16. Non-Linear Control

Non-Linear Control

The non-linear equations of motions are:

\ddot{z} = g - \frac{u_1}{m}\cos\phi

\ddot{y} = \frac{u_1}{m}\sin\phi

\ddot{\phi} = \frac{u_2}{I_x}

Solving them for u_1, \phi_{\text{command}}, u_2 (where \bar{u_1} ==\ddot{z} ; \bar{u_2} == \ddot{\phi} ; \phi_{\text{command}} == \phi ) gives us:

u_1 = \frac{m(g - \bar{u_1})}{\cos\phi}

\phi_{\text{command}} = \sin^{-1}\left(\frac{m\ddot{y}_{\text{target}}}{u_1}\right)

u_2 = I_x \bar{u}_2