I have a mobile robot with four mecanum wheels with the following configuration:
I need to define a dynamic model and I was starting using the Lagrange equation to evaluate the kinetic and potential energy.
$$\dfrac{\mathrm{d}}{\mathrm{d}t}\left(\dfrac{\partial L}{\partial\dot{\vartheta}_\mathrm{iw}}\right)- \dfrac{\partial L}{\partial\vartheta_\mathrm{iw}}=\tau_i $$
Then, in order to calculate the kinetic energy, I was using:
$$K=\frac12 m(v_x^2+v_y^2)+\frac12 J_zw_z^2+\frac12 J_w(\dot{\vartheta}_{1w}^2+\dot{\vartheta}_{2w}^2+\dot{\vartheta}_{3w}^2+\dot{\vartheta}_{4w}^2)$$
by considering all the four wheels. I found out that some researchers do not consider all the four wheels, but only three because of the limited degrees of freedom (it should be $3$ dof).
Is it correct to consider the contributions of all wheels or do I just need to consider only three wheels? Can you please explain it better to me?
Is the format of the first equation correct or do I need to add some other contribute?
Thank you.
EDIT: This is the coordinates system that I'm using to define the model:
