In terms of the $$xs$,$ys

$,$zs coordinates of a fixed space frame $\{$s$\},$

the frame $\{$a$\}$ has its $$xa$-$axis pointing in the direction $(0,0,1)$ and its $$ya

$-$axis

pointing in the direction $(1,0,0),$ and the frame $\{$b$\}$ has its $$xb$-$axis pointing

in the direction $(1,0,0)$ and its $$yb

$-$axis pointing in the direction $(0,0,1).$

$($a$)$ Draw by hand the three frames, at different locations so that they are easy

to see.

$($b$)$ Write down the rotation matrices Rsa and Rsb$.$

$($c$)$ Given Rsb$,$ how do you calculate R

$1$

sb without using a matrix inverse?

Write down R

$1$

sb and verify its correctness using your drawing.

$($d$)$ Given Rsa and Rsb$,$ how do you calculate Rab $($again without using matrix inverses$)?$ Compute the answer and verify its correctness using your

drawing.

$($e$)$ Let R $=$ Rsb be considered as a transformation operator consisting of

a rotation about $$x by $90$

$.$ Calculate R$1=$ RsaR, and think of Rsa

as a representation of an orientation, R as a rotation of Rsa, and R$1$ as

the new orientation after the rotation has been performed. Does the new

orientation R$1$ correspond to a rotation of Rsa by $90$ about the worldfixed $$xs$-$axis or about the body$-$fixed $$xa$-$axis? Now calculate R$2=$ RRsa.

Does the new orientation R$2$ correspond to a rotation of Rsa by $90$

about the world$-$fixed $$xs$-$axis or about the body$-$fixed $$xa$-$axis?

$($f$)$ Use Rsb to change the representation of the point pb $=(1,2,3)($which is

in $\{$b$\}$ coordinates$)$ to $\{$s$\}$ coordinates.

$($g$)$ Choose a point p represented by ps $=(1,2,3)$ in $\{$s$\}$ coordinates. Calculate

p

$0=$ Rsbps and p

$00=$ RT

sbps$.$ For each operation, should the result be

interpreted as changing coordinates $($from the $\{$s$\}$ frame to $\{$b$\})$ without

moving the point p or as moving the location of the point without changing

the reference frame of the representation?

$($h$)$ An angular velocity w is represented in $\{$s$\}$ as $\backslash$omega s $=(3,2,1).$ What is its

representation $\backslash$omega a in $\{$a$\}?$

$($i$)$ By hand, calculate the matrix logarithm $[\backslash$omega $]\backslash$theta of Rsa. $($You may verify your

answer with software.$)$ Extract the unit angular velocity $\backslash$omega and rotation

amount $\backslash$theta $.$ Redraw the fixed frame $\{$s$\}$ and in it draw $\backslash$omega $.$

$($j$)$ Calculate the matrix exponential corresponding to the exponential coordinates of rotation $\backslash$omega $\backslash$theta $=(1,2,0).$ Draw the corresponding frame relative

to $\{$s$\},$ as well as the rotation axis $\backslash$omega