Position Control by means of a Flexible Transmission

Figure 8.36. Position step response of the flexible transmission (controller B)



Figure 8.37. Flexible transmission – rejection of a position disturbance (controller B)


Figure 8.36. Position step response of the flexible transmission (controller B)



Figure 8.39. Flexible transmission – rejection of a position disturbance (controller C)
Table 8.8. The parameters of the RST controllers (B and C) for the flexible transmission

Controller B				Controller C
Coefficients of polynomial R(q^-1)
R(0) = 0.497161 R(1) = -1.009731 R(2) = 0.622142	R(3) = -0.186272 R(4) = 0.285405		R(0) = 0.307895 R(1) = -0.515832 R(2) = -0.037637			R(3) = 0.452544 R(4) = -0.165905 R(5) = 0.167641
Coefficients of polynomial S(q^-1)
S(0) = 1.000000 S(1) = -0.376884 S(2) = -0.256739	S(3) = -0.239733 S(4) = -0.126644		S(0) = 1.000000 S(1) = -0.376884 S(2) = -0.256739			S(3) = -0.181950 S(4) = -0.109785 S(5) = 0.074642
Coefficients of polynomial T(q^-1) (identical for B and C)
T(0) = 1.429225 T(1) = -2.839061 T(2) = 3.181905		T(3) = -2.687952 T(4) = 1.594832 T(5) = -0.576414			T(6) = 0.118277 T(7) = -0.012656 T(8) = 5.488E-04
Coefficients of polynomials Bm(q^-1) and Am (q^-1) (identical for B and C)
Bm(0) = 0.124924 Bm(1) = 0.087209				Am(0) = 1.000000 Am(1) = -1.129301 Am(2) = 0.341434

The characteristics of this controller (C) are summarized in Table 8.7. The corresponding frequency characteristics of the sensitivity functions are shown in Figures 8.34 and 8.35 (curves C) and the real time results are presented in Figures

8.38 and 8.39 (to be compared with Figures 8.36 and 8.37).
The parameters of the controllers B and C are given in Table 8.8.

Control of a 360° Flexible Robot Arm

Figure 8.40 gives a view of the 360° flexible robot arm.

It is constituted by two aluminium sheets, each one is 1m long and 10cm wide, with a thickness of 0.7mm. The two sheets are coupled every 10cm by a rigid
frame. The system is very flexible and presents many low damped vibration
modes. The energy is essentially concentrated in the first three vibration modes.
The sampling frequency (20 Hz) has been chosen such that these three vibration modes lie between 0 and 0.5 f_s . Data acquisition is performed through anti aliasing filters. The block diagram of the control scheme is given in Figure 8.41. One of the extremities of the arm is directly coupled to the axis of a DC motor. The corresponding local position loop contains a cascade control of motor current, speed and position (measured by a potentiometer type transducer). The band pass





Figure 8.40. 360° Flexible robot arm (Laboratoire d’Automatique de Grenoble, INPG/CNRS/UJF)
of this loop is higher than the frequency of the first vibration mode.
The output of the system is the position of the free end of the arm. The
measurement of the position of the free end is done by combining information
upon the position of the motor axis (provided by an incremental transducer) and
those provided by a carried on measurement device (including a light beam and a mirror), which gives the angular position with respect to the motor axis position
(for details concerning the measurement device see Landau et al. 1996).




Figure 8.41. Position control scheme for the 360° flexible robot arm

The measurement system allows one to cover a rotation from 0 to 360°. The

control signal provided by the computer is the reference position for the motor
axis.
The identified and validated model for the case without load is (Langer and
Landau 1999)


A(q ^1 )  1  2.1049 q ^1  1.04851 q ^2  0.33836 q ^3  0.46 q ^4
 1 .5142 q ^5  0.7987 q ^6
B (q ^1 )  0.0064 q ^1  0.0146 q ^{ 2}  0.0697 q ^3  0.044 q ^{ 4}
 0 .0382 q ^5  0.007 q ^6
d  0
The frequency characteristic of this model is shown in Figure 8.42.
This model is characterized by three very low damped vibration modes
(₁ = 2.6173, ₁ = 0.018; ₂ = 14.4027, ₂ = 0.025; ₃ = 48.1169, ₃ = 0.038).
The pole – zero map is shown in Figure 8.43. One notes the presence of unstable
zeros. The unstable zeros with positive real part correspond to continuous time
unstable zeros (non-minimum phase system). Since the model has unstable zeros
the pole placement strategy will be used for the controller design.