Modeling of Flexible Gantry Robot

5.2 Flexible Gantry Robot

5.2.2 System Modeling

5.2.2.2 Modeling of Flexible Gantry Robot

for the step response to reach 63.2% of its fi nal steady-state magnitude is about 32 ms.

The ER clutch actuator has somewhat slow response due to the inertia effect of the inner cylinder and the viscous friction torque induced from the antagonistic rotating motion of two outer cylinders. By considering the dynamics of the ER clutch actuator, the fi eld- dependent output torque of the governing model Equation 5.21 can be modifi ed as

β ⎛ ⎛ ⎞⎞

= ± π α ⋅ −⎜⎝ ⎜⎝−τ⎟⎠⎟⎠

2

elec 2 i cl 1 exp t

T r l E (5.22)

where τ denotes the time constant. Consequently, transfer function from the input electric fi eld, E, to the output torque, T_elec, is obtained in Laplace domain as follows:

2 elec( ) 2 i cl

( ) ( 1)

T s r l

E s^β s

π ⋅α

= ± τ + (5.23)

Figure 5.9b shows the measured Bode plot of the ER clutch actuator. It represents actuator’s frequency response characteristics for various control input frequencies (from 0.5 to 25 Hz) fi xing the magnitude at 2.5 kV/mm. As seen from Figure 5.9b, the ER clutch actuator has the bandwidth of about 15.6 Hz, and this result implies that the actuator can operate adequately in this bandwidth to control the motion of the X–Y table system.

where

εc is the induced strain in the piezoceramic due to the effect of the voltage applied to the piezoceramic

dn is the distance from the bottom of the piezofi lm sensor to the neutral axis The strain component and the distance are given by

c V r t d( , ) 31/tc

ε = ⋅ (5.26)

f f f a a a f a c c c

n

f f a a c c

(2 ) (2 2 )

2( )

t E t t t E t t t t E

d t E t E t E

+ + + + +

= + + (5.27)

In Equation 5.25, c is a constant implying the bending moment per volt. This constant is determined by the geometrical and material properties of the fl exible arm.

Upon assuming Euler–Bernoulli beam theory, small elastic defl ections, and neglect- ing axial defl ections, the kinetic energy and the potential energy are given as follows:

{ } { } { }

= + ρ

∫

¹ + ρ

∫

+

1

2 2 2

2

k t a t

0

2 ( , ) d ( , ) d ( , )

l L

l

T M x u r t r u r t r m u L t (5.28)

{ } { }

=

∫

¹ ⋅ ′′ − +

∫

′′

1

2 2

e a a

0

2 1 ( , ) ( ) d ( , ) d

l L

V

l

V EI w r t M t r E I w r t r

EI (5.29)

Piezofilm

MV

x x(t)

w(r,t)

r O

A

A

Piezoceramic Aluminum tf ta tc

dn

b

Section A-A mt

FIGURE 5.10 The fl exible gantry robot arm.

where

m_t and M_t are the tip mass and the total moving mass (ball screw housing, moving part, shaft, etc.) except the mass of the fl exible arm, respectively

ρ is the effective mass per unit length of the arm

EI is the effective bending stiffness of the arm bonded with a piezoceramic actuator and a piezofi lm sensor

These are derived from the neutral axis, and hence given by

c a f

ρ = ρ + ρ + ρ

{ }

{ }

{ }

= + + + −

+ + + −

+ + −

3 2

c c c f a c n

3 2

a a a f a n

3 2

f f f f n

12 ( 2 )

12 ( 2 )

12 ( 2 )

EI E bt bt t t t d

E bt bt t t d

E bt bt t d (5.30)

The virtual work done by the nonconservative external force is given by

a( )

W F t x

δ = ⋅δ (5.31)

where F_a is the axial direction force generated by the transmitted torque from the ER clutch.

Using the assumed mode-summation method, Equation 5.24 can be rewritten as

∞

=

= +

∑

₁φ ( , ) ( ) _i( ) ( )_i

i

u r t x t r q t (5.32)

where

ϕi(r) is the mode shape function q_i(t) is the modal coordinate

Now, substituting Equations 5.28, 5.29, and 5.31 into Lagrange’s equations, and augment- ing proportional damping, a couple of ordinary differential equations are derived by

{ }

^∞

=

ρ + ρ − + + ⋅ + ⋅ =

⋅ + + ζ ω + ω =φ′ ⋅ = ∞

∑

…

1 a 1 t t a

1 2 1

( ) ( ) ( )

( ) ( ) 2 ( ) ( ) ( ) ( ) for 1, 2, ,

i i

i

i i

i i i i i i V

di di

l L l m M x t m q t F

m l

x t q t q t q t M t i

m m (5.33)

where

ζ i is the damping ratio

ω i is the natural frequency of the fl exible arm

The generalized mass, m_di, and the coeffi cient, m_i, are given by

= ρφ + ρ φ + φ

= ρφ + ρ φ + φ = ∞

∫ ∫

∫ ∫

^…

1 1

1 1

2 2 2

a t

0 0

a t

0 0

( ) d ( ) d ( )

( ) d ( ) d ( ) for 1, 2, ,

l l

di i i i

l l

i i i i

m r r r r m L

m r r r r m L i

(5.34)

Now, by considering the mechanism of ball screw, one can obtain the governing equation of motion for each axis. As the control torque is transmitted from the ER clutch, the ball screw rotates to activate the radial direction force F_r, and this can be converted into the axial direction force F_a. The relationship between F_r and F_a is obtained by considering force equilibrium in the ball screw axis as follows:

r a( )

tan( )

F t = F

Φ + ϕ (5.35)

Here Φ and φ are the lead angle and friction angle of the ball screw, respectively. The equation of rotational motion of the table system can be expressed by

ci bs cp c elec r fric

(J +J +J ) ( )θ + θ =t C ( )t T ( )t −rF t( )−T ( )t (5.36) where

J_ci, J_bs, and J_cp are the moment of inertia of the clutch inner cylinder, the ball screw, and the coupling, respectively

C_c is the viscous damping coeffi cient

T_fric is the unmodeled total frictional torque including T_f

rF_r is the torque used to move the moving part of the table system

Considering the relationship between the angular displacement, θ, and the axial displacement, x, and substituting Equation 5.35 into Equation 5.36, the governing equations of motion given by Equation 5.33 is reconstructed with a fi nite number of n control modes as follows:

⎧ρ + ρ − + + + π + + ⎫⋅ + π ⋅ = +

⎨ Φ + ϕ ⎬ Φ + ϕ Φ + ϕ

⎩ ⎭

+ ζ ω + ω = ⋅φ′ ⋅ + =

…

ci bs cp c elec

1 a 1 t t

1 2

2 ( ) 2 ( )

( ) ( ) ( ) ( )

tan( ) tan( ) tan( )

( ) 2 ( ) ( ) ⁱ( ) ( ) ( ) for 1,2, ,

i i i i i i i

di

J J J C T t

l L l m M x t x t D t

rl rl r

q t q t q t c l V t d t i n

m (5.37)

where r and l are the radius and lead of the ball screw, respectively. The disturbances D(t) and d_i(t) are given by

=

⎧ ⎫

⎪ ⎪

= −⎨⎪⎩ ⋅ + 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)