Controller Formulation

5.2 Flexible Gantry Robot

5.2.3 Controller Formulation

=

⎧ ⎫

⎪ ⎪

= −⎨⎪⎩ ⋅ + Φ + ϕ ⎬⎪⎭

⎛ ⎞

= −⎜⎝ ⎟⎠⋅ =

∑

…

fric 1

( ) ( )

tan( )

( ) ( ) for 1, 2, ,

n

i i

i

i i

di

D t m q t T

r

d t m x t i n

m

(5.38)

The disturbance D(t) has an effect on the moving table system from the oscilla- tion of the fl exible robot arm and the unavoidable frictional torque. The disturbance d_i(t) induced from the acceleration of the moving part infl uences the fl exible arm to be oscillated. By treating the coupling terms as disturbances, one can consider one multi-input multi-output (MIMO) system as two single-input single-output (SISO) systems. In Equation 5.37, the fi rst one represents the relationship between the input torque T_elec and the output displacement x, while the other between the input voltage V and the output tip defl ection w of the fl exible arm. Therefore, two transfer func- tions are obtained in Laplace domain as follows:

=

Φ + ϕ

= +

⋅φ ⋅φ′

=

∑

+ ζ ω + ω

2

elec eq eq

1

2 2

1

( ) 1 / { tan( )}

( )

( , ) ( ) ( ) /

( ) 2

n

i i di

i i i

i

x s r

T s M s C s

w L s c L l m

V s s s

(5.39)

where the equivalent mass of table system M_eq and the equivalent viscous damping coeffi cient C_eq are given by

π + +

= ρ + ρ − + + +

Φ + ϕ

π η

= π⋅ =

Φ + ϕ ⋅ Φ + ϕ

ci bs cp

eq 1 a 1 t t

2 3

c i cl

eq

2 ( )

( )

tan( )

2 8

tan( ) tan( )

J J J

M l L l m M

rl

C r l

C rl h rl

(5.40)

additive uncertainties on the normalized left coprime factors of the nominal plant G_t and G_f can be expressed by

Δ −Δ Δ

−

−Δ Δ

−

=

=

π⋅ α Φ + ϕ

= + Δ + Δ = ±

τ + ⋅ + Δ ⋅ + + Δ ⋅

=

= + Δ + Δ = + Δ

+ ζ + Δζ ω + Δω + ω + Δω

∑

t t

f f

1

t t t

2 i cl 1

t t 2

eq eq eq eq

1

f f f

1

f f 2 2

1

( )

2 / { tan( )}

( ) ( )

( 1) {( ) ( ) }

( )

( )

( ) ( )

2( )( ) ( )

M N

n

M N

i i i i i i

i

G s M N

r l r

M N

s M M s C C s

G s M N

k k

M N

s s

(5.42)

In the above, [Δ Δ_Nt, _Mt] and [Δ Δ_Nf, _Mf] represent the coprime factor uncertainties induced from the variations of system parameters such as the total moving mass, M_t, and the viscosity of ER fl uid, η, which can be easily altered by temperature, and the natural frequencies and damping ratios of a fl exible arm with end-effector loading conditions. The coeffi cient k represents a plant gain that also varies with respect to the variation of the loading conditions of the fl exible robot arm. In order to identify the parameter variations of natural frequency and damping ratio, frequency responses of the fl exible arm are experimentally obtained. The measured parameters are as follows: the fi rst mode natural frequency varies from 4.563 to 3.187 Hz by adding the tip mass of 10 g. The corresponding damping ratio changes from 0.0242 to 0.0158. The fl exible arm without tip mass is adopted as the nominal plant, and the other arm that has the tip mass of 10 g as the perturbed plant. Also, ±30% variations are applied in parameters such as the total moving mass and the viscous coeffi cient of the ER clutch actuator. The singular-value plots of the nominal plants, G_t, G_f, and the perturbed plants, G_tΔ, G_fΔ, are presented in Figure 5.11a and b. It is clearly seen that the perturbed plant has a different magnitude and bandwidth in the table system, and a different magnitude and natural frequency in the fl exible arm.

Now, to guarantee the robust stability and performance of the system, loop shap- ing is carried out using frequency dependent pre-compensators W_t and W_f until the shaped plants G_tW_t and G_fW_f satisfy the desired open loop shapes. The inspection of the maximum singular-value plot of the nominal plant for moving table system indicates that considerable additional gain is required to improve performance char- acteristics, especially the closed loop bandwidth. The nominal open loop cross-over frequency at 0.0156 rad/s (see Figure 5.11a) implies a very low closed loop bandwidth.

Gain of 75 × 10⁴ to input channel gives an open loop gain cross-over frequency of about 4.928 rad/s. In addition, the pole is used for high-frequency roll off rate that guarantees robust stability of the table system. Consequently, the pre-compensator W_t for moving table system is designed as follows:

t

100 7500 W 1000

= ×s

+ (5.43)

On the other hand, pre-compensator W_f for fl exible robot arm is selected as a simple PI compensator to ensure zero steady-state output and quickly suppress the oscilla- tion of the robot arm due to the disturbance induced from the moving table accelera- tion. The additional gain is also added with this compensator to improve closed loop performance. Thus, the W_f is designed as follows:

f

5 s 40

W s

= × + (5.44)

It is noted that both pre-compensators are designed as fi rst-order functions to reduce the order of the total control systems.

As a second step, robust stabilization of normalized left coprime factorizations for shaped plants G_tWt and G_fWf is performed. The following corollary demonstrates the conditions for controller K that robustly stabilizes the system with coprime factor uncertainty [32].

10^–3 10^–2 10^–1 10⁰ 10¹ 10² 10³ –200

–100 0 100 200

10^–2 10^–1 10⁰ 10¹ 10² 10³ 10⁴ –160

–120 –80 –40 0 40 80 120

10^–210^–1 10⁰ 10¹ 10² 10³ 10⁴ –200

–150 –100 –50 0 50 100 150

10^–2 10^–1 10⁰ 10¹ 10² 10³ 10⁴ –150

–100 –50 0 50 100

10^–410^–310^–210^–110⁰10¹10²10³ 10⁴ 0

25 50 75 100

10^–410^–310^–210^–110⁰10¹10²10³10⁴ –50

0 50 100 150

Singular values (rad/s)

Frequency (rad/s)

Nominal plant Perturbed plant Shaped plant GtWt Loop gain Gt Kt

(a)

Nominal plant Perturbed plant Shaped plant G_fW_f Loop gain Gf Kf

Singular values (dB)

Frequency (rad/s) (b)

Sensitivity function Complement sensitivity fuction

Singular values (dB)

Frequency (rad/s) (c)

Sensitivity function Complement sensitivity fuction

Singular values (dB)

Frequency (rad/s) (d)

Singular values (dB)

Frequency (rad/s) (e)

Singular values (dB)

Frequency (rad/s) (f )

FIGURE 5.11 Singular value plots of the control system. (From Han, S.S. et al., J. Robot.

Syst., 16, 581, 1999. With permission.)

Corollary 5.1

K stabilizes G_Δ for all [ΔN, ΔM] ∈ D_S if and only if 1. K stabilizes G

2. K ( )^{1 1}

I GK M

I

− −

∞

⎡ ⎤ − ≤ γ

⎢ ⎥⎣ ⎦

Here D_S denotes a set of stable bounded (||Δ||_∞ < γ⁻¹) perturbations, and condition (2) can be proved by using the small gain theorem. Therefore, what we have to do in this stage is to fi nd the optimal solution, γmin, to the normalized left coprime factor robust stabiliza- tion of the shaped plant by following the relation given in McFarlane and Glover [32]:

−

− −

∞

⎡ ⎤ − =⎧⎨ − ⎫⎬

⎢ ⎥ ⎩ ⎭

⎣ ⎦

= γ

^{2 1}

1 1

s s s s

min

inf ( ) 1 [ , ]

K H

K I G K M N M

I

(5.45) where N˜

s, and M˜

s denote the normalized left coprime factors of the shaped plant G_tW_t and/or G_fW_f. At this stage, γ can be viewed as a design indicator: if the loop shaping has been well carried out, a suffi ciently small value will be obtained for γmin.

By choosing a suitable γ a little larger than γmin, γt = 2.17 and γf = 2.59 were obtained for the moving table system and the fl exible arm, respectively. The singular-value plots of the loop gains and the shaped plants are also presented in Figure 5.11a and b. As seen from the fi gures, the loop gain and the shaped plant are well accorded with each other. These results imply that the successful loop shaping is achieved. In addition, Figure 5.11c and d presents the sensitivity and complementary sensitivity functions of both systems. A small magnitude in the low frequency range and 0 dB in the high- frequency range for the sensitivity function plots is observed. This result implies that the designed controller for each system guarantees the desired performance and it can effectively reject the external disturbances. Also, from the complementary sensitiv- ity function plots, the sensor noise suppression is well guaranteed because they show small magnitude in the high-frequency range and 0 dB in the low frequency range.

Now, by using previously obtained value of γ, one can constitute the fi nal H_∞ con- troller by combining the suboptimal controller, K_∞, with pre-compensators as follows:

∞

∞

− × + × + × + × + ×

= =

+ × + × + × + × + ×

× + × + × + ×

= =

+ × +

13 4 9 3 12 2 13 14

t t t 5 3 4 6 3 8 2 10 11

5 3 6 2 7 8

f f f 4 3 3

4.547 10 1.150 10 1.192 10 4.169 10 1.744 10

2.842 10 2.722 10 9.171 10 3.803 10 4.477 10

1.023 10 4.357 10 3.313 10 8.988 10

8.619 10 5

s s s s

K W K

s s s s s

s s s

K W K

s s .023 10× ^{5 2}s +1.074 10× ⁷s

(5.46)

where K_t and K_f are controllers for the moving table system and the fl exible arm, respectively. Figure 5.11e and f shows the singular-value plots of the designed con- trollers, and Figure 5.12 presents the block diagram of the H_∞ control system.