TECHNICAL RESEARCH REPORT

Transcription

1 TECHNICAL RESEARCH REPORT Orientation Control of a Satellite by G.C. Walsh, R. Montgomery, S.S. Sastry T.R ISR INSTITUTE FOR SYSTEMS RESEARCH Sponsored by the National Science Foundation Engineering Research Center Program, the University of Maryland, Harvard University, and Industry

2 Orientation Control of a Satellite Gregory C. Walsh Institute for Systems Research A.V. Williams Bldg.(115) University of Maryland College Park, Md 04. S. Shankar Sastry Electronics Research Laboratory University of California Berkeley, CA 940. February 8, 1995 Richard Montgomery Department of Mathematics University of California, Santa Cruz, CA Abstract In this paper we consider the problem of controlling the orientation of a satellite with thrusters which provide two control torques. We derive open loop steering control laws for steering the satellite from one orientation to another. Then we develop closed loop regulation control laws. For a \symmetric satellite, (i.e. one with two moments of inertia equal) we stabilize a ve dimensional subset of the states. The regulator for the asymmetric satellite, which iscontrollable, renders an arbitrary conguration asymptotically stable. Anovel choice of \Listing coordinates for SO() proves to be useful in deriving these control laws. Research supported in part by NSF grant IRI , in part by NASA under grant NAG -4, and also by NSF grant DCDR The authors would like to thank Carlos Canudas DeWit for suggesting this problem. 1

3 1 Introduction The problem of controlling the attitude of a rigid body has a long and rich history in control. Early work includes the work of Meyer [1], Brockett [], Baillieul [1] and Crouch [8], on controllability and stabilization. Controllability was discussed for models of the satellite with fewer than three inputs both in [1, 8], but work on stabilization by Meyer [1] and others [] considered the fully actuated satellite. Our paper focuses on the control of a satellite with only two input torques. Control laws stabilizing a subset of the states of the system of a satellite with two inputs can be found in []. Despite the wealth of ne work, some interesting questions about stabilization remain open and are worth reconsidering in the light of recent work on the control of nonholonomic systems [, 4, ]. Recent results of Coron [] indicate that it is possible to nd time-periodic and smooth control laws which stabilize the origin. One of our contributions is the construction of such stabilizing control laws. Indeed, there has been renewed interest in this direction, see for example [5, 0]. The contributions of this paper are: 1. a set of open{loop planners for steering a satellite with two inputs,. a novel set of coordinates for the orientation of the satellite related to Listing's law and the Hopf bration,. and, nally, control laws which stabilize the angular velocities and orientation of the satellite. The control laws derived by us are smooth time varying control laws. It has been conjectured, however, that somewhat faster convergence for the regulators may be attainable by use of nonsmooth feedback control laws (see [1]). After we announced [] our results, two other papers have presented dierent control laws for stabilization of the asymmetric satellite. The one by Morin, Samson, Pomet amd Jiang [15] employ techniques close to those described in this paper, with dierent choices of output function. The paper of Coron [] uses dierent techniques. He transforms the system into a locally quadratic form and using a dilation technique demonstrates Lyapunov stability. We consider a rigid body model of the satellite with thrusters, providing input torques about the rst two principal axes. This assumption is without loss of generality: what follows is easily modied for other locations of the thrusters. The satellite model obeys the Euler equations for the rigid body. The moments of inertia through the unit axes e i, i =1; ; will be denoted I i and the input torques are denoted 1 ;. Euler's equations for the model of the satellite are given by: 4 _! 1 _! _! 5 = 4 _R = R^! I,I I 1!! I,I 1 I!! 1 I 1,I I! 1! where ^! refers to the skew-symmetric matrix given by ^! = 4 0,!!! 0,! 1,!! : 1 I 1 I 0 5 The state of the satellite is characterized by the orientation matrix R SO(), and the angular velocity vector! <. SO() denotes the group of rotations on <. As noted in the literature,

4 the dynamics simplify after the following input transformation, 1 = I 1 u 1 + I, I!! I 1 to = I u + I 1, I I 4 _! 1 _! _! 5 = 4 u 1 u! 1!!! 1 5 _R = R^! (1) The constant := I 1,I I. The analysis of the system divides into two cases: the symmetric satellite, with I 1 = I, i.e. =0,and the asymmetric satellite with = 0. A control law for pointing the symmetric satellite, that is, stabilizing 4 of the states, may be found in [0]. (The four variables that are stabilized are Re, a point on the two-sphere, and! 1 ;!. An alternative to this law is rederived in Section.1 as a starting point for our nal goal.) The organization of the paper is as follows: In Section, we present the open{loop path planning algorithms for steering the satellite between a given initial and nal orientation. Separate solutions are given for the symmetric and asymmetric satellite. The symmetric and asymmetric satellite are qualitatively dierent, because in the symmetric case it is impossible to control the angular velocity about the symmetry axes. In Section, we construct feedback controls for stabilizing the satellite. The method of exact input{output linearization is employed, using a clever choice of the output functions and coordinates. Finally, in the second part of this section we derive the stabilizing control law, and prove that it works using a Lyapunov argument. Once again, the symmetric and asymmetric cases must be treated separately. Finally, we provide simulation results for both cases. Open{Loop Steering In this section, we focus on the problem of constructively generating control laws for steering the satellite of (1) from an initial orientation R i SO(), initial angular velocity! i < at time 0 to a nal orientation R f SO() and nal angular velocity! f < at time T. We solve the problem at rst for a symmetric satellite, that is I 1 = I, with! (0) = 0. Owing to the symmetry,! is constant for all control laws and so we will need to choose! (T ) = 0, as well. Proposition 1 (Steering the Symmetric Satellite) Consider the dynamics of the satellite on SO()< described byequation (1) with I 1 = I, that is =0and! 0. Given R i ;R f SO(), an initial angular velocity! i and a time T > 0, there exists a piecewise constant control law, u() dened on [0;T], which steers the symmetric satellite from R i ;! i at time 0 to R f ;! f =0at time T. Proof: The proof is constructive and involves an algorithm. The function atan(y; x) = arg(x + iy). 1. System Halt, (0; T ). 5 For T seconds apply the controls: 5 [u 1 ;u ]=[,5! 1(0) ;,5! (0) ]. T T The system halt maneuver brings the possibly non-zero initial! 1,! to zero.

5 . Pointing, ( T 5 ; T 5 ). Dene ; by performing the following computation: x d = R( T 5 ),1 R f e = atan(,x d ();x d ()). = arcsin(x d (1)), with [0;]. Apply each of the following control laws for T seconds: 0 0 [u 1 ;u ]=[ T ; 0] 0 [u 1 ;u ]=[, T ; 0] 0 ] [u 1 ;u ]=[0; T ] [u 1 ;u ]=[0;, The maneuver is composed of two rotations: The angles ; have been chosen so that: 0 T R( T 5 ) = R( T 5 ) Exp(^e 1) Exp(^e ): () R,1 ( T 5 )R f e = Exp(^e 1 ) Exp(^e )e : () By using relation () in (), we nd R f e = R( T 5 )e. We are now pointing in the desired direction, R f e.. Measurement, ( T ). 5 Compute: R d = R,1 ( T 5 )R f. = atan(,r d (1; );R d (1; 1)). If = 0, STOP. In the next step, the residual rotation error is computed. Because R( T 5 )e = R f e we see that R d e = e. This implies that R d = Exp( ^e ) and further, = atan(,r d (1; );R d (1; 1)). If = 0we have achieved the nal orientation, otherwise the rest of the algorithm must be executed. 4. Outward Leg, ( T 5 ; T 5 ). Apply each for T seconds: 10 [u 1 ;u ]=[ 10 T ; 0] [u 1 ;u ]=[, ; 0] 5. Correction, ( T 5 ; 4T 5 ). 10 T Apply each for T seconds: ] [u 1 ;u ]=[0; T ] [u 1 ;u ]=[0;, 10 T 4

6 . Return Leg ( 4T 5 ;T). Remarks Apply each for T seconds: 10 [u 1 ;u ]=[, 10 T ; 0] [u 1 ;u ]=[ ; 0] 10 T To see the justication of these last three steps, examine the product Exp( ^e 1) Exp( ^e ) Exp(, ^e 1). Recall that R Exp(^a)R,1 is equal to Exp last three steps is Exp( Exp( d ^e 1)e )= Exp( ^e )R d. cra. Thus, the net motion of the 1. Area Form. If we set! = 0 and think of the remaining angular velocities! 1 ;! as the controls then the resulting system on SO() is equivalent to the equations of parallel transport for a unit vector tangent to the two-sphere S. Consequently, it follows from the Gauss-Bonnet theorem that the rotation about the e axis induced by the last three steps is equal to the area enclosed by the path traced by R(t)e on S. For a detailed proof of this fact, see [1, 5]. This suggests that the dynamics of the system is related to an area form which should have a simple expression in the right set of coordinates. This expression is found in x.1.. Regulator from the Planner. There is little hope of converting this algorithm into a regulator since large deviations are needed to correct for small errors in rotation about e. The resulting regulator would not be stable in the sense of Lyapunov. We can, however, linearize the control system about the trajectory and construct a feedback control law (see [] for details) which will stabilize to the open loop path in the face of noise and small modeling errors.. Symmetric satellite with! = 0: Since! is constant, it is useful to solve the problem for R r (t) =R(t)Exp(,! t^e ). The dynamics in this new set of coordinates is given by: _R r (t) = R r ^! r! r (t) = 4 cos(! t), sin(! t) 0 sin(! t) cos(! t) ! 1! 0 5 This is veried by dierentiating R r. Now dening u =[u 1 ;u ] T, and v =[v 1 ;v ] T, set! u =! cos(! t) sin(! + t) v,! 1!, sin(! t) cos(! t) It may now be veried that under this input transformation _! r 1 = v 1 and _! r = v. An open{loop planner, similar to the one found in Proposition 1, may now be used. 4. Asymmetric Satellite. Notice that at each point after system halt in the algorithm of Proposition 1, either! 1 or! is zero. This ensures that _! remains zero. Therefore, as long as the algorithm of Proposition 1 is given an asymmetric satellite with! =0,it will work as designed. An initial non-zero! can be removed by ramping up and down both! 1 and! the proper amounts. 5

7 Feedback Control In the following section, we apply geometric nonlinear control to capture the dynamics of four of the six states of either the symmetric or asymmetric satellite. Exact linearization methods require coordinates and so impose Euclidean structure on the fundamentally non-euclidean space SO()<. It is not possible to avoid coordinate singularities since SO()< is not <. However, a judicious choice of output functions connes the singularities to an S 1 subset of SO(). This is achieved through the use of a non-standard coordinate chart related to Listing's law [11] of eye movement. This chart also appeared in the attitude control of a model of a falling cat with a no-twist joint [14]. It is closely related to the Hopf bration SO()! S which takes R to Re. In this section, controllers stabilizing the additional states of the zero dynamics as well as the four linear states are presented..1 Input-Output Linearization In this subsection we input-output linearize [10] the system (1) for both the symmetric and asymmetric satellite. A natural choice of output is the pointing direction R(t)e, a variable vector conned to the surface of the unit sphere S. The compicated control dynamics involving! and rotations about e do not appear in these outputs and their derivatives. We use a stereographic parameterization of S, which has only one singularity. Our region of interest will be the identity matrix which projects to the north pole of the sphere. Consequently, wechoose the stereographic projection to have its singularity at the south pole. The resulting output functions are h 1 (R;!) = h (R;!) = e T 1 Re 1+e T Re e T Re 1+e T Re : (4) We will show that the outputs have vector relative degree, [10] { that is to say, the controls are recaptured upon taking two derivatives of the h i. Two states that are rendered unobservable by feedback can be expressed explicitly by noting that for all R SO() of the form R = Exp( ^e ), h 1 and h are zero. It follows that the h i do not capture the, and _ =! dynamics. Proposition (Input{Output Linearization) Given the control system for a satellite on SO() < described by equation (1), the outputs h(r;!) of equation (4) have vector relative degree, for all khk < 1. Proof: Dierentiating the outputs for i =1;, we get _h i = et i R^!e, et i Re e T R^!e 1+e T Re (1 + e T Re ; ) which may be written _h1 _h = A(R)! 1! : (5)

8 If we denote r ij = e T i Re j, the ij th element of the matrix R, the matrix A(R) is,1 (1 + r A(R) = )r 1,(1 + r )r 11 + r 1 r 1 (1 + r ) (1 + r )r, r r,(1 + r )r 1 + r r 1 : Note that these equations contain no explicit dependence on!. The determinant of the matrix A(R), the so-called decoupling matrix [10], is det (A(R)) = 1 (1 + r ) : The determinant is bounded for every h 1 ;h since r =,1 corresponds to the south pole of S. In terms of the outputs h 1 ;h, the determinant 1 of the decoupling matrix is 1(1 + 4 h 1 + h ),thus the chosen outputs thus have vector relative degree,. The second derivative is h1 h = f 0 (!; R)+A(R) u 1 u The choice of inputs u = A,1 (R)(v, f 0 (!; R)) input-output linearizes the system: : h 1 = v 1 h = v : () If the outputs were zero, then h 1 = h _ 1 = h = h _ =0. By equation (5),! 1 =! =0,thus from equation (1) we see that _! = 0. The system will be rotating at the constant rate! about the e axis. Thus, the zero dynamics manifold is TS 1. Furthermore, the trajectories are bounded for all time. To solve for the zero dynamics, we require a coordinate chart for SO() which will keep track of this rotation about e as measured in the body frame. Euler angles, although a favorite of the literature [1,, 8], are not advisable for this system. The x, y, z Euler angle chart, when projected down to the S, is not symmetric. Any Euler angle chart which is symmetric, for example the z, y, z, is singular at the north pole. Listing, a psychologist studying the movement of the eye, noted that the eye moves in a way which minimally twists the optic nerve. Listing's law [9, 11] describes the subset of SO() swept out by such an eye. This subset is <P <P = SO(). The same subset of SO() describes the conguration space of a cat which does not twist its spinal cord [1]. It can be thought of as the image under the exponential map of a two dimensional subspace of so(). We will use polar coordinates on < in order to parameterize the subspace. Thus R`(; ) = Exp((cos()e 1 d+ sin()e )) for any ; T. The formula for R`(; ) is cos ()(1, cos()) + cos() cos() sin()(1, cos()) sin() sin() cos() sin()(1, cos()) sin 4 ()(1, cos()) + cos(), sin() cos(), sin() sin() sin() cos() cos() 5 : 1 In the standard Euler angle parameterization, the determinant is 1 (1+cos() cos()).

9 The complete chart for SO() is R(; ; ) = R`(; ) Exp( ^e ). It may be veried that the equation for the decoupling matrix has the following pleasing form in these coordinates. A(R) = (1 + h 1 + h ), sin( ), cos( ) cos( ), sin( ) Proposition (Zero Dynamics) Consider the control system of the satellite of equation (1) with the input-output linearizing control law of Proposition. The zero dynamics for these outputs, parametrized by and!, are given by _ = _! = (1 + h 1 + h )(h _ h 1, h 1 _ h )+!, (1 + h + sin( )( h _ 1 h ), h _ 1) + cos( ) h _ 1 h _ where is measured as in the Listing parameterization. Proof: The derivative of! may be computed directly using equation (5). To compute the derivative of, dierentiate the Listing coordinate chart R(; ; ). This implies _R = R^! = R` ^!` Exp( ^e )+ _ R^e = R d ( Exp(, ^e )^!`)+ _ R^e! =!` + _ : The quantity!` may be computed directly from the derivative of the matrix R`(; ). The equation for _ is _ =!, (1, cos()) _ : The angles, and their derivatives depend only on h and _ h. It maybeveried that these dynamics are Remarks _ = _! = (1 + h 1 + h )(h _ h 1, h 1 _ h )+!, (1 + h + sin( )( h _ 1 h ), h _ 1) + cos( ) h _ 1 h _ : () 1. Stabilization Strategy The four states h 1 ;h ; _ h 1 and _ h constitute a controllable linear system and are therefore easy to stabilize to a point. The challenge rests with the remaining dynamics,! and. It is impossible to aect the dynamics of! in the symmetric ( = 0) case so we are forced to consider the symmetric and asymmetric cases separately. The controllers we propose in the next two subsection are composite: a linear system controller for the h variables will be perturbed by a high order control law to stabilize what would normally be the zero dynamics. 8

10 . Use of Euler Angles Suppose we use x, y, z Euler angles instead of this nonstandard chart. We set R(; ; )= Exp (^e 1 ) Exp (^e ) Exp ( ^e ). It may be veried that the zero dynamics take on the deceptively simple form: _ =, sin() _ +! This essentially says that evolves according to area swept out on the two-sphere whose spherical coordinates are (; ). Because this is such a special equation, one might expect that it would have a simple expression in h 1 ;h coordinates. However, this dierential equation is ill dened at the equator of S. The singularity and the asymmetry are artifacts of the coordinate chart we have chosen for SO().. Use of Quaternions Hamilton's quaternions parameterize SO() in a way which avoids singularities (see for example [1]). Specically, the unit quaternions form a Lie group which is dieomorphic to the three-sphere S inside the space < 4 of all quaternions, and this group comes with a canonical double covering map S! SO(). One composes this double cover with our map R! Re to form the standard Hopf bration S! S. The splitting of the orientation space of our two-torque satellite into a `direction' Re S and a rotational angle about e, when pulled up to the quaternions, corresponds to the base and ber of the standard Hopf bration.. The Regulator for the Symmetric Satellite We can control at most ve states of the symmetric satellite since! is a constant for all inputs. We will consider the case! 0 here. The case of! =0is similar. The notion of a saturation function is useful (see, for example, [18]). This is a C function () : <! < chosen so that (x) =x for jxj <and (x) = sgn(x) for jxj >. Here is a small positive number. Note ( ) = 0 for all j j >. The new output functions use these saturation functions: that y = h + ( ) cos(t) ( ) sin(t) It may beveried that the second derivative of the output y is y = B(h; _ h; )v + f 1 (h; _ h; ) : For suciently small, the matrix B() is invertible for all h; h, _ and. With this notation, the matrix B becomes Its determinant is B(h; _ h; ) = ( ) ( ) cos(t)h 1+h 1 +h ( ) sin(t)h 1+h 1 +h det(b) = 1+ 4 ( ) ( ) cos(t)h 1+h 1 + h :,4 ( ) ( ) cos(t)h 1 1+h 1 +h 1, ( ) sin(t)h 1 1+h 1 +h, ( ) sin(t)h 1 1+h 1 + h 5 : : (8) Thanks to the higher order terms in the denominators of the second and third terms, this determinant is bounded away from zero by a careful selection of () and. d(, sin d) =, cos d ^ d is the area form on the two-sphere. 9

11 Proposition 4 (Regulator for the Symmetric Satellite) Consider the control system of the symmetric satellite given by equation (1) with = 0 and! = 0, Then, all smooth control laws v(r;!; t) of the form v =,B,1 (h; h; _ ) f 1 (h; h; _ )+k 1 y + k _y with k 1 ;k > 0 render the equilibrium point R = I;! = 0asymptotically stable for > 0 chosen suciently small,. Proof: From the denition of y, it follows that (h; h; _ )! (y; _y; ) is an invertable change of coordinates. In the y-coordinates we have that y =,k 1 y, k _y, with k i > 0. This implies that y; _y are exponentially stable. Now apply center manifold theory [4]. The derivative of is higher order in y, so and t parameterize the center manifold. The variables y are exponentially stable with no higher order perturbation so on the center manifold we have y = _y =0. To nish the proof, we have to solve for the dynamics on the center manifold. Using y = _y =0,wemay solve for h and h. _ To reduce the burden of the notation, dene b( ;t)=h, y; thus, restricted to the center manifold _h = b + b t _ (9) Substituting for _ h,we nd that: _ 0 ( ) sin(t) ( ) 1, 1+h + 1 h 1 A =, ( ) 1+h 1 + h For small enough, the term 1, ( ) ( ) sin(t) is greater than zero. Thus, the dynamics are 1+h 1 +h asymptotically stable. By the center manifold theorem, the equilibrium point (R;!) = (I;0) is stable. Figure 1 shows simulations of the trajectory of the system in the h 1 ;h space and resulting time evolution of the coordinate. h Phase plot in projective plane h1 ψ Plot of ψ versus time resulting time Figure 1. The trajectory in the phase plot of h 1 ;h on the left spirals into the origin as the error along the \ber direction is reduced. The resulting time plot of is on the right. The version of the theorem that we use is for periodic time-varying systems, see for example, []. 10

12 . The Asymmetric Satellite Regulator In this sub-section we present the regulator control law for the asymmetric satellite. To begin with we rewrite the model in a more convenient form. Write, sin( ), cos( ) S = S( )= cos( ), sin( ) and! =! 1! Recall that h; u <. The control system is _h = (1 + h 1 + h ) _! = u S( )! _! =! 1! _ =! + h 1+h + h1 _, h 1 h _ 1 h (10) Note that S,1 = Now we dene the new outputs y by:, sin( ) cos( ), cos( ), sin( ) y =! + S( ),1 h + `( ;! ;t) (11) where ` = `1 ` = ( +! ) cos(t),k 1 ( +! ) cos(t) (1) (We are now working locally near =0so we may think of as a real instead of as an angular coordinate. The reader may replace the occurrences of +! in the denition of ` with any function which agrees with +! up to terms of order. In particular, this function can be taken to be periodic in.) Dierentiate the output to nd: u _y = 1 + f (h; y; ;! ;t) u Proposition 5 (Regulator for the Asymmetric Satellite) Consider the satellite with dynamics described by equation (1) with = 0. All smooth control laws u(r;!; t) of the form u =,(k 0 y + f (h; y; ;! ;t)) with y as dened in (11), k 0 > 0, sgn(k 1 )=sgn() and jk 1 j > 0:4, locally asymptotically stabilize the equilibrium point R = I;! = 0. 11

13 Proof: We begin by writing the dynamical system resulting from our control law in terms of the variables (h; y;! ; ). The system has a time-periodic center manifold [19, ] passing through the origin which we approximate. We express the dynamics on this center manifold to the appropriate order. Then we use the method of averaging together with a Lyapunov function to prove stability. The control law u =,(k 0 y + f (h; y; ;! ;t)) leads to closed loop dynamics given by: _h = (1 + h 1 + h ),h + S( ),1 (y, `) _y =,k 0 y _! = (y 1, ( +! ) cos(t) + sin( )h 1, cos( )h ) y + k 1 ( +! ) cos(t) + cos( )h 1 + sin( )h _ =! + 1+h + 1 h The linearization of this dierential equation is h h1 _, h 1 h h =,h + S(0),1 (y, (`1( ;! ;t); 0) T ) _y =,k 0 y (1) _! = 0 _ =! (14) It follows that the h; y directions belong to the stable eigen-directions, and that the ;! plane corresponds to the (generalized) eigenspace of the 0 eigenvalue. Also the subspace y = 0 is preserved by the dynamics. It follows that there is a time dependent center manifold passing through 0 and contained in the subspace y = 0. To calculate it we will use a coordinate transformation ~h = h + ( ;! ;t), to eliminate the time-varying dependence found in the equation for h _. In terms of these new variables, the linearized ~ h dynamics will be _ ~ h =, ~ h. Consequently, the periodic time-varying, center manifold [19, ] is parameterized by ( ;! ;t) and can be expressed in the form y = 0 and ~ h =( ;! ;t). We now solve for the lower order terms of h on the center manifold. Since y 0 on the center manifold, we have (see (11)): We know that h _ = A(h; )! with A = 1+h 1 +h S. Consequently! =,`, S,1 h (15) which we expand out: _h = 1+h 1 + h S! _h 1 = 1+h 1 + h _h = 1+h 1 + h (, sin( )! 1, cos( )! ) By substituting (15) into (1) we compute the the dynamics of h, _h 1 = 1+h 1 + h _h = 1+h 1 + h (cos( )! 1, sin( )! ) : (1),h 1 + sin( )( +! ) cos(t), k 1 cos( )( +! ) cos(t),h, cos( )( +! ) cos(t), k 1 sin( )( +! ) cos(t) 1 :

14 Now we solve for the coordinates ~ h. We require that the dynamics of ~ h 1 have no linear time-varying terms and that the nonlinear terms of ~ h dynamics be strictly third order. The resulting center manifold equation ~ h i = i (t; ;! ), i = 1;, will be of order two and three respectively. The general form of such a transformation is: ~h 1 = h 1 +( ! + 1! ) cos(t)+( ! + 1! ) sin(t) ~h = h +( 1 +! ) cos(t)+( + 4! ) sin(t) (1) Solving for the vectors 1 =( 11 ; 1 ; 1 ; 14 ) T < 4 ; =( 1 ; ; ; 4 ; 5 ; ) T < we get, for given k 1, 1 = (0:; 0:44; 0:4; 0:08) T = (,0:+0:k 1 ;,0:8 + 0:88k 1 ; 0:44, 0:04k 1 ;,0:4+0:4k 1 ; 0:4 + 0:1k 1 ; 0:448, 0:8k 1 ) T : (18) Knowing the center manifold to second order, we can express the dynamics of and! on the center manifold to third order. We will denote time independent polynomials by P i (), time-average zero terms by O i (t; ), and remainder terms by R i (t; ). The functions O i ;R i are time periodic. _ =! (1 + P 1 ( ;! )) + O 1 (t; ;! )+R 1 (t; ;! ) _! = P ( ;! )+O (t; ;! )+R (t; ;! ) The terms O i are third order, while the terms R i are fourth or higher order. The polynomials P 1 and P are P 1 ( ;! ) = 1! +! + P ( ;! ) = 41! + 4! + 4! + 44 ; (19) with the vectors < and 4 < 4 being = (k 1, 1) (0:11; 0:5; 0:0) T 4 =,k 1 0: :019 ; 0:45 + 0:19 ; 0:5 + 0:1 T ; 0:4 : (0) k 1 k 1 k 1 We apply an averaging transformation [4], ( ;! )=x 1 + (x 1 ;t) where x 1 < and where is third order in x 1 and is time-average zero. The linearization of such a transformation at x 1 =0 is full rank hence a local dieomorphism. The dynamics of x 1, computed by dierentiation, are then, _x 1 = x 1 (1 + P 1 (x 1 )) + O 1 (x 1 ;t)+ 1 1, x t 1 x R 11 (x 1 ;t) P (x 1 )+O (x 1 ;t)+, x t 1 x 11 + R 1 (x 1 ;t) where R jk (x 1 ;t) is at least order four and x 1 denotes the second component of the vector x 1. In the standard averaging theory, one sets t =, O 1 (x 1 ;t) O (x 1 ;t) (1) () 1

15 resulting in a time periodic, time average zero and removing lower order time dependent terms. In our case, averaging will have to be repeatedly applied to obtain this result. With the choice of satisfying (), we have _x 1 = x 1 (1 + P 1 (x 1 )) + O 11 (x 1 ;t) P (x 1 )+x 1 O 1 (x 1 ;t) + R 1 (x 1 ;t) where R 1 (x 1 ;t) is of order four and higher, O 11 (x 1 ;t) is third order and is time-average zero, and O 1 (x 1 ;t) is second order and time-average zero. The i th iteration of averaging has the form, x i,1 = x i + (x i ;t). It maybe veried that the dynamics of the i th iteration are given by _x i = x i (1 + P 1 (x i )) + x i,1 i O i1(x 1 ;t) P (x 1 )+x i i O i(x i ;t) + R i (x i ;t) with R i (x i ;t) fourth and higher order, O i1 (x i ;t) order 4, i for i (1; 4), and O i (x i ;t) order, i for i (1; ). At the fourth iteration, third order time dependent terms drop out of the equation for _x i and in the fth iteration third order time dependent terms drop out of the equation for _x i1. Relabel the state variables of the fth interation by z <. In () below, the remainder terms, written as R i (t; z), are time periodic terms of order 4 and higher. The polynomials P 1 ;P are as dened earlier. _z 1 = z (1 + P 1 (z)) + R (t; z) _z = P (z)+r 4 (t; z) () The following Lyapunov function is locally positive denite if k 1 > 0 and 0 is suciently small. V (t; z) = 0:1k 1 z :5z + k 1 0 z 1 z + z 4 1 z g 1 (t)+ z z 5 1 z g (t)+ 4 z 1 (4) The functions g i (t) are periodic and time average zero. They will be specied later, as will the constants i. We now compute the derivative of the Lyapunov function, keeping track of terms up to order six. The order four terms of _ V are: z P (z)+0:4k 1 z 1 z +k 1 0 z 1 z : The Lyapunov function has been chosen so that terms involving z 1 z cancel. Thus the fourth order term may be written as z P (z), where P (z) = 51 z + 5 z z z 1 (5) is a quadratic with coecients: 5 =,k 1 0:14808; 0:45 + 0:19 ; 0:04 + 0:1 T, 0 : () k 1 k 1 The order ve terms are z R (t; z)+z 4 1 z g 1 (t) +5 z 4 1 t z +z 1 z g 1(t): Due to the periodic nature of the vector eld, the order 5 part of the rst term may be written z R 5(t; z)+z z 4 1 ( (t)), where 11 (t) is periodic with time average zero and R 5 (t; z) is order 14

16 . This calculation does not take into account the th order terms in z R (t; z). For this reason we have to consider it again amongst the sixth R order terms. Choose =, 10 so as to cancel the 5 t constant (averaged) part and choose g 1 (t) = 0 11()d so as to cancel the time dependent average zero part of the the z z 4 1 coecient of z R. Consequently, the order 5 terms are now of the form z ~ R (t; z). The order six terms are as follows. k 1 0 z 1 z P 1(z)+k 1 0 z 1 P (z)+ 4 z 5 1 z +5z 4 1 z g (t)+z 5 1 z g (t) + z R (t; z) : As with the order four terms, the periodic nature of the vector eld implies that it may be written z R 4(t; z)+z z 5 1( (t)). As before, 4 and g (t) can be chosen as to cancel all the z 5 1 z term, leaving the th order terms in the form,0:4(k 1 ) 0 z 1 + z R (t; z). All other terms are order or higher. The derivative of the Lyapunov function is now t _V = z (P (z)+r 8 (t; z)), 0:4(k 1 ) 0 z 1 + R 9 (t; z) : () We must check that this is negative denite for z small. The constant 0 is chosen arbitrarily small and positive. The polynomial P (z) is quadratic. R 8 (t; z) is cubic and higher in z and R 9 is seventh order and higher. So it suces to show that the quadratic P (z) is negative denite. The coecients of z 1 and z occurring in P are both negative, since and k 1 are taken positive. So it remains to show that the discriminant D = b,4ac = 5, of P is negative for appropriate 0 ;k 1. This is easily seen from the numerical expression for the 5i. (In fact, D =(k 1 ) times, approximately,:18, :01=k 1 + :0=k, 1 1: 0 which is negative for all 1=k 1 and 0 suciently small.) The precise calculation of the critical gain 0:4 is left to the interested reader. Figure shows simulations of the control law for the asymmetric satellite. 0. Phase plot in projective plane 0.4 Plot of ψ and ω versus time h ω ψ h time Figure. After an initial transient wherein the large error in h is corrected, note that the trajectory in the phase plot of h 1 ;h on the left wobbles back and forth about the origin, unlike the symmetric regulator phase plane trajectory. The plot on the right shows the resulting evolution of and!, which converge at a slower than linear rate. 4 Conclusion Our main contribution is the construction of stabilizing control laws for regulation of both the symmetric and asymmetric satellite. We also contribute both open loop control laws for large 15

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