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Advances in Space Research 53 (2014) 97–109

Inverse free steering law for small satellite attitude control andpower tracking with VSCMGs

M.S.I. Malik ⇑, Sajjad Asghar

Department of EE, International Islamic University, H-10 Campus, Islamabad, Pakistan

Received 31 March 2013; received in revised form 27 September 2013; accepted 1 October 2013Available online 11 October 2013

Abstract

Recent developments in integrated power and attitude control systems (IPACSs) for small satellite, has opened a new dimension tomore complex and demanding space missions. This paper presents a new inverse free steering approach for integrated power and attitudecontrol systems using variable-speed single gimbal control moment gyroscope. The proposed inverse free steering law computes theVSCMG steering commands (gimbal rates and wheel accelerations) such that error signal (difference in command and output) in feed-back loop is driven to zero. H1 norm optimization approach is employed to synthesize the static matrix elements of steering law for astatic state of VSCMG. Later these matrix elements are suitably made dynamic in order for the adaptation. In order to improve theperformance of proposed steering law while passing through a singular state of CMG cluster (no torque output), the matrix elementof steering law is suitably modified. Therefore, this steering law is capable of escaping internal singularities and using the full momentumcapacity of CMG cluster. Finally, two numerical examples for a satellite in a low earth orbit are simulated to test the proposed steeringlaw.� 2013 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Attitude control; Inverse free steering law; Control moment gyroscope; Optimization; Singularity avoidance

1. Introduction

The new generation of Earth observing small satellitesequipped with precise observing sensors has recently beenincreasing its ability to capture data from desired groundtargets at much faster rates due to advanced attitude con-trol systems. The use of small satellites (less than 500 kg)for more complex space missions is rapidly increasing.The reduction in masses is achieved by combining subsys-tems into more complex subsystems.

With the advent of miniature single gimbal controlmoment gyroscopes (CMGs), the small satellite’s fast

0273-1177/$36.00 � 2013 COSPAR. Published by Elsevier Ltd. All rights rese

http://dx.doi.org/10.1016/j.asr.2013.10.001

Abbreviations: IF, inverse free; IPACS, integrated power attitude co-ntrol system; MRP, modified rodrigues parameters; MSI, modified sing-ularity index; DMSI, modified singularity index matrix⇑ Corresponding author. Tel.: +92 3009897316.

E-mail addresses: msi_id@yahoo.com, shahid.phdcs75@iiu.edu.pk(M.S.I. Malik), sajadoon@hotmail.com (S. Asghar).

maneuvering capability is greatly increased. The single gim-bal CMGs, both conventional and variable speed have tor-que amplification property and VSCMGs have an extradegree of freedom which can be used to encounter singular-ity and can also be used as a mechanical battery to storeenergy. This energy is then used to meet power require-ments of the satellites. The concept of using flywheels orVSCMGs for simultaneously controlling power and atti-tude of satellite is known as Integrated Power and AttitudeControl Systems IPACS, the term originally coined byAnderson and Keckler, 1973.

The study of IPACS has caught interest of manyresearchers in last few decades. Roes (1961) introduced theidea of storing energy in flywheels instead of batteries. How-ever, he did not mention the possibility of simultaneouslyusing momentum stored in wheels for attitude controlof satellite. Many researchers had studied high-speed fly-wheels and momentum wheels for integrated attitude andpower control (Tsiotras et al., 2001; Hall, 1997, 2000;

rved.

98 M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109

Lappas et al., 2010). Generally torque requirements of agilesatellites cannot be met by flywheels, so a number ofresearchers had proposed usage of VSCMGs in IPACSapplications (Yoon and Tsiotras, 2002; Richie, 2008; Richieet al., 2007, 2009). The singularities associated with VSCMGscluster in a pyramid configuration have been analyzed and dis-cussed in detailed in Yoon and Tsiotras (2004). The adaptivecontrol techniques have been used for VSCMGs in IPACSapplication (Yoon and Tsiotras, 2002; Xiyuan et al. (2009)).The steering law in these studies is based on some variantsof pseudo-inverse of Jacobian matrix and adding null motionfor avoiding singularities of VSCMGs.

Similarly for conventional CMGs, the steering laws areusually based on variants of pseudo-inversion of Jacobianmatrix (Bedrossian et al., 1990; Schaub et al., 1998). Thesesteering laws have limitations due to presence of singulari-ties. Recently Hirohisa (2013) has proposed an adaptive-skew pyramid type CMG system to handle singularityissues. A simpler configuration of twin CMGs for attitudecontrol of satellite has been studied in Shinya et al. (2013).Different control techniques were employed to attitudecontrol problem using CMGs (Abdelrahman and Park,2013; Kim et al., 2011; Chen et al., 2011; Yavuzogluet al., 2011; Chen et al., 2010; Xu and Chu, 2011;Kai et al., 2010; Wu and Wei, 2010; Sands et al., 2009etc.). Different issues related to spacecraft attitude controlwere addressed in Xie et al. (2008), Nagi et al. (2013). Kimet al. (2009), Silvio et al. (2012), Jia et al. (2012). The steeringalgorithms and control laws for CMGs were tested on exper-imental platforms on ground and in space (Mendoza-Barce-nas et al., 2012; Blackwell, 2012; Richie et al., 2009).

Pechev (2007) proposed the CMG steering law as afeedback loop transfer function containing Jacobian

Fig. 1. Attitude contr

matrix and a dynamic control matrix. The H1 controltheory was used to design this dynamic control matrix.The implementation of steering law does not involvematrix inversion. A similar steering law is proposed inJingwen et al. (2011).

So far, pseudo-inverse based steering laws for VSCMGshave been proposed for IPACS applications. Majority ofthese techniques perform matrix inversion, whereas we pro-pose an inverse free steering law. In this paper, H1 controlbased steering law is designed for VSCMGs performingcombined attitude and power tracking. The adaptationand singularity escaping features are introduced to pro-posed law for making it more efficient. This steering lawcomputes gimbal rates and wheel accelerations by minimiz-ing the error (between command & output) in a feedbackloop, thus it does not need to compute matrix inversion.The saturation limits of actuators are explicitly incorpo-rated in synthesis of dynamic matrix element of steeringlaw, singularity avoidance and escape features are alsoaddressed.

A simultaneous IPACS closed loop block diagram ispresented in Fig. 1. The commanded torque (TC) anddesired power (PC) are combined in commanded vector(TCP). Similarly the output torque (TO) and outputpower (Pa) are combined in output vector (TOP). Theerror signal (e = TCP � TOP) is used as input tothe proposed steering law (4-states controller), whichcomputes steering commands (wheel acceleration andgimbal rates). The VSCMGs cluster convert these com-mands into physical motion and satellite experiencesoutput torque and power being applied on it. The atti-tude angles and body angular rates are measured bysensors.

ol block diagram.

M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109 99

The paper is organized as follows: in Section 2 we havepresented the mathematical formulation of attitude motionand mechanical power storage for a satellite equipped witha cluster of VSCMGs. The proposed steering law is pre-sented in Section 3, with detailed discussion on its struc-ture. In Section 4, matrix elements of proposed steeringlaw are designed based on H1 control theory. The pro-posed steering law is slightly modified to improve its singu-larity escaping behavior by adding a singular matrix nearsingularity in Section 5. The stability analysis of feedbackloop involved in steering law is presented in Section 6. Adesign example for small satellite data is presented in Sec-tion 7. In Section 8, proposed law is tested in simulationsfor small satellite performing attitude maneuver and fulfill-ing the power demand simultaneously.

2. System model

We consider a satellite equipped with N VSCMGs. Theangular speed of spinning wheel of ith VSCMG is denotedby Xi. It is gimbaled along a fixed axis gi expressed in satel-lite body frame. The spin axis of ith VSCMG is representedby si, which is function of gimbal angular displacement di.When spinning wheel is rotated about gimbal axis then itcauses a torque vector exerted by ith VSCMG on satellitein ti direction orthogonal to both spin and gimbal axes.Thus orthogonal unit vectors associated with ith VSCMGexpressed in satellite body frame are summarized as (asshown in Fig. 2)

ti ¼ torque axis vector

gi ¼ gimbal axis vector

si ¼ angular momentum directional unit vectorðspin axisÞ

Here we are neglecting moments of inertia of gimbalframes because gimbal rates do not drive the total angularmomentum as they have much smaller values than wheelspeeds (Yoon and Tsiotras, 2004). With these assumptions,

Fig. 2. Spacecraft body with a single VSCMG.

the magnitude of angular momentum of ith wheel is hi = -Jwi Oi. Where i = 1 . . .N, Jwi is the moment of inertia of ithVSCMG’s wheel about its spin axis and angular momen-tum vector of each wheel is (hi = Jwi Oi si) as in Yoonand Tsiotras (2004). The total angular momentum ofVSCMG cluster is the vector sum of individual CMGmomentum

H d1; . . . ; dN; X1; . . . ;XNð Þ ¼XN

i¼1

hi ð1Þ

Now the total angular momentum h of spacecraft equippedwith VSCMGs cluster is defined as

h ¼ Hþ Jx ð2Þwhere x is a vector of body rates, J is the inertia matrix ofsatellite. The dynamic equation of satellite is derived bytaking time derivative of equation (2) by assuming inertiaof spacecraft to be constant.

J _xþ x� h ¼ � _Hþ Te ð3Þwhere Te is the external disturbance torque and _H is theoutput torque of VSCMG cluster. We take time derivativeof equation (1) to obtain the expression for output torque( _H or TO) of VSCMG cluster as

_H ¼XN

i¼1

hi_diti þ

XN

i¼1

J wi_Xisi ¼ TO ð4Þ

We define

DðX; dÞ,½J w1X1t1; . . . ; J wNXN tN � ð5ÞEðdÞ,½J w1s1; . . . ; J wN sN� ð6Þwhere d , (d1, . . .,dN)T e [0, 2p)N and X , (O1, . . .,ON)T eRN. Therefore Eq. (4) can be re-written as

½DðX; dÞ EðdÞ �_d_X

" #¼ TO ð7Þ

The expression of output torque in Eq. (7) can be made fur-ther compact by combining gimbal angles and wheel speedsin a single 2N � 1 vector g as

g ¼d

X

� �ð8Þ

And by defining a 3 � 2N matrix Q as

Q ¼ ½D : E� ð9ÞNow Eq. (7) can be written as

Q _g ¼ To ð10Þ

2.1. Power tracking

The formulation for simultaneous attitude control andpower tracking using VSCMGs was presented in (Yoonand Tsiotras, 2002). The total rotational kinetic energystored in wheels of N VSCMG is defined as

100 M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109

Er ¼XN

j¼1

1

2XT

j J wjXj ¼ XTJwX ð11Þ

Then, the output power can be defined as time-rate ofchange of this kinetic energy given as

P ¼ XTJw_X ¼ ½ 0 XT Jw �

_d_X

" #ð12Þ

The Eqs. (10) and (12) are combined together to form anintegrated output torque and power equation for VSCMGcluster, as

B _g ¼ TOP ð13Þ

where

Bðd; XÞ4�2N ¼D3�N E3�N

01�N ðXT J wÞ1�N

� �

TOP ¼TO

P

� �

2.2. Attitude and power control

Here we formulate attitude control law for re-orienta-tion of a satellite to a desired attitude. We use quaternionrepresentation for attitude of satellite. The attitude kine-matic equation is given below

_q ¼ 1

2GqðqÞx; GqðqÞ ¼

�q1 �q2 �q3

q0 �q3 q2

q3 q0 �q1

�q2 q1 q0

26664

37775 ð14Þ

The desired attitude is achieved by applying a commandedtorque TC. A quaternion feedback controller (proportionalderivative) is used to compute commanded torque, de-scribed as (Pechev (2007))

TC ¼ K1qe þ K2x ð15Þ

where K1 and K2 are gain-matrices suitably designed toachieve better performance. The power requirement de-pends on size of satellite, orbital period, eclipse periodand sunlight period. The required power profile is functionof time and it is generated for a small spacecraft example inthe simulation result section. We represent it by Pc in theformulation.

For simultaneous attitude and power control, com-manded torque and commanded power are combinedtogether to form a commanded vector as

TCP ¼TC

P C

� �

The steering equation equates the VSCMG output vector(torque and power) to commanded vector (torque andpower) for developing a steering law. The output vectorof VSCMG given in Eq. (13) is equated to the commandedvector as shown below

B _g ¼ TCP ð16ÞIn next section, proposed steering law is developed usingthis steering equation.

3. Development of inverse free steering law

The steering Eq. (16) is used to develop the proposedsteering law. Inverse free (IF) steering law is derived usingcontrol theory, it applies H1 control technique to minimizethe difference (or error) between commanded vector (TCP)and output vector (TOP) in a feedback loop. This logic isdescribed in Fig. 4. The proposed inverse free steeringlaw is

_g ¼ KðsÞe ð17Þwhere K(s) is transfer function matrix which maps the errorbetween commanded and output vector (e = TCP � TOP)to VSCMG steering commands. By substituting expressionfor error e in Eq. (17), we get

_g ¼ KðsÞ½Tcp � Top� ð18Þ

Using definition of VSCMG output vector (torque andpower) from Eq. (13), we get

_g ¼ KðsÞ½Tcp � B _g�

Finally

_g ¼ KðsÞ½BKðsÞ þ I��1TCP ð19Þ

The design of the control transfer function matrix K(s)based on H1 will be discussed in next section. In orderto avoid singularity, K(s) should be full transfer matrixso that it can generate the desired output vector (TOP)when B (d,X) encounters rank deficiency. The relationshipbetween output vector and commanded vector is derivedby substituting Eq. (19) in Eq. (13) as

TOP ¼ BKðsÞ½BKðsÞ þ I��1TCP ð20Þ

At steady state, K(s) will satisfy the following identity.

BKðsÞ½BKðsÞ þ I��1 ¼ I ð21ÞThe proposed steering law (Eq. (17)) is implemented bypresenting the control matrix K(s) in its state-space repre-sentation as

KðsÞ : _z ¼ Akzþ Bke _g ¼ Ckz ð22Þwhere Ak e R4�4, Bk e R4�4 and Ck e R2N�4 are state matri-ces and z e R4�1 represents the states of the control transferfunction matrix. These matrices are designed using H1control theory. The control matrix in transfer functionform can be written as

KðsÞ ¼ CkðsI� AKÞ�1BK ð23Þ

4. H‘ Design of K(s)

The control matrix K(s) is designed for plant given inEq. (13). In this nonlinear model the matrix B (d,X)

Fig. 3. 4-VSCMG configuration with standard pyramid.

M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109 101

depends on gimbal angles and wheel speeds. We can com-pute matrix B along a predefined path of gimbal anglesand wheel speeds and choose a suitable static matrix B0

(time invariant) for design of control matrix K(s). Thendesign of control matrix is adapted with dynamic matrixB (d,X) so that steering law generates proper steeringcommands. Therefore design process is performed intwo steps.

The control matrix K(s) maps the error vector(e = TCP � TOP) to steering commands (gimbal ratesand wheel accelerations) as given in Eq. (17). The gimbalrates and wheel acceleration are constrained due to prac-tical limitations. These steering constraints are explicitlyincorporated in control matrix synthesis. Hence controllaw is designed in optimal H1 sense (Pechev, 2007; Doyleet al., 1989) that solves the following sensitivity minimiza-tion problem

minKðsÞw1ðsÞ½Iþ BoKðsÞ��1

w2KðsÞ½Iþ BoKðsÞ��1

����������1

ð24Þ

The weighted function w1(s) determines the bandwidth ofthe steering law and is derived from the dynamics ofactuator. Whereas the weighted function w2(s) formulatesthe constraints in terms of upper bounds on the gimbalrates and wheel acceleration ð _gÞ. If all directions ofactuator are weighted equally, then weight w1(s) ischosen as

w1ðsÞ ¼b

sþ aI4�4 ð25Þ

where b is the bandwidth of the steering law and a is thegain of sensitivity function at steady state. The maximumupper bound (gimbal rates and wheel acceleration) con-straints are implemented in w2(s).

w2ðsÞ ¼1

_dmaxIN�N 0N�N

0N�N 1_Xmax

IN�N

" #ð26Þ

where _dmax and _Xmax are maximum gimbal rates and max-imum wheel acceleration respectively.

By apply the standard optimization algorithm(Doyle et al., 1989), from weight w1(s), matrix Ak trans-forms to Ak = �a I4�4 and the matrix Bk reduces toBk = d I4�4. The state-feedback gain of the controller isdescribed as

Ck ¼ BTdX ð27Þ

where X > 0, X = XT is the solution of the Riccati equationassociated with the state feedback design of K(s) (Doyleet al., 1989). Where d is the desired bandwidth in the steer-ing law. The matrix Ck is updated at each time interval withthe current gimbal angles (d) and wheel speeds (X). Aftersubstituting values of Ak, Bk and Ck in Eq. (24), we get

KðsÞ ¼ d2BT X

sþ að28Þ

5. Singularity avoidance

In order to avoid singularity, K(s) should be full ranktransfer matrix so that it can generate the desired outputvector (TOP) when B (d,X) encounters rank deficiency.So far we have assumed a cluster of N VSCMG units.These units can be arranged in different configurations(pyramid, roof-type etc.) depending on application require-ment. Here we considered the standard pyramid redundantconfiguration of four VSCMGs with skew angle b chosenas (b � 54.74 deg) (Fig. 3). The momentum envelope of thisconfiguration is approximately spherical (Bedrossian et al.,1990).

Moreover, the VSCMG cluster has extra degrees of free-dom namely variation in wheel speeds. Therefore, thematrix B will always generate the desired torque providednone of the wheel speeds becomes zero. Due to torqueamplification property of CMG the output torque gener-ated by gimbal movement in ti direction is much largerthan torque generated by wheel speed variation in si direc-tion. Thus we can define the singularity of VSCMG clusterin terms of rank deficiency of matrix D which correspondsto torque generated by gimbals movement. In singularstate, a CMG cluster cannot produce torque in certaindirection as all the spin axes si of CMGs are projected inthat direction and torque directions ti become coplanarmaking matrix D of rank 2. The singularity of VSCMGscluster is similar to the singularity of the conventionalCMGs (Yoon and Tsiotras, 2004). The singularities ofVSCMGs can also badly affect the performance of attitudecontrol system. Therefore, we can modify our steering lawin neighborhood of singularity to improve its singularityavoiding/escaping performance.

A matrix Ds is computed for elliptic singularity indesired maneuver direction and added to Jacobian matrix

Fig. 4. Inverse free controller block diagram.

102 M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109

(D) when it approaches singularity. Here we illustrate thisscheme for roll maneuver example. For a four CMGs pyr-amid configuration the Jacobian matrix defined in Eq. (5)can be written as

D X; dð Þ ¼ J w

�X1cb cos d1 �X1 sin d1 X1sb cos d1

X2 sin d2 �X2cb cos d2 X2sb cos d2

X3cb cos d3 X3 sin d3 X3sb cos d3

�X4 sin d4 X4cb cos d4 X4sb cos d4

26664

37775

T

ð29Þwhere cb = cos (b) and sb = sin (b) respectively. This con-figuration has elliptic singularity at gimbal angles [�pi/2, 0,pi/2, 0] for roll axis maneuver. Eq. (30) for these singulargimbal angles results in following matrix

DSðXÞ ¼ J w

0 X1 0

0 �X2cb X2sb

0 X3 0

0 X4cb X4sb

26664

37775

T

ð30Þ

Here, we can observe that, matrix DS depends only onwheel speeds of CMG cluster. This matrix is multipliedby a positive weighting parameter k and then added toJacobian matrix. The weighting parameter k is defined as

k ¼ k0expð�l detðDDT ÞÞ ð31Þwhere k0 and l are positive scalars. The weight (k) has va-lue k0 when Jacobian matrix is singular and has zero valuewhen Jacobian matrix is far from singularity. Therefore,Eq. (28) of proposed steering law is modified as

Ck ¼ B^TdX ð32Þwhere

B^ðd;XÞ4�2N ¼D3�N þDS3�N E3�N

01�N ðXT J wÞ1�N

� �

6. Stability analysis

The stability of feedback loop involving proposed steer-ing law is analyzed in this section. Here we shall establishboth internal and BIBO stability of closed loop system. Aclosed loop transfer function can be written as

GðsÞ ¼ BKðsÞ½BKðsÞ þ I��1 ð33Þ

Substituting K(s) from Eq. (29) and after simplifying we get

GðsÞ ¼ d2BBTX½sIþ aIþ d2BBTX��1 ð34Þ

It can be noted that the matrix ðd2BBTXþ aIÞ > 0 is posi-tive definite as a > 0; d2 > 0; X > 0 and BBT > 0, so itsEigen values will have positive real parts. Therefore polesof closed loop transfer function given in Eq. (35) lies in lefthalf of complex plane. However it does not complete theproof of stability as this closed loop system involves a timevarying matrix ðBBTÞ. But it can be shown that this time-varying matrix is bounded as

0 < cminI 6 BBT6 cmaxI <1 ð35Þ

where cmin ¼ lminJ 2wkXk

2min > 0 and cmax ¼ lmaxJ 2

wkXk2max

> 0 represent lower and upper bounds of matrix BBT

respectively. These limits are dictated by physical con-straints of the VSCMG cluster. Now we may re-writeEq. (35) as impulse response matrix

Gðt; sÞ ¼ d2BBTXUðt; sÞ ð36Þ

where Uðt; sÞ is state transition matrix for the given systemand is defined as

Uðt; sÞ ¼ exp �Z t

sd2BBTXþ aI� �

ds

� �

¼ e�aðt�sÞexp �Z t

sd2BBTXds

� �:

A time varying system is asymptotically stable if norm ofstate transition matrix Uðt; 0Þ decreases to zero as t!1.Here we define norm of a matrix as the largest absolute Ei-gen value of that matrix. The bounds on matrix BBT givenin Eq. (36) can be translated into bounds on the state tran-sition matrix as

0 6 e�atexp �d2cmaxXt� �

6 Uðt; 0Þ6 e�atexp �d2cminXt

� �6 I ð37Þ

Therefore, norm of the state transition matrix is boundedby 0 6 kUðt; 0Þk 6 1. And kUðt; 0Þk ! 0 as t!1. There-fore, system is asymptotically stable.

Similarly this system will be BIBO stable ifR t0kGðt; sÞkds <1 for all t > 0 Now for bounded state

transition matrix Uðt; sÞ the upper bound of impulseresponse matrix is defined as

Gðt; sÞ 6 d2cmaxXe�aðt�sÞexp �d2cminXðt � sÞ� �

6 d2cmaxXe�a t�sð Þ <1 ð38Þ

M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109 103

Finally norm of the impulse response matrix is integratedasZ t

0

kGðt; skÞds 6d2cmax

akXkð1� e�atÞ 6 d2cmax

akXk <1

ð39Þ

Therefore system is BIBO stable. Hence it completes theproof of stability.

7. Steering law synthesis – an example

The H1 synthesis of dynamic control matrices involvedin the steering law is illustrated in this section. The perfor-mance weights are designed. As all components of torquevector are weighted equally, resulting in a common pole.The weight w1(s) incorporates a tracking bandwidth ofapproximately 455 rad/s. The weight w2 limits the maxgimbal rates to 1 rad/s2 (for all gimbal angles) and wheelacceleration to 35 rad/s2. This results in following perfor-mance weights for four VSCMGs pyramid cluster.

w1ðsÞ ¼455

sþ 0:05I4�4w2 ¼

135

I4�4 04�4

04�4 11I4�4

" #ð40Þ

We select d0 = [p, p/4, �p/4, p] and X0 = [20,000,19,000, 18,000, 16,000]T RPM for the linearization of thematrix B in Eq. (13), where skew angle b = 54.74�.

Table 1Numerical parameters.

Parameter Value Units

K1 1.5 � I(3 � 3)

K2 6.0 � I(3 � 3)

J 10 � I(3 � 3) Kg m2

x (0) [0,0,0]T rad/sO (0) [20000, 19000, 18000, 16000] rpmJw 1.7 � 10�4 Kg m2

k0 0.445l 1000

Dðd0;X0Þ EðX0Þ0 XT Jw

� �¼

0:5773 0 �0:8165 0

0:7071 �0:4082 0:5774 0

0:4082 �0:7071 0:5774 0

0 �0:5773 �0:8165 0

0 �1 0 0:3560

�0:7071 �0:4082 0:5774 0:3382

�0:4082 �0:7071 �0:5774 0:3204

�1 0 0 0:2848

266666666666664

377777777777775

T

ð41Þ

Solving the optimization problem (Doyle et al., 1989), theRiccati equation solution is given by

X ¼

12:2230 9:1904 �0:1990 0

9:1904 12:2230 0:1990 0

�0:1990 0:1990 2:3899 0

0 0 0 4:946

26664

37775 ð42Þ

Then H1 synthesis results in following forms of the matri-ces of Eq. (23)

Ak ¼

�0:05 0 0 0

0 �0:05 0 0

0 0 �0:05 0

0 0 0 �0:05

26664

37775

Bk ¼

1:971 0 0 0

0 1:971 0 0

0 0 1:971 0

0 0 0 1:971

26664

37775 ð43Þ

The output matrix element of the dynamic control matrixcan be written as

Ck ¼ ð1:971ÞBT X ð44Þ

8. Simulation & discussion

In this section, the combined attitude and power controlof a small satellite in low Earth orbit is simulated to vali-date the performance of proposed steering law. In simula-tions, the attitude dynamics and kinematics of spacecraftequipped with VSCMGs cluster are modeled by differentialequations (3) and (14) respectively. The output of satellitedynamics model are angular velocity vector and attitudequaternion. The commanded torque vector is computedusing Eq. (15) and it is augmented with openloop powerrequirement profile to form a commended vector. Thesteering law Eq. (23) with modified definition of matrixCk from Eq. (33) are used to generate gimbal rates andwheel acceleration commands for commanded vectordescribed in Eq. (16). The numerical parameters used insimulations are listed in Table 1.

The power requirements depends on size of satellite,orbital period, eclipse and sunlight periods. The requiredpower profile for a small spacecraft with moment of inertia10 kg m2 on each axis is taken to be 5 W instantaneouspeak power during 90% of the eclipse time and 40 Winstantaneous peak power during 10% of eclipse time.The orbital period is 5750 s. During one orbit, sunlightexposure time is 3750 s and eclipse time is 2000 s as indi-cated in Fig. 5g, the first sunlight period runs from 0 to3750 s followed by eclipse until 5750 s. At the end of eacheclipse period (250 s long) 10% more energy is drained fromthe wheels as compared to beginning or even middle of theeclipse. The time of second orbit is from 5750 to 11,500 sand it also contains peak eclipse period of 250 s. At theend, the satellite remains in the sunlight from 11,500 s tothe 12,000 s.

In Section 7 we have used the maximum upper boundfor wheel acceleration as 35 rad/s2 for the synthesis of state

0 5 10 15 20 25 30 35 40 45 50-200

-150

-100

-50

0

50

100

150

Gim

bal A

ngle

s [de

g]

IF Steering Law, Roll Maneuver

Fig. 5a. Roll maneuver, gimbal angles.

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

gim

bal r

ates

[rad

/s]

IF Steering Law, Roll Maneuver

Fig. 5b. Roll maneuver, gimbal rates.

0 2000 4000 6000 8000 10000 120001

2

3

4

5

6

7

8x 10

4

Whe

el S

peed

[RPM

]

IF Steering Law, Roll Maneuver

Sunlight Time Sunlight Time Eclipse TimeEclipse Time

Fig. 5c. Roll maneuver, wheel speed.

104 M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109

matrices of steering law. The minimum value for the powerrequirement during eclipse time is �40 W. Wheel accelera-tion can be calculated from Eq. (12) as

P ¼ XT Jw_X

_X ¼ ðXT JwÞ�1

P

Substituting value of P = �40w, Jw = 1.7 � 10�4 andX = [20,000 19,000 18,000 16,000]T RPM, the value of

wheel acceleration is computed as 33.5 rad/s2. Therefore,wheel acceleration limit of 35(rad/s2) seems prudent choice.We consider two cases to test the performance of steeringlaw with different maneuver requirements.

8.1. Example 1: 40� roll maneuver

A rest–rest 40 degree roll maneuver is demanded withinitial CMG gimbal angles [0, 0, 0, 0]. The proposed

0 2000 4000 6000 8000 10000 12000-35

-30

-25

-20

-15

-10

-5

0

5

Whe

el A

ccel

erat

ion

[rad

/s2]

IF Steering Law, Roll Maneuver

Fig. 5d. Roll maneuver, wheel acceleration.

IF Steering Law, Roll Maneuver

Fig. 5e. Roll maneuver, singularity index.

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

Att

itude

Qua

tern

ion

IF Steering Law, Roll Maneuver

Fig. 5f. Roll maneuver, attitude quaternion.

M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109 105

steering law successfully avoids the elliptic singularity asshown in Fig. 5a. All four gimbal angles do not trappedin the internal elliptic singularity; successfully generatesoutput (3-axis) torque while singularity index remainsbelow five throughout the maneuver. The condition no(the ratio of the largest singular value of D to the smallest)is plotted as singularity index and is shown in Fig. 5e. Thelarge singularity index indicates nearly singular matrix.

Initially the singularity index increases as gimbal anglesapproach the elliptic singular configurations. After thatthe steering law simply avoids the singular configurationsand maintains singularity index below value 3.

The results of gimbal rates are shown in Fig. 5b. Asdepicted in Fig. 5b, the steering law breaks the symmetryin the generation of gimbal rates as many steering laws gen-erate symmetric profiles of gimbal rates leading CMG

0 2000 4000 6000 8000 10000 12000-50

-40

-30

-20

-10

0

10

- Desired - ActualPo

wer

Pro

file

[Wat

ts]

IF Steering Law, Roll Maneuver

Fig. 5g. Roll maneuver, power profile.

0 10 20 30 40 50 60 70 80-250

-200

-150

-100

-50

0

50

100

150

200

250

Gim

bal A

ngle

s [de

g]

IF Steering Law, started from Preffered Angles

Fig. 6a. Large maneuver, gimbal angle.

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

gim

bal r

ates

[rad

/s]

IF Steering Law, started from Preffered Angles

Fig. 6b. Large maneuver, gimbal rates.

106 M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109

cluster to singular gimbal configuration. The non-symmet-ric profiles of gimbal rates generated by proposed steeringlaw helps in avoiding and escaping singular states. Thewheel speeds for VSCMGs spin up during exposure tosun light to store energy in mechanical form in wheels asshown in Fig. 5c. The sunlight and eclipse periods areclearly separated by lines and labels. The figure also shows

the maximum spin up and minimum spin down speed val-ues during the maneuver. The wheels spin down to provideenergy in eclipse time period for the spacecraft but theynever de-spin during the maneuver. The magnitude ofwheel accelerations are small during sunlight exposureand grows larger in eclipse cycles as more energy is neededat the end of each eclipse time period as shown in Fig. 5d.

0 2000 4000 6000 8000 10000 120001

2

3

4

5

6

7

8x 10

4

Whe

el S

peed

[RPM

]

IF Steering Law, started from Preffered Angles

Sunlight Time Sunlight TimeEclipse Time Eclipse Time

Fig. 6c. Large maneuver, wheel speed.

0 2000 4000 6000 8000 10000 12000-35

-30

-25

-20

-15

-10

-5

0

5

Whe

el A

ccer

atio

n [r

ad/s2

]

IF Steering Law, started from Preffered Angles

Fig. 6d. Large maneuver, wheel acceleration.

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

45

Sing

ular

ity In

dex

IF Steering Law, started from Preffered Angles

Fig. 6e. Large maneuver, singularity index.

M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109 107

As described earlier, the maximum upper bound forwheel accelerations are taken to be 35 rad/s2. The steeringlaw keeps acceleration value below maximum limit in thede-spin cycles. The response of attitude quaternion is pre-sented in Fig. 5f. Although it is a single axis maneuver,but small motion is observed on other two axes of satellite.It will cause small pointing inaccuracy during tracking

maneuvers, but will not affect the performance duringrest-to-rest reorientation maneuvers. The actual powerresponse closely tracks the desired power profile as shownin Fig. 5g. Therefore, proposed steering law can success-fully perform combined attitude and power control for asatellite equipped with VSCMGs in this roll maneuverexample.

0 10 20 30 40 50 60

-0.5

0

0.5

1

Att

itude

Qua

tern

ion

IF Steering Law, started from Preffered Angles

Fig. 6f. Large maneuver, attitude quaternion.

0 2000 4000 6000 8000 10000 12000-50

-40

-30

-20

-10

0

10

- Desired

--- ActualPow

er P

rofil

e [W

atts

]

IF Steering Law, started from Preffered Angles

Fig. 6g. Large maneuver, power profile.

108 M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109

8.2. Example 2: large maneuver

In this example, a three axes maneuver is performed tobring satellite from initial attitude characterized by quater-nion [�0.5, 0.5, �0.5, 0.5] to final quaternion state [1, 0, 0,0]. The initial gimbal angles are chosen to be preferred gim-bal angles [p/4, �p/4, �p4, p/4]. The power requirementsfor this maneuver are same as presented in case 1.

The response of gimbal angles plotted in Fig. 6a showsthat CMG cluster encounters some singular state in thestart of maneuver for the short time but then escapes it suc-cessfully. The singularity escaping can be confirmed fromsingularity index plotted in Fig. 6e. It can be seen singular-ity index value quickly approaches 43 in the start of themaneuver but due to singularity escaping feature of thesteering law it comes down to a small value in a short time.Hence it is proved that steering law successfully generatesthe desired torque in singular states and efficiently escapesthese singularities. The response of gimbal rates are shownin Fig. 6b. The steering law also breaks the symmetry ingimbal rates as in 40 deg roll maneuver case. The non-sym-metric profiles of gimbal rates for this maneuver show thatthe steering law has a capability to efficiently avoid andescape internal singularities and thus generates the desiredoutput torque. The magnitude of all gimbal rates are within

the limits imposed in synthesis of state matrices of controlelement of steering law as depicted in Fig. 6b. The wheelspeeds of VSCMGs similarly spins up in exposure to sun-light period and de-spins to generate energy used by thespacecraft in the eclipse period as in example 1. The sun-light and eclipse time periods are labeled separately asshown in Fig. 6c.

The response of wheel acceleration is presented inFig. 6d. The magnitude of wheel acceleration is small dur-ing exposure to sunlight period because spacecraft requiresapproximately 5 W peak power but it has large magnitudevalues in eclipse time periods due to more energydemanded by the spacecraft at the end of each eclipse cycle(40 W). The profile of attitude quaternion for the space-craft is shown in Fig. 6f. As described in the start of thissection, this is three axes maneuver and steering law suc-cessfully achieved the desired re-orientation objective.The power requirements for this maneuver are same as inexample 1. The actual power response closely tracks thedesired power profile as shown in Fig. 6g.

9. Conclusions

In this paper, a new inverse free steering law for IPACS

is presented. The proposed steering law is derived using H1

M.S.I. Malik, S. Asghar / Advances in Space Research 53 (2014) 97–109 109

norm optimization approach. The saturation limits on gim-bal rates and wheel acceleration are incorporated in H1design via weight matrices. The steering law is modifiedto add adaptation and singularity escaping feature forenhancing its performance. The stability of steering law isalso discussed. The simulation results for two maneuverexamples show that the performance of steering law intracking required power profile and commanded torque issatisfactory. Moreover, the proposed steering law canescape the internal singularities.

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