The robot consists of:

1- Hydraulic oil with bulk modulus=1.5x10⁹ Pa, kinematic viscosity= 200 cSt, Specific gravity= 0.9, in 13 mm diameter rusted steel pipes with 5 standard 90^o elbows at the pressure line and other 5 at the return line.

2- Hydraulic pump at M1 with volume Vp=0.01 m3 , leakage coefficient=5x10^-11 m5 /N/s, pump speed = 1400 rpm, y = 0.1, Kp=4x10^-3 , Pump power=1 KW

3- Reversible hydraulic motor at M1 with volume VM1=0.01 m3 , head=1.5 m, displacement=2 l/radian, max. motor speed = 200 rpm and motor acceleration = 1.2 rad/s2

4- Reversible hydraulic motor at M2 with volume VM2=0.01 m3 , head=1.5 m, displacement=2 l/radian, max. motor speed = 200 rpm and motor acceleration = 1.2 rad/s2

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 L1=0.6 m and mass m1=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

Determine

1- The volume of the hydraulic reservoir (Tank).

2- The pressure in the piston needed to lift the load when the pump is off, in the case when the piston rod is in up position and when the rod is down position.

3- The pressure in the piston when the pump is on.

4- Build a block diagram that contains the pump, the hydraulic motors and the hydraulic piston with the input as the displacement of the pump and the output as the displacement of the load.

5- Consider that the hydraulic motor M1 forms a hydrostatic transmission system with the pump and find the transfer function between the angular speed of motor M1 and pump displacement (y). find the natural frequency, hydraulic stiffness, and damping ratio, and draw the bode plot for this transfer function. (Take leakage and compressibility into account).

6- Build a three-degrees-of-freedom dynamic model taking the two hydraulic motors at M1 and M2 and the piston at M3. (Neglect the pump). The inputs are the toque T1 for motor M1 and torque T2 for motor M2 and pressure P for piston M3. The outputs are angular displacements M1 and M2 and linear displacement M3 and velocity for M3 and acceleration for M3. Use the state space approach selecting the displacement and velocity at M1, displacement and velocity at M2, and displacement and velocity at M3 as state variables. Use the oil hydraulic stiffness and damping calculated in (5). Write the equations of motion and derive the four matrices of state space (A, B, C, D).