satellite attitude control and power tracking with energy

JOURNAL OF GUIDANCE CONTROL AND DYNAMICS

Vol 24 No 1 JanuaryndashFebruary 2001

Satellite Attitude Control and Power Trackingwith EnergyMomentum Wheels

Panagiotis Tsiotras curren and Haijun Shendagger

Georgia Institute of Technology Atlanta Georgia 30332-0150and

Chris HallDagger

Virginia Polytechnic Institute and State University Blacksburg Virginia 24061-0203

A control law for an integrated powerattitude control system (IPACS) for a satellite is presented Four or moreenergymomentum wheels in an arbitrary noncoplanar con guration and a set of three thrusters are used toimplement the torque inputs The energymomentum wheels are used as attitude-control actuators as well as anenergy storage mechanism providing power to the spacecraft In that respect they can replace the currently usedheavy chemical batteries The thrusters are used to implement the torques for large and fast (slew) maneuversduring the attitude-initialization and target-acquisition phases and to implement the momentum managementstrategies The energymomentum wheels are used to provide the reference-tracking torques and the torques forspinning up or down the wheels for storing or releasing kinetic energy The controller published in a previous workby the authors is adopted here for the attitude-tracking function of the wheels Power tracking for charging anddischarging the wheels is added to complete the IPACS framework The torques applied by the energymomentumwheels are decomposed into two spaces that are orthogonal to each other with the attitude-control torques andpower-tracking torques in each space This control law can be easily incorporated in an IPACS system onboarda satellite The possibility of the occurrence of singularities in which no arbitrary energy pro le can be trackedis studied for a generic wheel cluster con guration A standard momentum management scheme is considered tonull the total angular momentum of the wheels so as to minimize the gyroscopic effects and prevent the singularityfrom occurring A numerical example for a satellite in a low Earth near-polar orbit is provided to test the proposedIPACS algorithm The satellitersquos boresight axis is required to track a ground station and the satellite is requiredto rotate about its boresight axis so that the solar panel axis is perpendicular to the satellitendashsun vector

Introduction

M OST spacecraft use chemical batteries (usually NiCd orNiH2) to store excess energy generated by the solar pan-

els during periods of exposure to the sun1 During an eclipse thebatteries are used to provide power for the spacecraft subsystemsThe batteries are recharged when the spacecraft is in the sunlightThe primary problem with this approach is the cycle life of batter-ies and the additional power system mass required for controllingthe charging and discharging cycles Chemical batteries have shal-low discharge depth (approximately 20ndash30 of their rated energy-storage capacity) large weight and require operation within stricttemperature limits (at or below 20degC in a low-Earth orbit) that oftendrive the entire spacecraft thermal design

An alternative to chemical batteries is the use of ywheels tostore energy The use of ywheels has the bene t of increased ef- ciency (up to 90 depth of discharge with essentially unlimitedlife) operation in a relatively hot (up to 40degC) environment andthe potential to combine the energy-storage and the attitude-controlfunctions into a single device thus increasing reliability and re-ducing overall weight and spacecraft size This concept termedthe integrated power and attitude-control system (IPACS) has beenstudied since the 1960s but it has been particularly popular sincethe 1980s In fact the use of ywheels instead of batteries to storeenergy on spacecraft was suggested as early as 1961 in the pa-per by Roes2 when a 17 Wcent hkg composite ywheel spinning at

Received 14 October 1999 revision received 4 March 2000 acceptedfor publication 17 March 2000 Copyright cdeg 2000 by the authors Pub-lished by the American Institute of Aeronautics and Astronautics Inc withpermission

curren Associate Professor School of Aerospace Engineering ptsiotrasaegatechedu Senior Member AIAA

daggerGraduate Student School of Aerospace Engineering gt7318dprismgatechedu Student Member AIAA

DaggerAssociate Professor Department of Aerospace and Ocean Engineeringchallaoevtedu Associate Fellow AIAA

10000ndash20000 rpm on magnetic bearings was proposed The con- guration included two counterrotating ywheels and the authordid not mention the possibility of using the momentum for attitudecontrol However because many spacecraft use ywheels (in theform of momentum wheels control moment gyros etc) to controlattitude the integration of these two functions is naturally of greatinterest Numerous studies of this integration have been conductedAnderson and Keckler3 originated the term IPACS in 1973 A studyby Cormack4 done for the Rockwell Corporation examined the useof an integrated IPACS Keckler and Jacobs5 presented a descrip-tion of the concept Will et al6 investigated the IPACS concept andperformed simulations by using (linearized) equations of motionNotti et al7 performed an extensive systems study and investigatedlinear control laws for attitude control Their study included tradestudies on the use of momentum wheels control moment gyrosand counterrotating pairs NASA and Boeing also conducted sep-arate studies on the IPACS concept89 Anand et al1011 discussedthe system design issues associated with using magnetic bearingsas did Downer et al12 Flatley13 studied a tetrahedron array of fourmomentum wheels and considered the issues associated with si-multaneously torquing the wheels for attitude control and energystorage Around the same time as Flatley OrsquoDea et al14 includedsimultaneous attitude determination in their study of a combined at-titude reference and energy-storage (CARES) system focusing ontechnology-related issues Oglevie and Eisenhaure1516 performed asystem-level study of IPACSs Reference 15 includes a substantiallist of references to earlier work Olmsted17 presented technology-related issues associated with a particular ywheel design Optimaldesign criteria associated with an integrated attitude-control andenergy-storage (ACES) system have been discussed by Studer andRodriguez18

Most of these previous investigations of the IPACS focus on gen-eral design issues The exact nonlinear equations of motions arenot considered even when the attitude-control results are providedIn this paper the exact nonlinear equations of motion are used todesign an attitude controller that tracks a reference attitude pro le

23

24 TSIOTRAS SHEN AND HALL

The wheel-control torque space is decomposed into two spaces thatare orthogonal to each other The attitude-control torques lie in oneof these two spaces and the torques in the other space are usedto spin up or down the energymomentum wheels to store or ex-tract kinetic energy In the following sections the system model isgiven rst then the reference attitude and energymomentum-wheelpower-tracking controllers are presented A numerical example con-sidering a iridium-type satellite19 in orbit is provided to illustrate theproposed IPACS methodology

It should be pointed out that ywheels used in an IPACS typicallyrotate at much higher speeds than standard momentum wheels astheir operation is driven primarily by the energy-storage functionrather than the attitude-control function In this paper we use theterm energymomentum wheels to emphasize this difference withcommonly used momentum wheels

System ModelDynamics

Consider a rigid spacecraft with an N -wheel cluster installed toprovide internal torques Let B denote the spacecraft body frameThen the rotational equations of motion for the spacecraft can beexpressed as

CcedilhB = h poundB J iexcl 1(hB iexcl Aha) + ge (1a)

Ccedilha = ga (1b)

where hB is the angular-momentum vector of the spacecraft in theB frame given by

hB = JB + Aha (2)

where ha is the N pound 1 vector of the axial angular momenta of thewheels B is the angular-velocity vector of the spacecraft in theB frame ge is the 3 pound 1 vector of external torques ga is the N pound 1vector of the internal axial torques applied by the platform to thewheels and A is the 3 pound N matrix whose columns contain the axialunit vectors of the N energymomentum wheels J is an inertialikematrix de ned as

J = I iexcl AIsAT (3)

where I is the moment of inertia of the spacecraft including thewheels and Is = diagIs1 Is2 Is N is a diagonal matrix withthe axial moments of inertia of the wheels

The axial angular-momentum vector of the wheels can be writtenas

ha = IsAT B + Is s (4)

where s = ( x s1 x s2 x s N )T is an N pound 1 vector denoting theaxial angular velocity of the energymomentum wheels with respectto the spacecraft We denote the total axial angular velocity of thewheels relative to the inertial frame as c = ( x c1 x c2 x cN )T Using this notation we can write Eq (4) as

ha = Is c (5)

where c = s + AT B Note that because typically the wheels spinat a much higher speed than the spacecraft itself s Agrave B and wehave that c frac14 s

In Eq (1) the external torques are assumed to include the controltorque that is due to thruster ring the gravity gradient torque andthe other disturbance torques ie

ge = gt + gg + gd (6)

where the subscripts t g and d denote the thruster gravity gradientand disturbance respectively The gravity gradient torque is givenby20

gg =iexcl3 l | R3

c

cent| c pound

3 Ic3 (7)

where c3 is the unit vector iexcl rc Rc [the same as zl the z axis ofthe local-vertical local-horizontal (LVLH) frame shown in Fig 1]

Fig 1 LVLH frame

expressed in the body frame rc is the vector from the Earthcenter to the spacecraft center of mass with Rc = j rc j and l =3986005 pound 105 km3 s iexcl 2 is the constant gravitational parameter Itshould be pointed out that although we restrict the discussion toexternal control torques that are due to thruster ring our approachremains the same for other choices of actuators such as magneto-torques etc

KinematicsThe so-called modi ed Rodrigues parameters (MRPs) given in

Refs 21 and 22 are chosen to describe the attitude kinematicserrorofthe spacecraft The MRPs are de ned in terms of the Euler principalunit vector e and angle u by

frac34 = e tan( u 4) (8)

The MRPs have the advantage of being well de ned for the wholerange for rotations ie u 2 [0 2 p ) The differential equation gov-erning the kinematics in terms of the MRPs is given by21

Ccedilfrac34 = G(frac34) (9)

where

G(frac34) = 121 + frac34 pound + frac34frac34T iexcl [(1 + frac34T frac34) 2]1 (10)

and 1 is the 3 pound 3 identity matrix

Tracking ControllerIn one of our previous works23 three control laws were presented

to track a reference attitude pro le by using coordinated action ofboth thrusters and momentum wheels Here we restate the con-troller II from Ref 23 that will be used for attitude tracking Thiscontrol law assumes that the reference frame dynamics and kine-matics are given by

CcedilhR = h poundR J iexcl 1hR + gR (11)

Ccedilfrac34R = G(frac34R )R (12)

TSIOTRAS SHEN AND HALL 25

where the subscript R stands for the reference frame to be tracked(referred to below as the desired or the target frame) and wherehR = JR

Given the reference attitude history to be tracked from Eqs (11)and (12) the angular-velocity tracking error in the body frame is

d = B iexcl CBR ( d frac34) R (13)

where CBR ( d frac34) is the rotation matrix from the reference frame R to

the body frame B and d frac34 is the MRP vector between the referenceframe and the body frame ie

CBR ( d frac34) = CB

N

poundCR

N

currenT(14)

and the attitude error satis es the following differential equation

d Ccedilfrac34 = G( d frac34) d (15)

A feedback control law to render d 0 and d frac34 0 is found withthe following Lyapunov function

V = 12d T J iexcl 1 d + 2k2 (1 + d frac34T d frac34) (16)

where k2 gt 0 This function is positive de nite and radially un-bounded21 in terms of the tracking errors d and d frac34 Taking thederivative of V and using Eqs (13) and (15) yields the time deriva-tive of V in terms of B and R and the tracking error d and d frac34ie

CcedilV = d Tpoundh pound

B J iexcl 1(hB iexcl Aha ) + gt + gg iexcl J poundB d

iexcl JCBR ( d frac34)J iexcl 1h pound

R J iexcl 1hR iexcl JCBR ( d frac34)J iexcl 1gR iexcl Aga + k2 d frac34

curren

(17)

The external torque gt is typically chosen to effect a desired large-angle attitude correction whereas the internal torques ga are chosento eliminate errors For any external torque if we select the internaltorques such that

Aga = h poundB J iexcl 1(hB iexcl Aha ) + gt + gg iexcl J pound

B d


R J iexcl 1hR iexcl JCBR ( d frac34)J iexcl 1gR + k1 d + k2 d frac34

(18)

with k1 gt 0 the derivative of the Lyapunov function is

CcedilV = iexcl k1 d T d middot 0 (19)

As shown in Ref 23 the tracking error system is asymptoticallystable ie B R and frac34B frac34R as t 1

Equation (18) allows for a variety of different control actions Inparticular once the thruster control action gt has been chosen theenergymomentum-wheel action is computed directly from Eq (18)Later on we will use this equation to design control laws duringtwo different phases of the satellite operation namely the target-acquisition (attitude-initialization ) phase and the attitudepower-tracking phase We also point out that in the absence of initial errors[ d (0) = d frac34(0) = 0] and for initial wheel axial momentum zero[ha (0) = 0] Eq (18) reduces to Aga = 0 and gt = gR iexcl gg

Given the torque action gt we denote the right-hand side ofEq (18) by f The internal torque provided by the wheels ga isgiven by the solution of the linear system Aga = f If N lt 3 thissystem is overdetermined and a solution may not exist if N = 3(and for noncoplanar wheels) the solution is uniquely determinedand if N gt 3 the system is underdetermined and there exist an in- nite number of solutions In particular in the latter case everysolution has the form ga = gr + gn where gr belongs to the rangespace R(AT ) of the matrix AT and gn belongs to the null spaceN (A) of the matrix A

It is seen that gn does not contribute to the attitude control input asAgn = 0 One can always choose the torque gr to ful ll the equationAgr = f and subsequently use gn to perform powerenergy-storagemanagement24 Note that this approach can be implemented as longas N (A) has nonzero dimension which is always true for a clusterwith more than three noncoplanar wheels

In the following section we consider a general momentum-wheelcluster and construct the torques in the null space of A so as notto disturb the attitude-control operation of the spacecraft and totrack a desired power pro le In other words the power- and theattitude-tracking operations are performed simultaneously and in-dependently of one another Power-tracking objectives do not inter-fere with attitude-tracking objectives and vice versa We insist onthis separation of objectives as it is unlikely that any IPACS thatcompromises either power or attitude-control requirement will beacceptable for use in routine spacecraft operations

Power TrackingAs shown in the preceding section the energymomentum-wheel

torque required for controlling the attitude is given by an expressionof the form

Aga = f (20)

where f is the 3 pound 1 required torque vector The general solution forga is given by

ga = A+ f + gn (21)

where A+ = AT (AAT ) iexcl 1 is the projection operator on the range ofAT and thus A+ f = gr 2 R(AT ) and where gn 2 N (A) ie

Agn = 0 (22)

Note that Aga = A(A+ f + gn) = f so gn does not affect the space-craft motion

The total kinetic energy stored in the wheels is

T = 12 T

c Isc (23)

The power (rate of change of the energy) is given by

dT

dt= P = T

c Is Ccedilc (24)

The objective here is to nd a controller gn in the null space of Ato provide the required power function P(t ) Equation (5) impliesthat ga = Ccedilha = Is Ccedilc so from Eq (24) we have

Tc ga = P (25)

Therefore simultaneous attitude control and power managementrequire a control torque vector ga that satis es the following set oflinear equations

sup3A

Tc

acutega =

sup3f

P

acute(26)

From Eq (21) we have that the torque gn in the null space of A hasto satisfy

Tc (A+ f + gn ) = P (27)

or let the modi ed power Pm = P iexcl Tc A+ f

Tc gn = Pm (28)

Because gn 2 N ( A) there always exists a vector ordm 2 RN such that

gn = PN ordm (29)

where PN = 1N iexcl AT (AAT ) iexcl 1A is the (orthogonal) projection onN ( A) Thus we have T

c PN ordm = Pm to which a minimum normsolution is given by

ordm = PN c

iexclT

c PN c

cent iexcl 1Pm (30)

and the energy management torque gn can then be chosen as

gn = PN c

iexclT

c PN c

cent iexcl 1Pm (31)


A solution of Eq (31) exists as long as PN c 6= 0 This impliesthat either c 6= 0 or that c 62 N (A) = R(AT ) The last require-ment is also evident from Eq (26) in which the matrix on theleft-hand side does not have full row rank if c 2 R(AT )

It should be pointed out that for satellite applications duringsunlight the solar panels provide enough power for the spacecraftequipment and the wheels spin up to absorb and store the excessenergy Because the sunlight period is longer than the eclipse pe-riod tracking a speci c power function is less signi cant during thesunlight than during the eclipse period when the power is solelyprovided by spinning down the wheels Some authors have there-fore chosen to discard power tracking altogether and simply spinup the wheels during sunlight to store the excess energy This is theapproach used for example in Ref 13 On the contrary unloadingof the wheels during the eclipse is more critical because the wheeldeceleration should be done at a certain rate in order to provide thenecessary power to the spacecraft bus

Singularity Avoidance and Momentum ManagementSo far we have found a controller ga that tracks desired power

pro les while controlling the spacecraft attitude In addition as wasmentioned already from Eq (28) we can see that if c lies in therange space of AT then because gn is in the null space of A we have

Tc gn = 0 (32)

This case is termed as singular which implies that the controllerloses the capability of tracking an arbitrary power function and fromEq (27) the only power the wheels can supply is T

c A+ f Howeverthe latter case is undesirable in many practical applications Forexample for a stabilized spacecraft when the torque f is small theamount of supplied power can be less than the required power levelduring the singularity

For a three-axis stabilized satellite if the angular momenta of thewheels are not distributed such that the total angular momentumof the wheels stays close to zero the gyroscopic effects will playa signi cant role increasing the required torque for the attitudecontrol Hence in the following a momentum management schemewill be included such that the total angular momentum of the wheelswill become zero when some external torques are applied to thespacecraft when needed It will be seen that this scheme will furtherreduce the occurrence of the singularity mentioned above

The ideal purpose of the momentum management is to makethe total angular momentum of the wheels zero or keep it withincertain limits Here it is assumed that zero total momentum is re-quired ie Aha = 0 which implies that ha 2 N (A) For a clus-ter with identical wheels (the typical case) Eq (5) implies thatc 2 N (A) Recall now that singularity occurs when c 2 R(AT )But N (A) = R(AT ) and after momentum management the wheelangular-velocity vector is orthogonal to the singularity subspaceR(AT ) The possibility of singularity has thus been reduced asmuch as possible This does not of course include the case in whichc = 0 which belongs to both N (A) and R(AT ) However becausefor a safety margin the energy stored in the wheels must alwaysexceed the minimum energy level necessary for the spacecraft op-eration the all-zero wheel angular velocity can happen only duringinitial satellite deployment when the wheels may be in a lockedposition In this case the wheel angular-velocity initialization isperformed after the deployment In doing so a torque in the nullspace of A is applied to the wheels to accelerate the wheel angularvelocities to their normal operation range

A simple momentum management scheme is adopted fromRef 25 The torque required for momentum management is

gt = iexcl k(Aha iexcl Ahan) (33)

where han denotes the nominal angular-momentum vector of thewheels and k gt 0 is a feedback control gain For the purpose ofmomentum unloading and singularity avoidance here we chooseAhan = 0

Numerical ExampleTo demonstrate the aforementioned algorithm for the attitude-

and power-tracking controller the following numerical example has

Table 1 Iridium 25578 orbital elements

n revday 1457788549M0 deg 2347460x deg 1255766X deg 1328782i deg 865318e 000216220Epoch 05231999 00161224

Fig 2 Con guration of energymomentum wheels

been performed A near-polar orbital satellite (orbital data chosenfrom the satellite Iridium 25578) is considered in this simulation19

The orbital elements are shown in Table 1 n is the orbital frequencyM0 is the mean anomaly at the epoch time x is the argument ofperigee X is the right ascension of the ascending node i is theorbital inclination and e is the eccentricity of the orbit The satelliteis assumed to have moment of inertia matrix in a principal axessystem

I =

2

4200 0 0

0 200 0

0 0 175

3

5 kg m2 (34)

The four-wheel cluster shown in Fig 2 is chosen for this simu-lation The wheels are assumed to have the same axial moments ofinertia The A matrix in this case is given by

A =

264

1 0 0p

33

0 1 0p

33

0 0 1p

33

375 (35)

The normal power requirement of this satellite is 680 W How-ever it is required to be able to provide an instantaneous peakpower of 4 kW for up to 5 min Considering the 34-min eclipsetime and assuming that the 4-kW peak power lasts 5 min we ndthat the wheels should store at least 072-kWh energy With a 100safety margin taken into account the wheels are required to store15-kWh energy when they are fully charged Suppose that the nom-inal speed for the fourth wheel when the wheels are fully chargedis iexcl 4000 rads ( iexcl 38197 rpm) and the speed for the other three is23094 rads (22053 rpm) These speeds render zero total angularmomentum of the wheels The energy and the wheel speed requirethat each wheel have an axial moment of inertia 0338 kg m2 Eachenergymomentum wheel is assumed to provide a maximum torqueof 1 Ncent m

The disturbance torque that is due to aerodynamics solar pres-sure and other environmental factors is assumed to be25

gd =

2

44 pound 10 iexcl 6 + 2 pound 10 iexcl 6 sin(nt )



3

5 N cent m (36)


Fig 3 Satellite mission illustration

Fig 4 Body frame of the satellite

The satellite is required toperform sun and ground station trackingas follows The symmetry axis should point to a ground stationwhile the satellite is rotated by this axis such that the solar panel isperpendicular to the vector from the satellite to the sun The groundstation in this example is chosen to be Cape Canaveral (longitude80467degW latitude 28467degN) The sun and the ground station areassumed to be always available to the satellite In the followingsubsection we brie y describe the mission and the algorithm forobtaining the reference attitude maneuver history

Mission De nitionThe example is shown in Fig 3 The inertial frame is chosen

to be the J2000 geocentric inertial coordinate system denoted bythe subscript n The vectors rs rt and rc denote the positions ofthe sun the ground station and the satellite respectively and thevectors ls and lt denote the vectors from the satellite to the sun andthe ground station The following coordinate systems are used inthe simulation besides the inertial frame shown in Figs 1 and 4Hereafter ˆ(cent ) denotes a unit vector and Ccedil(cent ) denotes the derivativewith respect to time taken in the inertial frame

1) LVLH orbital frame oxl yl zl with the zl axis pointing towardthe Earth center the yl axis normal to the orbital plane and the xl

axis completing the orthogonal frame

2) body frame oxb yb zb with the zb axis along the boresight axisof the satellite and the yb axis pointing along the solar panels

The mission requires that at each moment along the orbit the zb

axis should track the ground station ie zb tracks the unit vectoralong lt In addition the satellite should also track an attitude suchthat the yb axis is perpendicular to ls

Computing the Reference Attitude Pro leThe satellite orbit is propagated by the orbit generator given in

Ref 26 From this we know rc Ccedilrc and rc in the inertial frame at anytime The sun position (rs ) velocity ( Ccedilrs ) and acceleration (rs ) in theinertial frame are computed by the algorithm given in Ref 27 Wecan compute the position (rt ) velocity ( Ccedilrt ) and acceleration (rt ) ofthe ground station in the inertial frame by converting the UniversalTime (UT) into Greenwich Sidereal Time26 (GST) From these weknow that

ls = rs iexcl rc Ccedills = Ccedilrs iexcl Ccedilrc ls = rs iexcl rc (37a)

lt = rt iexcl rc Ccedillt = Ccedilrt iexcl Ccedilrc lt = rt iexcl rc (37b)

In the following subsection the attitude angular velocity and angu-lar acceleration of the reference frame which are the desired attitudeangular velocity and angular acceleration for the body frame willbe computed The desired or target reference frame is denoted byoxr yr zr and is shown in Fig 3

Attitude ReferenceThe rotation matrix CL

N from the inertial frame to the LVLH framecan be readily computed by the orbital elements Once the LVLHframe is known we can write

lt = (lt cent xl )xl + (lt cent yl )yl + (lt cent zl)zl (38)

so that

zr = [(lt cent xl )xl + (lt cent yl )yl + (lt cent zl )zl ] t (39)

where lt = t lt Because the yr axis is perpendicular to ls and zr itcan be computed by

yr =zr pound ls

j zr pound ls j(40)

and the xr axis is then given by

xr = yr pound zr (41)


From the unit vectors xr yr and zr we can compute CRN the

rotation matrix from the inertial frame to the target reference frameThe value of g s = ls cent yb is used as a condition for sun tracking ieg s = 0 implies that the sun is being tracked In addition the valueg t = j lt pound zb j is used as a condition for ground station tracking ieg t = 0 implies that the ground station is being tracked

Angular-Velocity and the Angular-Acceleration ReferencesWe proceed to compute the angularvelocity of the reference frame

with respect to the inertial frame expressed in the target referenceframe R = [ x r x x ry x r z]T The algorithm for computing x r x andx r y can be found in Ref 28

The derivative of the vector lt in the inertial frame can be writtenin the desired reference frame as

Ccedillt = ( Ccedillt cent xr )xr + ( Ccedillt cent yr )yr + ( Ccedillt cent zr )zr (42)

Fig 5 Error in angular velocity of the satellite during target acquisition

Fig 6 Attitude error during target acquisition

Notice that Ccedillt can also be written as

Ccedillt =Rdltdt

+ R pound lt =R dltdt

+ R pound ( t zr ) (43)

where Rdlt dt is the derivative of lt taken in the target referenceframe A comparison of Eqs (42) and (43) then yields

x r x = iexcl Ccedillt cent yr t (44a)

x r y = Ccedillt cent xr t (44b)

dR lt

dt= Ccedillt cent zr (44c)

Because yr cent ls = 0 its time derivative is

Ccedilyr cent ls + yr cent Ccedills = 0 (45)


Expanding the derivative terms in the above equation we end upwith

x rz =x r x ls cent zr + yr cent Ccedills

ls cent xr

(46)

We can obtain the angular-acceleration vector Ccedil R = ( Ccedilx r x Ccedilx r y Ccedilx r z)T by taking the derivatives of x r x x ry and x r z By doing sowe get

Ccedilx r x = iexcl ( lt cent yr + 2x r x Ccedil t ) t + x r y x r z (47a)

Ccedilx r y = ( lt cent xr iexcl 2x r y Ccedil t ) t iexcl x r x x r z (47b)

Ccedilx rz = ( CcedilMN iexcl M CcedilN ) N 2 (47c)

Fig 7 Sun tracking condition during target acquisition (acutes = Atildels cent Atildeyb )

Fig 8 Ground station tracking condition during target acquisition (acutet = j Atildelt pound Atildezb j )

where

M = x r x ls cent zr + yr cent Ccedills (48)

N = ls cent xr (49)

CcedilM = Ccedilx rx ls cent zr + x r x Ccedills cent zr + x r x ls cent ( R pound zr )

+ R pound yr cent Ccedills + yr cent ls (50)

CcedilN = Ccedills cent xr + ls cent (R pound xr ) (51)

Given R and CcedilR we compute the reference torque gR fromEq (11)


Simulation ResultsWith the reference satellite attitude pro le computed in the pre-

ceding section we can apply the attitude- and power-tracking con-troller Two simulations are conducted in sequence First a target-acquisition maneuver is performed with the thrusters in which thesatellite is maneuvered from the LVLH frame to the required sunand ground station tracking attitude Then the attitude-control andenergy-storage functions are switched on and the energymomentumwheels are used to keep tracking the sun and the ground sta-tion Here we assume that the wheels are spun up to their nomi-nal angular velocities before deployment This can be done withthe power provided by the launch vehicle If the wheels are in thelocked position (not spinning) after separation a constant torquega = [1

p3 1

p3 1

p3 iexcl 1]T (Ncent m) which is in the null space

of the matrix A can be used to spin the wheels up without disturbingthe satellite motion The power for this initial spinup of the wheels

Fig 9 Required thruster torque for the target acquisition (ga = 0)

Fig 10 Angular-velocity error during tracking

can be provided by the solar panels of the satellite A problem mayoccur in case the solar panels do not face the sun at deployment Withbody-mounted solar cells it should not be a major concern as someof the cells will be facing the sun any time the spacecraft is in thesunlight Another approach is to have a primary (nonrechargable)battery for the initial deploymentacquisition stage

In the simulations quaternions are used to describe the attitudefrom the inertial frame to the body and reference frames due to thelarge-angle orbital maneuver and MRPs are used to describe thedifference between the body and the reference frame The resultsare presented in the following two subsections

Target AcquisitionBefore the satellite can be operated to track the sun and the ground

station target acquisition has to be performed so that the satellite ismaneuvered to obtain the right attitude in order to start continuous


tracking The simulation starts with the satellite body frame alignedwith the LVLH frame Because this initial target-acquisition ma-neuver is usually a fast large-angle (slew) maneuver and the wheelsare of fairly low control authority (1 Ncent m in this example) externalthrusters have to be used to issue the required torque ie in Eq (17)we choose

Aga = 0 (52)

gt = iexcl h poundB J iexcl 1(hB iexcl Aha ) iexcl gg + J pound

B d + JCBR ( d frac34)J iexcl 1h pound

R J iexcl 1hR

+ JCBR ( d frac34)J iexcl 1gR iexcl k1 d iexcl k2 d frac34 (53)

The controller gains are chosen as k1 = 24 and k2 = 27 Figure 5shows the angular-velocity tracking error of the satellite and Fig 6shows the quaternions of the body frame and the reference frameIt is seen that after raquo 70 s the satellite attitude tracks the refer-

Fig 11 Attitude error during tracking

Fig 12 Sun tracking condition during tracking (acutes = ls cent Atildeyb )

ence In Fig 7 it is shown that the sun tracking condition valueg s goes to zero after raquo 90 s In Fig 8 it is shown that the groundstation tracking condition value g t goes to zero also after raquo 90 s Fig-ure 9 shows the torque required for performing the target-acquisitionmaneuver

Continuous TrackingAfter the satellite tracks the sun and the ground station the

energymomentum-wheel attitude- and power-tracking control isswitched on so that the satellite will keep tracking the sun and theground station In this case the wheels will provide both the target-tracking torques and the energy-storage torques The thrusters willonly be used to issue the necessary momentum management torqueie in Eq (17) we choose for every two orbits

gt = iexcl k(Aha iexcl Ahan ) (54)



B d


R J iexcl 1hR iexcl JCBR ( d frac34)J iexcl 1gR

+ k1 d + k2 d frac34 (55)

The simulation orbit starts from 022399 07593228 and lasts for25000 s (approximately 4 orbits) The controller gains are chosenas k1 = 24 k2 = 27 and k = 0005 During the eclipse the nominalpower requirement during eclipse is 680 W with an additional re-quirement of 4-kW power for 5 min During sunlight the wheelsare charged with a power level of 1 kW until the total energy storedin the wheels reaches 15 kWh After the wheels are charged themomentum management is switched on every two orbits Figures 10and 11 show the angular velocity and attitude error during tracking

Fig 13 Ground station tracking condition during tracking (acutet = j lt pound Atildezb j )

Fig 14 Sunlighteclipse indication and power pro le

The angular-velocity errors are practically zero for the whole pe-riod of the maneuver The largest errors occur at the times of thehighest wheel momenta compare with Fig 15 The attitude error inFig 11 corresponds to a pointing error of less than 01 deg about allthree axes Figures 12 and 13 show that the sun and ground stationtracking conditions are being satis ed Figure 14 shows the powerpro le with the sunlight and eclipse indication where sunlight isindicated by 1 and eclipse is indicated by iexcl 1 The correspondingtorque applied by the wheels is shown in Fig 15 along with thetotal angular momentum of the wheels As expected the wheelscharge during sunlight and discharge during the eclipse Every twoorbits (at approximately 06 pound 104 and 18 pound 104 s in Fig 15) thetotal angular momentum of the wheels Aha goes to zero because ofproper momentum management Finally Fig 16 shows the wheelspeeds


Fig 15 Axial torques and the total angular momentum of the energymomentum wheels

Fig 16 Angular velocities of the energymomentum wheels

These plots indicate that there are no major theoretical problemsin using a cluster of four or more energymomentum wheels in asuitable geometric con guration to successfully control the attitudeand power requirements of a satellite in a typical Earth orbit ex-ample In reality several issues need to be addressed before theimplementation of such a control law for an IPACS For instancevibration suppression algorithms for high-speed ywheels properrotor sizing and containment issues are of great interest in that re-spect These and other similar issues are left for future investigation

ConclusionsIn this paper we develop an algorithm for controlling the space-

craft attitude while simultaneously tracking a desired power pro leby using a clusterof more than three noncoplanar energymomentumwheels The torque is decomposed into two perpendicular spacesOne is the null space of this matrix whose columns are the unit

vectors along the axes of each wheel The torque in this space isused to track the required power level of the wheels and the torquein the space perpendicular to the null space of this matrix is usedto control the attitude of the satellite The torque decomposition isbased on solving a set of linear equations Singularities may occurin case the coef cient matrix does not have full row rank In thiscase no arbitrary power pro le can be tracked A momentum man-agement scheme is considered to null the total angular momentumof the wheels in order to minimize the gyroscopic effects and alsoprevent the occurrence of singularities A numerical example basedon a realistic example for an iridium-type satellite demonstrates theef cacy of the proposed algorithm

References1Wertz J and Larson W (eds) Space Mission Analysis and Design

Kluwer Academic Boston 1991 pp 349ndash369


2Roes J B ldquoAn Electro-Mechanical Energy Storage System for SpaceApplicationrdquoProgress in Astronautics and Rocketry Vol 3 Academic NewYork 1961 pp 613ndash622

3Anderson W W and Keckler C R ldquoAn Integrated PowerAttitude Control System (IPACS) for Space Applicationrdquo Proceedings ofthe 5th IFAC Symposium on Automatic Control in Space Pergamon NewYork 1973 pp 81 82

4Cormack III A ldquoThree Axis Flywheel Energy and Control SytemsrdquoNASA TN-73-GampC-8 1973

5Keckler C R and Jacobs K L ldquoA Spacecraft Integrated PowerAttitude Control Systemrdquo Proceedings of the 9th Intersociety Energy Con-version Engineering Conference American Society of Mechanical Engi-neers New York 1974 pp 20ndash25

6Will R W Keckler C R and Jacobs K L ldquoDescription and Sim-ulation of an Integrated Power and Attitude Control System Concept forSpace-Vehicle Applicationrdquo NASA TN D-7459 1974

7Notti J E Cormack III A Schmill W C and Klein W J ldquoIntegratedPowerAttitude Control System (IPACS) Study Volume IImdashConceptual De-signsrdquo NASA CR-2384 1974

8Rodriguez G E Studer P A and Baer D A ldquoAssessment of FlywheelEnergy Storage for Spacecraft Power Systemsrdquo NASA TM-85061 1983

9Gross S ldquoStudy of Flwheel Energy Storage for Space Stationsrdquo NASACR-171780 1984

10Anand D Kirk J A and Frommer D A ldquoDesign Considerationsfor Magnetically Suspended Flywheel Systemsrdquo Proceedings of the 20thIntersociety Energy Conversion Engineering Conference Vol 2 Society ofAutomotive Engineers Warrendale PA 1985 pp 449ndash453

11Anand D Kirk J A Amood R B Studer P A and RodriguezG E ldquoSystem Considerations for Magnetically Suspended Flywheel Sys-temsrdquo Proceedings of the 21st Intersociety Energy Conversion EngineeringConference American Chemical Society Washington DC Vol 3 1986 pp1829ndash1833

12Downer J Eisenhaure D Hockney R Johnson B and OrsquoDea SldquoMagnetic Suspension Design Options for Satellite Attitude Control andEnergy Storagerdquo Proceedings of the 20th Intersociety Energy ConversionEngineering Conference Society of Automotive Engineers Warrendale PAVol 2 1985 pp 424ndash430

13Flatley T ldquoTetrahedron Array of Reaction Wheels for Attitude Controland Energy Storagerdquo Proceedings of the 20th Intersociety Energy Conver-sion Engineering Conference Society of Automotive Engineers Warren-dale PA Vol 2 1985 pp 2353ndash2360

14OrsquoDea S Burdick P Downer J Eisenhaure D and Larkin LldquoDesign and Development of a High Ef ciency Effector for the Control of

Attitude and Power in Space Systemsrdquo Proceedings of the 20th Interso-ciety Energy Conversion Engineering Conference Society of AutomotiveEngineers Warrendale PA Vol 2 1985 pp 353ndash360

15Oglevie R E and Eisenhaure D B ldquoAdvanced Integrated Powerand Attitude Control System (IPACS) Technologyrdquo NASA CR-3912 Nov1985

16Oglevie R E and Eisenhaure D B ldquoIntegated Power and AttitudeControl System (IPACS) TechnologyrdquoProceedings of the 21st IntersocietyEnergy Conversion Engineering Conference American Chemical SocietyWashington DC Vol 3 1986 pp 1834ndash1837

17Olmsted D R ldquoFeasibility of Flywheel Energy Storage in SpacecraftApplicationsrdquo Proceedings of the 20th Intersociety Energy Conversion En-gineering Conference Society of Automotive Engineers Warrendale PAVol 2 1985 pp 444ndash448

18Studer P andRodriguez E ldquoHigh SpeedReactionWheels for SatelliteAttitude Control and Energy Storagerdquo Proceedings of the 20th IntersocietyEnergy Conversion Engineering Conference Vol 2 Society of AutomotiveEngineers Warrendale PA 1985 pp 349ndash352

19Fossa C E Raines R A Gunsch G H and Temple M A ldquoAnOverview of the IRIDIUM Low Earth Orbit (LEO) Satellite Systemrdquo Pro-ceedings of the IEEE National Aerospace and Electronics Conference Instof Electrical and Electronics Engineers Piscataway NJ 1998 pp 152ndash159

20Hughes P Spacecraf t Attitude Dynamics Wiley New York 1986pp 233ndash248

21Tsiotras P ldquoStabilization and Optimality Results for the Attitude Con-trol Problemrdquo Journal of Guidance Control and Dynamics Vol 19 No 41996 pp 772ndash777

22Shuster M D ldquoA Survey of Attitude Representationsrdquo Journal of theAstronautical Sciences Vol 41 No 4 1993 pp 439ndash517

23Hall C Tsiotras P and Shen H ldquoTracking Rigid Body Motion UsingThrusters and Reaction Wheelsrdquo AIAA Paper 98-4471 Aug 1998

24Hall C ldquoHigh-Speed Flywheels for Integrated Energy Storage andAttitude Controlrdquo Proceedings of the American Control Conference Vol 3IEEE Publications Piscataway NJ 1997 pp 1894ndash1898

25Sidi M Spacecraf t Dynamics and Control A Practical EngineeringApproach Cambridge Univ Press New York 1997 pp 241 242

26Wertz J (ed) Spacecraf t Attitude Determination and Control ReidelBoston 1997 pp 132ndash138

27Meeus J Astronomical Algorithms Willmann-Bell Richmond VA1991 Chaps 21 24 25 31 and Appendices

28Hablani H ldquoDesign of a Payload Pointing Control system for TrackingMoving Objectsrdquo Journal of Guidance Control and Dynamics Vol 12No 3 1989 pp 365ndash374


The wheel-control torque space is decomposed into two spaces thatare orthogonal to each other The attitude-control torques lie in oneof these two spaces and the torques in the other space are usedto spin up or down the energymomentum wheels to store or ex-tract kinetic energy In the following sections the system model isgiven rst then the reference attitude and energymomentum-wheelpower-tracking controllers are presented A numerical example con-sidering a iridium-type satellite19 in orbit is provided to illustrate theproposed IPACS methodology

It should be pointed out that ywheels used in an IPACS typicallyrotate at much higher speeds than standard momentum wheels astheir operation is driven primarily by the energy-storage functionrather than the attitude-control function In this paper we use theterm energymomentum wheels to emphasize this difference withcommonly used momentum wheels

System ModelDynamics

Consider a rigid spacecraft with an N -wheel cluster installed toprovide internal torques Let B denote the spacecraft body frameThen the rotational equations of motion for the spacecraft can beexpressed as

CcedilhB = h poundB J iexcl 1(hB iexcl Aha) + ge (1a)

Ccedilha = ga (1b)

where hB is the angular-momentum vector of the spacecraft in theB frame given by

hB = JB + Aha (2)

where ha is the N pound 1 vector of the axial angular momenta of thewheels B is the angular-velocity vector of the spacecraft in theB frame ge is the 3 pound 1 vector of external torques ga is the N pound 1vector of the internal axial torques applied by the platform to thewheels and A is the 3 pound N matrix whose columns contain the axialunit vectors of the N energymomentum wheels J is an inertialikematrix de ned as

J = I iexcl AIsAT (3)

where I is the moment of inertia of the spacecraft including thewheels and Is = diagIs1 Is2 Is N is a diagonal matrix withthe axial moments of inertia of the wheels

The axial angular-momentum vector of the wheels can be writtenas

ha = IsAT B + Is s (4)

where s = ( x s1 x s2 x s N )T is an N pound 1 vector denoting theaxial angular velocity of the energymomentum wheels with respectto the spacecraft We denote the total axial angular velocity of thewheels relative to the inertial frame as c = ( x c1 x c2 x cN )T Using this notation we can write Eq (4) as

ha = Is c (5)

where c = s + AT B Note that because typically the wheels spinat a much higher speed than the spacecraft itself s Agrave B and wehave that c frac14 s

In Eq (1) the external torques are assumed to include the controltorque that is due to thruster ring the gravity gradient torque andthe other disturbance torques ie

ge = gt + gg + gd (6)

where the subscripts t g and d denote the thruster gravity gradientand disturbance respectively The gravity gradient torque is givenby20

gg =iexcl3 l | R3

c

cent| c pound

3 Ic3 (7)

where c3 is the unit vector iexcl rc Rc [the same as zl the z axis ofthe local-vertical local-horizontal (LVLH) frame shown in Fig 1]

Fig 1 LVLH frame

expressed in the body frame rc is the vector from the Earthcenter to the spacecraft center of mass with Rc = j rc j and l =3986005 pound 105 km3 s iexcl 2 is the constant gravitational parameter Itshould be pointed out that although we restrict the discussion toexternal control torques that are due to thruster ring our approachremains the same for other choices of actuators such as magneto-torques etc

KinematicsThe so-called modi ed Rodrigues parameters (MRPs) given in

Refs 21 and 22 are chosen to describe the attitude kinematicserrorofthe spacecraft The MRPs are de ned in terms of the Euler principalunit vector e and angle u by

frac34 = e tan( u 4) (8)

The MRPs have the advantage of being well de ned for the wholerange for rotations ie u 2 [0 2 p ) The differential equation gov-erning the kinematics in terms of the MRPs is given by21

Ccedilfrac34 = G(frac34) (9)

where

G(frac34) = 121 + frac34 pound + frac34frac34T iexcl [(1 + frac34T frac34) 2]1 (10)

and 1 is the 3 pound 3 identity matrix

Tracking ControllerIn one of our previous works23 three control laws were presented

to track a reference attitude pro le by using coordinated action ofboth thrusters and momentum wheels Here we restate the con-troller II from Ref 23 that will be used for attitude tracking Thiscontrol law assumes that the reference frame dynamics and kine-matics are given by

CcedilhR = h poundR J iexcl 1hR + gR (11)

Ccedilfrac34R = G(frac34R )R (12)








N

poundCR

N

currenT(14)










curren

(17)



B d



(18)










Aga = f (20)


ga = A+ f + gn (21)


Agn = 0 (22)



T = 12 T

c Isc (23)


dT

dt= P = T

c Is Ccedilc (24)


Tc ga = P (25)


sup3A

Tc

acutega =

sup3f

P

acute(26)


Tc (A+ f + gn ) = P (27)


Tc gn = Pm (28)


gn = PN ordm (29)



ordm = PN c

iexclT

c PN c

cent iexcl 1Pm (30)


gn = PN c

iexclT

c PN c

cent iexcl 1Pm (31)






Tc gn = 0 (32)















I =

2

4200 0 0

0 200 0

0 0 175

3

5 kg m2 (34)


A =

264

1 0 0p

33

0 1 0p

33

0 0 1p

33

375 (35)



gd =

2




3

5 N cent m (36)




















so that



yr =zr pound ls

j zr pound ls j(40)













Ccedillt =Rdltdt


+ R pound ( t zr ) (43)




dR lt







ls cent xr

(46)







where


N = ls cent xr (49)








p3 1

p3 1











Aga = 0 (52)



R J iexcl 1hR











B d



+ k1 d + k2 d frac34 (55)


















































N

poundCR

N

currenT(14)










curren

(17)



B d



(18)










Aga = f (20)


ga = A+ f + gn (21)


Agn = 0 (22)



T = 12 T

c Isc (23)


dT

dt= P = T

c Is Ccedilc (24)


Tc ga = P (25)


sup3A

Tc

acutega =

sup3f

P

acute(26)


Tc (A+ f + gn ) = P (27)


Tc gn = Pm (28)


gn = PN ordm (29)



ordm = PN c

iexclT

c PN c

cent iexcl 1Pm (30)


gn = PN c

iexclT

c PN c

cent iexcl 1Pm (31)






Tc gn = 0 (32)















I =

2

4200 0 0

0 200 0

0 0 175

3

5 kg m2 (34)


A =

264

1 0 0p

33

0 1 0p

33

0 0 1p

33

375 (35)



gd =

2




3

5 N cent m (36)




















so that



yr =zr pound ls

j zr pound ls j(40)













Ccedillt =Rdltdt


+ R pound ( t zr ) (43)




dR lt







ls cent xr

(46)







where


N = ls cent xr (49)








p3 1

p3 1











Aga = 0 (52)



R J iexcl 1hR











B d



+ k1 d + k2 d frac34 (55)
















































Tc gn = 0 (32)















I =

2

4200 0 0

0 200 0

0 0 175

3

5 kg m2 (34)


A =

264

1 0 0p

33

0 1 0p

33

0 0 1p

33

375 (35)



gd =

2




3

5 N cent m (36)




















so that



yr =zr pound ls

j zr pound ls j(40)













Ccedillt =Rdltdt


+ R pound ( t zr ) (43)




dR lt







ls cent xr

(46)







where


N = ls cent xr (49)








p3 1

p3 1











Aga = 0 (52)



R J iexcl 1hR











B d



+ k1 d + k2 d frac34 (55)






























































so that



yr =zr pound ls

j zr pound ls j(40)













Ccedillt =Rdltdt


+ R pound ( t zr ) (43)




dR lt







ls cent xr

(46)







where


N = ls cent xr (49)








p3 1

p3 1











Aga = 0 (52)



R J iexcl 1hR











B d



+ k1 d + k2 d frac34 (55)





















































Ccedillt =Rdltdt


+ R pound ( t zr ) (43)




dR lt







ls cent xr

(46)







where


N = ls cent xr (49)








p3 1

p3 1











Aga = 0 (52)



R J iexcl 1hR











B d



+ k1 d + k2 d frac34 (55)














































ls cent xr

(46)







where


N = ls cent xr (49)








p3 1

p3 1











Aga = 0 (52)



R J iexcl 1hR











B d



+ k1 d + k2 d frac34 (55)














































p3 1

p3 1











Aga = 0 (52)



R J iexcl 1hR











B d



+ k1 d + k2 d frac34 (55)













































Aga = 0 (52)



R J iexcl 1hR











B d



+ k1 d + k2 d frac34 (55)













































B d



+ k1 d + k2 d frac34 (55)











































satellite attitude control and power tracking with energy

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