In this paper about Backstepping controll of a quadrotor helicopter an algorithm for control is described, but I have hit a dead end.
In equation 15 it is described the part of state space for the angular and translation motion of a rigid body.
The author states $R_r$ is the rotation velocity matrix between Earth fixed reference frame and Body fixed reference frame. I assume the "between" means a rotation from body coordinates to earth coordinates.
$R_t$ is the translation velocity matrix between Earth fixed reference frame and Body fixed reference frame. I also assume the "between" means a rotation from body coordinates to earth coordinates.
One question is how do I calculate the value of the angular acceleration described by the partial derivative of $\dot{\phi}$ and $\dot{\theta}$. The author does not state and I would like to know if it is numerically or if it is analytically possible.
The other question is in which referential is $\dot{\zeta}$ and why did the author make a rotation and "derotation" on $Kt$? From the paper $G$ is a vector with the $z$ element set to $g=9.81$.
Last question, more like a curiosity, would anybody give me a pointer to state-space formulation? I do not follow how the author composed the state space system.
Best Answer
If $\vec{\omega} =\vec{ \omega}(\phi,\theta,\dot{\phi},\dot{\theta})$ then
$$ \vec{\alpha} = \frac{\partial \vec{\omega}}{\partial \phi} \dot{\phi} + \frac{\partial \vec{\omega}}{\partial \dot{\phi}} \ddot{\phi} + \frac{\partial \vec{\omega}}{\partial \theta} \dot{\theta} + \frac{\partial \vec{\omega}}{\partial \dot{\theta}} \ddot{\theta}$$
Note also that the derivative of the 3×3 rotation matrix $R$ is
$$ \frac{\partial}{\partial t} R = \vec{\omega} \times R $$
and that means that if $R=R_1(\hat{x},\phi) R_2(\hat{z},\theta)$ then
$$ \dot{R} = \vec{\omega}\times R = (\hat{x} \dot{\phi} \times R_1) R_2 + R_1 ( \hat{z} \dot{\theta} \times R_2) \\ = \hat{x} \dot{\phi} \times R + (R_1 \hat{z} \dot{\theta}) \times R$$ which is how you derive the rotational kinematic relationships
$$ \vec{\omega} =\hat{x} \dot{\phi} + R_1(\hat{x},\phi) \hat{z} \dot{\theta} $$