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
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