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

Equations in terms of state variables

From Equation (27)

(30)

(31)

Substituting from (27) in (28) yields

(32)

Substituting from (27) in (29) yields

(33)

Rearranging equations (30), (31), (32) and (33)





(34)

If we consider that the angle of the payload swing θ and the displacement of the cart X from a fixed reference, the output vector will be

(35)

It should be noted here that each of the two variables θ and X are measurable quantities, and thus the two equations (34) and (35) can be placed in a vector-matrix form

(36)

(37)

Equations (36), (37) can be put into a compact form

(38)