I'm trying to model a robot which is composed of a non-holonomic and a holonomic system overall is non-holonomic. Basically it's a robot arm + mobile platform. So far I have found a similar work like this https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=293207 before the Simulation section, they say they ignore the dynamics of the manipulator over the mobile platform, so they are treated independently. I cannot ignore it. So far I got this:

$$ Vel_{mobilePlataform} = \begin{bmatrix} \dot x_1 \\ \dot y_1 \\ \dot z_1\end{bmatrix} = \begin{bmatrix} f_x(yaw, pitch, roll) \\ f_y(yaw, pitch, roll) \\ f_z (yaw)\end{bmatrix} $$ thanks to this: A tratise on Analytical Dyanmics - L.A.Pars example, the rolling penny.

So the fx and fy are non integrables, so they are constraints, so far so good. But attaching a single joint on the penny's radios direction. I will get this velocities on the joints $$ Vel_{joint} = \begin{bmatrix} \dot x_2 \\ \dot y_2 \\ \dot z_2 \end{bmatrix} = \begin{bmatrix}g_x \\ g_y \\ g_z \end{bmatrix} + \begin{bmatrix} \dot x_1 \\ \dot y_1 \\ \dot z_1 \end {bmatrix} $$

since gz is not integrable, z2 also becomes a constraint. So, the whole system is non-holonomic, right? I'm stuck here. Then, the equation of motion should be for every Q like this? $$ Q_{x1}, Q_{y1}, Q_{yaw}, Q_{pitch}, Q_{roll}, Q_{x2}, Q_{y2}, Q_{z2}, Q_{joint-angle} $$ So, then for computing the lagrange multipliers will be like this

$$ Q_{x1}\delta x_1 + Q_{y1}\delta y_1+ Q_{yaw}\delta(yaw)+ Q_{pitch}\delta(pitch)+ Q_{roll}\delta(roll)+ Q_{x2}\delta x_2+ Q_{y2}\delta y_2+ Q_{z2} \delta z_2+ Q_{joint-angle}\delta(joint-angle) = \lambda_1 (\delta x_1 - f_x) + \lambda_2(\delta y_2 - f_y)+\lambda_3(\delta x_2-g_x-f_x) +\lambda_4(\delta y_2 -g_y-f_y) + \lambda_5(\delta z_2-g_z-f_z) $$

by grouping terms I can have the values of all lambdas, for example:

$$ Q_{x1} = \lambda_1 \\ ... \\ Q_{x2} = \lambda_3 $$

Am I doing ok? Does this mean the trajectory of the linked masses can be independent of each other?


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