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	The homogeneous position vector of point P with respect to Frame A is</p>
<p>
	<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>p</mi><mrow><mi>A</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0.7071</mn></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.7071</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.7071</mn></mrow></mtd><mtd><mrow><mn>0.7071</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mn>0.5</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0.866</mn></mrow></mtd><mtd><mrow><mn>0.5</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn></mrow></mtd><mtd><mrow><mn>0.866</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0.866</mn></mrow></mtd><mtd><mrow><mn>0.433</mn></mrow></mtd><mtd><mrow><mn>0.25</mn></mrow></mtd><mtd><mrow><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn></mrow></mtd><mtd><mrow><mn>0.75</mn></mrow></mtd><mtd><mrow><mn>0.433</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn></mrow></mtd><mtd><mrow><mn>0.866</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mrow></mstyle></math> </span>​</p>@
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	&nbsp;</p>
<div>
	A robot arm, consisting of three segments, has end A fixed to the ground and end P free to perform tasks. The arm has four joints and each one has its own frame of reference (see figure below). The orientation of Frame B with respect to Frame A is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><mtable rowalign="baseline" columnalign="center" groupalign="{left}" rowspacing="1.0ex"><mtr><mtd><mrow><mn>0.7071</mn></mrow></mtd><mtd><mrow><mo>−</mo><mn>0.7071</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.7071</mn></mrow></mtd><mtd><mrow><mn>0.7071</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd></mtr></mtable></mfenced></mrow></math> ​, the position vector of the origin of Frame B with respect to Frame A is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><mtable rowalign="baseline" columnalign="center" groupalign="{left}" rowspacing="1.0ex"><mtr><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.5</mn></mrow></mtd></mtr></mtable></mfenced></mrow></math> ​, the orientation of Frame C with respect to Frame B is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><mtable rowalign="baseline" columnalign="center" groupalign="{left}" rowspacing="1.0ex"><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0.866</mn></mrow></mtd><mtd><mrow><mn>0.5</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mo>−</mo><mn>0.5</mn></mrow></mtd><mtd><mrow><mn>0.866</mn></mrow></mtd></mtr></mtable></mfenced></mrow></math> ​,&nbsp;the position vector of the origin of Frame C with respect to Frame B is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><mtable rowalign="baseline" columnalign="center" groupalign="{left}" rowspacing="1.0ex"><mtr><mtd><mrow><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd></mtr></mtable></mfenced></mrow></math> ​, the orientation of Frame D with respect to Frame C is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><mtable rowalign="baseline" columnalign="center" groupalign="{left}" rowspacing="1.0ex"><mtr><mtd><mrow><mn>0.866</mn></mrow></mtd><mtd><mrow><mn>0.433</mn></mrow></mtd><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mn>0.5</mn></mrow></mtd><mtd><mrow><mn>0.75</mn></mrow></mtd><mtd><mrow><mn>.433</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mo>−</mo><mn>0.5</mn></mrow></mtd><mtd><mrow><mn>.866</mn></mrow></mtd></mtr></mtable></mfenced></mrow></math> <span>, the position vector of the origin of frame D with respect to Frame C is <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><mtable rowalign="baseline" columnalign="center" groupalign="{left}" rowspacing="1.0ex"><mtr><mtd><mrow><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd></mtr></mtable></mfenced></mrow></math> </span> ​&nbsp;​and the position vector of the point P with respect to Frame D is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><mtable rowalign="baseline" columnalign="center" groupalign="{left}" rowspacing="1.0ex"><mtr><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.25</mn></mrow></mtd></mtr></mtable></mfenced></mrow></math> ​.</div>
<div>
	&nbsp;</div>
<div>
	<img alt="" src="__BASE_URI__4Frames.PNG" style="width: 500px; height: 342px;"></div>
<div>
	Find the position vector of point P with respect to frame A. (Give answers with 3 digits after the decimal place)</div>
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	What is the x-component of the position vector of the point P with respect to Frame A?</p>
<div>
	&nbsp;</div>@
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	What is the y-component of the position vector of point P with respect to Frame A?</p>@
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	What is the z-component of the position vector of point P with respect to Frame A?</p>@
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qu.1.2.name=Position and orientation. Q1 Cascading rotations@
qu.1.2.comment=<p>
	The rotation matrix of the original frame with respect to the new frame is the transpose of</p>
<p>
	<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>0.3</mn></mrow></mfenced></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>0.3</mn></mrow></mfenced></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>0.3</mn></mrow></mfenced></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>0.3</mn></mrow></mfenced></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>0.5</mn></mrow></mfenced></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>0.5</mn></mrow></mfenced></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>0.5</mn></mrow></mfenced></mrow></mtd><mtd><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>0.5</mn></mrow></mfenced></mrow></mtd></mtr></mtable></mfenced></mrow></mrow></mstyle></math> </span>​</p>
<p>
	&nbsp;</p>@
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qu.1.2.question=<p>
	A new frame of reference is obtained by rotating a frame of reference about its x-axis by&nbsp;0.5 rad&nbsp;and then about the original y-axis by 0.3 rad.&nbsp;</p>
<p>
	<img alt="" src="__BASE_URI__2rotations.PNG" style="width: 600px; height: 343px;"></p>
<p>
	What is the rotation matrix of the original frame with respect to the new frame?</p>
<p>
	Use three decimal places and provide the answer as a set of three vectors in the form: ((r11,r21,r31),(r12,r22,32),(r13,r23,r33)).</p>@
qu.1.2.answer=((0.955,0.142,0.259),(0,0.878,-0.479),(-0.296,0.458,0.838))@

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qu.1.3.name=Position and orientation. Q2. Roll, pitch and yaw angles@
qu.1.3.comment=<p>
	The roll, pitch and yaw angles can be found using the following equations:</p>
<p>
	Pitch:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&beta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>sin</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>a</mi><mrow><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>1</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math> </span> ​</p>
<p>
	Yaw:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>tan</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msub><mi>a</mi><mrow><mn>2</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>1</mn></mrow></msub><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&beta;</mi></mrow></mfenced></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><msub><mi>a</mi><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mn>1</mn></mrow></mrow></msub><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&beta;</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced></mrow></mstyle></math> </span> ​</p>
<p>
	Roll:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&gamma;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>tan</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msub><mi>a</mi><mrow><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2</mn></mrow></msub><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&beta;</mi></mrow></mfenced></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><msub><mi>a</mi><mrow><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>3</mn></mrow></msub><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&beta;</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced></mrow></mstyle></math> </span> ​</p>
<p>
	where&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>a</mi><mrow><mi>i</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>j</mi></mrow></msub></mrow></mstyle></math> </span>​is the element in the i<sup>th&nbsp;</sup>row and j<sup>th&nbsp;</sup>column of the rotation matrix.&nbsp;</p>@
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qu.1.3.difficulty=0.0@
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qu.1.3.school=7f01c0fb-116a-4f9d-bafc-e91de0b62105@
qu.1.3.attributeAuthor=true@
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qu.1.3.numberOfAttemptsLeft=1@
qu.1.3.numberOfTryAnother=0@
qu.1.3.numberOfTryAnotherLeft=0@
qu.1.3.question=<p>
	The rotation matrix of a frame with respect to a reference frame is</p>
<p>
	&nbsp;</p>
<p>
	<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><mtable rowalign="baseline" columnalign="center" groupalign="{left}" rowspacing="1.0ex"><mtr><mtd><mrow><mn>0.216</mn></mrow></mtd><mtd><mrow><mn>0.128</mn></mrow></mtd><mtd><mrow><mn>0.968</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.216</mn></mrow></mtd><mtd><mrow><mn>0.960</mn></mrow></mtd><mtd><mrow><mo>−</mo><mn>0.176</mn></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mn>0.952</mn></mrow></mtd><mtd><mrow><mn>0.248</mn></mrow></mtd><mtd><mrow><mn>0.180</mn></mrow></mtd></mtr></mtable></mfenced></mrow></math> ​.</p>
<p>
	&nbsp;</p>
<p>
	Find the roll, pitch and yaw angles.</p>
<p>
	&nbsp;</p>
<p>
	Provide the answer in radians, round the values to the first decimal place and use the following format: (roll,pitch,yaw).</p>@
qu.1.3.answer=(0.9,1.3,0.8)@

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qu.1.4.name=Position and orientation. Q4. Joints@
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qu.1.4.question=<p>
	Match the joint with its description:</p>
<p>
	&nbsp;</p>
@
qu.1.4.term.1=Revolute Joint@
qu.1.4.term.1.def.1=Allows rotation about one axis.@
qu.1.4.term.2=Spherical Joint@
qu.1.4.term.2.def.1=Allows three degrees of rotational freedom about the center of the joint.@
qu.1.4.term.3=Prismatic Joint@
qu.1.4.term.3.def.1=Allows relative translation about one axis.@

qu.1.5.mode=Multiple Choice@
qu.1.5.name=Forward Kinematics Q1. DH intersecting axes@
qu.1.5.comment=<p>
	The&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi mathvariant="italic">i</mi><mrow><mi mathvariant="italic">th</mi></mrow></msup></mrow></math> </span> <span>&nbsp;link length <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="italic">a</mi><mrow><mi mathvariant="italic">i</mi></mrow></msub></mrow></math> </span> </span> ​&nbsp;is ​equal to the length of the common normal between the&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi mathvariant="italic">i</mi><mrow><mi mathvariant="italic">th</mi></mrow></msup></mrow></math> </span> ​and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mfenced open="(" close=")" separators=","><mrow><mi mathvariant="italic">i</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant="italic">th</mi></mrow></msup></mrow></math> </span> ​joint axes. If these two axes&nbsp;intersect, then the length of the common normal between them is equal to zero i.e.&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><msub><mi>a</mi><mrow><mi>i</mi></mrow></msub></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math> </span> ​.</p>@
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qu.1.5.difficulty=0.0@
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qu.1.5.question=<p>
	<span style="font-size:12px;">If the&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi mathvariant="italic">i</mi><mrow><mi mathvariant="italic">th</mi></mrow></msup></mrow></math>&nbsp; ​and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mfenced open="(" close=")" separators=","><mrow><mi mathvariant="italic">i</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant="italic">th</mi></mrow></msup></mrow></math>&nbsp; ​joint axes of a robot manipulator intersect, then which of the following has to be true:</span></p>
<p>
	&nbsp;</p>
@
qu.1.5.answer=1@
qu.1.5.choice.1=the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>i</mi><mrow><mi>th</mi></mrow></msup></mrow></mstyle></math> link length <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>a</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math> is equal to zero@
qu.1.5.choice.2=the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>i</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>1</mi></mrow></mfenced><mrow><mi>th</mi></mrow></msup></mrow></mstyle></math> link length <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>a</mi><mrow><mi>i</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></msub></mrow></mstyle></math> is equal to zero@
qu.1.5.choice.3=the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>i</mi><mrow><mi>th</mi></mrow></msup></mrow></mstyle></math> joint offset <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>d</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math> is equal to zero@
qu.1.5.choice.4=the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>i</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mrow><mi>th</mi></mrow></msup></mrow></mstyle></math> joint offset <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>d</mi><mrow><mi>i</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></msub></mrow></mstyle></math> is equal to zero@
qu.1.5.fixed=@

qu.1.6.mode=Multipart@
qu.1.6.name=Forward Kinematics Q2. DH conv X2@
qu.1.6.comment=<p>
	&nbsp;</p>
<div>
	The coordinate system of the ground link can be chosen based on convenience as long as its z-axis is aligned with the axis of joint 1.</div>
<div>
	&nbsp;</div>
<div>
	The coordinate system of the end effector, also called the hand coordinate system, can also be chosen based on convenience as long as its x-axis is normal to the last joint axis.</div>
<div>
	&nbsp;</div>@
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qu.1.6.question=<p>
	True or False?</p>
@
qu.1.6.weighting=1,1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.editing=useHTML@
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qu.1.6.part.1.choice.2=False@
qu.1.6.part.1.choice.1=True@
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qu.1.6.part.1.question=<p>
	According to the DH convention, the hand coordinate system of a robot manipulator is selected such that its x-axis is normal to the axis of the last joint.&nbsp;</p>
<p>
	&nbsp;</p>@
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qu.1.6.part.1.name=Forward Kinematics Q2. DH conv #1 - clone@
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qu.1.6.part.2.choice.1=True@
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qu.1.6.part.2.question=<p>
	According to the DH convention, the coordinate system of the base link of a robot manipulator is selected such that its x-axis is aligned with the axis of joint 1.&nbsp;</p>
<p>
	&nbsp;</p>@
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qu.1.6.part.2.name=Forward Kinematics Q2. DH conv #1@

qu.1.7.mode=Ntuple@
qu.1.7.name=Forward Kinematics Q3.@
qu.1.7.comment=@
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qu.1.7.question=<p>
	The following table contains the DH parameters of a serial manipulator:</p>
<p>
	&nbsp;</p>
<p>
	<img alt="" src="__BASE_URI__Tableofparams.PNG" style="width: 300px; height: 324px;"></p>
<p>
	&nbsp;</p>
<p>
	The position vector of a point with respect to the hand coordinate system is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><mtable rowalign="baseline" columnalign="center" groupalign="{left}" rowspacing="1.0ex"><mtr><mtd><mrow><mn>0.1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0.3</mn></mrow></mtd></mtr></mtable></mfenced></mrow></math> ​. Find the position vector of this point with respect to the base coordinate system.</p>
<p>
	&nbsp;</p>
<p>
	Provide the answer using two decimal places for each component of the vector.</p>
<p>
	&nbsp;</p>@
qu.1.7.answer=(1.55,0.30,1.85)@

