I am trying to derive a (linears) state space representation for a system of a cart with 2 weights and a Force applied.
I have derived the following equation for the different Forces in the system: $$ F+(M+m)q''+ml[-\theta_1'' \cos(\theta_1)+\theta_1'^2 \sin(\theta_1)+\theta_2'' \cos(\theta_2) - \theta_2'^2 \sin(\theta_2)] = 0 $$ I am not sure how to include gravity in this but even if I did and I did linearize it around (q=0, q'=0, q''=0, F=0, $\theta_1=0$, $\theta_2=0$, $\theta_1'=0$, $\theta_2'=0$, $\theta_1''=0$, $\theta_2''=0$) I do not see how I would get a valid Dynamics matrix from the equation as it relates $\theta_1$ and $\theta_2$ to q'' instead of q'' to its own terms.
