A single link robotic manipulator

The dynamic equations governing the behavior of a single link robot with flexible joint are traditionally obtained from Lagrangian dynamics considerations. The simple robot system under study is shown in the attached image. Let $x_1=\theta_m$ be the motor angular position, the corresponding angular velocity $x_2 = d\theta/dt$, the elastic force $x_3 = k_s(\theta_t - \theta_m)$ and $x_4 = \{ d\theta_l/dt - d\theta_m/dt\}/\rho$, where $\rho^2=1/k_s$. Then the state variable representation is:

$$\begin{align*} \dot{x}_1(t) &= x_2(t)\\ \dot{x}_2(t) &= -a_5x_2(t)+a_1x_3(t)+a_1u(t)\\ \dot{x}_3(t) &= x_4(t)/\rho\\ \dot{x}_4(t) &= \{ -a_2a_3\sin{[\rho^2x_3(t)+x_1(t)]}-a_4x_3(t)-a_7x_2(t)-a_6\rho x_4(t) - a_1u(t)\}/\rho \end{align*}$$

with $a_1=1/J_m$, $a_2=1/J_l$, $a_3=mgl$,$a_4=a_1+a_2$,$a_5=B_m/J_m;a_6=B_l/J_l$,$a_7=a_6-a_5$ and $u(t)=\tau(t)$.