AAS03-558

Mixed Control Moment Gyro and Momentum Wheel Attitude ControlStrategies

C. Eugene Skelton II† and Christopher D. Hall‡

Department of Aerospace & Ocean EngineeringVirginia Polytechnic Institute & State University

Blacksburg, Virginia 24061

This paper develops control strategies that use single-gimbal control mo-ment gyros for coarse large angle slew maneuvering and momentum wheelsfor precise control and error reduction due to initial conditions. Lyapunovand feedback linearization control laws are developed. Numerical simu-lations of the derived mixed control moment gyro and momentum wheelcontrol laws are evaluated.

INTRODUCTION

A spacecraft can use either internal or external actuators to control its attitude. Generally externalactuators such as thrusters are used for large fast slewing maneuvers. However thrusters are notideal for precision attitude control due to their discontinuous nature. Internal actuators can bemomentum exchange devices, such as momentum wheels and control moment gyros, or non-movingdevices like magnetic torquers. A momentum wheel is a high inertia flywheel mounted on a motorthat can provide one axis control by changing the relative momentum of the vehicle. Momentumwheels can perform precise maneuvers and maintain attitude; however the varying wheel speedstend to excite structural dynamics. Momentum wheels require torques proportionate to the desiredoutput torque which makes them non-ideal for rapid slewing. A control moment gyro has a flywheelmounted on a motor that spins at a constant relative speed. The flywheel and motor are mountedto a gimbal motor that can rotate about an axis perpendicular to the spin axis of the flywheel.Control moment gyros (CMGs) can be double-gimballed or single-gimballed [1]. For the purpose ofthis paper a CMG refers to a single gimballed CMG. A control moment gyro can create high torquesfor slewing, that are dependant on the flywheel speed, inertia, and the gimbal angle rate, based onmomentum exchange.

Many control laws have been developed for control moment gyro applications such as the Interna-tional Space Station [2, 3]. The disadvantage of CMGs is that certain sets of the gimbal angles cancause singularities where the CMG cluster is unable to provide torque about a given axis. Many pa-pers have explored control laws that have various methods of avoiding CMG singularities includinguse of gimballed momentum wheels [4] or implementing optimization techniques to steer away fromsingularity configurations.

Oh and Vadali [5] presented a Lyapunov control law for attitude and velocity control with CMGs.They argued that commanding gimbal velocities were ideal for attitude control of the body in orderto take advantage of the so called torque amplification factor, rather than commanding gimbalaccelerations which lead to undesirable gimbal rates. In Ref [6] Ford presented an extension ofRef. [5] by including the possibility of gimballed momentum wheels in the notation that we use inthis research. Schaubet.al [7] presented an extension of the control law of Ref. [5] by changing theattitude representation from Euler Parameters to Modified Rodrigues Parameters. Both changes

†Graduate Research Assistant, cskelton@vt.edu. Student Member ASME.‡Professor, cdhall@vt.edu. Associate Fellow AIAA, Member AAS.

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were made for singularity avoidance. The advantage of Ford’s equations are that the actual motortorques are explicit for ease of implementation.

Control laws developed to use thrusters for coarse slew control and momentum wheels for precisecontrol or to reduce errors due to initial conditions have been proven to be effective [8]. Thispaper develops control strategies that use single-gimbal control moment gyros for coarse large slewmaneuvering and momentum wheels for precise control and error reduction due to initial conditions.Mixing control moment gyros and momentum wheels for attitude control is believed to be a novelapproach.

The governing equations for a rigid body with control moment gyros and momentum wheels arepresented much like that of Ref. [6]. Lyapunov control laws are developed from the governingequations. Feedback linearization techniques are used to transform the state equations and thenchoose a control input to cancel out the nonlinearities of the system dynamics. Simulations of thederived mixed control moment gyro and momentum wheel control laws are evaluated. The goal ofthis paper is to develop control laws that use control moment gyros and momentum wheels for largeslew maneuvers.

EQUATIONS OF MOTION

The governing equations of a rigid body with gimballed momentum wheels have been developed inRef. [6]. These equations are easily used to represent a body with any combination of control momentgyros, momentum wheels, and gimballed momentum wheels (GMW). We modify the equationsslightly to represent a rigid body with only control moment gyros and momentum wheels. Thegoverning equations represent the absolute angular momentum of the body about its center of mass,the absolute angular momenta of the actuators about their control axes, the gimbal angles, andthe Modified Rodrigues Parameters from the body fixed frame to the inertial frame. These arethe equations that are used to derive the mixed control laws. In this section we briefly review theequations derived by Ford [6] and the notation. Refs. [4, 6] contain complete derivations of theseequations. We then use Ford’s notation to present the governing equations for a rigid body withCMGs and MWs.

GMW Equations of Motion

The equations that describe the motion of a rigid body with N GMWs are

h = −ω×h + ge (1)hswa = gw (2)hga = diag((It − (Isg + Isw))AT

s ω− IswΩ)(ATt ω) + gg (3)

δ = I−1g hga −AT

g ω (4)

where

h = (IB + AtItATt + AsIsgAT

s )ω + Ashswa + Aghga (5)hswa = IswAT

s ω + AsΩ (6)

hga = IgATg ω + Igδ (7)

(8)

The absolute angular momentum of the system is h. The skew symmetric matrix is defined as

ω× =

0 ω3 −ω2

−ω3 0 ω1

ω2 −ω1 0

(9)

The external torques on the system are ge. The absolute axial angular momentum of the GMWflywheel is hswa and the torque on the flywheel about the same axis is gw. Equation 3 describes

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the change in absolute angular momentum of the GMW about its gimbal axis. The matrices Is, It,and Ig are diagonal matrices that represent the inertias of the GMWs about their spin, transverse,and gimbal axes, respectively. The spin axis inertia matrix is split into Isw and Isg to represent theinertia of the flywheel and GMW frame, respectively. The angular velocity of between the body-fixedreference frame and the inertial reference frame is ω. The relative speeds of the momentum wheelsare Ω, and the gimbal angle rates are δ, where both are N×1. A matrix containing all of the gimbalangles is δ. The torques that the gimbal motors exert on the GMWs are gg. The matrices As, At,and Ag contain the orientation of the GMWs spin, transverse, and gimbal axes, respectively. Notethat As and At change with gimbal angle, while Ag remains constant regardless of gimbal angle.The inertia of the body including transverse inertias of the momentum wheels are IB .

CMG and Momentum Wheel Equations of MotionWe wish to use this notation to define the equations of motion for a rigid body with N controlmoment gyros, and M momentum wheels. The momentum wheels are defined in a similar mannerto the GMWs. Each momentum wheel has a reference frame FWk

(k = 1...M) that is fixed in thebody. The vector components of the unit axes (~csk,~ctk,~cgk) are assumed to be given in Fb. Wedefine a matrix

Cs = [~cs1,~cs2, ...,~csk] (10)

to describe the orientation of the spin axis in the body. We form an additional inertia like matrixfor the momentum wheels formed from the flywheels spin inertia as,

IswMW = diag(IswMW1, IswMW2, ..., IswMWM ) (11)

The spin axis angular momentum of the k-th MW is not contained in the body angular momentumterms, so we form an M × 1 matrix

hswMWa = IswMW CTs ω + IswMW ΩMW (12)

where ΩMW is an M × 1 matrix formed with the relative angular velocities of the flywheels abouttheir spin axis. We define the momentum wheel’s change in absolute angular momentum about itsspin axes as

hswMWa = gMW (13)

The equations that describe the motion of each CMG are similar to the equations for a GMW.We exclude the time rate of change of absolute angular momentum of the CMG spin axis becausean independent controller ensures that the angular velocity of the flywheel about its spin axis isconstant relative to the CMG frame.

KinematicsFor this research we choose to use Modified Rodrigues Parameters (MRPs) because singularities canbe avoided with use of the shadow set [9]. Euler’s Theorem states that any rotation about a singlefixed point can be described by a single rotation about a fixed axis through that point. The axisand angle are called Euler axis and Euler angle, respectively. It is common to define e as the Euleraxis and Φ as the Euler angle. The three element set of MRPs is defined as

σ = e tan(Φ/4) (14)

As presented in Ref. [9] we use a shadow set of MRPs

σS = − 1σT σ

σ (15)

to avoid singularities. The shadow set replaces the original MRPs when σT σ > 1. The magnitude ofthe MRP that switches to the shadow set is user defined. The conversion from MRPs to a directioncosine matrix is

R = 1 +4(1−σT σ)(1 + σT σ)2

σ× +8

(1 + σT σ)2(σ×)2 (16)

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Knowing the angular velocity as a function of time we can determine the attitude by integrating

σ = G(σ)ω (17)

where ω is the angular velocity between the two frames and

G(σ) =12

(1 + σ× + σσT − 1 + σT σ

21)

(18)

Summary

The equations that describe the motion of a rigid body with N CMGs and M momentum wheelsare

h = −ω×h + ge (19)hga = diag((It − (Isg + Isw))AT

s ω− IswΩ)(ATt ω) + gg (20)

hswMWa = gMW (21)

δ = I−1g hga −AT

g ω (22)σ = G(σ)ω (23)

where

h = (IB + AtItATt + AsIsgAT

s )ω + Ashswa + Aghga + CshswMWa (24)hswa = IswAT

s ω + AsΩ (25)

hga = IgATg ω + Igδ (26)

hswMWa = IswMW CTs ω + IswMW ΩMW (27)

CONTROL LAWS

We wish to define control laws that use the torque amplification of CMGs for large rapid slew andthe fine tuning abilities of momentum wheels. In this section we present the reference body dynamicsand develop the state equations for this problem. We present a CMG feedback control law developedby Oh and Vadali [5]. This control law allows us to find the open loop CMG commands for thetracking maneuver. Derivation of Lyapunov and feedback linearization control laws conclude thissection.

Reference Body

The use of a reference body for tracking follows that of Ref. [8]. We start by defining a referencetrajectory. Consider a rigid reference body that has a body-fixed reference frame FR. The centerof mass of the reference body is coincident with the origin of FR and is fixed in inertial space. Theorientation of FR with respect to the inertial reference frame FI is expressed by the direction cosinesmatrix RRI . The MRPs that describe RRI are

RRI = 1 +4(1−σT

RσR)(1 + σT

RσR)2σ×R +

8(1 + σT

RσR)2(σ×R)2 (28)

The equations that describe the motion of the reference body are

hR = −ω×RIBωR + gR (29)σR = G(σR)ωR (30)

Note that the reference body has the same inertia of the controlled body without the actuatorinertias.

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State Equations

With the reference body and the actual body fully defined we can form the terms δω and δσ whichare the difference between the body and reference angular velocity and attitude, respectively:

δσ =(1−σT

RσR)σ− (1−σT σ)σR + 2σ×σR

1 + (σTRσR)(σT σ) + 2σRσ

(31)

δω = ω−RBRωR (32)

From these equations we defineδh = h− IBRBRωR (33)

the difference between the angular momentum of the bodies, where RBR = R(δσ).

Substituting equation 24 into δh and simplifying gives

δh = IBω + (AtItATt + AsIsgAT

s )ω + Ashswa + Aghga (34)+CshswMWa − IBRBRωR

= IBδω + (AtItATt + AsIsgAT

s )ω + Ashswa + Aghga (35)+CshswMWa

Taking the time derivative of equation 33 yields

δh = h− IBdRBR

dtωR − IBRBRωR (36)

Hall et.al. [8] showed that

IBdRBR

dtωR = IBω×δω (37)

Taking the time derivative of Equation 35 gives

δh = IBδω + Ashswa + Ashswa + Aghga + CshswMWa (38)

+(AtItATt + AtItAT

t + AsIsgATs + AsIsgAT

s )ω + (AtItATt + AsIsgAT

s )ω

The state equations are the attitude and angular velocity error rates. We use Equation 38 andEquation 17 to express the states as

δσ = G(δσ)δω (39)δω = I−1

B [δh− Ashswa −Ashswa −Aghga − (AtItATt + AsIsgAT

s )ω (40)

−(AtItATt + AtItAT

t + AsIsgATs + AsIsgAT

s )ω−CsgMW ]

Open Loop CMG Control

We use a gimbal velocity law first presented by Oh and Vadali [5] to find the CMG control maneuverfor a large rapid slew. Presented in this notation the gimbal torques are

gg = Igδ + IgATg ω− diag((It − Is)AT

s ω− IswΩ)ATt ω (41)

where

δ = kδ(δdes − δ) (42)

δdes = D†(k3δω + k4δσ− JRBRωR − J−1ω×(Jω + Ashswa + Aghga)) (43)D = −Atdiag(IswΩ) (44)

+1/2[(as1aTt1 + at1aT

s1)(ω + RBRωR) (as2aTt2 + at2aT

s2)(ω + RBRωR)......(asNaT

tN + atNaTsN )(ω + RBRωR)](It − Is)

The coefficients kδ, k3, and k4 are positive constant scalars.

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Lyapunov Controller

We have developed the equations that we need to derive the control law. Consider a Lyapunovcandidate function

V =12δωT K1δω + 2k2 ln(1 + δσT δσ) (45)

where K1 = KT1 > 0 and k2 > 0. The derivative of V is

V = HT I−1B K1δω + k2δσ

T δω (46)

where

H = δh− Ashswa −Ashswa −CshswMWa −Aghga (47)

−(AtItATt + AtItAT

t + AsIsgATs + AsIsgAT

s )ω−(AtItAT

t + AsIsgATs )ω

Equation 47 can be expanded to

H = −ω×((IB + AtItATt + AsIsgAT

s )ω + Ashswa + Aghga (48)+CshswMWa) + ge − IBω×δω− IBRBRI−1

B [−ω×RIωR + gR]

−Ashswa −Asgw −CsgMW

−Ag(diag((It − Is)ATs ω− IswΩ)(AT

t ω) + gg)

−(AtItATt + AtItAT

t + AsIsgATs + AsIsgAT

s )ω−(AtItAT

t + AsIsgATs )ω

With H in expanded form we identify the control torques for the body ge, gw, gg, and gMW . Forany CMG to maintain a constant velocity of the flywheel about its spin axis relative to the bodythe spin torque is

gw = Isw(ATs ω + AT

s ω) (49)

For this problem we consider ge, gw, and gg as known parameters. With the gMW input we caninsure that V < 0 when δω 6= 0 and δσ 6= 0.

By choosing K1 = IB equation 46 becomes

V = δωT (H + k2δσ) (50)

We choose the control torque gMW to be

CsgMW = δh− Ashswa −Asgw −Aghga − (AtItATt + AsIsgAT

s )ω (51)

−(AtItATt + AtItAT

t + AsIsgATs + AsIsgAT

s )ω+k2δσ + k1δω

which yields

V = −k1δωT δω ≤ 0 (52)

since k1 > 0. From this result we see that

limt→∞

δω(t) = 0 (53)

With Equations 35, 38, and 51, we have

IBδω = −k2δσ− k1δω (54)

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Due to Equation 53 we havelim

t→∞Jδω = − lim

t→∞k2δσ (55)

so that δσ → 0. From Equation 39 we know that δσ → 0, so that the limit of δσ is a constant,therefore

limt→0

δσ = 0 (56)

With LaSalle’s Theorem [10], the attitude and angular velocity error due to tracking with this controllaw is globally asymptotically stable.

Input-State Feedback Linearization Controller

We now take a different approach to controlling this problem. For a certain class of nonlinearsystems we can use a state feedback control and a change of variables that transforms the nonlinearsystem into an equivalent linear system [10]. We investigate input-state linearization of this nonlinearsystem.

We begin with the state equations 39 and 40 which we transform with

z1 = δσ (57)z2 = G(δσ)δω = δσ (58)

The transformed state equations are

z1 = z2 (59)z2 = Gδω + Gδω (60)

(61)

where G = G(δσ). Substitution of Equation 40 allows the state equations to be written as

z1 = z2 (62)z2 = Gδω + G(I−1

B (δh− Ashswa −Ashswa −Aghga (63)

−(AtItATt + AtItAT

t + AsIsgATs + AsIsgAT

s )ω−(AtItAT

t + AsIsgATs )ω−CsgMW )

(64)

In this form we can choose CsgMW to cancel the nonlinear terms. We choose the momentum wheelcontrol torques to be

CsgMW = δh− Ashswa −Ashswa −Aghga (65)

−(AtItATt + AtItAT

t + AsIsgATs + AsIsgAT

s )ω−(AtItAT

t + AsIsgATs )ω + IBν

whereν = k5δω + k6δσ (66)

The constants k5 and k6 are positive. The system with control is

z =([

0 10 G

]+

[k6G 0

0 k5G

])z (67)

With the system represented linearly we can use linear stability analysis. From linear theory we canguarantee asymptotic stability if the coefficient matrix is Hurwitz [10].

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NUMERICAL SIMULATION

We present the results of numerical simulations of the derived control laws for illustration. We firstpresent the feed forward control inputs as calculated by the closed loop control moment gyro controllaw. Then we show how slight variations in initial conditions make the system diverge (as expected)from the desired reference conditions with only CMG feed forward inputs. Implementation of theclosed loop momentum wheel control laws show that initial errors are rejected with minimal torques.

We define the actuator orientations in FB which is aligned with the principal inertia reference frame.The principal inertia matrix is IB = diag(6, 5, 10) kg m2. Three CMGs are used for the feed forwardcontrol aligned such that

As0 =

1 0 00 1 00 0 1

(68)

Ag =

0 −1 00 0 −1−1 0 0

(69)

(70)

All three CMGs have the same inertia properties, Isw = 0.01 kg m2, Isg = 0.01 kg m2, It = 0.01kg m2, and Ig = 0.01 kg m2. The nominal relative angular velocity of each CMG flywheel is 2000rpm. Three momentum wheels are aligned with the principal axes with Isw = 0.04 kg m2 for eachmomentum wheel. The transverse inertias of the momentum wheels along with inertias of the torquedevices for the momentum wheels are included in the body inertia.

The reference motion to be tracked is a 90o slew about the 3 axis of the body. The reference motionis generated with bang-bang external torques to simulate the motion of a body with thrusters. TheModified Rodrigues Parameter representation of the reorientation of the reference body is shown inFigure 1.

The gains for determining the control moment gyro maneuvering for the reorientation are kδ = 100,k4 = 100, and k3 =

√2Iiik4. Figure 2 shows the attitude and velocity error for the body with

CMG feedback control. We now take the exact inputs to the CMGs that produced the previousresult and implement them as feed forward inputs. Figure 3 shows the feed forward CMG controlwith an initial error of σ = [0.01, 0.02,−0.01]. Clearly without feedback control, any initial error orperturbation will result in errors in tracking and final orientation.

We apply the Lyapunov momentum wheel feedback control laws. The gains are k1 = 54 and k2 = 47.Figure 4 shows that the momentum wheels correct for initial error while allowing the majority of thetorque for the slew to come from the CMGs. Figure 5 shows the momentum wheel applied torquesthroughout the maneuver.

Numerical simulations of the feedback linearization control law were performed. The linear gainswere chosen to be k5 = 5 and k6 = 4 for this simulation. Figure 6 is the tracking error. The controltorques that the momentum wheels apply to the system are shown in Figure 7.

CONCLUSIONS

The objective of this research is to derive and validate mixed CMG and momentum wheel controllaws. A control moment gyro gimbal rate feed forward control law provides output torques requiredfor large rapid angular slew maneuvers. Lyapunov and feedback linearization techniques are usedto develop momentum wheel control laws for initial condition and tracking error rejection. Thecontrollers are derived from the nonlinear state equations for attitude and angular velocity error.Both control laws guarantee asymptotic convergence. Numerical simulations are preformed to verifyconvergence of the system with the derived control laws. Momentum wheel control torques for eachcontroller are small in comparison to the torques required for the angular slew maneuver.

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Figure 1: MRP to be tracked

Future work will include implementation of the mixed actuator attitude control laws on the Dis-tributed Spacecraft Attitude Control System Simulator (DSACSS) at Virginia Tech. Exploration ofmomentum wheel feedback laws that do not have knowledge of the open loop CMG dynamics will bepreformed. Power studies of mixed CMG and momentum wheel attitude control versus momentumwheel feedback control must be conducted to verify the advantage of the mixed actuator attitudecontrol concept.

References

[1] J. R. Wertz and W. J. Larson, Space Mission Analysis and Design. New York, New York:Microcosm Press, 1999.

[2] S. J. P. Robert H. Bishop and J. W. Sunkel, “Adaptive control of space station with controlmoment gyros,” IEEE Control Systems, vol. 12, pp. 23–28, October 1992.

[3] S. N. Singh and T. C. Bossart, “Exact feedback linearization and contol of space station usingCMG,” IEEE Transactions on Automatic Control, vol. 38, pp. 184–187, January 1993.

[4] K. Ford and C. Hall, “Singular direction avoidance steering for control-moment gyros,” Journalof Guidance, Control, and Dynamics, vol. 23, pp. 648–656, July 2000.

[5] H. Oh and S. Vadali, “Feedback control and steering laws for spacecraft using single gimbalcontrol moment gyros,” Journal of the Astronautical Sciences, vol. 38, pp. 183–203, April 1991.

[6] K. A. Ford, Reorientations of Flexible Spacecraft Using Momentum Exchange Devices. PhDthesis, Air Force Institute of Technology, 1997.

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Figure 2: Tracking Reference Body with CMG feedback

[7] H. Schaub, S. R. Vadali, and J. L. Junkins, “Feedback control law for variable speed controlmoment gyros,” Journal of the Astronautical Sciences, vol. 46, pp. 307–328, July 1998.

[8] C. D. Hall, P. Tsiotras, and H. Shen, “Tracking rigid body motion using thrusters and momen-tum wheels,” The Journal of the Astronautical Sciences, vol. 50, no. 3, 2002.

[9] M. D. Shuster, “A survey of attitude representations,” The Journal of the Astronautical Sci-ences, vol. 41, no. 4, pp. 439–518, 1993.

[10] H. K. Khalil, Nonlinear Systems. New York, New York: Macmillian, 1992.

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Figure 3: Tracking Reference Body with CMG Feed Forward and Initial Error

Figure 4: Tracking Reference Body with CMG Feed Forward,Initial Error, and MW Lyapunov feed-back control

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Figure 5: MW Lyapunov feedback control torques

Figure 6: Tracking Reference Body with CMG Feed Forward, Initial Error, and MW feedback lin-earization

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Figure 7: MW feedback linearization torques

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