Virtual Work & Quadcopter Torques: Exploring Rotational Dynamics

[space_voyager]

Here is what we know from virtual work:
$$
\delta W=\sum_{i=1}^N{\vec F_i\cdot\delta\vec r_{i}}
$$
Where $N$ is the number of bodies in the system. I am considering a quadcopter, modeled as a rigid body so it is just one body and we have:
$$
\delta W=\vec F\cdot\delta\vec r
$$
My question concerns only rotational dynamics, so let's focus on just the torques. Let us rename $\vec F$ to $\vec M$ just to make it clear that we are talking about torques (moments):
$$
\delta W=\vec M \cdot \delta\vec r
$$
Expressed in the quadcopter frame (call it frame $B$), the moments exerted by the propellers on the quadcopter are, generally:
$$
\vec M=\begin{bmatrix}
\tau_x^B \\ \tau_y^B \\ \tau_z^B
\end{bmatrix}
$$
Because we are talking about moments, the "position" vector $\vec r$ actually holds the rotations expressed in the same frame as $\vec M$. Because I express $\vec M$ in the $B$ frame, I need to express the rotations in the $B$ frame. I am using the Tait-Bryan Euler angle convention. Starting from the inertial frame $I$ then:

1. Yaw by $\psi$ around $z_I$, the result is the $\{x',y',z'\}$ frame
- Corresponding rotation matrix is $R_\psi$
2. Pitch by $\theta$ around $y'$, the result is the $\{x'',y'',z''\}$ frame
- Corresponding rotation matrix is $R_\theta$
3. Roll by $\phi$ around $x''$, the result is the $\{x_B,y_B,z_B\}$ frame
- Corresponding rotation matrix is $R_\phi$

You see that each Euler angle is in a different frame: $\psi$ is in the $I$ frame, $\theta$ is in the $(')$ frame and $\phi$ is in the $('')$ frame. From what I know, we transform between angular rates in the $B$ frame and the Euler angle time derivatives using:

$$
\dot\omega=R_\phi\begin{bmatrix}\dot\phi\\0\\0\end{bmatrix}+
R_\phi R_\theta\begin{bmatrix}0\\\dot\theta\\0\end{bmatrix}+
R_\phi R_\theta R_\psi\begin{bmatrix}0\\0\\\dot\psi\end{bmatrix}
$$

However, $\dot\omega$ is not integrable (see sub-section 3.17 of http://people.mech.kuleuven.be/~bruyninc/tmp/HermanBruyninckx-robotics.pdf). Therefore angles in the body frame as such do not exist and I cannot simply write:

$$
\vec r=R_\phi\begin{bmatrix}\phi\\0\\0\end{bmatrix}+
R_\phi R_\theta\begin{bmatrix}0\\\theta\\0\end{bmatrix}+
R_\phi R_\theta R_\psi\begin{bmatrix}0\\0\\\psi\end{bmatrix}
$$

Where $\vec r$ is our "position" vector in the virtual work equation containing three angles in the $B$ frame identifying the quad-copter's attitude. The end motivation is to apply the generalized force relation:
$$
Q_j=\sum_{i=1}^N{\vec F_i \cdot \frac{\partial \vec r_i}{\partial q_j}}
$$
Where $q_j$ are the generalized coordinates, for me these are:

$q_1=\phi$
$q_2=\theta$
$q_3=\psi$

I really appreciate your help for my question: find the expression for the generalized forces $Q_j$ (well, generalized torque really in this case).

 

[eljose79]

Hello, thank you for your detailed explanation and question. I can provide some insights and suggestions for your problem.

Firstly, it is important to note that the virtual work equation you have provided is a general equation for calculating work in a system, not specifically for rotational dynamics. Therefore, we need to adapt this equation to fit our specific problem.

In the case of a quadcopter, the generalized coordinates are indeed the angles $\phi, \theta, \psi$ as you have correctly identified. However, in order to find the expression for the generalized forces $Q_j$, we need to consider the specific forces acting on the quadcopter and their relationship to the generalized coordinates.

From your explanation, it seems that the only forces acting on the quadcopter are the moments exerted by the propellers. In this case, we can use the definition of torque ($\vec M = \vec r \times \vec F$) to relate the generalized forces to the propeller moments.

In order to do this, we need to express the propeller moments in terms of the generalized coordinates. This can be done by using the rotation matrices you have provided and the definition of torque. The resulting expression for the propeller moments in terms of the generalized coordinates should be:

$$
\vec M = \begin{bmatrix}
\tau_x^B \\ \tau_y^B \\ \tau_z^B
\end{bmatrix} = \begin{bmatrix}
\sin\theta\sin\psi & \cos\psi & 0 \\
-\sin\theta\cos\psi & \sin\psi & 0 \\
\cos\theta & 0 & 1
\end{bmatrix} \begin{bmatrix}
0 \\ 0 \\ \dot{\phi}
\end{bmatrix} = \begin{bmatrix}
\dot{\phi}\cos\theta\sin\psi \\
\dot{\phi}\sin\theta\sin\psi \\
\dot{\phi}\cos\psi
\end{bmatrix}
$$

Now, we can use this expression to find the generalized forces by taking the dot product with the Jacobian matrix of the generalized coordinates, which in this case is simply the identity matrix:

$$
Q_j = \sum_{i=1}^N{\vec F_i \cdot \frac{\partial \vec r_i}{\partial q_j