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Aerospace Science and Technology 39 (2014) 109–119

Contents lists available at ScienceDirect

Aerospace Science and Technology

www.elsevier.com/locate/aescte

Optimal uses of reaction wheels in the pyramid conﬁguration using a

new minimum inﬁnity-norm solution

Hyungjoo Yoon ∗, Hyun Ho Seo, Hong-Taek Choi

Satellite Control System Department, Korea Aerospace Research Institute, Daejeon 305-806, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:

Received 2 May 2014

Received in revise d form 30 July 2014

Accepted 6 September 2014

Avail abl e online 16 September 2014

Keywords:

Minimum inﬁnity-norm solution

Time optimal maneuver

Reaction wheel array

Momentum envelope

In this study, simple methods are presented to improve the agility performance of a spacecraft with

four reaction wheels in the pyramid conﬁguration. A new and simple method is proposed to determine

the momentum and the torque envelopes, which are deﬁned as the maximum momentum and torque

capacities of the wheel array, respectively. Then, based on the geometry of the envelopes, the best

shape of the pyramid conﬁguration needed to deliver optimal agility performance is discussed. In this

paper, new methods are also proposed to optimally distribute three-dimensional torque and momentum

commands, to the individual reaction wheels. The developed methods are based on the use of novel

algorithms to solve minimum inﬁnity-norm, or L∞-norm, problems. These algorithms can easily be

implemented with minimal modiﬁcation of conventional ones, but yield considerable improvement of

agility performance in numerical examples.

©2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

Agility performance has become one of the key factors in de-

veloping/operating modern satellite systems, especially for Earth-

imaging satellites, because it determines the number of available

imaging targets within the duration of a given pass. Agility per-

formance can be improved in various ways, but it is mainly deter-

mined by the maximum torque and momentum capacities of the

actuators. A modern satellite is generally equipped with an array

of at least three, possibly more, reaction wheels for redundancy.

Hence, the total combined capacity of the array, which is referred

to as an ‘envelope’, should be considered. This envelope is deter-

mined not only by the capacity of individual wheels, but also by

the conﬁguration of the wheel array. In this paper, we discuss how

to determine the wheel conﬁguration needed for optimal agility

performance.

Another closely related problem that should be considered, is

the eﬃcient distribution of three-dimensional torque and momen-

tum commands to individual reaction wheels. Because there are

generally more than three reaction wheels, it becomes necessary to

solve an under-determined linear-equations system, which in gen-

eral has an inﬁnite number of solutions.

*Corresponding author. Tel.: +82 10 3324 3660.

E-mail addresses: drake.yoon@gmail.com (H. Yoon), seo2h@kari.re.kr (H.H. Seo),

hongtaek@kari.re.kr (H.-T. Choi).

In fact, this problem can be considered a special case of the

control allocation problem. There have been a lot of investigations

of control allocation problems for aircraft (see Ref. [7] and the ref-

erences therein) and a few on spacecraft attitude control (e.g., see

Ref. [3]). However, the focus of these studies was not the best use

of wheel array capacity to achieve optimal agility, which is the

main topic of this paper.

Conventionally, the minimum L2-norm solution, which mini-

mizes the square sum of the individual torque/momentum, is used

because it minimizes the total power/energy of the wheel array.

However, as will be shown later, this method does not fully utilize

the envelopes of the wheel array. On the other hand, the minimum

L∞-norm (or ‘inﬁnity-norm’) solution, which minimizes the maxi-

mum absolute value of the individual torque/momentum, may be a

better choice for higher agility performance. While the minimum

L2-norm solution can easily be obtained using a pseudo-inverse

matrix, it is well known that the minimum L∞-norm solution can-

not be expressed in a simple closed form, and thus needs more

sophisticated algorithms to ‘search’ for it.

Cadzow proposed just such an algorithm [1,2]. His algorithm

is eﬃcient and is applicable to under-determined problems in any

number of dimensions, but it is subject to the Haar condition [5].

Moreover, the algorithm does not solve the problem itself but, in

fact, solves its ‘dual optimization problem’. For this reason, it does

not help much to understand the nature of the problem. Grav-

agne and Walker [4] proposed an algorithm also based on the dual

problem, and showed how the solution could be applied to multi-

link robot controls. Markley et al. [8] ﬁrst related the minimum

http://dx.doi.org/10.1016/j.ast.2014.09.002

1270-9638/©2014 Elsevier Masson SAS. All rights reserved.

110 H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119

Fig. 1. Reaction wheel array in the pyramid conﬁguration.

inﬁnity-norm problem to spacecraft attitude control with reaction

wheels, and provided the motivation for the present paper. They

presented the results from a study of the nature of the torque

and momentum envelopes, and provided a scheme to deﬁne them.

They also proposed an attitude control loop based on their min-

imum inﬁnity-norm solution algorithm. Their paper clearly re-

vealed the geometric aspects of the envelopes, and their results are

widely applicable to general conﬁguration (e.g., in terms of num-

ber of wheels, sizes, and axis directions) of a reaction wheel array.

Their method, however, is based upon a purely geometric approach

without considering coding eﬃciency, meaning that there is room

for improvement in terms of coding and computational eﬃciency.

Verbin and Ben-Asher [9] also proposed an algorithm for a case

with four reaction wheels.

In this paper, simpler and computationally more eﬃcient meth-

ods are proposed than those presented in the earlier works [8,

9]. Our works is focused on the pyramid conﬁguration with four

identical reaction wheels, which is in fact the industry standard

owing to its minimum number of redundancies and its symmet-

ric capacities. We propose a simple new method which deﬁnes the

momentum/torque envelopes. Then we provide a scheme to opti-

mize the pyramid conﬁguration for the inertia properties of a given

spacecraft, using the relationship between its moment of inertia,

and the envelope under consideration. For the distribution of the

torque/momentum, we herein propose a new algorithm to obtain

the minimum inﬁnity-norm solution, that also runs much faster

than that of Ref. [8].

Another distinct feature of the present work is that it provides

another new algorithm which calculates an optimal momentum

distribution with the wheel speeds minimally diverging from a

nominal set value, in the sense of the L∞-norm distance. This al-

gorithm can be used to make the wheel speeds stay as close to

the (non-zero) nominal speeds as possible, even when the total

angular momentum of the array is zero. This feature is useful in

practice to keep the wheels from operating at, or crossing zero rpm

(at which wheel characteristics become nonlinear due to static and

Coulomb friction). This nonlinear behavior near zero rpm may in-

stantaneously increase the attitude control error and degrade the

mechanical and electrical reliability of the wheel. So, in some space

programs, it is preferable to avoid the zero rpm operation com-

pletely, if possible.

Finally, comparative numerical simulations are provided to

show the effectiveness of the proposed methods. It will be shown

that a control loop using the proposed methods, yields superior

agility performance over the conventional L2-norm method, and

also successfully leads the wheels to a given non-zero nominal

speed after completing the maneuver.

2. Momentum and torque envelope

2.1. The pyramid conﬁguration

In this section, we propose a new means of composing the mo-

mentum and the torque envelopes of a reaction wheel array. This

paper mainly deals with a four-wheel array in a pyramid conﬁgu-

ration of the type shown in Fig. 1, which is the most common in

practice. (The direction of each spin axis can be ﬂipped, but it is

deﬁned intentionally, as shown in Fig. 1, to make the null space

vector according to Eq. (4). The reason will be explained in Sec-

tion 4.3.)

Let us denote the wheel spin direction vectors by W=

[ˆ

w1, ···, ˆ

w4]3×4; the total angular momentum Ht∈R3and the

angular momenta vector of the array Hw=[H1, ···, H4]T∈R4are

then related according to

H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119 111

Fig. 2. Momentum envelope of a reaction wheel array in the pyramid conﬁguration with Hi∈[−Hmax, Hmax ]. (For interpretation of the references to color in this ﬁgure

legend, the reade r is referred to the web versio n of this article.)

Ht=WHw,(1)

and the torque relationship can be written as

Tt=WTw,(2)

where Tt∈R3is the total torque and Tw=[T1, ···, T4]T∈R4is

the torque vector of the wheel array. With the speciﬁc conﬁgura-

tion shown in Fig. 1, the matrix Wcan be expressed as

W=cos β1cos β2−cos β1cos β2−cos β1cos β2cos β1cos β2

sin β1cos β2sin β1cos β2−sin β1cos β2−sin β1cos β2

−sin β2sin β2−sin β2sin β2

(3)

The momentum of each wheel Hi, where i =1, ···, 4, is assumed

to be limited as Hi∈[−Hmax, Hmax], and the wheel torque Ti

(i =1, ···, 4)as Ti∈[−Tmax , Tmax ]. It should be noted that in the

pyramid conﬁguration, the total momentum and the total torque

become zero when the four individual wheels have the same

wheel momentum or torque, respectively. In other words, Whas a

null vector

Hw,n=α[1,1,1,1]T,(4)

which yields Ht=WHw,n=0.

2.2. Momentum and torque envelopes

The momentum envelope is deﬁned as the maximum momen-

tum capacity of Htin three-dimensional space, which can be pro-

duced by the array of reaction wheels, each with momentum con-

straints. (The torque envelope can be similarly deﬁned.) Markley

et al. [8] presented a detailed study of the envelope and pro-

posed a method to visualize it, but their method is based on a

geometric approach and thus may not intended to be computa-

tionally eﬃcient. Here, a new and simpler method is presented to

obtain the envelopes of the four-wheel pyramid conﬁguration. Ac-

cording to earlier studies [5] and [8], the total angular momentum

reaches the envelope only if at least two out of four wheels have

the min/max angular momentum (i.e., ±Hmax). In other words, the

surface of the envelope consists of the locus facets of the total an-

gular momentum with two wheel speeds saturated and the other

wheel speeds set free, within their speed limits. (See also Neces-

sary Condition 1, presented later.) This is, however, a necessary but

not a suﬃcient condition; hence, some of the locus facets are not

on the envelope surfaces, but lie inside them.

For example, let us assume that the No. 1 and 2 wheels are

saturated, and that the No. 3 and 4 wheels are free. We can then

obtain four locus facets,

(H1,H2)=(−Hmax,−Hmax ), (H1,H2)=(Hmax ,Hmax)(5a)

(H1,H2)=(Hmax,−Hmax ), (H1,H2)=(−Hmax ,Hmax)(5b)

with H3and H4being free within their limits. Among these com-

binations, the ﬁrst two facets (with Eq. (5a)) contain the null

angular momentum cases Hw=[−Hmax, −Hmax , −Hmax , −Hmax ]T

and Hw=[Hmax , Hmax , Hmax , Hmax]T. This implies that these two

facets are obviously not on the envelope surface. The other facets

with Eq. (5b) may still be parts of the envelope surface. Therefore,

we can select 12 facets (=C(4, 2) ×2) in this way, which can be

obtained via

(Hi,Hj)=(Hmax,−Hmax )and Hk∈[−Hmax,Hmax ]∀k= i,j

(Hi,Hj)=(−Hmax,Hmax )and Hk∈[−Hmax,Hmax ]∀k= i,j

(6)

where i, j =1, ···, 4and i = j. These facets actually complete the

momentum envelope, as shown in Fig. 2 (red lines are spacecraft

body axes, green lines are spin axes of the reaction wheels). This

derivation may not seem very rigorous, but it certainly yields a

correct result and is suitable for practical purposes.

3. Wheel conﬁguration for optimal agility performance

The envelope is not a sphere but a polyhedron which can be

skewed depending on the conﬁguration of the wheel array, as

shown in Fig. 2. This fact indicates that the magnitudes of the

available torque/momentum may also var y in different directions.

112 H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119

Fig. 3. Momentum envelope (blue-lined, semi-transparent) and momentum ellipsoid (gray, gridded) with the spacecraft moment of inertia Ixx :Iyy :Izz =2 :3 :1. (For

interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

In addition, spacecraft are not inertially symmetric in general, im-

plying that the magnitudes of the available body rate and ac-

celeration also vary along the rotational direction. Thus, in the

present paper, we deﬁne the optimality of agility performance in

the sense that the available body rate and the acceleration vectors

along the ‘worst’ direction (along which the vectors have minimum

magnitude) have the largest magnitude (i.e., we deﬁne a maximin

problem). It is also assumed that the maneuver is executed as an

eigenaxis rotation in which the body rate and the acceleration vec-

tors have a common, body-ﬁxed direction during the maneuver.

This rotation can be achieved with control logic available in the

literature [10,11].

In most cases, the conﬁguration angle β1is set to β1=45◦for

symmetry between the xand yaxes. Another conﬁguration angle

β2, which is referred to as a skew (or cant) angle, is commonly

set to β2=35.26◦(tan β2=1/√2) [8]. These values give the to-

tal torque/momentum along the spacecraft body axes the same

magnitude, while also maximizing the minimum distance from the

origin to the envelope surface [6,8]. This choice is made, however,

without consideration of the inertia property of the spacecraft.

Another choice of conﬁguration angles can be made with con-

sideration of the spacecraft inertia. For simplicity, let us consider

a spacecraft whose body axes are the principal axes, that is, the

products of inertia are negligible and the spacecraft’s matrix of in-

ertia is I=diag([Ixx, Iyy, Izz]). This set of optimal β1and β2values

can be calculated easily by equating a body rate or an acceleration,

along each of the spacecraft body axes, to each other. From the

following relationships

ωx,max =Hx,max

Ixx =4

Ixx

Hmax cos β1cos β2(7a)

ωy,max =Hy,max

Iyy =4

Iyy

Hmax sin β1cos β2(7b)

ωz,max =Hz,max

Izz =4

Izz

Hmax sin β2(7c)

and

ωx,max =ωy,max =ωz,max,(8)

where H•,max, (• =x, y, z) is the maximum total momentum along

the spacecraft body axes, the optimal conﬁguration angles β1and

β2can be obtained as

tan β1=Iyy

Ixx

(9a)

tan β2=Izz

I2

xx +I2

yy

,(9b)

and the maximum momentum along each body axis is

ωx,max =ωy,max =ωz,max =4Hmax

I2

xx +I2

yy +I2

zz

.(10)

Here, it should be noted that the maximum slew rates in Eq. (10)

cannot be obtained simultaneously; instead, each of them is ob-

tained when the total momentum is aligned along the correspond-

ing body axis.

The above-mentioned method is very convenient; however, as

far as the authors are aware, it has yet to be proved that Eq. (9)

is actually the optimal solution which yields the maximum worst

body rate. Here, we present a discussion of the optimality by intro-

ducing a momentum ellipsoid and its relationship with the momen-

tum envelope. The momentum ellipsoid is deﬁned as the locus of

the angular momentum vector required for the spacecraft to have

an angular rate with a given magnitude ω. The ellipsoid has

a shape which is obviously determined by the spacecraft inertial

property, and is scaled by the given value of ω. The ellipsoid has

three semi-axes along the spacecraft body axes, and each length

is proportional to the principal moment of inertia (Iii) about each

body axis, and the corresponding slew rate ω, as shown in Fig. 3.

In rectangular coordinates, the equation of the ellipsoid is:

x2

I2

xx +y2

I2

yy +z2

I2

zz =ω2.(11)

To maximize the available slew rate ωalong the worst rota-

tional direction, we need to ﬁnd the optimal conﬁguration of the

wheel array such that the momentum ellipsoid inscribed in the

momentum envelope, has a maximum value of ω. Due to sym-

metry, only the three facets of the envelope which make tangential

contact with the ellipsoid should be considered. From some ge-

ometric relationships and calculations, it can be shown that the

magnitude of the spacecraft slew rates, which put the ellipsoid

tangentially into contact with Planes I, II, and III (see Fig. 3) are:

ω2

I=16 sin2β1cos2β1cos2β2

I2

xx sin2β1+I2

yy cos2β1

(12a)

H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119 113

Fig. 4. Square of the slew rates ωI(red), ωII (blue), and ωIII (green) and the optimal

conﬁguration. (For interpretation of the references to color in this ﬁgure legend, the

reader is referred to the web version of this article.)

ω2

II =16 cos2β1sin2β2cos2β2

I2

xx sin2β2+I2

zz cos2β1cos2β2

(12b)

ω2

III =16 sin2β1sin2β2cos2β2

I2

yy sin2β2+I2

zz sin2β1cos2β2

(12c)

respectively. Next, we can obtain the optimal conﬁguration angles

which maximize the minimum among ωI, ωII, and ωIII . Fig. 4 shows

the square of the slew rates versus the conﬁguration angles us-

ing Eq. (12). It can be shown that the optimal conﬁguration angles

can be obtained when ω2

I=ω2

II =ω2

III. In other words, the opti-

mal conﬁguration can be obtained when the momentum ellipsoid

simultaneously comes into contact with all the facets of the enve-

lope. From Eqs. (12) and this condition, one can obtain Eq. (9).

There are twelve worst directions in total, along which the

maximum available body rate is minimized. They are parallel with

the following vectors:

[±Ixx ,±Iyy,0]T,(13a)

[±Ixx ,0,±Izz]T,(13b)

[0,±Iyy,±Izz]T.(13c)

The maximum worst body rate with the optimal conﬁguration can

also be expressed as

ωI=ωII =ωIII =2√2Hmax

I2

xx +I2

yy +I2

zz

,(14)

which is only 1/√2(≈70 %) of the maximum body rate along the

body axes in Eq. (10).

Note that the conventional choice of conﬁguration angles (i.e.,

β1=45◦and tan β2=1/√2) is optimal only when the spacecraft

is inertially symmetric (i.e., when Ixx =Iyy =Izz).

4. Optimal torque/momentum distribution

The aforementioned optimal conﬁguration would not be of

much use, unless the attitude control law fully utilizes its opti-

mized capacities. Therefore, such an attitude control law should be

employed to achieve optimal agility performance.

An attitude control law calculates the attitude error and then

calculates the required torque and/or the momentum command in

the three-dimensional body frame. A command distribution logic

is then used to calculate the distribution of the three-dimensional

commands to the individual reaction wheels. In this section, we

present new algorithms for the optimal torque/momentum distri-

bution.

4.1. The minimum L2-norm (or conventional) method

The most commonly used solution in practice is the minimum

L2-norm solution, which is given by

Tw,2=W+Tt(15)

where W+is the pseudo-inverse matrix. The L2-norm is deﬁned

as the sum of the squares of the individual elements, i.e., x2=

x2

1+x2

2+···+x2

N=√xTx, for a vector x =[x1, ···, xN]T. Be-

cause the L2-norm can be interpreted as the total energy or power

of the vector signal, this method yields optimal power-eﬃciency.

However, for the power eﬃciency, this method allocates as large

a command as possible to a wheel whose spin axis is closest to

the total commanded torque/momentum. This leads the wheel to

be easily saturated even when the total torque/momentum does

not reach the envelopes yet. Therefore, it can be concluded that

this method does not fully utilize the capacities of the wheel ar-

ray, and is thus not optimal in terms of agility performance.

4.2. The minimum L∞-norm method

Because each wheel has limited torque/momentum capacities,

the total angular momentum and the torque are also constrained

within the envelopes. Therefore, optimal maneuvering perfor-

mance can be accomplished by delaying wheel speed saturation

as much as possible. This can be achieved by minimizing the max-

imum value, or L∞-norm, of the set of individual wheel momen-

tum/torque values [8]. The L∞-norm of a vector x =[x1, ···, xN]T

is deﬁned as

x∞=max|x1|,|x2|,···,|xN|.(16)

The minimum L∞-norm method enables the wheel array to

fully utilize its torque/momentum capacities. Unlike the minimum

L2-norm solution, the minimum L∞-norm solution cannot be ex-

pressed in a simple closed-form. Therefore, a sophisticated algo-

rithm which searches for the solution is necessary. Markley et

al. [8] presented such an algorithm based on the geometric prop-

erties of the solution.

4.3. New algorithm for the minimum L∞-norm solution

In this section, a new, computationally eﬃcient algorithm is

proposed to ﬁnd the minimum L∞-norm solution. For a given

under-determined linear equation (2), all of the solutions can be

written in a general form as

Tw=Tw,2+Tnull (17)

where Tw,2=W+Ttis the minimum L2-norm solution and Tnull is

the null vector of the matrix W, which satisﬁes WTnull =0.

At this point, a case is considered in which there are four

wheels and the nullity of Wis 1. (The wheel sizes and the spin

directions may not be symmetric at this point.) The null vector

Tnull can be expressed as

Tnull =α[Tn,1,Tn,2,Tn,3,Tn,4]T(18)

where αis any real scalar number and where the vector

[Tn,1, Tn,2, Tn,3, Tn,4]Tis the basis of the null space of W. Note

that Tn,i= 0(i =1, ···, 4)when any combination of three col-

umn vectors (or the spin axes) of Wspans a three-dimensional

114 H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119

space, which is true for the pyramid conﬁguration. This can eas-

ily be proved by showing that the assumption of Tn,j=0for any

jyields Tn,i=0, ∀i. The general form of the solution, Eq. (17), can

then be written as

⎡

⎢

⎢

⎣

T1

T2

T3

T4

⎤

⎥

⎥

⎦=⎡

⎢

⎢

⎣

T2,1

T2,2

T2,3

T2,4

⎤

⎥

⎥

⎦+⎡

⎢

⎢

⎣

αTn,1

αTn,2

αTn,3

αTn,4

⎤

⎥

⎥

⎦

.(19)

At this point, we will use some necessary conditions for the min-

imum L∞-norm solution to derive the problem solving algorithm.

The ﬁrst is well known and is given as follows.

Necessary Condition 1. At least two elements of the minimum

L∞-norm solution have the same absolute value as the L∞-norm of the

solution itself. (Of course, the absolute value of these elements is greater

than, or equal to, the others.)

Proof. The proof is omitted (see Refs. [5,8]). 2

Necessary Condition 1is closely related to the fact that at least

two wheels are saturated at the momentum envelope, as shown in

the previous section. Before introducing the second condition, we

make an assumption to simplify the derivation process.

Assumption 1. All of the elements of the null space basis, Tn,i, (i =

1, ···, 4)have the same sign (e.g., positive), such that Tn,i>0.

This assumption does not lose any generality; one can ﬂip the

sign of any Tn,iby ﬂipping the direction of ˆ

wi. Note that the spin

directions deﬁned in Fig. 1 satisfy this assumption. The second nec-

essary condition is presented below.

Necessary Condition 2. All of the elements in Necessary Condition 1

(whose absolute values are identical to the L∞-norm), do not have the

same sign.

Proof. Let Tbe the minimum L∞-norm solution and assume that

its elements, of which the absolute values are identical to the min-

imum L∞-norm, have a same sign. Then, because the elements of

the null basis vector, Tn,i, also have a same sign (from Assump-

tion 1), there always exists a null vector Tnull which gives the

solution T +Tnull, a smaller L∞-norm. This violates the assump-

tion that Tis the minimum L∞-norm solution. 2

From these necessary conditions, an algorithm can be derived

to calculate the solution. There are six (=C(4, 2)) possible candi-

dates of αwhich may make Twthe solution, as follows:

α=−T2,1+T2,2

Tn,1+Tn,2↔1,2,

α=−T2,1+T2,3

Tn,1+Tn,3↔1,3,

α=−T2,1+T2,4

Tn,1+Tn,4↔1,4,

α=−T2,2+T2,3

Tn,2+Tn,3↔2,3,

α=−T2,2+T2,4

Tn,2+Tn,4↔2,4,

α=−T2,3+T2,4

Tn,3+Tn,4↔3,4.

(20)

In these equations, p, qindicates that the corresponding value of

αis derived from the condition that the p-th and the q-th ele-

ments have the same absolute value but have different signs. For

example, the ﬁrst condition comes from T2,1+αTn,1=−(T2,2+

αTn,2). Then, the minimum L∞-norm solution can be obtained

Fig. 5. Minimum L2-norm (top) and minimum L∞-norm (bottom) solutions in the

pyramid conﬁguration.

by comparing the maximum absolute values of the elements of

the general solution in Eq. (19) with the six candidates of α, as

given in Eq. (20), and then selecting the one with the lowest

value. The newly developed algorithm can be implemented with

far fewer code lines and can run much faster (typically more than

ﬁve times faster with MATLAB) than the previous method pro-

posed by Markley et al. [8].

4.4. A special case: the symmetric pyramid conﬁguration

For the symmetric pyramid conﬁguration, described in Sec-

tion 2.1, for which the wheels are identical and spin axes are

symmetric (as shown in Fig. 1), the null basis vector can be written

as [1, 1, 1, 1]T. From the symmetry of this conﬁguration, an even

simpler algorithm can be derived.

Let us assume that the minimum L2-norm solution, Tw,2, is

calculated as shown at the top of Fig. 5. Because the null vector

can be written as Tnull =α[1, 1, 1, 1]T, the addition of this null

vector Tnull to Tw,2can be geometrically interpreted as a shift of

the solution with α, or an equivalent shift of the ‘zero’-line with

−α. Therefore, it can be intuitively concluded that the minimum

L∞-norm solution, Tw,∞, can be obtained as

Tw,∞=Tw,2+α[1,1,1,1]T(21)

where −αis the average of the min/max values of Tw,2; that is,

α=−min(Tw,2)+max(Tw,2)

2(22)

where min(·)and max(·)are the minimum and the maximum

values of the elements, respectively. Owing to the symmetry of

the pyramid conﬁguration, the solution can be obtained without

comparing the norms of the multiple candidates (as in the previ-

ous section). This algorithm can easily be implemented by adding

only few code lines to the conventional L2-norm method. For the

symmetric pyramid conﬁguration, the minimum L∞-norm solution

probably cannot be simpler than the newly proposed one.

4.5. Minimum L∞-distance solution from nominal values

The wheel momentum distribution using the minimum L∞-

norm solution can be successfully used for slew maneuver con-

trol. However, this method will make the speed of each individual

wheel zero when the total angular momentum command is zero,

because this state obviously has the minimum L∞-norm value.

H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119 115

Fig. 6. Combined torque and modiﬁed momentum control loops.

Thus, if a spacecraft has zero-bias momentum, the wheel speeds

become zero when the spacecraft completes a maneuver and is at

rest. As noted in the introduction, it is not preferable for a reac-

tion wheel to operate near zero rpm due to its nonlinear behavior.

This behavior can cause considerable attitude errors, and thus may

hinder performance of assigned tasks. Moreover, if the wheels are

kept in low rpm range for long, there might be some impact on

wheel reliability. (For this reason, some wheel manufacturers set

limits on the allowable number of turns at low rpm.)

The present section proposes a new algorithm with which the

wheel speed vector has the minimum L∞-distance from a given

(non-zero) nominal speed. This algorithm can be derived in a man-

ner similar to that in Section 4.3. It can easily be shown that

the solution with minimum L∞-distance from a nominal vector,

Hnom =[¯

H1, ¯

H2, ¯

H3, ¯

H4]T, has at least two elements which have

the same absolute difference from the nominal momentum ele-

ments, and this absolute difference is greater than, or equal to,

those of the other elements. Hence, we can have six possible can-

didates of αas below.

α=−(H2,1+H2,2)−(¯

H1+¯

H2)

Hn,1+Hn,2↔1,2

α=−(H2,1+H2,3)−(¯

H1+¯

H3)

Hn,1+Hn,3↔1,3

α=−(H2,1+H2,4)−(¯

H1+¯

H4)

Hn,1+Hn,4↔1,4

α=−(H2,2+H2,3)−(¯

H2+¯

H3)

Hn,2+Hn,3↔2,3

α=−(H2,2+H2,4)−(¯

H2+¯

H4)

Hn,2+Hn,4↔2,4

α=−(H2,3+H2,4)−(¯

H3+¯

H4)

Hn,3+Hn,4↔3,4

(23)

where Hn,i(i =1, ···, 4)are the elements of the null vector of W.

Then we can choose αwhich makes the solution have the mini-

mum L∞-distance from Hnom.

If the null basis vector is [1, 1, 1, 1]T(as in the symmetric pyra-

mid conﬁguration), and if the nominal wheel momentum of each

wheel also has the same value (for instance Hnom ), it becomes

possible to obtain the solution more easily. Because the nominal

wheel momentum vector Hnom =Hnom[1, 1, 1, 1]Tis itself a null

vector of W, the sum of this vector and the minimum L∞-norm

solution Hw,∞, that is,

Hn

w,∞=Hw,∞+Hnom,(24)

also becomes a solution for Eq. (1). Moreover, because Hw,∞has

the minimum L∞-distance from the zero vector [0, 0, 0, 0], the

solution Hn

w,∞has the minimum L∞-distance from the nomi-

nal momentum vector Hnom[1, 1, 1, 1]T, and thus is the minimum

L∞-distance solution. It is noticeable that the solution Eq. (24) also

can be obtained from Eq. (23) with Hn,i=1 and ¯

Hi=Hnom for ∀i

and a proper choice of α.

Fig. 6 shows the attitude control loop, which consists of the

minimum L∞-norm torque distribution and the modiﬁed min-

imum L∞-distance momentum distribution with respect to the

nominal speed. Its structure is nearly identical to that of Ref. [8]

except for the use of the modiﬁed minimum L∞-distance momen-

tum distribution. Using this control loop, it is now possible to lead

the wheel speeds to the (non-zero) nominal set value when the

spacecraft is at rest.

It should be noted that this method (especially with a large

nominal set value) may cause the wheels to become saturated

easily and thus may hinder optimal use of the full momentum ca-

pacity. Therefore, the nominal value should be selected with great

care: it should not be too large (reduction of the available momen-

tum capacity) or too small (attitude control performance at rest

and degraded reliability of the wheels). One may use a variable

nominal value, which is zero when the total momentum is large

(to fully utilize the momentum capacity) and is suﬃciently large

when the total momentum is small (to keep the wheel speeds

away from zero rpm).

In some actual space programs, for instance, those involving

imaging satellites, interruption of an imaging mission due to wheel

speed zero-crossing should be avoided. In addition, it may be a

case that reliability of the wheels has higher priority than agility

performance. For such cases, one may force the wheels to oper-

ate only within the half of the speed range without a sign change,

that is, with Hi∈[0, Hmax]. This scheme can be implemented by

setting the nominal speed to half the maximum speed, and set-

ting saturation limits at zero and the maximum rpm. Of course, as

the momentum envelope is reduced to half the original one, some

agility performance should be sacriﬁced. However, the torque ca-

pacity remains the same, and it is also a dominant factor for agility

performance, especially in small angle slews, so the sacriﬁce of

agility performance may not be that great, and may be tolerable

depending on the mission requirements.

5. Numerical simulation

Comparative simulation examples are given in this section. The

spacecraft inertia property is I=diag[Ixx, Iyy, Izz] =

diag[1000, 1500, 500]kg m2, and for each reaction wheel, the iner-

tia is 0.4kgm

2. The maximum wheel torque is 2Nm, and the

maximum wheel speed is 600 rpm. (These values are roughly

based on actual satellites developed and operated by Korea

Aerospace Research Institute.) It is assumed that the spacecraft

is initially at rest and that the total angular momentum of the

wheel array is initially zero (i.e., zero-momentum bias). The space-

craft is assumed to be commanded to perform a slew maneuver of

60 deg. The rotational axis is set to −I−1ˆ

w2, where ˆ

w2is the spin

axis of Wheel #2, so that the required total angular momentum

vector is aligned with ˆ

w2. This choice will distinctly show the dif-

ference between the conventional L2-norm method and the newly

116 H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119

Fig. 7. Attitude error quaternions.

Fig. 8. Spacecraft body rate magnitude.

proposed L∞-norm method (will be shown later). For the given in-

ertial property, the optimal conﬁguration angles are computed to

be β1=56.3◦and β2=15.5◦using Eq. (9). These values are used

during the comparison of the performance of the L2-norm and the

L∞-norm methods. For the attitude feedback loop, we developed

a PD-control algorithm which leads the spacecraft to rotate about

a ﬁxed Euler rotational axis without excessive transient overshoot.

This is a modiﬁcation of an earlier scheme proposed in Ref. [10].

When using the minimum L∞-distance control (from the nomi-

nal set value) presented in Section 4.5, the nominal speed is set

to 100 rpm when the total momentum magnitude is smaller than

half of the maximum momentum of one wheel; otherwise, it is set

to zero rpm. During the simulation, the individual wheel torque is

forced to be zero if the wheel speed is saturated (with a 30 rpm

margin for practical reasons) and the commanded torque is out-

ward the limit. This scheme keeps the wheel speeds within their

allowable speed range.

Figs. 7 and 8show the improved agility performance of the

newly developed L∞-norm method over the conventional L2-norm

method. The maneuver time (in which all of the attitude errors

in each body axis become less than 0.01◦) is shortened from

52 to 41 sec (reduction of about 21%) in this speciﬁc scenario,

by the proposed method. More speciﬁcally, the maximum body

rate was increased from 2.0 to 2.5 deg/sec, and the accelera-

tion from 0.14 and 0.4 (before and after the saturation of Wheel

#2 at 10 sec, respectively) to 0.19 deg/sec2. These results show

that the proposed method successfully improved agility perfor-

mance by more fully utilizing the momentum and torque capac-

ities.

The superiority of the proposed method is more clearly un-

derstood in Figs. 9 and 10. These display the different histo-

ries of reaction wheel speeds and the trajectory of the total

angular momentum in the three-dimensional space, respectively.

With both methods, the wheel speeds, which are initially at

the nominal speed of 100 rpm, are operated within the given

speed limits (−600 ∼+600 rpm). However, with the conventional

L2-norm method, Wheel #2 becomes saturated early, while the

other wheels are still far from the speed limit. This occurs because

H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119 117

Fig. 9. Reaction wheel speeds.

Fig. 10. Angular momentum trajectories.

the required momentum is along the axis of Wheel #2 and thus

the minimum L2-norm method uses this wheel as much as possi-

ble. After Wheel #2 is saturated, angular acceleration is decreased

signiﬁcantly, as shown in Fig. 8(a). On account of the reduced

acceleration, the total angular momentum could not reach the en-

velope and return to zero, as shown in Fig. 10, to complete the

60-degree maneuver. This implies that the control law did not fully

utilize the momentum capacity. Moreover, because a sort of ‘sym-

metry’ is broken when the Wheel #2 becomes saturated, the wheel

speeds do not return to nominal (100 rpm) after the maneuver

(see the wheel speeds after stabilization in Fig. 9(a)). The wheel

speeds may converge to an unpredictable value after each maneu-

ver, and thus may become too large or too small after a number of

maneuvers. Therefore, additional operations and/or control logics

may be needed to return the speeds to nominal value after ma-

neuvers.

On the other hand, the minimum L∞-norm method delays

wheel speed saturation until the total momentum reaches the en-

velope so that the spacecraft can maneuver with a larger body

rate. It is noteworthy that, because the required total momentum

118 H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119

Fig. 11. Total power of reaction wheel array.

is along ˆ

w2in this scenario, all of the wheels are used uniformly

to generate this momentum. (Note that all of the wheels should

be saturated at the vertex (the black dot in Fig. 2) which lies on

ˆ

w2.) As shown in Fig. 10(b), the total momentum reached the en-

velope, which implies that the proposed method fully utilized the

momentum capacity of the array. It also can be observed that the

wheel speeds returned to nominal after the maneuver, owing to

the minimum L∞-distance momentum distribution scheme.

Fig. 11 shows the time histories of the total power of the wheel

array. As anticipated in the discussions in Introduction and Sec-

tion 4.1, the proposed L∞-norm method shows lower power ef-

ﬁciency than the conventional L2-norm method, as the cost for

its superior agility performance. However, energy consumption for

attitude maneuvers takes only a small portion of the total en-

ergy budget, because the time duration of maneuvers is very short

compared to the whole ﬂight time. Therefore, the lower power ef-

ﬁciency is generally tolerable in practice.

Finally, we conducted another simulation in which the wheel

speed zero-crossing condition would be absolutely avoided. These

results are shown in Fig. 12. In Fig. 12(b), it can be seen that the

wheel speeds, which were initially set to a (ﬁxed) nominal value

of 300 rpm (half the maximum rpm), did not cross the zero-rpm

point, and operated between 0 and +600 rpm, later returning to

the initial value. The momentum capacity, and thus the maximum

body rate (about 1.2 de g /sec), are each only half the original ones,

but the maneuver time was 59 sec, an increase of 41%. This in-

crease may be acceptable in some space programs, in return for

better attitude pointing stability and wheel reliability.

6. Conclusions

To make the best use of a reaction wheel array, we combined

geometric, mathematical, and algorithmic approaches. These meth-

ods were developed with an emphasis on computational eﬃciency

and ease of implementation, and are therefore preferable for prac-

tical applications. The newly developed methods yield results iden-

tical to those of Markley et al. but can be implemented much eas-

ier and run faster. In addition, we proposed a new method which

allows the wheels to operate as close to their nominal speed as

possible. We also successfully demonstrated the validity and ef-

fectiveness of all the methods introduced here using numerical

simulations.

The developed methods are applicable to a case with four re-

action wheels, especially in the pyramid conﬁguration. Therefore,

extending the proposed algorithms for use with more general con-

ﬁgurations would be worthy of further study. The authors also

hope that this work promotes studies and applications of the min-

imum inﬁnity-norm methods in applications other than agile ma-

neuver. For instance, the developed methods could be applied to

determine the wheel speciﬁcation requirements. It could also be

applied in design of the momentum management logic required Fig. 12. Maneuver without wheel speed zero-crossing.

H. Yoon et al. / Aerospace Science and Technology 39 (2014) 109–119 119

to compensate for external disturbances (e.g., solar pressure or air

drag) applied to spacecraft. Finally, it should be mentioned that

the agility performance also depends on the design of the atti-

tude feedback control. So optimal attitude feedback design should

be considered for even better agility performance, along with the

methods presented in this work.

Conﬂict of interest statement

None declared.

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