The “sky crane” was a novel solution to the problem of landing the 2000 lb Curiosity rover on the surface of Mars. Curiosity hangs from the descent stage by 60-ft long nylon tethers (Figure 1 (a)).
The descent stage, weighing 1500 lb, uses its thrusters to hover as the rover is lowered to the
surface. Thus, the rover behaves like a pendulum whose base is moving horizontally. The side
thruster force is not constant but is controlled to keep the descent stage from deviating left or right
from the desired vertical path.
Such a control system effectively acts like a spring and a damper, as shown in Figure 1 (b). The
rover mass is mr

, the descent stage mass is md, and the net horizontal component of the thruster
forces is f = kx + cẋ. Among other simplifications, this model neglects vertical motion and any
rotational motion.
Derive the equations of motion of the system in terms of the angle θ and the displacement x,
using: (5+5)
a) Hamiltonian (mechanical energy) Method
b) Lagrangian Method