2。Mathematical model of electro-hydraulic systems

An double rod EHS designed for a quadruped robot by the Korea Institute of Industrial Technology is depicted in Fig。 1。 This system

maximum  stroke  of  the  cylinder,  respectively。  And  then, the

total volume of fluid in both chambers, Vt is defined as follows:

Vt ¼ VA þ VB ¼ 2V 0: ð2Þ

Although nonlinear model of the mechanical behavior of servo- valve can be developed from the relationship between the spool position and the input current the valve dynamics can be approx- imated as the second order linear dynamics [33] by

€xv  þ 2fxn x_ v  þ x2xv  ¼ Kv x2 i; ð3Þ

n n

where xv is the spool position of the servo-valve [m], i is the input current of the torque motor [mA], f is the damping ratio [rad], xn

Fig。 1。  Representative EHS。

768 W。 Kim et al。 / Mechatronics 22 (2012)   766–777

is the natural frequency [rad/s], and Kv is the torque motor gain [m/ mA]。 Assume that the valve orifices are matched and symmetrical, the orifice areas are linear with valve stroke, and the compressibil- ity flows are zero。 The control flow equation of the hydraulic valve

for the load flow rate can be written   as

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

3。

Observer design

For position feedback, we design a high gain observer to esti- mate the full state and reduce the nonlinearity of the EHS [32]。

Eq。 (9) can be rewritten  as

x_  ¼ Ax þ /ðxÞþ Bu; 10

QL ¼ Cdwxv

q ðPs — sgnðxv ÞPL Þ; ð4Þ

y ¼ Cx; ð

where QL is the load flow rate [m3/s], Cd is the discharge coefficient, w is the area gradient of the servo-valve spool [m], Ps is the supply pressure of the pump [N/m2], and q is the density of hydraulic oil [kg/m3]。 The function sgn is defined as

where

2 0   1   0 0 0 3

6 0    0  AP

6 7

8 1ðx > 0Þ

A ¼ 6 0

6

0    0    h1pffiPffiffisffiffi

0 7;

7

sgn x ><

0ðx ¼ 0Þ

: ð5Þ

6 0   0   0 0 1 7

4 5

0    0   0 0 0

>: —1ðx < 0Þ

Applying the law of continuity to each actuator chamber,  the 2 0 3

load flow continuity equation is given as (6) [1] 1 1 2

6 m ð—kx  — bx Þ 7

6

s —