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New control laws for stabilization of a rigid body motion using rotors system
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 33 (2006) 818–829 www.elsevier.com/locate/mechrescom
New control laws for stabilization of a rigid body motion using rotors system Awad El-Gohary
*
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Available online 27 April 2006
Abstract This paper presents a new class of globally asymptotic stabilizing control laws for dynamics and kinematics attitude motion of a rotating rigid body. The rigid body motion is controlled with the help of a rotor system with internal friction. The Lyapunov technique is used to prove the global asymptotic properties of the stabilizing control laws. The obtained control laws are given as functions of the angular velocity, Cayley–Rodrigues and Modified-Rodrigues parameters. It is shown that linearity and nonlinearity of the control laws depend not only upon the Lyapunov function structure but also the rotors friction. Moreover, some of the results are compared with these obtained in the literature by other methods. Numerical simulation is introduced. 2006 Published by Elsevier Ltd. Keywords: Rigid body; Rotors system with friction; Global stabilization; Cayley–Rodrigues and Modified-Rodrigues parameters
1. Introduction In recent years a considerable effort has been devoted to the design of control laws for challenging dynamical systems, such as robot manipulators, high-performance rigid spacecraft, satellite and space vehicles. The problem of description and control of the attitude motion of rigid bodies is investigated in Tsiotras (1996, 1994) and Rotea (1998). A lot of literatures use Euler angles for the orientation of a rigid body and the other use the direction-cosine (see, for example, El-Gohary, 2005b,c, 2001b; Krementulo, 1977). Unfortunately, neither the Euler angles nor the direction-cosine is the optimal using for the kinematic attitude of a rigid body motion. Since the Euler angles have singularities in the 3–1–3 Euler angles when h = 0 or p. If instead we had used 3–1–2 Euler angles, these would have been well behaved at h = 0 or p but would have shown singular behavior when h = ±p/2 (see Moore, 1994). Because of the singularity difficulty of the Euler angles and large number of singularities of
* Present address: Department of Statistics and OR, Faculty of Science King Saud University, P.O. Box 2455, Riyadh, Saudi Arabia. Fax: +9 661 4676274. E-mail addresses: [email protected], [email protected]
0093-6413/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.mechrescom.2006.03.008
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elements in the direction-cosine matrix, we will use another representation of a rigid body kinematics attitude such as Euler parameters, Cayley–Rodrigues and Modified-Rodrigues parameters. The results in this paper complement and extend similar results published recently in terms of Euler parameters kinematic parameterizations (El-Gohary, 2002, 2001a, 1996, 2005a; Tsiotras, 1994). Euler parameters, Cayley–Rodrigues and Modified-Rodrigues parameters, can be employed to prove global asymptotic stability of a rigid body motion without imposing conditions on the control gains and system parameters. This motivated us to use these parameters for obtaining a new class of control laws that globally asymptotically stabilize a rigid body motion about its equilibrium state. Euler parameters were employed in El-Gohary (2002, 2001a) to study the global asymptotic stabilization of a rigid body rotational motion with the help of rotor and moving masses systems respectively, using Lyapunov technique. Moreover, these parameters also were employed in the paper (El-Gohary, 1996, 2000) for describing the rigid body programmed motion and proving the global asymptotic stabilization of the rigid body rotational motion. Also, Euler parameters are used in El-Gohary (1997) to study the global asymptotic stabilization of the relative programmed motion of a satellite-gyrostat. The Modified-Rodrigues parameters as a stereographic projections of Euler parameters are used in El-Gohary (2005c) to study the problem of global optimal stabilization of a rigid body rotational motion about the equilibrium state. The problem of controlling the rotational motion of a rigid body using three independent control torques is introduced in Tsiotras (1994) and Rotea (1998). The current paper is organized as follows. Section 2 of the paper introduces the equations of motion of a rigid body carrying a rotor system taking into consideration the internal rotors friction. Section 3 contains some important first integral of both dynamical and kinematical equations. The asymptotic stability feedback control law is obtained as function of angular velocity and Euler parameters in Section 4. The main results of this paper are concentrated in Sections 5 and 6, since both of Cayley–Rodrigues and Modified-Rodrigues parameters are employed to study the problem of global stabilization of a rigid body motion about its equilibrium state with linear and nonlinear feedback control input using Lyapunov technique. 2. Formulation the problem and equations of motion Consider a mechanical system S which consists of a rigid body P (platform) whose center of mass with a fixed point O and 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 mass of the system S. To describe the motion of the system, two systems of coordinates are introduced. The first Ongf is an inertial system and the second Ox1x2x3 is fixed in the body and coincides with the principal axes of the body at O. ~ and the principal Let xi and Ci (i = 1, 2, 3) be the components of the rigid body angular velocity vector x moments of inertia of the system relative to the principal axes Ox1x2x3, respectively. The resulting system dynamics are given by ~_ þ x ~ ¼~ ~G G L;
ð2:1Þ
~ and ~ where G L are the system total angular momentum and the external moments vectors referring to the body fixed frame Ox1x2x3. The components of the system total angular momentum referring to the body principal axes Ox1x2x2 are given by Gi ¼ C i xi þ I i U_ i
ði ¼ 1; 2; 3Þ;
where Ii and Ui (i = 1, 2, 3) are the ith rotor axial moment of inertia and the ith rotor angle of rotation referring to the body axes Ox1x2x2. Eq. (2.1) should be augmented by the equations the rotors motion relative to the body coordinates system Ox1x2x2. That is € 1 Þ ¼ u1 l1 U_ 1 I 1 ðx_ 1 þ U
ð123Þ;
ð2:2Þ
where the symbol (123) means that the two other equations can be obtained from the given equation by cyclic permutations of the indices 1 ! 2 ! 3, the dot denotes the differentiation with respect to time, ui (i = 1, 2, 3)
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are the control moments applied to the rotors and developed by electric motors that rigidly mounted on the body and the terms li U_ i (i = 1, 2, 3) denote the frictional moment due to the ith rotor friction. Note that the coefficients li (i = 1, 2, 3) depend upon various factors such as temperature, rotors angular velocities and other factors. Hence, an exact model of the rotors friction is very complicated. In particular, the control moments ui will be employ to make the motion of the rigid body is global asymptotical stable about one of its equilibrium states. In addition to the dynamics equation (2.1), which provides the time history of the rigid body angular veloc~, the orientation of the body is given by the kinematic equations this because of the direct integraity vector x tion of Eq. (2.1) do not provide any useful information about the rigid body orientation. If R(t) is the rotation matrix which relates both the inertial Ongf and the body Ox1x2x3, coordinate systems, then the time derivative of this matrix is given by R_ ¼ Tð~ xÞR;
ð2:3Þ
where Tð~ xÞ is the skew-symmetric matrix 2 3 0 x3 x2 6 7 T ¼ 4 x3 0 x1 5: x2 x1 0 The system of Eqs. (2.1) and (2.3) will be used for studying the global asymptotic stabilization of the rigid body motion about its equilibrium state with the help of the control moments ui (i = 1, 2, 3) using Lyapunov function technique. 3. Euler parameters and first integrals Euler’s principal rotation theorem (Tsiotras, 1996; Lure, 1961) states that a complete angular displacement of a rigid body about a fixed point can be accomplished by a single rotation through an angle U about a vector ^l, which is fixed in both the body and inertial frames. Using this rotation theorem we can define the Euler parameters (k0, k1, k2, k3) as follows: U U k0 ¼ cos ; ki ¼ li sin ði ¼ 1; 2; 3Þ; ð3:1Þ 2 2 where ^l ¼ ðl1 ; l2 ; l3 Þ are the components of principal vector of rotation ^l referred to the body axes. Further, it is easy to verify that the Euler parameters k ¼ ðk0 ; k1 ; k2 ; k3 Þ ¼ ðk0 ; ~ kÞ satisfy the following set of ordinary differential equations: _ ~x ~ ~ k ¼ k0 x 2~ k;
~: kx 2k_ 0 ¼ ~
ð3:2Þ
Of prime importance is the fact that the kinematic equation (3.2) are linear in the elements of both ~ x and k. This makes the numerical integration of the kinematic equations simple. The direction cosine transformation matrix ½CðkÞ between an arbitrary body fixed frame and inertial is defined by 2 3 k2 0 k3 6 7 2 ð3:3Þ ½C ¼ ðk2 j~ kj ÞI 44 þ 2~ k~ kT 2k0 ½~ k; ½~ k ¼ 4 k3 0 k1 5; 0
k2
k1
0
where I4·4 is the identity matrix of order four. In fact, if the inertial coordinate system Ongf, is initially coincident with the body principal coordinate system Ox1x2x3, then the direction cosines l1, l2, l3 of the unit vector ^l are identical in both systems of coordinate. From (3.1) one can easily establishes the following Euler parameters constraint: k k ¼ k20 þ k21 þ k22 þ k23 ¼ 1:
ð3:4Þ
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Eq. (3.4) represents the first integral of the system (3.2) which can be used to reduce the number of the kinematic parameters from four to three parameters. Inserting Eq. (2.2) into Eq. (2.1), the equations of the rigid body dynamics take the form ðC 1 I 1 Þx_ 1 ¼ G2 x3 G3 x2 u1 þ L1 þ l1 U_ 1
ð123Þ;
ð3:5Þ
where Li (i = 1, 2, 3) are the components of the external moments referring to the body coordinates. In particular, we will restrict our attention in this paper to the cases in which these moments are very small compared with the artificial internal control moments ui or there are no external moments affected on the body. Note that these moments usual ignored in the control design (Levine, 1995). It is easy to verify that, in the absence of the external moments ~ L, the system (2.1) admits the following first integral: ~0 ¼ const: ~G ~ ¼ G2 ðtÞ þ G2 ðtÞ þ G2 ðtÞ ¼ G G 1 2 3
ð3:6Þ
It follows that if G1(t), G2(t) and G3(t) are bounded at t = 0, then these functions are always bounded. In general, reducing to zero all the variables U_ i , xi (i = 1, 2, 3) is impossible. It would be possible only in a very particular case when the motion of the rigid body takes place in the subspace of initial states _ xi0 ¼ C 1 i I i Ui0
ði ¼ 1; 2; 3Þ:
ð3:7Þ
Consequently, the systems (2.2) and (3.5) cannot be stabilized with respect to all the variables U_ i , xi (i = 1, 2, 3). This property does negate the conditions of stability of these systems. Thus, we shall attempt to investigate the problem of global asymptotic stabilization of the rotational motion of the rigid body alone, with arbitrary functions G1(t), G2(t) and G3(t) being bounded and satisfying the conditions (3.5) and (3.6). The third first integral of our system can be obtained as follows. Denote by hi (i = 1, 2, 3) the components of the total angular momentum of the system referring to the inertial Ongf, we find that when ~ L ¼~ 0, the system (2.1) admits the following three first integrals: G1 ðtÞci1 ðkÞ þ G2 ðtÞci2 ðkÞ þ G3 ðtÞci3 ðkÞ ¼ hi
ði ¼ 1; 2; 3Þ;
ð3:8Þ
which can be used to eliminate the rotors angular velocities U_ i (i = 1, 2, 3) as follows: " # 3 X 1 U_ i ¼ I i hs csi ðkÞ C i xi ði ¼ 1; 2; 3Þ;
ð3:9Þ
s¼1
where cij(k) (i, j = 1, 2, 3) are the elements of the transformation matrix that appointed by (3.3) and the components hi (i = 1, 2, 3) of the system total angular momentum referring to the inertial axes Ongf are regarded as constants. Now we can rapidly be eliminated the rotors angular velocities U_ i (i = 1, 2, 3) from Eq. (3.5) to get " # 3 3 3 X X X ðC 1 I 1 Þx_ 1 ¼ x3 hs cs2 ðkÞ x2 hs cs3 ðkÞ þ l1 I 1 hs cs1 ðkÞ C 1 x1 u1 ð123Þ: ð3:10Þ 1 s¼1
s¼1
s¼1
In the absence of the external control moments the systems (3.2) and (3.10) admit the following special solution: xi ¼ 0;
ki ¼ 0;
ui ¼ li U_ i0 ¼ li hi0 I 1 i ¼ const:;
k0 ¼ 1
ði ¼ 1; 2; 3Þ:
ð3:11Þ
In particular, this solution represents unstable equilibrium state of the rigid body and the control moments ui will be used to make this state is globally asymptotically stable. By global asymptotic stabilization we mean that all of the trajectories of the closed-loop systems (3.2) and (3.10) remain bounded and tend to zero for arbitrary initial state. 4. Global stabilization and control laws In this section, we will discuss the complete feedback control law in terms of Euler parameters that stabilizes the equilibrium state (3.11) of the rigid body. The global stabilization of the system (3.2) and (3.10) about
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the equilibrium state (3.11) means that with suitable choice of the control moments ui, all of the trajectories of the of this system remain bounded and tend to zero for arbitrary initial state. Here, our goal is to select the control law which make the systems (3.2) and (3.10) is global asymptotic stable about the solution (3.11). Theorem 4.1. The nonlinear control law 1 ui ¼ ðk i li C i I 1 i Þxi þ cki þ li I i
3 X
hs csi ðkÞ ði ¼ 1; 2; 3Þ
ð4:1Þ
i¼1
globally asymptotically stabilizes the rigid body about the equilibrium state (3.11), where, both of ki (i = 1, 2, 3) and c are positive control gains. Proof. The proof of this theorem is based on the following quadratic Lyapunov function: 2V ¼
3 X 2 ðC i I i Þx2i þ 2ck2i þ 2cðk0 1Þ :
ð4:2Þ
i¼1
This function represents the sum of two quadratic terms, the first of that involves only the angular velocity and the second term involves the kinematic parameters. With this choice of the Lyapunov function the control law is linear if the rotors friction coefficients are zero and nonlinear if these coefficients different from zero. The time of the Lyapunov function along the trajectories of the closed-loop systems (3.2), (3.10) and (4.1) we get V_ ¼
3 X
k i x2i
ð4:3Þ
i¼1
this implies that the equilibrium state (3.11) is stable but not necessary asymptotic. Now V_ 6 0 for all t, but this is not enough to prove the asymptotic stability. Since V_ ¼ 0 will vanish whenever x1 ¼ x2 ¼ x3 ¼ 0
ð4:4Þ
whether k0 ¼ 1;
k1 ¼ k2 ¼ k3 ¼ 0
ð4:5Þ
or not. If for some instant, say t1, V_ ðt1 Þ ¼ 0, and if further V_ ðtÞ ¼ 0 for all t P t1, then V(t) = const., but not necessarily zero, for all t P t1. In fact, the obtained control law is successful after the following analysis. Assume that, V_ ðt1 Þ ¼ 0, we must prove that this contingency cannot occur, unless V_ ðtÞ cannot remain zero for all t P t1. It is clear that V_ ðtÞ ¼ 0, if and only if xi = 0 (i = 1, 2, 3). But in this case, the system of Eq. (3.10) reduces to ðC i I i Þx_ i ¼ cki
ði ¼ 1; 2; 3Þ;
ð4:6Þ
since Ci Ii > 0 by definition and if ki = 0 (i = 1, 2, 3), then x_ i ¼ 0 (i = 1, 2, 3) and V_ ¼ 0 because of the body in the desired equilibrium state. Assume that xi = 0 (i = 1, 2, 3) and at least one of ki (i = 1, 2, 3) is different from zero, then from (4.6), at least one of x_ i (i = 1, 2, 3) is also non-zero. Hence xi = 0 (i = 1, 2, 3) only instantaneously, after which at least one of x_ i becomes non-zero. Thus V_ is non-zero also. But if V_ is non-zero, it can only be negative, which means that V continuously to decrease. This proves the global asymptotic stability of the equilibrium state (3.11) of the rigid body in the Lyapunov. Thus the theorem is proved. h The physical meaning of the results can be explained using the angular momentum theorem. The stabilization of the rotational a rigid body motion about a fixed point by means of rotors is possible since every directional motion of the rotors causes a directional motion of the rigid body. Also, since every rotor is driven by an electric motor then the control of the body is reduced to the control of the voltage transmitted to the
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driving motors. Hence, under the action of the control law (4.1) the rigid body is globally asymptotically to equilibrium state (3.11) starting from arbitrary initial state. Also, note that the control law (4.1) reduces to linear function if the rotors friction is neglected. Also we can easy verify that as t ! 1 the first integral (3.6) has the following limiting value: lim G21 ðtÞ þ G22 ðtÞ þ G23 ðtÞ ¼ I 21 U_ 210 þ I 22 U_ 220 þ I 23 U_ 230 : t!1
Therefore, existence of this limiting value of the first integral (3.6) confirms the conclusion that the rotors motion is not controllable. As soon as the body arrives at the desired equilibrium state (3.11) the stabilizing control law reduces to u1 ¼ l1 U_ 10 ;
u2 ¼ l2 U_ 20 ;
u3 ¼ l3 U_ 30
ð4:7Þ
and the rotors will move under the inertial moments only and their motion is governed by €1 ¼ U €2 ¼ U € 3 ¼ 0: U
ð4:8Þ
This means that, the rotors rotate around their axes with constant angular velocities U_ i0 , i = 1, 2, 3. In this case, as soon as the body arrives at the desired equilibrium state the rotors become deactivated because of at this state there is no change in the rotors angular momentum. Now we will introduce some pervious result as a first approximation of the obtained results. The direction cosines bij (i, j = 1, 2, 3) between the inertial and body principal axes relate with Euler parameters ks (s = 0,1, 2, 3) by the following relations (Lure, 1961): 1 b32 b23 ¼ 4k0 k1 ¼ 4k1 1 ðk21 þ k22 þ k23 Þ þ ’ 4k1 ð123Þ: ð4:9Þ 2 Substituting from (4.9) into (4.1), keeping it to the second order term and setting li = 0, we get u1 ’ aðb31 b13 Þ þ k 1 x1 ;
a ¼ c=4
ð123Þ:
ð4:10Þ
This result approves with result introduced in El-Gohary (2001b) and Krementulo (1977) when the rotation of the body is infinitesimal about the equilibrium state. This confirms the effectiveness of our solution. Further the results introduced in Tsiotras (1994) can be obtained by assuming ki = 1 (i = 1, 2, 3), c = 1 of the control gains and the friction coefficients li (i = 1, 2, 3) to be zero in (4.1). These special cases confirm that the preset results generalize and unify many pervious results. Through this paper the numerical values of the rigid body and rotors moments of inertia Ci, Ii, control gains ki (i = 1, 2, 3), c, and the initial state are chosen to be C 1 ¼ 15 kg m2 ; 2
C 2 ¼ 22 kg m2 ; 2
I 1 ¼ 1:5 kg m ; I 2 ¼ 2:2 kg m ; ðU_ 10 ; U_ 20 ; U_ 30 Þ ¼ ð0:3; 0:8; 0:4Þ:
C 3 ¼ 17 kg m2 ; 2
I 3 ¼ 1:7 kg m ;
k 1 ¼ k 2 ¼ k 3 ¼ 5;
c ¼ 10;
~ð0Þ ¼ ð0:2; 0:6; 0:4Þ; x
Fig. 1. (a) Rigid body angular velocity, (b) Euler parameters and (c) rotor angular velocities.
ð4:11Þ
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Fig. 1 displays the rigid body angular velocity vector, the Euler attitude parameters, and rotors angular velocities using feedback control law (4.1) for the initial values k ¼ p1ffiffi2 ð1; 1; 0; 0Þ of Euler parameters. 5. Cayley–Rodrigues parameters The four parameterizations of (3.1) introduce a redundant parameters in order to achieve a non-singular description of the rigid body orientation. It is well-known that the least number of parameters required to describe the orientation of a rotating rigid body is three. Such this case is Cayley–Rodrigues parameters or Modified-Rodrigues parameters can be used. Cayley–Rodrigues parameters can be used to eliminate the constraint (3.4) associated with the Euler parameters, further these parameters reduce the number of coordinates that describe the rigid body orientation from four to three. This fact can be established by defining so-called Cayley–Rodrigues parameters as follows: mi ¼
ki k0
ði ¼ 1; 2; 3Þ:
ð5:1Þ
Using the definition (5.1), then the kinematic equations (3.2) take the form 2_m1 ¼ x1 x2 m3 þ x3 m2 þ m1
3 X
mi x i
ð123Þ:
ð5:2Þ
i¼1
m which defined by The Cayley–Rodrigues parameters mi (i = 1, 2, 3) are also components of the Gibbes vector ~ the following form: U ~ m ¼ ^l tan : 2
ð5:3Þ
Obviously from (5.3) Cayley–Rodrigues parameters cannot be used to describe eigenaxis of rotations more than 180. At this end our aim is to employ the control moments ui (i = 1, 2, 3) to make the rigid body motion is global asymptotical stable about its equilibrium state that determined by xi ¼ 0;
mi ¼ 0;
ui ¼ li U_ i0 ¼ li hi0 I 1 ði ¼ 1; 2; 3Þ: i ¼ const:
ð5:4Þ
Theorem 5.1. The nonlinear control law ui ¼ ðk i li C i I 1 i Þxi þ c 1 þ
3 X s¼1
! m2s mi þ li I 1 i
3 X
hs csi ð~ qÞ ði ¼ 1; 2; 3Þ
ð5:5Þ
s¼1
globally asymptotically stabilizes the rigid body about the equilibrium state (5.4). Proof. The proof of this theorem is based on the use of the following Lyapunov function: 2V ¼
3 X ðC i I i Þx2i þ 2cm2i :
ð5:6Þ
i¼1
The total time derivative of the function V along the trajectories of the closed-loop system (3.10), (5.2) and (5.5) takes the form V_ ¼
3 X
k i x2i
ð5:7Þ
i¼1
this implies that the rigid body motion is only stable about the equilibrium state (5.4) in the Lyapunov sense. Similarly the prove of asymptotic stability is identical to theorem (4.1). h
A. El-Gohary / Mechanics Research Communications 33 (2006) 818–829
825
Fig. 2. (a) Rigid body angular velocity, (b) Cayley–Rodrigues parameters and (c) rotor angular velocities.
Fig. 2 displays the rigid body angular velocity vector, the Cayley–Rodrigues parameters, and the rotors angular velocities using feedback control law (5.5) for the values ~ mð0Þ ¼ p1ffiffi2 ð1; 1; 0Þ of Cayley–Rodrigues parameters. In the case of the rotors friction is neglected the control law (5.5) reduces to ! 3 X u1 ¼ k 1 x1 þ cm1 1 þ m2i ; ð5:8Þ i¼1
which is also nonlinear function of Cayley–Rodrigues parameters. If we choose a different Lyapunov function, we can easy establish the fact that in the absence of the rotors friction a linear control law suffices to provide global asymptotic stability of the rigid body about the equilibrium state (5.4). Theorem 5.2. The nonlinear control law 1 ui ¼ ðk i li C i I 1 i Þxi þ cmi þ li I i
3 X
hs csi ð~ mÞ
ði ¼ 1; 2; 3Þ
ð5:9Þ
s¼1
globally asymptotically stabilizes the rigid body about the equilibrium state (5.4). Proof. The proof of this theorem is based on the use of the following Lyapunov function: ! 3 3 X X 2V ¼ ðC i I i Þx2i þ 2c ln 1 þ m2i : i¼1
ð5:10Þ
i¼1
The time derivative of the Lyapunov function (5.10) along the trajectories of the closed-loop system (3.10), (5.2) and (5.9) takes the form V_ ¼
3 X
k i x2i
ð5:11Þ
i¼1
this implies that the rigid body motion is only stable about the equilibrium state (5.4) in the Lyapunov sense. Also the prove of the asymptotic stability is identical to theorem (4.1). h In particular, if the rotors friction is neglected then the control law (5.9) reduces to ui ¼ k i xi þ cmi
ði ¼ 1; 2; 3Þ;
ð5:12Þ
which is a linear function of Cayley–Rodrigues parameters. Fig. 3 displays the rigid body angular velocity vector, the Cayley–Rodrigues parameters, and the rotors angular velocities using feedback control law (5.9) for the values ~ mð0Þ ¼ ð1; 1; 0Þ of Cayley–Rodrigues parameters. Using the relations between the direction cosines bij (i, j = 1, 2, 3) and Cayley–Rodrigues parameters and keeping it to the second order term we get the following relations (Lure, 1961):
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Fig. 3. (a) Rigid body angular velocity, (b) Cayley–Rodrigues parameters and (c) rotor angular velocities.
m1 ð1 þ m21 þ m22 þ m23 Þ ¼ k0 k1 ½1 k21 k22 k23
2
1 ’ k0 k1 ¼ ðb32 b23 Þ ð123Þ: 4
ð5:13Þ
Substituting (5.13) into (5.12), and keeping it to the second order term we can get c ð123Þ: ð5:14Þ u1 ’ aðb13 b31 Þ þ k 1 x1 ; a ¼ 4 Thus the result approves the result introduced in El-Gohary (2001b) and Krementulo (1977). This confirms the effectiveness of our solution. Moreover the results introduced in Tsiotras (1994) can be achieved by the assuming ki = 1 (i = 1, 2, 3), c = 1 of control gains in (5.12). This proves that the present results generalize and unify many pervious results. At this end from (5.8) and (5.12), we conclude that, the linearity and nonlinearity of the control law depend upon both of the Lyapunov function and rotors friction. 6. Modified-Rodrigues parameters Although the Cayley–Rodrigues parameters eliminate the constraint (3.4) imposes on the Euler parameters but they have the disadvantage of becoming unbounded when k0 = 0 and they do not allow eigenaxis rotations greater than 180. For these causes we will define the following Modified-Rodrigues parameters: qi ¼
ki 1 þ k0
ði ¼ 1; 2; 3Þ:
ð6:1Þ
The kinematics equation (3.2) in terms of these parameters are given by 3 X 1 2q_ 1 ¼ x3 q2 x2 q3 þ x1 q þ q1 xi qi 2 i¼1
ð123Þ;
ð6:2Þ
where q ¼ 1 q21 q22 q23 . Indeed we can easily prove that the parameters qi (i = 1, 2, 3) can be viewed as components of the vector U ~ q ¼ ^l tan ; 4
ð6:3Þ
which is well-defined for all eigenaxis rotations in the range 0 6 U < 360. Now, we wish to employ the control moments ui (i = 1, 2, 3) to make the rigid body motion is global asymptotical stable about its equilibrium state that determined by xi ¼ 0;
qi ¼ 0;
ui ¼ li U_ i0 ¼ li hi0 I 1 ði ¼ 1; 2; 3Þ: i ¼ const:
Theorem 6.1. The nonlinear control law ui ¼ ðk i
li C i I 1 i Þxi
þ cqi 1 þ
3 X s¼1
! q2s
þ I 1 i li
3 X
hs csi ð~ qÞ;
i ¼ 1; 2; 3
s¼1
globally asymptotically stabilizes the rigid body motion about the equilibrium state (6.4).
ð6:4Þ
ð6:5Þ
A. El-Gohary / Mechanics Research Communications 33 (2006) 818–829
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Proof. The proof of this theorem is based on the use of the following Lyapunov function: 2V ¼
3 X
ðC i I i Þx2i þ 4cq2i :
ð6:6Þ
i¼1
The time derivative of the Lyapunov function (6.6) along the trajectories of the closed-loop system (3.10), (6.2) and (6.5) takes the form V_ ¼
3 X
k i x2i 6 0
ð6:7Þ
i¼1
this implies that the rigid body motion is only stable about the equilibrium state (6.4) in the Lyapunov sense. Similarly the asymptotic stability can be proved as in theorem (4.1). h Fig. 4 displays the rigid body angular velocity vector, the Euler attitude parameters, and the rotors angular velocities using feedback control law (6.5) for the values ~ qð0Þ ¼ p1ffiffi2 ð1; 1; 0Þ of Modified-Rodrigues parameters. When the rotors friction is neglected, the control law (6.5) reduces to ! 3 X 2 ui ¼ k i xi þ cqi 1 þ qs ; i ¼ 1; 2; 3; ð6:8Þ s¼1
which is a nonlinear function of Modified-Rodrigues parameters. Theorem 6.2. The nonlinear control law 1 ui ¼ ðk i li C i I 1 i Þxi þ cqi þ I i li
3 X
hs csi ð~ qÞ;
i ¼ 1; 2; 3
ð6:9Þ
s¼1
globally asymptotically stabilizes the rigid body about the equilibrium state (6.4). Proof. The proof of this theorem is based on the use of the following Lyapunov function: ! 3 3 X X 2 2 2V ¼ ðC i I i Þxi þ 2c ln 1 þ qi : i¼1
ð6:10Þ
i¼1
The total time derivative of the Lyapunov function (6.10) along the trajectories of the closed-loop system (3.10), (6.2) and (6.9) takes the form V_ ¼
i X
k i x2i
ð6:11Þ
i¼1
this implies that the rotational motion of the body is only stable in the Lyapunov sense. Also the asymptotic stability can be proves using the same method of theorem (4.1). h
Fig. 4. (a) Rigid body angular velocity, (b) Modified-Rodrigues parameters and (c) rotor angular velocities.
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Fig. 5. (a) Rigid body angular velocity, (b) Modified-Rodrigues parameters and (c) rotor angular velocities.
When the rotors friction is neglected, the control law (6.9) reduces to ui ¼ k i xi þ cqi ;
i ¼ 1; 2; 3;
ð6:12Þ
which is a linear function of Modified-Rodrigues parameters. This results support the fact that the linearity and nonlinearity of the control law depend not only on the Lyapunov function structure but also the rotors friction. Fig. 5 displays the rigid body angular velocity vector, the Euler attitude parameters, the rotors angular velocities using feedback control law (6.9) for the values ~ qð0Þ ¼ ð0; 1; 1:2Þ of Modified-Rodrigues parameters. At this end we conclude that when the rotors friction is neglected the linearity and nonlinearity of the control law depend upon the choice of the Lyapunov function as shown from (5.8) and (5.12); (6.8) and (6.12). Also as an expected results the rotors friction does not affect on the asymptotic stability of the rigid body motion since this forces consider internal forces of prescribed mechanical system. 7. Conclusion A new class of global asymptotic stabilizing control laws that stabilize the rigid about one of its equilibrium states is obtained in terms of Euler parameters, Cayley–Rodrigues parameters and Modified-Rodrigues parameters. The linearity and nonlinearity of the stabilizing control law depend on the Lyapunov function only if the rotors friction is neglected. Many special cases of the obtained results are presented. Also, numerical simulations of the controlled rigid body and rotor system are introduced. References El-Gohary, A., 1996. On the stability of an equilibrium position and rotational motion of a gyrostat. Mech. Res. Commun. 24 (4), 457– 462. El-Gohary, A., 1997. On the stability of relative programmed motion of Satellite-gyrostat. Mech. Res. Commun. 25 (4), 371–379. El-Gohary, A., 2000. On the control of programmed motion of a rigid body containing moving masses. Int. J. Non-Linear Mech. (35), 27– 35. El-Gohary, A., 2001a. Global stability of the rotational motion of a rigid body containing moving masses. Int. J. Non-Linear Mech. (36), 663–669. El-Gohary, A., 2001b. Optimal stabilization of the equilibrium positions of a rigid body using rotors. Chaos Solitons Fractals (12), 2007– 2014. El-Gohary, A., 2002. Global stabilization of a rotational motion of a rigid body using rotors system. Appl. Math. Comput. 133, 297–307. El-Gohary, A., 2005a. Optimal control of the rotational motion of a rigid body using Euler parameters with the help of rotors system. Eur. J. Mech. A—Solids 24, 111–221. El-Gohary, A., 2005b. Optimal control of an equilibrium position of a rigid body using rotors system with friction forces. Chaos Solitons Fractals 23, 1585–1597. El-Gohary, A., 2005c. Optimal control of rigid body motion with the help of rotors using stereographic coordinates. Chaos Solitons Fractals 25 (5), 1229–1244. Krementulo, V., 1977. Stabilization of Steady Motion of a Rigid Body using Rotating Masses. Nauka Press, Moscow (in Russian). Levine, W., 1995. The Control Handbook, A CRC Handbook Published in Cooperation with IEEE Press. Lure, L., 1961. Analytical Mechanics. Fizmatgiz, Moscow. Moore, R., 1994. Fundamentals of Space Systems. Oxford University Press.
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Rotea, M., 1998. Suboptimal control of rigid body motion with quadratic cost. Dyn. Control 8, 55–80. Tsiotras, P., 1994. New control laws for the attitude stabilization of rigid bodies. In: 13th IFAC Symposium on Automatic Control in Aerospace, September 12–16, Palo Alto, CA, pp. 316–312. Tsiotras, P., 1996. Stabilization and optimality results for the attitude control problem. J. Guidance Control Dyn. 19 (4).
Mechanics Research Communications 33 (2006) 818–829 www.elsevier.com/locate/mechrescom
New control laws for stabilization of a rigid body motion using rotors system Awad El-Gohary
*
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Available online 27 April 2006
Abstract This paper presents a new class of globally asymptotic stabilizing control laws for dynamics and kinematics attitude motion of a rotating rigid body. The rigid body motion is controlled with the help of a rotor system with internal friction. The Lyapunov technique is used to prove the global asymptotic properties of the stabilizing control laws. The obtained control laws are given as functions of the angular velocity, Cayley–Rodrigues and Modified-Rodrigues parameters. It is shown that linearity and nonlinearity of the control laws depend not only upon the Lyapunov function structure but also the rotors friction. Moreover, some of the results are compared with these obtained in the literature by other methods. Numerical simulation is introduced. 2006 Published by Elsevier Ltd. Keywords: Rigid body; Rotors system with friction; Global stabilization; Cayley–Rodrigues and Modified-Rodrigues parameters
1. Introduction In recent years a considerable effort has been devoted to the design of control laws for challenging dynamical systems, such as robot manipulators, high-performance rigid spacecraft, satellite and space vehicles. The problem of description and control of the attitude motion of rigid bodies is investigated in Tsiotras (1996, 1994) and Rotea (1998). A lot of literatures use Euler angles for the orientation of a rigid body and the other use the direction-cosine (see, for example, El-Gohary, 2005b,c, 2001b; Krementulo, 1977). Unfortunately, neither the Euler angles nor the direction-cosine is the optimal using for the kinematic attitude of a rigid body motion. Since the Euler angles have singularities in the 3–1–3 Euler angles when h = 0 or p. If instead we had used 3–1–2 Euler angles, these would have been well behaved at h = 0 or p but would have shown singular behavior when h = ±p/2 (see Moore, 1994). Because of the singularity difficulty of the Euler angles and large number of singularities of
* Present address: Department of Statistics and OR, Faculty of Science King Saud University, P.O. Box 2455, Riyadh, Saudi Arabia. Fax: +9 661 4676274. E-mail addresses: [email protected], [email protected]
0093-6413/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.mechrescom.2006.03.008
A. El-Gohary / Mechanics Research Communications 33 (2006) 818–829
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elements in the direction-cosine matrix, we will use another representation of a rigid body kinematics attitude such as Euler parameters, Cayley–Rodrigues and Modified-Rodrigues parameters. The results in this paper complement and extend similar results published recently in terms of Euler parameters kinematic parameterizations (El-Gohary, 2002, 2001a, 1996, 2005a; Tsiotras, 1994). Euler parameters, Cayley–Rodrigues and Modified-Rodrigues parameters, can be employed to prove global asymptotic stability of a rigid body motion without imposing conditions on the control gains and system parameters. This motivated us to use these parameters for obtaining a new class of control laws that globally asymptotically stabilize a rigid body motion about its equilibrium state. Euler parameters were employed in El-Gohary (2002, 2001a) to study the global asymptotic stabilization of a rigid body rotational motion with the help of rotor and moving masses systems respectively, using Lyapunov technique. Moreover, these parameters also were employed in the paper (El-Gohary, 1996, 2000) for describing the rigid body programmed motion and proving the global asymptotic stabilization of the rigid body rotational motion. Also, Euler parameters are used in El-Gohary (1997) to study the global asymptotic stabilization of the relative programmed motion of a satellite-gyrostat. The Modified-Rodrigues parameters as a stereographic projections of Euler parameters are used in El-Gohary (2005c) to study the problem of global optimal stabilization of a rigid body rotational motion about the equilibrium state. The problem of controlling the rotational motion of a rigid body using three independent control torques is introduced in Tsiotras (1994) and Rotea (1998). The current paper is organized as follows. Section 2 of the paper introduces the equations of motion of a rigid body carrying a rotor system taking into consideration the internal rotors friction. Section 3 contains some important first integral of both dynamical and kinematical equations. The asymptotic stability feedback control law is obtained as function of angular velocity and Euler parameters in Section 4. The main results of this paper are concentrated in Sections 5 and 6, since both of Cayley–Rodrigues and Modified-Rodrigues parameters are employed to study the problem of global stabilization of a rigid body motion about its equilibrium state with linear and nonlinear feedback control input using Lyapunov technique. 2. Formulation the problem and equations of motion Consider a mechanical system S which consists of a rigid body P (platform) whose center of mass with a fixed point O and 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 mass of the system S. To describe the motion of the system, two systems of coordinates are introduced. The first Ongf is an inertial system and the second Ox1x2x3 is fixed in the body and coincides with the principal axes of the body at O. ~ and the principal Let xi and Ci (i = 1, 2, 3) be the components of the rigid body angular velocity vector x moments of inertia of the system relative to the principal axes Ox1x2x3, respectively. The resulting system dynamics are given by ~_ þ x ~ ¼~ ~G G L;
ð2:1Þ
~ and ~ where G L are the system total angular momentum and the external moments vectors referring to the body fixed frame Ox1x2x3. The components of the system total angular momentum referring to the body principal axes Ox1x2x2 are given by Gi ¼ C i xi þ I i U_ i
ði ¼ 1; 2; 3Þ;
where Ii and Ui (i = 1, 2, 3) are the ith rotor axial moment of inertia and the ith rotor angle of rotation referring to the body axes Ox1x2x2. Eq. (2.1) should be augmented by the equations the rotors motion relative to the body coordinates system Ox1x2x2. That is € 1 Þ ¼ u1 l1 U_ 1 I 1 ðx_ 1 þ U
ð123Þ;
ð2:2Þ
where the symbol (123) means that the two other equations can be obtained from the given equation by cyclic permutations of the indices 1 ! 2 ! 3, the dot denotes the differentiation with respect to time, ui (i = 1, 2, 3)
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are the control moments applied to the rotors and developed by electric motors that rigidly mounted on the body and the terms li U_ i (i = 1, 2, 3) denote the frictional moment due to the ith rotor friction. Note that the coefficients li (i = 1, 2, 3) depend upon various factors such as temperature, rotors angular velocities and other factors. Hence, an exact model of the rotors friction is very complicated. In particular, the control moments ui will be employ to make the motion of the rigid body is global asymptotical stable about one of its equilibrium states. In addition to the dynamics equation (2.1), which provides the time history of the rigid body angular veloc~, the orientation of the body is given by the kinematic equations this because of the direct integraity vector x tion of Eq. (2.1) do not provide any useful information about the rigid body orientation. If R(t) is the rotation matrix which relates both the inertial Ongf and the body Ox1x2x3, coordinate systems, then the time derivative of this matrix is given by R_ ¼ Tð~ xÞR;
ð2:3Þ
where Tð~ xÞ is the skew-symmetric matrix 2 3 0 x3 x2 6 7 T ¼ 4 x3 0 x1 5: x2 x1 0 The system of Eqs. (2.1) and (2.3) will be used for studying the global asymptotic stabilization of the rigid body motion about its equilibrium state with the help of the control moments ui (i = 1, 2, 3) using Lyapunov function technique. 3. Euler parameters and first integrals Euler’s principal rotation theorem (Tsiotras, 1996; Lure, 1961) states that a complete angular displacement of a rigid body about a fixed point can be accomplished by a single rotation through an angle U about a vector ^l, which is fixed in both the body and inertial frames. Using this rotation theorem we can define the Euler parameters (k0, k1, k2, k3) as follows: U U k0 ¼ cos ; ki ¼ li sin ði ¼ 1; 2; 3Þ; ð3:1Þ 2 2 where ^l ¼ ðl1 ; l2 ; l3 Þ are the components of principal vector of rotation ^l referred to the body axes. Further, it is easy to verify that the Euler parameters k ¼ ðk0 ; k1 ; k2 ; k3 Þ ¼ ðk0 ; ~ kÞ satisfy the following set of ordinary differential equations: _ ~x ~ ~ k ¼ k0 x 2~ k;
~: kx 2k_ 0 ¼ ~
ð3:2Þ
Of prime importance is the fact that the kinematic equation (3.2) are linear in the elements of both ~ x and k. This makes the numerical integration of the kinematic equations simple. The direction cosine transformation matrix ½CðkÞ between an arbitrary body fixed frame and inertial is defined by 2 3 k2 0 k3 6 7 2 ð3:3Þ ½C ¼ ðk2 j~ kj ÞI 44 þ 2~ k~ kT 2k0 ½~ k; ½~ k ¼ 4 k3 0 k1 5; 0
k2
k1
0
where I4·4 is the identity matrix of order four. In fact, if the inertial coordinate system Ongf, is initially coincident with the body principal coordinate system Ox1x2x3, then the direction cosines l1, l2, l3 of the unit vector ^l are identical in both systems of coordinate. From (3.1) one can easily establishes the following Euler parameters constraint: k k ¼ k20 þ k21 þ k22 þ k23 ¼ 1:
ð3:4Þ
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Eq. (3.4) represents the first integral of the system (3.2) which can be used to reduce the number of the kinematic parameters from four to three parameters. Inserting Eq. (2.2) into Eq. (2.1), the equations of the rigid body dynamics take the form ðC 1 I 1 Þx_ 1 ¼ G2 x3 G3 x2 u1 þ L1 þ l1 U_ 1
ð123Þ;
ð3:5Þ
where Li (i = 1, 2, 3) are the components of the external moments referring to the body coordinates. In particular, we will restrict our attention in this paper to the cases in which these moments are very small compared with the artificial internal control moments ui or there are no external moments affected on the body. Note that these moments usual ignored in the control design (Levine, 1995). It is easy to verify that, in the absence of the external moments ~ L, the system (2.1) admits the following first integral: ~0 ¼ const: ~G ~ ¼ G2 ðtÞ þ G2 ðtÞ þ G2 ðtÞ ¼ G G 1 2 3
ð3:6Þ
It follows that if G1(t), G2(t) and G3(t) are bounded at t = 0, then these functions are always bounded. In general, reducing to zero all the variables U_ i , xi (i = 1, 2, 3) is impossible. It would be possible only in a very particular case when the motion of the rigid body takes place in the subspace of initial states _ xi0 ¼ C 1 i I i Ui0
ði ¼ 1; 2; 3Þ:
ð3:7Þ
Consequently, the systems (2.2) and (3.5) cannot be stabilized with respect to all the variables U_ i , xi (i = 1, 2, 3). This property does negate the conditions of stability of these systems. Thus, we shall attempt to investigate the problem of global asymptotic stabilization of the rotational motion of the rigid body alone, with arbitrary functions G1(t), G2(t) and G3(t) being bounded and satisfying the conditions (3.5) and (3.6). The third first integral of our system can be obtained as follows. Denote by hi (i = 1, 2, 3) the components of the total angular momentum of the system referring to the inertial Ongf, we find that when ~ L ¼~ 0, the system (2.1) admits the following three first integrals: G1 ðtÞci1 ðkÞ þ G2 ðtÞci2 ðkÞ þ G3 ðtÞci3 ðkÞ ¼ hi
ði ¼ 1; 2; 3Þ;
ð3:8Þ
which can be used to eliminate the rotors angular velocities U_ i (i = 1, 2, 3) as follows: " # 3 X 1 U_ i ¼ I i hs csi ðkÞ C i xi ði ¼ 1; 2; 3Þ;
ð3:9Þ
s¼1
where cij(k) (i, j = 1, 2, 3) are the elements of the transformation matrix that appointed by (3.3) and the components hi (i = 1, 2, 3) of the system total angular momentum referring to the inertial axes Ongf are regarded as constants. Now we can rapidly be eliminated the rotors angular velocities U_ i (i = 1, 2, 3) from Eq. (3.5) to get " # 3 3 3 X X X ðC 1 I 1 Þx_ 1 ¼ x3 hs cs2 ðkÞ x2 hs cs3 ðkÞ þ l1 I 1 hs cs1 ðkÞ C 1 x1 u1 ð123Þ: ð3:10Þ 1 s¼1
s¼1
s¼1
In the absence of the external control moments the systems (3.2) and (3.10) admit the following special solution: xi ¼ 0;
ki ¼ 0;
ui ¼ li U_ i0 ¼ li hi0 I 1 i ¼ const:;
k0 ¼ 1
ði ¼ 1; 2; 3Þ:
ð3:11Þ
In particular, this solution represents unstable equilibrium state of the rigid body and the control moments ui will be used to make this state is globally asymptotically stable. By global asymptotic stabilization we mean that all of the trajectories of the closed-loop systems (3.2) and (3.10) remain bounded and tend to zero for arbitrary initial state. 4. Global stabilization and control laws In this section, we will discuss the complete feedback control law in terms of Euler parameters that stabilizes the equilibrium state (3.11) of the rigid body. The global stabilization of the system (3.2) and (3.10) about
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the equilibrium state (3.11) means that with suitable choice of the control moments ui, all of the trajectories of the of this system remain bounded and tend to zero for arbitrary initial state. Here, our goal is to select the control law which make the systems (3.2) and (3.10) is global asymptotic stable about the solution (3.11). Theorem 4.1. The nonlinear control law 1 ui ¼ ðk i li C i I 1 i Þxi þ cki þ li I i
3 X
hs csi ðkÞ ði ¼ 1; 2; 3Þ
ð4:1Þ
i¼1
globally asymptotically stabilizes the rigid body about the equilibrium state (3.11), where, both of ki (i = 1, 2, 3) and c are positive control gains. Proof. The proof of this theorem is based on the following quadratic Lyapunov function: 2V ¼
3 X 2 ðC i I i Þx2i þ 2ck2i þ 2cðk0 1Þ :
ð4:2Þ
i¼1
This function represents the sum of two quadratic terms, the first of that involves only the angular velocity and the second term involves the kinematic parameters. With this choice of the Lyapunov function the control law is linear if the rotors friction coefficients are zero and nonlinear if these coefficients different from zero. The time of the Lyapunov function along the trajectories of the closed-loop systems (3.2), (3.10) and (4.1) we get V_ ¼
3 X
k i x2i
ð4:3Þ
i¼1
this implies that the equilibrium state (3.11) is stable but not necessary asymptotic. Now V_ 6 0 for all t, but this is not enough to prove the asymptotic stability. Since V_ ¼ 0 will vanish whenever x1 ¼ x2 ¼ x3 ¼ 0
ð4:4Þ
whether k0 ¼ 1;
k1 ¼ k2 ¼ k3 ¼ 0
ð4:5Þ
or not. If for some instant, say t1, V_ ðt1 Þ ¼ 0, and if further V_ ðtÞ ¼ 0 for all t P t1, then V(t) = const., but not necessarily zero, for all t P t1. In fact, the obtained control law is successful after the following analysis. Assume that, V_ ðt1 Þ ¼ 0, we must prove that this contingency cannot occur, unless V_ ðtÞ cannot remain zero for all t P t1. It is clear that V_ ðtÞ ¼ 0, if and only if xi = 0 (i = 1, 2, 3). But in this case, the system of Eq. (3.10) reduces to ðC i I i Þx_ i ¼ cki
ði ¼ 1; 2; 3Þ;
ð4:6Þ
since Ci Ii > 0 by definition and if ki = 0 (i = 1, 2, 3), then x_ i ¼ 0 (i = 1, 2, 3) and V_ ¼ 0 because of the body in the desired equilibrium state. Assume that xi = 0 (i = 1, 2, 3) and at least one of ki (i = 1, 2, 3) is different from zero, then from (4.6), at least one of x_ i (i = 1, 2, 3) is also non-zero. Hence xi = 0 (i = 1, 2, 3) only instantaneously, after which at least one of x_ i becomes non-zero. Thus V_ is non-zero also. But if V_ is non-zero, it can only be negative, which means that V continuously to decrease. This proves the global asymptotic stability of the equilibrium state (3.11) of the rigid body in the Lyapunov. Thus the theorem is proved. h The physical meaning of the results can be explained using the angular momentum theorem. The stabilization of the rotational a rigid body motion about a fixed point by means of rotors is possible since every directional motion of the rotors causes a directional motion of the rigid body. Also, since every rotor is driven by an electric motor then the control of the body is reduced to the control of the voltage transmitted to the
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driving motors. Hence, under the action of the control law (4.1) the rigid body is globally asymptotically to equilibrium state (3.11) starting from arbitrary initial state. Also, note that the control law (4.1) reduces to linear function if the rotors friction is neglected. Also we can easy verify that as t ! 1 the first integral (3.6) has the following limiting value: lim G21 ðtÞ þ G22 ðtÞ þ G23 ðtÞ ¼ I 21 U_ 210 þ I 22 U_ 220 þ I 23 U_ 230 : t!1
Therefore, existence of this limiting value of the first integral (3.6) confirms the conclusion that the rotors motion is not controllable. As soon as the body arrives at the desired equilibrium state (3.11) the stabilizing control law reduces to u1 ¼ l1 U_ 10 ;
u2 ¼ l2 U_ 20 ;
u3 ¼ l3 U_ 30
ð4:7Þ
and the rotors will move under the inertial moments only and their motion is governed by €1 ¼ U €2 ¼ U € 3 ¼ 0: U
ð4:8Þ
This means that, the rotors rotate around their axes with constant angular velocities U_ i0 , i = 1, 2, 3. In this case, as soon as the body arrives at the desired equilibrium state the rotors become deactivated because of at this state there is no change in the rotors angular momentum. Now we will introduce some pervious result as a first approximation of the obtained results. The direction cosines bij (i, j = 1, 2, 3) between the inertial and body principal axes relate with Euler parameters ks (s = 0,1, 2, 3) by the following relations (Lure, 1961): 1 b32 b23 ¼ 4k0 k1 ¼ 4k1 1 ðk21 þ k22 þ k23 Þ þ ’ 4k1 ð123Þ: ð4:9Þ 2 Substituting from (4.9) into (4.1), keeping it to the second order term and setting li = 0, we get u1 ’ aðb31 b13 Þ þ k 1 x1 ;
a ¼ c=4
ð123Þ:
ð4:10Þ
This result approves with result introduced in El-Gohary (2001b) and Krementulo (1977) when the rotation of the body is infinitesimal about the equilibrium state. This confirms the effectiveness of our solution. Further the results introduced in Tsiotras (1994) can be obtained by assuming ki = 1 (i = 1, 2, 3), c = 1 of the control gains and the friction coefficients li (i = 1, 2, 3) to be zero in (4.1). These special cases confirm that the preset results generalize and unify many pervious results. Through this paper the numerical values of the rigid body and rotors moments of inertia Ci, Ii, control gains ki (i = 1, 2, 3), c, and the initial state are chosen to be C 1 ¼ 15 kg m2 ; 2
C 2 ¼ 22 kg m2 ; 2
I 1 ¼ 1:5 kg m ; I 2 ¼ 2:2 kg m ; ðU_ 10 ; U_ 20 ; U_ 30 Þ ¼ ð0:3; 0:8; 0:4Þ:
C 3 ¼ 17 kg m2 ; 2
I 3 ¼ 1:7 kg m ;
k 1 ¼ k 2 ¼ k 3 ¼ 5;
c ¼ 10;
~ð0Þ ¼ ð0:2; 0:6; 0:4Þ; x
Fig. 1. (a) Rigid body angular velocity, (b) Euler parameters and (c) rotor angular velocities.
ð4:11Þ
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Fig. 1 displays the rigid body angular velocity vector, the Euler attitude parameters, and rotors angular velocities using feedback control law (4.1) for the initial values k ¼ p1ffiffi2 ð1; 1; 0; 0Þ of Euler parameters. 5. Cayley–Rodrigues parameters The four parameterizations of (3.1) introduce a redundant parameters in order to achieve a non-singular description of the rigid body orientation. It is well-known that the least number of parameters required to describe the orientation of a rotating rigid body is three. Such this case is Cayley–Rodrigues parameters or Modified-Rodrigues parameters can be used. Cayley–Rodrigues parameters can be used to eliminate the constraint (3.4) associated with the Euler parameters, further these parameters reduce the number of coordinates that describe the rigid body orientation from four to three. This fact can be established by defining so-called Cayley–Rodrigues parameters as follows: mi ¼
ki k0
ði ¼ 1; 2; 3Þ:
ð5:1Þ
Using the definition (5.1), then the kinematic equations (3.2) take the form 2_m1 ¼ x1 x2 m3 þ x3 m2 þ m1
3 X
mi x i
ð123Þ:
ð5:2Þ
i¼1
m which defined by The Cayley–Rodrigues parameters mi (i = 1, 2, 3) are also components of the Gibbes vector ~ the following form: U ~ m ¼ ^l tan : 2
ð5:3Þ
Obviously from (5.3) Cayley–Rodrigues parameters cannot be used to describe eigenaxis of rotations more than 180. At this end our aim is to employ the control moments ui (i = 1, 2, 3) to make the rigid body motion is global asymptotical stable about its equilibrium state that determined by xi ¼ 0;
mi ¼ 0;
ui ¼ li U_ i0 ¼ li hi0 I 1 ði ¼ 1; 2; 3Þ: i ¼ const:
ð5:4Þ
Theorem 5.1. The nonlinear control law ui ¼ ðk i li C i I 1 i Þxi þ c 1 þ
3 X s¼1
! m2s mi þ li I 1 i
3 X
hs csi ð~ qÞ ði ¼ 1; 2; 3Þ
ð5:5Þ
s¼1
globally asymptotically stabilizes the rigid body about the equilibrium state (5.4). Proof. The proof of this theorem is based on the use of the following Lyapunov function: 2V ¼
3 X ðC i I i Þx2i þ 2cm2i :
ð5:6Þ
i¼1
The total time derivative of the function V along the trajectories of the closed-loop system (3.10), (5.2) and (5.5) takes the form V_ ¼
3 X
k i x2i
ð5:7Þ
i¼1
this implies that the rigid body motion is only stable about the equilibrium state (5.4) in the Lyapunov sense. Similarly the prove of asymptotic stability is identical to theorem (4.1). h
A. El-Gohary / Mechanics Research Communications 33 (2006) 818–829
825
Fig. 2. (a) Rigid body angular velocity, (b) Cayley–Rodrigues parameters and (c) rotor angular velocities.
Fig. 2 displays the rigid body angular velocity vector, the Cayley–Rodrigues parameters, and the rotors angular velocities using feedback control law (5.5) for the values ~ mð0Þ ¼ p1ffiffi2 ð1; 1; 0Þ of Cayley–Rodrigues parameters. In the case of the rotors friction is neglected the control law (5.5) reduces to ! 3 X u1 ¼ k 1 x1 þ cm1 1 þ m2i ; ð5:8Þ i¼1
which is also nonlinear function of Cayley–Rodrigues parameters. If we choose a different Lyapunov function, we can easy establish the fact that in the absence of the rotors friction a linear control law suffices to provide global asymptotic stability of the rigid body about the equilibrium state (5.4). Theorem 5.2. The nonlinear control law 1 ui ¼ ðk i li C i I 1 i Þxi þ cmi þ li I i
3 X
hs csi ð~ mÞ
ði ¼ 1; 2; 3Þ
ð5:9Þ
s¼1
globally asymptotically stabilizes the rigid body about the equilibrium state (5.4). Proof. The proof of this theorem is based on the use of the following Lyapunov function: ! 3 3 X X 2V ¼ ðC i I i Þx2i þ 2c ln 1 þ m2i : i¼1
ð5:10Þ
i¼1
The time derivative of the Lyapunov function (5.10) along the trajectories of the closed-loop system (3.10), (5.2) and (5.9) takes the form V_ ¼
3 X
k i x2i
ð5:11Þ
i¼1
this implies that the rigid body motion is only stable about the equilibrium state (5.4) in the Lyapunov sense. Also the prove of the asymptotic stability is identical to theorem (4.1). h In particular, if the rotors friction is neglected then the control law (5.9) reduces to ui ¼ k i xi þ cmi
ði ¼ 1; 2; 3Þ;
ð5:12Þ
which is a linear function of Cayley–Rodrigues parameters. Fig. 3 displays the rigid body angular velocity vector, the Cayley–Rodrigues parameters, and the rotors angular velocities using feedback control law (5.9) for the values ~ mð0Þ ¼ ð1; 1; 0Þ of Cayley–Rodrigues parameters. Using the relations between the direction cosines bij (i, j = 1, 2, 3) and Cayley–Rodrigues parameters and keeping it to the second order term we get the following relations (Lure, 1961):
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Fig. 3. (a) Rigid body angular velocity, (b) Cayley–Rodrigues parameters and (c) rotor angular velocities.
m1 ð1 þ m21 þ m22 þ m23 Þ ¼ k0 k1 ½1 k21 k22 k23
2
1 ’ k0 k1 ¼ ðb32 b23 Þ ð123Þ: 4
ð5:13Þ
Substituting (5.13) into (5.12), and keeping it to the second order term we can get c ð123Þ: ð5:14Þ u1 ’ aðb13 b31 Þ þ k 1 x1 ; a ¼ 4 Thus the result approves the result introduced in El-Gohary (2001b) and Krementulo (1977). This confirms the effectiveness of our solution. Moreover the results introduced in Tsiotras (1994) can be achieved by the assuming ki = 1 (i = 1, 2, 3), c = 1 of control gains in (5.12). This proves that the present results generalize and unify many pervious results. At this end from (5.8) and (5.12), we conclude that, the linearity and nonlinearity of the control law depend upon both of the Lyapunov function and rotors friction. 6. Modified-Rodrigues parameters Although the Cayley–Rodrigues parameters eliminate the constraint (3.4) imposes on the Euler parameters but they have the disadvantage of becoming unbounded when k0 = 0 and they do not allow eigenaxis rotations greater than 180. For these causes we will define the following Modified-Rodrigues parameters: qi ¼
ki 1 þ k0
ði ¼ 1; 2; 3Þ:
ð6:1Þ
The kinematics equation (3.2) in terms of these parameters are given by 3 X 1 2q_ 1 ¼ x3 q2 x2 q3 þ x1 q þ q1 xi qi 2 i¼1
ð123Þ;
ð6:2Þ
where q ¼ 1 q21 q22 q23 . Indeed we can easily prove that the parameters qi (i = 1, 2, 3) can be viewed as components of the vector U ~ q ¼ ^l tan ; 4
ð6:3Þ
which is well-defined for all eigenaxis rotations in the range 0 6 U < 360. Now, we wish to employ the control moments ui (i = 1, 2, 3) to make the rigid body motion is global asymptotical stable about its equilibrium state that determined by xi ¼ 0;
qi ¼ 0;
ui ¼ li U_ i0 ¼ li hi0 I 1 ði ¼ 1; 2; 3Þ: i ¼ const:
Theorem 6.1. The nonlinear control law ui ¼ ðk i
li C i I 1 i Þxi
þ cqi 1 þ
3 X s¼1
! q2s
þ I 1 i li
3 X
hs csi ð~ qÞ;
i ¼ 1; 2; 3
s¼1
globally asymptotically stabilizes the rigid body motion about the equilibrium state (6.4).
ð6:4Þ
ð6:5Þ
A. El-Gohary / Mechanics Research Communications 33 (2006) 818–829
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Proof. The proof of this theorem is based on the use of the following Lyapunov function: 2V ¼
3 X
ðC i I i Þx2i þ 4cq2i :
ð6:6Þ
i¼1
The time derivative of the Lyapunov function (6.6) along the trajectories of the closed-loop system (3.10), (6.2) and (6.5) takes the form V_ ¼
3 X
k i x2i 6 0
ð6:7Þ
i¼1
this implies that the rigid body motion is only stable about the equilibrium state (6.4) in the Lyapunov sense. Similarly the asymptotic stability can be proved as in theorem (4.1). h Fig. 4 displays the rigid body angular velocity vector, the Euler attitude parameters, and the rotors angular velocities using feedback control law (6.5) for the values ~ qð0Þ ¼ p1ffiffi2 ð1; 1; 0Þ of Modified-Rodrigues parameters. When the rotors friction is neglected, the control law (6.5) reduces to ! 3 X 2 ui ¼ k i xi þ cqi 1 þ qs ; i ¼ 1; 2; 3; ð6:8Þ s¼1
which is a nonlinear function of Modified-Rodrigues parameters. Theorem 6.2. The nonlinear control law 1 ui ¼ ðk i li C i I 1 i Þxi þ cqi þ I i li
3 X
hs csi ð~ qÞ;
i ¼ 1; 2; 3
ð6:9Þ
s¼1
globally asymptotically stabilizes the rigid body about the equilibrium state (6.4). Proof. The proof of this theorem is based on the use of the following Lyapunov function: ! 3 3 X X 2 2 2V ¼ ðC i I i Þxi þ 2c ln 1 þ qi : i¼1
ð6:10Þ
i¼1
The total time derivative of the Lyapunov function (6.10) along the trajectories of the closed-loop system (3.10), (6.2) and (6.9) takes the form V_ ¼
i X
k i x2i
ð6:11Þ
i¼1
this implies that the rotational motion of the body is only stable in the Lyapunov sense. Also the asymptotic stability can be proves using the same method of theorem (4.1). h
Fig. 4. (a) Rigid body angular velocity, (b) Modified-Rodrigues parameters and (c) rotor angular velocities.
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A. El-Gohary / Mechanics Research Communications 33 (2006) 818–829
Fig. 5. (a) Rigid body angular velocity, (b) Modified-Rodrigues parameters and (c) rotor angular velocities.
When the rotors friction is neglected, the control law (6.9) reduces to ui ¼ k i xi þ cqi ;
i ¼ 1; 2; 3;
ð6:12Þ
which is a linear function of Modified-Rodrigues parameters. This results support the fact that the linearity and nonlinearity of the control law depend not only on the Lyapunov function structure but also the rotors friction. Fig. 5 displays the rigid body angular velocity vector, the Euler attitude parameters, the rotors angular velocities using feedback control law (6.9) for the values ~ qð0Þ ¼ ð0; 1; 1:2Þ of Modified-Rodrigues parameters. At this end we conclude that when the rotors friction is neglected the linearity and nonlinearity of the control law depend upon the choice of the Lyapunov function as shown from (5.8) and (5.12); (6.8) and (6.12). Also as an expected results the rotors friction does not affect on the asymptotic stability of the rigid body motion since this forces consider internal forces of prescribed mechanical system. 7. Conclusion A new class of global asymptotic stabilizing control laws that stabilize the rigid about one of its equilibrium states is obtained in terms of Euler parameters, Cayley–Rodrigues parameters and Modified-Rodrigues parameters. The linearity and nonlinearity of the stabilizing control law depend on the Lyapunov function only if the rotors friction is neglected. Many special cases of the obtained results are presented. Also, numerical simulations of the controlled rigid body and rotor system are introduced. References El-Gohary, A., 1996. On the stability of an equilibrium position and rotational motion of a gyrostat. Mech. Res. Commun. 24 (4), 457– 462. El-Gohary, A., 1997. On the stability of relative programmed motion of Satellite-gyrostat. Mech. Res. Commun. 25 (4), 371–379. El-Gohary, A., 2000. On the control of programmed motion of a rigid body containing moving masses. Int. J. Non-Linear Mech. (35), 27– 35. El-Gohary, A., 2001a. Global stability of the rotational motion of a rigid body containing moving masses. Int. J. Non-Linear Mech. (36), 663–669. El-Gohary, A., 2001b. Optimal stabilization of the equilibrium positions of a rigid body using rotors. Chaos Solitons Fractals (12), 2007– 2014. El-Gohary, A., 2002. Global stabilization of a rotational motion of a rigid body using rotors system. Appl. Math. Comput. 133, 297–307. El-Gohary, A., 2005a. Optimal control of the rotational motion of a rigid body using Euler parameters with the help of rotors system. Eur. J. Mech. A—Solids 24, 111–221. El-Gohary, A., 2005b. Optimal control of an equilibrium position of a rigid body using rotors system with friction forces. Chaos Solitons Fractals 23, 1585–1597. El-Gohary, A., 2005c. Optimal control of rigid body motion with the help of rotors using stereographic coordinates. Chaos Solitons Fractals 25 (5), 1229–1244. Krementulo, V., 1977. Stabilization of Steady Motion of a Rigid Body using Rotating Masses. Nauka Press, Moscow (in Russian). Levine, W., 1995. The Control Handbook, A CRC Handbook Published in Cooperation with IEEE Press. Lure, L., 1961. Analytical Mechanics. Fizmatgiz, Moscow. Moore, R., 1994. Fundamentals of Space Systems. Oxford University Press.
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Rotea, M., 1998. Suboptimal control of rigid body motion with quadratic cost. Dyn. Control 8, 55–80. Tsiotras, P., 1994. New control laws for the attitude stabilization of rigid bodies. In: 13th IFAC Symposium on Automatic Control in Aerospace, September 12–16, Palo Alto, CA, pp. 316–312. Tsiotras, P., 1996. Stabilization and optimality results for the attitude control problem. J. Guidance Control Dyn. 19 (4).