Optimization Vector

In Free-Floating Rigid Body Dynamics we expressed the equations of motion as an affine function of our optimization variable, $\optvar$ . Here, we look at each component in $\optvar$ and detail its meaning, position in the overall vector, and dimensions.

$\optvar = \bmat{ \jsrd_{fb}\\ \jsrd_{j}\\ \torque_{fb}\\ \torque_{j}\\ \we_{0}\\ \vdots\\ \we_{n}\\ }$

$\jsrd_{fb}$ : Floating base joint acceleration ( $6 \times 1$ )
$\jsrd_{j}$ : Joint space acceleration ( $n_{\dof} \times 1$ )
$\torque_{fb}$ : Floating base joint torque ( $6 \times 1$ )
$\torque_{j}$ : Joint space joint torque ( $n_{\dof} \times 1$ )
$\we_{n}$ : External wrench ( $6 \times 1$ )

Each of these variables are termed Control Variables in ORCA and are used to define every task and constraint.

These variables can of course be combined for convenience:

$\jsrd$ : Generalised joint acceleration, concatenation of $\jsrd_{fb}$ and $\jsrd_{j}$ ( $6+n_{\dof} \times 1$ )
$\torque$ : Generalised joint torque, concatenation of $\torque_{fb}$ and $\torque_{j}$ ( $6+n_{\dof} \times 1$ )
$\we$ : External wrenches ( $n_{\text{wrenches}} 6 \times 1$ )
$\optvar$ : The whole optimization vector ( $6 + n_{\dof} + 6 + n_{\dof} + n_{wrenches}6 \times 1$ )

With our optimization varible well defined, we can now formulate the optimization problem.