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Chaotic attitude motion of a magnetic rigid spacecraft in a circular orbit near the equatorial plane
Journal of the Franklin Institute 339 (2002) 121–128
Brief communication
Chaotic attitude motion of a magnetic rigid spacecraft in a circular orbit near the equatorial plane Li-Qun Chen*, Yan-Zhu Liu, Gong Cheng Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200030, China
Abstract This paper deals with chaotic attitude motion of a magnetic rigid spacecraft in a circular orbit near the equatorial plane of the earth. The dynamical model of the problem is derived. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behavior is 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 break of torus as the torque of the magnetic forces is increased. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Chaos; Melnikov’s method; Numerical simulation; Spacecraft attitude dynamics
1. Introduction Attitude dynamics of spacecraft is a scientific research subject with great significance [1–3]. As chaos is widely and deeply investigated, much attention has been paid on chaotic attitude motion of spacecraft. It not only provides a definite engineering background for exploring chaos, but also offers a new viewpoint for analyzing spacecraft. It has been shown that there exists chaotic attitude motion in some models of spacecraft, such as spinning satellites in a circular orbit, gyrostat satellites in the gravitational field, and tethered satellites [4,5]. However, almost all researchers only considered spacecraft moving in the gravitational field, except Beletsky, Pivovarov and Starostin who studied numerically a magnetic rigid spacecraft in a circular orbit in the *Corresponding author. E-mail address: [email protected] (L.-Q. Chen). 0016-0032/02/$22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 2 ) 0 0 0 1 7 - 0
122
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absence of the gravitational torque [6]. In present work, the planar libration of nonspinning spacecraft in both the gravity and magnetic field of the earth are considered. The spacecraft under the investigation is supposed to be in a circular orbit near the equatorial plane of the earth. Based on its dynamical model, both the Melnikov analysis and numerical simulation studies are carried out. 2. System description and problem formulation Consider a magnetic rigid spacecraft moving in a circular orbit with the orbital angular velocity oc in the gravitational and magnetic field of the earth. The inertial reference frame ðOe X0 Y0 Z0 Þ has the origin Oe in the mass center of the Earth, with the polar axis of the Earth as axis Z0 and the line from Oe to the ascending node as axis X0 : The base vectors of ðOe X0 Y0 Z0 Þ are i 0 ; j 0 ; k0 : The principal coordinate frame ðO xyzÞ is fixed with the mass center O of the rigid spacecraft as the origin, whose base vectors are i 0 ; j 0 ; k0 : Principal moments of inertia of the arbitrarily shaped spacecraft are A; B and C: Suppose that B > A: The orbital coordinate frame ðO XYZÞ is fixed with line from Oe to O as axis, line normal to orbital plane XY as axis z (see Fig. 1 in which y ¼ 0). The base vectors of ðO XYZÞ are i; j; k: Denote j as the libration angle in the orbital plane. Then the spacecraft is subject to the torque of the gravitational field [3] M g ¼ 3o2c ðB AÞ sin j cos j k:
ð1Þ
For an orbit near the equatorial plane of the earth, the orbital inclination angle is i51: In this case, projecting i 0 ; j 0 ; k0 onto the axes of ðO XYZÞ gives i 0 ¼ cosðoc tp Þi sinðoc tp Þj; j 0 ¼ sinðoc tp Þi þ cosðoc tp Þj ik; k0 ¼ i sinðoc tp Þi þ i cosðoc tp Þj þ k:
Fig. 1. Reference frames.
ð2Þ
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128
123
The magnetic field is assumed to be that of a dipole with the magnetic moment whose axis coincides the earth’s axis. For i51; the magnetic induction at a given time tp is [3] 3m 3m m i sinð2oc tp Þi 0 m ið1 cosð2oc tp ÞÞj 0 þ m k0 ; ð3Þ Hm ¼ m 2r3 2r3 r3 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 ¼ Ii 0 ¼ Iðcos ji þ sin jjÞ:
ð4Þ
The torque of the magnetic force from the interaction of the earth’s magnetic field with the permanent magnet mounted on the spacecraft is M m ¼ H m I: The angular momentum of the rigid spacecraft about its mass center is dj L ¼ C oc þ k: dtp Substituting Eqs. (1)–(6) into the law of moment of momentum dL ¼ Mg þ Mm dtp
ð5Þ
ð6Þ
ð7Þ
and projecting the result onto axis z leads to C
d2 j ¼ 3o2c ðB AÞ sin j cos j mm iIr3 ð2 sin j sinðoc tp Þ dt2p þ cos j cosðoc tp ÞÞ:
ð8Þ
Here only the rotation around an axis normal to the orbit plane is considered, and equilibrium and stability around the axis in the orbit plane are assumed as guaranteed by a control system. Introducing the dimensionless time t ¼ oc tp and denoting K ¼ 3ðB AÞ=2C and a ¼ iImm =Co2c r3 ; one obtains the desired differential equation j. þ K sin 2j þ að2 sin j sin t þ cos j cos tÞ ¼ 0;
ð9Þ
where the derivative is with respect to the dimensionless time t:
3. Melnikov analysis Assume the magnetic parameter a to be small and let a ¼ ea1 (0oe51). Then Eq. (9) is an integrable Hamiltonian system under small perturbations: j. þ K sin 2j ¼ ea1 ð2 sin j sin t þ cos j cos tÞ:
ð10Þ
For e ¼ 0; the unperturbed planar Hamiltonian system (6) has first integrals of motion 1 2 ’ 2j
þ K sin2 j ¼ H:
ð11Þ
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128
124
When H ¼ K; the integrable system (9) has two hyperbolic saddle points ð7p=2; 0Þ; whose unstable manifolds and stable manifolds constitute a heteroclinic cycle. The pffiffiffiffiffiffiffi heteroclinic orbits G7 started at ð0; 7 2K Þ are pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ð12Þ ðj7 ðnÞ; j ’ 7 ðnÞÞ ¼ ð7arcsinðtanhð 2K nÞÞ; 7 2K ð 2K nÞÞ: For ea0; if the Melnikov functions Z þN a1 ð2 sin j7 ðtÞsinðt þ tÞ þ cos j7 ðtÞcosðt þ tÞÞ j M7 ðtÞ ¼ ’ 7 ðtÞ dt ð13Þ N
have a simple zero, then the Poincare map of Eq. (10) admits transverse heteroclinic cycle [6]. Substituting Eq. (12) into Eq. (13), one evaluates the Melnikov functions for the heteroclinic orbits Gþ and G to yield ! ! 1 1 p þ pffiffiffiffiffiffiffi csch pffiffiffiffiffiffiffi sin t: ð14Þ M7 ðtÞ ¼ a1 p 2K 2K 2 2K
10-7 0.04 10-8
()
0.02
0.00
10-9
-0.02 10-10 -0.02 1250
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t
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(b)
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(c)
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0
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Fig. 2. Quasi-periodic motion (a ¼ 0:01): (a) the time history, (b) power spectrum, (c) the Poincare map and (d) the largest Lyapunov exponent.
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128
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Hence Mþ ðtÞ and M ðtÞ have simple zeros for all aa0: Thus, there exists transverse heteroclinic cycles in the Poincare map of system (9). The existence of such cycles implies that Smale’s horseshoe and a complicated nonwandering Cantor set occur. 4. Computational analysis The effect of the magnetic parameter a upon the spacecraft attitude motion is now investigated numerically by integrating Eq. (9). Eq. (9) is rewritten as x’ 1 ¼ x2 ; x’ 2 ¼ K sin 2x1 að2 sin x1 sin x3 þ cos x1 cos x3 Þ;
ð15Þ
x’ 3 ¼ 1: The fourth-order Runge-Kutta routine with fixed step-size 0.01 and accuracy 104 is used for numerical integration. Various tools including the time history, Poincare
1.0
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0.0
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1000 2000 3000 4000 5000 6000 7000
t
Fig. 3. Breakup of the torus ða ¼ 0:12Þ: (a) the time history, (b) power spectrum, (c) the Poincare map and (d) the largest Lyapunov exponent.
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128
126
map, power spectrum and Lyapunov exponents are used to identify the dynamical behavior. The Lyapunov exponents are calculated using the algorithm derived by Wolf et al. [7]. The Jacobian matrix 0
0
1
B @ 2K cos 2x1 að2 cos x1 sin x3 sin x1 cos x3 Þ 0 0 0
1
0
C að2 sin x1 cos x3 cos x1 sin x3 Þ A ð16Þ 0
characterizes the linearized system of Eq. (15), which is necessary in the computations of the Lyapunov exponents. The characteristic dynamical behavior are investigated by varying the magnetic parameter a; while the gravitational parameter K is kept constant at K ¼ 0:75: The numerical integration begin from the initial conditions described by ðx1 ; x2 ; x3 ÞT ¼ ð0; 0; 0ÞT : For a small enough, the system experiences quasi-periodic motion. For a ¼ 0:01; the time history, the Poincare map, power spectrum and the largest Lyapunov exponent are depicted in Fig. 2. The Lyapunov exponents are all very close to zero. For a relatively big, the quasi-periodic torus begin to break. In the case that a ¼ 0:12 and 0:1275; the time history, the Poincare map, power spectrum and the
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0.1
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0
1000 2000 3000 4000 5000 6000 7000
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Fig. 4. Breakup of the torus ða ¼ 0:12175Þ: (a) the time history, (b) power spectrum, (c) the Poincare map and (d) the largest Lyapunov exponent.
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128 4
127
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()
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10-10 -4
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d /dt
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-2 -4
-2
(c)
0
2
4
(d)
-0.02 -1000
0
1000 2000 3000 4000 5000 6000 7000
t
Fig. 5. Chaotic motion ða ¼ 0:12176Þ: (a) the time history, (b) power spectrum, (c) the Poincare map and (d) the largest Lyapunov exponent.
largest Lyapunov exponent are, respectively, depicted in Figs. 3 and 4. Even if these Poincare maps are almost the same as in Fig. 2, the power spectrums show the change. The Lyapunov exponents are 0.00, 0.00, 0.00 and 0.01, 0.00, 0.01. For adequately big a; the quasi-periodic torus breaks completely. Chaotic motion occurs in the system. In the case that a ¼ 0:1276; the time history, the Poincare map, power spectrum and the largest Lyapunov exponent are depicted in Fig. 5. The Lypunov exponents are 0.12, 0.00, 0.12.
5. Conclusions The planar libration of a magnetic rigid spacecraft subjected to both the gravitational and magnetic torque in a circular orbit near the equatorial plane of the earth is described by differential Eq. (9). Using the Melnikov method proves the existence of chaos in the sense of the Smale horseshoe. Numerical methods including time history, Poincare map, power spectrum and Lyapunov characteristic exponents are employed to demonstrate the transition from quasi-periodic motion to chaos as the increase of the torque of the magnetic field.
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Acknowledgements The research is supported by the National Natural Science Foundation of China (Project No. 10082003) and the Shanghai Municipal Development Foundation of Science and Technology (Project No. 98JC14032).
References [1] V.V. Beletsky, Motion of an artificial satellite about its center of mass, Israel Program for Scientific Translations, Jerusalem, 1966. [2] F.J.P. Rimrott, Introductory Attitude dynamics, Springer, Berlin, etc., 1989. [3] Y.-Z. Liu, Spacecraft Attitude Dynamics, Nat. Defense Indu. Press, Beijing, 1995 (in Chinese). [4] Y.-Z. Liu, L.-Q. Chen, Nonlinear problems in spacecraft attitude dynamics, in: Chien Wei-zang (Ed.), Proceedings of the Third International Conference on Nonlinear Mechanics, Shanghai University Press, Shanghai, 1998, pp. 80–86. [5] Y.-Z. Liu, L.-Q. Chen, G. Cheng, X.-S. Ge, Stability, bifurcations and chaos in spacecraft attitude dynamics, Adv. Mech. 30 (3) (2000) 351–357 (in Chinese). [6] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (corrected fifth printing), Springer, New York, 1997. [7] A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vasano, Determining Lyapunov exponents from a time series, Physica D 16 (1985) 285–317.
Brief communication
Chaotic attitude motion of a magnetic rigid spacecraft in a circular orbit near the equatorial plane Li-Qun Chen*, Yan-Zhu Liu, Gong Cheng Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200030, China
Abstract This paper deals with chaotic attitude motion of a magnetic rigid spacecraft in a circular orbit near the equatorial plane of the earth. The dynamical model of the problem is derived. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behavior is 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 break of torus as the torque of the magnetic forces is increased. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Chaos; Melnikov’s method; Numerical simulation; Spacecraft attitude dynamics
1. Introduction Attitude dynamics of spacecraft is a scientific research subject with great significance [1–3]. As chaos is widely and deeply investigated, much attention has been paid on chaotic attitude motion of spacecraft. It not only provides a definite engineering background for exploring chaos, but also offers a new viewpoint for analyzing spacecraft. It has been shown that there exists chaotic attitude motion in some models of spacecraft, such as spinning satellites in a circular orbit, gyrostat satellites in the gravitational field, and tethered satellites [4,5]. However, almost all researchers only considered spacecraft moving in the gravitational field, except Beletsky, Pivovarov and Starostin who studied numerically a magnetic rigid spacecraft in a circular orbit in the *Corresponding author. E-mail address: [email protected] (L.-Q. Chen). 0016-0032/02/$22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 2 ) 0 0 0 1 7 - 0
122
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128
absence of the gravitational torque [6]. In present work, the planar libration of nonspinning spacecraft in both the gravity and magnetic field of the earth are considered. The spacecraft under the investigation is supposed to be in a circular orbit near the equatorial plane of the earth. Based on its dynamical model, both the Melnikov analysis and numerical simulation studies are carried out. 2. System description and problem formulation Consider a magnetic rigid spacecraft moving in a circular orbit with the orbital angular velocity oc in the gravitational and magnetic field of the earth. The inertial reference frame ðOe X0 Y0 Z0 Þ has the origin Oe in the mass center of the Earth, with the polar axis of the Earth as axis Z0 and the line from Oe to the ascending node as axis X0 : The base vectors of ðOe X0 Y0 Z0 Þ are i 0 ; j 0 ; k0 : The principal coordinate frame ðO xyzÞ is fixed with the mass center O of the rigid spacecraft as the origin, whose base vectors are i 0 ; j 0 ; k0 : Principal moments of inertia of the arbitrarily shaped spacecraft are A; B and C: Suppose that B > A: The orbital coordinate frame ðO XYZÞ is fixed with line from Oe to O as axis, line normal to orbital plane XY as axis z (see Fig. 1 in which y ¼ 0). The base vectors of ðO XYZÞ are i; j; k: Denote j as the libration angle in the orbital plane. Then the spacecraft is subject to the torque of the gravitational field [3] M g ¼ 3o2c ðB AÞ sin j cos j k:
ð1Þ
For an orbit near the equatorial plane of the earth, the orbital inclination angle is i51: In this case, projecting i 0 ; j 0 ; k0 onto the axes of ðO XYZÞ gives i 0 ¼ cosðoc tp Þi sinðoc tp Þj; j 0 ¼ sinðoc tp Þi þ cosðoc tp Þj ik; k0 ¼ i sinðoc tp Þi þ i cosðoc tp Þj þ k:
Fig. 1. Reference frames.
ð2Þ
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128
123
The magnetic field is assumed to be that of a dipole with the magnetic moment whose axis coincides the earth’s axis. For i51; the magnetic induction at a given time tp is [3] 3m 3m m i sinð2oc tp Þi 0 m ið1 cosð2oc tp ÞÞj 0 þ m k0 ; ð3Þ Hm ¼ m 2r3 2r3 r3 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 ¼ Ii 0 ¼ Iðcos ji þ sin jjÞ:
ð4Þ
The torque of the magnetic force from the interaction of the earth’s magnetic field with the permanent magnet mounted on the spacecraft is M m ¼ H m I: The angular momentum of the rigid spacecraft about its mass center is dj L ¼ C oc þ k: dtp Substituting Eqs. (1)–(6) into the law of moment of momentum dL ¼ Mg þ Mm dtp
ð5Þ
ð6Þ
ð7Þ
and projecting the result onto axis z leads to C
d2 j ¼ 3o2c ðB AÞ sin j cos j mm iIr3 ð2 sin j sinðoc tp Þ dt2p þ cos j cosðoc tp ÞÞ:
ð8Þ
Here only the rotation around an axis normal to the orbit plane is considered, and equilibrium and stability around the axis in the orbit plane are assumed as guaranteed by a control system. Introducing the dimensionless time t ¼ oc tp and denoting K ¼ 3ðB AÞ=2C and a ¼ iImm =Co2c r3 ; one obtains the desired differential equation j. þ K sin 2j þ að2 sin j sin t þ cos j cos tÞ ¼ 0;
ð9Þ
where the derivative is with respect to the dimensionless time t:
3. Melnikov analysis Assume the magnetic parameter a to be small and let a ¼ ea1 (0oe51). Then Eq. (9) is an integrable Hamiltonian system under small perturbations: j. þ K sin 2j ¼ ea1 ð2 sin j sin t þ cos j cos tÞ:
ð10Þ
For e ¼ 0; the unperturbed planar Hamiltonian system (6) has first integrals of motion 1 2 ’ 2j
þ K sin2 j ¼ H:
ð11Þ
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128
124
When H ¼ K; the integrable system (9) has two hyperbolic saddle points ð7p=2; 0Þ; whose unstable manifolds and stable manifolds constitute a heteroclinic cycle. The pffiffiffiffiffiffiffi heteroclinic orbits G7 started at ð0; 7 2K Þ are pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ð12Þ ðj7 ðnÞ; j ’ 7 ðnÞÞ ¼ ð7arcsinðtanhð 2K nÞÞ; 7 2K ð 2K nÞÞ: For ea0; if the Melnikov functions Z þN a1 ð2 sin j7 ðtÞsinðt þ tÞ þ cos j7 ðtÞcosðt þ tÞÞ j M7 ðtÞ ¼ ’ 7 ðtÞ dt ð13Þ N
have a simple zero, then the Poincare map of Eq. (10) admits transverse heteroclinic cycle [6]. Substituting Eq. (12) into Eq. (13), one evaluates the Melnikov functions for the heteroclinic orbits Gþ and G to yield ! ! 1 1 p þ pffiffiffiffiffiffiffi csch pffiffiffiffiffiffiffi sin t: ð14Þ M7 ðtÞ ¼ a1 p 2K 2K 2 2K
10-7 0.04 10-8
()
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10-9
-0.02 10-10 -0.02 1250
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t
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(c)
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0
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Fig. 2. Quasi-periodic motion (a ¼ 0:01): (a) the time history, (b) power spectrum, (c) the Poincare map and (d) the largest Lyapunov exponent.
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128
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Hence Mþ ðtÞ and M ðtÞ have simple zeros for all aa0: Thus, there exists transverse heteroclinic cycles in the Poincare map of system (9). The existence of such cycles implies that Smale’s horseshoe and a complicated nonwandering Cantor set occur. 4. Computational analysis The effect of the magnetic parameter a upon the spacecraft attitude motion is now investigated numerically by integrating Eq. (9). Eq. (9) is rewritten as x’ 1 ¼ x2 ; x’ 2 ¼ K sin 2x1 að2 sin x1 sin x3 þ cos x1 cos x3 Þ;
ð15Þ
x’ 3 ¼ 1: The fourth-order Runge-Kutta routine with fixed step-size 0.01 and accuracy 104 is used for numerical integration. Various tools including the time history, Poincare
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Fig. 3. Breakup of the torus ða ¼ 0:12Þ: (a) the time history, (b) power spectrum, (c) the Poincare map and (d) the largest Lyapunov exponent.
L.-Q. Chen et al. / Journal of the Franklin Institute 339 (2002) 121–128
126
map, power spectrum and Lyapunov exponents are used to identify the dynamical behavior. The Lyapunov exponents are calculated using the algorithm derived by Wolf et al. [7]. The Jacobian matrix 0
0
1
B @ 2K cos 2x1 að2 cos x1 sin x3 sin x1 cos x3 Þ 0 0 0
1
0
C að2 sin x1 cos x3 cos x1 sin x3 Þ A ð16Þ 0
characterizes the linearized system of Eq. (15), which is necessary in the computations of the Lyapunov exponents. The characteristic dynamical behavior are investigated by varying the magnetic parameter a; while the gravitational parameter K is kept constant at K ¼ 0:75: The numerical integration begin from the initial conditions described by ðx1 ; x2 ; x3 ÞT ¼ ð0; 0; 0ÞT : For a small enough, the system experiences quasi-periodic motion. For a ¼ 0:01; the time history, the Poincare map, power spectrum and the largest Lyapunov exponent are depicted in Fig. 2. The Lyapunov exponents are all very close to zero. For a relatively big, the quasi-periodic torus begin to break. In the case that a ¼ 0:12 and 0:1275; the time history, the Poincare map, power spectrum and the
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Fig. 4. Breakup of the torus ða ¼ 0:12175Þ: (a) the time history, (b) power spectrum, (c) the Poincare map and (d) the largest Lyapunov exponent.
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0
2
4
(d)
-0.02 -1000
0
1000 2000 3000 4000 5000 6000 7000
t
Fig. 5. Chaotic motion ða ¼ 0:12176Þ: (a) the time history, (b) power spectrum, (c) the Poincare map and (d) the largest Lyapunov exponent.
largest Lyapunov exponent are, respectively, depicted in Figs. 3 and 4. Even if these Poincare maps are almost the same as in Fig. 2, the power spectrums show the change. The Lyapunov exponents are 0.00, 0.00, 0.00 and 0.01, 0.00, 0.01. For adequately big a; the quasi-periodic torus breaks completely. Chaotic motion occurs in the system. In the case that a ¼ 0:1276; the time history, the Poincare map, power spectrum and the largest Lyapunov exponent are depicted in Fig. 5. The Lypunov exponents are 0.12, 0.00, 0.12.
5. Conclusions The planar libration of a magnetic rigid spacecraft subjected to both the gravitational and magnetic torque in a circular orbit near the equatorial plane of the earth is described by differential Eq. (9). Using the Melnikov method proves the existence of chaos in the sense of the Smale horseshoe. Numerical methods including time history, Poincare map, power spectrum and Lyapunov characteristic exponents are employed to demonstrate the transition from quasi-periodic motion to chaos as the increase of the torque of the magnetic field.
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Acknowledgements The research is supported by the National Natural Science Foundation of China (Project No. 10082003) and the Shanghai Municipal Development Foundation of Science and Technology (Project No. 98JC14032).
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