Spacecraft Angular Velocity Stabilization Using A Single-Gimbal Variable Speed Control Moment Gyro

Transcription

1 Spacecraft Angular Velocity Stabilization Using A Single-Gimbal Variable Speed Control Moment Gyro Ancil Marsall and Panagiotis Tsiotras Georgia Institute of Tecnology, Atlanta, GA Feedback controllers for te stabilization of te angular velocity vector of a rigid spacecraft using a single-gimbal Variable Speed Control Moment Gyro (VSCMG) are presented. Linearization of te equations of motion sow tat complete attitude stabilization is not possible via linear metods. Noneteless, it is sown tat te linearized angular velocity equations are controllable, and a simple LQR control law is used to locally asymptotically stabilize te angular velocity vector. A Lyapunov-based approac is subsequently used to derive a state feedback control law tat globally asymptotically stabilizes te nonlinear angular velocity system. Introduction Stabilization of te angular velocity equations of a rigid spacecraft wit less tan tree control torques ave been addressed in several papers using various tecniques. In Ref. 1 it was sown, via Lyapunov metods, tat te angular velocity equations can be made locally asymptotically stable about te origin by means of two torques applied along two principal axes. Te control law proposed in Ref. 1 was nonlinear. Reference 2 complemented tese results by sowing asymptotic stability via te construction of a center manifold. A new control law was proposed, and te control law of Ref. 1 was verified. Reference 3 continued tis avenue of researc by sowing tat one external torque, aligned wit a principal axis, could stabilize te angular velocity vector about te origin. Moreover, it was sown tat te controller was robust relative to canges in te parameters defining te In Ref. 4, global asymptotic stability of te angular velocity was proved using a single, linear control law, provided tat te spacecraft as no symmetries. It was also sown tat a single control torque aligned wit a principal axis cannot asymptotically stabilize te system. Reference 5 furter proved tat a body wit an axis of symmetry can be globally asymptotically stabilized using one control torque. Te resulting control law must necessarily be nonlinear. However, no controller was presented in Ref. 5. Reference 6 verified tat te results of Refs. 4 Graduate Student, Scool of Aerospace Engineering. Member AIAA. ancil marsall@ae.gatec.edu Associate Professor, Scool of Aerospace Engineering. Associate FellowAIAA. p.tsiotras@ae.gatec.edu, Tel: (44) , Fax: (44) Corresponding autor. Copyrigt c 23 by P. Tsiotras and A. Marsall. Publised by te American Institute of Aeronautics and Astronautics, Inc. wit permission. and 5 follow easily as an application of te Jurdjevic- Quinn approac. It also included an explicit nonlinear control law wic provided global stability for te axisymmetric case. Reference 7 sowed tat te angular velocity of an axi-symmetric rigid body can be globally asymptotically stabilized by means of a linear feedback wen two control torques act on te body. Oter approaces used to develop globally asymptotically stabilizing controllers for a rigid spacecraft wit two torques include te general metodology of nonlinear zero dynamics in Ref. 8, and te energy tecniques of Ref. 9. On te same token, te autors of Ref. 1 addressed te angular velocity stabilization of an almost axi-symmetric rigid spacecraft wit partial attitude stabilization using two external torques. In te previously mentioned references, te control torques are assumed to be provided by gas jets. Alternatively, internal torques can be generated by momentum (or reaction) weels or control moment gyroscopes (CMGs). Te spin axis of a momentum weel is fixed in te body frame, and te spin rate of te flyweel is varied to produce a torque along te spin axis. In te CMG case te weel speed of te flyweel is kept constant. A gimbal assembly canges te spin axis of te flyweel, tus producing a torque wic is ortogonal to bot te spin and gimbal axes of te CMG. It is well known tat te primary advantage of single-gimbal CMGs over oter momentum excange devices is teir torque amplification property, tat is, te output torque produced from te rotation of te angular momentum vector is muc larger tan te one required for gimbal rotation. Several references discuss te use of CMGs for spacecraft attitude control. See, for example, Refs A complete controllability analysis of te spacecraft equations as been reported in Ref. 15. Tere, it is

2 sown tat te system is not controllable wit less tan tree reaction weels. Krisnan et al 16 provided a control law using two momentum weels for te restricted case of zero angular momentum. Reference 17 developed a control law to stabilize te spin axis of a rigid spacecraft about a specified inertia axis using two reaction weels. Finally, te autors in Ref. 18 applied modern nonlinear control tecniques for detumbling of a spacecraft wit a single momentum weel aligned along one of te spacecraft principal axes. Te use of Variable Speed Control Moment Gyros (VSCMGs) for spacecraft stabilization as received 19, 2 attention recently. A VSCMG can be tougt of as a ybrid device comprised of a momentum (or reaction) weel and a CMG. In particular, te weel speed of a VSCMG is allowed to vary, tus producing an additional torque over a conventional CMG. Tis torque is not fixed in te spacecraft body frame, as in te case of a momentum weel; rater, te direction of te spin axis of te VSCMG is allowed to rotate via a gimbal. An additional torque, perpendicular to te spin and gimbal axes is tus generated, as in te conventional CMG case. Tis additional degree of freedom can be utilized to avoid te gimbal lock singularity tat as plagued traditional CMG clusters. 19 Te use of te additional torque of VSCMGs as also been utilized for attitude control (and energy storage) of spacecraft in Refs In bot te CMG and VSCMG cases presented in te literature to date, a cluster of actuators as been used to provide a sufficient number of torques to acieve complete attitude stabilization (and possibly energy storage). In tis paper, we consider te case of control (stabilization) of a spacecraft via a single VSCMG actuator. Te outline of tis paper is as follows. First, we present te complete equations of motion of a spacecraft wit one VSCMG in an arbitrary orientation. Tese equations are composed of te dynamic and kinematic equations. Next, we linearize te equations of motion about an equilibrium point. Linearization sows tat te spacecraft attitude is uncontrollable wit only one VSCMG. However, te angular velocity equations remain controllable. A simple LQR feedback law is designed to acieve local asymptotic stability at te origin of te linearized angular velocity system. Next, we examine te exact angular velocity equations. We derive a nonlinear control law tat ensures global asymptotic stability of te angular velocity of te spacecraft about te origin using only one VSCMG. Several numerical examples are included to demonstrate te viability of te control algoritms proposed. Equations of Motion Te dynamic equations of motion of a spacecraft wit a cluster of VSCMGs ave been fully derived in te literature Herein, we will use te equations as derived by Ricie et al 21 and Yoon and Tsiotras. 2 In Ref. 2 it is assumed tat te center of mass of eac VSCMG weel coincides wit tat of te gimbal structure; te spacecraft, weels, and gimbal structure are rigid; te flyweels and gimbals are balanced; and te spacecraft rotational motion is decoupled from its translational motion. Figure 1 sows a spacecraft wit a single VSCMG. Te origin of te body frame B, is located at te center of mass of te entire spacecraft. Te gimbal frame G, represented by te ortonormal set of unit vectors, and, is arbitrarily located in te spacecraft platform. Fig. 1 Body Frame O Spacecraft wit a single VSCMG Dynamics in te body frame Specializing te dynamical equations of motion presented in Yoon and Tsiotras 2 to a single VSCMG, we get J ω+ Jω+A g I cg γ +A t Ω γ +A s Ω+ω = (1) were, := Jω + A g I cg γ + A s Ω (2) J(γ) := I B B + A s (γ)i cs A T s (γ)+a t (γ)i ct A T t (γ) +A g I cg A T g (3) Te argument in J( ) is included to denote explicitly te dependence of te spacecraft inertia matrix on te gimbal axis angle, γ. For notational simplicity, in te sequel we will often drop te argument wen it is clear from te context. As a result of (3) it follows tat J = γa t (I cs I ct )A T s + γa s (I cs I ct )A T t (4) Notice tat J is linear in te gimbal rate. In (2) te (column) vector IR 3,istetotal angular momentum of te spacecraft wit respect to te inertial

3 frame, expressed in te body frame. Similarly, ω IR 3 is te angular velocity vector of te spacecraft wit respect to te inertial frame, expressed in te body frame of te spacecraft. Te quantities Ω, and Ω are te magnitudes of te angular velocity and angular acceleration of te VSCMG weel, respectively, about te gimbal spin axis, wit respect to te gimbal frame. Also, γ and γ are te magnitudes of te gimbal rate and gimbal acceleration, respectively, about te gimbal axis, wit respect to te spacecraft platform. In our analysis we will assume, as usual, gimbal rate commands. Tis is also te case for standard CMG s in order to take full advantage of te torque amplification property. Tis implies a velocity steering law for te gimbal. Tus, our control inputs are Ω and γ and we can write γ = u 1, Ω =u2 (5) Often we use u := [u 1, u 2 ] T IR 2 for te combined control vector. Te matrix-valued function J : [, 2π) IR 3 3 provides te inertia matrix of te entire spacecraft, i.e. te spacecraft platform, including te weel and gimbal structure of te VSCMG, given te gimbal angle γ. Notice tat te matrix J(γ) is positive definite for all values of te gimbal angle γ [, 2π). Te constant matrix I B B is te sum of te inertia of te spacecraft platform, and te inertias of te mass centers of te weel, gimbal and spacecraft platform, about te equivalent mass center of te entire spacecraft. I c represents te sum of te inertia scalars of te weel and gimbal structure, i.e. I c = I w + I g, were = {s, t, g}. Te column vectors A IR 3, were = {s, t, g} are te body frame representations of te gimbal frame unit vectors,, and. Observe tat A s and A t are functions of te gimbal angle γ, as follows A s (γ) = A so c γ + A to s γ (6a) A t (γ) = A so s γ + A to c γ (6b) were A so and A to are te values of A s and A t at some initial time, and c γ := cos γ and s γ := sin γ. Finally, for any vector v =[v 1, v 2 v 3 ] T IR 3,te notation v IR 3 3 represents te skew symmetric matrix v 3 v 2 v = v 3 v 1. v 2 v 1 For te details of te derivation of equations (1)-(3), as well as te notation used in tis paper, te interested reader may refer to Ref. 2. Kinematics Witout loss of generality, Euler angles will be used to represent te attitude of te spacecraft. For a Euler angle sequence, te kinematic equations are given by 22 φ = ω 1 + ω 2 sin φ tan θ + ω 3 cos φ tan θ (7a) θ = ω 2 cos φ ω 3 sin φ (7b) ψ = ω 2 sin φ sec θ + ω 3 cos φ sec θ (7c) Linear System Analysis In tis section, we linearize te full nonlinear equations of motion, given by (1), (5) and (7), and examine teir controllability properties. We also present a linear control law wic stabilizes te angular velocity of te linearized system. Te only mild assumption made ere is tat te gimbal acceleration γ is negligible. Linearization Te equilibrium points of Eqs. (1), (5) and (7) are given by ω = φ = θ = ψ = and γ = γ o, Ω = Ω o, were γ o and Ω o are arbitrary constants. From equations (6) we get were A s A sf (γ o )+A tf (γ o ) γ (8a) A t A tf (γ o ) A sf (γ o ) γ, (8b) A sf (γ o ) := A so c γo + A to s γo, (9a) A tf (γ o ) := A so s γo + A to c γo, (9b) and were ( ) represents a small perturbation in te variable from its equilibrium value. Similarly, eac term of Eq. (1) results, to first order, in te following terms J ω J f ω (1a) Jω (1b) A t Ω γ Ω o A tf γ (1c) A s Ω Iws A sf Ω (1d) ω Ω o A sf ω (1e) were J f (γ o ):=I B B +A sf I cs A T sf +A tfi ct A T tf +A gi cg A T g. Te linearization of (1) tus yields ω = A 1 ω + B 1 γ + B 2 Ω (11) were te matrices A 1 IR 3 3, B 1 IR 3 1,andB 2 IR 3 1 are given by A 1 (γ o, Ω o ) := J 1 f Ω o A sf (12a) B 1 (γ o, Ω o ) := J 1 f Ω o A tf (12b) B 2 (γ o, Ω o ) := J 1 f A sf (12c) Note tat tese matrices depend on te equilibrium/reference values γ o and Ω o. Defining te new

4 state variable as x := [ ω T, φ, θ, ψ] T IR 6 and te control as u := [ γ, Ω] T IR 2,wecan express te linearized equations in te familiar form ẋ = A x + B u (13) were te matrices A IR 6 6 and B IR 6 2 are given by [ ] [ ] A1 A := 3 3 B1 B, B := 2 (14) were 1 is te identity matrix. Controllability of Linearized System Here we give two results on te controllability of te linearized complete system in Eqs. (13) and of te linearized angular velocity equations in Eq. (16). Proposition 1 Te linearized system described by Eqs. (13) and (14) is uncontrollable for any γ o [, 2π) and Ω o IR. Te result follows by sowing tat te controllability matrix 23 C o := [B ABA 2 BA 3 BA 4 BA 5 B] (15) as rank five for all γ o [, 2π) and Ω o IR. Since te state dimension is six, te pair (A, B) is uncontrollable. Tis result implies tat it is not possible to use linear tecniques to stabilize te complete attitude of te spacecraft using a single VSCMG. It leaves, owever, open te possibility tat te complete system of equations are controllable in te nonlinear sense. In te sequel we restrict our attention to te angular velocity subsystem. To tis end, define à := A 1 and B := [B 1 B 2 ], and rewrite Eq. (11) as ω = à ω + B u (16) Linearizing te kinematic equations, we get φ = ω 1, θ = ω 2, ψ = ω 3 (17) Proposition 2 Te linearized angular velocity system of Eq. (16) is controllable for all γ o [, 2π), and Ω o. Proof. From Teorem 3.1 of Ref. 23, (Ã, B) iscontrollable if te matrix [à λ1 B] as full row rank for all λ IR. In particular, tis must be true for all te eigenvalues λ i (i =1, 2, 3) of te matrix Ã. Calculating te row-reduced ecelon form of Ã, one obtains tat for λ = λ i (i =1, 2, 3) and Ω o, 1 [à λ1 B] = 1 (18) 1 Te rank of te matrix C o was calculated using te Symbolic Toolbox of MATLAB. 24 were ( ) are algebraic expressions in terms of te components of à tat do not affect te row rank of te matrix. Since te rank of te above matrix is always tree, te linearized angular velocity system of Eq. (11) is controllable for all γ o [, 2π), and Ω o. LQR Controller for te Angular Velocity Subsystem Given te controllable system in Eq. (16), we can find a linear control law via LQR metods. For example, we can determine a static full-state feedback law u = K ω suc tat te performance cost J = [ ω T Q ω + u T R u]dt (19) is minimized subject to te dynamics (16). Te matrix Q must be positive semi-definite wile R must be positive definite. Te control gain matrix K is given by K = R 1 BT P. Te matrix P = P T is positive semi-definite and satisfies te Algebraic Riccati Equation ÃT P + P à P BR 1 BT P + Q =. LQR optimal control designs is by now folklore. Details of can be found, for instance, in Ref. 23. Nonlinear System Analysis Te LQR controller of te previous section ensures asymptotic stability only locally about te equilibrium ω = and for gimbal angles and weel speeds close to teir reference values γ o and Ω o, respectively. Te last restriction is particularly troublesome, since stabilization of te angular velocity vector sould not inge upon γ and Ω being close to γ o and Ω o. As a matter of fact, significant control autority may tend to produce large deviations of te gimbal angle and weel speed from teir reference values; see Eqs. (5). In realistic cases, it is not reasonable to expect tat te states γ and Ω (wose values are of no particular interest, tus are not penalized in (19)) will remain small. For a more compreensive analysis of te stabilization problem, it is terefore necessary to work wit te exact, nonlinear equations of motion. In te sequel we improve on te previous results by finding a control law tat ensures global asymptotic stability for te nonlinear system. We tus also avoid te issue of te restricted (local) validity of te linearized equations due to potentially large deviation of te gimbal angle and weel speed from teir reference values. As in te linear case, in te sequel we assume tat te gimbal acceleration γ is negligible, and te control inputs are γ and Ω. Te dynamic equations of motion are tus given by J ω + Jω + A t Ω γ + A s Ω+ω = (2)

5 were and J are as in (2) and (3). To derive a stabilizing control law for (2), we consider te positive definite, continuously differentiable Lyapunov function V (ω) := 1 2 ωt Jω. Te derivative of V along te trajectories of te system is V (ω) =ω T J ω ωt Jω = ω T J ω + ω T Jω 1 2 ωt Jω = ω T (J ω + Jω) 1 2 ωt Jω = ω T (A t Ω γ + A s Ω+ω ) 1 2 ωt Jω Rewriting J = Φ γ were Φ := A t (I cs I ct )A T s + A s (I cs I ct )A T t and using te fact tat ω T ω = yields V (ω) = ω T A t Ω γ 1 2 ωt Φω γ ω T A s Ω = ω t Ω γ ω s ω t (I cs I ct ) γ ω s Ω were ω s = ω T A s and ω t = ω T A t are te components of te body angular velocity vector ω along te spin and transverse axes of te gimbal frame, respectively, i.e. ω = ω s + ω t + ω g. Proposition 3 Consider te following control law ( ( )) Ics I ct γ = k 1 (Ω) ω t Ω ω s ( ) 2 Ics I ct Ω =k 2 ω s + k 3 ω t k 4 (Ω) + k 1 (Ω) ωt 2 ω s were k 1 : R IR + is any function suc tat k 1 (Ω)Ω 2 is bounded for all Ω IR, k 4 (Ω) := Ωk (Ω), k 2 > and 2 k 2 >k 3. Tis control law globally asymptotically stabilizes te system given by Eq. (2) for all Ω(). Proof. Substituting tis control law in te expression for V (ω) leads to ( ( )) V (ω) = ωt 2 Ics I ct Ω k 1 (Ω) Ω ω s ( ( )) ω s ωt 2 Ics I ct (I cs I ct )k 1 (Ω) Ω ω s ω s (k 2 ω s + k 3 ω t k 4 (Ω) ( ) 2 Ics I ) ct + k 1 (Ω) ωt 2 ω s = ωt 2 Ω 2 k 1 (Ω) k 2 ωs 2 k 3 ω s ω t k 4 (Ω) [ ] [ ] ωt = ωt ω s G(Ω,ω,γ) ω s were te matrix G(Ω,ω,γ) is given by [ G(Ω,ω,γ):= ] 2 sgn(ω t )k 4 (Ω) k 3 Ω 2 k 1 (Ω) k 3 2 sgn(ω t )k 4 (Ω) k 2 It can be easily sown tat G(Ω,ω,γ) for all (Ω,ω,γ) R 2 [, 2π) andg(ω,ω,γ) > forω. It follows tat V. Te last inequality sows tat V, and ence ω is bounded. Terefore, γ and Ω as well as γω are bounded. Moreover, ω is bounded from (2). It follows tat ω, γ and Ω are uniformly continuous and tus V is uniformly continuous as well. From Barbalat s Lemma 25 it follows tat V. Tis implies tat ω s and Ωω t ast. Assume now tat Ω() and tat ω t ω t. Since ω s we ten ave tat after a sufficiently long time, Ω k 3 ω t Ωk (Ω) and te equilibrium Ω = is unstable. Hence, Ωω t, a contradiction. It follows tat, necessarily, ω t ast. Assume now tat ω s =Ωω t. It follows tat γ = Ω = and from Eq. (2) J ω + ω (Jω + A s Ω) = (21) wic, wen expressed in te gimbal frame, becomes J 13 ω g J 23 ωg 2 = (22a) J 23 ω g + J 13 ωg 2 + Ωω g = (22b) J 33 ω g = (22c) From Eq. (22c), we get ω g =. From Eq. (22a) or Eq. (22b), we conclude tat ω g =. Tus, te largest invariant set in {ω : V (ω) = } is te set ω =. Asymptotic stability follows from LaSalle s teorem and global asymptotic stability follows from te radial unboundedness of te function V and te fact tat te previous analysis olds for all initial conditions ω IR 3. Acceleration Steering Law In te actual spacecraft te gimbal control input is a torque (or gimbal acceleration) command, rater tan a gimbal velocity command. Te derived velocity command as to be implemented via an internal servo control loop. A simple implementation of tis idea is to use, say γ = K p ( γ d γ) (23) were K p > and were γ d as in Proposition 3. Tis (proportional) control law will ensure tat te actual gimbal velocity γ approaces te desired command γ d, as t. In practice K p as to be sufficiently large in order for te convergence to take place in a sort interval of time.

6 Numerical Examples In tis section we give some illustrative examples of te control design metods for te angular velocity subsystem using bot te linear and te nonlinear analysis of te previous sections. Bot examples applied te control laws developed earlier to te complete equations of motion in Eqs. (1)-(4) using te acceleration steering law in Eq. (23). Tis was done in order to compare eac control law individually and in relation to eac oter using a realistic evaluation model. Table 1 summarizes te values te moments of inertia and gimbal used in te simulations. Tese values rougly correspond to te spacecraft simulator described in Ref. 26. Te controller gains and te initial conditions are given in Table 2. Te first example corresponds to te LQR control design metod wic was developed from te linearized angular velocity system. Te results are sown in Fig. 2. Te weigting matrices Q and R in (19) were cosen by trial and error to stabilize te system quickly wit suitable damping. Teir values are sown in Table 2. Figures 3 and 4 sow te values of te gimbal angle and weel speed as well as teir rates. Table 1 Moments of inertia values. Symbol Value Units I B B kg m kg m 2 I wt,i wg.24 kg m 2 I gs.93 kg m 2 I gt,i gg.54 kg m 2 A so [ 1,, ] T A to [,.8161,.5779] T A go [,.5779,.8161] T Te second example corresponds to te nonlinear control law of Proposition 3. Te function k 1 (Ω) was cosen as k 1 (Ω) = µ/(1 + Ω 2 ). Te angular velocity istories wit te nonlinear control law are sown in Fig. 5. Te time istory of γ and Ω aresownin Fig. 6. Te time istories of te gimbal angles and te weel speed velocity are sown in Fig. 7. Notice tat te nonlinear controller is more aggressive resulting in larger values for te gimbal angle and weel speed. Since in a pysical system te weel and te gimbal rate commands saturate, it is imperative to modify te nonlinear control law so as to take into account tese saturation effects. Tis is left for future investigation. On te oter and, one may coose to use te nonlinear controller only if te initial conditions be- ω [rad/sec] Time istory of Angular Velocities ω 1 ω 2 ω 3.25 Fig. 2 Numerical simulations wit te LQR control law. γ [deg/sec] Ω [rpm/sec] Time istory of γ and Ω 25 Fig. 3 Time istory of γ and Ω wit te LQR Table 2 Controller gains and initial conditions. Symbol Linear Nonlinear Units ω() [.2,.1,.2] T [.1,.1,.1] T rad/sec γ() γ o = 2 2 deg γ() deg Ω() Ω o = rpm Q diag{1e 4, 1e 4, 1e 4 } R diag{1e 2, 1} µ 8 sec 1 k 2 4 sec 1 k 3 1 sec 1 2 K p 1 1 sec 1 come too large. After te trajectories reac a small neigborood of te origin (and witin te region of attraction of te linear controller), one can ten switc to te LQR controller, wose local performance can be pre-assigned via te optimization criterion (19). In

7 5 Time istory of γ and Ω 15 Time istory of γ and Ω 4 1 γ [deg] γ [deg/sec] Ω[rpm] Fig. 4 Time istory of γ and Ω wit te LQR ω [rad/sec] Time istory of Angular Velocities ω 1 ω 2 ω 3.2 Fig. 5 Numerical simulations wit te nonlinear tis sense, te linear controller acieves (local) performance and stability, wereas te nonlinear controller acts as a safety net to protect te system from large initial conditions. For comparison, in Fig. 8 we sow te results from te numerical simulations of te LQR wit initial conditions ω() = [.1.1.1] T. For tese (large) initial conditions, te LQR does not stabilize te nonlinear system. Finally, Fig. 9 sows a series of snapsots of a spacecraft wit one VSCMG undergoing a detumbling maneuver using te nonlinear control law of Proposition 3. Note tat, as expected, te final orientation of te spacecraft is suc tat te spin axis of te VSCMG is aligned wit te total angular momentum vector, wic remains constant in inertial frame at all times. Conclusions In tis paper, we ave addressed te stabilization problem of a rigid spacecraft wit a single-gimbal variable-speed control moment gyro (VSCMG). Since Ω [rpm/sec] Fig. 6 Time istory of γ and Ω wit te nonlinear γ [deg] Ω[rpm] Time istory of γ and Ω 15 Fig. 7 Time istory of γ and Ω wit te nonlinear no external control torques act on te system, reorientation of te spacecraft is acieved via momentum transfer between te spacecraft platform and te VSCMG. We sowed tat te complete attitude equations are not linearly controllable. Te angular velocity equations are, noneteless controllable. A simple LQR controller was used to locally asymptotically stabilize te angular velocity equations for an arbitrary gimbal frame orientation. Abandoning te restrictive assumptions made in te linear case we developed a control law for te nonlinear system wic ensures global asymptotic stability of te angular velocity equations. Acknowledgement: Support for tis work as been provided by AFOSR troug award no. F References 1 Brockett, R., Asymptotic Stability and Feedback Stabilization, Differential Geometric Control Teory, 1983,

8 ω ω ω ω Time =. sec, γ =. deg Time = 5. sec, γ = 2.4 deg Time = 1. sec, γ = 2. deg Time = 15. sec, γ = 31.5 deg Time = 2. sec, γ = 39.4 deg Time = 3. sec, γ = 49.1 deg Time = 4. sec, γ = 54.2 deg Time = 1. sec, γ = 57.9 deg Fig. 9 Series of snapsots of a detumbling maneuver of a spacecraft wit one VSMG. Te gimbal frame unit vectors,,and, te angular momentum vector, and te angular velocity vector ω are sown (not drawn to scale). Te time of eac snapsot and te gimbal angle γ are also depicted. y Time istory of γ and Ω.2 Fig. 8 Response of LQR controller wit large initial conditions. pp , (R.W. Brockett, R.S. Millman, and H.J. Sussman, eds.). 2 Aeyels, D., Stabilization of a class of nonlinear systems by a smoot feedback control, Systems & Control Letters, Vol. 5, 1985, pp Aeyels, D., Stabilization by smoot feedback of te angular velocity of a rigid body, Systems & Control Letters, Vol. 5, 1985, pp Aeyels, D. and Szarfanski, M., Comments on te stability of te angular velocity of a rigid body, Systems and Control Letters, Vol. 1, 1988, pp Sontag, E. and Sussmann, H., Furter comments on te stabilizability of te angular velocity of a rigid body, Systems & Control Letters, Vol. 12, 1988, pp w1 w2 w3 6 Outbib, R. and Sallet, G., Stabilizability of te angular velocity of a rigid body revisited, Systems & Control Letters, Vol. 18, 1992, pp Andriano, V., Global feedback stabilization of te angular velocity of a symmetric rigid body, Systems & Control Letters, Vol. 2, 1993, pp Brynes, C. and Isidori, A., NewResults and examples in nonlinear feedback stabilization, Systems & Control Letters, Vol. 12, 1989, pp Astolfi, A. and Ortega, R., Energy-Based Stabilization of Angular Velocity of Rigid Body in Failure Configuration, AIAA Journal of Guidance, Control and Dynamics, Vol.25, No. 1, 22, pp Tsiotras, P. and Scleicer, A., Detumbling and Partial Attitude Stabilization of a Rigid Spacecraft Under Actuator Failure, AIAA Guidance, Navigation and Control Conference, Denver, CO, 2, AIAA Paper O, S. and Vadali, S., Feedback Control and Steering Laws for Spacecraft Using Single Gimbal Control Moment Gyros, Journal of te Astronautical Sciences, Vol. 39, No. 2, 1991, pp Sing, S. N. and Bossart, T., Exact Feedback Linearization and Control of Space Station Using CMG, IEEE Transactions on Automatic Control, Vol. 38, No. 1, 1993, pp Ford, K. and Hall, C., Singular Direction Avoidance Steering for Control-Moment Gyros, Journal of Guidance, Control and Dynamics, Vol. 23, No. 4, 2, pp Heiberg, C. and Bailey, D., Precision Spacecraft Pointing Using Single-Gimbal Control Moment Gyroscopes wit Disturbance, Journal of Guidance, Control, and Dynamcis, Vol. 23, No. 1, 2, pp Crouc, P. E., Spacecraft attitude control and stabilization: Applications of geometric control teory to rigid body models, IEEE Transactions on Automatic Control, Vol. 29, No. 4, 1984, pp

9 16 Krisnan, H., McClamroc, H., and Reyanoglu, M., Attitude Stabilization of a Rigid Spacecraft Using Two Momentum Weel Actuators, Journal of Guidance, Control, and Dynamics, Vol. 18, No. 2, 1995, pp Kim, S. and Kim, Y., Spin-Axis Stabilization of a Rigid Spacecraft Using Two Reaction Weels, Journal of Guidance, Control, and Dynamics, Vol. 24, No. 5, 21, pp Bang, H., Myung, H.-S., and Tak, M.-J., Nonlinear Momentum Transfer Control of Spacecraft by Feedback Linearization, Journal of Spacecraft and Rockets, Vol. 39, No. 6, Scaub, H., Vadali, S. R., and Junkins, J. L., Feedback Control Lawfor Variable Speed Control Moment Gyro, Journal of te Astronautical Sciences, Vol. 46, No. 3, 1998, pp Yoon, H. and Tsiotras, P., Spacecraft Adaptive Attitude Control and Power Tracking Variable Speed Control Moment Gyros, Journal of Guidance, Control and Dynamics, Vol. 25, No. 6, 22, pp Ricie, D., Tsiotras, P., and Fausz, J., Simultaneous Attitude Control and Energy Storage using VSCMGs: Teory and Simulation, Proceedings of te American Control Conference, 21, pp , Arlington, VA. 22 Huges, P. C., Spacecraft Attitude Dynamics, Jon Wiley & Sons, NewYork, 1986, pp Zou, K. and Doyle, J., Essentials of Robust Control, Prentice Hall, 1998, pp Matworks, MATLAB: Symbolic Mat Toolbox v 2.1.2, Natwick, MA, Slotine, J. and Li, W., Applied Nonlinear Control, Prentice-Hall Inc., Englewood Cliffs, 1991, pp Jung, D. and Tsiotras, P., A 3-DoF Experimental Test- Bed for Integrated Attitude Dynamics and Control Researc, AIAA Guidance, Navigation and Control Conference, Austin, TX, 23, AIAA Paper