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

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? ...

1. 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:

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

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:

• You've ask this question here, and Mathematica, and Math, and Physics, but you haven't really labeled anything here! Which link is distal? Which one is the base? Which end is stationary and which end moves? The pink and orange links both have a series of pins. Starting on one end it looks like the numbering goes $O$, $C_2$, $C_3$, but on the other end it goes $A_i$, $B_i$. Jul 12 at 20:40
• You've got lots of thetas in your first question, but I can't tell what those are, or honestly what the omega or Omega are supposed to be either. The diagram has a $\psi$ that isn't mentioned in your questions and conversely there's no $\Omega$ in your diagram. Regarding your second question, the moments of inertia are joint-angle-dependent just like in a robot arm. No difference here. For the third question, I can't tell what you mean with direct kinematics - forward or inverse. I'm not understanding how fixing all the joint angles still gets you 8 platform orientations. Jul 12 at 20:43
• I would suggest splitting this up into distinct questions, and then leaving out all the extra information that isn't directly related to each question. For example, your first question here is mentioning Lagrangians, but really it looks like you're asking for joint conventions or labeling help or something. Jul 12 at 20:49
• @Chuck youtu.be/kN09M5QGGBk
– dtn
Jul 12 at 20:51
• That looks like a totally different mechanism. Are the joints in the diagram here not driven? Part of your question states, "$\theta_1, \theta_2, \theta_3$ - drive angle of rotation" but then I'm not seeing any other thetas and it also isn't apparent that the orange plate rotates. Jul 12 at 21:00