Problem $3$: There has been considerable engineering effort directed at finding ways to perform

manipulative operations in space$-$for example, assembling a space station and acquiring target

satellites. To perform such tasks, space shuttles carry a remote manipulator system $($RMS$)$ in the

cargo bay. The RMS has proven its effectiveness on recent shuttle missions, but now a new

design approach can be considered$-$a manipulator with inflatable arm segments. Such a design

might reduce manipulator weight by a factor of four while producing a manipulator that, prior to

inflation, occupies only one$-$eighth as much space in the cargo bay as the present RMS$.$

The use of an RMS for constructing a space structure in the shuttle bay is shown in Figure $2$a and

a model of the flexible RMS arm is shown in Figure $2b$ where $J$ is the inertia of the drive motor

and $L$ is the distance to the center of gravity of the load component. Write the equations of

motion in state variable matrix form. Your answer should be in the form of vec$(x{)}^{}=$Avec$(x)+$Bvec$(u).$

Note: Use a small angle approximation $sin{q}_1$~~$q_1$ to simplify equations.

$($a$)$

$($b$)$

Figure $2.$ Remote manipulator system and model for Problem $3.$ Dorf P$3\mathrm{.}29.$