Helical motion of a rigid body
I want to show that a rigid body, with two components of its angular velocity vector and one component of its linear velocity vector, in the absence of external forces and torques, has helical trajectories. This is usually taken for granted in various papers I have read (see for example this p.4 section 4, or this p.206 first paragraph).
I consider a moving rigid body whose centre of mass is located at a point $\boldsymbol{r}{0}(t)$ in the laboratory frame of reference defined by the basis vectors $\boldsymbol{e}{x},\boldsymbol{e}{y},\boldsymbol{e}{z}$. I attach a moving frame of reference $\left{ \boldsymbol{r}{0}(t);\boldsymbol{e}{x}^{\prime}(t),\boldsymbol{e}{y}^{\prime}(t),\boldsymbol{e}{z}^{\prime}(t)\right} $ to the body; this frame is centred at $\boldsymbol{r}{0}(t)$ and its axes rotate with the frame of the body. The rotating triad $\left(\boldsymbol{e}{x}^{\prime},\boldsymbol{e}{y}^{\prime},\boldsymbol{e}{z}^{\prime}\right)$ can be characterised by three Euler angles $\theta_{1},\theta_{2},\theta_{3}$ and the following transformation rules:
$$ \mathbf{e}{i}^{\prime}=\boldsymbol{L}\left(\theta{1},\theta_{2},\theta_{3}\right)\cdot\mathbf{e}_{i} $$
I adopt the Tait-Bryan angle convention, with an $y-x^{\prime\prime}-z^{\prime}$ intrinsic definition:$\theta_{1}$ is the yaw (anticlockwise around $e_{y}$), $\theta_{2}$ the pitch (anticlockwise around $e_{x}^{\prime}$) and $\theta_{3}$ the bank (roll, anticlockwise around $e_{z}^{\prime}$). 101
For brevity, I define the vector $\boldsymbol{\theta}=\left(\theta_{1},\theta_{2},\theta_{3}\right)$ of the three independent Euler angles. The matrix of the transformation is $$\boldsymbol{L}\left(\boldsymbol{\theta}\right)=\boldsymbol{R}{Z}\left(\theta{3}\right)\boldsymbol{R}{X}\left(\theta{1}\right)\boldsymbol{R}{Y}\left(\theta{2}\right)$$ where the $\bf R_i$ are the rotation matrices in $\mathbb{R}^3$. I assume that the body has linear velocity $\boldsymbol{U}(t)=U\boldsymbol{e}{z}^{\prime}(t)$ oriented along its anterior-posterior axis $\boldsymbol{e}{z}^{\prime}(t)$, and rotational velocity vector $\boldsymbol{\omega}(t)$. In the chosen representation, the components of angular velocity in the body's frame of reference are
\begin{align} \omega_{x} & =\dot{\theta}{2}\cos\theta{1}-\dot{\theta}{3}\sin\theta{1}\cos\theta_{2}\ \omega_{y} & =\dot{\theta}{1}+\dot{\theta}{3}\sin\theta_{2}\ \omega_{z} & =\dot{\theta}{2}\sin\theta{1}+\dot{\theta}{3}\cos\theta{1}\cos\theta_{2}. \end{align}
These can be found by noticing that $$\boldsymbol{L}^{T}\dot{\boldsymbol{L}}=\left(\begin{array}{ccc} 0 & -\omega_{z} & \omega_{y}\ \omega_{z} & 0 & -\omega_{x}\ -\omega_{y} & \omega_{x} & 0 \end{array}\right)$$
The trajectory of the body is therefore given by the curve traced by its centre of mass $\boldsymbol{r}{0}$ as it moves through space. 804