I studied engineering 10 years ago but am struggling to remember how to find the equations of motion for this project of mine.
I have a two wheeled inverted pendulum that I want to balance.
$M_w = $ wheel mass, $I_w = $ wheel inertia, $r = $ wheel radius
$M_b = $ body mass, $I_b = $ body inertia, $l = $ body length to center of mass
$x = $ displacement along ground
$\theta = $ angle from vertical
Using Lagrange I have.
$T = \frac{1}{2}M_{w}v_{w}^{2} + \frac{1}{2}I_{w}\omega _{w}^{2} + \frac{1}{2}M_{b}v_{b}^{2} + \frac{1}{2}I_{b}\omega _{b}^{2}$
$V = m_{b}gl\cos(\theta)$
where
$v_w = \dot{x}$
$\omega_w = \dot{x}/r$
$v_b = (\dot{x} + l\dot{\theta}cos(\theta))_\hat{i} - (l\dot{\theta}sin(\theta))_\hat{j}$
$v_b . v_b = (\dot{x} + l\dot{\theta}cos(\theta))_\hat{i}^2 - (l\dot{\theta}sin(\theta))_\hat{j}^2$
so
$L = \frac{1}{2}M_{w}\dot{x}^{2} + \frac{1}{2}I_{w}\frac{\dot{x}^2}{r^2} + \frac{1}{2}M_{b}v_{b}.v_{b} + \frac{1}{2}I_{b}\dot{\theta}^{2} -m_{b}gl\cos(\theta)$
Look ok so far?