New Singularity Escape/Avoidance SteeringLogic for Control Moment Gyro Systems

Bong Wie∗

Arizona State UniversityTempe, AZ 85287-6106

AbstractThe control moment gyro (CMG) steering logic of U.S.

Patent 6,039,290 is a simple yet effective way of passingthrough, and also escaping from, any internal singular-ities. However, it is unable to escape the external satu-ration singularities of certain CMG configurations evenwhen CMG momentum desaturation is requested. Con-sequently, a new steering logic which eliminates such aproblem of being trapped in saturation singularities isdeveloped. Furthermore, the new steering logic providesa simple means of avoiding troublesome, internal ellipticsingularities which are commonly encountered by mostother pseudoinverse-based steering logic. Various CMGsystems, such as a pyramid array of four single-gimbalCMGs, two and three parallel single-gimbal CMG con-figurations, and four parallel double-gimbal CMG con-figuration are used as illustrative examples.

1. Introduction

Control moment gyros (CMGs), as applied to space-craft attitude control and momentum management, havebeen extensively studied during the past three decades(Refs. 1-8) and more recently in Refs. 9-17. They havebeen successfully employed for a variety of space mis-sions, such as the Skylab, the MIR station, and the In-ternational Space Station. However, CMGs have neverbeen used in commercial satellites because their highertorque capabilities are not needed by most commercialsatellites and also because CMGs are much more expen-sive and mechanically complex than reaction wheels.

A CMG contains a spinning rotor with large, constantangular momentum, but whose angular momentum vec-tor direction can be changed with respect to the space-craft by gimballing the spinning rotor. The spinningrotor is mounted on a gimbal (or a set of gimbals), andtorquing the gimbal results in a precessional, gyroscopicreaction torque orthogonal to both the rotor spin andgimbal axes. The CMG is a torque amplification devicebecause small gimbal torque input produces large con-trol torque output on the spacecraft. Because the CMGsare capable of generating large control torques and stor-ing large angular momentum over long periods of time,

∗Professor, Dept. of Mechanical & Aerospace Engineering,bong.wie@asu.edu, (480) 965-8674, Fax (480) 965-1384.

they have been employed for attitude control and mo-mentum management of large space vehicles, such as theInternational Space Station (ISS). Four parallel mounteddouble-gimbal CMGs with a total weight of about 2400lb and with a design life of 10 years are employed onthe ISS. In particular, Kennel’s steering law (Ref. 4) hasbeen implemented on the double-gimbaled CMG systemof the ISS.

Although CMGs have never been used in commercialsatellites, the next-generation Earth imaging satelliteswill require rapid rotational maneuverability for high-resolution images. Rather than sweep the imaging sys-tem from side-to-side, the whole spacecraft body willturn rapidly. Pointing the entire spacecraft allows theimaging system with a narrow field of view to achievea higher definition and improves the resolution for itsimages. The overall cost and effectiveness of such agilespacecraft is greatly affected by the average retargetingtime. Thus, the development of a low-cost attitude con-trol system employing smaller and inexpensive CMGs,called mini-CMGs, is of practical importance for devel-oping future agile scientific spacecraft (Refs. 18-21) aswell as small agile satellites (Ref. 22).

However, the use of CMGs necessitates the develop-ment of CMG steering logic which generates the CMGgimbal rate commands in response to the CMG torquecommands. One of the principal difficulties in usingCMGs for spacecraft attitude control and momentummanagement is the geometric singularity problem inwhich no control torque is generated for the commandedcontrol torques along a particular direction. At such asingularity, CMG torque is available in all but one di-rection.

The CMG singularity problem, studied previously byMargulies and Aubrun (Ref. 2) and Bedrossian et al.(Refs. 5 and 6), has been further examined recentlyin Ref. 23 to characterize and visualize the physical aswell as mathematical nature of the singularities, singularmomentum surfaces, and other singularity related prob-lems.

A simple yet effective way of passing through, and alsoescaping from, any internal singularities, as applied toagile spacecraft pointing control, has been developed inRefs. 13-16. The patented steering logic of Wie, Bai-ley, and Heiberg (Ref. 14) is mainly intended for typicalreorientation maneuvers in which precision pointing or

1

AIAA Guidance, Navigation, and Control Conference and Exhibit11-14 August 2003, Austin, Texas

AIAA 2003-5659

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

x

αi

β i

ith CMG Momentum Vector

Inner Gimbal Axis

(−α , β = 0)axis

Outer Gimbal Axis(β , α = 0)axis

Rotor Axis(α = β = 0)

H i

z

y

tracking is not required during reorientation maneuvers,and it fully utilizes the available CMG momentum spacein the presence of any singularities. Although there arespecial missions in which prescribed attitude trajectoriesare to be “exactly” tracked in the presence of internalsingularities, most practical cases, however, will requirea tradeoff between robust singularity transit/escape andthe resulting, transient pointing errors.

Because the singularity robust steering logic devel-oped in Refs. 13-16 is based on the minimum two-norm,pseudoinverse solution, it does not explicitly avoid sin-gularity encounters but it rather approaches and rapidlytransits unavoidable singularities whenever needed. Iteffectively generates deterministic dither signals whenthe system becomes near singular. Any internal singu-larities can be escaped for any nonzero constant torquecommands using the singularity robust steering logic.

However, this paper will show that the patented CMGsteering logic of Wie, Bailey, and Heiberg (Refs. 13-16)is unable to escape the saturation singularities of cer-tain CMG configurations, which can be problematic forCMG momentum desaturation. Consequently, a newsteering logic is developed to eliminate such a saturationsingularity escape problem of the CMG steering logic de-veloped in Refs. 13-16. Furthermore, the new steeringlogic provides an effective means of explicitly avoiding,instead of passing through, the internal elliptic singu-larities which are commonly encountered by most otherpseudoinverse-based steering logic.

The remainder of this paper is briefly outlined as fol-lows. In Section 2, a summary of various CMG systemswill be presented. Section 3 will provide a brief summaryof the standard pseudoinverse-based steering logic. Sec-tion 4 will present the new singularity escape/avoidancesteering logic. In Section 5, the practicality and effec-tiveness of the new steering logic will be demonstratedusing various CMG systems, such as a pyramid array offour single-gimbal CMGs, two and three parallel single-gimbal CMG configurations, and four parallel double-gimbal CMG configuration of the International SpaceStation.

2. Control Moment Gyro Systems

This section provides a summary of several represen-tative CMG systems. These CMG systems will be usedin Section 5 to demonstrate the simplcity, practicality,and effectiveness of a new singularity escape/avoidancesteering logic (Ref. 24).

Pyramid Array of Four Single-Gimbal CMGs(see Ref. 23 for details)

Two and Three Parallel Single-Gimbal CMGConfigurations (see Ref. 23 for details)

Four Parallel Double-Gimbal CMGsFor a double-gimbal control moment gyro (DGCMG),

Figure 1: Inner gimbal angle αi and outer gimbal angleβi of the ith double-gimbal CMG.

the rotor is suspended inside two gimbals, and conse-quently the rotor momentum can be oriented on a spherealong any direction provided no restrictive gimbal stops.For the different purposes of redundancy managementand failure accommodation, several different arrange-ments of DGCMGs have been developed, such as threeorthogonally mounted DGCMGs used in the Skylab andfour parallel mounted DGCMGs employed for the Inter-national Space Station.

As shown by Kennel (Ref. 4), mounting of DGCMGsof unlimited outer gimbal angle freedom with all theirouter gimbal axes parallel allows drastic simplificationof the CMG steering law development in the redun-dancy management and failure accommodation and inthe mounting hardware.

Consider such a parallel mounting arrangement of fourdouble-gimbal CMGs with the inner and outer gimbalangles, αi and βi of the ith CMG, as defined in Figure 1.The total CMG momentum vector H = (Hx, Hy, Hz) isexpressed in the (x, y, z) axes as

H =

∑sinαi∑

cos αi cos βi∑cos αi sinβi

(1)

where a constant unit momentum is assumed for eachCMG. The time derivative of H becomes

H =

∑cos αiαi∑

(− sinαi cos βiαi − cos αi sinβiβi)∑(− sinαi sinβiαi + cos αi cos βiβi)

(2)

Note that the x-axis torque component is not a func-tion of the outer gimbal βi motions. Consequently,in Kennel’s CMG steering law implemented on the In-ternational Space Station, the inner gimbal rate com-mands, αi, are determined first for the commanded x-axis torque, then the outer gimbal rate commands, βi,for the commanded y- and z-axis torques.

Typical singularities of a system of four paralleldouble-gimbal CMGs are illustrated in Figure 2. The

2

y y

(a) 4H Saturation Singularity (b) 2H Internal Singularity

z z

y

(c) 0H Internal Singularity

z

y

(d) No singularity

z

Figure 2: Singularities of a system of four paralleldouble-gimbal CMGs.

4H saturation singularity is an elliptic singularity whichcannot be escaped by null motion. The 2H and 0H sin-gularities, shown in Figure 2(b) and (c) respectively, arehyperbolic singularities which can be escaped by nullmotion. A nonsingular configuration but with a zeromomentum is also shown in Figure 2(d).

The various CMG systems described in this sectionwill be used later to demonstrate the simplicity and ef-fectiveness of a new singularity escape/avoidance logicdeveloped in this paper.

3. Pseudoinverse Steering Logic

Ignoring the effect of spacecraft angular motion, wemay define the instantaneous torque vector, �τ , generatedby CMG gimbal motion xi as

�τ =d �H

dt=

n∑i=1

d�hi

dxixi (3)

or, in matrix form, as

τ =dHdt

=n∑

i=1

dhi

dxixi = Ax (4)

where A is a 3×n Jacobian matrix and x = (x1, · · · xn).For the given control torque command τ , the gimbal

rate command x, often referred to as the pseudoinversesteering logic, is then obtained as

x = A+τ (5)

whereA+ = AT (AAT )−1 (6)

This pseudoinverse is the minimum two-norm solutionof the following constrained minimization problem:

minx

||x||2 subject to Ax = τ (7)

where ||x||2 = xT x. Most CMG steering laws deter-mine the gimbal rate commands with some variant ofthe pseudoinverse of the form (6).

The pseudoinverse is a special case of the weightedminimum two-norm solution

x = A+τ where A+ = Q−1AT [AQ−1AT ]−1 (8)

of the following constrained minimization problem:

minx

||x||2Q subject to Ax = τ (9)

where ||x||2Q = xT Qx and Q = QT > 0. Later in thispaper, the significance of choosing Q �= I, where I is anidentity matrix, will be demonstrated.

If rank(A) < m for certain sets of gimbal angles, orequivalently rank(AAT ) < m, when A is an m × nmatrix, the pseudoinverse does not exist, and it is saidthat the pseudoinverse steering logic encounters singularstates. This singular situation occurs when all individ-ual CMG torque output vectors are perpendicular to thecommanded torque direction. Equivalently, the singularsituation occurs when all individual CMG momentumvectors have extremal projections onto the commandedtorque τ .

Since the pseudoinverse, A+ = AT (AAT )−1, is theminimum two-norm solution of gimbal rates subject tothe constraint Ax = τ , the pseudoinverse steering logicand all other pseudoinverse-based steering logic tend toleave inefficiently positioned CMGs alone causing thegimbal angles to eventually “hang-up” in anti-parallelsingular arrangements. That is, they tend to steer thegimbals toward anti-parallel singular states. Despite thisdeficiency, the pseudoinverse steering logic, or some vari-ant of pseudoinverse, is commonly employed for mostCMG systems because of its simplicity for onboard, real-time implementation.

4. Singularity Escape and Avoid-ance Steering Logic

A simple yet effective way of passing through, and alsoescaping from, any internal singularities was developedby Wie, Bailey, and Heiberg (Refs. 13-16). Such a sin-gularity robust CMG steering logic is mainly intendedfor typical reorientation maneuvers in which precisionpointing or tracking is not required during reorienta-tion maneuvers, and it fully utilizes the available CMGmomentum space in the presence of any singularities.Although there are special missions in which prescribedattitude trajectories are to be “exactly” tracked in thepresence of internal singularities, most practical cases,

3

however, will require a tradeoff between robust singular-ity transit/escape and the resulting, transient pointingerrors.

The singularity robust steering logic described inRefs. 13-14 has the following form: x = A#τ where

A# = [AT PA + λI]−1AT P

= AT [AAT + λE]−1 (10)

and

P−1 ≡ E =

1 ε3 ε2ε3 1 ε1ε2 ε1 1

> 0 (11)

The positive scalar λ and the off-diagonal elements εi

are to be properly selected such that A#τ �= 0 for anynonzero constant τ .

Note that there exists always a null vector of A# sincerank(A#) < 3 for any λ and εi when the Jacobian matrixA is singular. Consequently, a simple way of guarantee-ing that A#τ �= 0 for any nonzero constant τ commandis to continuously modulate εi, for example, as follows:

εi = ε0 sin(ωt + φi) (12)

where the amplitude ε0, the modulation frequency ω,and the phases φi need to be appropriately selected. Thescalar λ may be adjusted as:

λ = λo exp(−µdet(AAT )) (13)

where λo and µ are constants to be properly selected.It is emphasized that the singularity robust inverse

of the form (10) is based on the mixed, two-norm andweighted least-squares minimization although the result-ing effect is somewhat similar to that of artificially mis-aligning the commanded control torque vector from thesingular vector directions. Because the singularity ro-bust steering logic is based on the minimum two-norm,pseudoinverse solution, it does not explicitly avoid sin-gularity encounters but it rather approaches and rapidlytransits unavoidable singularities whenever needed. Thesteering logic effectively generates deterministic dithersignals when the system becomes near singular. Anyinternal singularities can be escaped for any nonzeroconstant torque commands using the singularity robuststeering logic.

The singularity robust steering logic of Wie, Baileyand Heiberg (Refs. 13-14) provides a simple yet effectiveway of passing through, and also escaping from, any in-ternal singularities. However, it has been recently foundthat a certain type of external saturation singularitiescannot be escaped for any choice of λ and εi.

Consequently, a new singularity escape/avoidancelogic (Ref. 24), which is capable of escaping/avoidingall types of singularities, is proposed as follows:

x = A#τ (14)

where

A# = [AT PA + Q]−1AT P

≡ Q−1AT [AQ−1AT + P−1]−1

≡ WAT [AWAT + V]−1 (15)

where W ≡ Q−1 and V ≡ P−1. For a 3 × n Jacobianmatrix A, [AT PA+Q] is an n×n matrix and [AWAT +V] is a 3 × 3 matrix.

The singularity robust inverse of the form (15) is thesolution of the well-known, mixed two-norm and least-squares minimization problem:

minx

(eT Pe + xT Qx) (16)

where e = Ax−τ is the torque error vector. Because Pand Q are positive definite matrices, [AT PA + Q] and[AWAT + V] are always nonsingular.

The weighting matrices, P and Q (equivalently, Wand V), must be properly chosen: (i) to obtain accept-able levels of torque errors and gimbal rates, (ii) to es-cape any internal as well as external singularities, and(iii) to pass through singularities or avoid singularity en-counters.

For example, P is chosen such that AT Pτ �= 0 for anytorque command τ , as follows:

P−1 ≡ V = λ

1 ε3 ε2ε3 1 ε1ε2 ε1 1

> 0 (17)

andλ = λo exp(−µdet(AAT )) (18)

εi = ε0 sin(ωt + φi) (19)

where the constant parameters, λ0, µ, ε0, ω, and φi,need to be appropriately selected.

Because the condition AT Pτ �= 0, which is a nec-essary condition, is not sufficient for escaping and/oravoiding singularities, the matrix Q also needs to beproperly chosen, as follows:

Q−1 ≡ W =

W1 λ λ λλ W2 λ λλ λ W3 λλ λ λ W4

> 0 (20)

Different values of Wi and/or nonzero off-diagonal ele-ments are needed to be able to escape all types of singu-larities, including the external saturation singularities.Furthermore, a proper choice of W �= I provides aneffective means for “explicitly” avoiding singularity en-counters (to be demonstrated in the next section).

Several examples of demonstrating the significance ofW �= I for escaping a certain type of external saturationsingularities for which the previous singularity robuststeering logic of Wie, Bailey and Heiberg (Refs. 13-14)fails will be presented in the next section. Examples

4

of avoiding an internal elliptic singularity for which theprevious singularity robust steering logic had to passthrough will also be presented in the next section.

5. Illustrative Examples

In this section, the simplicity and effectiveness of thenew steering logic will be demonstrated using variousCMG systems, such as two and three parallel single-gimbal CMG configurations, a pyramid array of foursingle-gimbal CMGs, and four parallel double-gimbalCMG configuration of the International Space Station.

Two Parallel Single-Gimbal CMGsFor a system of two parallel single-gimbal CMGs, we

have the CMG momentum vector H as

H = 2[

cos α cos βsin α cos β

](21)

and the Jacobian matrix A as

A = 2[

− sinα cos β − cos α sinβcos α cos β − sinα sinβ

](22)

where α is called the “rotation” angle and β the “scissor”angle.

For the internal anti-parallel singularity at x =(α, β) = (0, π/2), the Jacobian matrix becomes

A =[

0 −20 0

]

with its singular vector of u = (0, 1); i.e., AT u = 0.A null space vector of A such that Ax = 0 is: x =null(A) = (1, 0).

Although null motion does exist from this singular-ity as discussed in Ref. 23, this hyperbolic singularitycannot be escaped by null motion because the singularconfiguration (β = π/2) remains undisturbed during thisnull motion along the null manifold (or a degenerate tra-jectory). Although null motion can be generated at thishyperbolic singularity, this example demonstrates thatthe mere existence of null motion does not guaranteeescape from a singularity (Refs. 5 and 23).

However, this type of internal singularity can be es-caped by the singularity robust steering logic of Refs. 13-14.

For the saturation singularity at x = (α, β) = (0, 0),the Jacobian matrix becomes

A =[

0 02 0

]

with its singular vector of u = (−1, 0). Its null spacevector is x = null(A) = (0, 1).

This external elliptic singularity cannot be escapedby null motion because null motion does not exist inthe vicinity of this momentum saturation singularity, asdiscussed in Refs. 5 and 23.

The singularity robust inverse of Refs. 13-14 with λ =0.1 and ε = 0.1 becomes

A# = AT [AAT + 0.1E]−1 =[

−0.0488 0.48790 0

]

⇒ x = A#

[−1

0

]=

[0.0488

0

]

Note that x = (α, β) �= 0, but β = 0; i.e., the systemremains singular while α changes. In fact, β = 0 for anytorque command vector τ and any εi. Consequently, thesingularity robust steering logic of Refs. 13-14 is unableto command a none-zero β for this special case.

However, this external singularity can be escapedby the singularity escape/avoidance steering logic, pro-posed in this paper, of the form: x = A#τ where

A# = WAT [AWAT + V]−1

As can be seen in Figure 3, the momentum saturationsingularity is escaped using the proposed steering logicwith

V = λ

[1 εε 1

], W =

[1 λλ 1

]

where

ε = 0.1 cos t

λ = 0.01 exp(−10 det(AAT ))

In Figure 3, (ux, uy) represent the actual torques gen-erated by the CMG system for the commanded torquesof (τx, τy) = (−1, 0) for 0 ≤ t ≤ 3 sec. The CMG mo-mentum is completely desaturated at t = 2 sec, but thedesaturation torque is commanded until t = 3 sec tofurther explore any problem of encountering an internalsingularity. (For all of the simulation results presentedin this paper, the commanded torque level as well asthe CMG momentum is normalized as one without lossof generality. Thus it should be appropriately adjusteddepending on the actual values of gimbal rate limit andCMG momentum magnitude.) As can be seen in Figure3, the proposed steering logic also provides a simple yeteffective way of passing through the 0H internal hyper-bolic singularity, but with a small transient torque errorin uy.

As discussed in Ref. 1, a steering logic based on thedirect inverse of A can be obtained for this simple 2× 2problem, as follows:

x = A−1τ (23)

where

A−1 =1

2|A|

[− sinα cos β − cos α sin βcos α cos β − sinα sinβ

]

=12

[− sinα/ cos β cos α/ cos β− cos α/ sinβ − sinα/ sin β

](24)

5

0 1 2 3 40

0.5

1

1.5

2

2.5

Time (sec)

α (

deg)

0 1 2 3 40

50

100

150

Time (sec)

β (d

eg)

0 1 2 3 4-1.5

-1

-0.5

0

Time (sec)

u x

0 1 2 3 4-0.05

0

0.05

0.1

0.15

Time (sec)

u y

0 1 2 3 40

1

2

3

4

Time (sec)

det(

AA

T)

0 1 2 3 40

0.5

1

1.5

2

Time (sec)

CM

G M

omen

tum

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

Time (sec)

λ

0 1 2 3 4-0.1

-0.05

0

0.05

0.1

Time (sec)

ε

Figure 3: Singularity escape (momentum desaturation)simulation results for two parallel single-gimbal CMGswith the proposed new steering logic.

Then we have

α =− sinα τx + cos α τy

2(cos β + λ1)(25)

β =− cos α τx − sin α τy

2(sinβ + λ2)(26)

where

λ1 = λo exp(−µdet(AAT )) sinβ (27a)λ2 = λo exp(−µdet(AAT )) cos β (27b)

which are included in the steering law to avoid dividingby zero when the system becomes singular.

As shown in Figure 4, the momentum saturation sin-gularity is escaped using this direct-inverse steering logicfor (τx, τy) = (−1, 0) for 0 ≤ t ≤ 3 sec with λ0 = 0.01and µ = 10. In Figure 4, (ux, uy) represent the ac-tual torques generated by the CMG system for the com-manded torques: (τx, τy) = (−1, 0) for 0 ≤ t ≤ 3 sec.

0 1 2 3 4-1

-0.5

0

0.5

1

Time (sec)

α (

deg)

0 1 2 3 40

50

100

150

Time (sec)

β (d

eg)

0 1 2 3 4-1.5

-1

-0.5

0

Time (sec)

u x

0 1 2 3 4-1

-0.5

0

0.5

1

Time (sec)

u y

0 1 2 3 40

1

2

3

4

Time (sec)

det(

AA

T)

0 1 2 3 40

0.5

1

1.5

2

Time (sec)

CM

G M

omen

tum

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

Time (sec)

λ 1

0 1 2 3 4-5

0

5

10x 10

-3

Time (sec)λ 2

Figure 4: Singularity escape (momentum desatura-tion) simulation results for two parallel single-gimbalCMGs with the direct-inverse steering logic of Crenshaw(Ref. 1).

It can be seen that this direct inverse steering logic pro-vides a very smooth transit through the 0H internal sin-gularity, without any torque transient error. However,it is not always possible to find such an exact, direct-inverse steering logic for a general system of redundantCMGs.

Three Parallel Single-Gimbal CMGsFor a system of three single-gimbal CMGs with par-

allel gimbal axes, W should be selected as W1 �=W2 �= W3 to escape the 3H saturation singularity at(x1, x2, x3) = (0, 0, 0) with (τx, τy) = (−1, 0). Simu-lation results of (τx, τy) = (−1, 0) for 0 ≤ t ≤ 3 secwith W = diag{1, 2, 3} are shown in Figure 5. Detailedsingularity analyses of such a system of three parallelsingle-gimbal CMGs can be found in Ref. 23.

6

0 1 2 3 4-10

0

10

20

Time (sec)

x 1 (de

g)

0 1 2 3 4-150

-100

-50

0

50

Time (sec)

x 2 (de

g)

0 1 2 3 40

50

100

150

Time (sec)

x 3 (de

g)

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

Time (sec)

λ

0 1 2 3 4-1.5

-1

-0.5

0

Time (sec)

u x

0 1 2 3 4-0.05

0

0.05

0.1

0.15

Time (sec)

u y

0 1 2 3 40

0.5

1

1.5

2

2.5

Time (sec)

det(

AA

T)

0 1 2 3 40

1

2

3

Time (sec)

CM

G M

omen

tum

Figure 5: Singularity escape (momentum desaturation)simulation results for three parallel single-gimbal CMGswith the proposed singularity escape steering logic ofW = diag{1, 2, 3}.

Pyramid Array of Four Single-Gimbal CMGsFirst consider a special case (β = π/2) of the pyramid

array of four single-gimbal CMGs. Such two orthogonalpairs of two parallel single-gimbal CMGs is of practicalimportance due to its simple arrangement of four single-gimbal CMGs for three-axis control applications (Refs. 1and 23).

Similar to the case of three parallel single-gimbalCMGs, W �= I is also needed to escape the saturationsingularity, as shown in Figure 6. For this simulation,W = diag{1, 2, 3, 4} was used to escape the saturationsingularity at (π/2, π/2, π/2, π/2) with (τx, τy, τz) =(0, 0,−1) for 0 ≤ t ≤ 4 sec. It can be seen in Figure6 that an internal singularity is encountered at t = 2sec, but it is rapidly passed through.

In Figure 7, simulation results are shown for a typicalpyramid array of four single-gimbal CMGs (β = 53.13

deg) with initial gimbal angles of (0, 0, 0, 0) and a torquecommand of (τx, τy, τz) = (1, 0, 0). For this simulation,W = diag{1, 1, 1, 1} was used; i.e., the patented steer-ing logic of Wie, Bailey and Heiberg (Refs. 13-14) wasused for this simulation. It can be seen that the well-known internal elliptic singularity at (−π/2, 0, π/2, 0) isencountered but it is successfully passed through. Theinevitable transient torque errors during the singularitytransit can be seen in Figure 7.

In Figure 8, simulation results with W =diag{10−4, 1, 1, 1} are shown for the previous case ofa typical pyramid array of four single-gimbal CMGs(β = 53.13 deg) with initial gimbal angles of (0, 0, 0, 0)and a torque command of (τx, τy, τz) = (1, 0, 0). It canbe seen that the well-known internal elliptic singularityat (−π/2, 0, π/2, 0) is not encountered. However, the in-evitable transient torque errors, caused by skirting sucha troublesome, impassable elliptic singularity, can alsobe seen in Figure 8.

For the simulations shown in Figures 7 and 8, the max-imum gimbal rate of each CMG was explicitly imposedas ±2 rad/sec. Such a gimbal rate saturation limit didnot affect the singularity-avoidance performance of theproposed steering logic.

One may compare these simulation results (Figures 7and 8 ) with the various simulation results of Bedrossianet al. (Ref. 6) and Paradiso (Ref. 8). The simplicityand practicality of the proposed steering logic are em-phasized here again, compared to the complexity andreal-time implementation problem of various “explicit”singularity-avoidance steering logic employing gradientmethods, global search, optimization, and/or null mo-tion.

Four Parallel Double-Gimbal CMGsThe proposed steering logic can also be employed for

a system of four parallel double-gimbal CMGs describedby Ax = τ where x = (α1, α2, α3, α4, β1, β2, β3, β4), τ =(τx, τy, τz), and A is a 3 × 8 Jacobian matrix which canbe easily constructed from (2).

The saturation singularity escape capability of theproposed steering logic is demonstrated in Figure 9. Thesimulation conditions for this case are:

Initial gimbal angles: αi = βi = 0 for all i

Commanded torque: (τx, τy, τz) = (0, 1, 0)W = diag{1, 1, 2, 2, 1, 1, 2, 2}λ = 0.01 exp(−det(AAT ))ε = 0.1 cos t

In Figure 9, we have α1 = α2 �= α3 = α4 andβ1 = β2 �= β3 = β4. For this case with W =diag{1, 1, 2, 2, 1, 1, 2, 2}, it escapes the saturation singu-larity but it passes through the 0H singularity, as shownin Figure 9. However, for W = I, this system remainssingular at its singular momentum envelope; i.e., thesteering logic of Refs. 13-14 is unable to escape the sat-uration singularity of this system of four double-gimbalCMGs.

7

0 2 4 690

92

94

96

98

100

Time (sec)

x 1 (de

g)

0 2 4 6-100

-50

0

50

100

Time (sec)

x 2 (de

g)

0 2 4 690

92

94

96

98

100

Time (sec)

x 3 (de

g)

0 2 4 6-100

-50

0

50

100

Time (sec)

x 4 (de

g)

0 2 4 60

0.5

1

1.5

2

2.5

Time (sec)

det(

AA

T)

0 2 4 60

1

2

3

4

Time (sec)

CM

G M

omen

tum

0 2 4 6-0.1

-0.05

0

0.05

0.1

0.15

Time (sec)

u x

0 2 4 6-0.05

0

0.05

0.1

0.15

Time (sec)

u y

0 2 4 6-1.5

-1

-0.5

0

Time (sec)

u z

0 2 4 60

0.002

0.004

0.006

0.008

0.01

Time (sec)

λ

Figure 6: Singularity escape (momentum desaturation)simulation for four single-gimbal CMGs (β = π/2) withW = diag{1, 2, 3, 4}.

0 1 2 3 4-150

-100

-50

0

Time (sec)

x 1 (de

g)

0 1 2 3 4-150

-100

-50

0

50

Time (sec)

x 2 (de

g)

0 1 2 3 40

20

40

60

80

100

Time (sec)x 3 (

deg)

0 1 2 3 40

10

20

30

40

Time (sec)

x 4 (de

g)

0 1 2 3 40

0.5

1

1.5

2

Time (sec)

det(

AA

T)

0 1 2 3 40

0.5

1

1.5

2

2.5

Time (sec)

CM

G M

omen

tum

0 1 2 3 40

0.5

1

1.5

Time (sec)

u x

0 1 2 3 4-0.05

0

0.05

0.1

0.15

0.2

Time (sec)

u y

0 1 2 3 4-0.4

-0.3

-0.2

-0.1

0

Time (sec)

u z

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

Time (sec)

λ

Figure 7: Singularity transit simulation for four single-gimbal CMGs (β = 53.13 deg) with W = diag{1, 1, 1, 1}.

8

0 1 2 3 4-80

-60

-40

-20

0

Time (sec)

x 1 (de

g)

0 1 2 3 4-50

0

50

100

150

200

Time (sec)

x 2 (de

g)

0 1 2 3 40

50

100

150

Time (sec)

x 3 (de

g)

0 1 2 3 4-80

-60

-40

-20

0

Time (sec)

x 4 (de

g)

0 1 2 3 40

0.5

1

1.5

2

Time (sec)

det(

AA

T)

0 1 2 3 40

1

2

3

4

Time (sec)

CM

G M

omen

tum

0 1 2 3 40

0.5

1

1.5

Time (sec)

u x

0 1 2 3 4-0.02

0

0.02

0.04

0.06

Time (sec)

u y

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

Time (sec)

u z

0 1 2 3 40

1

2

3

4x 10

-4

Time (sec)

λ

Figure 8: Singularity avoidance simulation for foursingle-gimbal CMGs (β = 53.13 deg) with W =diag{10−4, 1, 1, 1}.

0 2 4 6-100

-50

0

50

100

Time (sec)

αi (

deg)

0 2 4 6-200

-100

0

100

200

Time (sec)

β i (de

g)

0 2 4 60

1

2

3

4

Time (sec)

CM

G M

omen

tum

0 2 4 60

5

10

15

Time (sec)

det(

AA

T)

0 2 4 6-0.15

-0.1

-0.05

0

0.05

0.1

Time (sec)

u x

0 2 4 60

0.5

1

1.5

Time (sec)

u y

0 2 4 6-0.15

-0.1

-0.05

0

0.05

Time (sec)

u z

0 2 4 60

0.002

0.004

0.006

0.008

0.01

Time (sec)λ

Figure 9: Momentum desaturation simulation resultsfor escaping the 4H saturation singularity but passingthrough the 0H singularity of four parallel double-gimbalCMGs with W = diag{1, 1, 2, 2, 1, 1, 2, 2}.

Furthermore, for W = diag{1, 2, 3, 4, 1, 2, 3, 4}, it canbe seen in Figure 10 that the 0H singularity is completelyavoided with no transient torque errors. The CMG mo-mentum is completely desaturated at t = 4 sec, but thedesaturation torque is continuously commanded even af-ter t = 4 sec to further check any problem of encounter-ing an internal singularity.

This example of a system of four double-gimbal CMGsdemonstrates that it is quite feasible to completely avoidsingularity encounters (with no transient torque errors)with the proper selection of W, in conjunction with thedeterministic dither signals generated in V.

For the parallel mounting arrangement of four double-gimbal CMGs, the x-axis torque component is not afunction of the outer gimbal βi motions, as can be seen in(2). Consequently, the inner gimbal rate commands canbe determined first for the commanded torque τx, then

9

0 2 4 6-200

-100

0

100

Time (sec)

αi (

deg)

0 2 4 6-200

-100

0

100

200

Time (sec)

β i (de

g)

0 2 4 60

1

2

3

4

Time (sec)

CM

G M

omen

tum

0 2 4 60

5

10

15

Time (sec)

det(

AA

T)

0 2 4 6-0.15

-0.1

-0.05

0

0.05

Time (sec)

u x

0 2 4 60

0.5

1

1.5

Time (sec)

u y

0 2 4 6-0.15

-0.1

-0.05

0

0.05

Time (sec)

u z

0 2 4 60

0.002

0.004

0.006

0.008

0.01

Time (sec)

λ

Figure 10: Momentum desaturation simulation resultsfor escaping the saturation singularity and also avoidingthe 0H singularity of four parallel double-gimbal CMGswith W = diag{1, 2, 3, 4, 1, 2, 3, 4}.

the outer gimbal rate commands for (τy, τz). In partic-ular, Kennel’s CMG steering law distributes the CMGangular momentum vectors such that all inner gimbalangles are equal which reduces the rate requirements onthe outer gimbals, as follows:

αi =τx∑cos αi

+ k(α∗ − αi) (28)

where k is called the inner gimbal angle distribution gainand α∗ is the desired inner gimbal angle for all CMGschosen as

α∗ =∑

(cos αi)αi∑cos αi

(29)

The proposed steering logic can then be incorporatedwith Kennel’s steering law to generate the outer gimbalrate commands, βi, using a reduced 2× 4 Jacobian ma-

trix, without requiring Kennel’s virtual CMG for avoid-ing 2H-singularity encounters.

6. Summary and Remarks

The proposed CMG steering logic is based on the well-known, mixed weighted two-norm and least-squares op-timization solution with the weighting matrices W andV. The proper use of W, in conjunction with the de-terministic dither signals in V, eliminates the saturationsingularity escape problem of the patented steering logicof Wie, Bailey, and Heiberg (Refs. 13-14). Either W(i.e., the weighted pseudoinverse solution) or V alonedoes not completely eliminate the singularity problem;i.e., both W and V are required for completely resolvingthe singularity problem.

As demonstrated in this paper, some internal ellip-tic singularities can be explicitly avoided or skirtedby the proper selection of the weighting matrix W.However, the proposed steering logic rather approachesand rapidly transits unavoidable singularities wheneverneeded.

A trial-and-error approach was used for choosingproper W’s of several examples in this paper. A system-atic or optimal way of determining the weighting matrixW, whose diagonal elements can be time-varying, needsto be further developed.

The proposed steering logic was not mathematicallyproved to be escaping and/or avoiding all types ofsingularities. However, the proposed steering logicis simple to implement, compared to many other ex-plicit singularity-avoidance algorithms employing gradi-ent methods, global search, optimization, and/or nullmotion.

The transient torque errors inherent to the proposedsteering logic during singularity escapes or transits maybe acceptable in most practical situations in whichprecision pointing or tracking is not required duringCMG momentum desaturation, CMG reconfiguration,or spacecraft reorientation maneuvers. Except specialmissions in which prescribed attitude trajectories areto be “exactly” tracked in the presence of singularityencounters, the transient pointing errors caused by sin-gularity escapes or transits may be acceptable in mostpractical situations with an outer attitude control loop(Ref. 16).

7. Conclusions

A new steering logic, which eliminates the saturationsingularity escape problem of the previously patentedsteering logic, has been presented. The proposed steer-ing logic utilizes an additional weighting matrix, W, inconjunction with the deterministic dither signals in theweighting matrix V of the patented steering logic. It canalso be employed to explicitly avoid singularity encoun-ters for which most other pseudoinverse-based steering

10

logic either fail or must transit through. The practi-cality, effectiveness, and simplicity of the proposed sin-gularity escape/avoidance logic, even in the presence ofgimbal-rate limit, were demonstrated for various CMGsystems, including the four parallel double-gimbal CMGsystem employed on the ISS.

References

[1] Crenshaw, J., “2-SPEED, A Single-Gimbal ControlMoment Gyro Attitude Control System,” Paper No. 73-895, presented at AIAA Guidance and Control Confer-ence, Key Biscayne, FL, August 20-22, 1973.[2] Margulies, G. and Aubrun, J. N., “Geometric Theoryof Single-Gimbal Control Moment Gyro Systems,” Jour-nal of the Astronautical Sciences, Vol. 26, No. 2, 1978,pp. 159-191.[3] Cornick, D. E., “Singularity Avoidance ControlLaws for Single Gimbal Control Moment Gyros,” Pa-per No. 79-1698, Proceedings of the AIAA Guidance andControl Conference, Boulder, CO, August 1979, pp. 20-33.[4] Kennel, H. F., “Steering Law for Parallel MountedDouble-Gimballed Control Moment Gyros: Revision A,”NASA TM-82390, January 1981.[5] Bedrossian, N. S., Paradiso, J., Bergmann, E. V., andRowell, D., “Redundant Single-Gimbal Control MomentGyroscope Singularity Analysis,” Journal of Guidance,Control, and Dynamics, Vol. 13, No. 6, 1990, pp. 1096-1101.[6] Bedrossian, N. S., Paradiso, J., Bergmann, E. V.,and Rowell, D., “Steering Law Design for RedundantSingle-Gimbal Control Moment Gyroscopes,” Journal ofGuidance, Control, and Dynamics, Vol. 13, No. 6, 1990,pp. 1083-1089 (also, N. S. Bedrossian’s M.S. thesis atMIT, August 1987).[7] Vadali, S. R., Oh, H., and Walker, S., “PreferredGimbal Angles for Single Gimbal Control Moment Gy-roscopes,” Journal of Guidance, Control, and Dynamics,Vol. 13, No. 6, 1990, pp. 1090-1095.[8] Paradiso, J. A., “Global Steering of Single GimballedControl Moment Gyroscopes Using a Direct Search,”Journal of Guidance, Control, and Dynamics, Vol. 15,No. 5, 1992, pp. 1236-1244.[9] Schaub, H. and Junkins, J. L., “Singularity Avoid-ance Using Null Motion and Variable-Speed Control Mo-ment Gyros,” Journal of Guidance, Control, and Dy-namics, Vol. 23, No. 1, 2000, pp. 11-16.[10] Ford, K. A. and Hall, C. D., “Singular DirectionAvoidance Steering for Control Moment Gyros,” Journalof Guidance, Control, and Dynamics, Vol. 23, No. 4,2000, pp. 648-656.[11] Heiberg, C. J., Bailey, D., and Wie, B., “PrecisionSpacecraft Pointing Using Single-Gimbal Control Mo-ment Gyros with Disturbance,” Journal of Guidance,Control, and Dynamics, Vol. 23, No. 1, 2000, pp. 77-85.[12] Wie, B., Space Vehicle Dynamics and Control,

AIAA Education Series, AIAA, Washington, DC, 1998,Chapter 7.[13] Wie, B., Heiberg, C., and Bailey, D., “Singular-ity Robust Steering Logic for Redundant Single-GimbalControl Moment Gyros,” Journal of Guidance, Control,and Dynamics, Vol. 24, No. 5, 2001, pp. 865-872.[14] Wie, B., Bailey, D., and Heiberg, C., “Robust Sin-gularity Avoidance in Satellite Attitude Control,” U.S.Patent 6,039,290, March 21, 2000.[15] Bailey, D., Heiberg, C., and Wie, B., “ContinuousAttitude Control That Avoids CMG Array Singulari-ties,” U.S. Patent 6,131,056, October 10, 2000.[16] Wie, B., Heiberg, C., and Bailey, D., “Rapid Multi-Target Acquisition and Pointing Control of Agile Space-craft,” Journal of Guidance, Control, and Dynamics,Vol. 25, No. 1, 2002, pp. 96-104.[17] Dominguez, J. and Wie, B., “Computation and Vi-sualization of Control Moment Gyroscope Singularities,”Paper No. 2002-4570, AIAA Guidance, Navigation, andControl Conference, Monterey, CA, August 5-8, 2002.[18] Roser, X., Sghedoni, M., “Control Moment Gyro-scopes (CMGs) and their Application in Future Scien-tific Missions,” Proceedings of 3rd ESA InternationalConference on Spacecraft Guidance, Navigation, andControl Systems, ESTEC, Noordwijk, The Netherlands,Nov. 26-29, 1996, pp. 523-528.[19] Busseuil, J., Llibre, M., and Roser, X., “High Pre-cision Mini-CMG’s and their Spacecraft Applications,”presented at AAS Guidance and Control-Advances inthe Astronautical Sciences, Breckenridge, CO., Febru-ary 1998.[20] Salenc, C. and Roger, X., “AOCS for Agile Sci-entific Spacecraft with mini-CMGs,” Proceedings of 4thESA International Conference on Spacecraft Guidance,Navigation, and Control Systems, ESTEC, Noordwijk,The Netherlands, Oct. 18-21, 1999, pp. 379-384.[21] Defendini, A, Lagadec, K., Guay, P., and Griseri,G., “Low Cost CMG-based AOCS Designs,” Proceed-ings of 4th ESA International Conference on SpacecraftGuidance, Navigation, and Control Systems, ESTEC,Noordwijk, The Netherlands, Oct. 18-21, 1999, pp. 393-398.[22] Lappas, V. J., Steyn, W. H., and Underwood, C. I.,“Attitude Control for Small Satellites Using Control Mo-ment Gyros,” presented at 52nd International Astronau-tical Federation Congress, Toulouse, France, Oct. 1-5,2001.[23] Wie, B., “Singularity Analysis and Visualization ofSingle-Gimbal Control Moment Gyro Systems,” PaperNo. 2003-5658, AIAA Guidance, Navigation, and Con-trol Conference, Austin, TX, August 11-14, 2003.[24] Wie, B., “Singularity Escape/Avoidance SteeringLogic for Control Moment Gyros,” Patent pending, In-vention Disclosure #M2-092, Arizona State University,June 13, 2002.

11