Singularity Analysis and Visualization ofSingle-Gimbal Control Moment Gyro Systems

Bong Wie∗

Arizona State UniversityTempe, AZ 85287-6106

AbstractThis paper examines the singularity problem inherent

to redundant single-gimbal control moment gyro (CMG)systems. It is intended to provide a comprehensivemathematical treatment of the CMG singularity prob-lem, expanding upon the previous work by Margulies,Aubrun, and Bedrossian. However, particular emphasisis placed on characterizing and visualizing the physicalas well as mathematical nature of the singularities, sin-gular gimbal angles, singular momentum surfaces, nullmotion manifolds, and degenerate null motions. Twoand three parallel single-gimbal CMG configurations anda typical pyramid array of four single-gimbal CMGs (in-cluding a special case of 90-deg skew angle) are examinedin detail throughout the paper to illustrate the variousconcepts and approaches useful for characterizing andvisualizing the CMG singularities.

1. Introduction

Control moment gyros (CMGs), as applied to space-craft attitude control and momentum management, havebeen extensively studied in the past (Refs. 1-7) and morerecently in Refs. 8-16. They have been successfully em-ployed mainly for large space vehicles, such as the Sky-lab, the MIR space station, and the International SpaceStation (ISS). However, CMGs have never been used incommercial satellites because their higher torque capa-bilities are not needed by most commercial satellites andalso because CMGs are much more expensive and me-chanically 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-

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

ing large angular momentum over long periods of time,they have been employed for attitude control and mo-mentum management of large space vehicles, such as theISS. Four parallel mounted double-gimbal CMGs with atotal weight of about 2400 lb and with a design life of10 years are employed on the ISS.

The next-generation Earth imaging satellites willrequire rapid rotational maneuverability for high-resolution images. Rather than sweep the gimbaledimaging system from side-to-side, the whole spacecraftbody (to which the imaging system rigidly attached) willturn rapidly. Pointing the entire spacecraft allows theimaging system to achieve a higher definition and im-proves the resolution for its images. Because the overallcost and effectiveness of such agile spacecraft are greatlyaffected by the average retargeting time, the develop-ment of a low-cost attitude control system employingsmaller and inexpensive CMGs, called mini-CMGs, is ofpractical importance for developing future agile scientificspacecraft (Refs. 17-20) as well as agile small satellites(Ref. 21).

However, the use of CMGs necessitates the develop-ment of CMG steering logic which generates the CMGgimbal rate commands for the commanded spacecraftcontrol torques. One of the principal difficulties in us-ing CMGs for spacecraft attitude control and momen-tum management is the well-known geometric singular-ity problem in which no control torque is generated forthe commanded control torque along a certain direction.At such a singularity, CMG torque is available in all butone particular direction.

As evidenced in Refs. 1 through 21, most CMG re-searchers have been focusing on the development ofsingularity-avoidance steering logic. However, this pa-per focuses on the singularity problem itself and is in-tended to provide its coherent, comprehensive mathe-matical treatment. This paper is also intended to pro-vide the reader with a summary of various different ap-proaches with many illustrative examples with signifi-cant new results. In particular, this paper will providethe comprehensive mathematical framework to CMG re-searchers with a renewed practical interest in developinga low-cost attitude control system employing smaller andinexpensive CMGs for small agile satellites (Refs. 17-21).

The remainder of this paper is outlined as follows. InSection 2, a summary of various single-gimbal CMG sys-

1

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

AIAA 2003-5658

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

SpacecraftReferenceFrame

CMG #1Gimbal Axis

CMG #2Gimbal AxisCMG #3

Gimbal Axis

CMG #4Gimbal Axis

23

4β

CMGPlatform

x

xx

x1

i

j

k

h 1

h2

h3

h 4

g2

4g 1g

3g

x

y

ztems will be presented. Such CMG systems will be usedthroughout the paper to illustrate the various conceptsand approaches useful for characterizing and visualiz-ing the CMG singularities. In Section 3, the singularmomentum surfaces will be characterized and visualizedusing a method introduced by Margulies and Aubrun(Ref. 2). Several illustrative examples with significantnew results will be presented. Section 4 will describe anew approach for analyzing the CMG singularity prob-lem. This new method is based on the Binet-Cauchyidentity. Its practicality as being applied to a pyra-mid array of four single-gimbal CMGs will be demon-strated. In Section 5, the CMG null motion problemwill be examined, expanding upon the previous work ofBedrossian et al. (Refs. 4 and 5). In particular, thedegenerate null motion will be examined in detail andseveral illustrative examples with significant new resultswill be presented.

2. Single-Gimbal CMG Systems

This section provides a summary of a few representa-tive single-gimbal CMG systems. These CMG systemswill be used as illustrative examples throughout the pa-per to illustrate the various concepts and approachesuseful for characterizing and visualizing the CMG sin-gularities.

Pyramid Array of Four Single-Gimbal CMGsA typical pyramid array of four single-gimbal CMGs

is illustrated in Figure 1. The total angular momentumvector, �H, of four single-gimbal CMGs is simply givenby

�H = �h1 + �h2 + �h3 + �h4 (1)

where �hi is the angular momentum vector of the ithCMG and it is assumed that |�hi| = 1 without loss ofgenerality. The ith gimbal angle, xi, describes rotationof �hi about the normalized gimbal-axis vector, �gi (|�gi| =1); i.e., �gi · �hi = 0 and �hi = �hi(xi).

For the pyramid mount of four single-gimbal CMGswith skew angle of β as illustrated in Figure 1, thegimbal-axis vectors can be simply represented as: �g1 =sinβ�i + cos β�k, �g2 = sinβ�j + cos β�k, �g3 = − sinβ�i +cos β�k, and �g4 = − sinβ�j + cos β�k.

The total CMG momentum vector �H is often ex-pressed in a spacecraft reference frame (x, y, z) with aset of orthogonal unit vectors {�i, �j, �k}, as follows:

�H = Hx�i + Hy

�j + Hz�k

=[

�i �j �k]

Hx

Hy

Hz

=

[�i �j �k

]H (2)

where H = (Hx, Hy, Hz) ≡ [Hx Hy Hz]T is the repre-sentation of the vector �H with respect to the basis vec-tors {�i, �j, �k}. Although the column vector H should bedistinguished from the vector �H itself, it is often called

Figure 1: Pyramid mounting arrangement of four single-gimbal CMGs.

a vector. However, the meaning should be clear fromthe context and in general we must be clear on what ismeant by a vector.

For the pyramid mount of four single-gimbal CMGswith skew angle of β, the total CMG momentum vectorcan be represented in matrix form as

H = h1(x1) + h2(x2) + h3(x3) + h4(x4)

=

−cβ sinx1

cos x1

sβ sinx1

+

− cos x2

−cβ sinx2

sβ sinx2

+

cβ sinx3

− cos x3

sβ sinx3

+

cos x4

cβ sinx4

sβ sinx4

where cβ ≡ cos β, sβ ≡ sinβ. Note that hi are periodicwith a period of 2π and that

d2hi

dx2i

= −hi ord2�hi

dx2i

= −�hi (3)

The differential of H becomes

dH = f1dx1 + f2dx2 + f3dx3 + f4dx4

= Adx (4)

where dx = (dx1, dx2, dx3, dx4), and A is the Jacobianmatrix defined as

A =[

f1 f2 f3 f4]

=

−cβ cos x1 sinx2 cβ cos x3 − sinx4

− sinx1 −cβ cos x2 sinx3 cβ cos x4

sβ cos x1 sβ cos x2 sβ cos x3 sβ cos x4

(5)

Equation (4) represents a linear mapping from dx =(dx1, dx2, dx3, dx4) to dH = (dHx, dHy, dHz). Conse-quently, we obtain

H = Ax (6)

where H = dH/dt and x = dx/dt.

2

i

j

h1

h2

x

y

x

y

= 0

= 90 deg

x

y

= 0

= 0

x1

x2

αβ

α

β

α

β

(a) 0H Anti-Parallel Singularity (b) 2H Parallel Singularity

2-SPEED Single-Gimbal CMG System (Ref. 1)A special case with β = π/2 was employed by

Crenshaw (Ref. 1) for the so-called 2-SPEED (TwoScissored Pair Ensemble, Explicit Distribution) single-gimbal CMG control system. For this configuration, wesimply have two orthogonal pairs of two parallel single-gimbal CMGs with a Jacobian matrix of the form

A =

0 sinx2 0 − sinx4

− sinx1 0 sinx3 0cos x1 cos x2 cos x3 cos x4

(7)

This special configuration is also of practical importance.Many other CMG configurations are in fact some vari-ants of this basic arrangement of two orthogonal pairsof two parallel CMGs, known as the 2-SPEED CMGsystem in the literature.

Two and Three Parallel Single-Gimbal CMGConfigurations

Two and three single-gimbal CMG configurations withparallel gimbal axes have also been studied for two-axiscontrol applications in Refs. 1-2. Consider first a casewith only two CMGs without redundancy. The momen-tum vectors, �H1 and �H2, move in the (x, y) plane normalto the gimbal axis, as shown in Figure 2. For such scis-sored single-gimbal CMGs, the total CMG momentumvector can be represented in matrix form as

H =[

cos x1 + cos x2

sinx1 + sin x2

](8)

where a constant unit momentum for each CMG is as-sumed.

Defining a new set of gimbal angles (α, β) as

α =x1 + x2

2, β =

x2 − x1

2(9)

where α is called the “rotation” angle and β the “scis-sor” angle (Ref. 1), we can express the CMG momentumvector as

H = 2[

cos α cos βsin α cos β

](10)

and we haveH = Ax (11)

where H = (Hx, Hy), x = (α, β), and A is the Jacobianmatrix defined as

A = 2[

− sin α cos β, − cos α sinβcos α cos β, − sinα sinβ

](12)

For a system of three single-gimbal CMGs with par-allel gimbal axes, the total CMG angular momentumvector is given by

H =[

cos x1 + cos x2 + cos x3

sin x1 + sinx2 + sin x3

](13)

Figure 2: Two single-gimbal CMGs with parallel gimbalaxes (Ref. 1).

and the Jacobian matrix for x = (x1, x2, x3) is

A =[

− sinx1 − sinx2 − sinx3

cos x1 cos x2 cos x3

](14)

The various CMG systems described in this sectionwill be used throughout the paper to illustrate the vari-ous concepts and approaches discussed in this paper.

3. Singularities and Singular Surfaces

This section briefly introduces a method developedby Margulies and Aubrun (Ref. 2) for analyzing and vi-sualizing the singular momentum surfaces. Illustrativeexamples are presented with significant new results.

Similar to �H expressed as in (2), an arbitrary vector�u can also be represented as

�u = ux�i + uy

�j + uz�k

=[

�i �j �k]

ux

uy

uz

=

[�i �j �k

]u (15)

where u = (ux, uy, uz) ≡ [ux uy uz]T .As introduced by Margulies and Aubrun (Ref. 2), let

�u be a unit vector of the punctured unit sphere definedas

S = {�u : |�u| = 1, �u �= ±�gi, i = 1, ..., n}where �gi is the gimbal-axis vector (|�gi| = 1). Such a unitvector along all possible directions in three-dimensional

3

space (except along the gimbal-axis directions) can beparameterized as

�u = ux�i + uy

�j + uz�k

= sin θ2�i − sin θ1 cos θ2

�j + cos θ1 cos θ2�k (16)

where θ1 and θ2 are the rotation angles of two-successiverotations about the x- and y-axes. The longitude andlatitude angles of spherical coordinates, commonly usedfor a unit vector description along all possible directionsin three-dimensional space, was found to be numericallyill-suited for visualization of the singular surfaces. (Notethat such a specific representation of �u as (16) was notprovided in Ref. 2.)

For a system of n single-gimbal CMGs, the differentialof the total CMG momentum vector becomes

d �H =n∑

i=1

d�hi =n∑

i=1

d�hi

dxidxi =

n∑i=1

�fidxi (17)

where �fi are unit tangent vectors defined as

�fi = �fi(xi) =d�hi

dxi= �gi × �hi (18)

Note that d �H and �fi are the equivalent vector repre-sentations of dH and fi, respectively, used in (4). Thethree vectors, {�fi, �gi,�hi}, form a set of orthogonal unitvectors rotating about each gimbal axis �gi. In Marguliesand Aubrun (Ref. 2), this set of orthonormal vectorsplays a major role in developing the geometric theory ofredundant single-gimbal CMGs.

The 3 × n Jacobian matrix A, introduced in the pre-ceding section, has maximum rank of 3. When the gim-bal axes are not arranged to be coplanar, the minimumrank of A is two. However, when rank(A) = 2, all �fi’sbecome coplanar and there exists a unit vector �u normalto that plane; i.e.,

�fi(xi) · �u ≡ fTi u = 0, i = 1, ..., n (19)

Consequently, for such a case, we have

d �H · �u = 0 (20)

and the CMG array cannot produce any momentum vec-tor change (or torque) along the direction of �u regard-less of the gimbal rates. Such a unit vector �u is calledthe singular vector, and a set of gimbal angles whenrank(A) = 2 is called the singular gimbal angles.

Because �fi is also orthogonal to �gi and |�fi| = 1, thesingularity condition (19) can be rewritten as

�fi = ± �gi × �u

|�gi × �u| (21)

Since �hi = �fi × �gi, the singularity condition for �hi(xi)can also be written as

�hi = ± (�gi × �u) × �gi

|�gi × �u| (22)

which leads to the following inner product of �hi and �u:

�hi · �u = ±|�gi × �u| (23)

The singular momentum vector, corresponding to thesingular vector �u and the singular gimbal angles x, isexpressed as

�H = �H(x(�u)) = �H(�u)

=∑

i

1si

(�gi × �u) × �gi

=∑

i

1si

[�u − �gi(�gi · �u)] (24)

where

si = �hi · �u = ±|�gi × �u| ≡ ±√

1 − (�gi · �u)2 (25)

Since �u = �u(θ1, θ2), we have �H = �H(θ1, θ2); i.e., �H isalso parameterized by θ1 and θ2. Singular momentumsurfaces can be directly obtained using (24) without re-course to singular gimbal angles.

Example 1: Two Single-Gimbal CMGsThe singular momentum vector of the two single-

gimbal CMGs with parallel gimbal axes (see Figure 2)can be found as

�H =2∑

i=1

1si

(�gi × �u) × �gi

= ± �u ± �u = 0 or ± 2�u (26)

since si = ±1 and �gi = �k for i = 1, 2. The singularmomentum vector is then described by

�H = Hx�i + Hy

�j

= ± 2�u = ±2(cos θ�i + sin θ�j); 0 ≤ θ ≤ 2π

orH2

x + H2y = 4 (27)

which represents a circle on the (Hx, Hy)-plane.

Example 2: 2-SPEED CMG SystemConsider a special case (β = 90 deg) of the pyramid

configuration of four single-gimbal CMGs, called a 2-SPEED system by Crenshaw (Ref. 1). For such twoorthogonal pairs of two parallel single-gimbal CMGs, thegimbal-axis vectors are simply represented by �g1 = �i,�g2 = �j, �g3 = −�i, and �g4 = −�j. Defining a unit vector as(16), we obtain the singular momentum vector as

�H = �H(θ1, θ2) =2(uy

�j + uz�k)√

u2y + u2

z

+2(ux

�i + uz�k)√

u2x + u2

z

(28)

where ux = sin θ2, uy = − sin θ1 cos θ2, and uz =cos θ1 cos θ2. The momentum saturation surface, also

4

-2

0

2

-2

0

2-4

-3

-2

-1

0

1

2

3

4

Hx

Hy

Hz

Figure 3: Momentum saturation surface (4H singularitysurface) for the 2-SPEED system of four single-gimbalCMGs (β = 90 deg). This momentum saturation surfacehas four circular holes (windows) and is flat along theedges of its circular windows.

often called the “4H” singularity surface, shown in Fig-ure 3, has four circular holes (windows) and is flat alongthe edges of its windows.

Similarly, we can also obtain the so-called “0H” sin-gularity surface as shown in Figure 4. The “0H” sur-face represents a singularity condition where two CMGsare aligned along the desired momentum vector direc-tion and the other two CMGs in the opposite direction.The 0H surface does not necessarily mean a zero totalmomentum. Only if they are anti-parallel, the total mo-mentum becomes zero (i.e., total H = 0). In a compositesingularity surface plot, consisting of both the 0H and4H surfaces, the four trumpet-like funnels of the 0H sur-face are smoothly patched to the 4H saturation surfacealong the edges of the circular windows.

There are also 2H singular curves consisting of twoperpendicular circles described by

H2x + H2

z = 4 (Hy = 0)H2

y + H2z = 4 (Hx = 0)

Example 3: Pyramid Array with Skew Angle ofβ

For a typical pyramid configuration of four single-gimbal CMGs with skew angle of β, we have

s1 = ±√

1 − (sβ ux + cβ uz)2

s2 = ±√

1 − (sβ uy + cβ uz)2

-2

0

2

-2

-1

0

1

2

-1.5

-1

-0.5

0

0.5

1

1.5

Hx

Hy

Hz

Figure 4: Internal anti-parallel 0H singularity surfacefor the 2-SPEED system of four single-gimbal CMGs(β = 90 deg).

s3 = ±√

1 − (−sβ ux + cβ uz)2

s4 = ±√

1 − (−sβ uy + cβ uz)2

Furthermore, analytic expressions for the singular mo-mentum surfaces, (Hx, Hy, Hz), can be obtained as

Hx = cβ(−sβ uz + cβ ux)/s1 + ux/s2

+ cβ(sβ uz + cβ ux)/s3 + ux/s4

Hy = uy/s1 − cβ(sβ uz − cβ uy)/s2 + uy/s3

+ cβ(sβ uz + cβ uy)/s4

Hz = sβ(−cβ ux + sβ uz)/s1 + sβ(sβ uz − cβ uy)/s2

+ sβ(sβ uz + cβ ux)/s3 + sβ(sβ uz + cβ uy)/s4

where ux = sin θ2, uy = − sin θ1 cos θ2, and uz =cos θ1 cos θ2.

For a case of β = 54.7 deg, the 4H, 2H, and 0H singu-larity surfaces can be obtained as provided in Ref. 16. Ina composite singularity surface plot, consisting of boththe 2H and 4H singularity surfaces, the eight trumpet-like funnels of the 2H surface are smoothly patched tothe 4H saturation surface along the edges of the circularwindows.

4. Singularity Analysis Usingthe Binet-Cauchy Identity

In the previous section, a method developed by Mar-gulies and Auburn (Ref. 2) was used to analyze and vi-sualize the singular momentum surfaces. A different ap-proach, called the cutting-plane method, for determin-ing the singular momentum surfaces can also be foundin Stocking and Meffe (Ref. 22). These two techniques

5

do not explicitly require the singular gimbal angle infor-mation to determine the singular momentum surfaces.

However, it is often preferred to determine all possiblesingular gimbal angles (i) to understand and characterizeall possible singularities of CMG systems, (ii) to be usedfor a direct singularity-avoidance logic (e.g., Refs. 3, 7),or (iii) to determine singular momentum surfaces indi-rectly (Ref. 16).

The singularity condition (19) can be rewritten in ma-trix form as

AT u = 0 (29)

In other words, the singular vector u of the Jacobian ma-trix A is the null space vector of AT ; i.e., u = null(AT ).Nontrivial solutions, u �= 0, exist for (AAT )u = 0 if andonly if AAT is singular. That is, we have the singularitycondition of the form

det(AAT ) = 0 (30)

In general, the singularity condition defines a set of sur-faces in x-space, or equivalently, in H-space. The sim-plest singular state is the momentum saturation singu-larity characterized by the momentum envelope which isa three-dimensional surface representing the maximumavailable angular momentum of CMGs along any givendirection. Any singular state for which the total CMGmomentum vector is inside the momentum envelope iscalled “internal.”

The singularity condition can also be written by usingthe Binet-Cauchy identity(Ref. 23), as follows:

det(AAT ) ≡n∑

i=1

M2i = 0 (31)

where Mi = det(Ai) are the Jacobian minors of order 3and Ai = A with ith column removed.

The singularity conditions, (31), for the pyramid arrayof four single-gimbal CMGs become

M1 = sβ[(s2s3c4 + c2s3s4) + cβ(c2c3s4 − s2c3c4)+ 2(cβ)2c2c3c4] = 0

M2 = sβ[(s3s4c1 + c3s4s1) + cβ(c3c4s1 − s3c4c1)+ 2(cβ)2c3c4c1] = 0

M3 = sβ[(s4s1c2 + c4s1s2) + cβ(c4c1s2 − s4c1c2)+ 2(cβ)2c4c1c2] = 0

M4 = sβ[(s1s2c3 + c1s2s3) + cβ(c1c2s3 − s1c2c3)+ 2(cβ)2c1c2c3] = 0

where si ≡ sinxi and ci ≡ cos xi. Although these sin-gularity conditions based on the Binet-Cauchy identityhave been discussed in the literature (e.g., Refs. 2, 4, and5), a new approach to computing the singularity momen-tum surfaces using the singularity conditions, Mi = 0,is presented here.

Because the minimum rank of the Jacobian matrix Ais two, the four conditions, M1 = M2 = M3 = M4 =

0, are not independent of one another and only two ofthem are independent. Consequently, any two of thesefour conditions may be used to find the singular gimbalangles, as follows.

When ci �= 0, the singularity conditions can be sim-plified as:

tan x3(tan x2 + tan x4) + cβ(tan x4 − tan x2) = −2(cβ)2

(32)

tan x4(tan x3 + tan x1) + cβ(tan x1 − tan x3) = −2(cβ)2

(33)

tan x1(tan x4 + tan x2) + cβ(tan x2 − tan x4) = −2(cβ)2

(34)

tan x2(tan x1 + tan x3) + cβ(tan x3 − tan x1) = −2(cβ)2

(35)

Because the minimum rank of the Jacobian matrix A istwo, only two of these four conditions are independent.Therefore, these four equations yield six singular gimbalangle combinations, as follows.Case 1: For all (x1, x3), determine (x2, x4) using (33)

and (35),

x2 = tan−1

(−2(cβ)2 − cβ tanx3 + cβ tanx1

tanx1 + tanx3

)

x4 = tan−1

(−2(cβ)2 − cβ tanx1 + cβ tanx3

tanx1 + tanx3

)

Case 2: For all (x2, x4), determine (x1, x3) using (32)and (34),

x1 = tan−1

(−2(cβ)2 − cβ tanx2 + cβ tanx4

tanx2 + tanx4

)

x3 = tan−1

(−2(cβ)2 − cβ tanx4 + cβ tanx2

tanx2 + tanx4

)

Case 3: For all (x1, x2), determine (x3, x4) using (34)and (35),

x3 = tan−1

(−2(cβ)2 − tanx1 tanx2 + cβ tanx1

tanx2 + cβ

)

x4 = tan−1

(−2(cβ)2 − tanx1 tanx2 − cβ tanx2

tanx1 − cβ

)

Case 4: For all (x2, x3), determine (x4, x1) using (32)and (35),

x1 = tan−1

(−2(cβ)2 − tanx2 tanx3 − cβ tanx2

tanx2 − cβ

)

x4 = tan−1

(−2(cβ)2 − tanx2 tanx3 + cβ tanx2

tanx3 + cβ

)

Case 5: For all (x3, x4), determine (x1, x2) using (32)and (33),

x1 = tan−1

(−2(cβ)2 − tanx3 tanx4 + cβ tanx3

tanx4 + cβ

)

x2 = tan−1

(−2(cβ)2 − tanx3 tanx4 − cβ tanx4

tanx3 − cβ

)

6

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Hx

Hz

Figure 5: Singular momentum projection on the(Hx, Hz)-plane for Case 7 with β = 54.7 deg.

Case 6: For all (x4, x1), determine (x2, x3) using (33)and (34),

x2 = tan−1

(−2(cβ)2 − tanx1 tanx4 + cβ tanx4

tanx1 + cβ

)

x3 = tan−1

(−2(cβ)2 − tanx1 tanx4 − cβ tanx1

tanx4 − cβ

)

There are additional six cases when ci = 0 (i.e. tanxi =±∞):

Case 7: sin(x2 + x4) = 0 when cos x1 = cos x3 = 0

Case 8: sin(x1 + x3) = 0 when cos x2 = cos x4 = 0

Case 9: tan x2 = − tan x4 = cβ when cos x1 = 0 (for all x3)

Case 10: tan x3 = − tan x1 = cβ when cos x2 = 0 (for all x4)

Case 11: tan x4 = − tan x2 = cβ when cos x3 = 0 (for all x1)

Case 12: tan x1 = − tan x3 = cβ when cos x4 = 0 (for all x2)

Singular surfaces in the three-dimensional vector mo-mentum space, H = (Hx, Hy, Hz), are then defined assurfaces mapped by:

Hx = − cβ sinx1 − cos x2 + cβ sin x3 + cos x4

Hy = cos x1 − cβ sinx2 − cos x3 + cβ sinx4

Hz = sβ sinx1 + sβ sinx2 + sβ sinx3 + sβ sinx4

As an illustrative example, the singular momentumprojection on the (Hx, Hz)-plane for Case 7 (with β =54.7 deg) is shown in Figure 5. For this case, the singu-larity conditions become:

(x1, x3) = (±π/2,±π/2)x2 + x4 = 0 or x2 + x4 = π

For x2 + x4 = 0, there are four line singularities passingthrough the four points denoted by ∗ as shown in Figure

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Hx

Hz

Figure 6: Complete singular momentum projection onthe (Hx, Hz)-plane for all 12 different cases with β =54.7 deg.

5. For x2 + x4 = π, the four ellipses on the (Hx, Hz)-plane with Hy ≡ 0 can be found as:

H2x +

(Hz ± 2 sinβ)2

sin2 β= 4

(Hx ± 2 cos β)2 +H2

z

sin2 β= 4

Note that Figure 5 is in fact a planar cross-section plotwith Hy = 0 (but not of the entire singular momentumenvelope). However, it provides the basic elliptical com-ponents of such an entire planar cross-section plot (e.g.,see Figure 9 in Ref. 2). The complete singular momen-tum projection on the (Hx, Hz)-plane for all 12 differentcases (with β = 54.7 deg) is shown in Figure 6. The fourellipses can also be noticed in this figure.

As demonstrated above, the proposed method pro-vides detailed information about singular gimbal anglesand their direct relationships to the resulting singularmomentum surfaces. The proposed method provides asystematic way of determining all possible singular gim-bal angles, while the method by Margulies and Aubrun(Ref. 2) does not. Detailed results for computing allpossible singular gimbal angles and also visualizing thesingular momentum surfaces using this new approachcan be found in Dominguez and Wie (Ref. 16).

5. CMG Null Motions

The so-called “null motions” of CMGs are definedas CMG gimbal motions that generate no net torquefrom the CMGs. One of the principal difficulties in us-ing CMGs for spacecraft attitude control and momen-tum management is the geometric singularity problemin which no control torque is generated for the com-manded control torque along a particular direction. Ata singularity, CMG torque is available in all but one di-

7

rection. In practice, the null motions are often employedto avoid and/or to escape such a singularity situation.

There are two types of singular states (Ref. 2): hyper-bolic states and elliptic states. The hyperbolic singularstates can sometimes be escaped through null motionwhereas the elliptic singular states cannot be escapedthrough any null motion. Although null motion can begenerated at the hyperbolic singularity, the mere exis-tence of null motion does not guarantee escape from thehyperbolic singularity. This so-called degenerate nullmotion problem was briefly mentioned in Bedrossian etal. (Refs. 4-5) without any specific example.

In this section, a mathematical framework for ana-lyzing such null motions as well as for determining thesingularity types is established by expanding upon theprevious work by Bedrossian et al. (Refs. 4-5). Thissection is also intended to provide the reader with somenew physical insights into the null motions (in partic-ular, the degenerate null motions). Several illustrativeexamples with significant new results are presented.

The virtual momentum vector, δ�hi, associated withthe ith CMG is defined such that

δ �H =n∑

i=1

δ�hi = 0 (36)

which is in fact the condition for null motion, also calledvirtual or zero-torque motion. The virtual momentumvector, δ�hi, is different from the actual momentum vec-tor change d�hi, but it must be compatible with the nullmotion constraint (36) irrespective of time.

The virtual (null motion) momentum vector, δ�hi, canbe expanded in a Taylor series, as follows:

δ �H =n∑

i=1

δ�hi

=n∑

i=1

[d�hi

dxiδxi +

12!

d2�hi

dx2i

δx2i +

13!

d3�hi

dx3i

δx3i + · · ·

]

=n∑

i=1

[�fiδxi −

12!

�hiδx2i −

13!

�fiδx3i + · · ·

]= 0 (37)

where δxi are the virtual (null motion) gimbal angle dis-placements from arbitrary gimbal angles xi, and �fi =d�hi/dxi.

The first-order necessary condition for null motion isthen given by

n∑i=1

�fiδxi = 0 (38)

which can be rewritten in matrix form asn∑

i=1

fiδxi = Aδx = 0 (39)

where δx = (δx1, ..., δxn) is called the null motion dis-placement vector and A is the Jacobian matrix previ-ously defined as in (5). In other words, the null vector

δx is the null-space vector of A; i.e., δx = n = null(A)and An = 0. A null-space vector can be obtained as(Ref. 2):

n = (C1, C2, C3, C4) = Jacobian null vectorCi = (−1)i+1Mi = order 3 Jacobian cofactorMi = det(Ai) = order 3 Jacobian minorAi = A with ith column removed

However, the mere existence of the local null vectors(the first-order necessary condition for null motion) isnot sufficient for an escape by null motion from singu-larity. The second-order necessary condition needs to bechecked.

To test whether null motion is possible at a given sin-gularity or to determine its type of singularity (i.e., el-liptic or hyperbolic), consider the null motion constraintexpressed in matrix form as

δH = H(x + δx) − H(x)

=n∑

i=1

[fiδxi −

12!

hiδx2i −

13!

fiδx3i + · · ·

]

= 0 (40)

which is the equivalent matrix representation of (37).Taking the inner product of δH with an arbitrary vec-

tor u, we obtain

uT δH = uT

{n∑

i=1

[fiδxi −

12!

hi(xi)δx2i −

13!

fiδx3i + · · ·

]}

= 0 (41)

Since uT fi = 0 (i = 1,...,4) when u is along the singularvector direction, we obtain the constraint equation fornull motion, as follows:

0 = uT

{n∑

i=1

[− 1

2!hiδx

2i +

14!

hiδxni + · · ·

]}

=n∑

i=1

(uT hi)(− 1

2!δx2

i +14!

δxni − · · ·

)

=n∑

i=1

si(cos δxi − 1) (42)

where si = uT hi = �u · �hi, which was previously definedin Section 3.

Considering only the second-order terms, we obtainthe second-order necessary condition for null motion(Refs. 2, 4), as follows:

n∑i=1

siδx2i = 0 (43)

which can be rewritten as:

δxT Sδx = 0 (44)

8

0

π/2

− π/2

2

2Η x

Η y

− 2

− 2

null manifold(degenerate motion)

not null trajectory

β

α

null manifold(degenerate motion)

where δx = (δx1, ..., δxn), S is a diagonal matrix de-fined as S = diag(si). If S is a sign-definite matrix,the only solution to (44) is δx = 0, and null motion isnot possible. However, the sign-definiteness of S is onlythe sufficient but not necessary condition for the trivialsolution, δx = 0, to (44).

The virtual null motion of gimbal angles can be ex-pressed using the first-order necessary condition, as fol-lows (Ref. 4):

δx =n−2∑i=1

cini = Nc (45)

where ci is the ith weighting coefficient, c =(c1, ..., cn−2), ni are the null space basis vectors of theJacobian matrix A such that Ani = 0 or N = null(A).Note that at a singularity, rank(A) = 2 and nullity(A) =n − rank(A).

Substituting (45) into (44), we obtain the second-order necessary condition of the form (Ref. 4):

cT Mc = 0 (46)

where M = NT SN.If M is a sign-definite matrix, the only solution to (46)

is c = 0, and null motion is not possible. This type ofsingularity is referred to as an elliptic singularity, andconsequently it cannot be escaped by null motion. Theother possibility for M is to be sign-indefinite (or sin-gular). This type of singularity is referred to as a hy-perbolic singularity. As to be illustrated for a system oftwo single-gimbal CMGs, however, the mere possibilityof null motion does not guarantee escape from singular-ity. Degenerate null motion solutions that do not affectthe rank of the Jacobian matrix must be excluded, asdiscussed in Ref. 4.

Example 1: Two Parallel Single-Gimbal CMGsA system of two parallel single-gimbal CMGs has a

Jacobian matrix A of the form

A = 2[

− sin α cos β, − cos α sinβcos α cos β, − sinα sinβ

](47)

where α is the “rotation” angle and β the “scissor” angleas shown in Figure 2. The singularity condition becomes

|A| = sinβ cos β = 0 (48a)⇒ β = 0,±π/2,±π ∀α (48b)

The singular momentum surface for β = 0 is simply acircle of the form: H2

x + H2y = 4. This also confirms

the singular momentum surface expression, (7), previ-ously found directly without recourse to singular gim-bal angles. An internal anti-parallel 0H singularity forβ = ±π/2 is located at the origin: (Hx, Hy) = (0, 0).

For the 0H singularity at x = (α, β) = (0, π/2), illus-trated in Figure 2, the Jacobian matrix becomes

A =[

0 −20 0

]

Figure 7: The (α, β) trajectories for two parallel single-gimbal CMGs.

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). This 0H (anti-parallel) singularity hasthe following null motion constraint:

0 =2∑

i=1

si(cos δxi − 1) = cos δx1 − cos δx2 (49)

Its null motion solution is δx1 = δx2 = arbitrary, whichcorresponds to β = π/2 for any α. This type of singular-ity with possible null motion is referred to as a hyper-bolic singularity. However, this hyperbolic singularitycannot be escaped by null motion, because the singu-lar configuration (β = π/2) remains undisturbed duringnull motion along the null manifold (or a degeneratenull trajectory), as illustrated in Figure 7. This exam-ple demonstrates that the mere existence of null motiondoes not guarantee escape from a hyperbolic singularity.

For the 2H saturation singularity with x = (α, β) =(0, 0), the Jacobian matrix becomes

A =[

0 02 0

]

with its singular vector of u = (−1, 0), i.e., AT u = 0.Its null space vector is: x = null(A) = (0, 1). Thissaturation singularity has the null motion constraint as

0 =2∑

i=1

si(cos δxi − 1) = cos δx1 + cos δx2 − 2 (50)

The only solution to this equation is δx1 = δx2 = 0;i.e., null motion does not exist. This type of singularitywith no possible null motion is referred to as an elliptic

9

(a) 2H Internal Elliptic Singularity (b) 0H Internal Hyperbolic Singularity

x

z

CMG #1CMG #3

CMG #4

CMG #2 x

z

CMG #1CMG #3

CMG #4

CMG #2

Figure 8: Singularity illustration for a pyramid mount-ing arrangement of four single-gimbal CMGs.

singularity. All saturation (external) singularities areelliptic; i.e., they cannot be escaped by null motion.

Example 2: Pyramid Array of Four CMGsConsider a pyramid array of four CMGs with β =

53.13 deg. It can be shown that a set of gimbal an-gles, x = (−π/2, 0, π/2, 0), shown in Figure 8(a), is aninternal elliptic singularity with its singular vector ofu = (1, 0, 0), as follows:

A =

0 0 0 01 −0.6 1 0.60 0.8 0 0.8

null(A) = N =

−0.7952 0−0.2774 −0.53920.4623 −0.64700.2774 0.5392

null(AT ) = u =[

1 0 0]T

S = diag(uT hi) = diag(0.6,−1.0, 0.6, 1.0)

M = NT SN =[

0.5077 −0.1795−0.1795 0.2512

]

eig(M) = 0.1588, 0.6

However, a set of gimbal angles, x = (−π/2, 0, π/2, π),shown in Figure 8(b), can be shown to be a hyperbolicsingularity with its singular vector of u = (1, 0, 0). It canbe further shown that this hyperbolic singularity can beescaped by null motion because its null motion is not adegenerate solution (Ref. 4).

Example 3: Three Parallel Single-Gimbal CMGsFor a system of three single-gimbal CMGs with par-

allel gimbal axes, described by (13) and (14), the singu-larity condition becomes

det(AAT ) = M21 + M2

2 + M23

= sin2(x3 − x2) + sin2(x3 − x1) + sin2(x2 − x1)= 0 (51)

and singular gimbal angles can be found as

x1 = x2 = x3 : External 3H Singularityxi = xj , xk = xi ± π : Internal 1H Singularity

-20

2-2

02

0

1

2

x (rad)y (rad)

det(

AA

T)

-20

2-2

02

1

2

3

x (rad)y (rad)

Mom

entu

m H

-2 0 2

-2

0

2

x (rad)

y (r

ad)

det(AAT) = constant

-2 0 2

-2

0

2

x (rad)

y (r

ad)

H = constant

Figure 9: Contour plots of det(AAT ) and H for a sys-tem of three parallel CMGs. Note that (x, y) are thenew transformed gimbal angle coordinates defined byEq. (53).

The singular momentum surfaces are then simply de-scribed by two circles.

The 1H singularity has the null motion constraint as

0 =3∑

i=1

si(cos δxi − 1)

= (cos δx1 − 1) + (cos δx2 − 1) − (cos δx3 − 1)= cos δx1 + cos δx2 − cos δx3 − 1 (52)

Its non-degenerate null motion solution can be foundas: δx2 = δx3 and δx1 = 0, and thus it is a hyperbolicsingularity which can be escaped by null motion.

A new set of orthogonal coordinates (x, y, z) for gim-bal angles, described by Margulies and Aubrun (but notexplicitly provided in Ref. 2), can be found as

xyz

=

1/√

2 −1/√

2 01/√

6 1/√

6 −2/√

61/√

3 1/√

3 1/√

3

x1

x2

x3

(53)

or

x1

x2

x3

=

1/√

2 1/√

6 1/√

3−1/

√2 1/

√6 1/

√3

0 −2/√

6 1/√

3

xyz

(54)

Using (54), we can then express det(AAT ) and the totalangular momentum H as (Ref. 2):

det(AAT ) = 2 − cos22x√

2− cos

2x√2

cos6y√

6

H2 = 1 + 4 cos2x√2

+ 4 cosx√2

cos3y√

6

10

which are independent of z.Contour plots of det(AAT ) and H vs. (x, y) are

shown in Figure 9. In the contour plot of H, one caneasily identify an elliptic singularity at the center and hy-perbolic singularities at the six saddle-like points. Theconstant-H lines of H = 1, connecting the internal hy-perbolic singularities are null motion trajectories. Thezero-momentum points are not singular points.

Conclusions

In this paper, particular emphasis has been placedon characterizing and visualizing the physical as well asmathematical nature of the singularities, singular mo-mentum surfaces, and null motions. It is hoped thatthe comprehensive mathematical treatment as well asseveral illustrative examples with significant new resultspresented in this paper can be utilized toward developinga low-cost attitude control system, employing smallerand inexpensive CMGs, for future agile imaging satel-lites.

References

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11