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Chaotic attitude motion of a magnetic rigid spacecraft and its control
International Journal of Non-Linear Mechanics 37 (2002) 493–504
Chaotic attitude motion of a magnetic rigid spacecraft and its control Li-Qun Chen ∗ , Yan-Zhu Liu Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China Received 5 September 2000; received in revised form 5 January 2001
Abstract This paper deals with chaotic attitude motion of a magnetic rigid spacecraft with internal damping in a circular orbit near the equatorial plane of the earth. The dynamical model of the problem is established. The Melnikov analysis is carried out to prove the existence of a complicated non-wandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincare map, power spectrum and Lyapunov exponents. Numerical simulations indicate that the onset of chaos is characterized by the intermittency as the increase of the torque of the magnetic forces and decrease of the damping. The input–output feedback linearization method is applied to control chaotic attitude motions to the given 6xed point and periodic motion. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Spacecraft; Chaos; Intermittency; Control; Feedback linearization
1. Introduction As chaos is widely and deeply investigated, much attention has been paid on chaotic attitude motion of spacecraft [1,2]. It not only provides a de6nite engineering background for exploring chaos, but also o
Corresponding author. E-mail address: [email protected] (L. Chen). 0020-7462/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 0 2 3 - 3
494
L.-Q. Chen, Y.-Z. Liu / International Journal of Non-Linear Mechanics 37 (2002) 493–504
In 1998 Chen and Liu investigated planar attitude motion of a spacecraft with both atmosphere drag and internal damping, and established a necessary condition of occurring chaos by using the Melnikov method. They also demonstrated numerically the chaotic motion [20]. In the present work, the planar libration of non-spinning spacecraft with internal damping in both the gravity and magnetic 6eld of the earth are treated. The spacecraft under the investigation is supposed in a circular orbit near the equatorial plane of the earth. The case without internal damping and in the absence of the gravitational torque has been numerically studied by Beletsky et al. [17,18]. Cheng and Liu also studied chaotic motion of a magnetic rigid spacecraft in a circular orbit near the equatorial plane of the earth [21], but they did not consider the damping. Based on its dynamical model, both the Melnikov analysis and the numerical studies are carried out. In recent years, controlling chaos has drawn great attention because of its theoretical importance and possible applications. There are some attempts to control chaotic attitude motion of spacecraft [13,22– 25]. Here the exact linearization in the non-linear system theory is employed to control chaotic attitude motion to the given 6xed point and periodic motion. 2. Equations of motion Consider a magnetic rigid spacecraft moving in a circular orbit with the orbital angular velocity !c in the gravitational and magnetic 6eld of the earth. The principal inertia moments of the arbitrarily shaped spacecraft are A; B and C. Without loss of generality, suppose that B ¿ A. Assume that the internal damping torque Md is proportional to angular velocity whose coeHcients is c. The spacecraft has a actuator that can provide the control torque Mc . The inertial reference frame (Oe -X0 Y0 Z0 ) is established with the mass centre Oe of the earth as the origin, the polar axis of the earth as axis Z0 , and line from Oe to ascending node as axis X0 . The base vectors of (Oe X0 Y0 Z0 ) are i0 ; j0 ; k0 . The principal coordinate frame (O-xyz) is established with the mass center O of the rigid spacecraft as the origin, whose base vectors are i ; j ; k . The orbital coordinate frame (O-XYZ) is established with a line from Oe to O as axis, a line normal to orbital place XY as axis z. The base vectors of (O-xyz) are i; j; k. Denote that the ’ is libration angle in the orbital plane. Then the spacecraft is subject to the torque of the gravitational 6eld [26] Mg = −3!c2 (B − A) sin ’ cos ’k:
(1)
For an orbit near the equatorial plane of the earth, the orbital inclination angle i1. In this case, projecting i0 ; j0 ; k0 onto the axes of (O-xyz) gives i0 = cos !c tp i − sin !c tp j; j0 = sin !c tp i + cos !c tp j − ik; k0 = i sin !c tp i + i cos !c tp j + k:
(2)
The magnetic 6eld is assumed to be that of a dipole with the magnetic moment whose axis coincides the earth’s axis. For i1, the magnetic induction at a given time tp is [26] m 3m 3m Hm = − 3 i sin 2!c tp i0 − 3 i(1 − cos 2!c tp )j0 + 3 k0 ; (3) 2r 2r r where r is the radius of the orbit. The constant magnetic moment I of the spacecraft is supposed to be directed along axis X , that is I = I i = I (cos ’i + sin’j):
(4)
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The torque of the magnetic force from the interaction of the earth’s magnetic 6eld with the permanent magnet mounted on the spacecraft is Mm = Hm × I :
(5)
Hence,
m iI (2 sin ’ sin !c tp + cos ’ cos !c tp ): r3 The angular momentum of the rigid spacecraft about its mass center is d’ k: L=C !+ dtp MmZ = −
The theorem of angular momentum about the center of mass gives dL = Mg + Mm + Md + Mc : dtp
(6) (7)
(8)
Here only the planar libration is considered. Substituting Eqs. (1), (6) and (7) into Eq. (8) and projecting the result onto axis z lead to d2 ’ C 2 = −3!c2 (B − A)sin ’ cos ’ dtp −m iIr −3 (2 sin ’ sin !c tp + cos ’ cos !c tp ) − c’˙ + Mc :
(9)
Introducing the dimensionless time t = !c tp and denoting iIm c 3(B − A) ; = ; = ; 2 3 2C C!c r C!c one obtains the desired di
Mc C!c2
(10)
’O + ’˙ + Ksin 2’ + (2 sin ’ sin t + cos ’ cos t) = u;
(11)
K=
u=
where the derivative with respect to dimensionless time t. 3. Melnikov analyses of the uncontrolled system For the case without control, u = 0. The equation of motion of the uncontrolled system is ’O + ’˙ + Ksin 2’ + (2 sin ’ sin t + cos ’ cos t) = 0:
(12)
Assume the magnetic parameter to be small and let =1 (0 ¡ 1). Then Eq. (12) is an integrable Hamiltonian system under small perturbations ’O + Ksin 2’ = −1 (2 sin ’ sin t + cos ’ cos t):
(13)
For = 0, the unperturbed planar Hamiltonian system ’O + Ksin 2’ = 0
(14)
has 6rst integral of motion 1 2 ˙ 2’
+ K sin2 ’ = H:
(15)
When H = K, the integrable system (15) has two hyperbolic saddle points (±( =2); 0), whose unstable manifolds and stable manifolds constitute a heteroclinic cycle. The heteroclinic orbits ± started at √ (0; ± 2K) are √ √ √ (’± (!); ’˙ ± (!)) = (±arcsin(th( 2K!)); ± 2K sech( 2K!)): (16)
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For = 0, if the Melnikov functions +∞ M± (") = [ − 1 ’˙ ± (t) − 1 (2 sin ’± (t)sin(t + ") −∞
+ cos ’± (t) cos(t + "))]’˙ ± (t) dt
(17)
have a simple zero, then the Poincare map of Eq. (13) has transverse heteroclinic cycles [27]. Substituting Eq. (16) into Eq. (17), one evaluates the Melnikov functions for the hetoroclinic orbits + and − to yield 1 1 +√ csch √ sin " − 41 K: (18) M± (") = −1 2K 2K 2 2K Hence M+ (") and M− (") have simple zeros on the condition of 1 8K 2 √ √ sh = ¿ : (19) 1 (1 + 2K) 2 2K If Eq. (19) holds, there exists transverse heteroclinic cycles in the Poincare map of the system (12). The existence of such cycles implies that a complicated non-wandering Cantor set occurs. Hence chaos in the sense of Smale’s horseshoe appears in the system. 4. Computational analyses of the uncontrolled system The e
(20)
The fourth order Runge–Kutta routine is used for numerical integration. Various tools including the time history, Poincare map, power spectrum and Lyapunov exponents are used to identify the dynamical behaviors. The Lyapunov exponents are calculated using the algorithm derived by Wolf et al. [28]. The linearized system of Eq. (20) is characterized by the Jacobian matrix 0 1 0 −2K cos 2x1 − (2 cos x1 sin x3 − sin x1 cos x3 ) − −(2 sin x1 cos x3 − cos x1 sin x3 ) : (21) 0 0 0 The characteristic dynamical behavior are investigated by varying the magnetic parameter and the damping parameter , respectively, while the gravitational parameter K is kept constant at K = 1:1. In this case, the Melnikov condition of chaos given by Eq. (19) is = ¿ 1:5738. The numerical integration begin from the initial conditions described by (x1 ; x2 ; x3 )T = (0; 0; 0). For a given , the intermittency transition accrues as the magnetic parameter is increased Fix = 0:2. The the Melnikov condition of chaos is ¿ 0:3148. The dynamical behaviors of the uncontrolled system at = 0:6984; 0:6985; 0:69855; 0:6986 are depicted in Figs. 1– 4. The corresponding Lyapunov exponents are shown in Table 1. As is increased, the periodic behavior become the regular motion intermittently interrupted by a 6nite duration burst of irregular motion. For the further increase of , the bursts become
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Fig. 1. Periodic motion ( = 0:6984; = 0:2). (a) time history (b) phase trajectory (c) Poincare map (d) the largest Lyapunov exponent.
Fig. 2. Intermittency ( = 0:6985; = 0:2). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
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Fig. 3. Intermittency ( = 0:69855; = 0:2). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
Fig. 4. Chaotic motion ( = 0:6986; = 0:2). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
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Table 1 Lyapunov exponents ( = 0:2)
#1
#2
#3
0.6984 0.6985 0.69855 0.6986
0.00 0.12 0.13 0.17
−0:01
−0:19 −0:32 −0:33 −0:37
0.00 0.00 0.00
so frequent that the regular behavior can no longer be distinguished, and the chaotic motion is developed. Due to the transient behavior, the largest Lyapunov exponent is positive at the beginning of the periodic motion in Fig. 1. During the intermittency transition shown in Figs. 2 and 3, the motions depend sensitively on initial conditions so that the largest Lyapunov exponents are still positive for the duration of the periodic motions between irregular bursts. For a given , the intermittency transition accrues as the damping parameter is decreased. Fix =0:7. Then the Melnikov condition is ¡ 0:448. The case that = 0:295; 0:290; 0:285; 0:280 are depicted in Figs. 5 –8. The corresponding Lyapunov exponents are listed in Table 2. It should be pointed out that there is certain di
(22)
where from Eq. (10)
x2 f (x1 ; x2 ; x3 ) = −K sin 2x1 − x2 − (2 sin x1 sin x3 + cos x1 cos x3 ) ; 1 +(x) = 1;
(23)
h(x1 ; x2 ; x3 ) = x1 :
Let Lg h(x) denote the Lie-derivative of a scalar function h(x) with respect to a vector 6eld g, then Lg Lf0 h(x) = 0; Lf2 h(x)
Lf1 h(x) = x2 ;
Lg Lf1 h(x) = 1;
= −K sin 2x1 − x2 − (2 sin x1 sin x3 + cos x1 cos x3 ):
(24)
If given the desired output yR (t), then the input–output feedback control law has the expression [29] u = yO R − (−K sin 2x1 − x2 − (2 sin x1 sin x3 + cos x1 cos x3 )) − c1 (y˙ − y˙ R ) − c0 (y − yR )
(25)
where coeHcients c1 and c0 can be determined by normal design principles such as pole placement, linearquadratic optimal regulator, or robust service regulator.
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Fig. 5. Periodic motion ( = 0:7; = 0:295). (a) time history (b) phase trajectory (c) Poincare map (d) the largest Lyapunov exponent.
Fig. 6. Intermittency ( = 0:7; = 0:290). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
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501
Fig. 7. Intermittency ( = 0:7; = 0:285). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
Fig. 8. Chaotic motion ( = 0:7; = 0:280). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
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Table 2 Lyapunov exponents ( = 0:7)
#1
#2
#3
0.295 0.290 0.285 0.280
0.00 0.05 0.10 0.12
0.00 0.00 0.00 0.00
−0:30 −0:34 −0:38 −0:40
Fig. 9. Control of chaos to goal (26).
Fig. 10. Control of chaos to goal (27).
For the chaotic motions shown in Figs. 4 and 8, the control goals are successively taken as a 6xed point and a period 2 motion. ’R1 (t) = 0;
(26)
’R2 (t) = sin 0:5t:
(27)
Choose c1 =2:8; c0 =4:0 according to pole placement. Activate control when t0 =1300. The control results are, respectively, shown in Figs. 9 and 10. The corresponding control signals u = u(t) are, respectively, shown in Figs. 11 and 12.
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Fig. 11. Control signal for goal (26).
Fig. 12. Control signal for goal (27).
6. Conclusions The planar libration of a magnetic rigid spacecraft with internal damping subjected to both the gravitational and magnetic torque in a circular orbit near the equatorial plane of the earth is described by the di
Acknowledgements The research is supported by the National Natural Science Foundation of China (Project No. 10082003), Shanghai Municipal Development Foundation of Science and Technology.
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References [1] Y.-Z. Liu, L.-Q. Chen, Nonlinear problems in spacecraft attitude dynamics, in: Wei-zang Chien (Ed.), Proceedings of the Third International Conference on Nonlinear Mech. Shanghai University Press, 1998, pp. 80 –86. [2] Y.-Z. Liu, L.-Q. Chen, G. Cheng, X.-S. Ge, Stability, bifurcation and chaos in spacecraft attitude dynamics, Adv. Mech. 30(3) (2000) 351–357 (in Chinese). [3] A. Guran, X. Tong, F.P.J. Rimrott, Instabilities in a spinning axi-symmetric rigid satellite, Mech. Res. Commun. 18 (5) (1991) 287–291. [4] X. Tong, F.P.J. Rimrott, Chaotic attitude motion of gyrostat satellites in a central force 6eld, Nonlinear Dynamics 4 (1993) 269–278. [5] X. Tong, B. Tabarrok, F.P.J. Rimrott, Chaotic motion of an asymmetric gyrostat in the gravitational 6eld, Int. J. Non-linear Mech. 30 (3) (1995) 191–203. [6] A.C. Or, Chaotic motions of a dual-spin body, J. Appl. Mech. 65 (1) (1998) 150–156. [7] J.-H. Peng, Y.-Z, Liu, Chaotic motion of a gyrostat with asymmetric rotor, Int. J. Non-linear Mech. 35 (3) (2000) 431–437. [8] G. Cheng, Y.-Z. Liu, Chaotic motion of an asymmetric gyrostat in the magnetic 6eld of the earth, Acta Mech. 141 (2000) 125–134. [9] J.-H. Peng, Y.-Z. Liu, Chaotic motion of the tethered satellite system, Tech. Meth. 16 (4) (1996) 327–331. [10] W. Steiner, Transient chaotic oscillations of a tethered satellite system, Acta. Mech. 127 (1998) 155–163. [11] P.A. Meehan, S.F. Asokanthan, Chaotic motion in a spinning spacecraft with circumferential nutational damper, Nonlinear Dynamics 12 (1997) 69–87. [12] C.L. Gray, A.P. Mazzoleni, D.R. Campbell, Analytical criterion for chaotic dynamics in Bexible satellites with nonlinear controller damping, J. Guid. Control Dynamics 21 (5) (1998) 558–565. [13] Z.-M. Ge, C.-I. Lee, H.-H. Chen, S.-C. Lee, Nonlinear dynamics and chaos control of a damped satellite with partially-6lled liquid, J. Sound Vibration 217 (1998) 807–825. [14] M. Seisl, A.L. Steindl, Chaoticsche Schwingungen von Satelliten, Z. Angew Math. Mech. 69 (1989) 352–354. [15] V.I. Gouliaev, A.L. Zubritskaya, V.L. Koshkin, Universal sequence of bifurcation of doubling of the oscillation period for a satellite in an elliptical orbit, Mech. Solids 24 (1989) 1–6. [16] V.I. Gouliaev, Zavrazhina, Universal similarities of transition to chaos of satellite oscillation in elliptic orbit. Int. J. Non-linear Mech. 32 (1998) 215 –226. [17] V.V. Beletsky, Regular und Chaotisch Bewegung Sturrer Korper, Teubner-Verlag, Stuttgart, 1995. [18] V.V. Beletsky, M.L. Pivoarov, E.L. Starostin, Regular and chaotic motions in applied dynamics of a rigid body, Chaos 6 (2) (1996) 155–166. [19] X. Tong, F.P.J. Rimrott, Numerical studies on chaotic planar motion of satellites in an elliptic orbit, Chaos Solitons Fractals 1 (1991) 179–186. [20] L.-Q. Chen, Y.-Z. Liu, Chaotic attitude motion of a class of spacecraft on an elliptic orbit, Tech. Mech. 18 (1) (1998) 41–44. [21] G. Cheng, Y.-Z. Liu, Chaotic motion of a magnetic rigid satellite in an orbit near the equatorial plane of the earth, Tech. Mech. 19 (2) (1999) 197–201. [22] L.-Q. Chen, Y.-Z. Liu, Chaotic attitude motion of non-spinning spacecraft and its parametric open-plus-closed-loop control, Acta Mech. Sinica 30(3) (1998) 363–369 (in Chinese). [23] P.A. Meehan, S.F. Asokanthan, Control of chaotic motion in a spinning spacecraft with a circumferential nutational damper, Nonlinear Dynamics 17 (3) (1998) 269–284. [24] L.-Q. Chen, Y.-Z. Liu, An inversion system control for chaotic oscillations, Tech. Mech. 19 (1) (1999) 1–4. [25] L.-Q. Chen, Y.-Z. Liu, Controlling chaotic attitude motion of spacecraft by the input–output linearization, Z. Angew Math. Mech. 80 (10) (2000) 701–704. [26] Y.-Z. Liu, Spacecraft Attitude Dynamics. Nat. Defense Indu. Press, Beijing, 1995 (in Chinese). [27] Y.-Z. Liu, L.-Q. Chen, Nonlinear Dynamics, Shanghai Jiaotong University Press, Shanghai, 2000 (in Chinese). [28] A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vasano, Determining Lyapunov exponents from a time series, Physica D 16 (1985) 285–317. [29] A. Isidori, Nonlinear Control Systems, Springer, Berlin, 1989.
Chaotic attitude motion of a magnetic rigid spacecraft and its control Li-Qun Chen ∗ , Yan-Zhu Liu Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China Received 5 September 2000; received in revised form 5 January 2001
Abstract This paper deals with chaotic attitude motion of a magnetic rigid spacecraft with internal damping in a circular orbit near the equatorial plane of the earth. The dynamical model of the problem is established. The Melnikov analysis is carried out to prove the existence of a complicated non-wandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincare map, power spectrum and Lyapunov exponents. Numerical simulations indicate that the onset of chaos is characterized by the intermittency as the increase of the torque of the magnetic forces and decrease of the damping. The input–output feedback linearization method is applied to control chaotic attitude motions to the given 6xed point and periodic motion. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Spacecraft; Chaos; Intermittency; Control; Feedback linearization
1. Introduction As chaos is widely and deeply investigated, much attention has been paid on chaotic attitude motion of spacecraft [1,2]. It not only provides a de6nite engineering background for exploring chaos, but also o
Corresponding author. E-mail address: [email protected] (L. Chen). 0020-7462/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 0 2 3 - 3
494
L.-Q. Chen, Y.-Z. Liu / International Journal of Non-Linear Mechanics 37 (2002) 493–504
In 1998 Chen and Liu investigated planar attitude motion of a spacecraft with both atmosphere drag and internal damping, and established a necessary condition of occurring chaos by using the Melnikov method. They also demonstrated numerically the chaotic motion [20]. In the present work, the planar libration of non-spinning spacecraft with internal damping in both the gravity and magnetic 6eld of the earth are treated. The spacecraft under the investigation is supposed in a circular orbit near the equatorial plane of the earth. The case without internal damping and in the absence of the gravitational torque has been numerically studied by Beletsky et al. [17,18]. Cheng and Liu also studied chaotic motion of a magnetic rigid spacecraft in a circular orbit near the equatorial plane of the earth [21], but they did not consider the damping. Based on its dynamical model, both the Melnikov analysis and the numerical studies are carried out. In recent years, controlling chaos has drawn great attention because of its theoretical importance and possible applications. There are some attempts to control chaotic attitude motion of spacecraft [13,22– 25]. Here the exact linearization in the non-linear system theory is employed to control chaotic attitude motion to the given 6xed point and periodic motion. 2. Equations of motion Consider a magnetic rigid spacecraft moving in a circular orbit with the orbital angular velocity !c in the gravitational and magnetic 6eld of the earth. The principal inertia moments of the arbitrarily shaped spacecraft are A; B and C. Without loss of generality, suppose that B ¿ A. Assume that the internal damping torque Md is proportional to angular velocity whose coeHcients is c. The spacecraft has a actuator that can provide the control torque Mc . The inertial reference frame (Oe -X0 Y0 Z0 ) is established with the mass centre Oe of the earth as the origin, the polar axis of the earth as axis Z0 , and line from Oe to ascending node as axis X0 . The base vectors of (Oe X0 Y0 Z0 ) are i0 ; j0 ; k0 . The principal coordinate frame (O-xyz) is established with the mass center O of the rigid spacecraft as the origin, whose base vectors are i ; j ; k . The orbital coordinate frame (O-XYZ) is established with a line from Oe to O as axis, a line normal to orbital place XY as axis z. The base vectors of (O-xyz) are i; j; k. Denote that the ’ is libration angle in the orbital plane. Then the spacecraft is subject to the torque of the gravitational 6eld [26] Mg = −3!c2 (B − A) sin ’ cos ’k:
(1)
For an orbit near the equatorial plane of the earth, the orbital inclination angle i1. In this case, projecting i0 ; j0 ; k0 onto the axes of (O-xyz) gives i0 = cos !c tp i − sin !c tp j; j0 = sin !c tp i + cos !c tp j − ik; k0 = i sin !c tp i + i cos !c tp j + k:
(2)
The magnetic 6eld is assumed to be that of a dipole with the magnetic moment whose axis coincides the earth’s axis. For i1, the magnetic induction at a given time tp is [26] m 3m 3m Hm = − 3 i sin 2!c tp i0 − 3 i(1 − cos 2!c tp )j0 + 3 k0 ; (3) 2r 2r r where r is the radius of the orbit. The constant magnetic moment I of the spacecraft is supposed to be directed along axis X , that is I = I i = I (cos ’i + sin’j):
(4)
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The torque of the magnetic force from the interaction of the earth’s magnetic 6eld with the permanent magnet mounted on the spacecraft is Mm = Hm × I :
(5)
Hence,
m iI (2 sin ’ sin !c tp + cos ’ cos !c tp ): r3 The angular momentum of the rigid spacecraft about its mass center is d’ k: L=C !+ dtp MmZ = −
The theorem of angular momentum about the center of mass gives dL = Mg + Mm + Md + Mc : dtp
(6) (7)
(8)
Here only the planar libration is considered. Substituting Eqs. (1), (6) and (7) into Eq. (8) and projecting the result onto axis z lead to d2 ’ C 2 = −3!c2 (B − A)sin ’ cos ’ dtp −m iIr −3 (2 sin ’ sin !c tp + cos ’ cos !c tp ) − c’˙ + Mc :
(9)
Introducing the dimensionless time t = !c tp and denoting iIm c 3(B − A) ; = ; = ; 2 3 2C C!c r C!c one obtains the desired di
Mc C!c2
(10)
’O + ’˙ + Ksin 2’ + (2 sin ’ sin t + cos ’ cos t) = u;
(11)
K=
u=
where the derivative with respect to dimensionless time t. 3. Melnikov analyses of the uncontrolled system For the case without control, u = 0. The equation of motion of the uncontrolled system is ’O + ’˙ + Ksin 2’ + (2 sin ’ sin t + cos ’ cos t) = 0:
(12)
Assume the magnetic parameter to be small and let =1 (0 ¡ 1). Then Eq. (12) is an integrable Hamiltonian system under small perturbations ’O + Ksin 2’ = −1 (2 sin ’ sin t + cos ’ cos t):
(13)
For = 0, the unperturbed planar Hamiltonian system ’O + Ksin 2’ = 0
(14)
has 6rst integral of motion 1 2 ˙ 2’
+ K sin2 ’ = H:
(15)
When H = K, the integrable system (15) has two hyperbolic saddle points (±( =2); 0), whose unstable manifolds and stable manifolds constitute a heteroclinic cycle. The heteroclinic orbits ± started at √ (0; ± 2K) are √ √ √ (’± (!); ’˙ ± (!)) = (±arcsin(th( 2K!)); ± 2K sech( 2K!)): (16)
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For = 0, if the Melnikov functions +∞ M± (") = [ − 1 ’˙ ± (t) − 1 (2 sin ’± (t)sin(t + ") −∞
+ cos ’± (t) cos(t + "))]’˙ ± (t) dt
(17)
have a simple zero, then the Poincare map of Eq. (13) has transverse heteroclinic cycles [27]. Substituting Eq. (16) into Eq. (17), one evaluates the Melnikov functions for the hetoroclinic orbits + and − to yield 1 1 +√ csch √ sin " − 41 K: (18) M± (") = −1 2K 2K 2 2K Hence M+ (") and M− (") have simple zeros on the condition of 1 8K 2 √ √ sh = ¿ : (19) 1 (1 + 2K) 2 2K If Eq. (19) holds, there exists transverse heteroclinic cycles in the Poincare map of the system (12). The existence of such cycles implies that a complicated non-wandering Cantor set occurs. Hence chaos in the sense of Smale’s horseshoe appears in the system. 4. Computational analyses of the uncontrolled system The e
(20)
The fourth order Runge–Kutta routine is used for numerical integration. Various tools including the time history, Poincare map, power spectrum and Lyapunov exponents are used to identify the dynamical behaviors. The Lyapunov exponents are calculated using the algorithm derived by Wolf et al. [28]. The linearized system of Eq. (20) is characterized by the Jacobian matrix 0 1 0 −2K cos 2x1 − (2 cos x1 sin x3 − sin x1 cos x3 ) − −(2 sin x1 cos x3 − cos x1 sin x3 ) : (21) 0 0 0 The characteristic dynamical behavior are investigated by varying the magnetic parameter and the damping parameter , respectively, while the gravitational parameter K is kept constant at K = 1:1. In this case, the Melnikov condition of chaos given by Eq. (19) is = ¿ 1:5738. The numerical integration begin from the initial conditions described by (x1 ; x2 ; x3 )T = (0; 0; 0). For a given , the intermittency transition accrues as the magnetic parameter is increased Fix = 0:2. The the Melnikov condition of chaos is ¿ 0:3148. The dynamical behaviors of the uncontrolled system at = 0:6984; 0:6985; 0:69855; 0:6986 are depicted in Figs. 1– 4. The corresponding Lyapunov exponents are shown in Table 1. As is increased, the periodic behavior become the regular motion intermittently interrupted by a 6nite duration burst of irregular motion. For the further increase of , the bursts become
L.-Q. Chen, Y.-Z. Liu / International Journal of Non-Linear Mechanics 37 (2002) 493–504
497
Fig. 1. Periodic motion ( = 0:6984; = 0:2). (a) time history (b) phase trajectory (c) Poincare map (d) the largest Lyapunov exponent.
Fig. 2. Intermittency ( = 0:6985; = 0:2). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
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Fig. 3. Intermittency ( = 0:69855; = 0:2). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
Fig. 4. Chaotic motion ( = 0:6986; = 0:2). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
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Table 1 Lyapunov exponents ( = 0:2)
#1
#2
#3
0.6984 0.6985 0.69855 0.6986
0.00 0.12 0.13 0.17
−0:01
−0:19 −0:32 −0:33 −0:37
0.00 0.00 0.00
so frequent that the regular behavior can no longer be distinguished, and the chaotic motion is developed. Due to the transient behavior, the largest Lyapunov exponent is positive at the beginning of the periodic motion in Fig. 1. During the intermittency transition shown in Figs. 2 and 3, the motions depend sensitively on initial conditions so that the largest Lyapunov exponents are still positive for the duration of the periodic motions between irregular bursts. For a given , the intermittency transition accrues as the damping parameter is decreased. Fix =0:7. Then the Melnikov condition is ¡ 0:448. The case that = 0:295; 0:290; 0:285; 0:280 are depicted in Figs. 5 –8. The corresponding Lyapunov exponents are listed in Table 2. It should be pointed out that there is certain di
(22)
where from Eq. (10)
x2 f (x1 ; x2 ; x3 ) = −K sin 2x1 − x2 − (2 sin x1 sin x3 + cos x1 cos x3 ) ; 1 +(x) = 1;
(23)
h(x1 ; x2 ; x3 ) = x1 :
Let Lg h(x) denote the Lie-derivative of a scalar function h(x) with respect to a vector 6eld g, then Lg Lf0 h(x) = 0; Lf2 h(x)
Lf1 h(x) = x2 ;
Lg Lf1 h(x) = 1;
= −K sin 2x1 − x2 − (2 sin x1 sin x3 + cos x1 cos x3 ):
(24)
If given the desired output yR (t), then the input–output feedback control law has the expression [29] u = yO R − (−K sin 2x1 − x2 − (2 sin x1 sin x3 + cos x1 cos x3 )) − c1 (y˙ − y˙ R ) − c0 (y − yR )
(25)
where coeHcients c1 and c0 can be determined by normal design principles such as pole placement, linearquadratic optimal regulator, or robust service regulator.
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Fig. 5. Periodic motion ( = 0:7; = 0:295). (a) time history (b) phase trajectory (c) Poincare map (d) the largest Lyapunov exponent.
Fig. 6. Intermittency ( = 0:7; = 0:290). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
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Fig. 7. Intermittency ( = 0:7; = 0:285). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
Fig. 8. Chaotic motion ( = 0:7; = 0:280). (a) time history (b) Poincare map (c) power spectrum (d) the largest Lyapunov exponent.
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Table 2 Lyapunov exponents ( = 0:7)
#1
#2
#3
0.295 0.290 0.285 0.280
0.00 0.05 0.10 0.12
0.00 0.00 0.00 0.00
−0:30 −0:34 −0:38 −0:40
Fig. 9. Control of chaos to goal (26).
Fig. 10. Control of chaos to goal (27).
For the chaotic motions shown in Figs. 4 and 8, the control goals are successively taken as a 6xed point and a period 2 motion. ’R1 (t) = 0;
(26)
’R2 (t) = sin 0:5t:
(27)
Choose c1 =2:8; c0 =4:0 according to pole placement. Activate control when t0 =1300. The control results are, respectively, shown in Figs. 9 and 10. The corresponding control signals u = u(t) are, respectively, shown in Figs. 11 and 12.
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Fig. 11. Control signal for goal (26).
Fig. 12. Control signal for goal (27).
6. Conclusions The planar libration of a magnetic rigid spacecraft with internal damping subjected to both the gravitational and magnetic torque in a circular orbit near the equatorial plane of the earth is described by the di
Acknowledgements The research is supported by the National Natural Science Foundation of China (Project No. 10082003), Shanghai Municipal Development Foundation of Science and Technology.
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