Non-constant angular velocity

Question

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.

Answer 1

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

$$ \vec{\omega} =\hat{x} \dot{\phi} + R_1(\hat{x},\phi) \hat{z} \dot{\theta} $$

Answer 2

The answer provided by ja72 pointed me to the correct direction but did not answer directly my questions. The answers are:

The author states Rr 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. Rt 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 is right the other not. According to equation $(1)$ $R_t$ is a transformation matrix from body frame to earth fixed frame.

$R_r$ on the other hand is a rotation matrix from body fixed frame to earth fixed frame so my assumption on $R_r$ was wrong.

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 answer to this question is through the gradient of the vector, which in matrix from, is similar to (but not) the Jacobian of the matrix. This can be obtained analytically taking the derivatives of $R_r$.

The other question is in which referential is ζ˙ and why did the author make a rotation and "derotation" on Kt?

According to $(1)$ the referential of $\zeta$ is earth fixed. The "derotation" is due to symbolic manipulation in $(15)$.

Regarding the state space formulation I still do not follow how it works out, but from $(9)$ I was able to get angular and linear acceleration:

$$\begin{cases} \ddot{\eta_{i}} & =\left(-I_{t}R_{r}\right)^{-1}\left(-T_{pb}+I_{t}\left(\frac{\partial R_{r}}{\partial\phi}\dot{\phi}+\frac{\partial R_{r}}{\partial\theta}\dot{\theta}+\frac{\partial R_{r}}{\partial\psi}\dot{\psi}\right)\dot{\eta}+K_{r\_aero}R_{r}\dot{\eta}+\left(R_{r}\dot{\eta}\right)\times\left(I_{t}R_{r}\dot{\eta}\right)\right)\\ \ddot{V}_{i} & =\left(-mR_{t}^{T}\right)^{-1}\left(K_{t}R_{t}^{T}\dot{V}_{i}+mR_{t}^{T}G-F_{pb}\right) \end{cases}$$

So far this formulation seems to be giving good results but I have not verified against another opinion, and I may be missing something.