PLEASE SHOW HOW EQUATION $4\mathrm{.}5$ WAS DETERMINED

EXAMPLE $4\mathrm{.}2$

Give a description of the subspace of ${?}_2^{0}T$ for the polar manipulator with two degrees

of freedom shown in Fig. $4\mathrm{.}6.$ We have

${?}^0{P}_{2\text{O}\text{R}\text{G}}=\left[\begin{array}{c}x\\ y\\ 0\end{array}\right],$

where $x$ and $y$ can take any values. The orientation is restricted because the ${?}^0$hat$(Z{)}_2$ axis

must point in a direction that depends on $x$ and $y.$ The ${?}^0$hat$(Y{)}_2$ axis always points down,

and the ${?}^0$hat$(x{)}_2$ axis can be computed as the cross product ${?}^0$hat$(Y{)}_2{?}^0$hat$(Z{)}_2.$ In terms of $x$ and

$y,$ we have

The subspace can therefore be given as

${?}_2^{0}T=\left[\begin{array}{cccc}\frac{y}{\sqrt[\phantom{2}]{x^2+y^2}}& 0& \frac{x}{\sqrt[\phantom{2}]{x^2+y^2}}& x\\ \frac{-x}{\sqrt[\phantom{2}]{x^2+y^2}}& 0& \frac{y}{\sqrt[\phantom{2}]{x^2+y^2}}& y\\ 0& -1& 0& 0\\ 0& 0& 0& 1\end{array}\right].$