I am working on Lagrangian derived equations of motions for a robot in matrix form.
Structure of such an equation is known:
$M(\alpha_1,\alpha_2,q)\ddot{q}+C(\alpha_1,\alpha_2,q,\dot{q})\dot{q}+G(\alpha_1,\alpha_2,q)=\tau$
$M(\alpha_1,\alpha_2,q)$, $C(\alpha_1,\alpha_2,q,\dot{q})$ are matrices, and $G(\alpha_1,\alpha_2,q)$ are vector.
$\ddot{q},\dot{q},q$ and $\tau$ - generalized coordinates and drive forces (all as vectors).
Problem: formulate an optimization problem and choose such parameters $\alpha_1,\alpha_2$ to minimize (for example) the maximum amount of drive forces.
But, in this case, we are manipulating matrices, and I don't know which functional on the matrix minimizes the effort?
Trace, norm, their maximum/minimum, their path integral, condition number, etc?
I asked this question here, but did not get a clear answer: https://robotics.stackexchange.com/questions/22427/equations-of-motion-in-matrix-form-and-energy-consumption