Problem $4$: Vehicle Jack

Given the vehicle jack with the vector loop diagram and equations shown below, derive the Jacobian, velocity, and acceleration equations in matrix form and determine any singular poses.

Vector loop equations:

vec$(r{)}_7+$vec$(r{)}_2+$vec$(r{)}_3-$vec$(r{)}_4=$vec$(0)$

vec$(r{)}_6+$vec$(r{)}_5+$vec$(r{)}_7-$vec$(r{)}_1=$vec$(0)$

Constraints:

${}_2+{}_4=$

${}_5+{}_1=$