D ynamical Modeling and Control of Motion System of the Gantry Crane to Minimize Swing Angle of the Payload

Simulation Results

The model and the controller were validated in simulations using MATLAB. The model parameters used in simulations are, M = 120 kg, m = 380 kg, L = 1 m in the equation (36)

(45)
The weighting matrices used in the simulations are shown below
(46)
The corresponding generated gains from the LQR are
(47)
Simulation results proved that starting from any initial conditions (initial inclination in the angle), the system was able to stabilize in less than 1s, the initial conditions used in the simulations are,

Fig. 6-a shows that the system response time to reach a zero-swing angle took just under a second. Of course, the angle does not reach zero completely, as Fig. 6-b shows, after zooming in the curve, that the angle stabilizes at the value 0.0025 rad or 0.14 degrees, which is very close to zero and is considered practically acceptable.

Fig. 6-a. The curve of the payload swing angle.

Fig. 6-b. Zoom in of the curve of the payload swing angle.

Fig. 6-c. The curve of the angular velocity of the payload swing.

As for the angular velocity, which starts from the value of 0 clockwise to reach the maximum negative value about (-0.8 rad) in less than a quarter of a second, and the positive rotation begins counterclockwise until it reaches the maximum positive value, which is less than (0.03 rad), then it reaches the zero bound in a little less than a second and stays there, which means no swing, as shown in Fig 6-c.

Fig. 6-d. shows the angular acceleration that starts negative and then increases until it reaches its highest value (about 3 rad/s²) at a time of a little more than quarter a second, after that it stabilizes at the zero bound in a little more than half a second and remains there.

Fig. 6-d. The curve of the angular Acceleration of the payload swing.

Fig. 6-e shows the payload displacement, where the payload begins to move positively with great acceleration in the first half of a second, then after that, the movement slows down in order to reduce the swing angle, then the slope of the curve is constant and the payload travels a distance of little more than two meters in five seconds.

Fig. 6-e. The curve of the displacement of the payload.

As shown in the Fig. 6-f. the cart starts moving at a velocity of zero from the reference point (origin), then the velocity increases to reach the highest value (2.75 m/s) in less than a quarter of a second, then it slows down to reach the lowest value at a time slightly greater than half a second, then its value (about 0.25 m/s) increases slightly to reach (0.25 m/s) and then stabilizes at this value, meaning it remains constant.

Fig. 6-f. The curve of the velocity of the cart.

Fig. 6-g. shows the acceleration of the cart, which starts at (55 m/s²) and then decreases rapidly until it becomes negative deceleration and reaches a value of (-10 m/s²) at a time about a quarter of a second, then starts to rise until it reaches zero at the time (0.8 s) and continues as well, meaning that the cart is moving with constant velocity after this time.

Fig. 6-g. The curve of the Acceleration of the cart.