Write the equations of motion and derive the four matrices of state space (A, B, C, D)

 

M1 inger arm outer arm M2 M3 The robot consists of: 1- Hydraulic oil with bulk modulus-1.5x10 Pa, kinematic viscosity= 200 cSt, Specific gravity=0.9, in 13 mm diameter rusted steel pipes with 5 standard 90 elbows at the pressure line and other 5 at the return line. 2- Hydraulic pump at M1 with volume Vp=0.01 m, leakage coefficient=5x10-11 m/N/s, pump speed = 1400 rpm, y = 0.1, Kp 4x10-, Pump power-1 KW 3- Reversible hydraulic motor at M1 with volume VM-0.01 m, head=1.5 m, displacement 2 1/radian, max. motor speed = 200 rpm and motor acceleration = 1.2 rad/s 4- Reversible hydraulic motor at M2 with volume VM2-0.01 m, head=1.5 m, displacement 2 1/radian, max. motor speed = 200 rpm and motor acceleration = 1.2 rad/s 5- Double acting hydraulic linear piston at M3 with piston diameter DM3=5 cm, rod diameter RM3= 2 cm, stroke=30 cm 6- Inner arm with length L=0.6 m and mass m=3 kg 7- Outer arm with length L2=0.5 m and mass m2=2 kg 8- Load at the end effector equals to 100 kg