Question 10.3.8: Controlling the Position of a Robot Arm Link The drive syste...
Controlling the Position of a Robot Arm Link
The drive system for one link of a robot arm is illustrated in Figure 10.3.13. The equivalent inertia of the link and all the drive components felt at the motor shaft is I_{e}. Gravity produces an opposing torque that is proportional to \sin θ but which we model as a constant torque T_{d} felt at the motor shaft (this is a good approximation if the change in θ is small). Neglect friction and damping in the system. Develop the block diagram of a proportional control system using an armature-controlled motor for this application. Assume that motor rotation angle θ_{m} is measured by a sensor and is related to the arm rotation angle θ by θ_{m} = N θ, where N is the gear ratio
The dynamics of the mechanical subsystem are described by
I_{e} \frac{d^{2} θ_{m}}{d t^{2}} = T − \frac{1}{N} T_{d} (1)
where the motor torque is T = K_{T} i_{a}. The system is like that shown in Figure 10.3.4. The block diagram can be obtained by modifying Figure 10.3.8 using equation (1) and collecting the various gains into one gain: K_{P} = K_{tach} K_{1} K_{a}. The resulting diagram is shown in Figure 10.3.14. Note that the system actually controls the motor rotation angle, so the arm angle command θ_{r} must be converted to the motor angle command θ_{mr}.
