- Email: [email protected]
Stabilization of a Class of 2-DOF Underactuated Mechanical Systems Via Lyapunov's Direct Approach
8th IFAC Symposium on Nonlinear Control Systems University of Bologna, Italy, September 1-3, 2010
Stabilization of a Class of 2-DOF Underactuated Mechanical Systems Via Lyapunov’s Direct Approach Turker Turker ∗ Haluk Gorgun ∗ Galip Cansever ∗ ∗
Department of Control and Automation Engineering, Yildiz Technical University, Istanbul, Turkey (e-mails: {turker,gorgun,cansever}@yildiz.edu.tr).
Abstract: We study a class of 2-DOF underactuated mechanical systems(UAMS) where the objective is to obtain a design framework for stabilization. We propose a stabilization procedure via Lyapunov’s direct approach to a class of UAMS employing partial feedback linearization and a change of coordinates. Asymptotic stability of unstable equilibrium point has been achieved in a large region of attraction and we make use of La Salle’s theorem to prove it. Designed controller scheme has been successfully illustrated through simulations on two different UAMS namely the pendulum on a cart and the inertia wheel pendulum. 1. INTRODUCTION Nonlinear control of fully actuated mechanical systems (FAMS) have been examined in past few decades and many control techniques have been developed in this area such as feedback linearization, passivity based control, Lyapunov based control etc. On the other hand, control of underactuated mechanical systems (UAMS) have been considered densely for last few years, and many researchers have worked on nonlinear control techniques which were inspired by the control schemes of FAMS. UAMS are defined as the systems with fewer independent actuators than degrees of freedom to be controlled. UAMS suffer from lack of very useful some system properties that FAMS have. The necessity of control of UAMS arises in many practical applications such as satellites, underwater vehicles, ships, walking robots etc. (see Fantoni et al. [2002] and OlfatiSaber [2001] for more details). There are also many UAMS which were designed for experiments in control laboratories. Pendulum on a cart, rotary pendulum, ball and beam, inertia wheel pendulum etc. can be counted in this group of systems. Due to the lack of control input to the unactuated coordinate(s), nonlinear control design for UAMS is very difficult comparing to FAMS. UAMS are not fully feedback linearizable and many well-known control methods can not be directly applied to these systems. In Spong [1996], collocated partial feedback linearization was introduced and after that control of a class of UAMS has achieved by energy based control. Olfati-Saber [2001] also used collocated partial feedback linearization and an additional change of coordinates to transform the system into the normal form. Then, the stabilization procedure was concluded by backstepping method. Another approach for the control of UAMS has introduced in Bloch et al. [2000, 2001] which is called controlled lagrangians. They proposed to modify form of kinetic and potential energy of uncontrolled dynamics to controlled Lagrangians. Ortega et al. [2002] (see also Estern et al. [2000], Ortega et al. [2002], Acosta et al. 978-3-902661-80-7/10/$20.00 © 2010 IFAC
[2005]) proposed a method namely interconnection and damping assignment passivity based control (IDA-PBC) for port-hamiltonian systems and applied this method to the UAMS. IDA-PBC can be basically interpreted as shaping the system’s total energy and injecting damping to the obtained system. To achieve energy shaping, the authors defined target dynamics and modify the inertia matrix for desired kinetic energy of the system, and shaped the potential energy of the system to assign the desired equilibrium point. A very interesting method for the global stabilization of a class of UAMS was introduced by Fantoni et al. [2000]. In this method the system simply brought to a certain homoclinic orbit to obtain a global asymptotic stability. More recently, Lyapunov’s direct method has been used to stabilize a class of UAMS in White et al. [2006, 2007]. In this approach, after proposing a candidate Lyapunov function, the stabilizing controller has been designed via proposed Lyapunov function. In order to obtain the time derivative of the Lyapunov function being negative semi definite, several differential equations have been solved which were formed by three matching conditions. Although (local) asymptotic stability has been achieved, the designed controller was dynamic. Ibanez [2009] has given a procedure to obtain a static feedback controller for the ball and beam system which was derived from following White and his coworkers approach. Under the direction of these methods, this paper addressed the problem of achieving general stabilization of a class of 2-DOF UAMS with a static feedback controller. Accordingly the stabilizing controller has been produced via partial feedback linearization and Lyapunov’s direct method with suitable and simplifying coordinate changes. 2. PROBLEM FORMULATION Dynamic equations of 2 degrees of freedom UAMS can be described as following.
1128
10.3182/20100901-3-IT-2016.00256
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
M11 (q)¨ q1 + M12 (q)¨ q2 + H1 (q, q) ˙ =0
(1)
q1 + M22 (q)¨ q2 + H2 (q, q) ˙ =u M12 (q)¨
(2)
where q ∈ R2 is generalized coordinates, u is the control input, Mij (q) are the elements of inertia matrix (M (q)) ˙ which is positive definite and symmetric and Hk (q, q) are the functions which can be aparted as Hk (q, q) ˙ = hkp (q, q) ˙ + hkq (q) for k = 1, 2 containing centrifugal and coriolis effects, and gravity terms. We will now try to change the structure of matrix M to a diagonal matrix (moreover to the identity) by change of coordinates beginning with partial feedback linearization (see Spong [1996]). In order to apply partial feedback linearization, q¨2 can be obtained from (1) as, 1 q¨2 = (−M11 q¨1 − H1 ). M12
We now define a transformation as follows to move along (see Olfati-Saber [1998]), (6)
By taking derivative of (6) twice, d q¨r = γ (q1 )¨ q1 + q¨2 + (γ " (q1 ))q˙1 . (7) dt ! q M11 (θ) T dθ, qs = q1 , q¯ = [qr qs ] , After defining γ(q1 ) = 0 1 M 12 (θ) and setting the control input as following, "
M11 M22 M22 )u1 + (H2 (q, q) ˙ − H1 (q, q)) ˙ (8) M12 M12
one can obtain the following equations from (4) and (5), q¨r −
q , q¯˙) d " h1 (¯ (γ (q1 ))q˙s + =0 dt M12 (¯ q) q¨s = u1
" # " # " # ( −h1q ηr ) " # p˙ r 0 c1 c2 p r + u = + M12 1 2 p˙s c3 c4 p s g2 $%&' $ %& ' $ %& ' C
G
where u1 is the new control input. In order to simplify further analysis, we denote the following change of coordinates.
(13)
F
where c1 and c2 can be determined as following, q , pr ) η˙r h1p (¯ − ηr pr M12 q , ps )ηr d(γ " (qs )) ηr h1p (¯ c2 = − . dt ηs ps M12 ηs
(14) (15)
We pursue from here to the procedure which aims to design stabilizing controller. 3. CONTROL DESIGN In order to find a stabilizing controller, the Lyapunov function is of the form V (¯ q , p) =
1 T p Kp + φ(¯ q) 2
(16)
where K = K T > 0 is a 2x2 constant matrix, φ is a scalar function that has to be positive definite with local minimum at the point which the system will be driven to. Taking time derivative of the Lyapunov function yields V˙ = pT K p˙ + pT η −1 ∇q¯φ.
(17)
After substituting (13) into (17), a straight forward calculation gives us V˙ = pT $%&' KC p + pT (KG + η −1 ∇q¯φ) +pT KF u2 . $ %& ' W
(18)
Z
If we achieve to compose a skew-symmetric W (¯ q , p) and Z(¯ q ) = 0, then the time derivative of the Lyapunov function becomes
(9) (10)
(12)
where c3 ,c4 ,g2 and u2 are design parameters which will be selected. After inserting it in (10), one can rewrite the dynamics of p as,
(3)
M11 H1 (q, q) ˙ q¨1 + q¨2 + = 0 (4) M12 M12 M11 M22 M22 )¨ q1 + (H2 (q, q) ˙ − H1 (q, q)) ˙ = u(5) (M12 − M12 M12
u = (M12 −
1 η˙ s ps (− + u 2 + c3 p r + c4 p s + g 2 ) ηs ηs
c1 =
Note that to proceed (3), M12 should be nonzero for all q. However, that is not an issue, because, if it is equal to zero that means inertia matrix is already in diagonal form. One can obtain the following equations by plugging (3) into (2), dividing (1) by M12 and rearranging the system equations.
qr = γ(q1 ) + q2 .
u1 =
V˙ = pT KF u2 ,
(19)
which easily can be made negative semi-definite by choosing an appropriate u2 , so that the system becomes at least stable. Notably, next challenge is finding η such that W is skew-symmetric and Z = 0. Defining K as following
p = η(¯ q )q¯˙
(11)
"
k k K= 1 2 k2 k3
T
q ) = diag[ηr (¯ q ); ηs (¯ q )], ηr and ηs where p = [pr ps ] , η(¯ denote non-zero functions which will be assigned later. By defining u1 in (10) as following,
one can compute
1129
#
(20)
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
W =
"
# k1 c1 + k2 c3 k1 c2 + k2 c4 . k2 c1 + k3 c3 k2 c2 + k3 c4
(21)
From (14), (15) and (21) with satisfying [W ]ii = 0, c3 and c4 can be calculated as k1 c1 k2 k2 c4 = − c2 . k3 c3 = −
(22) (23)
Once c3 and c4 calculated they can be plugged into the other entries of W , in order to find η that satisfies [W ]12 + [W ]21 = 0 which constitutes a differential equation with two unknowns of η c2 c1 (k1 k3 − k22 )( − ) = 0. k3 k2
(24)
This differential equation has two unknowns namely ηr and ηs one of which can be selected as a free parameter to solve it. Let’s consider now the vector Z satisfying
Z=
"
k1 k2 k2 k3
# ( −h1q ηr ) M12 g2
∂φ 1 0 r + ηr 1 ∂q ∂φ . 0 ηs ∂qs
1 h1q ηr k1 1 ∂φ (− + ). k2 M12 ηr ∂qr
(25)
a¨ q1 − b cos(q1 )¨ q2 − bg sin(q1 ) = 0
d ! a sin(qs )q˙s (γ (qs )) = − . dt b cos2 (qs )
4. EXAMPLES In order to show effectiveness of the proposed stabilization procedure, we applied it on two different well-known 2DOF UAMS namely, pendulum on a cart and inertia wheel pendulum in this section.
η˙ r a sin(qs )q˙s ηr + = 0. k2 ηr k3 b cos2 (qs )ηs
ml ml cos(q1 )¨ g sin(q1 ) = 0 (28) q2 − 2 2 ml ml cos(q1 )¨ sin(q1 )q˙12 = u (29) q1 + (m + M )¨ q2 + − 2 2 J q¨1 −
where q1 and q2 denote pendulum angle from the up vertical and cart position . And J, m, l, M, g denote inertia,
(31)
(32)
(33)
Equation (33) contains two design parameters namely ηr and ηs , and a solution to that differential equation can be found via assigning one of them freely and then solving the obtained equation. By taking ηs =
1 cos2 (qs )
(34)
equation (33) turns out
4.1 Pendulum on a Cart The dynamic equations of the pendulum on a cart (Fig.1) can be given in the form of (1) and (2) as,
=u
(30)
Substituting (32) in (24) we get following ordinary differential equation
k3 h1q ηr k1 1 ∂φ 1 ∂φ h1q ηr k2 − (− + )+ = 0. (27) M12 k2 M12 ηr ∂qr ηs ∂qs
After solving (24) and (26) under the constraints φ(¯ q) > 0 and ηr (¯ q ), ηs (¯ q ) $= 0 with local minimum at the desired equilibrium, the control signal can be computed using (8) and (12).
b sin(q1 )q˙12
Note that M12 = −b cos(q1 ) has to be nonzero. So that, the pendulum angle has to be in the interval of (−π/2, π/2) which constructs the region of attraction. It can be easily observed that from (30) ! q anda (31), h1p = 0, h1q = −bg sin(qs ) and γ = 0 s − b cos(θ) dθ from which we can get the following
(26)
A partial differential equation is obtained by using (26) in the second row of Z −
mass and length of the pendulum, mass of the cart, acceleration of gravity, respectively. By defining a = J, b = ml 2 , c = m + M , (28) and (29) can be rewritten as,
−b cos(q1 )¨ q1 + c¨ q2 +
In order to set Z to be zero, g2 can be calculated from the first row of (25) g2 = −
Fig. 1. Pendulum on a Cart
η˙ r a sin(qs )ηr q˙s + = 0. k2 ηr bk3
(35)
Since the differential equation obtained has values depending only on qs , ηr can be selected as depending on qs . If (35) is rewritten under that fact, !
ηr ps a sin(qs )ηr ps + =0 k2 ηr ηs bk3 ηs and whose solution can be obtained as
1130
(36)
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
bk3 . ak2 cos(qs )
(37)
1.5 1 pendulum angle (rad)
ηr = −
Using (37) in (27) results a partial differential equation k3 ∂φ g(k1 k3 − k22 ) sin(qs )ηr 1 ∂φ − + = 0. k2 cos(qs ) k2 ηr ∂qr ηs ∂qs
0.5 0 ï0.5 ï1
(38)
ï1.5
φ(¯ q ) can be solved from (38) as,
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t (sec)
12
14
16
18
20
0
cart position (m)
ï2
bg(k1 k3 − k22 ) φ(¯ q ) =d2 + 3ak3 cos3 (qs ) + d3 (qr −
d2 = −
bgk3 (k1 k3 − k22 ) 3ak22
bgk3 (k1 k3 − k22 ) 1 − 1) ( 3 3ak22 cos (qs ) 2aArctanh(tan( q2s )) 2 + d3 (qr − ) . b
(
ï12
Fig. 2. Pendulum Angle and Cart Position of the Pendulum on a Cart System u2 = −kF T Kp
V˙ = −kpT KF F T Kp (41)
Ωc = {(q, p) ∈ ((−π/2, π/2) × R3 |V (q, p) < c)}
(42)
(44)
which is negative semi-definite. Thus, V (¯ q , p) is nonincreasing implying q¯ and p are bounded in the region of attraction. Note that q is also bounded, since qr and γ(q1 ) is bounded in (q, p) ∈ ((−π/2, π/2) × R3 ). Let’s define the set Ωc¯ ∈ ((−π/2, π/2) × R3 ) as follows
cg sin(q1 ) ac + (−b cos(q1 ) + ) cos(q1 ) c cos(q1 )
1 (−ηs" q˙12 + c3 pr + c4 ps + g2 + u2 )). ηs
(43)
for some constant k > 0, one can obtain the time derivative of the Lyapunov function as
The control input to the system can be constructed as u = b sin(q1 )q˙12 −
ï8
(40)
φ(¯ q ) can be rewritten as
φ(¯ q) =
ï6
ï10
2ak32 Arctanh(tan( q2s )) 2 ) (39) b
with an integral constants d2 which need to be assigned to assure φ(¯ q ) > 0 and d3 > 0. By choosing
ï4
(45)
with c¯ = sup{c > 0 : V (q, p) < c|Ωc is bounded}.
where
(46)
To proceed, we need to show the largest invariant set in Ωc¯.Let’s concentrate again on (44) which can be rewritten by using (11) and (6) as,
1 k1 ηs = , c3 = − tan(qs )q˙s , c4 = − tan(qs )q˙s , cos2 (qs ) k2 k1 k3 bg sin(qs ) g2 = − ak22 cos2 (qs ) 2aArctanh(tan( q2s )) 2d3 a cos(qs ) ). (qr − + bk3 b
V˙ = −k(k2 pr + k3 ps )2 = −k((k2 γ " (q1 ) + k3 ηs )q˙1 + k2 q˙2 ). (47)
and u2 to be assigned. We are ready to state our result now. Proposition 1. The pendulum on a cart system (30), (31) in closed loop with the control input given by (42) guarantees asymptotically stable equilibrium at zero with the region of attraction containing the set (q, p) ∈ ((−π/2, π/2)xR3 ).
It can be concluded from (47) that V˙ is equal to 0 only ! when q˙2 = (k2 γ (qk1 2)+k3 ηs ) q˙1 which is satisfied when
Proof. In order to prove the asymptotic stability of the closed loop system with q1 ∈ (−π/2, π/2), we invoke La Salle’s invariant set principle(see Khalil [1998]). The Lyapunov function (16) is positive definite in the region of attraction with φ(¯ q ) given in (41) and positive definite and symmetric matrix K ∈ R2×2 defined in (20). We assign 1131
i. q˙1 $= q˙2 In this case either k2 γ " (q1 ) + k3 ηs = 0 or q˙2 = (k2 γ ! (q1 )+k3 ηs ) q˙1 has to be provided. Note that any k2 equilibrium point with q1 $= 0 in the region of attraction can be obtained with only the control input compensates the gravity force, however, that means a force on q2 which causes q¨2 $= 0. So that, V˙ = 0 with q˙1 $= q˙2 can only be true for a moment which implies V˙ can not stay equal to 0 when q˙1 $= q˙2 . ii. q˙1 = q˙2 This case is only provided under the velocities of both coordinates are equal to 0 which implies also
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
−
k3 ∂φ 1 ∂φ mgl(k1 k3 − k22 ) sin(qs )ηr − + = 0, (51) k2 I2 k2 ηr ∂qr ηs ∂qs
whose solution can be found by fixing ηr and ηs to 1 as, φ(¯ q ) = d4 −
mgl(k1 k3 − k22 ) cos(qs ) k2 +d5 ( qr +qs )2 . (52) k2 I2 k3
We define d4 =
q¨1 = q¨2 = 0. Recalling the dynamic equations (30) and (31) with the control input given by (42) and (43), one can observe that these conditions only true when q1 = q2 = 0. Finally it is easy to show that from (42) and (43), the control input u is also bounded under the considerations above. This concludes the proof. In order to test the performance of the stabilization procedure, we simulated the designed controller for the R pendulum on a cart system on Matlab Simulink% . The parameters a,b,c of the pendulum on a cart model have chosen as 1 for the sake simplicity. The design parameters of the control signal k and d3 have been set to 100 and the entries of the constant matrix [K] were fixed to k1 = 10,k2 = 1, and k3 = 2. Initial values for (q1 (0), q2 (0), q˙1 (0), q˙2 (0)) were set to (π/2 − π/36, 0, 0, 0) which is very close to horizontal for the pendulum with zero cart position and zero velocities. Figure 2 shows the results for pendulum angle and cart position. 4.2 Inertia Wheel Pendulum The dynamic equations of the inertia wheel pendulum (Fig.3) can be given as,
I2 q¨1 + I2 q¨2 = u
φ(¯ q) =
(49)
Consequently the control input to the system is designed directly as u = mgl sin(q1 ) + I1 (g2 + u2 ). (55) Proposition 2. The inertia wheel pendulum system (48,49) in closed loop with the control input given by (55) guarantees asymptotically stable equilibrium at zero with the region of attraction the whole state space minus a set of initial conditions of q2 . Proof. The proof follows the same argument as in the proof of Proposition 1. The control input u2 u2 = −kF T Kp
(56)
for some constant k > 0, provides the time derivative of the Lyapunov function to be negative semi-definite. Since V (¯ q , p) is non-increasing, q¯ and p are bounded, and resulting q is also bounded, satisfies k2 (I1 + I2 ) V˙ = −k(( + k3 )q˙1 + k2 q˙2 ). I2
(57)
It is clear that V˙ is equal to zero only when q˙2 = − k12 ( k2 (II12+I2 ) + k3 )q˙1 is satisfied. We have concluded that it is satisfied only when q˙1 = q˙2 = 0 which was already discussed in the proof of Proposition 1. Notably, pendulum has two equilibrium at q1 = 0 and q1 = π. The task is now to prove that q = 0 is the only equilibrium point in the invariant set that we can claim asymptotic stability with La Salle. We plug (26) into (55), and we get bounded control input mglk1 I1 sin(q1 ) k2 I2 2d5 I1 k2 (I1 + I2 ) + k3 I2 k2 − ( q1 + q2 ). k3 k3 I2 k3
(50)
evidently, ηr is constant. On the other hand ηs does not have any effect so far. Thus we can use it, if it is needed in partial differential equation which is obtained from (27),
mgl(k1 k3 − k22 ) k2 (1 − cos(qs ))+ d5 ( qr + qs )2 (54) k2 I2 k3
which is clearly positive definite with respect to q¯ with d5 > 0 and k2 > 0.
(48)
where q1 and q2 denote pendulum angle from the up vertical and angular position of the wheel. I1 , I2 , m, l and g denote, positive inertia constants of the pendulum and wheel, mass and length of the pendulum, acceleration of gravity, respectively. Note that M12 = I2 and it is nonzero. It can be easily observed that from (48) and (49), h1p = 0, 2 h1q = −mgl sin(qs ) and γ = I1I+I q1 which is affine on q1 . 2 By recalling (24) one can obtain, η˙r =0 k3 ηr
(53)
φ(¯ q ) can be rewritten as,
Fig. 3. Inertia Wheel Pendulum
(I1 + I2 )¨ q1 + I2 q¨2 − mgl sin(q1 ) = 0
mgl(k1 k3 − k22 ) , k2 I2
u = mgl sin(q1 ) + I1 u2 −
(58)
When q˙1 = q˙2 = 0 and q1 = π, from the dynamics of the inertia wheel pendulum and u, we can realize that
1132
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
pendulum angle (rad)
4 3 2 1 0 ï1
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t (sec)
12
14
16
18
20
2
wheel angle (rad)
0 ï2 ï4 ï6 ï8
Fig. 4. Pendulum and Wheel Angle of the Inertia Wheel Pendulum System the system can not stay at this equilibrium point unless 2 )+k3 I2 )π. Based on this finding, with q2 (0) = − kk32 ( k2 (I1 +I k3 I2 any q2 (0) ∈ R (except some values which makes true the above equality), q1 = π is not an equilibrium with the proposed control design. With the same arguments of the proof of Proposition 1, the other equilibrium point where q˙1 = q˙2 = 0 and q1 = 0 is asymptotically stable from dynamics of the system. That concludes the proof. Designed controller for the inertia wheel pendulum has R been simulated on Matlab Simulink% . System parameters has been assigned as I1 = 1 and I2 = 2. The design parameters of the control signal k and d5 have been set to -10 and 10 respectively, and the entries of [K] have been fixed to k1 = 10,k2 = 1, and k3 = 2. Initial values for (q1 (0), q2 (0), q˙1 (0), q˙2 (0)) were set to (π, −2π, 0, 0) which is a stable equilibrium point for the open loop system. Figure 4 shows the results for pendulum and cart angles. 5. CONCLUSION A stabilization procedure for a class of 2-DOF UAMS has been presented with two examples namely pendulum on a cart and inertia wheel pendulum, and simulation results has shown the effectiveness of the proposed method. It should be noted that the region of attraction of the closed loop system is shaped by the Mij of inertia matrix for 2DOF UAMS. For future work, this study can be extended to more general class of UAMS. ACKNOWLEDGEMENTS The authors gratefully acknowledges YTU, BAPK for supporting this study with the project number 2010-0402-KAP01.
A.M. Bloch , D.E. Chang, N.E. Leonard, J.E. Marsden. Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping. IEEE Transactions on Automatic Control, volume 46, issue 10,pages 1556– 1571, 2001. C.A. Ibanez. The Lyapunov direct method for the stabilisation of the ball on the actuated beam International Journal of Control, volume 82, issue 12,pages 2169– 2178, 2009. F.G. Estern, R. Ortega, F.R. Rubio, J. Aracil. Stabilization of a Class of Underactuated Mechanical Systems Via Total Energy Shaping. Proceedings of the IEEE Conference on Decision and Control, pages 1137–1143, 2001. H. Khalil. Nonlinear Systems, Upper Saddle River, NJ: Prentice-Hall, 2002. I. Fantoni, R. Lozano, M.W. Spong. Energy based control of the Pendubot IEEE Transactions on Automatic Control, volume 45, issue 4,pages 725–729, 2000. I. Fantoni, R. Lozano. Nonlinear control for underactuated mechanical systems, Springer-Verlag London, Communications and Control Engineering Series, 2002. J. Acosta, R. Ortega, A. Astolfi and A. Mahindrakar. Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one IEEE Transactions on Automatic Control, volume 50, issue 12,pages 1936–1955, 2005. M.W. Spong. Energy Based Control of a Class of Underactuated Mechanical Systems IFAC World Congress, 1996. R. Olfati-Saber. Nonlinear Control of Underactuated Systems with Application to Robotics and Aerospace Vehicles Massachusetts Institute of Technology, Department of Electrical and Computer Science, Ph.D. thesis, 2001. R. Olfati-Saber. Nonlinear Control and Reduction of Underactuated Systems with Symmetry I: Actuated Shape Variables Case Proceedings of the IEEE Conference on Decision and Control, pages 4158–4163, 2001. R. Ortega, A. van der Schaft, B. Maschke and G. Escobar. Interconnection and damping assignment passivitybased control of port-controlled hamiltonian systems Automatica, volume 38, no 4,pages 1137–1143, 2002. R. Ortega, M. Spong, F. Gomez and G. Blankenstein. Stabilization of underactuated mechanical systems via interconnection and damping assignment IEEE Transactions on Automatic Control, volume 47, issue 8,pages 1218–1233, 2002. W.N. White, M. Foss, and G. Xin. A Direct Lyapunov Approach for a Class of Underactuated Mechanical Systems Proceedings of the American Control Conference, pages 14–16, 2006. W.N. White, M. Foss, and G. Xin. A Direct Lyapunov Approach for Stabilization of Underactuated Mechanical Systems Proceedings of the American Control Conference, pages 4817–4822, 2007.
REFERENCES A.M. Bloch , N.E. Leonard, J.E. Marsden. Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem. IEEE Transactions on Automatic Control, volume 45, issue 12,pages 2253– 2270, 2000. 1133
Stabilization of a Class of 2-DOF Underactuated Mechanical Systems Via Lyapunov’s Direct Approach Turker Turker ∗ Haluk Gorgun ∗ Galip Cansever ∗ ∗
Department of Control and Automation Engineering, Yildiz Technical University, Istanbul, Turkey (e-mails: {turker,gorgun,cansever}@yildiz.edu.tr).
Abstract: We study a class of 2-DOF underactuated mechanical systems(UAMS) where the objective is to obtain a design framework for stabilization. We propose a stabilization procedure via Lyapunov’s direct approach to a class of UAMS employing partial feedback linearization and a change of coordinates. Asymptotic stability of unstable equilibrium point has been achieved in a large region of attraction and we make use of La Salle’s theorem to prove it. Designed controller scheme has been successfully illustrated through simulations on two different UAMS namely the pendulum on a cart and the inertia wheel pendulum. 1. INTRODUCTION Nonlinear control of fully actuated mechanical systems (FAMS) have been examined in past few decades and many control techniques have been developed in this area such as feedback linearization, passivity based control, Lyapunov based control etc. On the other hand, control of underactuated mechanical systems (UAMS) have been considered densely for last few years, and many researchers have worked on nonlinear control techniques which were inspired by the control schemes of FAMS. UAMS are defined as the systems with fewer independent actuators than degrees of freedom to be controlled. UAMS suffer from lack of very useful some system properties that FAMS have. The necessity of control of UAMS arises in many practical applications such as satellites, underwater vehicles, ships, walking robots etc. (see Fantoni et al. [2002] and OlfatiSaber [2001] for more details). There are also many UAMS which were designed for experiments in control laboratories. Pendulum on a cart, rotary pendulum, ball and beam, inertia wheel pendulum etc. can be counted in this group of systems. Due to the lack of control input to the unactuated coordinate(s), nonlinear control design for UAMS is very difficult comparing to FAMS. UAMS are not fully feedback linearizable and many well-known control methods can not be directly applied to these systems. In Spong [1996], collocated partial feedback linearization was introduced and after that control of a class of UAMS has achieved by energy based control. Olfati-Saber [2001] also used collocated partial feedback linearization and an additional change of coordinates to transform the system into the normal form. Then, the stabilization procedure was concluded by backstepping method. Another approach for the control of UAMS has introduced in Bloch et al. [2000, 2001] which is called controlled lagrangians. They proposed to modify form of kinetic and potential energy of uncontrolled dynamics to controlled Lagrangians. Ortega et al. [2002] (see also Estern et al. [2000], Ortega et al. [2002], Acosta et al. 978-3-902661-80-7/10/$20.00 © 2010 IFAC
[2005]) proposed a method namely interconnection and damping assignment passivity based control (IDA-PBC) for port-hamiltonian systems and applied this method to the UAMS. IDA-PBC can be basically interpreted as shaping the system’s total energy and injecting damping to the obtained system. To achieve energy shaping, the authors defined target dynamics and modify the inertia matrix for desired kinetic energy of the system, and shaped the potential energy of the system to assign the desired equilibrium point. A very interesting method for the global stabilization of a class of UAMS was introduced by Fantoni et al. [2000]. In this method the system simply brought to a certain homoclinic orbit to obtain a global asymptotic stability. More recently, Lyapunov’s direct method has been used to stabilize a class of UAMS in White et al. [2006, 2007]. In this approach, after proposing a candidate Lyapunov function, the stabilizing controller has been designed via proposed Lyapunov function. In order to obtain the time derivative of the Lyapunov function being negative semi definite, several differential equations have been solved which were formed by three matching conditions. Although (local) asymptotic stability has been achieved, the designed controller was dynamic. Ibanez [2009] has given a procedure to obtain a static feedback controller for the ball and beam system which was derived from following White and his coworkers approach. Under the direction of these methods, this paper addressed the problem of achieving general stabilization of a class of 2-DOF UAMS with a static feedback controller. Accordingly the stabilizing controller has been produced via partial feedback linearization and Lyapunov’s direct method with suitable and simplifying coordinate changes. 2. PROBLEM FORMULATION Dynamic equations of 2 degrees of freedom UAMS can be described as following.
1128
10.3182/20100901-3-IT-2016.00256
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
M11 (q)¨ q1 + M12 (q)¨ q2 + H1 (q, q) ˙ =0
(1)
q1 + M22 (q)¨ q2 + H2 (q, q) ˙ =u M12 (q)¨
(2)
where q ∈ R2 is generalized coordinates, u is the control input, Mij (q) are the elements of inertia matrix (M (q)) ˙ which is positive definite and symmetric and Hk (q, q) are the functions which can be aparted as Hk (q, q) ˙ = hkp (q, q) ˙ + hkq (q) for k = 1, 2 containing centrifugal and coriolis effects, and gravity terms. We will now try to change the structure of matrix M to a diagonal matrix (moreover to the identity) by change of coordinates beginning with partial feedback linearization (see Spong [1996]). In order to apply partial feedback linearization, q¨2 can be obtained from (1) as, 1 q¨2 = (−M11 q¨1 − H1 ). M12
We now define a transformation as follows to move along (see Olfati-Saber [1998]), (6)
By taking derivative of (6) twice, d q¨r = γ (q1 )¨ q1 + q¨2 + (γ " (q1 ))q˙1 . (7) dt ! q M11 (θ) T dθ, qs = q1 , q¯ = [qr qs ] , After defining γ(q1 ) = 0 1 M 12 (θ) and setting the control input as following, "
M11 M22 M22 )u1 + (H2 (q, q) ˙ − H1 (q, q)) ˙ (8) M12 M12
one can obtain the following equations from (4) and (5), q¨r −
q , q¯˙) d " h1 (¯ (γ (q1 ))q˙s + =0 dt M12 (¯ q) q¨s = u1
" # " # " # ( −h1q ηr ) " # p˙ r 0 c1 c2 p r + u = + M12 1 2 p˙s c3 c4 p s g2 $%&' $ %& ' $ %& ' C
G
where u1 is the new control input. In order to simplify further analysis, we denote the following change of coordinates.
(13)
F
where c1 and c2 can be determined as following, q , pr ) η˙r h1p (¯ − ηr pr M12 q , ps )ηr d(γ " (qs )) ηr h1p (¯ c2 = − . dt ηs ps M12 ηs
(14) (15)
We pursue from here to the procedure which aims to design stabilizing controller. 3. CONTROL DESIGN In order to find a stabilizing controller, the Lyapunov function is of the form V (¯ q , p) =
1 T p Kp + φ(¯ q) 2
(16)
where K = K T > 0 is a 2x2 constant matrix, φ is a scalar function that has to be positive definite with local minimum at the point which the system will be driven to. Taking time derivative of the Lyapunov function yields V˙ = pT K p˙ + pT η −1 ∇q¯φ.
(17)
After substituting (13) into (17), a straight forward calculation gives us V˙ = pT $%&' KC p + pT (KG + η −1 ∇q¯φ) +pT KF u2 . $ %& ' W
(18)
Z
If we achieve to compose a skew-symmetric W (¯ q , p) and Z(¯ q ) = 0, then the time derivative of the Lyapunov function becomes
(9) (10)
(12)
where c3 ,c4 ,g2 and u2 are design parameters which will be selected. After inserting it in (10), one can rewrite the dynamics of p as,
(3)
M11 H1 (q, q) ˙ q¨1 + q¨2 + = 0 (4) M12 M12 M11 M22 M22 )¨ q1 + (H2 (q, q) ˙ − H1 (q, q)) ˙ = u(5) (M12 − M12 M12
u = (M12 −
1 η˙ s ps (− + u 2 + c3 p r + c4 p s + g 2 ) ηs ηs
c1 =
Note that to proceed (3), M12 should be nonzero for all q. However, that is not an issue, because, if it is equal to zero that means inertia matrix is already in diagonal form. One can obtain the following equations by plugging (3) into (2), dividing (1) by M12 and rearranging the system equations.
qr = γ(q1 ) + q2 .
u1 =
V˙ = pT KF u2 ,
(19)
which easily can be made negative semi-definite by choosing an appropriate u2 , so that the system becomes at least stable. Notably, next challenge is finding η such that W is skew-symmetric and Z = 0. Defining K as following
p = η(¯ q )q¯˙
(11)
"
k k K= 1 2 k2 k3
T
q ) = diag[ηr (¯ q ); ηs (¯ q )], ηr and ηs where p = [pr ps ] , η(¯ denote non-zero functions which will be assigned later. By defining u1 in (10) as following,
one can compute
1129
#
(20)
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
W =
"
# k1 c1 + k2 c3 k1 c2 + k2 c4 . k2 c1 + k3 c3 k2 c2 + k3 c4
(21)
From (14), (15) and (21) with satisfying [W ]ii = 0, c3 and c4 can be calculated as k1 c1 k2 k2 c4 = − c2 . k3 c3 = −
(22) (23)
Once c3 and c4 calculated they can be plugged into the other entries of W , in order to find η that satisfies [W ]12 + [W ]21 = 0 which constitutes a differential equation with two unknowns of η c2 c1 (k1 k3 − k22 )( − ) = 0. k3 k2
(24)
This differential equation has two unknowns namely ηr and ηs one of which can be selected as a free parameter to solve it. Let’s consider now the vector Z satisfying
Z=
"
k1 k2 k2 k3
# ( −h1q ηr ) M12 g2
∂φ 1 0 r + ηr 1 ∂q ∂φ . 0 ηs ∂qs
1 h1q ηr k1 1 ∂φ (− + ). k2 M12 ηr ∂qr
(25)
a¨ q1 − b cos(q1 )¨ q2 − bg sin(q1 ) = 0
d ! a sin(qs )q˙s (γ (qs )) = − . dt b cos2 (qs )
4. EXAMPLES In order to show effectiveness of the proposed stabilization procedure, we applied it on two different well-known 2DOF UAMS namely, pendulum on a cart and inertia wheel pendulum in this section.
η˙ r a sin(qs )q˙s ηr + = 0. k2 ηr k3 b cos2 (qs )ηs
ml ml cos(q1 )¨ g sin(q1 ) = 0 (28) q2 − 2 2 ml ml cos(q1 )¨ sin(q1 )q˙12 = u (29) q1 + (m + M )¨ q2 + − 2 2 J q¨1 −
where q1 and q2 denote pendulum angle from the up vertical and cart position . And J, m, l, M, g denote inertia,
(31)
(32)
(33)
Equation (33) contains two design parameters namely ηr and ηs , and a solution to that differential equation can be found via assigning one of them freely and then solving the obtained equation. By taking ηs =
1 cos2 (qs )
(34)
equation (33) turns out
4.1 Pendulum on a Cart The dynamic equations of the pendulum on a cart (Fig.1) can be given in the form of (1) and (2) as,
=u
(30)
Substituting (32) in (24) we get following ordinary differential equation
k3 h1q ηr k1 1 ∂φ 1 ∂φ h1q ηr k2 − (− + )+ = 0. (27) M12 k2 M12 ηr ∂qr ηs ∂qs
After solving (24) and (26) under the constraints φ(¯ q) > 0 and ηr (¯ q ), ηs (¯ q ) $= 0 with local minimum at the desired equilibrium, the control signal can be computed using (8) and (12).
b sin(q1 )q˙12
Note that M12 = −b cos(q1 ) has to be nonzero. So that, the pendulum angle has to be in the interval of (−π/2, π/2) which constructs the region of attraction. It can be easily observed that from (30) ! q anda (31), h1p = 0, h1q = −bg sin(qs ) and γ = 0 s − b cos(θ) dθ from which we can get the following
(26)
A partial differential equation is obtained by using (26) in the second row of Z −
mass and length of the pendulum, mass of the cart, acceleration of gravity, respectively. By defining a = J, b = ml 2 , c = m + M , (28) and (29) can be rewritten as,
−b cos(q1 )¨ q1 + c¨ q2 +
In order to set Z to be zero, g2 can be calculated from the first row of (25) g2 = −
Fig. 1. Pendulum on a Cart
η˙ r a sin(qs )ηr q˙s + = 0. k2 ηr bk3
(35)
Since the differential equation obtained has values depending only on qs , ηr can be selected as depending on qs . If (35) is rewritten under that fact, !
ηr ps a sin(qs )ηr ps + =0 k2 ηr ηs bk3 ηs and whose solution can be obtained as
1130
(36)
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
bk3 . ak2 cos(qs )
(37)
1.5 1 pendulum angle (rad)
ηr = −
Using (37) in (27) results a partial differential equation k3 ∂φ g(k1 k3 − k22 ) sin(qs )ηr 1 ∂φ − + = 0. k2 cos(qs ) k2 ηr ∂qr ηs ∂qs
0.5 0 ï0.5 ï1
(38)
ï1.5
φ(¯ q ) can be solved from (38) as,
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t (sec)
12
14
16
18
20
0
cart position (m)
ï2
bg(k1 k3 − k22 ) φ(¯ q ) =d2 + 3ak3 cos3 (qs ) + d3 (qr −
d2 = −
bgk3 (k1 k3 − k22 ) 3ak22
bgk3 (k1 k3 − k22 ) 1 − 1) ( 3 3ak22 cos (qs ) 2aArctanh(tan( q2s )) 2 + d3 (qr − ) . b
(
ï12
Fig. 2. Pendulum Angle and Cart Position of the Pendulum on a Cart System u2 = −kF T Kp
V˙ = −kpT KF F T Kp (41)
Ωc = {(q, p) ∈ ((−π/2, π/2) × R3 |V (q, p) < c)}
(42)
(44)
which is negative semi-definite. Thus, V (¯ q , p) is nonincreasing implying q¯ and p are bounded in the region of attraction. Note that q is also bounded, since qr and γ(q1 ) is bounded in (q, p) ∈ ((−π/2, π/2) × R3 ). Let’s define the set Ωc¯ ∈ ((−π/2, π/2) × R3 ) as follows
cg sin(q1 ) ac + (−b cos(q1 ) + ) cos(q1 ) c cos(q1 )
1 (−ηs" q˙12 + c3 pr + c4 ps + g2 + u2 )). ηs
(43)
for some constant k > 0, one can obtain the time derivative of the Lyapunov function as
The control input to the system can be constructed as u = b sin(q1 )q˙12 −
ï8
(40)
φ(¯ q ) can be rewritten as
φ(¯ q) =
ï6
ï10
2ak32 Arctanh(tan( q2s )) 2 ) (39) b
with an integral constants d2 which need to be assigned to assure φ(¯ q ) > 0 and d3 > 0. By choosing
ï4
(45)
with c¯ = sup{c > 0 : V (q, p) < c|Ωc is bounded}.
where
(46)
To proceed, we need to show the largest invariant set in Ωc¯.Let’s concentrate again on (44) which can be rewritten by using (11) and (6) as,
1 k1 ηs = , c3 = − tan(qs )q˙s , c4 = − tan(qs )q˙s , cos2 (qs ) k2 k1 k3 bg sin(qs ) g2 = − ak22 cos2 (qs ) 2aArctanh(tan( q2s )) 2d3 a cos(qs ) ). (qr − + bk3 b
V˙ = −k(k2 pr + k3 ps )2 = −k((k2 γ " (q1 ) + k3 ηs )q˙1 + k2 q˙2 ). (47)
and u2 to be assigned. We are ready to state our result now. Proposition 1. The pendulum on a cart system (30), (31) in closed loop with the control input given by (42) guarantees asymptotically stable equilibrium at zero with the region of attraction containing the set (q, p) ∈ ((−π/2, π/2)xR3 ).
It can be concluded from (47) that V˙ is equal to 0 only ! when q˙2 = (k2 γ (qk1 2)+k3 ηs ) q˙1 which is satisfied when
Proof. In order to prove the asymptotic stability of the closed loop system with q1 ∈ (−π/2, π/2), we invoke La Salle’s invariant set principle(see Khalil [1998]). The Lyapunov function (16) is positive definite in the region of attraction with φ(¯ q ) given in (41) and positive definite and symmetric matrix K ∈ R2×2 defined in (20). We assign 1131
i. q˙1 $= q˙2 In this case either k2 γ " (q1 ) + k3 ηs = 0 or q˙2 = (k2 γ ! (q1 )+k3 ηs ) q˙1 has to be provided. Note that any k2 equilibrium point with q1 $= 0 in the region of attraction can be obtained with only the control input compensates the gravity force, however, that means a force on q2 which causes q¨2 $= 0. So that, V˙ = 0 with q˙1 $= q˙2 can only be true for a moment which implies V˙ can not stay equal to 0 when q˙1 $= q˙2 . ii. q˙1 = q˙2 This case is only provided under the velocities of both coordinates are equal to 0 which implies also
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
−
k3 ∂φ 1 ∂φ mgl(k1 k3 − k22 ) sin(qs )ηr − + = 0, (51) k2 I2 k2 ηr ∂qr ηs ∂qs
whose solution can be found by fixing ηr and ηs to 1 as, φ(¯ q ) = d4 −
mgl(k1 k3 − k22 ) cos(qs ) k2 +d5 ( qr +qs )2 . (52) k2 I2 k3
We define d4 =
q¨1 = q¨2 = 0. Recalling the dynamic equations (30) and (31) with the control input given by (42) and (43), one can observe that these conditions only true when q1 = q2 = 0. Finally it is easy to show that from (42) and (43), the control input u is also bounded under the considerations above. This concludes the proof. In order to test the performance of the stabilization procedure, we simulated the designed controller for the R pendulum on a cart system on Matlab Simulink% . The parameters a,b,c of the pendulum on a cart model have chosen as 1 for the sake simplicity. The design parameters of the control signal k and d3 have been set to 100 and the entries of the constant matrix [K] were fixed to k1 = 10,k2 = 1, and k3 = 2. Initial values for (q1 (0), q2 (0), q˙1 (0), q˙2 (0)) were set to (π/2 − π/36, 0, 0, 0) which is very close to horizontal for the pendulum with zero cart position and zero velocities. Figure 2 shows the results for pendulum angle and cart position. 4.2 Inertia Wheel Pendulum The dynamic equations of the inertia wheel pendulum (Fig.3) can be given as,
I2 q¨1 + I2 q¨2 = u
φ(¯ q) =
(49)
Consequently the control input to the system is designed directly as u = mgl sin(q1 ) + I1 (g2 + u2 ). (55) Proposition 2. The inertia wheel pendulum system (48,49) in closed loop with the control input given by (55) guarantees asymptotically stable equilibrium at zero with the region of attraction the whole state space minus a set of initial conditions of q2 . Proof. The proof follows the same argument as in the proof of Proposition 1. The control input u2 u2 = −kF T Kp
(56)
for some constant k > 0, provides the time derivative of the Lyapunov function to be negative semi-definite. Since V (¯ q , p) is non-increasing, q¯ and p are bounded, and resulting q is also bounded, satisfies k2 (I1 + I2 ) V˙ = −k(( + k3 )q˙1 + k2 q˙2 ). I2
(57)
It is clear that V˙ is equal to zero only when q˙2 = − k12 ( k2 (II12+I2 ) + k3 )q˙1 is satisfied. We have concluded that it is satisfied only when q˙1 = q˙2 = 0 which was already discussed in the proof of Proposition 1. Notably, pendulum has two equilibrium at q1 = 0 and q1 = π. The task is now to prove that q = 0 is the only equilibrium point in the invariant set that we can claim asymptotic stability with La Salle. We plug (26) into (55), and we get bounded control input mglk1 I1 sin(q1 ) k2 I2 2d5 I1 k2 (I1 + I2 ) + k3 I2 k2 − ( q1 + q2 ). k3 k3 I2 k3
(50)
evidently, ηr is constant. On the other hand ηs does not have any effect so far. Thus we can use it, if it is needed in partial differential equation which is obtained from (27),
mgl(k1 k3 − k22 ) k2 (1 − cos(qs ))+ d5 ( qr + qs )2 (54) k2 I2 k3
which is clearly positive definite with respect to q¯ with d5 > 0 and k2 > 0.
(48)
where q1 and q2 denote pendulum angle from the up vertical and angular position of the wheel. I1 , I2 , m, l and g denote, positive inertia constants of the pendulum and wheel, mass and length of the pendulum, acceleration of gravity, respectively. Note that M12 = I2 and it is nonzero. It can be easily observed that from (48) and (49), h1p = 0, 2 h1q = −mgl sin(qs ) and γ = I1I+I q1 which is affine on q1 . 2 By recalling (24) one can obtain, η˙r =0 k3 ηr
(53)
φ(¯ q ) can be rewritten as,
Fig. 3. Inertia Wheel Pendulum
(I1 + I2 )¨ q1 + I2 q¨2 − mgl sin(q1 ) = 0
mgl(k1 k3 − k22 ) , k2 I2
u = mgl sin(q1 ) + I1 u2 −
(58)
When q˙1 = q˙2 = 0 and q1 = π, from the dynamics of the inertia wheel pendulum and u, we can realize that
1132
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
pendulum angle (rad)
4 3 2 1 0 ï1
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t (sec)
12
14
16
18
20
2
wheel angle (rad)
0 ï2 ï4 ï6 ï8
Fig. 4. Pendulum and Wheel Angle of the Inertia Wheel Pendulum System the system can not stay at this equilibrium point unless 2 )+k3 I2 )π. Based on this finding, with q2 (0) = − kk32 ( k2 (I1 +I k3 I2 any q2 (0) ∈ R (except some values which makes true the above equality), q1 = π is not an equilibrium with the proposed control design. With the same arguments of the proof of Proposition 1, the other equilibrium point where q˙1 = q˙2 = 0 and q1 = 0 is asymptotically stable from dynamics of the system. That concludes the proof. Designed controller for the inertia wheel pendulum has R been simulated on Matlab Simulink% . System parameters has been assigned as I1 = 1 and I2 = 2. The design parameters of the control signal k and d5 have been set to -10 and 10 respectively, and the entries of [K] have been fixed to k1 = 10,k2 = 1, and k3 = 2. Initial values for (q1 (0), q2 (0), q˙1 (0), q˙2 (0)) were set to (π, −2π, 0, 0) which is a stable equilibrium point for the open loop system. Figure 4 shows the results for pendulum and cart angles. 5. CONCLUSION A stabilization procedure for a class of 2-DOF UAMS has been presented with two examples namely pendulum on a cart and inertia wheel pendulum, and simulation results has shown the effectiveness of the proposed method. It should be noted that the region of attraction of the closed loop system is shaped by the Mij of inertia matrix for 2DOF UAMS. For future work, this study can be extended to more general class of UAMS. ACKNOWLEDGEMENTS The authors gratefully acknowledges YTU, BAPK for supporting this study with the project number 2010-0402-KAP01.
A.M. Bloch , D.E. Chang, N.E. Leonard, J.E. Marsden. Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping. IEEE Transactions on Automatic Control, volume 46, issue 10,pages 1556– 1571, 2001. C.A. Ibanez. The Lyapunov direct method for the stabilisation of the ball on the actuated beam International Journal of Control, volume 82, issue 12,pages 2169– 2178, 2009. F.G. Estern, R. Ortega, F.R. Rubio, J. Aracil. Stabilization of a Class of Underactuated Mechanical Systems Via Total Energy Shaping. Proceedings of the IEEE Conference on Decision and Control, pages 1137–1143, 2001. H. Khalil. Nonlinear Systems, Upper Saddle River, NJ: Prentice-Hall, 2002. I. Fantoni, R. Lozano, M.W. Spong. Energy based control of the Pendubot IEEE Transactions on Automatic Control, volume 45, issue 4,pages 725–729, 2000. I. Fantoni, R. Lozano. Nonlinear control for underactuated mechanical systems, Springer-Verlag London, Communications and Control Engineering Series, 2002. J. Acosta, R. Ortega, A. Astolfi and A. Mahindrakar. Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one IEEE Transactions on Automatic Control, volume 50, issue 12,pages 1936–1955, 2005. M.W. Spong. Energy Based Control of a Class of Underactuated Mechanical Systems IFAC World Congress, 1996. R. Olfati-Saber. Nonlinear Control of Underactuated Systems with Application to Robotics and Aerospace Vehicles Massachusetts Institute of Technology, Department of Electrical and Computer Science, Ph.D. thesis, 2001. R. Olfati-Saber. Nonlinear Control and Reduction of Underactuated Systems with Symmetry I: Actuated Shape Variables Case Proceedings of the IEEE Conference on Decision and Control, pages 4158–4163, 2001. R. Ortega, A. van der Schaft, B. Maschke and G. Escobar. Interconnection and damping assignment passivitybased control of port-controlled hamiltonian systems Automatica, volume 38, no 4,pages 1137–1143, 2002. R. Ortega, M. Spong, F. Gomez and G. Blankenstein. Stabilization of underactuated mechanical systems via interconnection and damping assignment IEEE Transactions on Automatic Control, volume 47, issue 8,pages 1218–1233, 2002. W.N. White, M. Foss, and G. Xin. A Direct Lyapunov Approach for a Class of Underactuated Mechanical Systems Proceedings of the American Control Conference, pages 14–16, 2006. W.N. White, M. Foss, and G. Xin. A Direct Lyapunov Approach for Stabilization of Underactuated Mechanical Systems Proceedings of the American Control Conference, pages 4817–4822, 2007.
REFERENCES A.M. Bloch , N.E. Leonard, J.E. Marsden. Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem. IEEE Transactions on Automatic Control, volume 45, issue 12,pages 2253– 2270, 2000. 1133