7030euler.fodt

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{bold nitalic A}
right]
=
left [
matrix
{
1 # 0 # 0 ##
0 # cos %theta_1x # -sin %theta_1x ##
0 # sin %theta_1x # cos %theta_1x

}
right]
left [
matrix
{
cos %theta_2y # 0 # sin %theta_2y ##
0 # 1 # 0 ##
-sin %theta_2y # 0 # cos %theta_2y
}
right]
left [
matrix
{
cos %theta_3z # -sin %theta_3z # 0 ##
sin %theta_3z # cos %theta_3z # 0 ##
0 # 0 # 1
}
right]
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matrix
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cos %theta_2y # 
0 # 
sin %theta_2y ##
sin %theta_1x sin %theta_2y # 
cos %theta_1x # 
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-cos %theta_1x sin %theta_2y # 
sin %theta_1x # 
cos %theta_1x cos %theta_2y
}
right]
left [
matrix
{
cos %theta_3z # -sin %theta_3z # 0 ##
sin %theta_3z # cos %theta_3z # 0 ##
0 # 0 # 1
}
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right]
=
left [
matrix
{
cos %theta_2y cos %theta_3z # 
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sin %theta_2y ##
left (
stack
{
cos %theta_1x sin %theta_3z #
+
sin %theta_1x sin %theta_2y cos %theta_3z 
}
right ) # 
left (
stack
{
cos %theta_1x cos %theta_3z #
-
sin %theta_1x sin %theta_2y sin %theta_3z 
}
right ) # 
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left (
stack
{
sin %theta_1x sin %theta_3z #
-
cos %theta_1x sin %theta_2y cos %theta_3z
}
right ) # 

left (
stack
{
sin%theta_1x cos %theta_3z #
+
cos %theta_1x sin %theta_2y sin %theta_3z 
}
right ) # 

cos %theta_1x cos %theta_2y
}
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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="Object36" text:anchor-type="as-char" svg:width="0.3047in" svg:height="0.2134in" draw:z-index="8"><draw:object>
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        <annotation encoding="StarMath 5.0">left[
{bold nitalic A}
right]
=
left [
matrix
{
0 # 
0 # 
1 ##
left (
stack
{
cos %theta_1x sin %theta_3z #
+
sin %theta_1x cos %theta_3z 
}
right ) # 
left (
stack
{
cos %theta_1x cos %theta_3z #
-
sin %theta_1x sin %theta_3z 
}
right ) # 
0 ##
left (
stack
{
sin %theta_1x sin %theta_3z #
-
cos %theta_1x cos %theta_3z
}
right ) # 

left (
stack
{
sin%theta_1x cos %theta_3z #
+
cos %theta_1x sin %theta_3z 
}
right ) # 

0
}
right]
</annotation>
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=
%pi over 2</annotation>
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         </mfrac>
        </mrow>
        <annotation encoding="StarMath 5.0">%theta_1z + %theta_3z
=
tan^-1
A_21 over A_22</annotation>
       </semantics>
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       <semantics>
        <mrow>
         <mrow>
          <msub>
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           <mn>13</mn>
          </msub>
          <mo stretchy="false">=</mo>
          <mi>sin</mi>
         </mrow>
         <mrow>
          <msub>
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          </msub>
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          <mrow>
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           <mn>1</mn>
          </mrow>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">A_13=sin %theta_2y=-1</annotation>
       </semantics>
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    </draw:frame>, then</text:p>
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       <semantics>
        <mrow>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">[</mo>
          <mrow>
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         <mrow>
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          <mrow>
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            <mtr>
             <mtd>
              <mrow>
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                  <mtd>
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             <mtd>
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                     <mrow>
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                    </msub>
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                    <msub>
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                    <mi>sin</mi>
                    <msub>
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                     <mrow>
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                     </mrow>
                    </msub>
                   </mrow>
                  </mtd>
                 </mtr>
                 <mtr>
                  <mtd>
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                     <mrow>
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                      <mi>x</mi>
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                    <msub>
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             <mtd>
              <mrow>
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               <mrow>
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                 <mtr>
                  <mtd>
                   <mrow>
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                    <msub>
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                    <msub>
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                   </mrow>
                  </mtd>
                 </mtr>
                 <mtr>
                  <mtd>
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         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">left[
{bold nitalic A}
right]
=
left [
matrix
{
0 # 
0 # 
-1 ##
left (
stack
{
cos %theta_1x sin %theta_3z #
-
sin %theta_1x cos %theta_3z 
}
right ) # 
left (
stack
{
cos %theta_1x cos %theta_3z #
+
sin %theta_1x sin %theta_3z 
}
right ) # 
0 ##
left (
stack
{
sin %theta_1x sin %theta_3z #
+
cos %theta_1x cos %theta_3z
}
right ) # 

left (
stack
{
sin%theta_1x cos %theta_3z #
-
cos %theta_1x sin %theta_3z 
}
right ) # 

0
}
right]
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=
tan^-1
A_21 over A_22</annotation>
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   <text:p text:style-name="Standard">Otherwise:</text:p>
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        <annotation encoding="StarMath 5.0">left[
{bold nitalic A}
right]
=
left [
matrix
{
cos %theta_2y cos %theta_3z # 
-cos %theta_2y sin %theta_3z # 
sin %theta_2y ##
left (
stack
{
cos %theta_1x sin %theta_3z #
+
sin %theta_1x sin %theta_2y cos %theta_3z 
}
right ) # 
left (
stack
{
cos %theta_1x cos %theta_3z #
-
sin %theta_1x sin %theta_2y sin %theta_3z 
}
right ) # 
-sin %theta_1x cos %theta_2y ##
left (
stack
{
sin %theta_1x sin %theta_3z #
-
cos %theta_1x sin %theta_2y cos %theta_3z
}
right ) # 

left (
stack
{
sin%theta_1x cos %theta_3z #
+
cos %theta_1x sin %theta_2y sin %theta_3z 
}
right ) # 

cos %theta_1x cos %theta_2y
}
right]
</annotation>
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       <semantics>
        <mtable>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>θ</mi>
              <mrow>
               <mn>1</mn>
               <mi>x</mi>
              </mrow>
             </msub>
             <mo stretchy="false">=</mo>
             <msup>
              <mi>tan</mi>
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              </mrow>
             </msup>
            </mrow>
            <mfrac>
             <mrow>
              <mo stretchy="false">−</mo>
              <msub>
               <mi>A</mi>
               <mn>23</mn>
              </msub>
             </mrow>
             <msub>
              <mi>A</mi>
              <mn>33</mn>
             </msub>
            </mfrac>
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          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>θ</mi>
              <mrow>
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              </mrow>
             </msub>
             <mo stretchy="false">=</mo>
             <msup>
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             </msup>
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            <mrow>
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                  <msub>
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        <annotation encoding="StarMath 5.0">&quot;&quot;
%theta_1x
=
tan^-1
{-A_23} over A_33

newline
&quot;&quot;
%theta_2y
=
tan^-1
{A_13} over {left(A_33 over {cos %theta_1x}right)}
=
tan^-1
{A_13} over {left(A_23 over {-sin %theta_1x}right)}

newline
&quot;&quot;
%theta_3z
=
tan^-1
{-A_12} over A_11
</annotation>
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   <text:p text:style-name="equation">Check that <draw:frame draw:style-name="fr1" draw:name="Object17" text:anchor-type="as-char" svg:width="0.7457in" svg:height="0.228in" draw:z-index="21"><draw:object>
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&gt;
0</annotation>
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   <text:p text:style-name="equation">If not</text:p>
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%theta_1x
=
tan^-1
{A_23} over {-A_33}

newline
&quot;&quot;
%theta_2y
=
tan^-1
{A_13} over {left(A_33 over {cos %theta_1x}right)}
=
tan^-1
{A_13} over {left(A_23 over {-sin %theta_1x}right)}

newline
&quot;&quot;
%theta_3z
=
tan^-1
{A_12} over {-A_11}
</annotation>
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    </draw:frame></text:p>
   <text:p text:style-name="equation">Check that <draw:frame draw:style-name="fr1" draw:name="Object20" text:anchor-type="as-char" svg:width="0.7457in" svg:height="0.228in" draw:z-index="23"><draw:object>
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       <semantics>
        <mrow>
         <mi>cos</mi>
         <mrow>
          <msub>
           <mi>θ</mi>
           <mrow>
            <mn>2</mn>
            <mi>y</mi>
           </mrow>
          </msub>
          <mo stretchy="false">&lt;</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">cos %theta_2y
&lt;
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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      </office:binary-data>
     </draw:image>
    </draw:frame>.</text:p>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">Angular Velocity,<draw:frame draw:style-name="fr1" draw:name="Object18" text:anchor-type="as-char" svg:width="0.2154in" svg:height="0.2043in" draw:z-index="24"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mstyle mathvariant="bold">
         <mi mathvariant="normal">ω</mi>
        </mstyle>
        <annotation encoding="StarMath 5.0">{bold nitalic %omega}</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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      </office:binary-data>
     </draw:image>
    </draw:frame></text:h>
   <text:p text:style-name="Standard">To be done.</text:p>
   <text:h text:style-name="Heading_20_1" text:outline-level="1">Converting Euler Parameters to zxz Euler Angles</text:h>
   <text:p text:style-name="Standard">Although direct formulas are derived below, it may be easier for software maintenance to convert via the rotation matrix <draw:frame draw:style-name="fr1" draw:name="Object23" text:anchor-type="as-char" svg:width="0.3055in" svg:height="0.2134in" draw:z-index="27"><draw:object>
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         <mo fence="true" form="postfix" stretchy="true">]</mo>
        </mrow>
        <annotation encoding="StarMath 5.0">left[
{bold nitalic A}
right]</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame>. The direct formulas are</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object21" text:anchor-type="as-char" svg:width="3.352in" svg:height="1.6465in" draw:z-index="25"><draw:object>
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       <semantics>
        <mtable>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
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              <mi>θ</mi>
              <mrow>
               <mn>1</mn>
               <mi>z</mi>
              </mrow>
             </msub>
             <mo stretchy="false">=</mo>
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             </msup>
            </mrow>
            <mrow>
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             <msup>
              <mi>tan</mi>
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             </msup>
            </mrow>
            <mfrac>
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             </msub>
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              <mi>E</mi>
              <mn>1</mn>
             </msub>
            </mfrac>
           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
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              <mi>θ</mi>
              <mrow>
               <mn>2</mn>
               <mi>x</mi>
              </mrow>
             </msub>
             <mo stretchy="false">=</mo>
             <mn>2</mn>
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             <mi>sin</mi>
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              <mn>1</mn>
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            </msup>
            <mrow>
             <mfrac>
              <msub>
               <mi>E</mi>
               <mn>2</mn>
              </msub>
              <mrow>
               <mi>sin</mi>
               <mrow>
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                <mrow>
                 <mfrac>
                  <mrow>
                   <msub>
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                    <mrow>
                     <mn>1</mn>
                     <mi>z</mi>
                    </mrow>
                   </msub>
                   <mo stretchy="false">−</mo>
                   <msub>
                    <mi>θ</mi>
                    <mrow>
                     <mn>3</mn>
                     <mi>z</mi>
                    </mrow>
                   </msub>
                  </mrow>
                  <mn>2</mn>
                 </mfrac>
                </mrow>
                <mo fence="true" form="postfix" stretchy="true">)</mo>
               </mrow>
              </mrow>
             </mfrac>
             <mo stretchy="false">+</mo>
             <mn>2</mn>
            </mrow>
            <msup>
             <mi>sin</mi>
             <mrow>
              <mo stretchy="false">−</mo>
              <mn>1</mn>
             </mrow>
            </msup>
            <mfrac>
             <msub>
              <mi>E</mi>
              <mn>1</mn>
             </msub>
             <mrow>
              <mi>cos</mi>
              <mrow>
               <mo fence="true" form="prefix" stretchy="true">(</mo>
               <mrow>
                <mfrac>
                 <mrow>
                  <msub>
                   <mi>θ</mi>
                   <mrow>
                    <mn>1</mn>
                    <mi>z</mi>
                   </mrow>
                  </msub>
                  <mo stretchy="false">−</mo>
                  <msub>
                   <mi>θ</mi>
                   <mrow>
                    <mn>3</mn>
                    <mi>z</mi>
                   </mrow>
                  </msub>
                 </mrow>
                 <mn>2</mn>
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               </mrow>
               <mo fence="true" form="postfix" stretchy="true">)</mo>
              </mrow>
             </mrow>
            </mfrac>
           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>θ</mi>
              <mrow>
               <mn>3</mn>
               <mi>z</mi>
              </mrow>
             </msub>
             <mo stretchy="false">=</mo>
             <msup>
              <mi>tan</mi>
              <mrow>
               <mo stretchy="false">−</mo>
               <mn>1</mn>
              </mrow>
             </msup>
            </mrow>
            <mrow>
             <mfrac>
              <msub>
               <mi>E</mi>
               <mn>3</mn>
              </msub>
              <msub>
               <mi>E</mi>
               <mn>4</mn>
              </msub>
             </mfrac>
             <mo stretchy="false">−</mo>
             <msup>
              <mi>tan</mi>
              <mrow>
               <mo stretchy="false">−</mo>
               <mn>1</mn>
              </mrow>
             </msup>
            </mrow>
            <mfrac>
             <msub>
              <mi>E</mi>
              <mn>2</mn>
             </msub>
             <msub>
              <mi>E</mi>
              <mn>1</mn>
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        <annotation encoding="StarMath 5.0">&quot;&quot;
%theta_1z
=
tan^-1
{E_3} over {E_4}
+
tan^-1
{E_2} over {E_1}

newline
&quot;&quot;
%theta_2x
=
2 sin^-1
{E_2} over 
{sin left({%theta_1z -%theta_3z} over {2}right)}
+
2 sin^-1
{E_1} over 
{cos left({%theta_1z -%theta_3z} over {2}right)}

newline
&quot;&quot;
%theta_3z
=
tan^-1
{E_3} over {E_4}
-
tan^-1
{E_2} over {E_1}</annotation>
       </semantics>
      </math>
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   <text:h text:style-name="Heading_20_1" text:outline-level="1">Converting zxz Euler Angles to Euler Parameters</text:h>
   <text:p text:style-name="Standard">Although direct formulas are derived below, it may be easier for software maintenance to convert via the rotation matrix <draw:frame draw:style-name="fr1" draw:name="Object24" text:anchor-type="as-char" svg:width="0.3055in" svg:height="0.2134in" draw:z-index="28"><draw:object>
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{bold nitalic A}
right]</annotation>
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    </draw:frame>. The direct formulas are</text:p>
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E_1
=
cos left({%theta_1z -%theta_3z} over {2}right)
sin left({%theta_2x} over {2}right)

newline
&quot;&quot;
E_2
=
sin left({%theta_1z -%theta_3z} over {2}right)
sin left({%theta_2x} over {2}right)

newline
&quot;&quot;
E_3
=
sin left({%theta_1z +%theta_3z} over {2}right)
cos left({%theta_2x} over {2}right)

newline
&quot;&quot;
E_4
=
cos left({%theta_1z +%theta_3z} over {2}right)
cos left({%theta_2x} over {2}right)
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