MME2151: A model of a laboratory gantry crane: Control of Dynamic Systems Assignment, SIT, Singapore

Problem Statement
A model of a laboratory gantry crane.

The cart is driven by a motor driving the two rear wheels and the four wheels ride on two parallel rails. The load is suspended by a cable, assumed taut during the motion.
The model of the motor producing the torque to drive the wheel resulting in a traction force F(s) (assuming no slip of the wheels on the rail) to drive the cart is given as follows:

J: total equivalent inertia of wheels, gears, and motor rotor (assuming damping B≈0)
Km: motor gain
m: motor time constant
r: radius of the wheel
V(s): input voltage to the motor

The time constant of motor, m = 1/2.4 sec

The equations of motion (linearized) are for the cart and the load for small  are respectively
(𝑀 + 𝑚)𝑥̈= 𝑓 − 𝑚𝑙𝜃̈
𝑙𝜃̈= −𝑥̈− 𝑔𝜃

Crane-load system:
• M = 5 kg, mass of cart / trolley
• m = 0.2 kg, mass of load
• l = 0.82 m, length of cable
• g = 9.81 m/s2,acceleration due to gravity

Using block diagrams and transfer functions, model the cart-load system. Develop the respective transfer functions for the motor, the pulley and the crane-load dynamics. Design a PID controller Gc(s) to move the crane a displacement x from an arbitrary reference O-XY.

Project requirement
Block diagram

(a) Use the signal builder in Part 1 to design the desired displacement Xr = 2 m. The cart as in Part 1 is to move forward, rest and then return in the given time intervals.
(b) Design the PID controller Gc(s). The PID controller should be designed such that X → Xr in the specified time. You may use the auto-tune to get a set of PID gains.
(c) Next, experiment with these gains, starting with the auto-tuned PID values, and varying each one at the time, starting with the tuned P term, with the other terms set at 0. Then keeping the P at the tuned value, vary the I-term and then the D-term. Discuss the effect of the P-, the I- and D-term
on the tracking of X → Xr within the specified time. Briefly indicate the effect also on the resulting angle  of oscillations of the load, which is not controlled.

Report
You have already formed your own groups. Each group will submit a report. The report shall contain
• The transfer functions of the linearized cart and load dynamics
• The Simulink model as shown above.
• Relevant simulation results with explanation and the controller.