$3.$ Compute that composite displacement matrix obtained by a rotation by angle $\backslash$alpha about the current

x$-$axis followed by a translation of b units along the current x$-$axis, followed by a translation of d

units along the current z$-$axis, followed by a rotation by angle $\backslash$theta about the current z$-$axis.

$4.$ Consider the diagram of Figure $1.$ A robot is set to $1$ meter from a table. The table top is $1$ meter

high and $1$ meter square. A frame $\backslash$Sigma $1$ is fixed to the edge of the table as shown. A cube measuring

$20$ cm on a side is placed in the center of the table with frame $\backslash$Sigma $2$ attached to the center of the cube

as shown. A camera is situated directly above the center of the block $2$ meters above the table top

with frame $\backslash$Sigma $3$ attached as shown.

$($a$)$ Find the displacement matrices relating these frames to base frame $\backslash$Sigma $0.$ Find the displacement

matrix relating the frame $\backslash$Sigma $2$ to $\backslash$Sigma $3.$

$($b$)$ Suppose after the camera is calibrated, it is rotated $90$ about z$3.$ Recompute above displacement matrices.

$($c$)$ If the block on the table is rotated $90$ about z$2$ and moved so that its center has coordinated

$[0\mathrm{.}1,0\mathrm{.}8,0\mathrm{.}1]>$ relative to $\backslash$Sigma $1.$ Compute the displacement matrices relating the block frame

to the camera frame and the block frame to the base frame.

$5.$ Programming Assignment:

$($a$)$ Using the Z$-$Y$-$Z $(\backslash$alpha $,\backslash$beta $,\backslash$gamma $)$ Euler angle convention, write a MATLAB program to calculate the

displacement matrix H when the user enters the Euler angles $\backslash$alpha $,\backslash$beta $,\backslash$gamma and position vector d$.$

Test for two examples: $($a$)\backslash$alpha $=10$

$,\backslash$beta $=20$

$,\backslash$gamma $=30$

$,$ d $=[123]>,$ and $($ii$)\backslash$alpha $=30$

$,\backslash$beta $=$

$90$

$,\backslash$gamma $=55$

$,$ d $=[301]>.$

$($b$)$ For Euler angles $\backslash$alpha $=0$

$,\backslash$beta $=20$

$,\backslash$gamma $=0$ associated with R$0$

$1$

$,$ O$0$

$1=[301]>,$ and p

$1=[101]>,$

use MATLAB to compute p

$0$

$.$ Demonstrate with a sketch that your results are correct. Also,

using the same numbers, demonstrate all three interpretations of the displacement matrix.