Spherical Parallel Manipulator Lagrangian problem

Question

I faced a very serious problem and I urgently need the help of specialists in robotics, mechanics, physics and mathematics.

I am trying to derive equations of motion from the Lagrangian of a spherical parallel manipulator based on work (3.2 Inverse dynamic modeling): https://www.sciencedirect.com/science/article/abs/pii/S0921889014001250

According to this article, the Lagrangian for the corresponding drive axle consists of two parts:

1. Lagrangian of distal and proximal links.

$L_i=\frac{1}{2}I_{l_1}\theta_i^2+\frac{1}{2}\boldsymbol{\omega_i}^T\boldsymbol{I}_{l_2}\boldsymbol{\omega_i}-m_{l_1}\chi_1\boldsymbol{g}^T\boldsymbol{h_i}-m_{l_2}\chi_2\boldsymbol{g}^T\boldsymbol{e_{ix}}$

where $\boldsymbol{h_i}=\frac{u_i+v_i}{||u_i+v_i||}$;$\boldsymbol{e_{ix}}=\frac{v_i+w_i}{||v_i+w_i||}$; $m_{l_1},m_{l_2},\chi_1,\chi_2,I_{l_1}$ - masses, center of masses of proximal and distal links and moment of inertia of proximal link; $u_i,v_i,w_i$ - design vectors depending on design parameters of the mechanism, angles of rotation of drives and something else? ...

2. Platform Lagrangian

$L_p=\frac{1}{2}\boldsymbol{\Omega}^T\boldsymbol{I}_{p}\boldsymbol{\Omega}-m_{p}R\cos(\beta)g^T\boldsymbol{p}$

where $\boldsymbol{p}$ - unit vector of platform direction; $\boldsymbol{I}_{p}$ - tensor of inertia of platform; $\boldsymbol{g}=[0;0;-9.81]^T$;$m_p,R,\cos(\beta)$ - mass, radius of sphere and angle beetwen platform and center of rotation;

In general, the Lagrangian is written as follows:

$L=L_1+L_2+L_3+L_p$

In the article, the authors took as generalized coordinates:

$\boldsymbol{q}=[\theta_1,\theta_2,\theta_3,\phi,\Theta,\sigma]$;

where - $\theta_1,\theta_2,\theta_3$ - drive angle of rotation; $\phi,\Theta,\sigma$ - platform angles of rotation;

Further, the authors derive the equations of motion according to the classical formula, but they do not give more visual and additional calculations:

$\frac{d}{dt}(\frac{dL}{d\dot{\boldsymbol{q}}})-\frac{dL}{d\boldsymbol{q}}=0$

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From that moment on, I encountered problems of the following nature, which I cannot solve on my own:

1. Lagrangian structure assumes that the vector of generalized coordinates $q$ proposed by the authors lacks the position and velocity vector for the distal link and platform, i.e. $q≠[\theta_1,\theta_2,\theta_3,\phi,\Theta,\sigma]$, but $q=[\theta_1,\theta_2,\theta_3,\boldsymbol{\omega},\boldsymbol{\Omega}]$. Their generalized coordinates do not allow deriving the equations of motion, since angular velocity of the distal link $\boldsymbol{\omega}$ in the article there is no corresponding angular position.
2. Moments of inertia of the platform $\boldsymbol{I}_{p}$ and the distal link $\boldsymbol{I}_{l_2}$ are nonstationary and, strictly speaking, depend on the angle of rotation of the drives, i.e. from $\theta_1,\theta_2,\theta_3$. There is absolutely no indication of this in the article, but this should be taken into account when deriving the equations of motion from $L$. But the question is that it is not clear what all these moments of inertia depend on - on the angle of rotation of the drives or on the angle of rotation of the platform, i.e. what to include in the Lagrangian, $\boldsymbol{I}_{p}(\theta_1,\theta_2,\theta_3)$ or $\boldsymbol{I}_{l_2}(\phi,\Theta,\sigma)$ ?
3. The third problem follows from the second and is connected with the fact that we can still find the coordinates of the design vectors $u_i$ and $v_i$ using the known design parameters and angles of rotation of the drive $\theta_1,\theta_2,\theta_3$, but the position of the vector $w_i$ is calculated iteratively using the problem of direct kinematics, and one position of the drives $\theta_1,\theta_2,\theta_3$ corresponds to 8 platform positions, i.e. it is impossible to establish a direct analytical connection between the angles of rotation of the drives $\theta_1,\theta_2,\theta_3$ with the position of the platform $\phi,\Theta,\sigma$ and the coordinates of the vector $w_i$, and therefore, it is impossible to differentiate the potential energy component of the Lagrangian $L$ by the generalized coordinate $q$.

Dear experts, please help me understand the principle of working with this Lagrangian $L$ as soon as possible. I would be grateful for your help in resolving these issues. I have Mathematica, Maple and Simulink at hand and I am ready to do the proposed calculations and present the results.

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Some excerpts from the article are presented below: