2300kineij.fodt

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          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">%theta_IeJei =tan^-1 left(s over c right) </annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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    </draw:frame></text:p>
   <text:p text:style-name="equation">Use arctan2(x,y) to get the angle<draw:frame draw:style-name="fr1" draw:name="Object96" text:anchor-type="as-char" svg:width="1.0228in" svg:height="0.2201in" draw:z-index="13"><draw:object>
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       <semantics>
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          <mo stretchy="false">−</mo>
          <mi>π</mi>
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         <mo stretchy="false">≤</mo>
         <msub>
          <mi>θ</mi>
          <mi mathvariant="italic">IeJei</mi>
         </msub>
         <mo stretchy="false">&lt;</mo>
         <mi>π</mi>
        </mrow>
        <annotation encoding="StarMath 5.0">-%pi  &lt;=  %theta_IeJei &lt; %pi </annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:p text:style-name="equation">Do not modify the signs of c and s.</text:p>
   <text:p text:style-name="equation">Bryant angles have singularities when<draw:frame draw:style-name="fr1" draw:name="Object98" text:anchor-type="as-char" svg:width="1.0161in" svg:height="0.2417in" draw:z-index="14"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi>cos</mi>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <msub>
             <mi>θ</mi>
             <mrow>
              <mi mathvariant="italic">IeJe</mi>
              <mn>2</mn>
              <mi>y</mi>
             </mrow>
            </msub>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">cos
left(
%theta_IeJe2y
right) 
= 
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object42" text:anchor-type="as-char" svg:width="0.3689in" svg:height="0.2201in" draw:z-index="1"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi>θ</mi>
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        </msub>
        <annotation encoding="StarMath 5.0">%theta_IeJei</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame>wrt<draw:frame draw:style-name="fr1" draw:name="Object51" text:anchor-type="as-char" svg:width="0.1846in" svg:height="0.2071in" draw:z-index="15"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mstyle mathvariant="bold">
         <mi mathvariant="normal">α</mi>
        </mstyle>
        <annotation encoding="StarMath 5.0">{bold nitalic %alpha}</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
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   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object52" text:anchor-type="as-char" svg:width="2.3661in" svg:height="0.7165in" draw:z-index="16"><draw:object>
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       <semantics>
        <mrow>
         <mrow>
          <mfrac>
           <mrow>
            <mo stretchy="false">∂</mo>
            <msub>
             <mi>θ</mi>
             <mi mathvariant="italic">IeJei</mi>
            </msub>
           </mrow>
           <mrow>
            <mo stretchy="false">∂</mo>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">α</mi>
            </mstyle>
           </mrow>
          </mfrac>
          <mo stretchy="false">=</mo>
          <mfrac>
           <mn>1</mn>
           <mrow>
            <mn>1</mn>
            <mo stretchy="false">+</mo>
            <msup>
             <mrow>
              <mo fence="true" form="prefix" stretchy="true">(</mo>
              <mrow>
               <mfrac>
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                <mi>c</mi>
               </mfrac>
              </mrow>
              <mo fence="true" form="postfix" stretchy="true">)</mo>
             </mrow>
             <mn>2</mn>
            </msup>
           </mrow>
          </mfrac>
         </mrow>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
            <mfrac>
             <mrow>
              <mo stretchy="false">∂</mo>
              <mi>s</mi>
             </mrow>
             <mrow>
              <mo stretchy="false">∂</mo>
              <mstyle mathvariant="bold">
               <mi mathvariant="normal">α</mi>
              </mstyle>
             </mrow>
            </mfrac>
            <mrow>
             <mfrac>
              <mn>1</mn>
              <mi>c</mi>
             </mfrac>
             <mo stretchy="false">−</mo>
             <mfrac>
              <mi>s</mi>
              <msup>
               <mi>c</mi>
               <mn>2</mn>
              </msup>
             </mfrac>
            </mrow>
            <mfrac>
             <mrow>
              <mo stretchy="false">∂</mo>
              <mi>c</mi>
             </mrow>
             <mrow>
              <mo stretchy="false">∂</mo>
              <mstyle mathvariant="bold">
               <mi mathvariant="normal">α</mi>
              </mstyle>
             </mrow>
            </mfrac>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{partial %theta_IeJei}
over
{partial {bold nitalic %alpha}}
=
{1} over {1+left(s over c right)^2}
left(
{partial s}
over
{partial {bold nitalic %alpha}}
{1}
over
{c}
-
{s}
over
{c^2}
{partial c}
over
{partial {bold nitalic %alpha}}
right)
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object47" text:anchor-type="as-char" svg:width="0.3689in" svg:height="0.2201in" draw:z-index="3"><draw:object>
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object49" text:anchor-type="as-char" svg:width="0.3689in" svg:height="0.2201in" draw:z-index="5"><draw:object>
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{partial^2 %theta_IeJei}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
=
{1} over {left( c^2+s^2 right)}
left(
{partial^2 s}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
c
+
{partial s}
over
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{partial c}
over
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-
{partial c}
over
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{partial s}
over
{partial {bold nitalic %beta}}
-
s
{partial^2 c}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
right)

newline
&quot;&quot;~~~~~~~~~
-
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left(
2c
{partial c}
over
{partial {bold nitalic %beta}}
+
2s
{partial s}
over
{partial {bold nitalic %beta}}
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{partial %theta_IeJei}
over
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                   <mi mathvariant="normal">β</mi>
                  </mstyle>
                 </mrow>
                </mfrac>
               </mrow>
              </mrow>
              <mo fence="true" form="postfix" stretchy="true">)</mo>
             </mrow>
            </mrow>
            <mfrac>
             <mrow>
              <mo stretchy="false">∂</mo>
              <msub>
               <mi>θ</mi>
               <mi mathvariant="italic">IeJei</mi>
              </msub>
             </mrow>
             <mrow>
              <mo stretchy="false">∂</mo>
              <mstyle mathvariant="bold">
               <mi mathvariant="normal">α</mi>
              </mstyle>
             </mrow>
            </mfrac>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">&quot;&quot;
{partial^2 %theta_IeJei}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
=
{1} over {left( c^2+s^2 right)}
left(
{partial^2 s}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
c
+
{partial s}
over
{partial {bold nitalic %alpha}}
{partial c}
over
{partial {bold nitalic %beta}}
-
{partial c}
over
{partial {bold nitalic %alpha}}
{partial s}
over
{partial {bold nitalic %beta}}
-
s
{partial^2 c}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
-
left(
2c
{partial c}
over
{partial {bold nitalic %beta}}
+
2s
{partial s}
over
{partial {bold nitalic %beta}}
right)

{partial %theta_IeJei}
over
{partial {bold nitalic %alpha}}
right)
</annotation>
       </semantics>
      </math>
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object68" text:anchor-type="as-char" svg:width="0.3689in" svg:height="0.2201in" draw:z-index="11"><draw:object>
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=
-A_IeJe23

newline
&quot;&quot;
c
=
A_IeJe33</annotation>
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   <text:p text:style-name="equation">Use this when<draw:frame draw:style-name="fr1" draw:name="Object97" text:anchor-type="as-char" svg:width="1.4807in" svg:height="0.4634in" draw:z-index="59"><draw:object>
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              </mrow>
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        <annotation encoding="StarMath 5.0">abs 
{
sin 
left(
{%theta}_IeJe1x
right)
}
&gt;
sin 
left(
%pi over 4
right)</annotation>
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          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">%theta_IeJe2y
=
tan^-1 
left(
{-sin({%theta}_IeJe1x)A_IeJe13} 
over 
{A_IeJe23} 
right)
</annotation>
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     </draw:image>
    </draw:frame></text:p>
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       <semantics>
        <mtable>
         <mtr>
          <mtd>
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             <mi>s</mi>
             <mo stretchy="false">=</mo>
             <mrow>
              <mo stretchy="false">−</mo>
              <mi>sin</mi>
             </mrow>
            </mrow>
            <mrow>
             <mo fence="true" form="prefix" stretchy="false">(</mo>
             <mrow>
              <msub>
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               <mrow>
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                <mn>1</mn>
                <mi>x</mi>
               </mrow>
              </msub>
             </mrow>
             <mo fence="true" form="postfix" stretchy="false">)</mo>
            </mrow>
            <msub>
             <mi>A</mi>
             <mrow>
              <mi mathvariant="italic">IeJe</mi>
              <mn>13</mn>
             </mrow>
            </msub>
           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
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             <mi>c</mi>
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             <msub>
              <mi>A</mi>
              <mrow>
               <mi mathvariant="italic">IeJe</mi>
               <mn>23</mn>
              </mrow>
             </msub>
            </mrow>
           </mrow>
          </mtd>
         </mtr>
        </mtable>
        <annotation encoding="StarMath 5.0">&quot;&quot;
s
=
-sin({%theta}_IeJe1x)A_IeJe13

newline
&quot;&quot;
c
=
A_IeJe23

</annotation>
       </semantics>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="equation">otherwise use this</text:p>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mi mathvariant="normal">θ</mi>
           <mrow>
            <mi mathvariant="italic">IeJe</mi>
            <mn>2</mn>
            <mi>y</mi>
           </mrow>
          </msub>
          <mo stretchy="false">=</mo>
          <msup>
           <mi>tan</mi>
           <mrow>
            <mo stretchy="false">−</mo>
            <mn>1</mn>
           </mrow>
          </msup>
         </mrow>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mfrac>
            <mrow>
             <mi>cos</mi>
             <mrow>
              <mo fence="true" form="prefix" stretchy="false">(</mo>
              <mrow>
               <msub>
                <mi mathvariant="normal">θ</mi>
                <mrow>
                 <mi mathvariant="italic">IeJe</mi>
                 <mn>1</mn>
                 <mi>x</mi>
                </mrow>
               </msub>
              </mrow>
              <mo fence="true" form="postfix" stretchy="false">)</mo>
             </mrow>
             <msub>
              <mi>A</mi>
              <mrow>
               <mi mathvariant="italic">IeJe</mi>
               <mn>13</mn>
              </mrow>
             </msub>
            </mrow>
            <msub>
             <mi>A</mi>
             <mrow>
              <mi mathvariant="italic">IeJe</mi>
              <mn>33</mn>
             </mrow>
            </msub>
           </mfrac>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">%theta_IeJe2y
=
tan^-1 
left(
{cos({%theta}_IeJe1x)A_IeJe13} 
over 
{A_IeJe33} 
right) </annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object94" text:anchor-type="as-char" svg:width="1.4453in" svg:height="0.4453in" draw:z-index="64"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mtable>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <mi>s</mi>
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             <mi>cos</mi>
            </mrow>
            <mrow>
             <mo fence="true" form="prefix" stretchy="false">(</mo>
             <mrow>
              <msub>
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               <mrow>
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                <mi>x</mi>
               </mrow>
              </msub>
             </mrow>
             <mo fence="true" form="postfix" stretchy="false">)</mo>
            </mrow>
            <msub>
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             <mrow>
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              <mn>13</mn>
             </mrow>
            </msub>
           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <mi>c</mi>
             <mo stretchy="false">=</mo>
             <msub>
              <mi>A</mi>
              <mrow>
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               <mn>33</mn>
              </mrow>
             </msub>
            </mrow>
           </mrow>
          </mtd>
         </mtr>
        </mtable>
        <annotation encoding="StarMath 5.0">&quot;&quot;
s
=
cos({%theta}_IeJe1x)A_IeJe13

newline
&quot;&quot;
c
=
A_IeJe33

</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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=
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=
-A_IeJe12

newline
&quot;&quot;
c
=
A_IeJe11</annotation>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Relative BryantAngle Velocity,<draw:frame draw:style-name="fr1" draw:name="Object1" text:anchor-type="as-char" svg:width="1.5965in" svg:height="0.2201in" draw:z-index="56"><draw:object>
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(t,
{bold nitalic q}_IE,{bold nitalic s}_I,
{bold nitalic q}_JE,{bold nitalic s}_J)</annotation>
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=
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left(
{dot s}
c
-
s
{dot c}
right)</annotation>
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(t,
{bold nitalic q}_IE,{bold nitalic s}_I,
{bold nitalic q}_JE,{bold nitalic s}_J)</annotation>
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=
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left(
{ddot s}
c
-
s
{ddot c}
-left(
2{c}
{dot c}
+
2{s}
{dot s}
right)
{dot %theta}_IeJei
right)
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