Reduced-Eort Control Laws for. Panagiotis Tsiotras and Jihao Luo y. University of Virginia, Charlottesville, VA

Transcription

1 Journal of Guiance, Control, an Dynamics ol. XX, No. X, July-August XXXX Reuce-Eort Control Laws for Uneractuate Rigi Spacecraft Panagiotis Tsiotras an Jihao Luo y University of irginia, Charlottesville, A 9-44 Abstract Recent results show that a nonsmooth, time-invariant feeback control law can be use to rotate an axisymmetric rigi spacecraft to the zero equilibrium using only two control torques. This metho, however, may require a signicant amountofcontrol eort, especially for initial conitions close to an equilibrium manifol corresponing to rotations about the unactuate principal axis. In this paper a control law is propose which reuces the control eort require to perform rest-to-rest maneuvers for initial conitions close to this equilibrium manifol. Specically, the phase space of the system is ivie into two parts, one corresponing to initial conitions proucing large control eort (the \ba" region) an the other corresponing to initial conitions proucing small control signals (the \goo" region). The propose control law then reners this unesirable equilibrium manifol unstable riving the trajectories of the close-loop system into the \goo" region, where the original control law is subsequently use. Numerical simulations inicate reuction of the control magnitue at the orer of 8-9 % for initial conitions close to the equilibrium manifol. Introuction The problem of stabilization of a rigi boy using less than three control inputs has receive consierable attention in the recent literature. Both the problems of the stabilization of the ynamics, an the stabilization of the kinematics have been treate in the literature {6. The stabilization problem of the complete system, i.e., the ynamics an the kinematics, has been aresse in Refs. 7{. The attitue stabilization of an axially symmetric rigi boy using two inepenent control torques was stuie by Krishnan, et al. 8,9 an Tsiotras et al.. If the uncontrolle principal axis is not the axis of symmetry, the system is strongly accessible an small time locally controllable 9. When the uncontrolle axis coincies with the axis of symmetry, the complete system fails to be controllable or even accessible. However, the system equations are strongly accessible an small time locally controllable in the case of zero spin rate. A nonlinear control approach was evelope in Ref. 8, which achieves arbitrary reorientation for this restricte case. In Refs. 4-5, the authors presente a new formulation of the attitue kinematics which was use in Ref. to solve the same problem avoiing the successive switchings of Ref. 8. References 8 an treate the axisymmetric case, whereas the non-symmetric case was ealt with in Refs. { an 6. In this paper, a moication of the control law presente in Ref. for the attitue stabilization of an axi-symmetric rigi boy using two inepenent control torques is propose. Because Brockett's necessary conition for smooth stabilizability is not satise for this system, any stabilizing (time-invariant) control law is necessarily nonsmooth. (Stabilizing time-varying Assistant Professor, Department of Mechanical, Aerospace an Nuclear Engineering. Member AIAA. y Grauate Stuent, Department of Mechanical, Aerospace an Nuclear Engineering. smooth control laws may still exist, however.) This nonsmoothness is evient in Ref. in the form of the nonierentiability of the control law at the origin. Because of the singularity at the origin, this control law may saturate the actuators, especially for initial conitions close to the equilibrium manifol. It is therefore esirable to moify the control law of Ref. to reuce the require control signals. Compare to the control law in Ref., the moie control law propose here remeies this large control input problem by riving the trajectories of the close-loop system away from the singular equilibrium manifol, towars a region in the state space where the \high authority" part of the control input remains small an boune. The proceure is simple an can be easily valiate using phase portrait consierations. A numerical example illustrates the control eort improvement using the new control law. The Uneractuate Spacecraft The ynamics of a rigi spacecraft with two controls can be written as _! = a!! + u (a) _! = a!! + u (b) _! = a!! (c) where a i are the inertia parameters satisfying a + a + a + a a a =. Herewe assume a boy-xe reference frame along the principal axes of inertia. Equations () escribe an uneractuate spacecraft with no control authority about the r principal axis. Notice in this case,! canbecontrolle only inirectly through juicious choice of the time histories of! (t) an! (t). In case of an axi-symmetric boy (about the -axis), a =ana = ;a = a an Eqs. () reuce

2 to ^ ^ i = i _! = a!! + u (a) _! = ;a!! + u (b) _! = (c) c z θ where! () =! is constant. Introucing the complex variables! =! +i! an u = u +iu (with i = p ;) the previous equations can be written as ^ b _! = ;ia!! + u () u^ Kinematics of the Attitue Motion a b ^ i The orientation of a rigi spacecraft can be specie using various parameterizations, for example, Eulerian Angles, Euler Parameters, Cayley-Rorigues Parameters, Cayley-Klein parameters, etc 7. Recently, a new parameterization using a pair of complex an real coorinates was introuce 4,5. Accoring to these results, the relative orientation between twogiven reference frames can be represente by two rotations, one corresponing to the real coorinate (z) an the other corresponing to the complex coorinate (w). Specifically, one can align an (inertial) reference frame to a boy-xe frame by rst performing an initial rotation of magnitue z about, say, the inertial -axis an then performing a secon rotation to move theintermei- ate -axis to the boy -axis. The secon rotation can be completely characterize by the complex coorinate w = w + i w. It is a rotation of magnitue = arccos about the unit vector ^u = w + w jwj ;jwj +jwj (4) i( w ; w) ^i + ^i jwj (5) This situation is epicte in Fig., where (^i ^i ^i ) is the intermeiate reference frame resulting from the rotation z about the inertial ^i axis, an where (a b c) enote the coorinates of the unit vector along the ^i axis in the boy frame, that is, ^i = a ^b + b ^b + c ^b (6) It can be shown 5 that the coorinates of the ^b axis in the ^i frame are also relate to (a b c) ^b = ;a^i ; b^i + c^i (7) With this notation, w represents the stereographic coorinates corresponing to the unit vector (a b c) ene by 8,5 w = b ; ia (8) +c Alternatively, the equations a = i (w ; w) +jwj b = w + w +jwj c = ;jwj +jwj (9) ^ i Fig. Attitue representation using (w z) coorinates. an can be use to n a b an c once w is known. Here jj enotes the absolute value of a complex number, i.e., w w = jwj, w C. The kinematic equations, which provie the geometric constraints of the motion an relate the rates of the kinematic parameters w an z to the angular velocity vector, can be written as follows,5 _w = ;i! w +! +! w (a) _z =! + Im(! w) (b) where! =! + i! an w = w + iw. Notice these equations take the convenient form t jwj = ( + jwj )Re(! w) (a) _z =! + Im(! w) (b) where bar enotes complex conjugate, Re() an Im() enote the real an imaginary parts of a complex number respectively. In Eq. (b) only the imaginary part of the prouct! w appears, while in Eq. (a) only the real part appears. This uality (or anti-symmetry) of Eqs. (a) an (b) is esirable an can be use to erive stabilizing control laws for the kinematics escribe by Eqs. (). Clearly, w = if an only if jwj =, an stabilization of the system in Eqs. () is equivalent to stabilization of the system in Eqs. (). References,9,8 inicate that the coorinates (w z) oer some signicant avantages for attitue analysis an control problems. 4 Problem Statement Consier an axi-symmetric boy with the applie torque vector in the plane which is perpenicular to the symmetry axis. In such a case, the system is escribe by Eqs. () an thus! remains constant. If initially! () 6=,nocontrol input can bring the system to the equilibrium. The system is not controllable

3 to the equilibrium, but it is controllable to the submanifol! = w = in the (!! w z)-space. For a more etaile iscussion on this issue, refer to Refs. 8{. Therefore, for an axi-symmetric boy, actively controlle rotation to the equilibrium for the system in Eqs. ()-() makes sense only if!. In this case, the system equations simplify to _! = u (a) _w =! +! w (b) _z = Im(! w) (c) This system can be stabilize to the origin, but any time-invariant stabilizing control law has to be necessarily nonsmooth, since Eqs. () fail Brockett's necessary conition for smooth stabilizability. Therefore we concentrate on using nonsmooth (albeit time-invariant) stabilizers for this system. Equations () represent a system in cascae form, with the kinematics (b)-(c) being the riven subsystem an the ynamics (a) being the riving subsystem. Methoology in Ref. use this structure to erive a non-smooth control law to stabilize Eqs. (). In essence, the controller esign consists of a two-step process. In the rst step, only stabilization of the kinematics is aresse, with the angular velocity treate as the control input. In the secon step, the control torque u is chosen to shape the esire velocity prole. Since the angular velocity in the rst step is (necessarily) a nonsmooth function of w an z, caution shoul be exercise when implementing this angular velocity in the secon step. The nonsmooth controller of Ref., along with its potential rawbacks, is summarize in the next section. 5 A Nonsmooth Controller for the Kinematics In Ref. a nonsmooth control law was propose for the kinematic system escribe by _w =! +! w (a) _z = Im(! w) (b) an was later implemente though the integrator in Eq. (a). The propose control law inref. was motivate by the ecoupling of these equations with respect to the prouct! w, as evient from the iscussion following Eqs. (). This control law is given by! = ;w ; i z w (4) where >= >. With this control law, the close loop system in terms of jwj an z is given by jwj t = ;( + jwj )jwj (5a) _z = ;z (5b) which is globally exponentially stable. As can be easily inferre by observing Eqs. (4) an (), the rst term in the control law (4) has an eect only on the ierential equation for w, whereas the secon term in Eq. (4) has an eect only on the ierential equation for z. Moreover, the secon term in Eq. (4) is a nonsmooth function of z an w. The main isavantage of the control law in Eq. (4) is that the last term, which involves the ratio z= w, may become unboune without careful choice of the gains. The previously impose gain conition >= ensures that the rate of ecay ofz is at least as large as the rate of ecay of w, such that their ratio remains boune. Actually, one can easily establish from Eqs. (5) that for >=, along the solutions of the system, z= w! as t!. Introucing the variable v = jwj, the system in Eqs. (5) takes the form _v = ;( + v)v (6a) _z = ;z (6b) This is a system which evolves on IR + IR. Typical trajectories an the vector el of the close-loop system in Eq. (6) for =an = are shown in Fig.. (Since z oes not change sign it suces to plot only the z> case.) Fig. Phase portrait of system in Eqs. (6). Although in Eq. (4), the ratio z= w, an hence the control eort!, remains boune by properchoice of control gains, the control input! may take large values in the region where w is small. From Eq. (5a), jw(t)j jw()j for all t an for small initial conitions w(), the control law may use a substantial amount of energy, especially in regions where jzj is large. In Fig., for example, the region which is close to the z axis is clearly unesirable as far as control expeniture is concerne. Moication of the control law in Eq. (4), such that the vector el close to the z axis points away from this axis, is highly esirable. In short, the iea is to ivie the (z v) phase space into two regions accoring to the value of the ratio = z jwj = z v (7)

4 This ratio is a irect inication of the relative magnitue between z an w. This ratio shoul be kept small in orer to avoi high control eort. Hence, if initially the states are in an unesirable region where attains large values, the feeback control strategy shoul rive the trajectories to a \safe" region in the state space where remains relatively small. Without loss of generality, choose as unesirable the region where jj >, leaving jj as the esirable region. These two regions, enote by D an D respectively, are therefore ene by D = f(z v) IR IR + : > jj > g (8a) D = f(z v) IR IR + : jj g (8b) These two regions are shown in Fig D D η = + η = D Fig. Regions D an D in (z v) phase space. 6 Main Results The propose moication to the control law in Eq. (4) is simple. Positive feeback is use for v when the trajectory is in region D, while z is ecreasing. This change will make the manifol v = (equivalently, w = ) unstable an the trajectories will move towars the region D an subsequently stay there. The control law in region D is essentially the same as in Eq. (4). Notice that, by enition, insie the region D we have jj, an since jzj=j wj = jjjwj we can ensure that! will not take excessive values as long as the trajectories remain in D. These statements will be mae more precise in the sequel. 6. Propose Control Law for Kinematics The propose control law for the system in Eqs. () is ene by! = ;()w ; i() z w where () an () are smooth functions satisfying (9) ; c () < () < c 8 (z v) D (a) () c c () c 8 (z v) D (b) an < c < c. One possible choice is, for example, () = c arctan ; ( ; ) (a) () = c arctan ; ( ; ) + c From Eqs. (), an are boune as (b) ; c () c an () c () for all IR. Moreover, notice that () < () for all (z v) D. The next theorem gives the main result of the paper. Theorem 6. Consier the system in Eqs. () an let the control law be as in Eqs. (9)-() with < c < c. Then for initial conitions (z() w()) IR (Cnfg), the following properties hol: (i) w(t) 6= 8 t. (ii) the trajectory(z() w()) is boune an lim (z(t) w(t)) = () t! (iii) the control law!() is boune an it has a boune erivative. With the control law in Eq. (9), the close-loop system takes the form _v = ;()( + v)v (4a) _z = ;()z (4b) where v = jwj an as in Eq. (7). From Eq. () we have that z ecays monotonically for all initial conitions, whereas v increases in the region D an ecreases in D. The result is that the trajectories of Eqs. (4) ten to D an then to the origin, as require. Before proving Theorem 6. we nee to establish the following two lemmas. Lemma 6. The region D is invariant for the system in Eqs. (4). Proof. The bounary of the set D is given by the two lines = (cf. Fig. ). On the bounary of D the feeback gains are () = an () = c=. The vector el on the bounary of D is therefore _v = (5a) _z = ; c z (5b) which points into the interior of D. Therefore trajectories in D cannot escape this region an thus it is invariant for the close-loop system in Eqs. (4). This lemma establishes that for initial conitions in D, the trajectories of the close-loop system remain in D for all times. Equivalently, if at some time t the trajectory enters D,itstays in D for all t t. Figure shows the vector el on the bounary of D. 4

5 Lemma 6. Consier the system in Eqs. (4). For all initial conitions (z v) D the trajectories enter the region D in nite time. As long as (z v) D, from Eq. (a) is Proof. boune as () < c=. This implies that z is boune. Actually, jz(t)jjz()j for all t. Note that z oes not change sign for all t. Without loss of generality, assume that z() (the case z() being similar). If (z() v()) D then, by enition () >. The erivative of in D is then _ = _z v ; z v _v = ;() + ()( + v) ;() (6) since () < anv> hence is boune in D. Let cl D enote the closure of D in IR, that is, cl D = D [f(z v) IR IR + : jj =g [f(z v) IR IR + : v =g (7) Then it is an easy exercise to show that _ 6= for all (z v) cl D nf( )g. Hence there exists >such that _ <; in D an consequently, monotonically ecreases. Thus, every trajectory starting in D will leave this set an enter D in nite time. Notice that the set f(z v) IR IR + : v = an z 6= g is an unstable manifol for the close-loop system. Figure shows the vector el on the bounary of D. The following corollary follows irectly from Lemmas 6. an 6.. Corollary 6. Consier the system in Eqs. (4). For all initial conitions (z() v()) IR (IR +nfg), is boune for all t. We are now reay to give the proof of Theorem 6.. Proof. [Theorem 6.] From Eqs. (4a) an () we have that _v ; c( + v)v (8) where c >. The solution of the ierential equation _x = ; c( + x)x x() = x > (9) is given by x(t) = () c e c t ; where c =(x +)=x. Clearly, x(t) 6= for all t an lim t! x(t) =. Therefore v() is boune below by the solutions of the ierential equation (9) subject to initial conition x = v(). Hence, jw(t)j 6= for all t an w() approaches the origin asymptotically. We now show that lim t!(z(t) v(t)) =. If (z() v()) D then accoring to Lemma 6. we have that (z(t) v(t)) D for all t an D is an invariant set for the close-loop system. Consier now the positive enite, raially unboune function :IR IR +! IR + given by (z v)= v + z 8 (z v) D () The erivative of along the trajectories of (4) is _ = ;()( + v)v ; ()z 8(z v) D () therefore, the trajectories are boune in D. Moreover, _ = if an only if ()( + v)v + ()z =. Using the enitions of () an() ind an recalling that v, one establishes that the last equality is not satise in D unless z = v =. By LaSalle's theorem, lim t!(z(t) v(t)) =, for all initial conitions in D.To nish the proof, recall from Lemma 6. that if (z() v()) D then jzj is boune by jz()j an there exist a time t > suchthat(z(t ) v(t )) D. This implies that for all t t the trajectories in D are boune, an are conne insie the strip jz(t)j jz()j. However, accoring to the previous iscussion, the trajectory with initial conition (z(t ) v(t )) satises lim t!(z(t) v(t)) =. Therefore, it has been shown that for all (z() v()) IR (IR +nfg) the trajectories remain boune an have the property that lim t!(z(t) v(t)) =. By the enition of v, this implies that lim (z(t) w(t)) = () t! In orer to show that! is boune, write the ratio z= w = w. From Eq. (9) one obtains j!j cjwj + cjjjwj (4) From Corollary 6., for all initial conitions (z() w()) IR (Cnfg) is boune. Since w is also boune, from Eq. (4) it follows that! is boune. From Eq. (a) it follows immeiately that _w is also boune. Moreover, since _ = ;() + ()( + v) (5) an () () v an are all boune, we have that _ is boune. The erivative of! is given by _! = ; _()w ; () _w ; i _() w ; i()_w ; i() _w (6) Using Eqs. () one has _() = ; 4c _() = ; c _ (7a) + ( ; ) _ (7b) + ( ; ) Since _ is boune, _() an _() are both boune. Finally, the bouneness of _! follows irectly from Eq. (6) an the fact all the terms in the right han sie of this equation are boune. The vector el an the corresponing trajectories of the close-loop system with the control law in Eq. (9) is shown in Fig. 4 (compare with Fig. ). Remark 6. Theorem 6. shows that for all initial conitions w() 6= the control law in Eq. (9) rives the system trajectories to the origin. This control law cannot be use if w() = (an z 6= ). Linearization of the 5

6 Fig. 4 Phase portrait of system in Eqs. (4). system represente by Eq. () about w =, however, shows that this system is controllable an choosing, for example, a constant control! =! c C, one can move away from the z-axis into the D region once in D, use of the control in Eq. (9) rives the system to the origin. Remark 6. Another choice of a feeback control for Eq. () is the sublinear control in terms of w,! = ; w +jwj ; i z w which reners the close-loop system (8) _v = ;v (9a) _z = ;z (9b) globally exponentially stable. The previous methoology can be applie mutatis mutanis to this control law, as well. Moreover, several other similar moications can be introuce to the control law in Eq. (4). It shoul be evient that the results in this section can be applie to these control laws with only minor moications. 6. Propose Control Law for Complete System The control law in Eq. (9) was shown to achieve lim t!(z(t) w(t)) =. Moreover, it is a boune controller with boune erivative. This allows one to implement this control through the ynamics in Eq. (a). To this en, ene the error e =! ;! (4) where! is the esire angular velocity prole given in Eq. (9). Consier the following feeback control u =_! ; (! + ()w + i()w) (4) where > an where _! is given in Eq. (6), along with Eqs. (7). The value of _ is now given by _ = ;()+()(+v)+im(e=w);(+v)re(e=w) (4) With the control law in Eq. (4) the close-loop system takes the form _e = ;e (4a) _v = ;()( + v)v +( + v)re(e w) (4b) _z = ;()z + Im(e w) (4c) Notice that for e = the system reuces to the one in Eqs. (4). For large enough, Eq. (4a) is essentially a bounary layer subsystem to the slow system given by Eqs. (4b)-(4c). Singular perturbation theory guarantees that as soon as the error becomes small enough, the (z v) trajectories of the system will follow the ones of Eqs. (4). Next we show that the control law in Eq.(4)is wellene, in the sense that it remains boune for all t. We show that with large enough w(t) 6= for all t, i.e., w(t) tens to zero only asymptotically for all initial conitions insie an a priori given compact set. Proposition 6. Consier the system in Eqs. (4) an the compact set N = f(! w z) W: jej[( + v)=v] g (44) where W = C (Cnfg) IR, an let c > c > an >( c + )=. Then for all initial conitions in N, jwj is boune belowbyanexponentially ecaying function. Proof. Equation (4b) can be re-written as t jwj = ;( + jwj )(()jwj ; Re(e w)) (45) Note that from Eq. (4a) je(t)j je()je ;t an using Eq. (44), je(t)j jw()j +jw()j Consier now the ierential equation e ;(c +)t= t (46) t j ^wj = ;( c + )( + j ^wj )j ^wj (47) The solution of this equation is given by j ^w(t)j = c e (c+)t= ; c; e ;(c+)t= (48) where c = (j ^w()j +)=j ^w()j. Comparison of Eqs. (46) an (48) implies that je(t)j j ^w(t)j 8 t (49) where j ^wj obeys Eq. (47) with j ^w()j = jw()j. 6

7 Notice now that since Re(e w) jejjwj, an using Eq. (49), one has from Eq. (45) that t jwj ;( + jwj )(()jwj + jejjwj) ;( + jwj )(()jwj + j ^wjjwj) (5) an since ; c () c nally, t jwj ;( + jwj )( cjwj + j ^wjjwj) (5) By comparing Eqs. (47) an (5) an since jw()j = j ^w()j, one obtains t jw()j t j ^w()j (5) Therefore there exist some t > such that jw(t)j j ^w(t)j for all t t. We claim that, actually, jw(t)j j^w(t)j for all t, an thus jwj is boune below by the exponentially ecaying function j ^wj. Assume that at some point t > we have that jw(t )j = j ^w(t )j an jw(t )j < j ^w(t )j see Fig. 5. t t Then the anger that w will move towars the z-axis before the control law in Eq. (4) becomes eective. This is one more reason which motivate the choice of the control law in Eq. (9). Namely, it is benecial for w to move away from the z-axis. This can reuce the value of the gain signicantly. In most situations it is not necessary to chose from Proposition 6.. Actually, asthenumerical simulations in the next section show, for most practical examples it suces to choose to be \suciently larger" than the gains c an c. From Eq. (4) it is also clear that shoul be at least as large as c=, in orer for e=w to remain boune. Remark 6. The rigi boy problem subject to two control inputs is only but one example of an uneractuate mechanical system. Systems of this form can be foun in the class of systems subject to nonholonomic, i.e., non-integrable constraints. Time-invariant control laws for these systems are necessarily nonsmooth an recently propose control laws {5 inclue singularities of the same form as in Eq. (4). It is therefore conceptually straightforwar to exten the results of this paper to this more general case. w ^ w 7 Numerical Example To illustrate the previous theoretical analysis, we have simulate the ierential equations () with the two control laws in Eqs. (4) an (9). The gains are chosen as c =:5 an c =. The value of the parameter =. The initial conitions were taken as w() = : ; i :5 an z() = :5. The results are shown in Figs. 6 an 7. Figure 6 shows the correspon- t Fig. 5 Time history of jwj an j ^wj. t.5.5 Ol Metho New Metho t jw(t )j = ;( + jw(t )j )( cjw(t )j + jw(t )j ) = ;( + j ^w(t )j )( c + )j ^w(t )j = t j ^w(t )j (5) which leas to a contraiction. Therefore jw(t)j j ^w(t)j an thus w(t) 6= for all t. In Ref. the control law in Eq. (4) was also implemente using the same methoology. That is, the control for the complete system was given by Eq. (4), where = c, = c an _ =_ =. The value of the gain increases with, which in turns increases as jwj ecreases. That is, when the initial conition is close to w = then a faster transient for! is require. This faster transient isachieve by taking large enough. A potential problem in the implementation of the control in Eq. (4) is now evient. If e oes not ecay \fast enough" so that!!! suciently fast, then there is Fig. 6 Close-loop trajectories for the two methos (kinematics only). ing close-loop trajectories, an Fig. 7 shows the magnitue of the angular velocity (control input for the kinematics) j!j. Soli lines correspon to the new control law in Eq. (9) an the ashe lines correspon to the previous control law given in Eq. (4). As it is evient 7

8 4 Ol Metho New Metho.5 α = α = 4 α = 8 ω u Time Fig. 7 Control eort for the two methos (kinematics only). from these gures, there is a substantial ecrease in control eort by using the control law in Eq. (9), especially uring the initial portion of the trajectory where z is large an jwj is small α = This control law was later implemente through the ynamics in Eq. (a). A rest-to-rest maneuver was consiere, thus!() =. Simulations for several values of are shown in Figs The trajectories in the (z v) space are very similar to the ones when! is the control input. In fact, for = the trajectories for the comα = 4 α =.5.5 Fig. 8 Close-loop trajectories for the complete system. plete system are essentially ientical to the ones with control law in Eq. (9). Figure 9 shows that increasing may increase the control eort, mainly because of the high-gain bounary layer part of the controller. At any rate, the corresponing control eort for the control law in Ref. is several orers of magnitue higher (not shown here). Moreover, for small such as = an = 4, the control eort for the controller in Ref. is not boune. In these cases the slow transients of Time Fig. 9 Control eort for the complete system. e allowe w to rift towars the z-axis before the control law in Eq. (4) is activate. On the other han, the controller in Eq. (9) forces the system trajectories away fromthez-axis, thus proviing enough time for the ynamic controller to \catch up." 8 Conclusions A nonsmooth control law has been constructe which stabilizes the kinematics of an uneractuate rigi spacecraft. It is shown that the propose control law is well ene an uses consierably less control effort than a previously erive control law. The main iea is to ivie the state space into two regions, one which inclues initial conitions resulting in high control expeniture an one which inclues initial conitions resulting in acceptable control input signals. The propose control law then forces all the close-loop system trajectories to leave the unesirable region of high control eort an subsequently use the original control law. Numerical examples inicate a signicant control eort reuction using the new control scheme. Because of the limite control torque on-boar a spacecraft, for practical situations this may be the ierence between feasibility an infeasibility of a particular reorientation maneuver. Acknowlegment: Financial support from the National Science Founation uner Grant CMS is gratefully acknowlege. References Aeyels, D., \Stabilization by SmoothFeeback ofthe Angular elocity of a Rigi Boy," Systems an Control Letters, ol. 6, No., 985, pp. 59{6. Aeyels, D. an Szafranski, M., \Comments on the Stabilizability of the Angular elocity of a Rigi Boy," Systems an Control Letters, ol., No., 988, 8

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