Table 1.

Recursive, base-to-tip algorithms for converting between the $\stackrel{.}{\theta }$ and v coordinates using Eq. 33 for serial-topology systems.

$v={\mathcal{D}}^{\frac{1}{2}}{[\mathrm{I}+\mathrm{H}\varphi \mathcal{K}]}^{\ast }\stackrel{.}{\theta }$	$\stackrel{.}{\theta }={[\mathrm{I}-\mathrm{H}\psi \mathcal{K}]}^{\ast }{\mathcal{D}}^{-\frac{1}{2}}v$
$\mathcal{V}(\mathcal{N}+1)=0$	$\mathcal{V}(\mathcal{N}+1)=0$
$\mathbf{for}k=\mathcal{N}\cdots 1$	$\mathbf{for}k=\mathcal{N}\cdots 1$
${\mathcal{V}}^{+}\left(k\right)={\varphi }^{\ast }(k+1,k)\mathcal{V}(k+1)$	${\mathcal{V}}^{+}\left(k\right)={\varphi }^{\ast }(k+1,k)\mathcal{V}(k+1)$
$v\left(k\right)={\mathcal{D}}^{\frac{1}{2}}\left(k\right)[\stackrel{.}{\theta }\left(k\right)+{\mathcal{G}}^{\ast }\left(k\right){\mathcal{V}}^{+}\left(k\right)]$	$\stackrel{.}{\theta }\left(k\right)={\mathcal{D}}^{-\frac{1}{2}}\left(k\right)v\left(k\right)-{\mathcal{G}}^{\ast }\left(k\right){\mathcal{V}}^{+}\left(k\right)$
$\mathcal{V}\left(k\right)={\mathcal{V}}^{+}\left(k\right)+{\mathrm{H}}^{\ast }\left(k\right)\stackrel{.}{\theta }\left(k\right)$	$\mathcal{V}\left(k\right)={\mathcal{V}}^{+}\left(k\right)+{\mathrm{H}}^{\ast }\left(k\right)\stackrel{.}{\theta }\left(k\right)$
end loop	end loop