Question: an intelligent robot arm which rotates in the horizontal plane...
Question details
An intelligent robot arm which rotates in the horizontal plane has a motor inertia of 0.01 kg-m^{2}, arm inertia 7.0 kg-m^{2}.
The motor shaft and bearing has a damping coefficient of 0.01 N-m/s. The arm and gear mechanism together has a damping coefficient of 3.5 N-m/s.
The intelligent robot arm system design is required to grasp masses ranging from 1 kg to 20 kg, and then rotates through an angle of 150 degrees with minimum oscillatory movements for the entire range of masses as well as shortest time to settle to within 1 degree of the final position.
The intelligent robot has a controller that compares set angle (${\theta}_{d}$ = 150 degrees) with the robot angle ${\theta}_{l}$ and outputs the motor control voltage v as:
$v={K}_{a}({\theta}_{d}-{\theta}_{l})+{K}_{b}{({\theta}_{d}-{\theta}_{l})}^{2}$
where
- ${\theta}_{d}$ = Desirable angle (rad)
- ${\theta}_{l}$ = Feedback angle from the arm position (rad)
- K_{a} = First order adjustable constant (V/rad)
- K_{b} = Second order adjustable constant (V/rad^{2})
The motor is a 12 VDC motor with a stall torque of 40 Nm, and a no load speed of 4000 RPM.
Which set of system parameters would you recommend?
Formulae applicable to this question:
The torque and rotational speed of a DC motor is given by:
${K}_{m}{T}_{m}+{K}_{o}{\omega}_{m}=v$
where
- v = Voltage (V)
- T_{m} = Motor torque (Nm)
- K_{m} = Motor torque constant (V/N/m)
- ${\omega}_{m}$ = Rotational speed (rad/s)
- K_{o} = Motor speed constant (Vs/rad)
Gear ratio n is given by:
$n=\frac{{\theta}_{l}}{{\theta}_{m}}$
- ${\theta}_{m}$ = Angle turned by the motor (rad)
- ${\theta}_{l}$ = Angle turned by the arm (rad)
The robot arm and motor dynamics are defined by Newton’s second law of motion.