamics at rest equilibria, that is, states in which thespacecraft and CMG gimbals do not rotate.

While the importance of stabilizability is widely acknowledged, the property ofSTLC is of practical importance too. This is because a lack of STLC at an equilibriumx implies that, given sufficiently small bounds on the steering time and gimbal rates,there exists a terminal state y arbitrarily close to x such that a trajectory startingfrom x cannot be steered to the terminal state y without violating at least one of thebounds. Thus, a lack of STLC could lead to a situation where an arbitrarily smallterminal reorientation requires either a relatively large amount of time, or relativelylarge gimbal rates, or both. On the other hand, STLC is the reasonable requirementthat terminal states that are sufficiently close to xmay be reached from x in arbitrarilysmall times using bounded gimbal rates.

In section 2, we identify a special class of singular configurations called criticallysingular configurations at which the magnitude of the CMG angular momentum hasa critical value. Equivalently, critically singular configurations are those singularconfigurations in which the CMG angular momentum and the singular direction arelinearly dependent. We give examples to emphasize that our notion of a criticallysingular configuration is different from the notion of an external singularity that isprevalent in the literature. In section 3, we show that the linearized dynamics atan equilibrium are controllable if and only if the corresponding CMG configurationis not critically singular. It immediately follows that the attitude dynamics of thespacecraft are STLC as well as smoothly stabilizable at those equilibria in whichthe CMG array is either in a nonsingular configuration or in a singular, but notcritically singular, configuration. Moreover, all equilibria on an angular momentumlevel set satisfy the condition mentioned above for almost every value of the systemangular momentum magnitude. We also consider reduced attitude dynamics obtainedby applying symplectic reduction [1, Chap. 4], [2, Appendix 5] to the full attitudedynamics, and show that these are STLC as well at equilibria of the full dynamics.

In section 4, we give sufficient conditions on critically singular CMG configu-rations for STLC to hold at equilibria involving those configurations. We use thesufficient condition for STLC given in [9] to show that STLC holds at every rest equi-librium if the total angular momentum of the spacecraft and the CMG array is zero.When the total angular momentum of the spacecraft and the CMG array is nonzero,STLC holds at those rest equilibria in which the CMG array is either in a criticallysingular configuration that is not an external singularity, or in a critically singularconfiguration such that the individual CMG spin angular momenta that are orthog-onal to the CMG angular momentum span a two-dimensional space. Interestingly,stronger sufficient conditions for STLC such as [8, Thm. 1.1], or its more widely usedinterpretation given in [23], do not apply in this case. The problem of STLC under

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CMG actuation thus represents a problem of practical significance that also possessessufficient complexity to necessitate the fullest use of available results for its resolution.To the best of our knowledge, our work represents the first instance where the resultsof [9] are used in a problem motivated by a practical application.

In section 5, we use the necessary condition given in [27] to show that STLC failsto hold at an equilibrium if the corresponding CMG configuration is critically singularand an external singularity. We exploit the STLC property of the reduced dynamicsto explicitly construct states that are not small-time reachable for the full dynamicsfrom an equilibrium of the kind mentioned above. As a corollary, we show that STLCfails to hold at an equilibrium if the magnitude of the CMG angular momentum hasa local maximum at that equilibrium. It immediately follows that STLC fails to holdat all equilibria if the spacecraft carries only one CMG.

In section 6, we show that equilibria in which the magnitude of the CMG an-gular momentum is either a local maximum or a nonzero local minimum cannot bestabilized using continuous time-invariant feedback. It immediately follows that noequilibrium is stabilizable using continuous time-invariant feedback if the spacecraftcarries only one CMG.

In summary, CMG configurations that are not critically singular pose no obstruc-tion to STLC and asymptotic stabilizability, regardless of whether they are externalor internal singularities. Even among critically singular configurations, a large class ofinternal singularities poses no obstruction to STLC, while external singularities ruleout STLC, and local extremizers of the CMG angular momentum magnitude rule outstabilizability.

2. Preliminaries. In this section, we introduce the necessary terminology, no-tation, and description of the dynamics.

2.1. Notation. Let SO(3) denote the Lie group of 3 × 3 special orthogonalmatrices. The Lie algebra so(3) of SO(3) is the set of 3 × 3 real skew-symmetricmatrices with the matrix commutator as the bracket operation. Define (·)× : R3 →so(3) by

a× =

⎡⎣ 0 −a3 a2

a3 0 −a1−a2 a1 0

⎤⎦, a ∈ R

3.

For every a ∈ R3, a× is simply the matrix representation in the standard basis of the

linear map b �→ (a × b) on R3, where × denotes the usual cross product on R

3. Themap (·)× is linear, bijective, and satisfies (a× b)× = a×b×− b×a× for every a, b ∈ R

3.The tangent space to SO(3) at R ∈ SO(3) is given by TRSO(3) = {RG : G ∈ so(3)} ={Rg× : g ∈ R

3}.We denote the unit circle by S1, the Euclidean norm on R

3 by ‖ · ‖, and thetwo-dimensional unit sphere {x ∈ R

3 : ‖x‖ = 1} by S2. Given an integer q > 0, we letIq denote the index set {1, . . . , q}, and T

q = S1 × · · · × S1 denote the q-dimensionaltorus, the Cartesian product of q copies of S1.

Given two C∞ vector fields f and g on a C∞ manifold N , we denote their Liebracket by [f, g]. If N is an embedded submanifold of a C∞ manifold M, and f andg are C∞ extensions to M of the vector fields f and g, respectively, then [f, g] is

the restriction of [f , g] to N . In particular, if M = Rn for some n, then, for every

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x ∈ N ⊆ M, the canonical identification between TxM and Rn yields

(2.1) [f, g](x) =d

dh

∣∣∣∣h=0

[g(x+ hf(x)) − f(x+ hg(x))].

As in [6, 7], we will find it convenient to apply the formula (2.1) to vector fields definedon SO(3)×T

q and S2×Tq for some q, which can be viewed as embedded submanifolds

of Rn for n = 9 + q and n = 3 + q, respectively.

2.2. Singular CMG configurations. Consider an array of q > 0 single-gimbalCMGs, each having a spin angular momentum of a constant, nonzero magnitude. Wedo not require all CMGs to be identical. In particular, we do not assume that the spinangular momenta of all the CMGs have the same magnitude. The body componentsof the spin angular momentum of the ith CMG give rise to a function νi : S

1 → R3.

We will denote the gimbal angle of the ith CMG by θi ∈ [0, 2π). Since each value ofthe gimbal angle determines a unique orientation of the gimbal, it will be convenientto view θi as an element of S1 itself.

For each i ∈ Iq, we denote by ν′i : S1 → R

3 and ν′′i : S1 → R3 the first and second

derivatives, respectively, of νi with respect to θi. Since νi(θi) is obtained by rotatingνi(0) about a fixed gimbal axis orthogonal to νi(0) through the angle θi, it is easy toshow that ν′i(θi) is orthogonal to νi(θi) while ν′′i (θi) = −νi(θi) for every i ∈ Iq andevery θi ∈ S1. Also, given i ∈ Iq, the vector νi(·)× ν′i(·) is a constant vector parallelto the gimbal axis of the ith CMG.

The body components of the total spin angular momentum of the CMG arraydefine a function ν : Tq → R

3 of the vector of gimbal angles θ = (θ1, . . . , θq) ∈ Tq

given by ν(θ) = ν1(θ1) + · · ·+ νq(θq). We refer to ν as the CMG angular momentum.The range ν(Tq) of ν is called the momentum volume of the CMG array, while thetopological boundary of ν(Tq) is referred to as the momentum envelope of the CMGarray. Each θ ∈ T

q represents a configuration of the CMG array. We define singularCMG configurations next.

Definition 2.1. A configuration θ ∈ Tq is a singular configuration of the CMG

array if the span of the vectors {ν′1(θ1), . . . , ν′q(θq)} has dimension less than three. Wedenote the set of singular configurations by S ⊂ T

q.It is clear that nonsingular configurations exist only if q ≥ 3.Given v ∈ S2, we let Sv = {θ ∈ S : vTν′i(θi) = 0, i ∈ Iq}. It is clear that, for every

θ ∈ S, there exists v ∈ S2 such that vTν′i(θi) = 0 for all i ∈ Iq. Thus S = ∪ν∈S2Sv.If θ ∈ S is such that θ ∈ Sv, then v is a singular direction corresponding to thesingular configuration θ. Except in the case where v ∈ S2 coincides with at least oneof the gimbal axes of the CMGs, Sv is finite, and this makes it possible to use S2 tolocally parametrize S near a singular configuration. Since we do not make use of thisparametrization in this paper, we refer the interested reader to [18] for details.

The magnitude of the CMG angular momentum gives rise to a smooth functionη : Tq → R defined by

(2.2) η(θ) = ‖ν(θ)‖2.

Next, we define a special class of singular configurations called critically singularconfigurations.

Definition 2.2. A configuration θ ∈ Tq of the CMG array is a critically singular

configuration if θ ∈ S and θ is a critical point of the smooth function η. We denotethe set of critically singular configurations by C ⊂ T

q.

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It is easy to verify that θ ∈ Tq represents a critically singular configuration if and

only if νT(θ)ν′i(θi) = 0 for all i ∈ Iq, and there exists v ∈ S2 satisfying vTν′i(θi) = 0for all i ∈ Iq. Hence the set C of critically singular configurations is given by C ={θ ∈ S : νT(θ)ν′i(θi) = 0, i ∈ Iq}. It is also easy to verify that, if θ ∈ T

q satisfies‖ν(θ)‖ = 0, then θ ∈ C if and only if θ ∈ Sv for v = ‖ν(θ)‖−1ν(θ) ∈ S2. As a result,critical points of the function η corresponding to nonzero critical values are criticallysingular configurations. If θ ∈ T

q satisfies ‖ν(θ)‖ = 0, then θ is a critical point ofthe map η, but θ ∈ C if and only if θ ∈ S holds additionally. Finally, it is evidentthat, in the case q = 1, every CMG configuration is critically singular. Hence singularconfigurations that are not critically singular exist only if q ≥ 2.

The results that we will present later involve only the notions introduced above.However, in order to examine how STLC relates to notions already used in the liter-ature, we will briefly introduce two more ways of qualifying singular configurations.

Consider v ∈ S2. The component of the CMG angular momentum along v givesrise to a function βv : Tq → R given by βv(θ) = vTν(θ). It is easy to see that θ ∈ Sv

if and only if θ is a critical point of the function βv. Furthermore, if θ ∈ Sv is suchthat

(2.3) min{vTνi(θi) : i ∈ Iq} > 0,

then the standard second-order sufficient condition for optimality implies that θ is astrict local maximizer for the function βv. In fact, it turns out that a θ ∈ Sv satisfying(2.3) is a strict global maximizer of βv. As a result, ν(θ) lies on the boundary ofthe momentum volume ν(Tq) if θ ∈ Sv satisfies (2.3). This property motivates thefollowing definition.

Definition 2.3. Given v ∈ S2, a singular configuration θ ∈ Sv is an externalsingularity if (2.3) holds. By contrast, θ ∈ Sv is an internal singularity if (2.3) failsto hold.

A singular configuration θ ∈ S is said to be inescapable (called impassable in [29])if for no ε > 0 there exists a differentiable curve γ : [0, ε] → T

q satisfying γ(0) = θ,ν(γ(t)) = ν(θ) for all t ∈ [0, ε], and γ(ε) /∈ S. Thus, a singular configuration θ isescapable (or passable) if and only if it is possible to reach a nonsingular configurationby starting from θ and using null gimbal motions, that is, gimbal motions that donot disturb the CMG angular momentum. The distinction between escapable andinescapable singularities is important when dealing with CMG steering algorithms,since null gimbal motions can be incorporated into singularity avoidance steeringlaws. However, as our results will show, the distinction is not important for STLC.Before moving on, we mention that all external singularities are inescapable [4, 18,32], although the converse is not true. A weaker sufficient condition for a singularconfiguration to be inescapable may be found in [4, 32].

The following example illustrates the definitions introduced above, and also showsthat the notion of a critically singular configuration introduced by us and the notionof an external singularity given in the literature are independent.

Example 2.4. A common CMG array used in practice is the so-called pyramidarray consisting of four single-gimbal CMGs arranged such that the gimbal axis ofeach CMG is orthogonal to one inclined face of a pyramid on a square base (see[4, 5, 7, 16, 32] for a figure). If we assume that the inclined faces of the pyramidmake an angle α ∈ (0, π2 ) with its base, each gimbal angle is defined to be zero whenthe spin angular momentum of the corresponding CMG points to the apex of thepyramid, each gimbal rotation is taken to be positive in the right-hand sense along

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the outward normal to the corresponding pyramid face, and each CMG possesses unitspin angular momentum, then, for a suitable choice of coordinate axes, the individualspin angular momentum vectors of the CMGs and their derivatives are given by

ν1(θ1) =[ − cosα cos θ1, − sin θ1, sinα cos θ1

]T,

ν2(θ2) =[ − sin θ2, cosα cos θ2, sinα cos θ2

]T,

ν3(θ3) =[cosα cos θ3, sin θ3, sinα cos θ3

]T,

ν4(θ4) =[sin θ4, − cosα cos θ4, sinα cos θ4

]T,(2.4)

ν′1(θ1) =[cosα sin θ1, − cos θ1, − sinα sin θ1

]T,

ν′2(θ2) =[ − cos θ2, − cosα sin θ2, − sinα sin θ2

]T,

ν′3(θ3) =[ − cosα sin θ3, cos θ3, − sinα sin θ3

]T,

ν′4(θ4) =[cos θ4, cosα sin θ4, − sinα sin θ4

]T.(2.5)

Consider the configurations(2.6)

θAdef=

⎡⎢⎢⎣π/2π/2π/2π/2

⎤⎥⎥⎦, θB def

=

⎡⎢⎢⎣

0−π/40π/4

⎤⎥⎥⎦, θC def

=

⎡⎢⎢⎣

0π0π

⎤⎥⎥⎦, θD def

=

⎡⎢⎢⎣

−π/2ππ/20

⎤⎥⎥⎦, θE def

=

⎡⎢⎢⎣

0000

⎤⎥⎥⎦.

The following assertions may be easily verified by using the expressions (2.4) and(2.5).

1. dim span{ν′1(θA1 ), . . . , ν′4(θA4 )} = 3. Thus, θA /∈ S. Also, ν(θA) = 0, so thatθA is a critical point of the function η. However, θA /∈ C since θA /∈ S.

2. vTν′i(θBi ) = 0 for v = [cosα sinα, 0, cosα]T and i ∈ I4. Thus θB ∈ S. More-

over, vTνi(θBi ) > 0 for all i ∈ I4, so that θB is an external and inescapable

singularity. Next, assume sin2 α = √2− 1. Then ν(θB)Tν′i(θ

Bi ) is nonzero for

i ∈ {2, 4}. Hence θB /∈ C. This shows that not all external or inescapablesingularities are critically singular configurations.

3. vTν′i(θCi ) = 0 for v = [0, 0, 1]T and i ∈ I4. Thus θC ∈ S. In addition,

ν(θC) = 0, so that θC ∈ C.4. ν(θD)Tν′i(θ

Di ) = 0 for all i ∈ I4, ν(θ

D)Tνi(θDi ) > 0 for all i ∈ {1, 3}, and

ν(θD)Tνi(θDi ) < 0 for all i ∈ {2, 4}, so that θD is an internal singularity

contained in C. This also shows that not all critically singular configurationsare external singularities.

5. ν(θE)Tν′i(θEi ) = 0 for all i ∈ I4, so that θE ∈ C. In addition, ν(θE)Tνi(θ

Ei ) > 0

for all i ∈ I4, so that θE is an external singularity contained in C. Moreover,if sin2 α > 1

2 , then the Hessian matrix of the function η at θE is negative-definite, and hence θE is a local maximizer of η.

2.3. Attitude dynamics. Consider a spacecraft carrying an array of q > 0single-gimbal CMGs in a torque-free environment. We describe the attitude of thespacecraft using a matrix R ∈ SO(3) such that the multiplication of the body compo-nents of a vector by R gives the components of that vector with respect to a referenceinertial frame. Since no external torques act on the spacecraft, the total inertialangular momentum of the spacecraft-CMG system, which we refer to as the system

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angular momentum, is a constant. We make two standard assumptions widely used inthe literature on attitude control using CMGs, namely that the inertias of the CMGassemblies are negligible compared to those of the spacecraft, and the contributionof gimbal motions to the system angular momentum is negligible. Under these as-sumptions, the actuation torque is given by (1.1), and the moment-of-inertia matrixof the spacecraft-CMG system may be assumed to be constant. The dynamics of thespacecraft on a level set of the system angular momentum may now be written as

(2.7) R(t) = R(t)(J−1{RT(t)μ− ν(θ(t))})×,

where J ∈ R3×3 is the symmetric, positive-definite, moment-of-inertia matrix of the

spacecraft-CMG system about the body-fixed frame, and μ ∈ R3 gives the constant

inertial components of the system angular momentum. See [7] for a detailed derivationof (2.7). Denoting the gimbal rate of the ith CMG by ui, we may write

(2.8) θi(t) = ui(t), i ∈ Iq.

Equations (2.7) and (2.8) describe the combined dynamics of the spacecraft andthe CMG array when the system angular momentum takes the constant value μ ∈ R

3,and define a μ-parametrized family of control systems of the form

(2.9) y(t) = fμ(y(t)) + g1(y(t))u1(t) + · · ·+ gq(y(t))uq(t)

on the (3+q)-dimensional manifoldN def= SO(3)×T

q , where y = (R, θ) ∈ N representsthe spacecraft attitude and the CMG configuration. The drift vector field fμ and thecontrol vector fields g1, g2, . . . , gq, are real analytic on N and are given by

(2.10) fμ(x) =(R(J−1

{RTμ− ν(θ)

})×, 0), gi(x) = (0, ei), x = (R, θ) ∈ N ,

where, for each i ∈ Iq, ei ∈ Rq is the ith column vector of the q × q identity matrix.

We will assume that the gimbal rates are measurable functions of time constrained totake values in the compact polydisk,

(2.11) Hρdef= {u ∈ R

q : |ui| ≤ ρi, i ∈ Iq},

for some constants ρi ∈ (0,∞), i ∈ Iq.It will be convenient to denote by RT (x) the set of states in N that may be

reached in time not greater than T by following all possible solutions of (2.9) thatstart at x ∈ N at t = 0 and result from using gimbal rates that take values inthe input constraint set (2.11). It is easy to see that, if T1, T2 > 0 and x ∈ N ,then RT1(x) ⊆ RT1+T2(x). If, in addition, x1 ∈ RT1(x) and x2 ∈ RT2(x1), thenx2 ∈ RT1+T2(x).

The dynamics (2.9) are strongly accessible at x ∈ N if RT (x) has a nonemptyinterior for each T > 0, and small-time locally controllable (STLC) at x ∈ N if x iscontained in the interior of RT (x) for each T > 0 [28].

We denote the set of uncontrolled equilibria of (2.9) by Eμ. It is easy to see bysetting ui ≡ 0 in (2.9) and using (2.10) that

(2.12) Eμ = {(R, θ) ∈ N : RTμ = ν(θ)}.

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Before moving on, we will briefly compare our formulation of the dynamics withprevious formulations appearing in the literature. The recent paper [3] considersSTLC of the dynamics described by (2.7) for a specific choice of a CMG array when theCMG gimbal angles are themselves treated as control inputs. Our formulation is takenfrom [7], and our choice of gimbal rates as inputs is consistent with a major portionof the CMG literature. The idea of eliminating the angular velocity and formulatingthe dynamics on a level set of the system angular momentum first appears in [14],although the formulation in [14] used quaternion variables instead of rotation matrices.The formulation of [10, 25, 26] includes only the CMG angular momentum among thestate variables, without accounting for the inner structure of the CMG array, and istherefore not suitable for investigating the effect of singular configurations on control-theoretic properties of the combined dynamics. We note that our formulation focusesonly on the dynamics on an angular momentum level set, and is therefore not suitablefor studying disturbance rejection problems involving external torques. Finally, wemention that the standard assumption of negligible gimbal inertias is not realisticespecially in the case of small spacecraft. Hence a realistic treatment of attitudecontrol using CMGs requires a more elaborate multibody dynamics model such as theone presented in the recent paper [24].

2.4. Reduced attitude dynamics. Although (2.7) involves only three dimen-sions instead of the six that one expects in rotational dynamics, it is important toremember that (2.7) does not represent a reduced dynamics model. On the con-trary, when taken together with the equation μ = 0 which we have suppressed, (2.7)constitutes a full description of the attitude dynamics. However, the overall systemdynamics are rotationally invariant, and the conservation of system angular momen-tum is but a manifestation of this invariance via Noether’s theorem [2]. Hence, therotational symmetry may be exploited to reduce the dynamics. Invariance and re-duction have been used in controllability studies before [13, 19] and will prove veryuseful for understanding the STLC of (2.9) as well. Therefore, we provide a brief andself-contained treatment of symmetry and reduction that is adapted to our setting.We refer the reader to the well-known sources [1, Chap. 4], [2, Appendix 5] for thegeneral theory of reduction in mechanical systems.

Throughout this subsection μ denotes a nonzero vector in R3. Let Iμ def

= {S ∈SO(3) : Sμ = μ} denote the isotropy group of μ. Every element of the group Iμrepresents a rotation about the vector μ and equals eαμ

×for a unique α ∈ [−π, π).

The group Iμ acts on the manifold N through the action Φμ : Iμ ×N → N definedby Φμ

S(x) = (SR, θ), where S ∈ Iμ and x = (R, θ) ∈ N . The dynamics given by(2.9) are invariant under the action Φμ in the sense that, if y(·) is a solution of (2.9)for some choice of inputs, then, for every S ∈ Iμ, Φμ

S(y(·)) is also a solution for thesame choice of inputs. This is easy to verify by direct substitution into (2.7). As animmediate consequence, we have

(2.13) RT (ΦμS(x)) = Φμ

S(RT (x)), T > 0, x ∈ N , S ∈ Iμ.

Next, let N r def= S2 × T

q, and define the continuous map φμ : N → N r byφμ(x) = (‖μ‖−1RTμ, θ) for x = (R, θ). Observe that, if x ∈ N and xr = φμ(x), then

(2.14) φ−1μ ({xr}) = {Φμ

S(x) : S ∈ Iμ}.

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As a result, the invariance of the dynamics (2.9) under the action Φμ implies that thedynamics (2.9) project nicely onto dynamics on the reduced state space N r under the

map φμ. The reduced dynamics are easily found by letting ξdef= ‖μ‖−1RTμ denote

the body components of the unit vector along the system angular momentum and byusing (2.7) to compute

(2.15) ξ(t) = ξ(t)× J−1{‖μ‖ξ(t)− ν(θ(t))}.

Equations (2.15) and (2.8) make up the reduced dynamics on N r and may be writtenin the standard form

(2.16) yr(t) = f rμ(y

r(t)) + gr1(yr(t))u1(t) + · · ·+ grq(y

r(t))uq(t),

where yr = (ξ, θ) ∈ N r, and f rμ and gri , i ∈ Iq, are real-analytic vector fields on N r

given by

f rμ(x

r) =(ξ(t)× J−1{‖μ‖ξ(t)− ν(θ(t))}, 0), gri(x

r) = (0, ei), xr = (ξ, θ) ∈ N r.

(2.17)

We denote by RrT (x) the set of states in N r that may be reached in time not

greater than T by following all possible solutions of (2.16) that start at xr ∈ N r att = 0 and result from using gimbal rates that take values in the input constraintset (2.11). Note that yr(·) is a solution of (2.16) for some choice of inputs if andonly if there exists a solution y(·) of (2.9) for the same choice of inputs such thatyr(·) = φμ(y(·)). This fact, along with (2.14) and the invariance of the dynamics (2.9)under the action Φμ, implies that

(2.18) RrT (φμ(x)) = φμ(RT (x)), T > 0, x ∈ N .

We denote by Erμ the set of uncontrolled equilibria of (2.16). It is easy to see that

φμ(Eμ) ⊆ Erμ. Moreover, xr = (ξ, θ) ∈ φμ(Eμ) if and only if ‖μ‖ξ = ν(θ). Solutions

of (2.9) that take values in φ−1μ (Er

μ) are called relative equilibria of (2.9) [1, sect. 4.3],[2, Appendix 5], [12, sect. 6.3]. Relative equilibria of (2.9) that are not equilibriumsolutions of (2.9) represent steady rotations of the spacecraft about the system angularmomentum with CMG gimbals locked.

3. Controllability of the linearization. In this section, we analyze the con-trollability of the linearization of (2.9) at its equilibrium points. Our main resultof this section depends on two lemmas, which we state next and prove later in theappendix. The first of these concerns CMG configurations that are singular, but notcritically singular.

Lemma 3.1. Suppose θ ∈ Tq and v ∈ S2 are such that θ ∈ Sv\C. Then, there

exist i, j ∈ Iq such that vT[J−1ν′i(θi)× ν(θ)] = 0 and ν′i(θi)× ν′j(θj) = 0.Before stating our next lemma, we list the expressions for a few brackets involving

the drift and control vector fields of (2.9) and (2.16). These brackets were computedby locally treating N and N r as embedded submanifolds of R3×3 × R

q and R3 ×R

q,respectively, and using (2.1). Recall that, given a C∞ vector field g on N , the Liebracket adnfµg is defined to be g for n = 0, and recursively defined by adnfµg =

[fμ, adn−1fµ

g] for n > 0. Given μ ∈ R3, x = (R, θ) ∈ N , xr = (ξ, θ) ∈ N r, and i, j ∈ Iq

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such that i = j, we have

[gi, gj](x) = 0, [fμ, gi](x) = (R(J−1ν′i(θi))×, 0),(3.1)

[gj , [fμ, gi]](x) = 0, [gi, [fμ, gi]](x) = (−R(J−1νi(θi))×, 0),(3.2)

[[fμ, gi], [fμ, gj ]](x) = (R(J−1ν′i(θi)× J−1ν′j(θj))×, 0),(3.3)

[[gi, [fμ, gi]], [fμ, gi]](x) = (R(J−1ν′i(θi)× J−1νi(θi))×, 0).(3.4)

ad2fµgi(x) = (R{J−1(J−1ν′i(θi)×RTμ) + J−1(RTμ− ν(θ)) × J−1ν′i(θi)}×, 0),(3.5)

[gi, ad

2fµgi

](x) = (−R{J−1(J−1νi(θi)×RTμ) + J−1(RTμ− ν(θ)) × J−1νi(θi)}×, 0),

(3.6)

[f rμ, g

ri ](x

r) = (ξ × J−1ν′i(θi), 0),(3.7)

ad2frµgri(x

r) =(ξ × [

J−1(‖μ‖ξ − ν(θ)) × J−1ν′i(θi)− ‖μ‖J−1{ξ × J−1ν′i(θi)}], 0).

(3.8)

If, in addition, μ = 0, then

(3.9)[ad1fµgi, ad

2fµgi

](x) = (R{J−1ν′i(θi)× (J−1ν′i(θi)× J−1ν(θ))}×, 0).

Lemma 3.2. Let μ ∈ R3, and suppose p = (R, θ) ∈ Eμ is such that θ ∈ C. Then

dim span{adnfµgi(p) : i ∈ Iq, n ≥ 1} ≤ 2 and span{adnfµgi(p) : i ∈ Iq, n ≥ 1} ⊆span{(R(J−1w)×, 0) : w ∈ R

3, wTν(θ) = 0}.Our first result, which we state next, asserts that the linearization of (2.9) at an

equilibrium is controllable if and only if the corresponding CMG configuration is notcritically singular.

Theorem 3.3. Let μ ∈ R3 and suppose p = (R, θ) ∈ Eμ. Then the linearization

of (2.9)–(2.10) at p is controllable if and only if θ /∈ C.Proof. Note that the linearization of (2.9) at the equilibrium p is controllable if

and only if the tangent vectors {adnfµgi(p) : n ∈ {0} ∪ Iq+2, i ∈ Iq} span TpN (see[15]).

To prove sufficiency, first suppose θ /∈ S. Then there exist distinct i, j, k ∈ Iq suchthat ν′i(θi), ν

′j(θj), and ν′k(θk) are linearly independent. It follows from (3.1) that

[fμ, gi](p), [fμ, gj](p), and [fμ, gk](p) are linearly independent. These three tangentvectors together with the control vector fields g1(p), . . . , gq(p) span TpN . As a result,the linearization of (2.9) at p is controllable.

Next, suppose θ ∈ S\C. Then there exists v ∈ S2 such that vTν′i(θi) = 0 forevery i ∈ Iq. Since θ /∈ C, v satisfies v × ν(θ) = 0. It follows from Lemma 3.1that there exist distinct i, j ∈ Iq such that ν′i(θi) × ν′j(θj) = 0 and vT[J−1ν′i(θi) ×ν(θ)] = 0. Because vTν′i(θi) = vTν′j(θj) = 0, it follows that ν′i(θi), ν

′j(θj), and

J−1ν′i(θi) × ν(θ) = J−1ν′i(θi) × RTμ are linearly independent. Equations (3.1) and(3.5) imply that the tangent vectors [fμ, gi](p), [fμ, gj](p), and [fμ, [fμ, gi]](p) arelinearly independent. These three tangent vectors when taken together with thecontrol vector fields g1(p), . . . , gq(p) span TpN . As a result, the linearization of (2.9)at p is controllable.

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To prove necessity, suppose θ ∈ C. It follows from Lemma 3.2 that

dim span{adnfµgi(p) : n ∈ {0} ∪ Iq+2, i ∈ Iq}= q + dim span{adnfµgi(p) : n ∈ Iq+2, i ∈ Iq} ≤ q + 2 < dim TpN .

Consequently, the linearization of (2.9) at p is not controllable.Since controllability of the linearization at an equilibrium implies STLC at that

equilibrium (see [12, Cor. 7.28]), it follows as a corollary of Theorem 3.3 that thedynamics (2.9) are STLC at an equilibrium in which the CMG array is not in acritically singular configuration. We state this corollary without proof.

Corollary 3.4. Let μ ∈ R3 and suppose p = (R, θ) ∈ Eμ is such that θ /∈ C.

Then the dynamics described by (2.9)–(2.10) are STLC at p when the gimbal rates areconstrained to take values in (2.11).

Our next corollary shows that STLC holds at all equilibria on “almost all” levelsets of the system angular momentum.

Corollary 3.5. If μ ∈ R3 is such that ‖μ‖2 is a regular value of the function η,

then the dynamics (2.9) are STLC at every equilibrium in Eμ. In particular, thereexists a set I ⊂ [0,∞) having zero Lebesgue measure such that the dynamics (2.9)are STLC at every equilibrium in Eμ whenever ‖μ‖2 ⊂ I and the gimbal rates areconstrained to take values in (2.11).

Proof. First, note that if (R, θ) ∈ Eμ, then ‖μ‖2 = ‖ν(θ)‖2, while if θ ∈ C, then‖ν(θ)‖2 is a critical value of η. As a result, if ‖μ‖2 is a regular value of η, then Eμcontains no point of the form (R, θ), where θ ∈ C. Hence the first statement followsfrom Corollary 3.4. The second statement follows from the first by using Sard’stheorem [20, p. 10] on the density of regular values of a smooth function.

Example 3.6. Consider a spacecraft carrying the CMG array described in Ex-ample 2.4. Let μ ∈ R

3 and suppose p = (R, θ) ∈ Eμ. Theorem 3.3 and Corollary3.4 imply that, if θ equals either the nonsingular configuration θA or the external,inescapable singularity θB introduced in Example 2.4, then the dynamics describedby (2.9)–(2.10) possess a controllable linearization at p and are STLC at p under theinput constraint (2.11). On the other hand, the linearized dynamics at p are notcontrollable if θ equals any of θC, θD, or θE. The results of subsequent sections willallow us to examine STLC at equilibria involving these three configurations.

The last result of this section deals with the linearization of the reduced dynamics(2.16).

Proposition 3.7. Suppose μ ∈ R3 and μ = 0. Let pr ∈ φμ(Eμ). Then the

linearization of the reduced dynamics (2.16)–(2.17) at pr are controllable. In partic-ular, the reduced dynamics (2.16)–(2.17) are STLC at pr when the gimbal rates areconstrained to take values in (2.11).

Proof. Let pr = (ξ, θ). Then pr ∈ φμ(Eμ) implies that ‖μ‖ξ = ν(θ) and ‖ν(θ)‖ =‖μ‖ = 0.

First, we claim that there exists i ∈ Iq such that ν(θ) × J−1ν′i(θi) = 0. If not,then all the vectors ν′i(θi), i ∈ Iq, are parallel to the nonzero vector Jν(θ). As aresult, all the vectors νi(θi), i ∈ Iq, as well as their sum ν(θ), are orthogonal to Jν(θ).However, the last conclusion contradicts the positive-definiteness of J , and this provesour claim.

Next, let i be as claimed above. Then the vectors ξ × J−1ν′i(θi) and ξ × J−1{ξ×J−1ν′i(θi)} are linearly independent. Indeed, a simple calculation using the propertiesof the cross product on R

3 shows that the cross product of these two vectors equals[{ξ × J−1ν′i(θi)}TJ−1{ξ × J−1ν′i(θi)}]ξ, which is seen to be nonzero from our claim

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above on recalling that ξ = ‖μ‖−1ν(θ). It now follows from the expressions (3.7)and (3.8) that the 2 + q vector fields [f r

μ, gri ], ad2fr

µgri , and grj , j ∈ Iq, are linearly

independent at pr. The first assertion now follows immediately. The second assertionfollows from [12, Cor. 7.28] as in the discussion preceding Corollary 3.4.

4. Sufficient conditions for STLC at critically singular configurations.In this section, we consider the STLC property at equilibria in which the CMG arrayis in a critically singular configuration. The next theorem is our main result.

Theorem 4.1. Let μ ∈ R3, and suppose p = (R, θ) ∈ Eμ is such that θ ∈ C.

If any one of the following three conditions holds, then the dynamics described by(2.9)–(2.10) are STLC at p for the input constraint set (2.11):

ν(θ) = 0,(4.1)

min{ν(θ)Tνi(θi) : i ∈ Iq} < 0,(4.2)

min{ν(θ)Tνi(θi) : i ∈ Iq} = 0, dim span{ν′i(θi) : i ∈ Iq, ν(θ)Tνi(θi) = 0} = 2.(4.3)

Theorem 4.1 implies that, if any one of (4.1)–(4.3) holds, then, given an arbitrarilysmall bound on the steering time, it is possible to steer the trajectory of (2.9)–(2.10)starting at p to all terminal states in a sufficiently small neighborhood of p within thestipulated time using gimbal motions satisfying the rate constraints (2.11). Beforeproceeding, it is worthwhile to consider the physical interpretations of the conditions(4.1)–(4.3). Since p = (R, θ) ∈ Eμ, condition (4.1) is equivalent to the condition thatthe system angular momentum is zero. Condition (4.2) implies that ν(θ) = 0 or,equivalently, μ = 0. In this case, θ ∈ Sv for v = ‖ν(θ)‖−1ν(θ), so that (4.2) impliesthat θ is an internal singularity. It is sufficient to consider (4.3) only in the caseν(θ) = 0. In this case, ν(θ)Tνi(θi) = 0 for some i ∈ Iq along with θ ∈ C implies thatν(θ) is colinear with the gimbal axis of the ith CMG. Hence (4.3) is equivalent tothe condition that the torques generated by all those CMGs whose gimbal axes arecolinear with the CMG angular momentum vector ν(θ) should span a two-dimensionalsubspace of R3. Note that Theorem 4.1 does not apply if μ = 0 and θ ∈ C is also anexternal singularity, since each of (4.1)–(4.3) is violated in that case.

We will use Proposition 5.1 of [9] to prove Theorem 4.1. A slightly more generalresult is Theorem 5.4 of [9], which is specialized to stationary trajectories in Theo-rem 7.20 of [12]. The application of these results will require us to introduce someadditional terminology and notation. The reader may also refer to section 3 of [9],subsection 7.1.5 of [12], or section 1 of [8].

Let Lie(ξ) be the free Lie algebra generated by the noncommutative indetermi-nates ξ = {ξ0, ξ1, . . . , ξq}. Elements of Lie(ξ) are linear combinations over R of formalLie brackets of the indeterminates ξ. Let Lie0(ξ) denote the smallest Lie subalgebraof Lie(ξ) containing {ξ1, . . . , ξn} that is closed under Lie brackets with ξ0. Given abracket B ∈ Lie(ξ) and i ∈ {0} ∪ Iq, we denote by |B|i the number of times theindeterminate ξi appears in the bracket B. A bracket B ∈ Lie0(ξ) is bad if |B|0 is oddand |B|i is even for every i ∈ Iq. Let B denote the subspace generated by all the badbrackets in Lie0(ξ).

An admissible weight vector is a vector l = [l0, l1, . . . , lq]T ∈ R

q+1 such that li ≥ 0for each i ∈ {0}∪Iq and [εl1−l0x1, . . . , ε

lq−l0xq ]T ∈ Hρ for every ε ∈ (0, 1) and x ∈ Hρ.

It is clear that, under our assumptions on the vector ρ, a vector l = [l0, l1, . . . , lq]T is

an admissible weight vector if and only if li ≥ l0 ≥ 0 for all i ∈ Iq.

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Next, suppose l is an admissible weight vector. The l-degree of a bracket B ∈Lie(ξ) is the sum

∑qi=0 li|B|i. The definition of the l-degree extends naturally to

l-homogeneous elements of Lie(ξ), that is, elements which can be written as a linearcombination of brackets having the same l-degree. Given a nonnegative k ∈ R, wewill denote by Vk the subspace of Lie0(ξ) generated by all brackets having l-degreenot greater than k. We denote by BS the set of elements of B that remain unchangedwhenever the indeterminate ξi is interchanged with ξj for every i, j ∈ Iq such thatli = lj . Finally, the set B∗

S of l-obstructions is the smallest Lie algebra containingBS that is closed under Lie brackets with ξ0. Equivalently, B∗

S ⊂ Lie0(ξ) is the Liealgebra generated by all elements of the form adnξ0B, where n ≥ 0 and B ∈ BS.

Next, consider a sequence h = {h0, h1, . . . , hq} of 1 + q vector fields on N . Given

B ∈ Lie(ξ), Evh(B) will denote the vector field on N obtained by substituting eachoccurrence of the indeterminate ξi in B with the vector field hi for every i ∈ {0}∪ Iq,and evaluating the resulting expression by interpreting the Lie bracket on Lie(ξ) as theLie bracket of vector fields on N . Further, if p ∈ N , then Evhp (B) denotes the tangent

vector in TpN obtained by evaluating the vector field Evh(B) at p. Similarly, for

each k ≥ 0 and p ∈ N , Vhk (p) ⊆ TpN denotes the subspace {Evhp (B) : B ∈ Vk}. An l-

homogeneous element B ∈ B∗S is said to be h-l-neutralized at p ∈ N if Evhp (B) ∈ Vh

k (p)for some k that is strictly less than the l-degree of B.

It was shown in [7] that the system (2.9)–(2.10) is strongly accessible on N forevery μ ∈ R

3. Hence the following result follows from Proposition 5.1 of [9] (equiva-lently, Theorem 7.20 of [12]).

Proposition 4.2. The dynamics (2.9)–(2.10) are STLC at p ∈ Eμ under theinput constraint (2.11) if there exists a nonnegative k and an admissible weight vectorl such that every l-homogeneous element of B∗

S having l-degree not greater than k ish-l-neutralized at p, and Vh

k (p) = TpN , with h = {fμ, g1, . . . , gq}.We will use Proposition 4.2 in the proof of Theorem 4.1. We mention, however,

that the condition in Proposition 4.2 is only a sufficient condition for the hypothesesof the results of [9] to hold, since the results of [9] allow the use of different weightvectors to neutralize different obstructions. On the other hand, the condition inProposition 4.2 is weaker than the condition stated in Theorem 1.1 of [8], whichrequires obstructions of all degrees to be neutralized. This latter condition, thoughmore widely used in applications (see, for instance, [23]), is too strong to be applicablein our case. Finally, we emphasize that the condition is structural in the sense that itdoes not involve the actual values of the positive constants ρi appearing in the inputconstraints (2.11).

Proof of Theorem 4.1. First, suppose (4.1) holds, and let h = {fμ, g1, . . . , gq}. Inthis case, ‖ν(θ)‖ = ‖RTμ‖ = 0. Choose li = 1 for all i ∈ {0} ∪ Iq. The two lowestpossible l-degrees for a bad bracket are 3 and 5. Hence, l-homogeneous elements ofB∗S having the three lowest possible l-degrees are of the form B, adξ0B, ad2ξ0B, and

C, where B and C are l-homogeneous elements of BS of degrees 3 and 5, respectively.Bad brackets having l-degree 3 are of the form [ξi, [ξ0, ξi]] for some i ∈ Iq. Hence

the l-homogeneous element of BS having the least l-degree 3 is B =∑q

i=1[ξi, [ξ0, ξi]].

Expression (3.2) along with 0 = ν(θ) =∑q

i=1 νi(θi) allows us to compute Evhp (B) = 0.

Since Evhp (ξ0) = fμ(p) = 0, the elements adξ0B and ad2ξ0B of B∗S also evaluate to

0 at p.There are only two types of bad brackets of the next lowest degree 5, namely,

those that contain ξ0 once and those that contain ξ0 thrice. Thus, a bad bracketCij of the first type has |Cij |0 = 1 and |Cij |i = |Cij |j = 2 for some i, j ∈ Iq.

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One can use (3.1) and (3.2) to show that Evh(Cij) = 0 if i = j. If i = j, then

Evhp (Cij) equals, to within a sign change, the tangent vector [gi, [gi, [gi, [fμ, gi]]]](p),whose expression can be easily calculated from (3.2) to be (R(J−1νi(θi))

×, 0). Since∑qi=1 νi(θi) = ν(θ) = 0, the l-homogeneous element C =

∑qi=1 Cii of BS evaluates

to 0 under Evhp . A bad bracket Ci of the second type has |Ci|0 = 3 and |Ci|i = 2

for some i ∈ Iq. For such a bracket, Evh(Ci) can be, to within a sign change, one ofc1 = [ad1fµgi, ad

2fµgi], c2 = [fμ, [gi, ad

2fµgi]], or c3 = [gi, ad

3fµgi]. We have c1(p) = 0

from (3.9). Using RTμ = ν(θ) = 0 in (3.6) gives [gi, ad2fµgi](p) = 0. Since fμ(p) = 0

also, it follows that c2(p) = 0. The Jacobi identity for Lie brackets of vector fieldsimplies that c3 = c2 − c1, and hence c3(p) = 0 as well.

It follows that every l-homogeneous element of B∗S having l-degree not greater than

5 evaluates to 0 at p, and is thus h-l-neutralized. Since the vectors J−1ν1(θ1) andJ−1ν′1(θ1) are linearly independent, it follows from (3.1), (3.2), and (3.4) that the 3+qtangent vectors [fμ, g1](p), [g1, [fμ, g1]](p), [[g1, [fμ, g1]], [fμ, g1]](p), and gi(p), i ∈ Iq,which are contained in Vh

5 (p), are linearly independent. Therefore, Vh5 (p) = TpN ,

and STLC at p follows from Proposition 4.2.Before continuing, we claim that if either (4.2) or (4.3) holds, then there exists

an index set I ⊆ Iq and positive constants λi > 0, i ∈ I, such that

(4.4) ν(θ)T

[∑i∈I

λiνi(θi)

]= 0,

and either span{ν′i(θi) : i ∈ I} has dimension 2, or there exists i ∈ I such thatν(θ)Tνi(θi) = 0. Indeed, suppose (4.2) holds. Then, it follows from 0 ≤ ν(θ)Tν(θ) =∑q

i=1 ν(θ)Tνi(θi) that there exist distinct i, j ∈ Iq such that ν(θ)Tνi(θi) < 0 <

ν(θ)Tνj(θj). Our claim then holds by letting I = {i, j}, λi = ν(θ)Tνj(θj), andλj = −ν(θ)Tνi(θi). On the other hand, if (4.3) holds, then our claim holds by lettingI = {i : ν(θ)Tνi(θi) = 0} and λi = 1, i ∈ I.

Next, assume that either (4.2) or (4.3) holds, and let I ⊆ Iq be as guaranteed byour claim above. We need only consider the case ν(θ) = 0 = μ. Consider the inputscaling ui =

√λiui, i ∈ I, ui = ui, i /∈ I. We may rewrite (2.9) as

(4.5) y(t) = fμ(y(t)) + g1(y(t))u1(t) + · · ·+ gq(y(t))uq(t),

where gi equals√λigi if i ∈ I and equals gi if i /∈ I. Clearly, (2.9) is STLC subject to

the input constraint (2.11) if and only if (4.5) is STLC subject to the input constraintu(t) ∈ Hρ for a suitably defined vector ρ of positive elements. Hence, it suffices toshow that (4.5) satisfies the sufficient condition given by Proposition 4.2. For thispurpose, we redefine h = {fμ, g1, . . . , gq}.

Let l0 = 1, li = 1.5 for i ∈ I, and li = 2.5 for i /∈ {0} ∪ I. The two lowestpossible l-degrees for a bad bracket are 4 and 6. As a result, each l-homogeneous ele-ment of B∗

S having l-degree not greater than 5 is one of the forms B or adξ0B, whereB ∈ BS has l-degree 4. Bad brackets having l-degree 4 are of the form [ξi, [ξ0, ξi]] forsome i ∈ I. Hence the l-homogeneous element of BS having the least l-degree 4 isB =

∑i∈I [ξi, [ξ0, ξi]]. We have Evh(B) =

∑i∈I [gi[fμ, gi]] =

∑i∈I λi[gi, [fμ, gi]]. We

also have Evhp (adξ0B) =∑

i∈I λi[fμ, [gi, [fμ, gi]]](p) =∑

i∈I λi[gi, ad2fµgi](p), where

the last equality uses the Jacobi identity. The condition (4.4) along with (3.2) and(3.6) implies that each of the tangent vectors Evhp (B) and Evhp (adξ0B) is of the form

(R(J−1u)×, 0), where u is orthogonal to RTμ = ν(θ). Consequently, u is contained in

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the span of the linearly independent vectors ν′i(θi) and J−1ν′i(θi)×RTμ for any arbi-

trarily chosen i ∈ I. Therefore, the expressions (3.1) and (3.5) show that Evhp (B)

and Evhp (adξ0B) are contained in the span of the tangent vectors [fμ, gi](p) and

[fμ, [fμ, gi]](p), both of which are contained in Vh3.5(p). Hence, all l-homogeneous

elements of B∗S having l-degree not greater than 5 are h-l-neutralized.

If the dimension of span{ν′i(θi) : i ∈ I} is 2, then there exist i, j ∈ I such thatν′i(θi) and ν

′j(θj) are linearly independent. In this case, the subspace Vh

5 (p) containsthe 3 + q tangent vectors [fμ, gi](p), [fμ, gj ](p), [[fμ, gi], [fμ, gj ]](p), and gk, k ∈ Iq,which are linearly independent by (3.1) and (3.3). On the other hand, if i ∈ I issuch that ν(θ)Tνi(θi) = 0, then the vectors ν′i(θi), J

−1ν′i(θi)× ν(θ), and νi(θi), whosescalar triple product has the absolute value |[ν′i(θi)TJ−1ν′i(θi)][ν(θ)

Tνi(θi)]| = 0, arelinearly independent. Hence, in this case, the subspace Vh

5 (p) contains the 3 + qlinearly independent tangent vectors [fμ, gi](p), [fμ, [fμ, gi]](p), [gi, [fμ, gi]](p), andgk(p), k ∈ Iq. In either case, Vh

5 (p) = TpN , and STLC follows from Proposition4.2.

Example 4.3. Consider a spacecraft carrying the CMG array described in Exam-ple 2.4. Let μ ∈ R

3 and suppose p = (R, θ) ∈ Eμ. If θ = θC introduced in Example2.4, then (4.1) holds. On the other hand, if θ = θD, then (4.2) holds. In either case,Theorem 4.1 implies that the dynamics (2.9)–(2.10) are STLC at p under the inputconstraint (2.11). Finally, if θ = θE, then none of the conditions (4.1)–(4.3) hold, andTheorem 4.1 cannot be applied. Our main result of the next section will allow us todetermine the STLC at equilibria involving the configuration θE.

5. Necessary conditions for STLC at critically singular configurations.In this section, we will use Theorem 1.2 of [27] to show that STLC fails to hold inone of the cases not covered by Theorem 4.1. First, we note that Theorem 1.2 of[27] gives a sufficient condition for an uncontrolled reference trajectory y(·) of (2.9)to lie on the boundary of the reachable set from the initial state y(0) of the referencetrajectory for some sufficiently small length of time. To apply the sufficient condition,we will choose the reference trajectory to be the constant trajectory starting froman uncontrolled equilibrium point p ∈ N of (2.9), so that, if the sufficient conditionholds, it will allow us to conclude that p lies on the boundary of the set of states thatare reachable from p in sufficiently small time using gimbal rates constrained to lie in(2.11), thus ruling out STLC at p. Since the result of [27] involves the adjoint systemcorresponding to the uncontrolled system

(5.1) y(t) = fμ(y(t)),

we will briefly introduce the notation necessary to investigate the adjoint system of(5.1) without performing cumbersome coordinate-based calculations.

The adjoint system of (5.1) is the Hamiltonian system on the cotangent bundleT∗N (together with its canonical symplectic structure [1, sect. 3.2]) whose Hamilto-nian function Hμ : T∗N → R is given by Hμ(Λ) = Λ(fμ(π

∗(Λ))) for every Λ ∈ T∗N ,where π∗ : T∗N → N is the canonical cotangent bundle projection.

The cotangent bundle T∗N may be identified with T∗SO(3)×T∗Tq in a natural

way. Since SO(3) is a Lie group, we may use left translations to identify TSO(3) =SO(3) × so(3) and T∗SO(3) = SO(3) × so(3)∗, where so(3)∗ is the vector space dual

to so(3) [1, sect. 4.4]. Using the inner product 〈v×1 , v×2 〉 def= 1

2 trace[(v×1 )

Tv×2 ] = vT1 v2on so(3), we may identify so(3)∗ with so(3), and hence T∗SO(3) with TSO(3). Thecotangent and tangent bundles of Tq may also be canonically identified with T

q ×Rq.

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Thus, an element Λ of T∗N may be identified with the 4-tuple (R, λ×R , θ, λθ), where(R, θ) = π∗(Λ) ∈ N , λR ∈ R

3, and λθ ∈ Rq. Similarly, an element v of TN may

be identified with the 4-tuple (R, v×R , θ, vθ), where (R, θ) = π(v) ∈ N , vR ∈ R3, and

vθ ∈ Rq, with π : TN → N being the canonical tangent bundle projection. Thus, in

the notation above, if Λ ∈ T∗N and v ∈ TN are such that π∗(Λ) = π(v), then

(5.2) Λ(v) = λTRvR + λTθ vθ.

Hence, the Hamiltonian function of the adjoint system of (5.1) may be written as

(5.3) Hμ(Λ) = λTRJ−1(RTμ− ν(θ)), Λ = (R, λ×R, θ, λθ) ∈ T∗N .

Our next lemma gives an equilibrium point of the adjoint system of (5.1).

Lemma 5.1. Suppose μ = 0, and (R, θ) ∈ Eμ is such that θ ∈ C. Then Λdef=

(R, (Jν(θ))×, θ, 0) ∈ T∗(R,θ)N is an equilibrium point of the adjoint system of (5.1).

Proof. Every critical point of a Hamiltonian function is an equilibrium point ofthe corresponding Hamiltonian system [1, Prop. 3.4.16]. Hence it suffices to showthat Λ is a critical point of the Hamiltonian function (5.3). Consider the curve γ :

[0, 1] → T∗N given by γ(t) = (Retw×1 , (Jν(θ)+tw2)

×, θ+tw3, tw4), where w1, w2 ∈ R3

and w3, w4 ∈ Rq are arbitrary. Note that γ(0) = Λ, and γ(0) represents an arbitrary

vector tangent to T∗N at Λ. Using (5.3), we may easily compute

(5.4) Hμ(γ(t)) = [νT(θ) + twT2 J

−1][e−tw×1 RTμ− ν(θ + tw3)], t ∈ [0, 1],

so that

(5.5)d

dt

∣∣∣∣t=0

Hμ(γ(t)) = νT(θ)

[−w1 ×RTμ−

q∑i=1

wi3ν

′i(θi)

]+ wT

2 J−1[RTμ− ν(θ)],

where wi3 denotes the ith component of w3. On recalling that (R, θ) ∈ Eμ and θ ∈ C,

so that RTμ = ν(θ) and νT(θ)ν′i(θi) = 0 for all i ∈ Iq, we immediately conclude thatthe right-hand side of (5.5) is zero. Since the choice of the representative curve γ wasarbitrary, we conclude that Λ is a critical point of the Hamiltonian function of theadjoint system of (5.1). This completes the proof.

Remark 5.2. One may use (5.2) along with Lemma 3.2 to show that, if θ ∈ Cand p = (R, θ) ∈ Eμ for some μ = 0, then Λ(adkfµgi(p)) = 0 for all k ∈ {0} ∪ Iq+2 and

i ∈ Iq. Thus, the local representative of Λ in any set of local coordinates on N is aleft null-vector of the controllability matrix of the linearization at p of (2.9) in thatset of local coordinates. Indeed, computations in suitably chosen local coordinatesshow that 0 is an uncontrollable eigenvalue of the linearization of (2.9) at p, and Λ isrelated to the corresponding left eigenvector of the state matrix of the linearization.

The main result of this section is based on Theorem 1.2 of [27], which we statebelow in the notation of this paper for the reader’s convenience. We note that Theorem1.2 of [27] also requires the condition that the control vector fields gi, i ∈ Iq, arelinearly independent. This condition is satisfied in our case, and is therefore excludedfrom the statement below.

Proposition 5.3 (see [27]). Suppose y : [0, T ] → N is a solution of (2.9) andγ : [0, T ] → T∗N is a solution of the adjoint system of (5.1) such that γ(t) ∈ T∗

y(t)N ,

t ∈ [0, T ]. Suppose, in addition, that the following conditions hold:1. γ(t)(h(y(t))) = 0, t ∈ [0, T ], for every vector field h of the form adkfµgi or

adkfµ [gi, gj] for some i, j ∈ Iq and some k ≥ 0.

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2. The q × q matrix L defined by Lij = γ(0) ([[fμ, gi], gj ](y(0))) is positive defi-nite.

3. γ(0) ([gi, [gj , gk]](y(0))) < 3λ/[2q2(q − 1)] for all i, j, k ∈ Iq, where λ is thesmallest eigenvalue of L.

Then, there exists T ∈ (0, T ] such that y(t) lies on the boundary of Rt(y(t)) for allt ∈ [0, T ].

The next theorem is the main result of this section.Theorem 5.4. Suppose μ = 0. Let θ ∈ C be such that

(5.6) min{ν(θ)Tνi(θi) : i ∈ Iq} > 0,

and let p = (R, θ) ∈ Eμ. Then the dynamics described by (2.9)–(2.10) are not STLCat p for inputs that take values in the constraint set (2.11). In particular, there exists

T > 0 and a sequence {αn}∞n=1 in [−π, π) converging to 0 such that (eαnμ×R, θ) /∈

RT (p).Proof. Let T > 0, and consider the constant reference trajectory y(t) ≡ p of the

uncontrolled system (5.1) on the interval [0, T ]. Let Λ = (R, (Jν(θ))×, θ, 0) be as inLemma 5.1, and consider the constant curve γ(t) ≡ Λ in T∗N on the interval [0, T ].

We trivially have γ(t) ∈ T∗y(t)N for all t ∈ [0, T ]. Moreover, by Lemma 5.1, γ is a

(constant) trajectory of the adjoint system of (5.1).We may use (5.2) along with Lemma 3.2 to show that, for all n ≥ 0, i ∈ Iq,

and t ∈ [0, T ], γ(t)(adkfµgi(y(t))) = Λ(adkfµgi(p)) = 0. Moreover, the vector fields

adkfµ [gi, gj] are identically zero. Hence condition (1) of Proposition 5.3 is satisfied.

Next, consider the matrix L ∈ Rq×q defined in condition (2) of Proposition 5.3.

The expressions (3.2) and (5.2) allow us to compute L as the diagonal matrix suchthat Lii = νT(θ)νi(θi) for each i ∈ Iq. The assumption (5.6) implies that L is positivedefinite. Hence condition (2) of Proposition 5.3 is satisfied. Finally, condition (3) ofProposition 5.3 holds trivially, since all brackets of the form [gi, [gj , gk]] are identically

zero. Hence Proposition 5.3 implies that there exists T ∈ (0, T ] such that y(t) = plies on the boundary of Rt(y(t)) = Rt(p) for all t ∈ [0, T ]. This completes the proofof the first assertion.

To prove the second assertion, let T > 0 and ε > 0 be such that p lies on theboundary of RT+ε(p). The existence of T and ε follows from our first assertion. Let{Un}∞n=1 be a sequence of open neighborhoods of p such that Un+1 ⊂ Un for each n and∩∞n=1Un = {p}. Since the drift and control vector fields in (2.9) are continuous and

gimbal rates take values in the compact set (2.11), there exists a sequence {Tn}∞n=1

in (0, ε) converging to zero such that RTn(p) ⊆ Un for each n.

Fix n. Let Wn denote the interior of RrTn

(pr), where prdef= φμ(p) ∈ φμ(Eμ).

By Proposition 3.7, the reduced dynamics are STLC at pr, and hence Wn is an openneighborhood of pr. Consequently, φ−1

μ (Wn)∩Un is an open neighborhood of p. There

exists xn = (Rn, θn) ∈ (φ−1

μ (Wn) ∩ Un)\RT+ε(p). Denote xrndef= φμ(xn), and note

that xrn ∈ Wn ⊆ RrTn

(pr). By (2.18), RrTn

(pr) = φμ(RTn(p)). Therefore, there existszn = (Vn, δ

n) ∈ RTn(p) ⊆ Un such that φμ(zn) = xrn = φμ(xn). As a result, δn = θn

and Sndef= RnV

Tn ∈ Iμ. Thus, there exists αn ∈ [−π, π) such that Sn = eαnμ

×. Also,

one easily checks that ΦμSn

(zn) = xn.Note that, by construction, xn, zn ∈ Un for each n. As a result, the sequences

{Rn}∞n=1 and {Vn}∞n=1 converge to R. Consequently, the sequence {Sn}∞n=1 convergesto the identity, and the sequence {αn}∞n=1 converges to 0.

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We claim that ΦμSn

(p) = (eαnμ×R, θ) /∈ RT (p) for each n. To arrive at a contra-

diction, suppose our claim does not hold for some n, so that

(5.7) ΦμSn

(p) ∈ RT (p).

By construction, zn = (Vn, δn) = (Vn, θ

n) ∈ RTn(p). By (2.13),

(5.8) ΦμSn

(zn) ∈ RTn(ΦμSn

(p)).

The inclusions (5.7) and (5.8) imply that xn = ΦμSn

(zn) ∈ RT+Tn(p) ⊆ RT+ε(p),which contradicts our construction of xn. This proves our claim and the secondassertion.

Note that for θ ∈ C, (5.6) is equivalent to the condition that θ is an externalsingularity in the sense of Definition 2.3. Thus Theorem 5.4 states that if the criticallysingular CMG configuration at the equilibrium p is an external singularity, then thereexist terminal states arbitrarily close to p that cannot be reached from p in smalltime using gimbal rates contained in (2.11). Moreover, some of these small-timeunreachable states can be obtained from p by locking the gimbals and rotating thespacecraft about the system angular momentum vector through any one of a sequenceof vanishingly small angles.

The proof of Theorem 5.4 makes an interesting use of the local controllabilityproperties of the reduced dynamics to construct states that are small-time unreachablefor the full dynamics. The second part of the proof is in the spirit of [19, Thm. 1],which relates the global controllability of a group-invariant system on a principal fiberbundle to that of the reduced or projected dynamics on the base space. Theorem 5.4also shows that it is not possible to have a result similar to [19, Thm. 1] for localcontrollability.

Critically singular configurations are critical points of the function η and includeas a special case local maximizers of η. The following corollary of Theorem 5.4 showsthat STLC fails at equilibria in which the CMG configuration is a local maximizer ofthe CMG angular momentum magnitude η. The proof involves showing that a localmaximizer of the CMG angular momentum magnitude is an external singularity.

Corollary 5.5. Suppose θ ∈ C is a local maximizer for the function η, andsuppose p = (R, θ) ∈ Eμ for some μ ∈ R

3, μ = 0. Then the spacecraft attitudedynamics described by (2.9)–(2.10) are not STLC at p for the input constraint set(2.11).

Proof. Since θ ∈ C is a local maximizer for the smooth function η, the q × qHessian matrix of η at θ is a negative-semidefinite matrix. In particular, the diagonalelements of the Hessian are nonpositive. The ith diagonal element of the Hessian of η

at θ is easily seen to be ∂2η∂θ2

i(θ) = 2‖ν′i(θi)‖2−2νT(θ)νi(θi). Applying the fact that the

diagonal elements of the Hessian are nonpositive leads to νT(θ)νi(θi) ≥ ‖ν′i(θi)‖2 > 0for all i ∈ Iq. Thus (5.6) holds, and the result follows from Theorem 5.4.

As another corollary of Theorem 5.4, we prove that STLC fails at all equilibria ifthe spacecraft carries only one CMG.

Corollary 5.6. Suppose q = 1. Let μ ∈ R3 and p = (R, θ) ∈ Eμ. Then, the

spacecraft attitude dynamics described by (2.9)–(2.10) are not STLC at p for the inputconstraint set (2.11).

Proof. In the case q = 1, the function η is a constant function. As a result, θ is alocal maximizer of the function η, and the conclusion follows from Corollary 5.5.

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Example 5.7. Continuing further with the CMG array introduced in Example

2.4, let μ ∈ R3 and R ∈ SO(3) be such that p

def= (R, θE) ∈ Eμ. Then (5.6) holds, and

hence Theorem 5.4 implies that the dynamics (2.9)–(2.10) are not STLC at p.Before closing the section, we remark that the conditions of Theorems 4.1 and

5.4 do not cover the case where ν(θ) = 0 and

(5.9) min{ν(θ)Tνi(θi) : i ∈ Iq} = 0, dim span{ν′i(θi) : i ∈ Iq, ν(θ)Tνi(θi) = 0} = 1.

It is easy to conceive of a CMG array possessing a critically singular configurationsatisfying (5.9). Hence, even though Theorems 3.3, 4.1, and 5.4 enable us to determineSTLC or the lack thereof for a large class of singular configurations, the problem ofdetermining STLC in the presence of singular configurations still remains partiallyopen.

6. Stabilizability. In this section, we analyze the feedback stabilizability ofequilibria of (2.9). Our first result below asserts that equilibrium points of (2.9) inwhich the CMG array is not in a critically singular configuration is locally asymptot-ically stabilizable using smooth static feedback.

Theorem 6.1. Let μ ∈ R3, and suppose p = (R, θ) ∈ Eμ is such that θ /∈ C.

Then p is locally asymptotically stabilizable by smooth static state feedback.Proof. Theorem 3.3 implies that the linearization of (2.9) at p is controllable, and

hence stabilizable (see [11], [12, Rem. 10.4.1], [33]). It now follows that (2.9) is locallyasymptotically stabilizable at p by smooth static state feedback [33, Prop. 1].

It is worthwhile to note that a corollary on stabilizability similar in spirit toCorollary 3.5 follows in an obvious manner from Theorem 6.1. In the interest of space,we omit stating such a corollary, and instead move on to our next result, which showsthat equilibrium points, at which the magnitude of the CMG angular momentumeither has a local maximum or a nonzero local minimum, cannot be stabilized usingcontinuous feedback. Note that, in both cases, the CMG array is in a critically singularconfiguration. The proof is based on showing that Brockett’s necessary condition[11, 33] for continuous stabilizability is violated.

Theorem 6.2. Let μ ∈ R3, and suppose p = (R, θ) ∈ Eμ is such that ν(θ) = 0,

and θ is either a local maximizer or a local minimizer of the function η. Then p isnot locally asymptotically stabilizable through continuous time-invariant feedback.

Proof. Since ν(θ)T(RTμ) > 0 and ν(θ)Tν(θ) > 0 hold for (R, θ) = (R, θ), we maychoose an arbitrarily small open neighborhood V ⊆ N of p such that ν(θ)T(RTμ) > 0

and ν(θ)Tν(θ) > 0 for all (R, θ) ∈ V . Suppose p is locally asymptotically stabilizablethrough continuous time-invariant feedback. Brockett’s necessary condition for con-tinuous stabilizability [11, Thm. 1], [33, Thm. 3] implies that the image of V × R

q

under the map

(6.1) (x, u) �→ fμ(x) + u1g1(x) + · · ·+ uqgq(x)

contains an open neighborhood in TN of the zero section over V (see the version givenby [12, Thm. 10.8]). In the notation of section 5, every element in the image of V×R

q

under the map (6.1) is of the form (R, h(R, θ)×, θ, u), where (R, θ) ∈ V , h : N → R3

is given by h(R, θ) = J−1{RTμ − ν(θ)}, and u ∈ Rq. Projecting onto the second

factor lets us conclude that the image of V under h contains an open neighborhood of0 ∈ R

3. Hence there exists δ > 0 such that, for every ε ∈ (−δ, δ), the vector εJ−1ν(θ)is contained in h(V).

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First, pick ε ∈ (−δ, 0), and choose (R, θ) ∈ V such that h(R, θ) = J−1{RTμ −ν(θ)} = εJ−1ν(θ). Since p ∈ Eμ, we have ν(θ) = RTμ, so that ‖ν(θ)‖2 = ‖μ‖2 =

‖Rμ‖2. Therefore, we have ‖ν(θ)‖2 − ‖ν(θ)‖2 = ‖RTμ‖2 − ‖ν(θ)‖2 = (RTμ +

ν(θ))T(RTμ − ν(θ)) = ε(RTμ + ν(θ))Tν(θ) < 0. This shows that θ is not a max-imizer of the map η over V . Similarly, by choosing ε ∈ (0, δ), it can be shown that θ isnot a minimizer of η over V . Since V can be chosen arbitrarily small, it follows that θis neither a local maximizer nor a local minimizer for η, and the result follows.

As a corollary of the above result, we show that no equilibrium of a spacecraftactuated by a single CMG is stabilizable through continuous feedback.

Corollary 6.3. Suppose q = 1. Let μ ∈ R3 and p = (R, θ) ∈ Eμ. Then p is not

locally asymptotically stabilizable through continuous time-invariant feedback.Proof. The result follows from Theorem 6.2 by noting that, for a single CMG, the

function η is a constant function, and hence every configuration is a local maximizerof the function η.

Example 6.4. Consider, once again, the CMG pyramid array described in Ex-ample 2.4. Let μ ∈ R

3 and suppose p = (R, θ) ∈ Eμ. Theorem 6.1 implies that pis locally asymptotically stabilizable through smooth feedback if θ is either the non-singular configuration θA or the external, inescapable singularity θB. On the otherhand, Theorem 6.2 implies that p is not locally asymptotically stabilizable throughcontinuous feedback if θ = θE and sin2 α > 1

2 .

7. Conclusions. The purpose of this paper was to investigate in what mannerthe presence of singular CMG configurations affects small-time local controllability(STLC) and stabilizability of the attitude dynamics of a spacecraft actuated by acontrol moment gyroscope (CMG) array. Our main achievement has been to identifya class of singular CMG configurations, called critically singular configurations, atwhich proving local controllability and stabilizability is problematic. The dynamicsnear those equilibria in which the configuration of the CMG array is possibly singular,but not critically singular, are linearly controllable and hence small-time locally con-trollable and stabilizable on a level set of the system angular momentum. Moreover,this condition is satisfied by all equilibria for almost all values of the system angularmomentum magnitude. In contrast, for equilibria in which the configuration of theCMG array is critically singular, STLC holds only under additional assumptions onthe CMG configuration. Specifically, STLC fails if the critically singular CMG con-figuration is an external singularity in the sense that we have defined. In particular,STLC and stabilizability fail at equilibria at which the magnitude of the CMG angularmomentum has a local maximum. The problem of determining STLC still remainsopen for certain critically singular configurations, while the problem of determiningstabilizability remains open for a large class of critically singular configurations.

Appendix. Proofs of lemmas.Proof of Lemma 3.1. To arrive at a contradiction, suppose that ν′i(θi)×ν′j(θj) = 0

for all i, j ∈ Iq. It follows that νi(θi)Tν′j(θj) = 0 for all i, j ∈ Iq. Hence, ν(θ)

Tν′j(θj) =0 for all j ∈ Iq. Therefore, θ ∈ C, which is a contradiction. The contradictionimplies that, for every k ∈ Iq, there exists an n ∈ Iq such that ν′k(θk) × ν′n(θn) = 0.In particular, the vectors {ν′1(θ1), . . . , ν′m(θq)} span the two-dimensional subspaceorthogonal to the vector v.

Since θ ∈ Sv\C, it follows that v×ν(θ) = 0. To arrive at a contradiction, supposethat vT[J−1ν′i(θi) × ν(θ)] = 0 for all i ∈ Iq. Using basic properties of the scalartriple product and noting that J = JT, it follows that ν′i(θi)

TJ−1[v × ν(θ)] = 0

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for all i ∈ Iq. Thus the nonzero vector J−1[v × ν(θ)] is orthogonal to every vectorin the two-dimensional subspace orthogonal to v. Hence there exists γ ∈ R\{0} suchthat v × ν(θ) = γJv. This implies that vTJv = 0, which contradicts the fact thatJ is positive definite. The contradiction implies that there exists i ∈ Iq such thatvT[J−1ν′i(θi) × ν(θ)] = 0. As shown in the previous paragraph, there exists j ∈ Iq

such that ν′i(θi)× ν′j(θj) = 0.Proof of Lemma 3.2. Choose i ∈ Iq. First, assume μ = 0. Then it follows that

ν(θ) = 0. We claim that, for every n ≥ 1, there exist functions hn,i : N → R3

and cn,i : N → R3 satisfying ν(θ)Thn,i(p) = 0 and cn,i(p) = 0 such that, for every

x = (R, θ) ∈ N ,

(A.1) adnfµgi(x) = (R(J−1hn,i(x) + cn,i(x))×, 0).

We prove this claim by induction on n.It follows from (3.1) that the claim is true for n = 1 with h1,i(x) = ν′i(θi) and

c1,i(x) = 0 for all x ∈ N . Next, suppose the claim is true for n = k. Consider

x = (R, θ) ∈ N , denote ω(x) = J−1(RTμ − ν(θ)), and note that ω(p) = 0. Analgebraic computation using (2.1) yields

(A.2) adk+1fµ

gi(x) = (R(J−1hk+1,i(x) + ck+1,i(x))×, 0),

where hk+1,i(x)def= J−1hk,i(x)× RTμ and ck+1,i(x)

def= ω(x)× [J−1hk,i(x)+ ck,i(x)]+

J−1[ck,i(x) × RTμ] + ψk+1,i(x) with

ψk+1,i(x) =d

ds

∣∣∣∣s=0

[J−1hk,i(x+ sfμ(x)) + ck,i(x+ sfμ(x))

].

Since fμ(p) = 0, it follows that ψk+1,i(p) = 0, and hence, by the induction hypoth-esis, ck+1,i(p) = 0. Furthermore, hk+1,i(p) equals J−1hk,i(p) × ν(θ) and satisfiesν(θ)Thk+1,i(p) = 0. Induction now implies that (A.1) holds for all n ≥ 1. Thus, forevery n ≥ 1 and every i ∈ Iq, the tangent vector adnfµgi(p) is contained in the two-

dimensional subspace {(R(J−1w)×, 0) : w ∈ R3, wTν(θ) = 0} of TpN . This completes

the proof in the case μ = 0.Next, assume μ = 0 so that ν(θ) = 0 as well. Equation (3.5) implies that

ad2fµgi(p) = 0. Since fμ(p) = 0, an easy computation shows that adk+1fµ

gi(p) = 0

whenever adkfµgi(p) = 0. Induction now implies that adnfµgi(p) = 0 for all n ≥ 2.Hence span{adnfµgi(p) : i ∈ Iq, n ≥ 1} = span{[fμ, gi](p) : i ∈ Iq}, which has dimen-

sion not greater than two since θ ∈ C. Moreover, {(R(J−1w)×, 0) : w ∈ R3, wTν(θ) =

0} = {(R(J−1w)×, 0) : w ∈ R3}, which proves the inclusion made in the assertion.

This completes the proof.

Acknowledgment. The authors thank Amit Sanyal for helpful discussions thatclarified the assumptions required to derive the attitude dynamics model used in thispaper.

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small-time local controllability and stabilizability of spacecraft attitude dynamics under cmg...

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