Optimal Manoeuvres of Underactuated Linear Mechanical Systems: The Case of Controlling Gantry Crane Operations

Figure 2

Histograms of (a) cart trajectory 𝑥, (b) swing angle 𝜃, and (c) force 𝐹_𝑎 (open loop).

(a) 1𝑥=√2(𝜂₁+𝜂₂)

(b) 𝜃=√2𝜂₂

(c) 𝐹_𝑎=√2𝑢₁