7020euler.fodt

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{bold nitalic A}
right]
=
left [
matrix
{
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sin %theta_1z # cos %theta_1z # 0 ##
0 # 0 # 1
}
right]
left [
matrix
{
1 # 0 # 0 ##
0 # cos %theta_2x # -sin %theta_2x ##
0 # sin %theta_2x # cos %theta_2x

}
right]
left [
matrix
{
cos %theta_3z # -sin %theta_3z # 0 ##
sin %theta_3z # cos %theta_3z # 0 ##
0 # 0 # 1
}
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Converting<draw:frame draw:style-name="fr1" draw:name="Object6" text:anchor-type="as-char" svg:width="0.3055in" svg:height="0.2134in" draw:z-index="7"><draw:object>
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=
left [
matrix
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left (
stack
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-
sin %theta_1z sin %theta_3z
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left (
stack
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-
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0 ##
left (
stack
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sin %theta_1z cos %theta_3z #
+
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}
right ) # 
left (
stack
{
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+
cos %theta_1z cos %theta_3z 
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0 # 
0 # 
1
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{bold nitalic A}
right]
=
left [
matrix
{
cos left( %theta_1z + %theta_3z right) # 
-sin left( %theta_1z + %theta_3z right) # 
0 ##
sin left( %theta_1z + %theta_3z right) # 
cos left( %theta_1z + %theta_3z right) # 
0 ##
0 # 
0 # 
1
}
right]
</annotation>
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tan^-1
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{bold nitalic A}
right]
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left [
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left (
stack
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+
sin %theta_1z sin %theta_3z
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right ) # 
left (
stack
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+
sin %theta_1z cos %theta_3z 
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right ) # 
0 ##
left (
stack
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sin %theta_1z cos %theta_3z #
-
cos %theta_1z sin %theta_3z 
}
right ) # 
left (
stack
{
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-
cos %theta_1z cos %theta_3z 
}
right ) # 
0 ##
0 # 
0 # 
-1
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{bold nitalic A}
right]
=
left [
matrix
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0 ##
sin left( %theta_1z - %theta_3z right) # 
cos left( %theta_1z - %theta_3z right) # 
0 ##
0 # 
0 # 
-1
}
right]
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        </mrow>
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=
tan^-1
A_21 over A_11</annotation>
       </semantics>
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   <text:p text:style-name="Standard">Otherwise:</text:p>
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       <semantics>
        <mtable>
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          <mtd>
           <mrow>
            <mtext/>
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             <msub>
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               <mn>1</mn>
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              </mrow>
             </msub>
             <mo stretchy="false">=</mo>
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            </mfrac>
           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>θ</mi>
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               <mn>3</mn>
               <mi>z</mi>
              </mrow>
             </msub>
             <mo stretchy="false">=</mo>
             <msup>
              <mi>tan</mi>
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            <mfrac>
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         <mtr>
          <mtd>
           <mrow>
            <mtext/>
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              <mi>θ</mi>
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             </msub>
             <mo stretchy="false">=</mo>
             <msup>
              <mi>tan</mi>
              <mrow>
               <mo stretchy="false">−</mo>
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              </mrow>
             </msup>
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            <mrow>
             <mfrac>
              <msub>
               <mi>A</mi>
               <mn>31</mn>
              </msub>
              <mrow>
               <mi>sin</mi>
               <msub>
                <mi>θ</mi>
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                 <mn>3</mn>
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             <msup>
              <mi>tan</mi>
              <mrow>
               <mo stretchy="false">−</mo>
               <mn>1</mn>
              </mrow>
             </msup>
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            <mfrac>
             <msub>
              <mi>A</mi>
              <mn>32</mn>
             </msub>
             <mrow>
              <mi>cos</mi>
              <msub>
               <mi>θ</mi>
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                <mn>3</mn>
                <mi>z</mi>
               </mrow>
              </msub>
              <msub>
               <mi>A</mi>
               <mn>33</mn>
              </msub>
             </mrow>
            </mfrac>
           </mrow>
          </mtd>
         </mtr>
        </mtable>
        <annotation encoding="StarMath 5.0">&quot;&quot;
%theta_1z
=
tan^-1
A_13 over A_23

newline
&quot;&quot;
%theta_3z
=
tan^-1
A_31 over A_32

newline
&quot;&quot;
%theta_2x
=
tan^-1
A_31 over {sin %theta_3z A_33}
=
tan^-1
A_32 over {cos %theta_3z A_33}</annotation>
       </semantics>
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newline
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newline
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newline
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   <text:h text:style-name="Heading_20_1" text:outline-level="1">Yaw Pitch Roll<draw:frame draw:style-name="fr1" draw:name="Object28" text:anchor-type="as-char" svg:width="0.9346in" svg:height="0.2508in" draw:z-index="29"><draw:object>
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%theta_2y ,
%theta_3x
right)</annotation>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Rotation Matrix,<draw:frame draw:style-name="fr1" draw:name="Object29" text:anchor-type="as-char" svg:width="0.3055in" svg:height="0.2134in" draw:z-index="30"><draw:object>
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   <text:p text:style-name="Standard">For any physical vector</text:p>
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{bold nitalic A}
right]
=
left [
matrix
{
cos %theta_1z cos %theta_2y # 
-sin %theta_1z  # 
cos %theta_1z sin %theta_2y ##
sin %theta_1z cos %theta_2y # 
cos %theta_1z # 
sin %theta_1z sin %theta_2y ##
-sin %theta_2y # 
0 # 
cos %theta_2y
}
right]
left [
matrix
{
1 # 0 # 0 ##
0 # cos %theta_3x # -sin %theta_3x ##
0 # sin %theta_3x # cos %theta_3x
}
right]
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        <annotation encoding="StarMath 5.0">left[
{bold nitalic A}
right]
=
left [
matrix
{
cos %theta_1z cos %theta_2y # 
left (
stack
{
-sin %theta_1z cos %theta_3x #
+
cos %theta_1z sin %theta_2y sin %theta_3x 
}
right ) # 
left (
stack
{
sin %theta_1z sin %theta_3x #
+
cos %theta_1z sin %theta_2y cos %theta_3x
}
right ) ##
sin %theta_1z cos %theta_2y # 
left (
stack
{
cos %theta_1z cos %theta_3x #
+
sin %theta_1z sin %theta_2y sin %theta_3x 
}
right ) # 
left (
stack
{
-cos %theta_1z sin %theta_3x #
+
sin %theta_1z sin %theta_2y cos %theta_3x
}
right ) ##
-sin %theta_2y # 
cos %theta_2y sin %theta_3x # 
cos %theta_2y cos %theta_3x
}
right]
</annotation>
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{
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-sin %theta_1z cos %theta_3x
}
right ) ##
1 # 
0 # 
0
}
right]
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        <annotation encoding="StarMath 5.0">left[
{bold nitalic A}
right]
=
left [
matrix
{
0 # 
left (
stack
{
-sin %theta_1z cos %theta_3x #

+cos %theta_1z sin %theta_3x 
}
right ) # 
left (
stack
{
sin %theta_1z sin %theta_3x #
+cos %theta_1z cos %theta_3x
}
right ) ##
0 # 
left (
stack
{
cos %theta_1z cos %theta_3x #
+sin %theta_1z sin %theta_3x 
}
right ) # 
left (
stack
{
-cos %theta_1z sin %theta_3x #
+sin %theta_1z cos %theta_3x
}
right ) ##
-1 # 
0 # 
0
}
right]
</annotation>
       </semantics>
      </math>
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            </mrow>
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           <msub>
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=
tan^-1
{-A_12} over A_22</annotation>
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   <text:p text:style-name="Standard">Otherwise:</text:p>
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          <mrow>
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                    </msub>
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                    <msub>
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             <mtd>
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        <annotation encoding="StarMath 5.0">left[
{bold nitalic A}
right]
=
left [
matrix
{
cos %theta_1z cos %theta_2y # 
left (
stack
{
-sin %theta_1z cos %theta_3x #
+
cos %theta_1z sin %theta_2y sin %theta_3x 
}
right ) # 
left (
stack
{
sin %theta_1z sin %theta_3x #
+
cos %theta_1z sin %theta_2y cos %theta_3x
}
right ) ##
sin %theta_1z cos %theta_2y # 
left (
stack
{
cos %theta_1z cos %theta_3x #
+
sin %theta_1z sin %theta_2y sin %theta_3x 
}
right ) # 
left (
stack
{
-cos %theta_1z sin %theta_3x #
+
sin %theta_1z sin %theta_2y cos %theta_3x
}
right ) ##
-sin %theta_2y # 
cos %theta_2y sin %theta_3x # 
cos %theta_2y cos %theta_3x
}
right]
</annotation>
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        <mtable>
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          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>θ</mi>
              <mrow>
               <mn>1</mn>
               <mi>z</mi>
              </mrow>
             </msub>
             <mo stretchy="false">=</mo>
             <msup>
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             </msup>
            </mrow>
            <mfrac>
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         <mtr>
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            <mrow>
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        <annotation encoding="StarMath 5.0">&quot;&quot;
%theta_1z
=
tan^-1
A_21 over A_11

newline
&quot;&quot;
%theta_2y
=
tan^-1
{-A_31} over {left(A_11 over {cos %theta_1z}right)}
=
tan^-1
{-A_13} over {left(A_21 over {sin %theta_1z}right)}

newline
&quot;&quot;
%theta_3x
=
tan^-1
A_32 over A_33
</annotation>
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