I am attempting to prove that some sequential series of rotation axes $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n\in\mathbb{R}^3$ is enough to generate all possible rotations when making a full rotation around each axis (rotate the angle of each axis from $0$ to $2\pi$), i.e. they are sufficient to generate all elements in $SO(3)$.
My train of thought:
3-axis rotation: We have $\mathbf{R}(\mathbf{v}_j,\theta_j)=e^{\hat{\mathbf{v}}_j\theta_j}\in SO(3)$, with $\mathbf{v}_j\in\mathbb{R}^3$, $\|\mathbf{v}_j\|_2=1$ and $\hat{\mathbf{v}}_j\in \mathfrak{so}(3)$.
Because $SO(3)$ is a group under matrix multiplication, I can construct (what I would call) a generator-candidate for $SO(3)$: \begin{equation} \mathbf{R}'=\prod_{j=1}^{3} {\mathbf{R}(\mathbf{v}_j,\theta_j)}\in SO(3). \end{equation}
I 'know' that for $\mathbf{R}'$ to be a generator that finds all elements in $SO(3)$, when parameterized by $\theta_j\in[0,2\pi)$ we need the following properties: \begin{align} \phi_1 = \arccos{(\mathbf{v}_1^T\mathbf{v}_2)}&=\pi/2 \\ \phi_2 = \arccos{(\mathbf{v}_2^T\mathbf{v}_3)}&=\pi/2 \end{align} This is a, sort of, elaborate way to state that to be able to get all 3D rotations (when $\theta_j$ goes from $0$ to $2\pi$ we actually get them all twice), the rotation axes need to be orthogonal.
4-axis rotation: This brings me to the more interesting case I want to discuss. Using the same notation as for the 3-axis case: \begin{equation} \mathbf{R}''=\prod_{j=1}^{4} {\mathbf{R}(\mathbf{v}_j,\theta_j)}\in SO(3). \end{equation}
I found (heuristically) that for $\mathbf{R}''$ to be a generator that finds all elements in $SO(3)$, we need the following angles: \begin{equation} \phi_i = \arccos{(\mathbf{v}_i^T\mathbf{v}_{i+1})},\;i=1,2,3 \end{equation} And if we write $\boldsymbol\Phi= [\phi_1,\;\phi_2,\;\phi_3]$, then we need the condition (which is necessary and sufficient): \begin{equation} \|\boldsymbol\Phi-\pi/2\|_{\ell 1} \leq \pi/2. \end{equation} Although I found this by numerical methods, I want to prove this (and possibly a more general case). (Actually, this captures the same condition for $\mathbf{R}'$ if we would take its $\phi_3=0$.)
$n$-axis rotation: More generally I could write \begin{align} \bar{\mathbf{R}}&=\prod_{j=1}^{n} {\mathbf{R}(\mathbf{v}_j,\theta_j)}\in SO(3) \\ \phi_i &= \arccos{(\mathbf{v}_i^T\mathbf{v}_{i+1})},\;i=1,2,\ldots,n-1 \\ \boldsymbol\Phi &= [\phi_1,\;\phi_2,\;\cdots,\;\phi_{n-1}] \\ \|\boldsymbol\Phi - \pi/2 \|_{\ell 1} &\leq c_n \end{align} So $c_3=0$, $c_4=\pi/2$, $c_5=?$, etc.
My attempt at proof:
According to (Chen2011) and (Yang2006) the 'volume' of $SO(3)$ is $8\pi^2$ (they only differ in the scaling factor 8). Therefore I assumed that by finding the Riemannian volume element belonging to my parameterization (as explained in Chen2011) I could integrate all $\theta_j,j=1,2,\ldots,n$ from $0$ to $2\pi$ for any series of rotation axes, and prove the 'orientation workspace volume' I get is $8\pi^2$. So: \begin{equation} \idotsint_V \sqrt{|\det{g}|}\,d\theta_1 \dots d\theta_n = 8\pi^2 \end{equation}
This approach proved futile, obviously, since there are many more ways to get a volume of $8\pi^2$ by 'counting' several elements in $SO(3)$ multiple times and others not at all. There is no way of excluding any volume element already integrated from re-integration.
My question:
Does anyone have any other ideas (or point me in the right direction) on how to prove that $\|\boldsymbol\Phi - \pi/2 \|_{\ell 1} \leq c_n$ holds with $c_3=0$ and $c_4=\pi/2$? (Or how to derive a method to explicitly calculate $c_n$, for $n\geq3$.)
References:
Chen2011: C. Chen and D. Jackson, "Parameterization and evaluation of robotic orientation workspace: a geometric treatment", IEEE Transactions on Robotics, vol. 27, no. 4, pp. 656–663, 2011. (http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5756487)
Yang2006: G. Yang, W. Lin, S. Mustafa, I.-M. Chen, and S. H. Yeo, "Numerical orientation workspace analysis with different parameterization methods", 2006 IEEE Conference on Robotics, Automation and Mechatronics, pp. 1–6, IEEE, 2006. (http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4018833)