# 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

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

1. Lagrangian of distal and proximal links.

$$L_i=frac{1}{2}I_{l_1}theta_i^2+frac{1}{2}boldsymbol{omega_i}^Tboldsymbol{I}_{l_2}boldsymbol{omega_i}-m_{l_1}chi_1boldsymbol{g}^Tboldsymbol{h_i}-m_{l_2}chi_2boldsymbol{g}^Tboldsymbol{e_{ix}}$$

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

$$L_p=frac{1}{2}boldsymbol{Omega}^Tboldsymbol{I}_{p}boldsymbol{Omega}-m_{p}Rcos(beta)g^Tboldsymbol{p}$$

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:

$$frac{d}{dt}(frac{dL}{ddot{boldsymbol{q}}})-frac{dL}{dboldsymbol{q}}=0$$

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: