IN SPACECRAFT applications, it is often expensive to determine

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 6, No., Septeber October Inertia-Free Spacecraft Attitude Control Using Reaction Wheels Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 Avishai Weiss, Ilya Kolanovsky, and Dennis S. Bernstein University of Michigan, Ann Arbor, Michigan 489-4 and Ait Sanyal New Mexico State University, Las Cruces, New Mexico 88-8 DOI:.4/.86 This paper extends the continuous inertia-free control law for spacecraft attitude tracking derived in prior work to the case of three axisyetric reaction wheels. The wheels are assued to be ounted in a known and linearly independent, but not necessarily orthogonal, configuration with an arbitrary and unknown orientation relative to the unknown spacecraft principal axes. Siulation results for slew and spin aneuvers are presented with torque and oentu saturation. I. Introduction IN SPACECRAFT applications, it is often expensive to deterine the ass properties with a high degree of accuracy. To alleviate this requireent, the control algoriths given in [ ] are inertia free in the sense that they require no prior odeling of the ass distribution. The control algoriths in [,] incorporate internal states that can be viewed as estiates of the oents and products of inertia; however, these estiates need not converge to the true values and in fact do not converge to the true values except under sufficiently persistent otion. The results of [] are based on rotation atrices [4] as an alternative to the quaternions as used in [,,6]. Quaternions provide a double cover of the rotation group SO() and, thus, when used as the basis of a continuous control algorith, cause unwinding that is unnecessary rotation away fro and then back to the desired physical attitude [7]. To avoid unwinding while using quaternions, it is thus necessary to resort to discontinuous control algoriths, which introduce the possibility of chatter due to noise as well as atheatical coplications [8 ]. On the other hand, rotation atrices allow for continuous control laws but introduce ultiple equilibria. Because the spurious equilibria of the closed-loop syste are saddle points, the attitude of the spacecraft converges alost globally (but not globally) to the desired equilibriu. Although the derivation of the inertia-free controller in [] and the present paper is based on rotation atrices, the relevant attitude error given by the S paraeter [see Eq. (4)] can be coputed fro any attitude paraeterization, such as quaternions or odified Rodrigues paraeters, and thus, the continuous inertia-free controllers presented in [] are not confined to rotation atrices. The inertia-free control laws in [ ] assue that three-axis input torques can be specified without onboard oentu storage, which iplies ipleentation in ters of thrusters. However, attitude control laws are typically ipleented with wheels, and thus, the onboard stored oentu varies with tie. To account for this effect, the contribution of this paper is the derivation of an inertia-free control law based on reaction-wheel actuation. Like the inertia-free control laws in [ ], the tuning of this control law requires no Presented as Paper -897 at the AIAA Guidance Navigation and Control Conference, Toronto, August ; received March ; revision received January ; accepted for publication 8 February ; published online 9 July. Copyright by Avishai Weiss. Published by the Aerican Institute of Aeronautics and Astronautics, Inc., with perission. Copies of this paper ay be ade for personal or internal use, on condition that the copier pay the $. per-copy fee to the Copyright Clearance Center, Inc., Rosewood Drive, Danvers, MA 9; include the code -884/ and $. in correspondence with the CCC. *Graduate Student, Departent of Aerospace Engineering. Professor, Departent of Aerospace Engineering. Meber AIAA. Assistant Professor, Departent of Mechanical and Aerospace Engineering. Meber AIAA. knowledge of the ass properties of the spacecraft, and this paper specifies the assuptions and odeling inforation concerning the reaction wheels and their placeent relative to the bus. The paper is organized as follows. In Sec. II, coordinate-free equations of otion for the spacecraft are derived, in which, unlike [], the present paper does not assue that the wheels are aligned with the principal axes of the spacecraft bus nor does it assue that the wheels are balanced with respect to the bus in order to preserve the location of its center of ass; in fact, the reaction wheels ay be ounted at any location and in any linearly independent configuration. In Sec. III, the control objectives are forulated, and in Sec. IV, the controller is developed. Siulation results are reported in Sec. V, in which the robustness to variations in the spacecraft inertia is deonstrated and controller perforance is exained under both torque and oentu saturation. Finally, concluding rearks are ade in Sec. VI. II. Spacecraft Model with Reaction Wheels This section derives the equations of otion for a spacecraft with reaction wheels, while highlighting the underlying assuptions on wheel geoetry, inertia, and attachent to the bus. Throughout the paper, the vector r q p denotes the position of point q relative to point p, the vector v q p X X r q p denotes the velocity of point q relative to point p with respect to frae F X, and the vector ω Y X denotes the angular velocity of frae F Y relative to frae F X. Note that denotes a coordinate-free (unresolved) vector. All fraes are orthogonal and right handed. Definition : Let F X be a frae, let B be a collection of rigid bodies B ; :::;B l, and let p be a point. Then, the angular oentu of B relative to p with respect to F X is defined by H B p X Xl H Bi p X () where, for i ; :::;l, the angular oentu H Bi p X of B i relative to p with respect to F X is defined by Z H Bi p X r d p v d p X d () B i The following properties of angular oentu are standard []. Lea : Let B be a rigid body, let F X and F Y be fraes, and let p be a point. Then, H B p X I B p ω Y X H B p Y () where the positive-definite coordinate-free inertia tensor I B p is defined by 4

46 WEISS ET AL. Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 Z I B p jr d p j U r d p rd p d (4) B and where U denotes the second-order identity tensor. Lea : Let B be a rigid body, let F X and F Y be fraes, let F Y be a body-fixed frae, and let p be a point that is fixed in B. Then, and H B p Y () H B p X I B p ω Y X (6) Lea : Let F X be a frae, let p be a point, let B be a rigid body with ass B, and let c be the center of ass of B. Then, H B p X H B c X r c p B v c p X (7) Consider a spacecraft sc actuated by three axisyetric wheels w, w, w attached to a rigid bus b in a known and linearly independent, but not necessarily orthogonal, configuration. Although the spacecraft is not a rigid body, the axial syetry of the wheels iplies that the center of ass c of the spacecraft is fixed in both the bus and the spacecraft. Because the inertia properties of the bus are assued to be unknown, the principal axes of the bus are unknown, and thus, the wheel configuration has an arbitrary and unknown orientation relative to the principal axes of the bus. Each wheel is ounted so that it rotates about one of its own principal axes passing through its own center of ass. It is not assued that the axis of rotation of each wheel passes through the center of ass of the bus, nor is it assued that the wheels are balanced with respect to the bus in order to preserve the location of its center of ass. Thus, the center of ass of the spacecraft and the center of ass of the bus ay be distinct points, both of which are unknown. Assue a bus-fixed frae F B ; three wheel-fixed fraes F W, F W, F W, for which the x axes are aligned with the rotation axes of w, w, w, respectively; and an Earth-centered inertial frae F E. The angular oentu of the spacecraft relative to its center of ass with respect to the inertial frae is given by Definition : H sc c E H b c E X H wi c E (8) where the angular oentu H b c E of the bus relative to c with respect to F E is given by Lea : H b c E I b c ω B E (9) where I b c is the positive-definite inertia tensor of the bus relative to the center of ass of the spacecraft, and ω B E is the angular velocity of F B with respect to F E. The angular oentu H wi c E of wheel i relative to the center of ass of the spacecraft with respect to the inertial frae is given by Lea : Lea : Lea : H wi c E I wi cω B E H wi c B I wi cω B E H wi c i B r ci c wi v ci c B I wi cω B E I wi c i ω Wi B () where I wi c is the inertia tensor of wheel i relative to the center of ass of the spacecraft, I wi c i is the inertia tensor of wheel i relative to the center of ass c i of the ith wheel, and ω Wi B is the angular velocity of wheel i relative to the bus. Thus, Eq. (8) is given by H sc c E I b c X I wi c ω B E X I wi c i ω Wi B () Resolving ω Wi B in F Wi yields ω Wi Bj Wi ψ i e () where e T and ψ i is the angular rate relative to F B of the ith wheel about the x axis of F Wi. Because F Wi is aligned with the principal axes of wheel i, it follows that I wi c i j Wi diagα i ; β i ; β i () Note that ω Wi B is an eigenvector of I wi c i with eigenvalue α i ; that is, I wi c i ω Wi B α i ω Wi B. A. Spacecraft Equations of Motion The equations of otion for a spacecraft with reaction wheels as described before are now derived. It follows fro Newton s second law for rotation that M sc c H E sc c E I b c X E z} { I wi c I wi c B z} { I b c X B z} { X ω B E ω B E X I wi c i ω Wi B ω B E X I wi c i ω Wi B I b c X I wi c B ω ω B E I b c X E z} { I wi c i ω Wi B ω B E I b c X I wi c ω B E B E X I wi c α i ω B W i B ω B E X α i ω Wi B To resolve Eq. (4) in F B, the following notation is used: J b I b c j B ; J wi I wi cj B ; J w X I wi c ; B (4) J sc J b J w ; ω ω B E j B ; _ω ω B B Ej B ; ν i ω Wi Bj B ; _ν i ω B W i Bj B ; τ dist M sc c j B The vector τ dist represents the disturbance torques, that is, all external torques applied to the spacecraft aside fro control torques. Disturbance torques ay be due to gravity gradients, solar pressure, atospheric drag, or the abient agnetic field. As in Eq. (), the angular acceleration _ν i of each wheel has one degree of freedo. In F Wi, Thus, B W ω i Bj Wi W i ω W i Bj Wi _ψ i e () _ν i ω B W i Bj B O B Wi ω B W i Bj Wi O B Wi _ψ i e (6)

WEISS ET AL. 47 Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 where the proper orthogonal atrix O B Wi R is the orientation atrix that transfors coponents of a vector resolved in F Wi into the coponents of the sae vector resolved in F B. With the preceding notation, resolving Eq. (4) in F B yields τ dist J b J w _ω X α i _ν i ω J b J w ω X α i ν i J sc _ω X α i O B W _ψ i e ω J sc ω X α i O B W ψ i e where J sc _ω J α _ν ω J sc ω J α ν (7) J α α O B W e α O B W e α O B W e (8) ν ψ ψ ψ T, and _ν _ψ _ψ _ψ T. Rearranging Eq. (7) and choosing the control input u to be _ν yields the equations of otion for a spacecraft with reaction wheels, which have the for J sc _ω J sc ω J α ν ω J α u τ dist (9) _ν u () In practice, a servo loop is closed around each reaction wheel in order to produce the desired wheel angular accelerations given in Eq. (). Instead of coanding wheel angular accelerations by ipleenting servo loops, otor torque coands can be used. To deterine the relationship between the desired angular acceleration and the required otor torque, the dynaic equations for each wheel ust be derived. It follows that M wi c i H E w i c i E E z} { I wi c i ω Wi E I wi c i ω W W i E ω Wi E I wi c i ω Wi E I wi c i ω B B E B ωw i B ω Wi B ω B E ω B E ω Wi B I wi c i ω B E ω Wi B () Resolving Eq. () in F B and projecting it along each otor axis yields τ otor;i e T i J w i c i _ω _ν i ν i ωων i J wi c i ω ν i () where J wi c i I wi c i j B. Although Eq. () can be used for torque control, the easureents of ω, _ω, ν i, and _ν i needed to ipleent it deonstrate why reaction wheels are typically angular-acceleration coanded and feedback controlled rather than torque coanded. B. Specialization: Principal-Axis Alignent As in [], the equations of otion (9) and () are now specialized by assuing that the principal axes of the bus are aligned with the rotational axes of the wheels; that the wheels are ass balanced relative to the center of ass of the bus so that the center of ass of the spacecraft coincides with the center of ass of the bus; and, finally, that the oents of inertia β, β, β of the wheels are luped into the bus inertia J b diagj b ;J b ;J b, where J b J b β β, J b J b β β and J b J b β β. In this configuration, O B W e " # ; O B W e " # " # ; O B W e () Therefore, J α J w diagα ; α ; α. Rewriting the equations of otion (9) and () as and siplifying yields J b _ω J b J α ω J α ν ω u τ dist (4) u J α _ω _ν () J b _ω J b J b ω ω α ω ω ν α ω ω ν u τ dist (6) J b _ω J b J b ω ω α ω ω ν α ω ω ν u τ dist (7) J b _ω J b J b ω ω α ω ω ν α ω ω ν u τ dist (8) which are Eqs. (7.9) and (7.6) of []. u α _ω _ν (9) u α _ω _ν () u α _ω _ν () III. Spacecraft Model, Assuptions, and Control Objectives For the control laws (44) and () given next, the assuptions presented in Sec. II.B are not invoked. The kineatics of the spacecraft are given by Poisson s equation _R Rω () which copleents Eqs. (9) and (). In Eq. (), ω denotes the skew-syetric atrix of ω, and R O E B R. Both rate (inertial) and attitude (noninertial) easureents are assued to be available. Copared to the case of thrusters treated in [], reaction-wheel actuation coplicates the dynaic equations due to the ter J α ν in Eq. (9), as well as the integrators () augented to the syste. The kineatic relation () reains unchanged. The torque inputs applied to each reaction wheel are constrained by current liitations on the electric otors and aplifiers as well as angular-velocity constraints on the wheels. These constraints are addressed indirectly in Sec. V. The objective of the attitude control proble is to deterine control inputs such that the spacecraft attitude given by R follows a coanded attitude trajectory given by a possibly tie-varying C rotation atrix R d t.fort, R d t is given by _R d t R d tω d t () R d R d (4)

48 WEISS ET AL. where ω d is the desired, possibly tie-varying angular velocity. The error between Rt and R d t is given in ters of the attitude-error rotation atrix which satisfies the differential equation ~R R T d R () _ ~ R ~R ~ω (6) where the angular velocity error ~ω is defined by Rewrite Eq. (9) in ters of ~ω as ~ω ω ~R T ω d (7) Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 J sc _ ~ω Jsc ~ω ~R T ω d J α ν ~ω ~R T ω d J sc ~ω ~R T ω d ~R T _ω d J α u τ dist (8) A. Attitude Error A scalar easure of attitude error is given by the eigenaxis attitude error, which is the rotation angle θt about the eigenaxis needed to rotate the spacecraft fro its attitude Rt to the desired attitude R d t. This angle is given by [] θt cos tr ~Rt (9) B. Spacecraft Inertia Because the control laws in this paper require no inertia odeling, exaples that span a range of possible inertia atrices are considered. The inertia of a rigid body is deterined by its principal oents of inertia, that is, the diagonal entries of the inertia tensor resolved in a principal body-fixed frae, in which case the inertia atrix is a diagonal atrix. If the inertia tensor is resolved in a nonprincipal body-fixed frae, then the diagonal entries are the oents of inertia and the offdiagonal entries are the products of inertia. The offdiagonal entries of the inertia atrix are thus a consequence of an unknown rotation between a principal body-fixed frae and the chosen body-fixed frae. Figure shows the triangular region of feasible principal oents of inertia of a rigid body. There are five cases that are highlighted for the principal oents of inertia λ λ λ >, where λ, λ, and λ satisfy the triangle inequality λ < λ λ. Let denote the ass of the rigid body. q The point λ λ λ corresponds to q a sphere of λ radius r, a cube whose sides have length l, and a p cylinder of length l and radius r; in which l r q and r λ. The point λ λ λ corresponds q to a cylinder of length l and λ radius r, in which l r and r. The point λ 6 λ λ, located at the centroid of the triangular q region, corresponds q to q a solid 8λ rectangular body with sides l 4λ >l λ >l. The reaining cases in Fig. are nonphysical, liiting cases. q The λ point λ λ λ corresponds to a thin disk of radius r and length l. The point λ λ and λ q corresponds to a thin λ cylinder of radius r and length l. Finally, each point along the line segent λ λ λ, in which λ > λ, corresponds to q a thin rectangular q plate with sides of length l >l. λ λ For all siulations of the inertia-free control laws, the principal axes are viewed as the noinal body-fixed axes, and thus, the noinal inertia atrix is a diagonal atrix whose diagonal entries are 6λ Fig. Feasible region of the principal oents of inertia λ, λ, and λ of a rigid body satisfying < λ λ λ, where λ < λ λ. The shaded region shows all feasible values of λ and λ in ters of the largest principal oent of inertia λ. The open dots and dashed line segent indicate nonphysical, liiting cases. the principal oents of inertia. To deonstrate robustness, the principal oents as well as the orientation of the body-fixed frae relative to the principal axes are varied. For convenience, λ is noralized to kg, and the inertia atrices J, J, J, J 4, and J are chosen to correspond to the points noted in Fig.. These atrices, which correspond to the sphere, cylinder with l r ; centroid, thin disk, and thin cylinder, respectively, are defined as J diag; ; ; J diag; ; ; J diag; ; ; J 4 diag; ; ; J diag; ;. (4) The inertia atrix J corresponding to the centroid of the inertia region serves as the noinal inertia atrix, whereas the inertia atrices J, J, J 4, and J are used as perturbations to deonstrate robustness of the control laws. A perturbation Jλ of J i in the direction of J j thus has the for Jλ λj i λj j (4) where λ ;. Finally, in order to facilitate nuerical integration of Euler s equation, note that J is chosen to be a nonsingular approxiation of the liiting inertia of a thin cylinder. IV. Controller Design Let I denote the identity atrix, for which the diensions are deterined by context, and let M ij denote the i, j entry of the atrix M: The following result is given in []. Lea : Let A R be a diagonal positive-definite atrix, and let R R be a rotation atrix. Then the following stateents hold: ) For all i; j ; ;, R ij ;. ) tra AR. ) tra AR if and only if R I. For convenience note that, if R is a rotation atrix and x, y R, then Rx Rx R T

WEISS ET AL. 49 Eigenaxis Attitude Error (rad) S.... 4 6 7 8 9 4 Angular Velocity Coponents (rad/sec)... ω ω ω Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 4 6 7 8 9 a) Top: Eigenaxis attitude error. Botto: Nor of the S paraeter Wheel Angular Velocity Coponents (rad/sec) 4 4 6 7 8 9 c) Angular rates of the reaction wheels and therefore, Next, introduce the notation Fig. Rx y Rx Ry J sc ω Lωγ ν ν ν 4 6 7 8 9 b) Spacecraft angular-velocity coponents Angular Acceleration Coponents (rad/sec ) 8 6 4 4 6 4 6 7 8 9 d) Angular accelerations of the reaction wheels Slew aneuver using the control law (44) with no disturbance. Letting ^γ, ~γ R 6 represent ^J sc, ~J sc, respectively, it follows that ~γ γ ^γ Likewise, let ^τ dist R denote an estiate of τ dist, and define the disturbance-estiation error: u u u where γ R 6 is defined by ~τ dist τ dist ^τ dist and γ J J J J J J T Lω 4 ω ω ω ω ω ω ω ω ω The assuptions upon which the following developent is based are now stated. Assuption : J sc is constant but unknown. Assuption : J α defined by Eq. (8) is constant, nonsingular, and known. That is, the spacecraft has three linearly independent, axisyetric wheels with known oents of inertia about their spin axes and known configuration relative to the bus. The controllers presented in [] are now extended to the case of reaction-wheel actuation. Next, let ^J sc R denote an estiate of J sc, and define the inertia-estiation error: ~J sc J sc ^J sc A. Control Law for Slew Maneuvers When no disturbances are present, the inertia-free control law given by Eq. (8) of [] achieves alost global stabilization of a constant desired attitude R d, that is, a slew aneuver that brings the

4 WEISS ET AL. Eigenaxis Attitude Error (rad).... Angular Velocity Coponents (rad/sec)... ω ω ω Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 a) Eigenaxis attitude error Wheel Angular Velocity Coponents (rad/sec) 4 c) Angular rates of the reaction wheels spacecraft to rest. The initial conditions of the slew aneuver ay be arbitrary; that is, the spacecraft ay have nonzero initial velocity. Given a, a, a R, define the vector easure of attitude error: S X ν ν ν a i ~R T e i e i (4) where, for i ; ;, e i denotes the ith colun of the identity atrix. When attitude easureents are given in ters of an alternative representation, such as quaternions, the corresponding attitude-error ~R defined by Eq. () can be coputed, and thus, Eq. (4) can be evaluated and used by the controller given in Theore next. Consequently, S can be coputed fro any attitude paraeterization. Theore : Let K p be a positive nuber and let A diaga ;a ;a be a diagonal positive-definite atrix. Then, the function. b) Spacecraft angular-velocity coponents Angular Acceleration Coponents (rad/sec ) 4 4 d) Angular accelerations of the reaction wheels Fig. Slew aneuver using the control law (44) with no disturbance. The acceleration of the reaction wheels is saturated at 4 rad s. Proof: It follows fro stateent of Lea that tra A ~R is nonnegative. Hence, V is nonnegative. Now, suppose that V. Then, ω, and it follows fro stateent of Lea that ~R I: Theore : Let K p be a positive nuber, let K v R be a positive-definite atrix, let A diaga ;a ;a be a diagonal positive-definite atrix with distinct diagonal entries, let R d be constant, define S as in Eq. (4), and define V as in Theore. Consider the control law Then, u u u u J α K p S K v ω (44) _Vω; ~R ω T K v ω (4) Vω; ~R ωt J sc ω K p tra A ~R (4) is positive definite; that is, V is nonnegative, and V if and only if ω and ~R I. is negative seidefinite. Furtherore, the closed-loop syste consisting of Eqs. (9), (), (6), and (44) is alost globally asyptotically stable [4], and for all initial conditions not in an ebedded subanifold of R SO R 6 R (see []), ω and ~R I as t.

WEISS ET AL. 4 Eigenaxis Attitude Error (rad).... Angular Velocity Coponents (rad/sec)... ω ω ω Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 Proof: Noting that then a) Eigenaxis attitude error Wheel Angular Velocity Coponents (rad/sec) 4 c) Angular rates of the reaction wheels d dt tra A ~R tra _ ~R tra ~Rω ω ~ d R X a i e T i ~Rω ω ~ d Re i X X X a i e T i X a i e T i a i e T i ~Rω ~R T ω d ~ Re i ~Rω ~R T ω d e i ~Re i ~ω a i e i ~R T e i T ~ω X T a i ~R T e i e i ~ω ~ω T S ν ν ν Angular Acceleration Coponents (rad/sec ). b) Spacecraft angular-velocity coponents.... d) Angular accelerations of the reaction wheels Fig. 4 Slew aneuver using the control law (44) with no disturbance. The acceleration of the reaction wheels is saturated at rad s. _Vω; ~R ω T J sc _ω K p ω T S ω T J sc ω J α ν ω J α uk p ω T S ω T K p S K v ωk p ω T S ω T K v ω The proof of the final stateent follows fro invariant set arguents that are siilar to those used in []. Note that J α is substituted for the input atrix B used in the inertia-free control law (8) of [], but otherwise, the controller requires no odification for the case of reaction-wheel actuation in order to achieve alost global stabilization of a constant desired attitude R d. B. Control Law for Attitude Tracking A control law that tracks a desired attitude trajectory in the presence of disturbances is given by Eq. () of []. This controller is based on an additional assuption. Assuption : Each coponent of τ dist is a linear cobination of constant and haronic signals, for which the frequencies are known but for which the aplitudes and phases are unknown. Assuption iplies that τ dist can be odeled as the output of the autonoous syste u u u _d A d d (46)

4 WEISS ET AL... ω ω ω Eigenaxis Attitude Error (rad)... Angular Velocity Coponents (rad/sec).. Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 4 6 7 8 9 a) Eigenaxis attitude error Wheel Angular Velocity Coponents (rad/sec) 4 6 7 8 9 c) Angular rates of the reaction wheels Fig. τ dist C d d (47) where d is the disturbance state, A d R n d n d and Cd R n d are known atrices, and the eigenvalues of A d are nonrepeated on the iaginary axis. In this odel, d is unknown, which is equivalent to the assuption that the aplitude and phase of each haronic coponent of the disturbance is unknown. The eigenvalues of A d are chosen to represent all frequency coponents that ay be present in the disturbance signal, in which the zero eigenvalue corresponds to a constant disturbance. By providing infinite gain at the disturbance frequencies, the controller asyptotically rejects the haronic disturbances. In particular, an integral controller provides infinite gain at zero frequency in order to reject constant disturbances. In the case of orbit-dependent disturbances, the frequencies can be estiated fro the orbital paraeters. Likewise, in the case of disturbances originating fro onboard devices, the spectral content of the disturbances ay be known. In other cases, it ay be possible to estiate the spectru of the disturbances through signal processing. Assuption iplies that A d can be chosen to be skew syetric, which is henceforth done. Let ^d R n d denote an estiate of d, and define the disturbance-state estiation error: ~d d ^d ν ν ν 4 6 7 8 9 b) Spacecraft angular-velocity coponents Angular Acceleration Coponents (rad/sec ) 8 6 4 4 6 4 6 7 8 9 d) Angular accelerations of the reaction wheels Slew aneuver using the control law (44) with no disturbance. The axiu rotation rate of each wheel is saturated at rad s. The attitude tracking controller in the presence of disturbances given in [] is odified next for reaction-wheel actuators. Theore : Let K p be a positive nuber, let K R, let Q R 6 6 and D R n d n d be positive-definite atrices, let A diaga ;a ;a be a diagonal positive-definite atrix, and define S as in Eq. (4). Then, the function V ~ω; ~R; ~γ; ~d ~ω K S T J sc ~ω K SK p tra A ~R u u u ~γt Q~γ ~ d T D ~d (48) is positive definite; that is, V is nonnegative, and V if and only if ~ω, ~R I, ~γ, and ~d. Proof: It follows fro stateent of Lea that tra A ~R is nonnegative. Hence, V is nonnegative. Now, suppose that V. Then, ~ω K S, ~γ, and ~d, and it follows fro stateent of Lea that ~R I, and thus, S. Therefore, ~ω. The following result concerns attitude tracking without knowledge of the spacecraft inertia. This control law does not regulate the speed of the wheels. Consequently, the function V defined by Eq. (48), which is used as a Lyapunov function in the proof of Theore 4 next, is not a positive-definite function of the angular rates of the wheels relative to the bus.

WEISS ET AL. 4.. ω ω Eigenaxis Attitude Error (rad).. Angular Velocity Coponents (rad/sec).. ω Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86. a) Eigenaxis attitude error Wheel Angular Velocity Coponents (rad/sec) c) Angular rates of the reaction wheels Fig. 6 Theore 4: Let K p be a positive nuber, let K v R, K R, Q R 6 6, and D R n d n d be positive-definite atrices, assue that A T d D DA d is negative seidefinite, let A diaga ;a ;a be a diagonal positive-definite atrix with distinct diagonal entries, define S and V as in Theore, and let ^γ and ^d satisfy ν ν ν. b) Spacecraft angular-velocity coponents Angular Acceleration Coponents (rad/sec ) 8 6 4 4 6 d) Angular accelerations of the reaction wheels Slew aneuver using the control law (44) with no disturbance. The axiu rotation rate of each wheel is saturated at rad s. where u J α v v v () v ^J sc ω J α ν ω ^J sc K _S ~ω ω ~R T _ω d (4) u u u _^γ Q L T ωω L T K _S ~ω ω ~R T _ω d ~ω K S (49) v ^τ dist () where and _S X a i ~R T e i ~ω e i () _^d A d ^d D C T d ~ω K S () and v K v ~ω K S K p S (6) Then, _V ~ω; ~R; ~γ; ~d ~ω K S T K v ~ω K S K p S T K S d ~ T A T d D DA d ~d (7) ^τ dist C d ^d () so that ^τ dist is the disturbance-torque estiate. Consider the control law is negative seidefinite. Furtherore, the closed-loop syste consisting of Eqs. (), (6), (8), and () is alost globally asyptotically stable, and for all initial conditions not in an ebedded subanifold of R SO R 6 R (see []), ~ω and ~R I as t.

44 WEISS ET AL. 9 7 8 8 Centroid to Thin Disk Centroid to Sphere Centroid to Thin Cylinder 6 Centroid to Thin Disk Centroid to Sphere Centroid to Thin Cylinder Settling 7 7 6 6 Settling 6 4 Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 4....4..6.7.8.9 4....4..6.7.8.9 λ λ a) b) Fig. 7 Settling tie as a function of λ for various cobinations (4) of inertia atrices resolved in principal fraes. Convergence is achieved for a) control law (44) and b) control law (). Each controller is ipleented in all cases with a single tuning. In all cases, the bus inertia J is unknown. Proof: _V ~ω; ~R; ~γ; ~d~ω K S T J sc _ ~ω Jsc K _S K p tra _ ~R ~γ T Q _^γ ~d T D _ ~d ~ω K S T J sc ω J α ν ω J sc ~ω ω ~R T _ω d J α u τ dist J sc K _S K p ~ω T S ~γ T Q _^γ ~d T D _ ~d ~ω K S T J sc ω J α ν ω J sc K _S ~ω ω ~R T _ω d v v v τ dist K p ~ω T S ~γ T Q _^γ ~d T D _ ~d ~ω K S T ~J sc ω ω ~J sc K _S ~ω ω ~R T _ω d ~ω K S T ~τ dist ~ω K S T K v ~ω K S K p ~ω K S T S K p ~ω T S ~γ T Q _^γ ~d T D _ ~d ~ω K S T Lω~γ ω LK _S ~ω ω ~R T _ω d ~γ ~ω K S T K v ~ω K S K p S T K S ~γ T Q _^γ ~d T C T d ~ω K S ~d T DA d ~d D C T d ~ω K S ~ω K S T K v ~ω K S K p S T K S ~γ T Q _^γ ~ω K S T ω LωLK _S ~ω ω ~R T _ω d ~γ ~ d T A T d D DA d ~d ~ω K S T K v ~ω K S K p S T K S ~γ T Q _^γ L T ωω L T K _S ~ω ω ~R T _ω d ~ω K S ~ d T A T d D DA d ~d ~ω K S T K v ~ω K S K p S T K S ~ d T A T d D DA d ~d The closed-loop spacecraft attitude dynaics Eq. (8) and the control law Eqs. ( 6) can be expressed as J _ ~ω Lω~γ ω L ~ω ~R T ω d R T _ω d ~γ LK _S^γ ~z d K v ~ω K S K p S (8) Fro Leas and 4 of [], the closed-loop syste consisting of Eqs. (49 ) and (8) has four disjoint equilibriu anifolds. These equilibriu anifolds in R SO R 6 R are given by E i f~ω; ~R; ~γ; ~d R SO R 6 R ~R R i ; ~ω ; ~γ; ~d Q i g (9) where, for all i f; ; ; g, Q i is the closed subset of R 6 R defined by Q i f~γ; ~d R 6 R LR T i ω d~γ R T i ω d LR T i ω d~γ C d ~d ; _ ~γ ; _ ~d A d ~dg Furtherore, the equilibriu anifold ~ω; ~R; ~γ; ~d ;I;Q of the closed-loop syste given by Eqs. (49 ) and (8) is locally asyptotically stable, and the reaining equilibriu anifolds given by ; R i ; Q i for i f; ; g are unstable. Finally, the set of all initial conditions converging to these equilibriu anifolds fors a lower-diensional subanifold of R SO R 6 R. V. Exaples Siulations are now provided to illustrate the inertia-free control laws (44) and (). To siulate slew and spin aneuvers, the following spacecraft paraeters are assued. The bus inertia atrix J b is noinally given by J, which corresponds to the centroid of the inertia region shown in Fig. with the body-fixed frae assued to be a principal body-fixed frae. The quantity J b is unknown to the controller. The axes of rotation of the reaction wheels are aligned with the spacecraft body-fixed frae unit vectors, and the wheel inertias are given by J w diagα ; β ; β kg, J w diagβ ; α ; β kg, and J w diagβ ; β ; α kg, where α α α. and β β β.7. The values of β i are unknown to the controller. Let K p be given by K p γ (6) tra

WEISS ET AL. 4 Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 and, as in [], let K v K v ω be given by K v η6 4 jω j jω j jω j Alternative choices of K v are given in []. 7 (6) A. Slew Maneuver Using Control Law (44) with No Disturbance Controller (44) is used for an aggressive slew aneuver, in which the objective is to bring the spacecraft fro the initial attitude R I and initial angular velocity ω. T rad s to rest (ω d ) at the desired final orientation R d diag; ;, which represents a rotation of 8 deg about the x axis. The reaction wheels are initially not spinning relative to the spacecraft; that is, v T rad s No disturbance is present, and the paraeters γ η are used in Eqs. (6) and (6). Figures a d show, respectively, the attitude error, angularvelocity coponents, angular rates of the wheels, and the control inputs, which are the angular accelerations of the wheels. The spacecraft attitude and angular-velocity coponents reach the coanded values in about s. The angular rates of the wheels approach constant values that are consistent with the initial, nonzero angular oentu. In practice, reaction wheels have a axiu instantaneous acceleration. Angular-acceleration saturation is enforced in Figs. and 4, in which convergence is slower than in Fig., although stability is aintained. Additionally, reaction wheels have a axiu rotational rate. Figure shows the effect of wheel-rate saturation at rad s, corresponding to about 4 rp. The reaction-wheel rates are saturated for up to s, although this does not ipact the control objective. Figure 6 shows plots for wheel-rate saturation at rad s or about 9 rp. Although this constraint on the rotation rate is too stringent to obtain zero steady-state error for the desired aneuver, the perforance of the controller degrades 8 8 7 x y z gracefully by achieving zero spacecraft angular velocity at an offset attitude. To evaluate perforance for slew aneuvers, define the settlingtie etric k in k> fk for all i f; :::;g; θk it s <. radg (6) where k is the siulation step, T s is the integration step size, and θkt s is the eigenaxis attitude error (9) at the kth siulation step. The etric k is thus the iniu tie such that the eigenaxis attitude error is less than. rad during the ost recent siulation steps. To illustrate the inertia-free property of the control laws (44) and (), the inertia of the spacecraft is varied using J b λ λj λj i (6) where λ ; and i ; 4;. Figure 7 shows how the settling tie depends on λ. Next, the robustness to isalignent of the reaction wheels relative to the principal axes is investigated. Here, the inertia atrix is rotated by an angle ϕ about one of the axes of frae F b. For a rotation about the x axis of F b, the inertia of the spacecraft is varied using J b ϕ O ϕj O T ϕ (64) where the proper orthogonal atrix O ϕ rotates vectors about the x axis by the angle ϕ. Siilar relations exist for rotations about the y and z axes. Figure 8 shows how a thruster isalignent angle ϕ affects the settling tie, in which ϕ is varied fro 8 to 8 deg. B. Slew Maneuver Using Control Law () Under Constant Disturbance The unknown constant disturbance torque τ dist.7. T is now considered. Note that the controller () is used in place of the controller (44), which lacks an integrator, and thus has a constant steady-state error bias due to the persistent disturbance. The paraeters of the controller () are chosen to be K I, A diag; ;, γ η, D I, and Q I 6. 6 6 x y z Settling 7 6 Settling 6 4 φ (deg) 4 φ (deg) a) b) Fig. 8 Settling tie as a function of principal-frae/body-frae rotation angle ϕ for rotations about each of the three principal axes of J. Convergence is achieved for a) control law (44) and b) control law ().

46 WEISS ET AL. Eigenaxis Attitude Error (rad).... Angular Velocity Coponents (rad/sec)..... ω ω ω Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 4 6 7 8 9 a) Eigenaxis attitude error Wheel Angular Velocity Coponents (rad/sec) 6 4 8 6 4 4 4 6 7 8 9 c) Angular rates of the reaction wheels. The spin rate grows unbounded due to the constant disturbance torque Disturbance Estiate Errors (N ) 8 6 4 4 6 4 6 7 8 9 e) Disturbance-estiate errors Fig. 9 ν ν ν 4 6 7 8 9 b) Spacecraft angular-velocity coponents Angular Moentu, X Angular Moentu, Y Angular Moentu, Z Total, x Bus, x Wheels, x 4 6 7 8 9 Total, y Bus, y Wheels, y 4 6 7 8 9 Total, z Bus, z Wheels, z 4 6 7 8 9 d) Angular oentu of the spacecraft relative to its center of ass with respect to the inertial frae resolved in the inertial frae. The total angular oentu is not conserved due to the constant disturbance torque in the bus-fixed frae Inertia Estiate Errors (kg ) Inertia Estiate Errors (kg ) Jtilde Jtilde Jtilde 4 6 7 8 9 Jtilde Jtilde Jtilde 4 6 7 8 9 f) Inertia-estiate errors Slew aneuver using the control law () under a disturbance that is constant with respect to the bus-fixed frae. Figures 9a 9f show, respectively, the attitude error, angular velocity coponents, angular rates of the wheels, angular oentu, disturbance-estiate errors, and inertia-estiate errors. The spacecraft attitude and angular velocity coponents reach the coanded values in about 8 s. Figure 9c indicates that the reaction-wheel rotational speed grows unbounded. Figure 9d shows that the total angular oentu of the spacecraft increases, which is consistent with the constant disturbance torque acting on the spacecraft. In practice, the spacecraft needs a ethod to dup the stored angular oentu so that the reaction wheel rates do not grow unbounded. Figure repeats the aneuver with axiu wheel saturation at rad s, corresponding to roughly rp. The controller brings the spacecraft to the desired orientation in about 6 s at which tie one of the angular rates of the reaction wheels reaches

WEISS ET AL. 47 Eigenaxis Attitude Error (rad).... Angular Velocity Coponents (rad/s) ω ω ω Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 4 6 7 8 9 Tie (s) a) Eigen axis attitude error Wheel Angular Velocity Coponents (rad/s) 4 6 7 8 9 Tie (s) c) Angular rates of the reaction wheels. The spin rate grows until reaching the saturation liit of rad/s Disturbance Estiate Errors (N ) 4 6 7 4 6 7 8 9 Tie (s) e) Disturbance-estiate errors Fig. each wheel is saturated at rad s. ν ν ν 4 6 7 8 9 Tie (s) b) Spacecraft angular-velocity coponents Angular Acceleration Coponents (rad/s ) 4 4 4 6 7 8 9 Tie (s) d) Angular accelerations of the reaction wheels Inertia Estiate Errors (kg ) Inertia Estiate Errors (kg ) 4 6 7 8 9 Tie (s) u u u Jtilde Jtilde Jtilde Jtilde Jtilde Jtilde 4 6 7 8 9 Tie (s) f) Inertia-estiate errors Slew aneuver using the control law () under a disturbance that is constant with respect to the bus-fixed frae. The axiu rotation rate of rad s, disturbance and inertia estiates diverge, and the syste is destabilized. C. Spin Maneuver Using Control Law () Consider a spin aneuver with the spacecraft initially at rest and R I. The desired attitude is deterined by R d I, and the coanded constant angular velocity is ω d... T rad s. Assue no disturbance. Figures a f show, respectively, the attitude errors, angular-velocity coponents, angular rates of the wheels, the control inputs, which are the angular accelerations of the wheels, angular oentu, and inertia-estiate errors. For this aneuver, the spin coand consists of a specified tie history of rotation about a specified body axis aligned in a specified inertial direction. The controller achieves the coanded spin within about s.

48 WEISS ET AL. Eigenaxis Attitude Error, rad.9.8.7.6..4... Angular Velocity Coponents, rad/sec... ω ω ω Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 4 6 7 8 9 Tie, sec a) Eigenaxis attitude error Wheel Angular Velocity Coponents, rad/sec Angular oentu, X Angular oentu, Y Angular oentu, Z 4 6 7 8 9 Tie, sec c) Angular rates of the reaction wheels 4 6 7 8 9 Tie, sec Total, y Bus, y Wheels, y ν ν ν Total, x Bus, x Wheels, x 4 6 7 8 9 Tie, sec Total, z Bus, z Wheels, z 4 6 7 8 9 Tie, sec e) Angular oentu of the spacecraft relative to its center of ass with respect to the inertial frae resolved in the inertial frae. 4 6 7 8 9 Tie, sec b) Spacecraft angular-velocity coponents Angular Acceleration Coponents (rad/sec ) 4 4 6 7 8 9 Tie, sec d) Angular accelerations of the reaction wheels Inertia Estiate Errors, kg Inertia Estiate Errors, kg 4 6 7 8 9 Tie, sec u u u Jtilde Jtilde Jtilde Jtilde Jtilde Jtilde 4 6 7 8 9 Tie, sec f) Inertia-estiate errors Fig. Spin aneuver using the control law (). VI. Conclusions Alost global stabilizability (that is, Lyapunov stability with alost global convergence) of spacecraft tracking is feasible without inertia inforation and with continuous feedback using three linearly independent reaction wheels, for which the axes of rotation are not necessarily aligned with the principal axes of the spacecraft bus, do not necessarily pass through the spacecraft s center of ass and are not necessarily ass balanced in order to preserve the location of the spacecraft s center of ass. In addition, asyptotic rejection of haronic disturbances (including constant disturbances as a special case) is possible with knowledge of the disturbance spectru but without knowledge of either the aplitude or phase. Under these assuptions, the adaptive control laws presented in this paper provide an alternative to previous controllers that ) require

WEISS ET AL. 49 Downloaded by UNIVERSITY OF MICHIGAN on Septeber 8, http://arc.aiaa.org DOI:.4/.86 exact or approxiate inertia inforation or ) are based on attitude paraeterizations such as quaternions that require discontinuous control laws or fail to be physically consistent (that is, specify different control torques for the sae physical orientation). A future extension will address spacecraft actuation using control oent gyroscopes. Acknowledgents The authors wish to thank Marc Cablor for generating Fig. and the reviewers for their helpful coents. References [] Junkins, J. L., Akella, M. R., and Robinett, R. D., Nonlinear Adaptive Control of Spacecraft Maneuvers, Journal of Guidance, Control, and Dynaics, Vol., No. 6, 997, pp. 4. doi:.4/.49 [] Ahed, J., Coppola, V. T., and Bernstein, D. S., Adaptive Asyptotic Tracking of Spacecraft Attitude Motion with Inertia Matrix Identification, Journal of Guidance, Control, and Dynaics, Vol., No., 998, pp. 684 69. doi:.4/.4 [] Sanyal, A., Fosbury, A., Chaturvedi, N., and Bernstein, D. S., Inertia- Free Spacecraft Attitude Tracking with Disturbance Rejection and Alost Global Stabilization, Journal of Guidance, Control, and Dynaics, Vol., No. 4, 9, pp. 67 78. doi:.4/.46 [4] Chaturvedi, N., Sanyal, A., and McClaroch, N. H., Rigid Body Attitude Control: Using Rotation Matrices for Continuous, Singularity- Free Control Laws, IEEE Control Systes Magazine, Vol., No.,, pp.. doi:.9/mcs..9449 [] Wie, B., and Barba, P. M., Quaternion Feedback for Spacecraft Large Angle Maneuvers, Journal of Guidance, Control, and Dynaics, Vol. 8, No., 98, pp. 6 6. doi:.4/.9988 [6] Joshi, S. M., Kelkar, A. G., and Wen, J. T., Robust Attitude Stabilization Using Nonlinear Quaternion Feedback, IEEE Transactions on Autoatic Control, Vol. 4, No., 99, pp. 8 8. doi:.9/9.467669 [7] Bhat, S. P., and Bernstein, D. S., A Topological Obstruction to Continuous Global Stabilization of Rotational Motion and the Unwinding Phenoenon, Systes and Control Letters, Vol. 9, No.,, pp. 6 7. doi:.6/s67-69(99)9- [8] Crassidis, J. L., Vadali, S. R., and Markley, F. L., Optial Variable- Structure Control Tracking of Spacecraft Maneuvers, Journal of Guidance, Control, and Dynaics, Vol., No.,, pp. 64 66. doi:.4/.468 [9] Mayhew, C. G., Sanfelice, R. G., and Teel, A. R., Quaternion-Based Hybrid Control for Robust Global Attitude Tracking, IEEE Transactions on Autoatic Control, Vol. 6, No.,, pp. 66. doi:.9/tac..849 [] Cortes, J., Discontinuous Dynaic Systes, IEEE Control Systes Magazine, Vol. 8, June 8, pp. 6 7. doi:.9/mcs.8.996 [] Levine, W. S., Control Systes Applications, CRC Press, Boca Raton, FL, 999, p.. [] Kasdin, N. J., and Paley, D.A., Engineering Dynaics, Princeton Univ. Press, Princeton, NJ,. [] Hughes, P. C., Spacecraft Attitude Dynaics, Wiley, 986; reprint, Dover, New York, 8, p. 7. [4] Koditschek, D. E., The Application of Total Energy as a Lyapunov Function for Mechanical Control Systes, Dynaics and Control of Multibody Systes, Vol. 97, edited by Marsden, J. E., AMS, Providence, RI, 989, pp. 7. [] Cruz, G., Yang, X., Weiss, A., Kolanovsky, I., and D. S. Bernstein, D. S., Torque-Saturated, Inertia-Free Spacecraft Attitude Control, AIAA Guidance, Navigation, and Control Conference, AIAA Paper - 67, portland, OR, Aug..