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

Kinematical analysis of the system
The cart is in a straight-line translational motion is represented by the variable X(t) measured from a fixed point. As for the payload, its movement is double in the plane, it moves with the cart and swings around an axis perpendicular to the plane through the point of suspension of the rope as shown in Fig. 2.

Displacement , velocity , and acceleration of the cart, measured from fixed reference (inertial fixed frame). As for the load, the vectors of displacement and , velocity and acceleration are as follows

(1)
Where: - Variable of cart motion measured from fixed point (Displacement of the cart), - Variable of payload motion measured from fixed point along x axis, - Variable of payload motion measured from x axis along y axis, L - the length of the rope.
By differentiating the two equations (1), we obtain the two velocity components
(2)

By differentiating the two equations (1), we obtain the two acceleration components

(3)
Where: - Angular velocity and acceleration of payload swing respectively.