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Exponential control of a rotational motion of a rigid body using quaternions
Applied Mathematics and Computation 137 (2003) 195–207 www.elsevier.com/locate/amc
Exponential control of a rotational motion of a rigid body using quaternions Awad EL-Gohary *, Ebrahim R. Elazab Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Abstract A new developed technique is used to study the problem of the exponential stabilization of the rotational motion of a rigid body about a fixed axis in the body with the help of rotors. In this technique, the exponential asymptotic stability of this motion is proved. The control moments leading to the rotational motion are obtained. The conditions that ensure the exponential asymptotic stabilization of this motion are used to obtain the stabilizing control law as non-linear functions of the phase coordinates of the system. Finally, many special cases and analyzing the results of the studied problem are investigated. Ó 2002 Published by Elsevier Science Inc. Keywords: Rigid body; Exponential stability; Rotational motion; Control; Quaternions
1. Introduction A series of publications has been proposed to solve the problem of controlling the angular motion of a rigid body using the Lyapunov method. In this paper, we will use an analytical method to investigate the stabilization of the rotational motion of a rigid body. In this method, the exponential asymptotic stabilization of this rotation is achieved using the control process. The second Lyapunov method is used by Mortensen [1] to construct the external control torques ensuring the asymptotic stabilization of the rotation of
* Corresponding author. Present address. Dept. of Statistics and O.R., Faculty of Science, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia. E-mail addresses: [email protected], [email protected] (A. EL-Gohary).
0096-3003/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 0 4 2 - 5
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a rigid body. Smirnov [2] has studied the active control of the rotational motion of a rigid body using the Lyapunov function. The problem of the exponential asymptotic stabilization of the attitude of a rigid spacecraft using two control torques is discussed in [3]. EL-Gohary [4] established the problem of the asymptotic stabilization of the permanent rotational motion and equilibrium position of a rigid spacecraft with the help of three control moments applied to its principal axes of inertia. Kelkar et al. [5] have studied the problem of global asymptotic stability of the attitude of a rigid spacecraft using quaternion feedback. In such study, a non-linear control law is derived with the help of the Lyapunov technique. Belikov [6] investigated the effect of gyroscopic forces on the stability of uniform rotation of a rigid body about its principal axis. The stability of uniform rotations of a rigid body with a fixed point, about its first principal axis is studied by Kovalev and Savchenko [7]. EL-Gohary [8] has employed three rotors to obtain the internal control moments ensuring the asymptotic stabilization of the permanent rotation of a rigid body rotating around a fixed point and also to stabilize the equilibrium position of a gyrostat in [9]. Zaremba [10] used rotating flywheels to study the problem of the exponential stabilization of the programmed motion of a rigid body. The necessary and sufficient conditions of stability of steady rotations of a gyrostat about its principal axes are obtained in [11]. Druzhinin proved in [12] that if the total angular momentum of a gyrostat is non-zero, the permanent rotations can only occur about its principal axes of inertia. Anchev [13] and Rumiantsev [14] showed that the stability of the permanent rotations of a rigid body requires some conditions on the mass distribution of the body. The application of quaternions to digital attitude control problems can reduce computation time by more than 40% over the equivalent direction cosine matrix technique [15]. The purpose of this paper is to present a new technique for studying the exponential asymptotic stabilization of the rotational motion of a rigid body. The present technique determines the control moments from the requirements that ensure the exponential asymptotic stability of the rotational motion of the body. The stabilizing control moments of this motion are derived as functions of the phase coordinates of the system. As particular cases, the rotation of the body about its third principal axis of inertia, and the equilibrium position of the body which occurs when the principal axes of inertia coincide with the inertial axes are proved to be exponentially asymptotically stable. The time needed for control is calculated numerically.
2. Equations of motion Consider a mechanical system consists of a rigid body with a fixed point O and three symmetrical rotors are attached to the principal axes of inertia of the
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body at O. The relative motion of the rotors does not change the mass distribution of the system. To describe the motion, we introduce two coordinate systems. The first system Ongf is an inertial coordinate system while the second system Oxyz is moving with the rigid body and coincides with the principal axes of inertia of the body at O. Let Ai and xi ði ¼ 1; 2; 3Þ be the principal moments of inertia of the system and the components of the angular velocity of the body referred to Oxyz, respectively. Also let Ii and Ui ði ¼ 1; 2; 3Þ be the axial moments of inertia of the rotors and the angles of rotation of the rotors around the principal axes of inertia Oxyz, respectively. The equations of motion for the mechanical system under considerations can be written in the following form [16]: € 1 þ I3 U_ 3 x2 I2 U_ 2 x3 ¼ 0 A1 x_ 1 þ ðA3 A2 Þx2 x3 þ I1 U
ð1 2 3Þ;
ð2:1Þ
where the symbol ð1 2 3Þ means that two other equations can be obtained from the given equation by cyclic permutations of the indices ð1 ! 2 ! 3Þ and the dot means differentiation with respect to time. ~ of the The components Gi ði ¼ 1; 2; 3Þ of the angular momentum vector G system referred to the principal axes of inertia can be calculated to be G1 ¼ A1 x1 þ I1 U_ 1
ð1 2 3Þ:
ð2:2Þ
Substituting from Eqs. (2.2) into (2.1), we can get the following system: G_ 1 þ x2 G3 x3 G2 ¼ 0
ð1 2 3Þ:
ð2:3Þ
Multiplying these equations by G1 ; G2 and G3 , respectively, and adding them, we obtain the first integral G21 þ G22 þ G23 ¼ const:
ð2:4Þ
Eq. (2.1) using (2.2) can be rewritten in the following equivalent form: € 1 þ x2 G3 x3 G2 ¼ 0 ð1 2 3Þ: A1 x_ 1 þ I1 U
ð2:5Þ
The equations of the relative motion of the rotors without internal friction are given by € 1 Þ ¼ L1 I1 ðx_ 1 þ U
ð1 2 3Þ;
ð2:6Þ
where Li ði ¼ 1; 2; 3Þ are the control moments that applied to the rotors. These moments will be determined from the conditions that ensure the exponential asymptotic stabilization of the rotational motion of the body. Using the quaternion parameters ki ði ¼ 0; 1; 2; 3Þ to determine the orientation of the body relative to the inertial axes, we obtain the kinematical equations for the angular orientation of the body in the following form [17]:
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2k_0 ¼ x1 k1 x2 k2 x3 k3 ; 2k_1 ¼ x1 k0 þ x3 k2 x2 k3 ð1 2 3Þ;
ð2:7Þ
where the quaternion parameters ki ði ¼ 0; 1; 2; 3Þ satisfy the geometrical condition 3 X
k2i ¼ 1:
ð2:8Þ
i¼0
Eqs. (2.5) and (2.7) together form a complete system of differential equations for studying the problem of the exponential stabilization of the rotational motion of the rigid body about an axis that is fixed in the body and directed along the unit vector ~ ‘.
3. Exponential stabilization of the rotational motion In this section, the exponential asymptotic stabilization of the rotational motion of the rigid body around a fixed axis in the body is proved. The control moments leading to this motion are obtained. The rotational motion of the body around an axis fixed in the body is given by xi ¼ x0 ‘i ; ki ¼ ‘i sinðx0 t=2Þ ði ¼ 1; 2; 3Þ; k0 ¼ cosðx0 t=2Þ;
ð3:1Þ
where ‘i ði ¼ 1; 2; 3Þ are the components of the unit vector ~ ‘ that the body rotates about it, referred to the principal axes of inertia and x0 is the value of the angular velocity of rotation of the body. Eqs. (2.5) and (2.7) will admit the particular solution (3.1), if the control moments Li ði ¼ 1; 2; 3Þ have the following form: ðrÞ ðrÞ ðrÞ L1 ¼ x0 ‘3 G2 ‘2 G3 ð1 2 3Þ; ð3:2Þ ðrÞ ~ at where the components Gi ði ¼ 1; 2; 3Þ of the angular momentum vector G the rotational state (3.1) are given by ðrÞ ðrÞ G1 ¼ x0 ‘1 A1 þ I1 U_ 1 ðrÞ Li ;
ðrÞ U_ i
ð1 2 3Þ
ð3:3Þ
and ði ¼ 1; 2; 3Þ are the values of the control moments and the angular velocities of rotation of the rotors at the rotational state (3.1), respectively. Now, we proceed to prove the exponential stabilization of the rotational motion (3.1). To do that, we will obtain the equations of the perturbed motion by using the following new variables:
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ni ¼ xi x0 ‘i ;
ðrÞ
Vi ¼ Li Li
gi ¼ ki ‘i sinðx0 t=2Þ;
199
ði ¼ 1; 2; 3Þ; ðrÞ
gi ¼ G i G i
ði ¼ 1; 2; 3Þ;
ð3:4Þ
g0 ¼ k0 cosðx0 t=2Þ; where g0 ; gi ; ni ði ¼ 1; 2; 3Þ describe the perturbed motion of the body about the rotational state (3.1). Substituting expression (3.4) into Eqs. (2.5) and (2.7) and using (2.6), we can get ðA1 I1 Þn_1 ¼ ðn3 þ x0 ‘3 Þg2 ðn2 þ x0 ‘2 Þg3 ðrÞ
ðrÞ
þ G2 n3 G3 n2 V1
ð1 2 3Þ
2g_ 0 ¼ ð‘1 n1 þ ‘2 n2 þ ‘3 n3 Þ sinðx0 t=2Þ x0 ð‘1 g1 þ ‘2 g2 þ ‘3 g3 Þ ðn1 g1 þ n2 g2 þ n3 g3 Þ; 2g_ 1 ¼ n1 cosðx0 t=2Þ þ ð‘2 n3 ‘3 n2 Þ sinðx0 t=2Þ þ x0 ð‘1 g0 þ ‘3 g2 ‘2 g3 Þ þ ðn1 g0 þ n3 g2 n2 g3 Þ
ð1 2 3Þ: ð3:5Þ
This system of differential equations can be used to find the control parameters Vi ði ¼ 1; 2; 3Þ that stabilize exponentially the rotational motion (3.1). Theorem 1. An arbitrary rotation of the rigid body can be exponentially damped to stabilize the body at the rotational state (3.1), if the control moments Li ði ¼ 1; 2; 3Þ are chosen in the form ðrÞ
ðrÞ
L1 ¼ ðn3 þ x0 ‘3 Þðg2 þ G2 Þ ðn2 þ x0 ‘2 Þðg3 þ G3 Þ ðA1 I1 Þ f½ðg0 þ cosðx0 t=2ÞÞ2 þ ðg1 þ ‘1 sinðx0 t=2ÞÞ2 W1 þ ½ðg1 þ ‘1 sinðx0 t=2ÞÞðg2 þ ‘2 sinðx0 t=2ÞÞ þ ðg0 þ cosðx0 t=2ÞÞ ðg3 þ ‘3 sinðx0 t=2ÞÞ W2 þ ½ðg1 þ ‘1 sinðx0 t=2ÞÞðg3 þ ‘3 sinðx0 t=2ÞÞ ðg0 þ cosðx0 t=2ÞÞðg2 þ ‘2 sinðx0 t=2ÞÞ W3 g =ðg0 þ cosðx0 t=2ÞÞ ð1 2 3Þ and cosðx0 t=2Þ 6¼ 0;
ð3:6Þ
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where W1 ¼ 2k1 g1 þ 2m1 a1 þ ð1=2Þðg1 þ ‘1 sinðx0 t=2ÞÞ ½ðn1 þ x0 ‘1 Þ2 þ ðn2 þ x0 ‘2 Þ2 þ ðn3 þ x0 ‘3 Þ2
ðx20 ‘1 =2Þ sinðx0 t=2Þ
ð1 2 3Þ;
2a1 ¼ n1 cosðx0 t=2Þ þ ð‘2 n3 ‘3 n2 Þ sinðx0 t=2Þ þ x0 ð‘1 g0 þ ‘3 g2 ‘2 g3 Þ þ ðn1 g0 þ n3 g2 n2 g3 Þ
ð1 2 3Þ ð3:7Þ
and mi ; ki < 0;
m2i þ 4ki < 0 ði ¼ 1; 2; 3Þ:
ð3:8Þ
Proof. After some non-difficult calculations, we can verify that the variables g1 ; g2 ; g3 of the system (3.5) satisfy the following system of differential equations: g_ 1 ¼ a1 ;
a_ 1 ¼ k1 g1 þ m1 a1
ð1 2 3Þ:
The solution of this system can be obtained in the following form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mi ðtt0 Þ=2 2 gi ðtÞ ¼ Ai e sin mi 4ki ðt t0 Þ=2 þ bi ði ¼ 1; 2; 3Þ;
ð3:9Þ
ð3:10Þ
where the constants Ai and bi ði ¼ 1; 2; 3Þ are given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i .h i 2
2g_ i ðt0 Þ mi gi ðt0 Þ m2i 4ki þ g2i ðt0 Þ ði ¼ 1; 2; 3Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi.h i bi ¼ tan1 gi ðt0 Þ m2i 4ki ði ¼ 1; 2; 3Þ 2g_ i ðt0 Þ mi gi ðt0 Þ
Ai ¼
ð3:11Þ and gi ðt0 Þ; g_ i ðt0 Þ are the values of the functions gi ðtÞ; g_ i ðtÞ ði ¼ 1; 2; 3Þ at t ¼ t0 , respectively. Using conditions (3.8), the solution (3.10) is exponentially asymptotically stable and tends to zero exponentially as ðt ! 1Þ with damping amplitude oscillations. This means that the motion (3.1) of the body is exponentially asymptotically stable with respect to gi ; ði ¼ 1; 2; 3Þ. Using solution (3.10) and Eq. (3.9), the functions ai ði ¼ 1; 2; 3Þ can be expressed as functions of time in the following form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ai ðtÞ ¼ Ai ki emi ðtt0 Þ=2 sin m2i 4ki ðt t0 Þ=2 þ bi þ li ði ¼ 1; 2; 3Þ;
ð3:12Þ
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where the constants li ði ¼ 1; 2; 3Þ are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 li ¼ tan ði ¼ 1; 2; 3Þ m2i 4ki =mi
201
ð3:13Þ
using conditions (3.8), we find that as ðt ! 1Þ, the functions ai ði ¼ 1; 2; 3Þ approach zero exponentially with damping amplitude oscillations. Now, using the geometrical relation (2.8) we can get ( )1=2 3 X 2 g0 ðtÞ ¼ cosðx0 t=2Þ cos ðx0 t=2Þ gi ð2‘i sinðx0 t=2Þ þ gi Þ ; i¼1
ð3:14Þ using solution (3.10), we can find that as ðt ! 1Þ, the function g0 ðtÞ tends to zero or 2 cosðx0 t=2Þ. Thus, if cosðx0 t=2Þ 6¼ 0 we can easily get jg0 ðtÞ þ cosðx0 t=2Þj > r ¼ const: > 0
8t P t0 :
ð3:15Þ
Using the last three equations of (3.5) with (3.9) and (3.12), the functions ni ði ¼ 1; 2; 3Þ can be derived as follows: n 2 n1 ðtÞ ¼ ½2a1 x0 ð‘1 g0 þ ‘3 g2 ‘2 g3 Þ ½ðg0 þ cosðx0 t=2ÞÞ 2
þ ðg1 þ ‘1 sinðx0 t=2ÞÞ þ ½2a2 x0 ð‘2 g0 þ ‘1 g3 ‘3 g1 Þ
½ðg1 þ ‘1 sinðx0 t=2ÞÞðg2 þ ‘2 sinðx0 t=2ÞÞ þ ðg0 þ cosðx0 t=2ÞÞ ðg3 þ ‘3 sinðx0 t=2ÞÞ þ ½2a3 x0 ð‘3 g0 þ ‘2 g1 ‘1 g2 Þ
½ðg1 þ ‘1 sinðx0 t=2ÞÞðg3 þ ‘3 sinðx0 t=2ÞÞ ðg0 þ cosðx0 t=2ÞÞ o. ð3:16Þ ðg2 þ ‘2 sinðx0 t=2ÞÞ ðg0 þ cosðx0 t=2ÞÞ ð1 2 3Þ using condition (3.15), we find that as ðt ! 1Þ, the functions ni ði ¼ 1; 2; 3Þ approach zero exponentially. Thus, the motion (3.1) is exponentially asymptotically stable with respect to g0 ; ni ði ¼ 1; 2; 3Þ. Hence, for an arbitrary motion of the rigid body starting from some initial states that lie in the neighborhood of solution g0 ¼ ni ¼ gi ¼ 0, the variables ni ; gi ði ¼ 1; 2; 3Þ tend to zero and the variable g0 ðtÞ ! 0 only if we exclude the values g0 ðtÞ ¼ 2 cosðx0 t=2Þ from the neighborhood containing the initial perturbations g0 ðt0 Þ; ni ðt0 Þ, and gi ðt0 Þ. Furthermore, the variables g0 ; ni ; gi ði ¼ 1; 2; 3Þ approach zero exponentially with oscillatory behavior. Consequently, the state g0 ðtÞ ¼ ni ðtÞ ¼ gi ðtÞ ¼ 0 of the body will be exponentially asymptotically stable in the Lyapunov sense and the body will be forced to approach the rotational motion (3.1) which completes the proof.
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4. Stabilizing control moments In this section, the control moments that must be applied to the rotors in order to impose the exponential asymptotic stabilization of the rotational motion (3.1) are derived. The system of equations (2.3) admits the following first integrals: G1 ci1 þ G2 ci2 þ G3 ci3 ¼ hi
ði ¼ 1; 2; 3Þ;
ð4:1Þ
where hi ði ¼ 1; 2; 3Þ are constants and cij ði; j ¼ 1; 2; 3Þ are the direction cosines between Oxyz and Ongf. Solving the system of equations (4.1) with respect to Gi ði ¼ 1; 2; 3Þ and using the relations between ki ; k0 ði ¼ 1; 2; 3Þ and the direction cosines cij ði; j ¼ 1; 2; 3Þ [17] we can easily get G1 ¼ ðk20 þ k21 k22 k23 Þh1 þ 2ðk1 k2 þ k0 k3 Þh2 þ 2ðk1 k3 k0 k2 Þh3 ð1 2 3Þ:
ð4:2Þ
Substituting from (4.2) into (3.6) and returning to the unperturbed variables we can get L1 ¼ ðA1 I1 Þfðk20 þ k21 Þ½2k1 ðk1 ‘1 sinðx0 t=2ÞÞ þ m1 ðx1 k0 þ x3 k2 x2 k3 x0 ‘1 cosðx0 t=2ÞÞ þ x2 k1 =2 ðx20 ‘1 =2Þ sinðx0 t=2Þ þ ðk1 k2 þ k0 k3 Þ½2k2 ðk2 ‘2 sinðx0 t=2ÞÞ þ m2 ðx2 k0 þ x1 k3 x3 k1 x0 ‘2 cosðx0 t=2ÞÞ þ x2 k2 =2 ðx20 ‘2 =2Þ sinðx0 t=2Þ þ ðk1 k3 k0 k2 Þ½2k3 ðk3 ‘3 sinðx0 t=2ÞÞ þ m3 ðx3 k0 þ x2 k1 x1 k2 x0 ‘3 cosðx0 t=2ÞÞ þ x2 k3 =2 ðx20 ‘3 =2Þ sinðx0 t=2Þ g=k0 þ x3 ½ðk20 þ k22 k21 k23 Þh2 þ 2ðk1 k2 k0 k3 Þh1 þ 2ðk2 k3 þ k0 k1 Þh3 x2 ½ðk20 þ k23 k21 k22 Þh3 þ 2ðk1 k3 þ k0 k2 Þh1 þ 2ðk2 k3 k0 k1 Þh2 ð1 2 3Þ;
ð4:3Þ
where x¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x22 þ x23 :
It is clear that the control moments (4.3) are non-linear functions of the phase coordinates xi ; ki ; k0 ði ¼ 1; 2; 3Þ of the system. If these moments are applied to the rotors, then the rotational motion (3.1) of the body will be exponentially asymptotically stable when the conditions (3.8) hold.
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5. Special cases In what follows, we will introduce different special cases of the investigated problem. (1) The rotational motion of the body about the third principal axis of inertia Oz with a constant angular velocity x0 is described by x1 ¼ x2 ¼ 0; x3 ¼ x0 ; k0 ¼ cosðx0 t=2Þ; k3 ¼ sinðx0 t=2Þ;
k1 ¼ k2 ¼ 0:
ð5:1Þ
This motion can be obtained by putting ‘1 ¼ ‘2 ¼ 0; ‘3 ¼ 1 in motion (3.1). This rotational motion can be exponentially asymptotically stabilized and the stabilizing control moments can be obtained by setting ‘1 ¼ ‘2 ¼ 0; ‘3 ¼ 1 in the formulas (4.3) L1 ¼ ðA1 I1 Þfðk20 þ k21 Þ½2k1 k1 þ m1 ðx1 k0 þ x3 k2 x2 k3 Þ þ x2 k1 =2
þ ðk1 k2 þ k0 k3 Þ½2k2 k2 þ m2 ðx2 k0 þ x1 k3 x3 k1 Þ þ x2 k2 =2
þ ðk1 k3 k0 k2 Þ½2k3 ðk3 sinðx0 t=2ÞÞ þ m3 ðx3 k0 þ x2 k1 x1 k2 x0 cosðx0 t=2ÞÞ þ x2 k3 =2 ðx20 =2Þ sinðx0 t=2Þ g=k0 þ x3 ½ðk20 þ k22 k21 k23 Þh2 þ 2ðk1 k2 k0 k3 Þh1 þ 2ðk2 k3 þ k0 k1 Þh3 x2 ½ðk20 þ k23 k21 k22 Þh3 þ 2ðk1 k3 þ k0 k2 Þh1 þ 2ðk2 k3 k0 k1 Þh2 ð1 2 3Þ:
ð5:2Þ
(2) The equilibrium position of the body which occurs when the principal axes of inertia Oxyz of the body coincide with the inertial axes Ongf, respectively, is given by x1 ¼ x2 ¼ x3 ¼ 0; k0 ¼ 1;
k1 ¼ k2 ¼ k3 ¼ 0:
ð5:3Þ
This position can be obtained from motion (3.1) by setting x0 ¼ 0. The control moments that stabilize exponentially asymptotically this position can be obtained from moments (4.3) by putting x0 ¼ 0 L1 ¼ ðA1 I1 Þfðk20 þ k21 Þ½2k1 k1 þ m1 ðx1 k0 þ x3 k2 x2 k3 Þ þ x2 k1 =2
þ ðk1 k2 þ k0 k3 Þ½2k2 k2 þ m2 ðx2 k0 þ x1 k3 x3 k1 Þ þ x2 k2 =2
þ ðk1 k3 k0 k2 Þ½2k3 k3 þ m3 ðx3 k0 þ x2 k1 x1 k2 Þ þ x2 k3 =2 g=k0 þ x3 ½ðk20 þ k22 k21 k23 Þh2 þ 2ðk1 k2 k0 k3 Þh1 þ 2ðk2 k3 þ k0 k1 Þh3
x2 ½ðk20 þ k23 k21 k22 Þh3 þ 2ðk1 k3 þ k0 k2 Þh1 þ 2ðk2 k3 k0 k1 Þh2
ð1 2 3Þ: ð5:4Þ
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6. Analyzing the results In the present section, we will be analyzed the results of the studied problem. (1) Using conditions (3.8) and cosðx0 t=2Þ 6¼ 0 which ensure the exponential asymptotic stabilization of the rotational motion (3.1) of the body and applying the control moments (3.6), we see that as ðt ! 1Þ, the functions ni ðtÞ; gi ðtÞ; g0 ðtÞ ði ¼ 1; 2; 3Þ tend to zero with oscillatory behavior. (2) The control moments (4.3) stabilize exponentially asymptotically the rotational motion (3.1) of the body without imposing any conditions on the mass distribution of the system. (3) If the control moments (3.2) are applied to the rotors, then the body performs the rotational motion (3.1). (4) When the body arrives at the rotational state (3.1), Eq. (2.3) takes the following form: ðrÞ ðrÞ ðrÞ G_ 1 ¼ x0 ‘3 G2 ‘2 G3 ð1 2 3Þ: ð6:1Þ The general solution of the system (6.1) is given by ðrÞ
G1 ¼ fð‘1 ‘2 ‘3 c2 þ ‘23 c3 Þ sinðx0 tÞ þ ð‘23 c2 ‘1 ‘2 ‘3 c3 Þ cosðx0 tÞ þ ‘1 ð1 ‘22 Þc1 g=f‘3 ð1 ‘22 Þg; ðrÞ
G2 ¼ fð‘3 c3 cosðx0 tÞ ‘3 c2 sinðx0 tÞÞ þ ‘2 c1 g=‘3 ;
ð6:2Þ
ðrÞ
G3 ¼ fð‘2 ‘3 c2 ‘1 c3 Þ sinðx0 tÞ ð‘1 c2 þ ‘2 ‘3 c3 Þ cosðx0 tÞ þ ð1 ‘22 Þc1 g=f1 ‘22 g; where ci ði ¼ 1; 2; 3Þ are constants. Substituting from formulas (6.2) into Eq. (3.3), we can obtain the angular velocities of the rotors in the following form: ðrÞ I1 U_ 1 ¼ fð‘1 ‘2 ‘3 c2 þ ‘23 c3 Þ sinðx0 tÞ þ ð‘23 c2 ‘1 ‘2 ‘3 c3 Þ cosðx0 tÞ
þ ‘1 ð1 ‘22 Þðc1 A1 x0 ‘3 Þg=f‘3 ð1 ‘22 Þg; ðrÞ
I2 U_ 2 ¼ fð‘3 c3 cosðx0 tÞ ‘3 c2 sinðx0 tÞÞ þ ‘2 ðc1 A2 x0 ‘3 Þg=‘3 ; ðrÞ I3 U_ 3
ð6:3Þ
¼ fð‘2 ‘3 c2 ‘1 c3 Þ sinðx0 tÞ ð‘1 c2 þ ‘2 ‘3 c3 Þ cosðx0 tÞ þ ð1 ‘22 Þðc1 A3 x0 ‘3 Þg=f1 ‘22 g:
From Eq. (6.3), we conclude that when the body arrives at the rotational state (3.1), the angular velocities of the rotors become periodic functions of time. (5) In what follows, some numerical examples will be introduced. In such examples different cases for the control constants ki and mi will be considered.
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Example 1. In the case, k1 ¼ m1 ¼ 1; k2 ¼ 2:5; m2 ¼ 1:4; k3 ¼ 4; m3 ¼ 3 and the initial perturbations are considered to be g1 ðt0 Þ ¼ g2 ðt0 Þ ¼ g3 ðt0 Þ ¼ 0:1; g_ 1 ðt0 Þ ¼ g_ 2 ðt0 Þ ¼ g_ 3 ðt0 Þ ¼ 0:8; ‘1 ¼ ‘2 ¼ 0; ‘3 ¼ 1; x0 ¼ 8 rad=s; t0 ¼ 0 then: (i) the behavior of the functions g0 ðtÞ; g1 ðtÞ; g2 ðtÞ; g3 ðtÞ is described by the curves shown in Fig. 1; (ii) the behavior of the functions n1 ðtÞ; n2 ðtÞ; n3 ðtÞ is described by the curves shown in Fig. 2; (iii) the time needed for control can be calculated to be 10 s approximately. Example 2. In the case, k1 ¼ 11; m1 ¼ m2 ¼ m3 ¼ 5; k2 ¼ 14; k3 ¼ 16 and the initial perturbations are considered to be g1 ðt0 Þ ¼ g2 ðt0 Þ ¼ g3 ðt0 Þ ¼ 0:1; g_ 1 ðt0 Þ ¼ g_ 2 ðt0 Þ ¼ g_ 3 ðt0 Þ ¼ 0:8; ‘1 ¼ ‘2 ¼ 0; ‘3 ¼ 1; x0 ¼ 8 rad=s; t0 ¼ 0 then: (i) the behavior of the functions g0 ðtÞ; g1 ðtÞ; g2 ðtÞ; g3 ðtÞ is described by
Fig. 1.
Fig. 2.
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Fig. 3.
Fig. 4.
the curves shown in Fig. 3; (ii) the behavior of the functions n1 ðtÞ; n2 ðtÞ; n3 ðtÞ is described by the curves shown in Fig. 4; (iii) the time needed for control can be calculated to be 2.5 s approximately. In Figs. 1–4 we find that the perturbed functions gi ðtÞ and ni ðtÞ approach zero exponentially with oscillatory behavior. Consequently, the control moðrÞ ments approach the rotational control moments Li exponentially with oscillatory behavior. These figures show that the time needed for control process in the second case is less than the first case. This comparison indicates that the time needed
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for control depends on the absolute values of the control constants ki and mi ði ¼ 1; 2; 3Þ such that the time decreases with decreasing these values. 7. Conclusion Under conditions (3.8) and cosðx0 t=2Þ 6¼ 0, the problem of the exponential asymptotic stabilization of the rotational motion of a rigid body has been studied using a new developed technique. In such a technique, the control moments are obtained as non-linear functions of the phase coordinates of the system and the angle of rotation x0 t. Many particular cases of the studied problem are investigated. Analyzing the results is presented. References [1] R.E. Mortensen, On system for automatic control of the rotation of a rigid body, Electronics Research Lab., University of California, Report No. 63, Issue No. 23, 1963. [2] E.Ya. Smirov, Active control the rotational motion of a rigid body, Izv. Akad. Nauk SSSR. MTT 9 (3) (1974) 5–10. [3] P. Morin, S. Samson, Time-varing exponential stabilization of a rigid spacecraft with two control torques, IEEE Trans. Automat. Control 42 (4) (1997) 528–534. [4] A.I. EL-Gohary, The control of a permanent rotational motion and equilibrium position of a spacecraft, Int. J. Non-linear Mech. 35 (2000) 987–995. [5] A.G. Kelkar, S.M. Joshi, J.T. Wen, Robust attitude stabilization of spacecraft using quaternion feedback, IEEE Trans. Automat. Control 40 (10) (1995) 1800–1803. [6] S.A. Belikov, Effect of gyroscopic forces on stability of uniform rotation of a rigid body about a principal axis, Izv. Akad. Nauk SSSR. MTT 16 (4) (1981) 3–10. [7] A.M. Kovalev, A.Ia. Savchenko, Stability of uniform rotations of a rigid body about a principal axis, J. Appl. Math. Mech. 39 (4) (1975) 650–660. [8] A.I. EL-Gohary, S.Z. Hassan, On the exponential stability of the permanent rotational motion of a gyrostat, J. Mech. Res. Commun. 26 (4) (1999) 479–488. [9] A.I. EL-Gohary, On the orientation of a gyrostat using internal rotors, Int. J. Mech. Sci. 43 (1) (2001) 225–235. [10] A.T. Zaremba, Stabilization of the programmed motions of a rigid body with uncertainty in the parameters of the equations of dynamics, J. Appl. Math. Mech. 61 (1) (1997) 33–39. [11] A.M. Kovalev, Stability of steady rotations of a heavy gyrostat about its principal axis, J. Appl. Math. Mech. 44 (6) (1982) 709–712. [12] E.I. Druzhinin, The permanent rotations of a balanced non-autonomous gyrostat, J. Appl. Math. Mech. 63 (5) (1999) 825–826. [13] A. Anchev, On the stability of permanent rotations of a heavy gyrostat, J. Appl. Math. Mech. 26 (1) (1962) 22–28. [14] V.V. Rumianstev, On the stability of motion of gyrostat, J. Appl. Math. Mech. 25 (1) (1961) 9–16. [15] B.P. Ickes, A new method for performing digital control system attitude computations using quaternions, AIAA J. 8 (1) (1970) 13–17. [16] A.I. EL-Gohary, The dynamics of relative motion of servo-constrained system, Ph.D. Dissertation, Russian People’s Friendship University, Moscow, 1994. [17] L.I. Lur’e, Analytical Mechanics, Fizmatigz, Moscow, 1961.
Exponential control of a rotational motion of a rigid body using quaternions Awad EL-Gohary *, Ebrahim R. Elazab Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Abstract A new developed technique is used to study the problem of the exponential stabilization of the rotational motion of a rigid body about a fixed axis in the body with the help of rotors. In this technique, the exponential asymptotic stability of this motion is proved. The control moments leading to the rotational motion are obtained. The conditions that ensure the exponential asymptotic stabilization of this motion are used to obtain the stabilizing control law as non-linear functions of the phase coordinates of the system. Finally, many special cases and analyzing the results of the studied problem are investigated. Ó 2002 Published by Elsevier Science Inc. Keywords: Rigid body; Exponential stability; Rotational motion; Control; Quaternions
1. Introduction A series of publications has been proposed to solve the problem of controlling the angular motion of a rigid body using the Lyapunov method. In this paper, we will use an analytical method to investigate the stabilization of the rotational motion of a rigid body. In this method, the exponential asymptotic stabilization of this rotation is achieved using the control process. The second Lyapunov method is used by Mortensen [1] to construct the external control torques ensuring the asymptotic stabilization of the rotation of
* Corresponding author. Present address. Dept. of Statistics and O.R., Faculty of Science, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia. E-mail addresses: [email protected], [email protected] (A. EL-Gohary).
0096-3003/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 0 4 2 - 5
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a rigid body. Smirnov [2] has studied the active control of the rotational motion of a rigid body using the Lyapunov function. The problem of the exponential asymptotic stabilization of the attitude of a rigid spacecraft using two control torques is discussed in [3]. EL-Gohary [4] established the problem of the asymptotic stabilization of the permanent rotational motion and equilibrium position of a rigid spacecraft with the help of three control moments applied to its principal axes of inertia. Kelkar et al. [5] have studied the problem of global asymptotic stability of the attitude of a rigid spacecraft using quaternion feedback. In such study, a non-linear control law is derived with the help of the Lyapunov technique. Belikov [6] investigated the effect of gyroscopic forces on the stability of uniform rotation of a rigid body about its principal axis. The stability of uniform rotations of a rigid body with a fixed point, about its first principal axis is studied by Kovalev and Savchenko [7]. EL-Gohary [8] has employed three rotors to obtain the internal control moments ensuring the asymptotic stabilization of the permanent rotation of a rigid body rotating around a fixed point and also to stabilize the equilibrium position of a gyrostat in [9]. Zaremba [10] used rotating flywheels to study the problem of the exponential stabilization of the programmed motion of a rigid body. The necessary and sufficient conditions of stability of steady rotations of a gyrostat about its principal axes are obtained in [11]. Druzhinin proved in [12] that if the total angular momentum of a gyrostat is non-zero, the permanent rotations can only occur about its principal axes of inertia. Anchev [13] and Rumiantsev [14] showed that the stability of the permanent rotations of a rigid body requires some conditions on the mass distribution of the body. The application of quaternions to digital attitude control problems can reduce computation time by more than 40% over the equivalent direction cosine matrix technique [15]. The purpose of this paper is to present a new technique for studying the exponential asymptotic stabilization of the rotational motion of a rigid body. The present technique determines the control moments from the requirements that ensure the exponential asymptotic stability of the rotational motion of the body. The stabilizing control moments of this motion are derived as functions of the phase coordinates of the system. As particular cases, the rotation of the body about its third principal axis of inertia, and the equilibrium position of the body which occurs when the principal axes of inertia coincide with the inertial axes are proved to be exponentially asymptotically stable. The time needed for control is calculated numerically.
2. Equations of motion Consider a mechanical system consists of a rigid body with a fixed point O and three symmetrical rotors are attached to the principal axes of inertia of the
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body at O. The relative motion of the rotors does not change the mass distribution of the system. To describe the motion, we introduce two coordinate systems. The first system Ongf is an inertial coordinate system while the second system Oxyz is moving with the rigid body and coincides with the principal axes of inertia of the body at O. Let Ai and xi ði ¼ 1; 2; 3Þ be the principal moments of inertia of the system and the components of the angular velocity of the body referred to Oxyz, respectively. Also let Ii and Ui ði ¼ 1; 2; 3Þ be the axial moments of inertia of the rotors and the angles of rotation of the rotors around the principal axes of inertia Oxyz, respectively. The equations of motion for the mechanical system under considerations can be written in the following form [16]: € 1 þ I3 U_ 3 x2 I2 U_ 2 x3 ¼ 0 A1 x_ 1 þ ðA3 A2 Þx2 x3 þ I1 U
ð1 2 3Þ;
ð2:1Þ
where the symbol ð1 2 3Þ means that two other equations can be obtained from the given equation by cyclic permutations of the indices ð1 ! 2 ! 3Þ and the dot means differentiation with respect to time. ~ of the The components Gi ði ¼ 1; 2; 3Þ of the angular momentum vector G system referred to the principal axes of inertia can be calculated to be G1 ¼ A1 x1 þ I1 U_ 1
ð1 2 3Þ:
ð2:2Þ
Substituting from Eqs. (2.2) into (2.1), we can get the following system: G_ 1 þ x2 G3 x3 G2 ¼ 0
ð1 2 3Þ:
ð2:3Þ
Multiplying these equations by G1 ; G2 and G3 , respectively, and adding them, we obtain the first integral G21 þ G22 þ G23 ¼ const:
ð2:4Þ
Eq. (2.1) using (2.2) can be rewritten in the following equivalent form: € 1 þ x2 G3 x3 G2 ¼ 0 ð1 2 3Þ: A1 x_ 1 þ I1 U
ð2:5Þ
The equations of the relative motion of the rotors without internal friction are given by € 1 Þ ¼ L1 I1 ðx_ 1 þ U
ð1 2 3Þ;
ð2:6Þ
where Li ði ¼ 1; 2; 3Þ are the control moments that applied to the rotors. These moments will be determined from the conditions that ensure the exponential asymptotic stabilization of the rotational motion of the body. Using the quaternion parameters ki ði ¼ 0; 1; 2; 3Þ to determine the orientation of the body relative to the inertial axes, we obtain the kinematical equations for the angular orientation of the body in the following form [17]:
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2k_0 ¼ x1 k1 x2 k2 x3 k3 ; 2k_1 ¼ x1 k0 þ x3 k2 x2 k3 ð1 2 3Þ;
ð2:7Þ
where the quaternion parameters ki ði ¼ 0; 1; 2; 3Þ satisfy the geometrical condition 3 X
k2i ¼ 1:
ð2:8Þ
i¼0
Eqs. (2.5) and (2.7) together form a complete system of differential equations for studying the problem of the exponential stabilization of the rotational motion of the rigid body about an axis that is fixed in the body and directed along the unit vector ~ ‘.
3. Exponential stabilization of the rotational motion In this section, the exponential asymptotic stabilization of the rotational motion of the rigid body around a fixed axis in the body is proved. The control moments leading to this motion are obtained. The rotational motion of the body around an axis fixed in the body is given by xi ¼ x0 ‘i ; ki ¼ ‘i sinðx0 t=2Þ ði ¼ 1; 2; 3Þ; k0 ¼ cosðx0 t=2Þ;
ð3:1Þ
where ‘i ði ¼ 1; 2; 3Þ are the components of the unit vector ~ ‘ that the body rotates about it, referred to the principal axes of inertia and x0 is the value of the angular velocity of rotation of the body. Eqs. (2.5) and (2.7) will admit the particular solution (3.1), if the control moments Li ði ¼ 1; 2; 3Þ have the following form: ðrÞ ðrÞ ðrÞ L1 ¼ x0 ‘3 G2 ‘2 G3 ð1 2 3Þ; ð3:2Þ ðrÞ ~ at where the components Gi ði ¼ 1; 2; 3Þ of the angular momentum vector G the rotational state (3.1) are given by ðrÞ ðrÞ G1 ¼ x0 ‘1 A1 þ I1 U_ 1 ðrÞ Li ;
ðrÞ U_ i
ð1 2 3Þ
ð3:3Þ
and ði ¼ 1; 2; 3Þ are the values of the control moments and the angular velocities of rotation of the rotors at the rotational state (3.1), respectively. Now, we proceed to prove the exponential stabilization of the rotational motion (3.1). To do that, we will obtain the equations of the perturbed motion by using the following new variables:
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ni ¼ xi x0 ‘i ;
ðrÞ
Vi ¼ Li Li
gi ¼ ki ‘i sinðx0 t=2Þ;
199
ði ¼ 1; 2; 3Þ; ðrÞ
gi ¼ G i G i
ði ¼ 1; 2; 3Þ;
ð3:4Þ
g0 ¼ k0 cosðx0 t=2Þ; where g0 ; gi ; ni ði ¼ 1; 2; 3Þ describe the perturbed motion of the body about the rotational state (3.1). Substituting expression (3.4) into Eqs. (2.5) and (2.7) and using (2.6), we can get ðA1 I1 Þn_1 ¼ ðn3 þ x0 ‘3 Þg2 ðn2 þ x0 ‘2 Þg3 ðrÞ
ðrÞ
þ G2 n3 G3 n2 V1
ð1 2 3Þ
2g_ 0 ¼ ð‘1 n1 þ ‘2 n2 þ ‘3 n3 Þ sinðx0 t=2Þ x0 ð‘1 g1 þ ‘2 g2 þ ‘3 g3 Þ ðn1 g1 þ n2 g2 þ n3 g3 Þ; 2g_ 1 ¼ n1 cosðx0 t=2Þ þ ð‘2 n3 ‘3 n2 Þ sinðx0 t=2Þ þ x0 ð‘1 g0 þ ‘3 g2 ‘2 g3 Þ þ ðn1 g0 þ n3 g2 n2 g3 Þ
ð1 2 3Þ: ð3:5Þ
This system of differential equations can be used to find the control parameters Vi ði ¼ 1; 2; 3Þ that stabilize exponentially the rotational motion (3.1). Theorem 1. An arbitrary rotation of the rigid body can be exponentially damped to stabilize the body at the rotational state (3.1), if the control moments Li ði ¼ 1; 2; 3Þ are chosen in the form ðrÞ
ðrÞ
L1 ¼ ðn3 þ x0 ‘3 Þðg2 þ G2 Þ ðn2 þ x0 ‘2 Þðg3 þ G3 Þ ðA1 I1 Þ f½ðg0 þ cosðx0 t=2ÞÞ2 þ ðg1 þ ‘1 sinðx0 t=2ÞÞ2 W1 þ ½ðg1 þ ‘1 sinðx0 t=2ÞÞðg2 þ ‘2 sinðx0 t=2ÞÞ þ ðg0 þ cosðx0 t=2ÞÞ ðg3 þ ‘3 sinðx0 t=2ÞÞ W2 þ ½ðg1 þ ‘1 sinðx0 t=2ÞÞðg3 þ ‘3 sinðx0 t=2ÞÞ ðg0 þ cosðx0 t=2ÞÞðg2 þ ‘2 sinðx0 t=2ÞÞ W3 g =ðg0 þ cosðx0 t=2ÞÞ ð1 2 3Þ and cosðx0 t=2Þ 6¼ 0;
ð3:6Þ
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where W1 ¼ 2k1 g1 þ 2m1 a1 þ ð1=2Þðg1 þ ‘1 sinðx0 t=2ÞÞ ½ðn1 þ x0 ‘1 Þ2 þ ðn2 þ x0 ‘2 Þ2 þ ðn3 þ x0 ‘3 Þ2
ðx20 ‘1 =2Þ sinðx0 t=2Þ
ð1 2 3Þ;
2a1 ¼ n1 cosðx0 t=2Þ þ ð‘2 n3 ‘3 n2 Þ sinðx0 t=2Þ þ x0 ð‘1 g0 þ ‘3 g2 ‘2 g3 Þ þ ðn1 g0 þ n3 g2 n2 g3 Þ
ð1 2 3Þ ð3:7Þ
and mi ; ki < 0;
m2i þ 4ki < 0 ði ¼ 1; 2; 3Þ:
ð3:8Þ
Proof. After some non-difficult calculations, we can verify that the variables g1 ; g2 ; g3 of the system (3.5) satisfy the following system of differential equations: g_ 1 ¼ a1 ;
a_ 1 ¼ k1 g1 þ m1 a1
ð1 2 3Þ:
The solution of this system can be obtained in the following form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mi ðtt0 Þ=2 2 gi ðtÞ ¼ Ai e sin mi 4ki ðt t0 Þ=2 þ bi ði ¼ 1; 2; 3Þ;
ð3:9Þ
ð3:10Þ
where the constants Ai and bi ði ¼ 1; 2; 3Þ are given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i .h i 2
2g_ i ðt0 Þ mi gi ðt0 Þ m2i 4ki þ g2i ðt0 Þ ði ¼ 1; 2; 3Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi.h i bi ¼ tan1 gi ðt0 Þ m2i 4ki ði ¼ 1; 2; 3Þ 2g_ i ðt0 Þ mi gi ðt0 Þ
Ai ¼
ð3:11Þ and gi ðt0 Þ; g_ i ðt0 Þ are the values of the functions gi ðtÞ; g_ i ðtÞ ði ¼ 1; 2; 3Þ at t ¼ t0 , respectively. Using conditions (3.8), the solution (3.10) is exponentially asymptotically stable and tends to zero exponentially as ðt ! 1Þ with damping amplitude oscillations. This means that the motion (3.1) of the body is exponentially asymptotically stable with respect to gi ; ði ¼ 1; 2; 3Þ. Using solution (3.10) and Eq. (3.9), the functions ai ði ¼ 1; 2; 3Þ can be expressed as functions of time in the following form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ai ðtÞ ¼ Ai ki emi ðtt0 Þ=2 sin m2i 4ki ðt t0 Þ=2 þ bi þ li ði ¼ 1; 2; 3Þ;
ð3:12Þ
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where the constants li ði ¼ 1; 2; 3Þ are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 li ¼ tan ði ¼ 1; 2; 3Þ m2i 4ki =mi
201
ð3:13Þ
using conditions (3.8), we find that as ðt ! 1Þ, the functions ai ði ¼ 1; 2; 3Þ approach zero exponentially with damping amplitude oscillations. Now, using the geometrical relation (2.8) we can get ( )1=2 3 X 2 g0 ðtÞ ¼ cosðx0 t=2Þ cos ðx0 t=2Þ gi ð2‘i sinðx0 t=2Þ þ gi Þ ; i¼1
ð3:14Þ using solution (3.10), we can find that as ðt ! 1Þ, the function g0 ðtÞ tends to zero or 2 cosðx0 t=2Þ. Thus, if cosðx0 t=2Þ 6¼ 0 we can easily get jg0 ðtÞ þ cosðx0 t=2Þj > r ¼ const: > 0
8t P t0 :
ð3:15Þ
Using the last three equations of (3.5) with (3.9) and (3.12), the functions ni ði ¼ 1; 2; 3Þ can be derived as follows: n 2 n1 ðtÞ ¼ ½2a1 x0 ð‘1 g0 þ ‘3 g2 ‘2 g3 Þ ½ðg0 þ cosðx0 t=2ÞÞ 2
þ ðg1 þ ‘1 sinðx0 t=2ÞÞ þ ½2a2 x0 ð‘2 g0 þ ‘1 g3 ‘3 g1 Þ
½ðg1 þ ‘1 sinðx0 t=2ÞÞðg2 þ ‘2 sinðx0 t=2ÞÞ þ ðg0 þ cosðx0 t=2ÞÞ ðg3 þ ‘3 sinðx0 t=2ÞÞ þ ½2a3 x0 ð‘3 g0 þ ‘2 g1 ‘1 g2 Þ
½ðg1 þ ‘1 sinðx0 t=2ÞÞðg3 þ ‘3 sinðx0 t=2ÞÞ ðg0 þ cosðx0 t=2ÞÞ o. ð3:16Þ ðg2 þ ‘2 sinðx0 t=2ÞÞ ðg0 þ cosðx0 t=2ÞÞ ð1 2 3Þ using condition (3.15), we find that as ðt ! 1Þ, the functions ni ði ¼ 1; 2; 3Þ approach zero exponentially. Thus, the motion (3.1) is exponentially asymptotically stable with respect to g0 ; ni ði ¼ 1; 2; 3Þ. Hence, for an arbitrary motion of the rigid body starting from some initial states that lie in the neighborhood of solution g0 ¼ ni ¼ gi ¼ 0, the variables ni ; gi ði ¼ 1; 2; 3Þ tend to zero and the variable g0 ðtÞ ! 0 only if we exclude the values g0 ðtÞ ¼ 2 cosðx0 t=2Þ from the neighborhood containing the initial perturbations g0 ðt0 Þ; ni ðt0 Þ, and gi ðt0 Þ. Furthermore, the variables g0 ; ni ; gi ði ¼ 1; 2; 3Þ approach zero exponentially with oscillatory behavior. Consequently, the state g0 ðtÞ ¼ ni ðtÞ ¼ gi ðtÞ ¼ 0 of the body will be exponentially asymptotically stable in the Lyapunov sense and the body will be forced to approach the rotational motion (3.1) which completes the proof.
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4. Stabilizing control moments In this section, the control moments that must be applied to the rotors in order to impose the exponential asymptotic stabilization of the rotational motion (3.1) are derived. The system of equations (2.3) admits the following first integrals: G1 ci1 þ G2 ci2 þ G3 ci3 ¼ hi
ði ¼ 1; 2; 3Þ;
ð4:1Þ
where hi ði ¼ 1; 2; 3Þ are constants and cij ði; j ¼ 1; 2; 3Þ are the direction cosines between Oxyz and Ongf. Solving the system of equations (4.1) with respect to Gi ði ¼ 1; 2; 3Þ and using the relations between ki ; k0 ði ¼ 1; 2; 3Þ and the direction cosines cij ði; j ¼ 1; 2; 3Þ [17] we can easily get G1 ¼ ðk20 þ k21 k22 k23 Þh1 þ 2ðk1 k2 þ k0 k3 Þh2 þ 2ðk1 k3 k0 k2 Þh3 ð1 2 3Þ:
ð4:2Þ
Substituting from (4.2) into (3.6) and returning to the unperturbed variables we can get L1 ¼ ðA1 I1 Þfðk20 þ k21 Þ½2k1 ðk1 ‘1 sinðx0 t=2ÞÞ þ m1 ðx1 k0 þ x3 k2 x2 k3 x0 ‘1 cosðx0 t=2ÞÞ þ x2 k1 =2 ðx20 ‘1 =2Þ sinðx0 t=2Þ þ ðk1 k2 þ k0 k3 Þ½2k2 ðk2 ‘2 sinðx0 t=2ÞÞ þ m2 ðx2 k0 þ x1 k3 x3 k1 x0 ‘2 cosðx0 t=2ÞÞ þ x2 k2 =2 ðx20 ‘2 =2Þ sinðx0 t=2Þ þ ðk1 k3 k0 k2 Þ½2k3 ðk3 ‘3 sinðx0 t=2ÞÞ þ m3 ðx3 k0 þ x2 k1 x1 k2 x0 ‘3 cosðx0 t=2ÞÞ þ x2 k3 =2 ðx20 ‘3 =2Þ sinðx0 t=2Þ g=k0 þ x3 ½ðk20 þ k22 k21 k23 Þh2 þ 2ðk1 k2 k0 k3 Þh1 þ 2ðk2 k3 þ k0 k1 Þh3 x2 ½ðk20 þ k23 k21 k22 Þh3 þ 2ðk1 k3 þ k0 k2 Þh1 þ 2ðk2 k3 k0 k1 Þh2 ð1 2 3Þ;
ð4:3Þ
where x¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x22 þ x23 :
It is clear that the control moments (4.3) are non-linear functions of the phase coordinates xi ; ki ; k0 ði ¼ 1; 2; 3Þ of the system. If these moments are applied to the rotors, then the rotational motion (3.1) of the body will be exponentially asymptotically stable when the conditions (3.8) hold.
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5. Special cases In what follows, we will introduce different special cases of the investigated problem. (1) The rotational motion of the body about the third principal axis of inertia Oz with a constant angular velocity x0 is described by x1 ¼ x2 ¼ 0; x3 ¼ x0 ; k0 ¼ cosðx0 t=2Þ; k3 ¼ sinðx0 t=2Þ;
k1 ¼ k2 ¼ 0:
ð5:1Þ
This motion can be obtained by putting ‘1 ¼ ‘2 ¼ 0; ‘3 ¼ 1 in motion (3.1). This rotational motion can be exponentially asymptotically stabilized and the stabilizing control moments can be obtained by setting ‘1 ¼ ‘2 ¼ 0; ‘3 ¼ 1 in the formulas (4.3) L1 ¼ ðA1 I1 Þfðk20 þ k21 Þ½2k1 k1 þ m1 ðx1 k0 þ x3 k2 x2 k3 Þ þ x2 k1 =2
þ ðk1 k2 þ k0 k3 Þ½2k2 k2 þ m2 ðx2 k0 þ x1 k3 x3 k1 Þ þ x2 k2 =2
þ ðk1 k3 k0 k2 Þ½2k3 ðk3 sinðx0 t=2ÞÞ þ m3 ðx3 k0 þ x2 k1 x1 k2 x0 cosðx0 t=2ÞÞ þ x2 k3 =2 ðx20 =2Þ sinðx0 t=2Þ g=k0 þ x3 ½ðk20 þ k22 k21 k23 Þh2 þ 2ðk1 k2 k0 k3 Þh1 þ 2ðk2 k3 þ k0 k1 Þh3 x2 ½ðk20 þ k23 k21 k22 Þh3 þ 2ðk1 k3 þ k0 k2 Þh1 þ 2ðk2 k3 k0 k1 Þh2 ð1 2 3Þ:
ð5:2Þ
(2) The equilibrium position of the body which occurs when the principal axes of inertia Oxyz of the body coincide with the inertial axes Ongf, respectively, is given by x1 ¼ x2 ¼ x3 ¼ 0; k0 ¼ 1;
k1 ¼ k2 ¼ k3 ¼ 0:
ð5:3Þ
This position can be obtained from motion (3.1) by setting x0 ¼ 0. The control moments that stabilize exponentially asymptotically this position can be obtained from moments (4.3) by putting x0 ¼ 0 L1 ¼ ðA1 I1 Þfðk20 þ k21 Þ½2k1 k1 þ m1 ðx1 k0 þ x3 k2 x2 k3 Þ þ x2 k1 =2
þ ðk1 k2 þ k0 k3 Þ½2k2 k2 þ m2 ðx2 k0 þ x1 k3 x3 k1 Þ þ x2 k2 =2
þ ðk1 k3 k0 k2 Þ½2k3 k3 þ m3 ðx3 k0 þ x2 k1 x1 k2 Þ þ x2 k3 =2 g=k0 þ x3 ½ðk20 þ k22 k21 k23 Þh2 þ 2ðk1 k2 k0 k3 Þh1 þ 2ðk2 k3 þ k0 k1 Þh3
x2 ½ðk20 þ k23 k21 k22 Þh3 þ 2ðk1 k3 þ k0 k2 Þh1 þ 2ðk2 k3 k0 k1 Þh2
ð1 2 3Þ: ð5:4Þ
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6. Analyzing the results In the present section, we will be analyzed the results of the studied problem. (1) Using conditions (3.8) and cosðx0 t=2Þ 6¼ 0 which ensure the exponential asymptotic stabilization of the rotational motion (3.1) of the body and applying the control moments (3.6), we see that as ðt ! 1Þ, the functions ni ðtÞ; gi ðtÞ; g0 ðtÞ ði ¼ 1; 2; 3Þ tend to zero with oscillatory behavior. (2) The control moments (4.3) stabilize exponentially asymptotically the rotational motion (3.1) of the body without imposing any conditions on the mass distribution of the system. (3) If the control moments (3.2) are applied to the rotors, then the body performs the rotational motion (3.1). (4) When the body arrives at the rotational state (3.1), Eq. (2.3) takes the following form: ðrÞ ðrÞ ðrÞ G_ 1 ¼ x0 ‘3 G2 ‘2 G3 ð1 2 3Þ: ð6:1Þ The general solution of the system (6.1) is given by ðrÞ
G1 ¼ fð‘1 ‘2 ‘3 c2 þ ‘23 c3 Þ sinðx0 tÞ þ ð‘23 c2 ‘1 ‘2 ‘3 c3 Þ cosðx0 tÞ þ ‘1 ð1 ‘22 Þc1 g=f‘3 ð1 ‘22 Þg; ðrÞ
G2 ¼ fð‘3 c3 cosðx0 tÞ ‘3 c2 sinðx0 tÞÞ þ ‘2 c1 g=‘3 ;
ð6:2Þ
ðrÞ
G3 ¼ fð‘2 ‘3 c2 ‘1 c3 Þ sinðx0 tÞ ð‘1 c2 þ ‘2 ‘3 c3 Þ cosðx0 tÞ þ ð1 ‘22 Þc1 g=f1 ‘22 g; where ci ði ¼ 1; 2; 3Þ are constants. Substituting from formulas (6.2) into Eq. (3.3), we can obtain the angular velocities of the rotors in the following form: ðrÞ I1 U_ 1 ¼ fð‘1 ‘2 ‘3 c2 þ ‘23 c3 Þ sinðx0 tÞ þ ð‘23 c2 ‘1 ‘2 ‘3 c3 Þ cosðx0 tÞ
þ ‘1 ð1 ‘22 Þðc1 A1 x0 ‘3 Þg=f‘3 ð1 ‘22 Þg; ðrÞ
I2 U_ 2 ¼ fð‘3 c3 cosðx0 tÞ ‘3 c2 sinðx0 tÞÞ þ ‘2 ðc1 A2 x0 ‘3 Þg=‘3 ; ðrÞ I3 U_ 3
ð6:3Þ
¼ fð‘2 ‘3 c2 ‘1 c3 Þ sinðx0 tÞ ð‘1 c2 þ ‘2 ‘3 c3 Þ cosðx0 tÞ þ ð1 ‘22 Þðc1 A3 x0 ‘3 Þg=f1 ‘22 g:
From Eq. (6.3), we conclude that when the body arrives at the rotational state (3.1), the angular velocities of the rotors become periodic functions of time. (5) In what follows, some numerical examples will be introduced. In such examples different cases for the control constants ki and mi will be considered.
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Example 1. In the case, k1 ¼ m1 ¼ 1; k2 ¼ 2:5; m2 ¼ 1:4; k3 ¼ 4; m3 ¼ 3 and the initial perturbations are considered to be g1 ðt0 Þ ¼ g2 ðt0 Þ ¼ g3 ðt0 Þ ¼ 0:1; g_ 1 ðt0 Þ ¼ g_ 2 ðt0 Þ ¼ g_ 3 ðt0 Þ ¼ 0:8; ‘1 ¼ ‘2 ¼ 0; ‘3 ¼ 1; x0 ¼ 8 rad=s; t0 ¼ 0 then: (i) the behavior of the functions g0 ðtÞ; g1 ðtÞ; g2 ðtÞ; g3 ðtÞ is described by the curves shown in Fig. 1; (ii) the behavior of the functions n1 ðtÞ; n2 ðtÞ; n3 ðtÞ is described by the curves shown in Fig. 2; (iii) the time needed for control can be calculated to be 10 s approximately. Example 2. In the case, k1 ¼ 11; m1 ¼ m2 ¼ m3 ¼ 5; k2 ¼ 14; k3 ¼ 16 and the initial perturbations are considered to be g1 ðt0 Þ ¼ g2 ðt0 Þ ¼ g3 ðt0 Þ ¼ 0:1; g_ 1 ðt0 Þ ¼ g_ 2 ðt0 Þ ¼ g_ 3 ðt0 Þ ¼ 0:8; ‘1 ¼ ‘2 ¼ 0; ‘3 ¼ 1; x0 ¼ 8 rad=s; t0 ¼ 0 then: (i) the behavior of the functions g0 ðtÞ; g1 ðtÞ; g2 ðtÞ; g3 ðtÞ is described by
Fig. 1.
Fig. 2.
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Fig. 3.
Fig. 4.
the curves shown in Fig. 3; (ii) the behavior of the functions n1 ðtÞ; n2 ðtÞ; n3 ðtÞ is described by the curves shown in Fig. 4; (iii) the time needed for control can be calculated to be 2.5 s approximately. In Figs. 1–4 we find that the perturbed functions gi ðtÞ and ni ðtÞ approach zero exponentially with oscillatory behavior. Consequently, the control moðrÞ ments approach the rotational control moments Li exponentially with oscillatory behavior. These figures show that the time needed for control process in the second case is less than the first case. This comparison indicates that the time needed
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for control depends on the absolute values of the control constants ki and mi ði ¼ 1; 2; 3Þ such that the time decreases with decreasing these values. 7. Conclusion Under conditions (3.8) and cosðx0 t=2Þ 6¼ 0, the problem of the exponential asymptotic stabilization of the rotational motion of a rigid body has been studied using a new developed technique. In such a technique, the control moments are obtained as non-linear functions of the phase coordinates of the system and the angle of rotation x0 t. Many particular cases of the studied problem are investigated. Analyzing the results is presented. References [1] R.E. Mortensen, On system for automatic control of the rotation of a rigid body, Electronics Research Lab., University of California, Report No. 63, Issue No. 23, 1963. [2] E.Ya. Smirov, Active control the rotational motion of a rigid body, Izv. Akad. Nauk SSSR. MTT 9 (3) (1974) 5–10. [3] P. Morin, S. Samson, Time-varing exponential stabilization of a rigid spacecraft with two control torques, IEEE Trans. Automat. Control 42 (4) (1997) 528–534. [4] A.I. EL-Gohary, The control of a permanent rotational motion and equilibrium position of a spacecraft, Int. J. Non-linear Mech. 35 (2000) 987–995. [5] A.G. Kelkar, S.M. Joshi, J.T. Wen, Robust attitude stabilization of spacecraft using quaternion feedback, IEEE Trans. Automat. Control 40 (10) (1995) 1800–1803. [6] S.A. Belikov, Effect of gyroscopic forces on stability of uniform rotation of a rigid body about a principal axis, Izv. Akad. Nauk SSSR. MTT 16 (4) (1981) 3–10. [7] A.M. Kovalev, A.Ia. Savchenko, Stability of uniform rotations of a rigid body about a principal axis, J. Appl. Math. Mech. 39 (4) (1975) 650–660. [8] A.I. EL-Gohary, S.Z. Hassan, On the exponential stability of the permanent rotational motion of a gyrostat, J. Mech. Res. Commun. 26 (4) (1999) 479–488. [9] A.I. EL-Gohary, On the orientation of a gyrostat using internal rotors, Int. J. Mech. Sci. 43 (1) (2001) 225–235. [10] A.T. Zaremba, Stabilization of the programmed motions of a rigid body with uncertainty in the parameters of the equations of dynamics, J. Appl. Math. Mech. 61 (1) (1997) 33–39. [11] A.M. Kovalev, Stability of steady rotations of a heavy gyrostat about its principal axis, J. Appl. Math. Mech. 44 (6) (1982) 709–712. [12] E.I. Druzhinin, The permanent rotations of a balanced non-autonomous gyrostat, J. Appl. Math. Mech. 63 (5) (1999) 825–826. [13] A. Anchev, On the stability of permanent rotations of a heavy gyrostat, J. Appl. Math. Mech. 26 (1) (1962) 22–28. [14] V.V. Rumianstev, On the stability of motion of gyrostat, J. Appl. Math. Mech. 25 (1) (1961) 9–16. [15] B.P. Ickes, A new method for performing digital control system attitude computations using quaternions, AIAA J. 8 (1) (1970) 13–17. [16] A.I. EL-Gohary, The dynamics of relative motion of servo-constrained system, Ph.D. Dissertation, Russian People’s Friendship University, Moscow, 1994. [17] L.I. Lur’e, Analytical Mechanics, Fizmatigz, Moscow, 1961.