EXPERIMENTAL VALIDATION ON THE TEST RIGCALLY, FINALLY LEADS TOTHE...

5. Experimental validation on the test rigcally, finally leads toThe benefits and the efficiency of the proposed control measures shall be pointed out by exper-y

_Sd

y

_Kd

(

t

)

, ˙y

_Kd

(

t

)

, ¨y

_Kd

(

t

)

,...y

_Kd

(

t

)

,y

⁽⁴⁾

_Kd

(

t

)

,y

⁽⁵⁾

_Kd

(

t

)

,κimental results obtained from the test set-up available at the Chair of Mechatronics, Universityv

_1d

of Rostock. For this purpose, a synchronous four times continuously differentiable desiredy

Kd

(

t

)

, ˙y

Kd

(

t

)

, ¨y

Kd

(

t

)

,.... (54)x

_d

(

t

) =

trajectory is considered for the position of the cage in both x- and y-direction. The desiredy

_Kd

(

t

)

,y

⁽⁴⁾

_Kd

(

t

)

,y

⁽⁵⁾

_Kd

(

t

)

,κy˙

Sd

trajectory is given by polynomial functions that comply with specified kinematic constraints,y

_Kd

(

t

)

, ˙y

_Kd

(

t

)

, ¨y

_Kd

(

t

)

,...v˙

_1d

which is achieved by taking advantage of time scaling techniques. The desired trajectoryshown in Figure 4 comprises a sequence of reciprocating motions with maximum velocities of2 m/s in horizontal direction and 1.5 m/s in vertical direction. The resulting tracking errorse

y

(

t

) =

y

_Kd

(

t

)

−y

_K

(

t

)

(55)

8

x 10

⁻³

and

6

e

x

(

t

) =

x

_Kd

(

t

)

−x

K

(

t

)

(56)

4

are depicted in Figure 5 and Figure 6. As can be seen, the maximum position error iny-direction during the movements is about 6 mm and the steady-state position error is smaller

2

than 0.2 mm, whereas the maximum position error inx-direction is approx. 4 mm. Figure 7

e

y

in m

0

0.015

−2

v

_1d

v

₁

−4

0.01

0

5

10

15

t in s

0.005

Fig. 5. Tracking errore

y

(

t

)

for the cage motion in horizontal direction.

v

1

in m

−0.005

5

x 10

⁻³

−0.01

3

−0.015

1

Fig. 7. Comparison of the desired valuesv

_1d

(

t

)

and the actual valuesv

₁

(

t

)

for the bendingdeflection.

e

x

in m

−1

shows the comparison of the bending deflection measured by strain gauges attached to theflexible beam with desired values. During the acceleration as well as the deceleration inter-vals, physically unavoidable bending deflections could be noticed. The achieved benefit is

−3

given by the fact the remaining oscillatons are negligible when the rack feeder arrives at itstarget position. This underlines both the high model accuracy and the quality of the activedamping of the first bending mode. Figure 8 depicts the disturbance rejection properties dueFig. 6. Tracking errore

x

(

t

)

for the cage motion in vertical direction.to an external excitation by hand. At the beginning, the control structure is deactivated, andthe excited bending oscillations decay only due to the very weak material damping. Afterapprox. 2.8 seconds, the control structure is activated and, hence, the first bending mode isactively damped. The remaining oscillations are characterised by higher bending modes thatdecay with material damping. In future work, the number of Ritz ansatz functions shall beJung, S. & Wen, J. (2004). Nonlinear model predictive control for the swing-up of a rotary in-verted pendulum,ASME J. of Dynamic Systems, Measurement and Control126(3): 666–

0.03