# Do symmetric angular velocities lead to symmetric axis of rotation observed in a fixed frame?

It has been observed that some insects with relatively small stroke amplitudes (such as mosquitos) take advantage of a translating axis of rotation to generate lift where insects with larger stroke amplitudes (such as house flies, or bees) do not Smart wing rotation and trailing-edge vortices enable high frequency mosquito flight - Bomphrey et al. 2017 (see figure 1d).

I'm working on a similar problem, where I need to first verify if simpler wing angle kinematics might result in similar motion of the axis of rotation observed by Bomphrey et al. 2017. They observed that in a fixed frame of motion, the axis of rotation traveled from the leading-edge of the wing to the trailing-edge during the pronation period of the wing motion (see figure 4a). By simpler wing kinematics I mean stroke $\phi$ and pitch $\psi$ angles defined below. These are defined using the same axes from Characterizing Mosquito Flight Using Measurement And Simulation - S. Iams 2014 (see figure 43). I've also included a plot of these angular positions and velocities I made using MatLab.

$$\phi(t)=A_1\sin\left(\frac{2\pi t}{T}\right)$$ $$\psi(t)=A_2\sin\left(\frac{2\pi t}{T}\right)$$

Now that I've given some introduction, my question is whether or not symmetric angular positions and velocities (see below) will automatically lead to symmetric motion of the axis of rotation? My intuition suggests that the answer is yes, but I'm not sure how to verify that, or whether I can take non-symmetric kinematics like those of real insects and make a similar judgement based on the kinematics alone regarding the symmetry of the motion of the axis of rotation. I'm also not entirely sure what the best method for defining the location of the axis of rotation would be.

Bomphrey et al. 2017 did not specify how they defined the axis of rotation, so after some thought I've come up with the following method as a workable option. Consider the vector $\vec{p}$ defined in the plane of the wing in the relative frame. The radial position $\vec{r}$ is a constant, and I will be considering different chordwise positions $\vec{c}$ but still keeping them constant over time.

$$\vec{p}=\begin{pmatrix} r\\ -c\cos(\psi)\\ -c\sin(\psi)\end{pmatrix}$$

To get the equation for the velocity of point $\vec{p}$ in the global frame, I use the classical transformation.

$$\dot{\vec{p}}=\dot{\vec{p}}_r+\vec{\Omega}\times \vec{p}$$

The angular velocity $\vec{\Omega}$ is given below. I asked a related question about how to derive the total angular velocity vector in this situation, which is identical to the vector sum of individual angular velocities in the rotating frame.

$$\vec{\Omega}=\begin{pmatrix} \dot{\psi}\\ 0\\ \dot{\phi}\end{pmatrix}$$

The velocity of point $\vec{p}$ in the global frame is given below.

$$\dot{\vec{p}}=\begin{pmatrix} c\cos(\psi)\dot{\phi}\\ 2c\sin(\psi)\dot{\psi}+r\dot{\phi}\\ -2c\cos(\psi)\dot{\psi}\end{pmatrix}$$

Finally, I determine the location of the axis of rotation by identifying the chordwise location $\vec{c}$ where the magnitude of $\vec{\dot{p}}$ is $0$ or minimum, here the magnitude is $\sqrt{(r^2+c^2\cos^2(\psi))\dot{\phi}^2+4c^2\dot{\psi}^2+2cr\sin(\psi)\dot{\psi}\dot{\phi}}$. Using this method, I think I can predict that location of the axis of rotation should also be symmetric in the half-stroke (first half-stroke $0<t<0.5$, and second half-stroke $0.5<t<1$) but I'm not sure if that is correct, especially since I'm not sure if using the magnitude is the appropriate method to define the axis of rotation. I have plotted this on a contour plot for $r=1$ and $-2<c<2$, where the black curve is the chordwise location of minimum $\dot{\vec{p}}$ (highlighted in white between $0<c<1$ i.e. between the leading and trailing-edge). EDIT: I calculated a 4th order Fourier series for the curve on the contour plot and shifted it forward by $2\pi$ to see if it was symmetric, it turns out that it IS symmetric. I also shifted it by $\pi$ and negated, and it IS also symmetric under this transformation. In general though, the wing angle kinematics are not symmetric as in S. Iams 2014 as both angles have a some phase (see figure below).

To summarize my question, so as to make this clear and not just a discussion.

• Does starting with wing angle kinematics that are symmetric in the half-stroke necessarily result in symmetric motion of the axis of rotation observed in the global frame? And how would I go about verifying that?
• A follow up question would be is the method I described above an accurate way to define the motion of the axis of rotation in general?