differential equations – Spherical Parallel Manipulator Lagrangian problem

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

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

  1. Lagrangian of distal and proximal links.


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? …

  1. Platform Lagrangian


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:


In the article, the authors took as generalized coordinates:


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:


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.

Some excerpts from the article are presented below:

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