I have a DH matrix (Denavit-Hartenberg) of a two link manipulator having differential equation of the form:
$$ \begin{bmatrix} \tau_1 \\ \tau_2 \end{bmatrix}= \begin{bmatrix} k_1+k_2\cos\theta_2 & k_3+k_4\cos\theta_2 \\ k_5+k_6\cos\theta_2 & k_7 \end{bmatrix} \begin{bmatrix} \ddot{\theta}_1 \\ \ddot{\theta}_2 \end{bmatrix}+ \begin{bmatrix} k_8\dot{\theta}_2^2 - k_9\dot{\theta}_1\dot{\theta}_2 \\k_{10}\dot{\theta}_1^2 \end{bmatrix}+f(\theta_1,\theta_2) $$
where $f$ is a nonlinear function. $k_i$ are constants
How can such a nonlinear coupled differential equation be expressed in state space form? How to decouple terms which contain product of both states?
Edit : I want to design a state observer for this, so wanted it to be represented in state space form.
You have an equation of the form $$ r=M(θ)\ddot θ+b(θ,\dot θ) $$ Introduce $\omega = \dot θ$ to find the first order system $$ \begin{bmatrix}\dot θ\\\dot ω\end{bmatrix} = \begin{bmatrix}ω\\M(θ)^{-1}(r-b(θ,ω))\end{bmatrix} $$ provided the matrix $M(θ)$ is invertible.