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Optimal stabilization of the rotational motion of a rigid body with the help of rotors
International Journal of Non-Linear Mechanics 35 (2000) 393}403
Optimal stabilization of the rotational motion of a rigid body with the help of rotorsq Awad El-Gohary Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Received 12 December 1998; accepted 29 March 1999
Abstract Optimal stabilization of the rotational motion of a symmetrical rigid body with the help of internal rotors is studied. In such a study the asymptotic stability of this motion is proved by using Barbachen and Krasovskii theorem. The optimal control moments which stabilize this motion are obtained using the conditions of ensuring the optimal asymptotic stability as non-linear functions of phase coordinates of the system. These moments stabilize asymptotically one type of the rotational motions of the rigid body. The other rotational motions are unstable in the Lyapunov sense. As a particular case of our problem, the equilibrium position of the rigid body, which occurs when the principal axes of inertia coincide with the inertial axes, is proved to be asymptotically stable. This study is characterized by that non-linear equations of motion which are used to prove the asymptotic stability and derivation of the control moments. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Optimal; Stabilization; Rigid body
1. Introduction The problems of optimal stabilization of the rotational motion of a rigid body and system of rigid bodies are very important for numerous applications such as dynamics of satellite-gyrostat, spacecraft, aircraft, robotics, biomechanics and the like. The problem of controlling the motion of a rigid body is processed by either external or internal torques. Mortensen [1] studied the problem of controlling the rotation of a free rigid body with the help of external torques by using the second Lyapunov method and Pontryagin principle. Krementulo [2] used an approximation method to study the optimal stabilization of the rotational motion of a rigid body with the help of internal rotors. Krementulo and Sokolova [3] studied the control of the motion of a rigid body using internal torques. Krementulo [4] solved the problem of optimal stabilization of the rotational motion of a gyrostat with the help of rotors. Vorotinikov [5] studied the control of the angular motion of a rigid body. He has also obtained the control parameters su$cient to ensure exponential asymptotic stability of the permanent rotation of the rigid body. Agafanov et al. [6] have considered the
q Contributed by W.F. Ames. E-mail address: [email protected] (A. El-Gohary)
0020-7462/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 9 ) 0 0 0 2 5 - 6
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control of the plane turn of the spacecraft using external forces. The problem of controlling the spherical motion of a rigid body, when the external controlling moments do not contain x-component and their axes do not coincide with the principal axes of inertia of the body, has been investigated by Lebedev [7]. He also discussed the control of the uniform rotation of the body about the principal axes of inertia and its orientation in inertial space. Vorotinikov [8] analysed the problem of controlling the reorientation of a symmetrical rigid body using a game-theoretic model of interference. Also he has obtained the non-linear control rules, which ensure stability of the body at the prescribed state in "nite time. Kovalev [9] obtained the necessary and su$cient conditions for partial control of linear autonomous system. Further he has studied the control of the rotational motion of a rigid body using a single rotor. Balaeva [10] examined the control of the motion of a rigid body in the absence of information about the angular velocity of the body. Sanchez de Alvarez [11] discussed stabilization of a rigid body dynamics using internal and external torques. El-Gohary [12] obtained the control moments su$cient to ensure asymptotic stability of the equilibrium position and the rotational motion of a gyrostat, using the Lyapunov function. Rymuntsev [13] has studied the problem of controlling and stabilization with respect to positional coordinates and impulse of the holonomic systems with ignorable coordinates with the help of controlling forces. The main object of this paper is to study the optimal stabilization of the rotational motion of a symmetrical rigid body, with the help of rotors using Barbachen and Krasovskii's theorem [14]. The optimal control moments, which ensure asymptotic stability of this motion, are obtained as functions of phase coordinates of the body. The obtained control moments stabilize one kind of the rotational motions of the rigid body. On the other hand, the other cases of the described rotational motion are proved to be unstable in the Lyapunov sense. As a special case of the studied problem, (i) it can be veri"ed that the equilibrium position of the rigid body is asymptotically stable; (ii) the optimal control moments ensure that they are obtained. In our study, the non-linear equations of motion are used to prove the asymptotic stability of the rotational motion and the derivation of the control moments as the non-linear functions of phase coordinates of the system.
2. Equations of motion Consider a mechanical system S comprised of a rigid body P (platform) with a "xed point O its centre of mass and a system of three symmetrical rotors R are attached to the principal axes of inertia of the system S, in such a way that the motion of the rotors does not modify the distribution of the masses of the system S. We introduce two coordinate systems (Fig. 1), Omgf be inertial coordinate system, and Ox x x , coordinate 1 2 3 system moving with the body axes coincident with the principal axes of inertia of the system S.u and i C , i"1, 2, 3 are the components of the angular velocity and principal moments of inertia of the body relative i to principal axes Ox x x , respectively. I , and ' , i"1, 2, 3 are the axial moments of inertia and the angles of 1 2 3 i i rotation of the rotors relative to the principal axes of inertia of the system. Equations of motion of the mechanical system can be written in the form of Krementulo [15,18]. C u5 #(C !C )u u #I 'G #I '0 u !I '0 u "0, 1 1 3 2 2 3 1 1 3 3 2 2 2 3 C u5 #(C !C )u u #I 'G #I '0 u !I '0 u "0, 2 2 1 3 1 3 2 2 1 1 3 3 3 1 C u5 #(C !C )u u #I 'G #I '0 u !I '0 u "0. 3 3 2 1 1 2 3 3 2 2 1 1 1 2 The equations of the relative motion of the rotors without internal friction are I (u5 #'G )"u , 1 1 1 1 I (u5 #'G )"u , 2 2 2 2 I (u5 #'G )"u , 3 3 3 3
(2.1)
(2.2)
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Fig. 1.
where u (i"1, 2, 3) are the control moments applied to the rotors, and dot means di!erentiation with respect i to time. These moments will be determined from the condition to ensure asymptotic stability of the rotational motion of the body. The Poisson kinematical equations for angular orientation of the rigid body in the inertial space are a5 "u a !u a , i1 3 i2 2 i3 a5 "u a !u a , i2 1 i3 3 i1 a5 "u a !u a , (i"1, 2, 3), i3 2 i1 1 i2
(2.3)
a (i, j"1, 2, 3) are the direction cosines between Omgf and the principal axes of inertia Ox x x , and satisfy ij 1 2 3 the geometrical condition
G
1, k"l 3 + a a " (k, l"1, 2, 3). ki li 0, kOl i/1
(2.4)
Consider a-symmetrical rigid body (C "C "C). Replace the coordinate system Ox (i"1, 2, 3) by a new 1 2 i semimoving coordinates system Oy (i"1, 2, 3) where semimoving coordinate axes Oy coincides with Ox , i 3 3 the axes Oy and Oy lie in the Ox x plane and are not involved in the intrinsic rotation of the body with 1 2 1 2 respect to the angle /. The direction cosines between Oy y y and the inertial axes Omgf are b (i, j"1, 2, 3). 1 2 3 ij Replacing u , u and u by the new variables ) , ) and ) which represent the components of instan1 2 3 1 2 3 taneous angular velocity of the trihedron Oy y y onto its axes Krementulo [15] (Fig. 2): it follows that 1 2 3 u ") cos /#) sin /, 1 1 2 u "!) sin /#) cos /, 2 1 2 u ") #/Q . 3 3
(2.5)
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Fig. 2.
The relations between direction cosines a and b are ij ij a "b cos /#b sin /, i1 i1 i2 a "!b sin /#b cos /, i2 i1 i2 a "b . i3 i3 Using transformation (2.5), the equations of motion (2.1) takes the following form: C)0 #(C !C)) ) #C ) /Q #g5 #g ) !g ) "0, 1 3 2 3 3 2 1 3 2 2 3 C)0 #(C!C )) ) !C ) /Q #gR #g ) !g ) "0, 2 3 1 3 3 1 2 1 3 3 1 C ()0 #/$ )#gR #g ) !g ) "0, 3 3 3 2 1 1 2 where g "I('0 cos /!'0 sin /), 1 1 2 g "I('0 sin /#'0 cos /), 2 1 2 g "I '0 , I "I "I. 2 3 3 3 1 The equations of the rotor motion becomes g5 #I)0 #(g #I) )/Q "w , 1 2 2 1 1 g5 #I)0 #(g #I) )/Q "w , 2 2 1 1 2 g5 #I (PQ #/$ )"w , 3 3 3 3 where w "u cos /!u sin /, 1 1 2 w "u sin /#u cos /, 2 1 2 w "u . 3 3
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
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The Poisson kinematical equations in the new variables takes the following form: bQ ") b !) b , i1 3 i2 2 i3 bQ ") b !) b , i2 1 i3 3 i1 (i"1, 2, 3). bQ ") b !) b i3 2 i1 1 i2
(2.11)
Eqs. (2.7) and (2.11) allow the stationary motion ) ") ") "0, w "w "w "0, 1 2 3 1 2 3 g "g "0, g "g(0)"const., /Q "u, 3 1 2 3 1, i"j (i, j"1, 2, 3), b " ij 0, iOj.
G
(2.12)
This special solution of Eqs. (2.7) and (2.11) represents an uniform rotation of the rigid body with the angular velocity /Q about the axis Of(Ox ). In such rotational motion, the rotors attached to the axes Ox and Ox are 3 1 2 at rest while the rotor attached to Ox rotates with an uniform angular velocity. 3 To eliminate the kinetic moments g (i"1, 2, 3) of the rotors from Eqs. (2.7) we will use the conservation i law of the kinetic moment of system (S) in the inertial axes Krementulo [15] (C) #g )b #(C) #g )b #[C () #/Q )#g ]b "h , 1 1 i1 2 2 i2 3 3 3 i3 i
(2.13)
where h are constants. i Substituting Eqs. (2.13) and (2.9) into Eq. (2.7), we get 3 3 (C!I))0 "!(C!I)u) #() #u) + h b ! ) + h b !w , 1 2 3 i i2 2 i i3 1 i/1 i/1 3 3 (C!I))0 "(C!I)) !() #)) + h b #) + h b !w , i i3 2 2 1 3 i i1 1 i/1 i/1 3 3 (C !I ))0 ") + h b !) + h b !w . 2 i i1 1 i i2 3 3 3 3 i/1 i/1
(2.14)
Using Eqs. (2.13) and (2.12), we "nd that h(0)"h(0)"0, 2 1
h(0)"C u#g(0) 3 3 3
(2.15)
for the stationary motion (2.12).
3. The stability problem To obtain the equations of the perturbed motion, we consider the stationary motion (2.12) to be the unperturbed motion, and for the perturbed motion we take new variables ) ") , d "b !1, b "b (i, j"1, 2, 3). i i ii ii ij ij
(3.1)
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Substituting Eq. (3.1) into Eqs. (2.7) and (2.11) the equations of the perturbed motion about stationary motion are A)0 ") [h b #h (d #1)#h b ]! ) [h b #h b #h (d #1)]#< , 1 3 1 12 2 22 3 32 2 1 13 2 23 3 33 1 0 A) ") [h b #h b #h (d #1)]! ) [h (d #1)#h b #h b ]#< , 2 1 1 13 2 23 3 33 3 1 11 2 21 3 31 2 0 B) ") [h (d #1)#h b #h b ]! ) [h b #h (d #1)#h b ]#< , 3 2 1 11 2 21 3 31 1 1 12 2 22 3 32 3 < "u[h b #h (d #1)#(h(0)#h )b ]# h(0)[) b !) (d #1)]!Au) !w , 1 1 12 2 22 3 3 32 3 3 32 2 33 2 1 < "!u[h (d #1)#h b #(h(0)#h )b ]# h(0)[) b !) (d #1)]#Au) !w , 3 3 31 1 33 1 1 3 3 31 2 1 11 2 21 < "h(0)[) b !) b ]!w , 3 2 31 1 32 3 3 A"C!I, B"C !I , (3.2) 3 3 dQ "b ) !b ) , 11 12 3 13 2 bQ "d ) !b ) #) , 21 22 3 23 2 3 bQ "b ) !d ) !) , (123). (3.3) 31 32 3 33 2 2 In what follows, we shall derive the stabilizing control parameters < (i"1, 2, 3) using a new developed i technique. In such a technique, the requirements to ensure optimal asymptotic stability of motion (2.12) are adopted. We formulate the problem of optimal stabilization of the rotational motion in the following manner: to choose the control parameters as functions of phase coordinates describing the motion of the rigid body about the "xed point O, so that the stationary motion (2.12) of system (3.2), (3.3) be asymptotically stable with respect to these coordinates and the functional
P
=
MC()2#)2)#C )2# 1 [k b !k b #< ]2# 1 [k b !k b #< ]2 1 2 3 3 4C 3 32 2 23 1 4C 1 13 3 31 2 0 # 1 3[k b !k b #< ]2N dt, (3.4) 4C 2 21 1 12 3 be minimum, such that the motion is attenuated resulting in evaluating the controlling parameters < (i"1, 2, 3). The constants k '0 (i"1, 2, 3) are positive. i i We select the optimal Lyapunov function in the form J"
2<"A()2#)2)#B)2#k (d2 #b2 #b2 )#k (b2 #d2 #b2 )#k (b2 #b2 #d2 ), 1 2 3 1 11 12 13 2 21 22 23 3 31 32 33 A"C!I'0, B"C !I '0. 3 3 Using the Krasovskii's theorem (1966), we have [19]
(3.5)
B[<; ) , ) , ) ; d , b ,2, d ; < , < , < ]"![) (k b !k b )#< )#) (k b , 1 2 3 11 12 33 1 2 3 1 2 23 3 32 1 2 3 31 !k b #< )#) (k b !k b )#< ) 1 13 2 3 1 12 2 21 3 3 ! + ) < ]#[C()2#)2)#C )2)] i i 1 2 3 3 i/1 # 1 (k b !k b #< )2# 1 (k b !k b #< )2 4C 3 32 4C 1 13 2 23 1 3 31 2 # 1 3 (k b !k b #< )2*0. (3.6) 4C 2 21 1 12 3
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From the conditions of optimal stabilization, we suppose that Eq. (3.6) has a zero when < "<(0), and the i i optimal control parameters are <(0)"!2(A#I)) #k b !k b , 1 1 2 23 3 32 <(0)"!2(A#I)) #k b !k b , 2 2 3 31 1 13 (3.7) <(0)"2(B#I )) #k b !k b . 3 3 3 1 12 2 21 Substituting Eq. (3.7) into Eq. (3.2) we get a system of di!erential equations which, together with the system (3.3) form a closed-loop system of di!erential equations, with the special solution (2.12) to be asymptotic stable. To prove the above statement, we "rst consider the Lyapunov function (3.5). This function is a positive-de"nite quadratic form, and its time derivative, using Eqs. (3.3), (3.4) and (3.6), gives (3.8)
(4.1)
) ") ") "0, d "0, b "0 (iOj, i, j"1, 2, 3), (4.2) 1 2 3 ii ij manifold (4.1) does not contain any complete trajectory, except that given by Eq. (4.2). This can be done when we make a suitable choice of the constants k in Eq. (3.2). This proves the asymptotic stability [14]. When i ) ") ") "0 then Eqs. (3.3), (3.4) together with Eq. (3.6) reduce to 1 2 3 k b "k b , k b "k b , 1 12 2 21 2 23 3 32 k b "k b , dQ "0, bQ "0. (4.3) 3 31 1 13 ii ij From relations (2.4), using Eq. (4.3), we have d2 #b2 #b2 #2d "0, 13 11 12 11 l2b2 #d2 #b2 #2d "0, 23 22 22 1 12 l2b2 #b2 #d2 #2d "0, 33 33 23 2 13 b (1#l #d #l d )#b b "0, 12 1 22 1 11 13 23 b (1#l #d #l d )#l b b "0, 13 2 33 2 11 3 12 23 b (1#l #d #l d )#l l b b "0, (4.4) 23 3 33 3 22 1 2 12 13 where l "k /k , l "k /k , l "k /k , l "l l . 1 1 2 2 1 3 3 2 3 2 13 Since, every element of the direction cosines matrix is equal to its algebraic complement, and using Eq. (4.3), we get b (1#l #l d )"l b b , 12 1 1 33 2 13 23 b (1#l #l d )"l b b , 13 2 2 22 2 12 23 b (l #l #l d )"l l b b . 23 1 2 2 11 1 2 12 13
(4.5)
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Substituting Eq. (4.5) into the latest three Eqs. (4.4), it follows that b (l l d #l d #l d #l l #l #l #1)"0, 12 1 2 11 2 22 1 33 12 1 2 b (l l d #l d #l d #l l #l #l #1)"0, 13 1 2 11 2 22 1 33 12 1 2 b (l l d #l d #l d #l l #l2#l #1)"0. (4.6) 23 1 2 11 2 22 1 33 12 1 2 The quantities l '0 and l '0 may be chosen to ensure that the "rst and second equations of Eq. (4.6) will 1 2 be satis"ed only for b "b "0. We will study in detail the equation 13 12 l l d #l d #l d #l l #l #l #1"0, (4.7) 1 2 11 2 22 1 33 12 1 2 which represents a plane in the space d (i"1, 2, 3) at a distance d from the origin, where ii (1#l )(1#l ) 1 2 . (4.8) d" Jl2#l2#l2l2 12 2 1 Hence, !(1#l )2$Jd2l2(1#l2)[((1#l )2/l2)#((1#l2)2/(1#l2))!d2] 1 1 1 1 1 1 1 . (4.9) l " 2 (1#l )2!d2(1#l2) 1 1 Since d (i"1, 2 ,3) vary in such a way that !2)d )0, then when d2*12, the points of plane (4.7) do not ii ii approach the initial point of coordinates. Furthermore, if it is assumed to be l determined from the condition 1 (1#l )2 (1#l2)2 1 "d2'12 (0(l (1), 1 # (4.10) 1 (1#l2) l2 1 1 then from condition (4.9) l must satisfy that 2 !(1#l2) 1 l " '0. (4.11) 2 (1#l )2!d2(1#l2) 1 1 It is easy to verify that, the "rst two equations of Eq. (4.6) will be satis"ed only when b "b "0, so that 12 13 the "rst equation of Eq. (4.4) becomes d2 #2d "0Nd "0, or d "!2. From the third equation of 11 11 11 11 Eq. (4.5) it follows that either b (l #l )"0 or b (l !l )"0. In both cases b "0. Thus, d "0 or !2 23 1 2 23 1 2 23 11 for l Ol . Also, from the second and third equations of Eq. (4.4), one may obtain from d and d the set of 1 2 22 33 values: d "0 or !2 and d "0 or !2. Now, if the initial values of the perturbed motion lie in the region 22 33 d2 #d2 #d2 (4, then the manifold
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5. Stabilizing control moments The main object of attaching the rotors (R) to the body (platform P) is to control its motion using the control moments [16]. This control will be satis"ed by regulating the motion of the rotors. The control moments which must be applied to the rotors in order to impose the asymptotic optimal stabilization of the rotational motion (4.12) can be obtained using the relations between the control parameters < and the i control moments u given by i u "M2C) #k b !k b !Au) #u[h b #h b #(h #h(0))b ] 3 32 1 1 3 32 2 23 2 1 12 2 22 3 #h(0)[) b !) b ]Ncos ut#M2C) #k b !k b #Au) 3 3 32 2 33 2 1 13 3 31 1 !u[h b #h b #(h #h(0))b ]#h(0)[) b !) b Nsin ut, 1 11 2 21 3 3 31 3 1 33 3 31+ u "!M2C) #k b !k b !Au) #u[h b #h b #(h #h(0))b ] 3 32 2 1 3 32 2 23 2 1 12 2 22 3 #h(0)[) b !) b ]Nsin ut#M2C) #k b !k b #Au) 3 3 32 2 33 2 1 13 3 31 1 !u[h b #h b #(h #h(0))b ]#h(0)[) b !) b Ncos ut, 3 1 33 3 31+ 3 31 1 11 2 21 3 (5.1) u "C ) #k b !k b #h(0)[) b !) b ]. 3 2 31 1 32 3 3 3 2 21 1 12 It is clear that, they are non-linear functions of phase coordinates of the system. If these moments are applied to the rotors, then the rotational motion Eq. (4.12) of the rigid body will be asymptotically stable in the Lyapunov sense. Furthermore, the equilibrium position of the rigid body occurs when the principal axes of inertia of the rigid body coincide with the inertial axes, and can be obtained by setting /"/Q "u"0 in (4.12): u "u "u "0, 1 2 3 '0 "'0 "'0 "0, 1 2 3 1, i"j (i, j"1, 2, 3), a " ij 0, iOj
G
(5.2)
is asymptotically stable. Moreover, the stabilizing control moments of this position can be obtained from Eq. (5.1) also by setting /"/Q "u"0: u "Cu !k a #k a , 1 1 2 23 3 32 u "Cu !k a #k a , 2 2 3 31 1 13 u "C u !k a #k a . (5.3) 3 3 3 1 12 2 21 This result, as a special case of the considered problem, agrees with the result deduced by Saakian [17]. We see that moments (5.1) stabilize only motion (4.12). Except this motion for the controlling moments (5.1), all motions of the rigid body which are given by ) ") ") "0, 1 2 3 $1, i"j (i, j"1, 2, 3), b " ij 0, iOj
G
(5.4)
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are unstable in the Lyapunov sense. For example, the motion ) ") ") "0, 1 2 3 b "!b "b "!1, 11 22 33 b "0, iOj (i, j"1, 2, 3) (5.5) ij is unstable in the Lyapunov sense. The equations of the perturbed motion about the solution (5.5) are A)0 ") [h b #h d #h b ]!) [h b #h b #h d ] 1 3 1 12 2 22 3 32 2 1 13 2 23 3 33 #h ) #h ) !2C) #k b !k b , 2 3 3 2 1 2 23 3 32 A)0 ") [h b #h b #h d ]!) [h d #h b #h b ] 2 1 1 13 2 23 3 33 3 1 11 2 21 3 31 !h ) #h ) !2C) #k b !k b , 3 1 1 3 2 3 31 1 13 B)0 ") [h d #h b #h b ]!) [h b #h d #h b ] 3 2 1 11 2 21 3 32 1 1 12 2 22 3 32 !h ) !h ) !2C ) #k b !k b 1 2 2 1 3 3 1 12 2 21 dQ "b ) !b ) , 11 12 3 13 2 bQ "d ) !b ) #) , 21 22 3 32 2 3 bQ "b ) !d ) #) (123), 31 32 3 33 2 2 d "b #1, d "b !1, d "b #1. 11 11 22 22 33 33 To prove that motion (5.5) is unstable in the Lyapunov sense, we consider the following function:
(5.6)
2'"A()2#)2)#B)2!k (d #b #b )2#k (b #d #b )2!k (b #b #d )2, (5.7) 3 1 11 12 13 2 21 22 23 3 31 32 33 2 1 the time derivative of this function, using Eq. (5.6) is given by (5.8) '0"!2[(A#I)()2#)2)#(B#I ))2]. 2 3 3 1 Assuming that consistency of the given initial motion ) P0, (b , b , b )P(!1, 0, 0) i 11 12 13 (b , b , b )P(0, 1, 0) (b , b , b )P(0, 0, !1), such that '(0 at t"t '0; then the function 21 22 23 31 32 33 0 ' does not grow. Hence at t"t '0, there is ')' (' -is the value of ' at t"t ). On the other side, 0 0 0 0 )2P0 as tPR for any motion of the systems (3.4), (3.3) and (3.6), so that the rigid body attains one of the i motions (5.4) or (4.12). Then b P0 (iOj, i, j"1, 2, 3). If b P!1, b P1, b P!1 as tPR then it ij 11 22 33 follows from Eq. (5.7) that 'P0. But from the inequality ')' (0 at all motion, the quantity 0 k (d #b #b )2#k (b #b #d )2'0 does not approch to zero at tPR. This means that the 1 11 12 13 3 31 32 33 integral curve beginning in a small neighborhood of motion deserts some "xedly neighborhood with growth of time. Hence this motion is unstable in the Lyapunov sense. Thus, under the conditions k '0, i"1, 2, 3; k "k l , k "k l , l "l l , and the quantities l , l deteri 1 21 2 33 2 13 1 2 mine relations (4.10) and (4.12), then when the moments (5.1) e!ect on the rotors, the rotational motion (2.12) will be asymptotically stable, while any other motion will not.
6. Conclusions Under the conditions (4.10) and (4.11) the optimal stabilization of the rotational motion of a symmetrical rigid body with the help of internal rotors is studied. The asymptotic stability of this motion is proved by using Krasovskii and Barbachen's theorem. The optimal control moments are obtained as non-linear
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functions of phase coordinates ) , a (i"1, 2, 3). These moments are obtained from the condition to ensure i i optimal asymptotic stability of this motion. The equilibrium position occurs when the principal axes of inertia coincide with the inertial axes is proved to be asymptotically stable. The stabilizing control moments of this position are obtained as functions of phase coordinates ) , a (i"1, 2, 3). If the control moments (5.1) i i are applied to the rotors, then rotational motion given by Eq. (4.12) will be the only asymptotically stable, while the rest of the motions are given by Eq. (5.4) are not. Our study is characterized by that the non-linear equations of motion are used to obtain the control moments as non-linear functions of phase coordinates of the body.
References [1] E. Mortensen, On system for control automatic control of the rotation of a rigid body, Electronics Research Laboratory, Repeat, .f. 63-23. Univ. of California, 1963. [2] V. Krementulo, About optimal stabilization of rotational motion of rigid body with "xed point with the help of the rotors, J. Appl. Math. Mech. 1 (1966) 42}50. [3] V. Krementulo, E. Sokolova, On optimal stabilization of rotational motion rigid body controllable-gyro, Izv. Akad. Nauk. SSSR MMT 5 (1969) 3}9. [4] V. Krementulo, On the optimal stabilization of rotational motion of gyrostat in Newtonian force "eld, J. Appl. Math. Mech. 5 (1970) 965}972. [5] V. Vorotninkov, The control of the angular motion of a rigid body, Izv. Akad. Nauk. SSSR MMT 6 (1986) 38}43. [6] S.A. Agafonov, K.B. Alekseyev, N.V. Nikolayev, Stability of the control of a plane turn of a spacecraft, Prk. Math. Mech. 1 (1991) 136}138. [7] D. Lebedev, Control of the motion of a rigid body rotating about its center mass, J. Appl. Math. Mech. 1 (1993) 13}18. [8] V. Vorotninkov, The control of the angular motion of a rigid body with interference a game-theoretic approach, J. Appl. Math. Mech. 3 (1994) 457}476. [9] M. Kovalev, The controllability of dynamical system with respect to part of variables, J. Appl. Math. Mech. 6 (1993) 995}1004. [10] I.A. Balaeva, On the problem of control in the motion of rigid body in the absence of information about angular velocity, Izv. Akad. Nauk. SSSR MMT 4 (1973) 15}21. [11] G. Sanchez de alvarez, Stabilization of rigid body dynamics by internal and external torques, Int. Fedration Aut. Control 4 (1992) 745}755. [12] A. El-Gohary, On the stability of an equilibrium position and rotational motion of a gyrostat, J. Mech. Res. Commun. 24 (11) (1997) 457}462. [13] V. Ryumantsev, On the control and stabilization of systems with ignorable coordinates, J. Appl. Math. Mech. 6 (1972) 966}976. [14] A. Barbachen, N. Krasovskii, About stability of motion in total, Dokl. Akad. Nauk. SSSR 86 (3) (1952) 453}456. [15] V. Krementulo, Stabilization of Stationary Motion of a Rigid Body, Nauk, Moscow, 1977. [16] H. Beghin, ED tude TheH orique des Compas Gyrostatiques Anchutz et Sperry, Paris imprimerie Nationle, 1921. [17] S. Saakian, Optimal stabilization of equilibrium position of rigid body with the help of rotors, Akad. Nauk. Armeni. SSR. 6 (1985) 35}41. [18] A.I. Lure'ye, Analytical Mechanics, Fizmatgiz, Moscow, 1961. [19] N. Krasovskii, Problems of stabilization of controlled motion in the book of I.G. Malkin, Theory of Stability of Motion, Nauka Press, Moscow, 1966.
Optimal stabilization of the rotational motion of a rigid body with the help of rotorsq Awad El-Gohary Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Received 12 December 1998; accepted 29 March 1999
Abstract Optimal stabilization of the rotational motion of a symmetrical rigid body with the help of internal rotors is studied. In such a study the asymptotic stability of this motion is proved by using Barbachen and Krasovskii theorem. The optimal control moments which stabilize this motion are obtained using the conditions of ensuring the optimal asymptotic stability as non-linear functions of phase coordinates of the system. These moments stabilize asymptotically one type of the rotational motions of the rigid body. The other rotational motions are unstable in the Lyapunov sense. As a particular case of our problem, the equilibrium position of the rigid body, which occurs when the principal axes of inertia coincide with the inertial axes, is proved to be asymptotically stable. This study is characterized by that non-linear equations of motion which are used to prove the asymptotic stability and derivation of the control moments. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Optimal; Stabilization; Rigid body
1. Introduction The problems of optimal stabilization of the rotational motion of a rigid body and system of rigid bodies are very important for numerous applications such as dynamics of satellite-gyrostat, spacecraft, aircraft, robotics, biomechanics and the like. The problem of controlling the motion of a rigid body is processed by either external or internal torques. Mortensen [1] studied the problem of controlling the rotation of a free rigid body with the help of external torques by using the second Lyapunov method and Pontryagin principle. Krementulo [2] used an approximation method to study the optimal stabilization of the rotational motion of a rigid body with the help of internal rotors. Krementulo and Sokolova [3] studied the control of the motion of a rigid body using internal torques. Krementulo [4] solved the problem of optimal stabilization of the rotational motion of a gyrostat with the help of rotors. Vorotinikov [5] studied the control of the angular motion of a rigid body. He has also obtained the control parameters su$cient to ensure exponential asymptotic stability of the permanent rotation of the rigid body. Agafanov et al. [6] have considered the
q Contributed by W.F. Ames. E-mail address: [email protected] (A. El-Gohary)
0020-7462/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 9 ) 0 0 0 2 5 - 6
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control of the plane turn of the spacecraft using external forces. The problem of controlling the spherical motion of a rigid body, when the external controlling moments do not contain x-component and their axes do not coincide with the principal axes of inertia of the body, has been investigated by Lebedev [7]. He also discussed the control of the uniform rotation of the body about the principal axes of inertia and its orientation in inertial space. Vorotinikov [8] analysed the problem of controlling the reorientation of a symmetrical rigid body using a game-theoretic model of interference. Also he has obtained the non-linear control rules, which ensure stability of the body at the prescribed state in "nite time. Kovalev [9] obtained the necessary and su$cient conditions for partial control of linear autonomous system. Further he has studied the control of the rotational motion of a rigid body using a single rotor. Balaeva [10] examined the control of the motion of a rigid body in the absence of information about the angular velocity of the body. Sanchez de Alvarez [11] discussed stabilization of a rigid body dynamics using internal and external torques. El-Gohary [12] obtained the control moments su$cient to ensure asymptotic stability of the equilibrium position and the rotational motion of a gyrostat, using the Lyapunov function. Rymuntsev [13] has studied the problem of controlling and stabilization with respect to positional coordinates and impulse of the holonomic systems with ignorable coordinates with the help of controlling forces. The main object of this paper is to study the optimal stabilization of the rotational motion of a symmetrical rigid body, with the help of rotors using Barbachen and Krasovskii's theorem [14]. The optimal control moments, which ensure asymptotic stability of this motion, are obtained as functions of phase coordinates of the body. The obtained control moments stabilize one kind of the rotational motions of the rigid body. On the other hand, the other cases of the described rotational motion are proved to be unstable in the Lyapunov sense. As a special case of the studied problem, (i) it can be veri"ed that the equilibrium position of the rigid body is asymptotically stable; (ii) the optimal control moments ensure that they are obtained. In our study, the non-linear equations of motion are used to prove the asymptotic stability of the rotational motion and the derivation of the control moments as the non-linear functions of phase coordinates of the system.
2. Equations of motion Consider a mechanical system S comprised of a rigid body P (platform) with a "xed point O its centre of mass and a system of three symmetrical rotors R are attached to the principal axes of inertia of the system S, in such a way that the motion of the rotors does not modify the distribution of the masses of the system S. We introduce two coordinate systems (Fig. 1), Omgf be inertial coordinate system, and Ox x x , coordinate 1 2 3 system moving with the body axes coincident with the principal axes of inertia of the system S.u and i C , i"1, 2, 3 are the components of the angular velocity and principal moments of inertia of the body relative i to principal axes Ox x x , respectively. I , and ' , i"1, 2, 3 are the axial moments of inertia and the angles of 1 2 3 i i rotation of the rotors relative to the principal axes of inertia of the system. Equations of motion of the mechanical system can be written in the form of Krementulo [15,18]. C u5 #(C !C )u u #I 'G #I '0 u !I '0 u "0, 1 1 3 2 2 3 1 1 3 3 2 2 2 3 C u5 #(C !C )u u #I 'G #I '0 u !I '0 u "0, 2 2 1 3 1 3 2 2 1 1 3 3 3 1 C u5 #(C !C )u u #I 'G #I '0 u !I '0 u "0. 3 3 2 1 1 2 3 3 2 2 1 1 1 2 The equations of the relative motion of the rotors without internal friction are I (u5 #'G )"u , 1 1 1 1 I (u5 #'G )"u , 2 2 2 2 I (u5 #'G )"u , 3 3 3 3
(2.1)
(2.2)
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395
Fig. 1.
where u (i"1, 2, 3) are the control moments applied to the rotors, and dot means di!erentiation with respect i to time. These moments will be determined from the condition to ensure asymptotic stability of the rotational motion of the body. The Poisson kinematical equations for angular orientation of the rigid body in the inertial space are a5 "u a !u a , i1 3 i2 2 i3 a5 "u a !u a , i2 1 i3 3 i1 a5 "u a !u a , (i"1, 2, 3), i3 2 i1 1 i2
(2.3)
a (i, j"1, 2, 3) are the direction cosines between Omgf and the principal axes of inertia Ox x x , and satisfy ij 1 2 3 the geometrical condition
G
1, k"l 3 + a a " (k, l"1, 2, 3). ki li 0, kOl i/1
(2.4)
Consider a-symmetrical rigid body (C "C "C). Replace the coordinate system Ox (i"1, 2, 3) by a new 1 2 i semimoving coordinates system Oy (i"1, 2, 3) where semimoving coordinate axes Oy coincides with Ox , i 3 3 the axes Oy and Oy lie in the Ox x plane and are not involved in the intrinsic rotation of the body with 1 2 1 2 respect to the angle /. The direction cosines between Oy y y and the inertial axes Omgf are b (i, j"1, 2, 3). 1 2 3 ij Replacing u , u and u by the new variables ) , ) and ) which represent the components of instan1 2 3 1 2 3 taneous angular velocity of the trihedron Oy y y onto its axes Krementulo [15] (Fig. 2): it follows that 1 2 3 u ") cos /#) sin /, 1 1 2 u "!) sin /#) cos /, 2 1 2 u ") #/Q . 3 3
(2.5)
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Fig. 2.
The relations between direction cosines a and b are ij ij a "b cos /#b sin /, i1 i1 i2 a "!b sin /#b cos /, i2 i1 i2 a "b . i3 i3 Using transformation (2.5), the equations of motion (2.1) takes the following form: C)0 #(C !C)) ) #C ) /Q #g5 #g ) !g ) "0, 1 3 2 3 3 2 1 3 2 2 3 C)0 #(C!C )) ) !C ) /Q #gR #g ) !g ) "0, 2 3 1 3 3 1 2 1 3 3 1 C ()0 #/$ )#gR #g ) !g ) "0, 3 3 3 2 1 1 2 where g "I('0 cos /!'0 sin /), 1 1 2 g "I('0 sin /#'0 cos /), 2 1 2 g "I '0 , I "I "I. 2 3 3 3 1 The equations of the rotor motion becomes g5 #I)0 #(g #I) )/Q "w , 1 2 2 1 1 g5 #I)0 #(g #I) )/Q "w , 2 2 1 1 2 g5 #I (PQ #/$ )"w , 3 3 3 3 where w "u cos /!u sin /, 1 1 2 w "u sin /#u cos /, 2 1 2 w "u . 3 3
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
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397
The Poisson kinematical equations in the new variables takes the following form: bQ ") b !) b , i1 3 i2 2 i3 bQ ") b !) b , i2 1 i3 3 i1 (i"1, 2, 3). bQ ") b !) b i3 2 i1 1 i2
(2.11)
Eqs. (2.7) and (2.11) allow the stationary motion ) ") ") "0, w "w "w "0, 1 2 3 1 2 3 g "g "0, g "g(0)"const., /Q "u, 3 1 2 3 1, i"j (i, j"1, 2, 3), b " ij 0, iOj.
G
(2.12)
This special solution of Eqs. (2.7) and (2.11) represents an uniform rotation of the rigid body with the angular velocity /Q about the axis Of(Ox ). In such rotational motion, the rotors attached to the axes Ox and Ox are 3 1 2 at rest while the rotor attached to Ox rotates with an uniform angular velocity. 3 To eliminate the kinetic moments g (i"1, 2, 3) of the rotors from Eqs. (2.7) we will use the conservation i law of the kinetic moment of system (S) in the inertial axes Krementulo [15] (C) #g )b #(C) #g )b #[C () #/Q )#g ]b "h , 1 1 i1 2 2 i2 3 3 3 i3 i
(2.13)
where h are constants. i Substituting Eqs. (2.13) and (2.9) into Eq. (2.7), we get 3 3 (C!I))0 "!(C!I)u) #() #u) + h b ! ) + h b !w , 1 2 3 i i2 2 i i3 1 i/1 i/1 3 3 (C!I))0 "(C!I)) !() #)) + h b #) + h b !w , i i3 2 2 1 3 i i1 1 i/1 i/1 3 3 (C !I ))0 ") + h b !) + h b !w . 2 i i1 1 i i2 3 3 3 3 i/1 i/1
(2.14)
Using Eqs. (2.13) and (2.12), we "nd that h(0)"h(0)"0, 2 1
h(0)"C u#g(0) 3 3 3
(2.15)
for the stationary motion (2.12).
3. The stability problem To obtain the equations of the perturbed motion, we consider the stationary motion (2.12) to be the unperturbed motion, and for the perturbed motion we take new variables ) ") , d "b !1, b "b (i, j"1, 2, 3). i i ii ii ij ij
(3.1)
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Substituting Eq. (3.1) into Eqs. (2.7) and (2.11) the equations of the perturbed motion about stationary motion are A)0 ") [h b #h (d #1)#h b ]! ) [h b #h b #h (d #1)]#< , 1 3 1 12 2 22 3 32 2 1 13 2 23 3 33 1 0 A) ") [h b #h b #h (d #1)]! ) [h (d #1)#h b #h b ]#< , 2 1 1 13 2 23 3 33 3 1 11 2 21 3 31 2 0 B) ") [h (d #1)#h b #h b ]! ) [h b #h (d #1)#h b ]#< , 3 2 1 11 2 21 3 31 1 1 12 2 22 3 32 3 < "u[h b #h (d #1)#(h(0)#h )b ]# h(0)[) b !) (d #1)]!Au) !w , 1 1 12 2 22 3 3 32 3 3 32 2 33 2 1 < "!u[h (d #1)#h b #(h(0)#h )b ]# h(0)[) b !) (d #1)]#Au) !w , 3 3 31 1 33 1 1 3 3 31 2 1 11 2 21 < "h(0)[) b !) b ]!w , 3 2 31 1 32 3 3 A"C!I, B"C !I , (3.2) 3 3 dQ "b ) !b ) , 11 12 3 13 2 bQ "d ) !b ) #) , 21 22 3 23 2 3 bQ "b ) !d ) !) , (123). (3.3) 31 32 3 33 2 2 In what follows, we shall derive the stabilizing control parameters < (i"1, 2, 3) using a new developed i technique. In such a technique, the requirements to ensure optimal asymptotic stability of motion (2.12) are adopted. We formulate the problem of optimal stabilization of the rotational motion in the following manner: to choose the control parameters as functions of phase coordinates describing the motion of the rigid body about the "xed point O, so that the stationary motion (2.12) of system (3.2), (3.3) be asymptotically stable with respect to these coordinates and the functional
P
=
MC()2#)2)#C )2# 1 [k b !k b #< ]2# 1 [k b !k b #< ]2 1 2 3 3 4C 3 32 2 23 1 4C 1 13 3 31 2 0 # 1 3[k b !k b #< ]2N dt, (3.4) 4C 2 21 1 12 3 be minimum, such that the motion is attenuated resulting in evaluating the controlling parameters < (i"1, 2, 3). The constants k '0 (i"1, 2, 3) are positive. i i We select the optimal Lyapunov function in the form J"
2<"A()2#)2)#B)2#k (d2 #b2 #b2 )#k (b2 #d2 #b2 )#k (b2 #b2 #d2 ), 1 2 3 1 11 12 13 2 21 22 23 3 31 32 33 A"C!I'0, B"C !I '0. 3 3 Using the Krasovskii's theorem (1966), we have [19]
(3.5)
B[<; ) , ) , ) ; d , b ,2, d ; < , < , < ]"![) (k b !k b )#< )#) (k b , 1 2 3 11 12 33 1 2 3 1 2 23 3 32 1 2 3 31 !k b #< )#) (k b !k b )#< ) 1 13 2 3 1 12 2 21 3 3 ! + ) < ]#[C()2#)2)#C )2)] i i 1 2 3 3 i/1 # 1 (k b !k b #< )2# 1 (k b !k b #< )2 4C 3 32 4C 1 13 2 23 1 3 31 2 # 1 3 (k b !k b #< )2*0. (3.6) 4C 2 21 1 12 3
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399
From the conditions of optimal stabilization, we suppose that Eq. (3.6) has a zero when < "<(0), and the i i optimal control parameters are <(0)"!2(A#I)) #k b !k b , 1 1 2 23 3 32 <(0)"!2(A#I)) #k b !k b , 2 2 3 31 1 13 (3.7) <(0)"2(B#I )) #k b !k b . 3 3 3 1 12 2 21 Substituting Eq. (3.7) into Eq. (3.2) we get a system of di!erential equations which, together with the system (3.3) form a closed-loop system of di!erential equations, with the special solution (2.12) to be asymptotic stable. To prove the above statement, we "rst consider the Lyapunov function (3.5). This function is a positive-de"nite quadratic form, and its time derivative, using Eqs. (3.3), (3.4) and (3.6), gives (3.8)
(4.1)
) ") ") "0, d "0, b "0 (iOj, i, j"1, 2, 3), (4.2) 1 2 3 ii ij manifold (4.1) does not contain any complete trajectory, except that given by Eq. (4.2). This can be done when we make a suitable choice of the constants k in Eq. (3.2). This proves the asymptotic stability [14]. When i ) ") ") "0 then Eqs. (3.3), (3.4) together with Eq. (3.6) reduce to 1 2 3 k b "k b , k b "k b , 1 12 2 21 2 23 3 32 k b "k b , dQ "0, bQ "0. (4.3) 3 31 1 13 ii ij From relations (2.4), using Eq. (4.3), we have d2 #b2 #b2 #2d "0, 13 11 12 11 l2b2 #d2 #b2 #2d "0, 23 22 22 1 12 l2b2 #b2 #d2 #2d "0, 33 33 23 2 13 b (1#l #d #l d )#b b "0, 12 1 22 1 11 13 23 b (1#l #d #l d )#l b b "0, 13 2 33 2 11 3 12 23 b (1#l #d #l d )#l l b b "0, (4.4) 23 3 33 3 22 1 2 12 13 where l "k /k , l "k /k , l "k /k , l "l l . 1 1 2 2 1 3 3 2 3 2 13 Since, every element of the direction cosines matrix is equal to its algebraic complement, and using Eq. (4.3), we get b (1#l #l d )"l b b , 12 1 1 33 2 13 23 b (1#l #l d )"l b b , 13 2 2 22 2 12 23 b (l #l #l d )"l l b b . 23 1 2 2 11 1 2 12 13
(4.5)
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Substituting Eq. (4.5) into the latest three Eqs. (4.4), it follows that b (l l d #l d #l d #l l #l #l #1)"0, 12 1 2 11 2 22 1 33 12 1 2 b (l l d #l d #l d #l l #l #l #1)"0, 13 1 2 11 2 22 1 33 12 1 2 b (l l d #l d #l d #l l #l2#l #1)"0. (4.6) 23 1 2 11 2 22 1 33 12 1 2 The quantities l '0 and l '0 may be chosen to ensure that the "rst and second equations of Eq. (4.6) will 1 2 be satis"ed only for b "b "0. We will study in detail the equation 13 12 l l d #l d #l d #l l #l #l #1"0, (4.7) 1 2 11 2 22 1 33 12 1 2 which represents a plane in the space d (i"1, 2, 3) at a distance d from the origin, where ii (1#l )(1#l ) 1 2 . (4.8) d" Jl2#l2#l2l2 12 2 1 Hence, !(1#l )2$Jd2l2(1#l2)[((1#l )2/l2)#((1#l2)2/(1#l2))!d2] 1 1 1 1 1 1 1 . (4.9) l " 2 (1#l )2!d2(1#l2) 1 1 Since d (i"1, 2 ,3) vary in such a way that !2)d )0, then when d2*12, the points of plane (4.7) do not ii ii approach the initial point of coordinates. Furthermore, if it is assumed to be l determined from the condition 1 (1#l )2 (1#l2)2 1 "d2'12 (0(l (1), 1 # (4.10) 1 (1#l2) l2 1 1 then from condition (4.9) l must satisfy that 2 !(1#l2) 1 l " '0. (4.11) 2 (1#l )2!d2(1#l2) 1 1 It is easy to verify that, the "rst two equations of Eq. (4.6) will be satis"ed only when b "b "0, so that 12 13 the "rst equation of Eq. (4.4) becomes d2 #2d "0Nd "0, or d "!2. From the third equation of 11 11 11 11 Eq. (4.5) it follows that either b (l #l )"0 or b (l !l )"0. In both cases b "0. Thus, d "0 or !2 23 1 2 23 1 2 23 11 for l Ol . Also, from the second and third equations of Eq. (4.4), one may obtain from d and d the set of 1 2 22 33 values: d "0 or !2 and d "0 or !2. Now, if the initial values of the perturbed motion lie in the region 22 33 d2 #d2 #d2 (4, then the manifold
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5. Stabilizing control moments The main object of attaching the rotors (R) to the body (platform P) is to control its motion using the control moments [16]. This control will be satis"ed by regulating the motion of the rotors. The control moments which must be applied to the rotors in order to impose the asymptotic optimal stabilization of the rotational motion (4.12) can be obtained using the relations between the control parameters < and the i control moments u given by i u "M2C) #k b !k b !Au) #u[h b #h b #(h #h(0))b ] 3 32 1 1 3 32 2 23 2 1 12 2 22 3 #h(0)[) b !) b ]Ncos ut#M2C) #k b !k b #Au) 3 3 32 2 33 2 1 13 3 31 1 !u[h b #h b #(h #h(0))b ]#h(0)[) b !) b Nsin ut, 1 11 2 21 3 3 31 3 1 33 3 31+ u "!M2C) #k b !k b !Au) #u[h b #h b #(h #h(0))b ] 3 32 2 1 3 32 2 23 2 1 12 2 22 3 #h(0)[) b !) b ]Nsin ut#M2C) #k b !k b #Au) 3 3 32 2 33 2 1 13 3 31 1 !u[h b #h b #(h #h(0))b ]#h(0)[) b !) b Ncos ut, 3 1 33 3 31+ 3 31 1 11 2 21 3 (5.1) u "C ) #k b !k b #h(0)[) b !) b ]. 3 2 31 1 32 3 3 3 2 21 1 12 It is clear that, they are non-linear functions of phase coordinates of the system. If these moments are applied to the rotors, then the rotational motion Eq. (4.12) of the rigid body will be asymptotically stable in the Lyapunov sense. Furthermore, the equilibrium position of the rigid body occurs when the principal axes of inertia of the rigid body coincide with the inertial axes, and can be obtained by setting /"/Q "u"0 in (4.12): u "u "u "0, 1 2 3 '0 "'0 "'0 "0, 1 2 3 1, i"j (i, j"1, 2, 3), a " ij 0, iOj
G
(5.2)
is asymptotically stable. Moreover, the stabilizing control moments of this position can be obtained from Eq. (5.1) also by setting /"/Q "u"0: u "Cu !k a #k a , 1 1 2 23 3 32 u "Cu !k a #k a , 2 2 3 31 1 13 u "C u !k a #k a . (5.3) 3 3 3 1 12 2 21 This result, as a special case of the considered problem, agrees with the result deduced by Saakian [17]. We see that moments (5.1) stabilize only motion (4.12). Except this motion for the controlling moments (5.1), all motions of the rigid body which are given by ) ") ") "0, 1 2 3 $1, i"j (i, j"1, 2, 3), b " ij 0, iOj
G
(5.4)
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are unstable in the Lyapunov sense. For example, the motion ) ") ") "0, 1 2 3 b "!b "b "!1, 11 22 33 b "0, iOj (i, j"1, 2, 3) (5.5) ij is unstable in the Lyapunov sense. The equations of the perturbed motion about the solution (5.5) are A)0 ") [h b #h d #h b ]!) [h b #h b #h d ] 1 3 1 12 2 22 3 32 2 1 13 2 23 3 33 #h ) #h ) !2C) #k b !k b , 2 3 3 2 1 2 23 3 32 A)0 ") [h b #h b #h d ]!) [h d #h b #h b ] 2 1 1 13 2 23 3 33 3 1 11 2 21 3 31 !h ) #h ) !2C) #k b !k b , 3 1 1 3 2 3 31 1 13 B)0 ") [h d #h b #h b ]!) [h b #h d #h b ] 3 2 1 11 2 21 3 32 1 1 12 2 22 3 32 !h ) !h ) !2C ) #k b !k b 1 2 2 1 3 3 1 12 2 21 dQ "b ) !b ) , 11 12 3 13 2 bQ "d ) !b ) #) , 21 22 3 32 2 3 bQ "b ) !d ) #) (123), 31 32 3 33 2 2 d "b #1, d "b !1, d "b #1. 11 11 22 22 33 33 To prove that motion (5.5) is unstable in the Lyapunov sense, we consider the following function:
(5.6)
2'"A()2#)2)#B)2!k (d #b #b )2#k (b #d #b )2!k (b #b #d )2, (5.7) 3 1 11 12 13 2 21 22 23 3 31 32 33 2 1 the time derivative of this function, using Eq. (5.6) is given by (5.8) '0"!2[(A#I)()2#)2)#(B#I ))2]. 2 3 3 1 Assuming that consistency of the given initial motion ) P0, (b , b , b )P(!1, 0, 0) i 11 12 13 (b , b , b )P(0, 1, 0) (b , b , b )P(0, 0, !1), such that '(0 at t"t '0; then the function 21 22 23 31 32 33 0 ' does not grow. Hence at t"t '0, there is ')' (' -is the value of ' at t"t ). On the other side, 0 0 0 0 )2P0 as tPR for any motion of the systems (3.4), (3.3) and (3.6), so that the rigid body attains one of the i motions (5.4) or (4.12). Then b P0 (iOj, i, j"1, 2, 3). If b P!1, b P1, b P!1 as tPR then it ij 11 22 33 follows from Eq. (5.7) that 'P0. But from the inequality ')' (0 at all motion, the quantity 0 k (d #b #b )2#k (b #b #d )2'0 does not approch to zero at tPR. This means that the 1 11 12 13 3 31 32 33 integral curve beginning in a small neighborhood of motion deserts some "xedly neighborhood with growth of time. Hence this motion is unstable in the Lyapunov sense. Thus, under the conditions k '0, i"1, 2, 3; k "k l , k "k l , l "l l , and the quantities l , l deteri 1 21 2 33 2 13 1 2 mine relations (4.10) and (4.12), then when the moments (5.1) e!ect on the rotors, the rotational motion (2.12) will be asymptotically stable, while any other motion will not.
6. Conclusions Under the conditions (4.10) and (4.11) the optimal stabilization of the rotational motion of a symmetrical rigid body with the help of internal rotors is studied. The asymptotic stability of this motion is proved by using Krasovskii and Barbachen's theorem. The optimal control moments are obtained as non-linear
A. El-Gohary / International Journal of Non-Linear Mechanics 35 (2000) 393}403
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functions of phase coordinates ) , a (i"1, 2, 3). These moments are obtained from the condition to ensure i i optimal asymptotic stability of this motion. The equilibrium position occurs when the principal axes of inertia coincide with the inertial axes is proved to be asymptotically stable. The stabilizing control moments of this position are obtained as functions of phase coordinates ) , a (i"1, 2, 3). If the control moments (5.1) i i are applied to the rotors, then rotational motion given by Eq. (4.12) will be the only asymptotically stable, while the rest of the motions are given by Eq. (5.4) are not. Our study is characterized by that the non-linear equations of motion are used to obtain the control moments as non-linear functions of phase coordinates of the body.
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