$6\mathrm{.}9[37]$ Derive the dynamic equations for the three$-$link manipulator with one

prismatic joint shown in Fig. $3\mathrm{.}9.$ Link $1$ has an inertia tensor given by

$C_{1}I=\left[\begin{array}{ccc}{I}_1& 0& 0\\ 0& I_{yy1}& 0\\ 0& 0& I_{zz1}\end{array}\right].$

Link $2$ has point mass $m_2$ located at the origin of its link frame. Link $3$ has an

inertia tensor given by

$C_{3}I=\left[\begin{array}{ccc}{I}_3& 0& 0\\ 0& I_{yy3}& 0\\ 0& 0& I_{zz3}\end{array}\right].$

Assume that gravity is directed opposite hat$(Z{)}_1$ and that viscous friction of magnitude

$v$ : is active at each ioint.

FIGURE $3\mathrm{.}9$: Manipulator having three degrees of freedom and one prismatic joint.