4.2 Three-Axis Active Mount
5.1.3 Controller Formulation
1 2
11 21
12 22
1 2
1 2
,
n n
q q
q q
q q
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
=⎢ ⎥ =⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
q q
In the above, bold means vector or matrix and ii is the unit vector given as follows:
T 1
1
1 0 0
=∂θ∂ = ⎡⎣ ⎤⎦
z
i (5.10)
It is clearly seen that the system model is highly nonlinear without neglecting nonlinear terms or linearization.
The control objective is to get θi(t) to track desired trajectories θid(t) that belong to the class of C1 function. In other words, the controller should force the tracking errors to zero asymptotically for any initial conditions. To accomplish this goal, the sliding mode con- troller is adopted, which features inherent robustness during sliding mode motion [13].
Now, one can defi ne sliding surfaces that guarantee the stability of the sliding mode system of the rigid two-link manipulator on the surfaces themselves by
1 11 11 12 21 11 12 12 22
2 21 11 22 21 21 12 22 22
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
s t c e t c e t mr e t mr e t s t c e t c e t mr e t mr e t
= + + +
= + + + (5.12)
where
cij is a time-varying parameter of the hyperplanes to be designed ei1 = θi − θid and ei2 = θ.
i − θ.
id are tracking errors
Then one can easily construct the following controller, Ti, which satisfi es the sliding condition; si s.
i < 0:
1 11 12 11 11 12 12 1 11 12 12
2 21 22 21 21 22 22 2 21 22 22
11 12 1d 1 1
21 22 2d 2 2
sgn( ) sgn( )
T c c e c c e fr mr mr e
T c c e c c e fr mr mr e
mr mr kr s
mr mr kr s
⎡ ⎤= −⎡ ⎤ ⎡ ⎤ ⎡− ⎤ ⎡ ⎤ ⎡ ⎤ ⎡+ − ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡ ⎤
⎡ ⎤ θ ⎡ ⎤
+⎢⎣ ⎥⎦⎢⎣θ ⎥⎦−⎢⎣ ⎥⎦
(5.13)
where kri > εi, i = 1,2. Here, εi are arbitrary small positive real values. In practice, it is not desirable to use the discontinuous control law, due to the chattering. Therefore the discontinuous control law is approximated by a continuous one inside the boundary layer [14].
To make the sliding surfaces guarantee the stability of the system, the surface parameter, cij, is designed as follows:
11 12 11 21 1
21 22 21 22 2
0 0
c c mr mr
c c mr mr
⎡ ⎤ ⎡= ⎤ ⎡λ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ λ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (5.14)
Then the error dynamics are asymptotically stable with the repeated eigenvalues, (λ1, λ1, λ2, λ2).
5.1.3.2 Constant Amplitude Controller
It is known that in a rigid-link robot, accurate tip position control can be achieved using joint angle measurement and an appropriate control scheme. However, tip posi- tion no longer has the simple fi xed relationship with the joint angle when the link is fl exible. This makes it diffi cult to control the end-point motion for a desired accuracy within an adequate time interval. The following constant amplitude controller (CAC) [15] is adopted for the piezoceramic actuators:
( ) sgn( f( )), 1, 2
i i i i
V t = − ⋅K c V t⋅ i= (5.15) where
Ki is a feedback gain V.
fi(t) is the time derivative of the output signal voltage, Vfi(t), from the distributed piezofi lm sensor bonded to the other surface of the fl exible link
The output voltage produced from the piezofi lm sensor is obtained by integrating the electric charge developed at a point on the piezofi lm along the entire length of the fi lm surface.
The feedback gain, Ki, of the controller (5.15) is to be chosen by considering the material property of the piezoceramic actuator as well as the geometrical prop- erty of the fl exible link. Furthermore, the feedback gain should be determined so that the system (5.8) is stable as follows. If there exists a small positive num- ber εi +2 such that θ ⋅i T ti( ) < εi+2, and the feedback gain, Ki, is chosen such that
( )
(
2 2 ( , ))
i i i i i i
K > ε+ c ∂ w l t 兾∂ ∂t x and the velocity terms are small, the stability of the system (5.8) with the amplitude controller (5.15) is guaranteed. To prove this, a positive defi nite Lyapunov functional is fi rst introduced, which is a measure of the potential and kinetic energy due to the oscillations of the link, given by
T T
1 1
2 2
F= z Mz + z Kz (5.16)
Taking the time derivative of Equation 5.16 becomes
T T T
T 3
2 2
1 1 2 2
1 1 2 2 1 1 2 2
1 2
1 2 ( )
( , ) ( , )
F t
O
w l t w l t
T T c V c V
t x t x
∂ = + +
∂
= +
∂ ∂
≅ θ + θ + +
∂ ∂ ∂ ∂
z Mz z Kz z Mz
z Bu z
(5.17)
Now, from the assumption imposed on the feedback gain, Ki, the time derivative of the Lyapunov functional of the closed loop system is negative defi nite as follows:
2 2
1 1 2 2
3 4 1 1 2 2
1 2
( , ) ( , )
F w l t w l t 0
K c K c
t t x t x
∂ < ε + ε − ∂ − ∂ <
∂ ∂ ∂ ∂ ∂ (5.18)
This satisfi es the Lyapunov stability condition and hence guarantees the stability of the distributed system (5.8).
The assumption imposed on the feedback gain physically implies that the motions of the hubs should be slow or in deceleration phase to make the system stable. In other words, the stability of the fl exible manipulator system can be violated by fast
motions of the hubs that in turn result in large oscillations of the fl exible links. It is remarked that if the motions of the hubs are completely stopped, the fl exible links can be treated as just cantilever beams. In this case, the fi rst two terms in Equation 5.17 disappear, and hence the Lyapunov stability is always satisfi ed by employing any positive feedback gains, Ki. From Equation 5.18, the magnitudes of the feedback gains, Ki, to satisfy the inequality depend upon the positive number, εi+2, and angu- lar velocities. However, it is very diffi cult to analytically calculate these quantities.
Thus, appropriate magnitudes of the feedback gains are normally determined in an empirical manner by investigating the hub motions of the motors and oscillation levels of the fl exible links.
In real implementation of the CAC controllers (5.14), the discontinuous property causes undesirable chattering associated with time delay and hardware limit. To effectively remove the chattering, one may use a so-called multistep amplitude con- troller (MAC) that proportionally tunes the magnitude of control voltage according to the output signal [16].
The angular displacements can be obtained by built-in optical encoders in the motors and the elastic defl ections by the distributed piezofi lm sensors. Therefore, no state estimator, which may be inevitably necessary in most conventional control methods, is needed for the implementation of the hybrid actuator control scheme.
This is also one of the major advantages of the control strategy.