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.
Best Answer
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.