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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Parallel Constraint<draw:frame draw:style-name="fr1" draw:name="Object1" text:anchor-type="as-char" svg:width="0.8311in" svg:height="0.2201in" draw:z-index="0"><draw:object>
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t
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   <text:p text:style-name="equation">Require two vectors<draw:frame draw:style-name="fr1" draw:name="Object2" text:anchor-type="as-char" svg:width="0.2626in" svg:height="0.2311in" draw:z-index="1"><draw:object>
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    </draw:frame>to be parallel as follows:</text:p>
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     </draw:image>
    </draw:frame>and<draw:frame draw:style-name="fr1" draw:name="Object6" text:anchor-type="as-char" svg:width="0.3161in" svg:height="0.2165in" draw:z-index="4"><draw:object>
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{bold nitalic s}
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=
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   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object8" text:anchor-type="as-char" svg:width="1.8311in" svg:height="0.2437in" draw:z-index="7"><draw:object>
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            <mi mathvariant="italic">Ie</mi>
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          <mrow>
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(
t,
{bold nitalic q},
{bold nitalic s}
)
=
{bold nitalic n}^T_{2IeO}
{bold nitalic t}_{JeO}</annotation>
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   <text:p text:style-name="equation">This method is simple if<draw:frame draw:style-name="fr1" draw:name="Object9" text:anchor-type="as-char" svg:width="0.3134in" svg:height="0.2165in" draw:z-index="8"><draw:object>
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         <mrow>
          <mn>1</mn>
          <mi mathvariant="italic">Je</mi>
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        </msub>
        <annotation encoding="StarMath 5.0">{bold nitalic vec n}_1Je</annotation>
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     </draw:object><draw:image>
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       <semantics>
        <msub>
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          <mover accent="true">
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         <mrow>
          <mn>2</mn>
          <mi mathvariant="italic">Je</mi>
         </mrow>
        </msub>
        <annotation encoding="StarMath 5.0">{bold nitalic vec n}_2Je</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame>are continuous.</text:p>
   <text:h text:style-name="Heading_20_3" text:outline-level="3">Parallel Constraint alternative<draw:frame draw:style-name="fr1" draw:name="Object11" text:anchor-type="as-char" svg:width="1.0098in" svg:height="0.2201in" draw:z-index="10"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">G</mi>
          </mstyle>
          <mtext>//</mtext>
         </msub>
         <mrow>
          <mo fence="true" form="prefix" stretchy="false">(</mo>
          <mrow>
           <mrow>
            <mi>t</mi>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">q</mi>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">s</mi>
            </mstyle>
            <mi>,</mi>
            <mi>u</mi>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="false">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic G}_&quot;//&quot;
(
t,
{bold nitalic q},
{bold nitalic s},
u
)</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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            <mrow>
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          <mo stretchy="false">=</mo>
          <mn>0</mn>
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        <annotation encoding="StarMath 5.0">{bold nitalic G}_&quot;//&quot;
(
{bold nitalic q},
{bold nitalic s},
u,
t
)
=
{bold nitalic t}_JeO
-
u
{bold nitalic t}_IeO
=
0</annotation>
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   <text:h text:style-name="Heading_20_4" text:outline-level="4">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object13" text:anchor-type="as-char" svg:width="0.2791in" svg:height="0.2201in" draw:z-index="12"><draw:object>
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over
{partial{bold nitalic q}_I}
=
-
u
{partial{bold nitalic t}_IeO}
over
{partial{bold nitalic q}_I}
</annotation>
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Constant Velocity Joint Constraint<draw:frame draw:style-name="fr1" draw:name="Object119" text:anchor-type="as-char" svg:width="0.8398in" svg:height="0.2201in" draw:z-index="73"><draw:object>
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   <text:p text:style-name="equation">Used in constant velocity joint only. </text:p>
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=
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A_Ie2Je1
=
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     </draw:image>
    </draw:frame></text:p>
   <text:h text:style-name="Heading_20_3" text:outline-level="3">Rack and Pinion Joint</text:h>
   <text:p text:style-name="equation">z axis of Ie is parallel to z axis of Je.</text:p>
   <text:p text:style-name="equation">x axis of Ie start <text:s/>parallel to x axis of Je.</text:p>
   <text:p text:style-name="equation">Pinion should be on the positive xy plane of Ie.</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr2" draw:name="Object67" text:anchor-type="as-char" svg:width="0.3807in" svg:height="0.2098in" draw:z-index="77"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi>r</mi>
         <mi mathvariant="italic">pitch</mi>
        </msub>
        <annotation encoding="StarMath 5.0">r_pitch</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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      </office:binary-data>
     </draw:image>
    </draw:frame>is pitch radius.</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr2" draw:name="Object66" text:anchor-type="as-char" svg:width="2.7256in" svg:height="0.2311in" draw:z-index="78"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>G</mi>
          <mi mathvariant="italic">rack</mi>
         </msub>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mrow>
             <mi>t</mi>
             <mi>,</mi>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">q</mi>
             </mstyle>
            </mrow>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mrow>
           <msub>
            <mi>r</mi>
            <mrow>
             <mi mathvariant="italic">IeJeIe</mi>
             <mn>1</mn>
            </mrow>
           </msub>
           <mo stretchy="false">+</mo>
           <msub>
            <mi>r</mi>
            <mi mathvariant="italic">pitch</mi>
           </msub>
          </mrow>
         </mrow>
         <mrow>
          <msub>
           <mi>θ</mi>
           <mrow>
            <mi mathvariant="italic">IeJe</mi>
            <mn>3</mn>
           </mrow>
          </msub>
          <mo stretchy="false">+</mo>
          <mi>f</mi>
         </mrow>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="false">(</mo>
           <mrow>
            <mi>t</mi>
           </mrow>
           <mo fence="true" form="postfix" stretchy="false">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">G_{rack}
left(
t,
{bold nitalic q}
right)
=
r_{IeJeIe1}+ r_pitch %theta_IeJe3
+f(t)
=
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_3" text:outline-level="3">Screw Joint</text:h>
   <text:p text:style-name="equation">z axis of Ie is parallel to z axis of Je.</text:p>
   <text:p text:style-name="equation">x axis of Ie start <text:s/>parallel to x axis of Je.</text:p>
   <text:p text:style-name="equation">Screw uses the right hand rule.</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr2" draw:name="Object54" text:anchor-type="as-char" svg:width="0.4008in" svg:height="0.2098in" draw:z-index="82"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi>d</mi>
         <mi mathvariant="italic">pitch</mi>
        </msub>
        <annotation encoding="StarMath 5.0">d_pitch</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAfsDAAAn
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      </office:binary-data>
     </draw:image>
    </draw:frame>is pitch of screw.</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr2" draw:name="Object53" text:anchor-type="as-char" svg:width="3.0484in" svg:height="0.2299in" draw:z-index="83"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>G</mi>
          <mi mathvariant="italic">screw</mi>
         </msub>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mrow>
             <mi>t</mi>
             <mi>,</mi>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">q</mi>
             </mstyle>
            </mrow>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mn>2</mn>
         </mrow>
         <mi>π</mi>
         <mrow>
          <msub>
           <mi>r</mi>
           <mrow>
            <mi mathvariant="italic">IeJeIe</mi>
            <mn>3</mn>
           </mrow>
          </msub>
          <mo stretchy="false">−</mo>
          <msub>
           <mi>d</mi>
           <mi mathvariant="italic">pitch</mi>
          </msub>
         </mrow>
         <mrow>
          <msub>
           <mi>θ</mi>
           <mrow>
            <mi mathvariant="italic">IeJe</mi>
            <mn>3</mn>
           </mrow>
          </msub>
          <mo stretchy="false">+</mo>
          <mi>f</mi>
         </mrow>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="false">(</mo>
           <mrow>
            <mi>t</mi>
           </mrow>
           <mo fence="true" form="postfix" stretchy="false">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">G_{screw}
left(
t,
{bold nitalic q}
right)
=
2%pi r_{IeJeIe3}- d_pitch %theta_IeJe3
+ f(t)
=
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:p>
   <text:h text:style-name="Heading_20_3" text:outline-level="3">Gear Constraint<draw:frame draw:style-name="fr1" draw:name="Object39" text:anchor-type="as-char" svg:width="0.9437in" svg:height="0.2201in" draw:z-index="76"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>G</mi>
          <mi mathvariant="italic">gear</mi>
         </msub>
         <mrow>
          <mo fence="true" form="prefix" stretchy="false">(</mo>
          <mrow>
           <mrow>
            <mi>t</mi>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">q</mi>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">s</mi>
            </mstyle>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="false">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">G_{gear}
(
t,
{bold nitalic q},
{bold nitalic s}
)</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AAIA/wMAAAAAAAAAAAD/AwAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAA
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       AQAAAQBzoQAAAAAA//8BAHMAjAABAAAAAACLAAEAAgAAAB8AigABADoAAAACADQAAAAKAFN0
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       AAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAHoIAACNAQAAAQAAAJ/ghwAA
       AAAA//8BAJ/gjAABAAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAAAjAABAAAAAAA=
      </office:binary-data>
     </draw:image>
    </draw:frame></text:h>
   <text:p text:style-name="Standard">Endframe I cannot be in line with endframe J z-axis. </text:p>
   <text:p text:style-name="Standard">Endframe J cannot be in line with endframe I z-axis.</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object42" text:anchor-type="as-char" svg:width="0.2063in" svg:height="0.2165in" draw:z-index="79"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi>z</mi>
         <mi>I</mi>
        </msub>
        <annotation encoding="StarMath 5.0">z_I</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAQsCAAAn
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       AAAAAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCG
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       AgAXAAAAWAEAAPcBAAABAElVAAAAAAD//wEASQCMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQAC
       AAAAAACMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame><text:s/>and <draw:frame draw:style-name="fr2" draw:name="Object43" text:anchor-type="as-char" svg:width="0.2161in" svg:height="0.2165in" draw:z-index="84"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi>z</mi>
         <mi>J</mi>
        </msub>
        <annotation encoding="StarMath 5.0">z_J</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAASUCAAAn
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       AAAAAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCG
       AAEABAAAAAAAAAByAAIAFwAAAIQAAAByAQAAAQB6oQAAAAAA//8BAHoAjAABAAAAAACLAAEA
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       AP8DAAAAAAAAAAAA/wMAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIA
       AgAXAAAAWAEAAPcBAAABAEpuAAAAAAD//wEASgCMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQAC
       AAAAAACMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame><text:s/>need not be parallel. Hence any kind of gears.</text:p>
   <text:p text:style-name="Standard">Create two relative orbit angles<draw:frame draw:style-name="fr2" draw:name="Object50" text:anchor-type="as-char" svg:width="0.3709in" svg:height="0.2201in" draw:z-index="80"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi>ϕ</mi>
         <mi mathvariant="italic">IeJe</mi>
        </msub>
        <annotation encoding="StarMath 5.0">%phi_IeJe</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAa4DAAAv
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       AAByAAIAIAAAAKcBAAD3AQAABABJZUplnwEAAAAA//8EAEkAZQBKAGUAjAABAAAAAACVAAEA
       BAAAAAAAAACWAAEAAgAAAAAAjAABAAAAAAA=
      </office:binary-data>
     </draw:image>
    </draw:frame>and <draw:frame draw:style-name="fr2" draw:name="Object51" text:anchor-type="as-char" svg:width="0.3717in" svg:height="0.2201in" draw:z-index="81"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi>ϕ</mi>
         <mi mathvariant="italic">JeIe</mi>
        </msub>
        <annotation encoding="StarMath 5.0">%phi_JeIe</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAa8DAAAv
       AgAAFQAAAIsAAQACAAAA//+BAAEAEAAAAAAAAAAAAAAArgMAAC4CAACVAAEABAAAAAAAAACW
       AAEAAgAAAAkAiwABAAIAAAAfAIoAAQA6AAAAAgA0AAAACgBTdGFyU3ltYm9sAAAAAAAApwEA
       AP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAIgAAQACAAAAAQCHAAEABQAAAP////8A
       hgABAAQAAAAAAAAAcgACABoAAABqAAAAjQEAAAEAAADL4DEBAAAAAP//AQDL4IwAAQAAAAAA
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       AAAAAgD/AwAAAAAAAAAAAP8DAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAA
       AAByAAIAIAAAAKcBAAD3AQAABABKZUllnwEAAAAA//8EAEoAZQBJAGUAjAABAAAAAACVAAEA
       BAAAAAAAAACWAAEAAgAAAAAAjAABAAAAAAA=
      </office:binary-data>
     </draw:image>
    </draw:frame>for marker I and J. If both the x axes are in line initially, the gear constraint is</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object69" text:anchor-type="as-char" svg:width="1.2626in" svg:height="0.4583in" draw:z-index="85"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mi>ϕ</mi>
           <mi mathvariant="italic">JeIe</mi>
          </msub>
          <mo stretchy="false">=</mo>
          <mrow>
           <mi>π</mi>
           <mo stretchy="false">−</mo>
           <mfrac>
            <msub>
             <mi>r</mi>
             <mi>I</mi>
            </msub>
            <msub>
             <mi>r</mi>
             <mi>J</mi>
            </msub>
           </mfrac>
          </mrow>
         </mrow>
         <msub>
          <mi>ϕ</mi>
          <mi mathvariant="italic">IeJe</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0"> %phi_JeIe = %pi - {r_I} over {r_J} %phi_IeJe</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       /wQASQBlAEoAZQCMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAAAACMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object41" text:anchor-type="as-char" svg:width="1.4917in" svg:height="0.4583in" draw:z-index="86"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mi>ϕ</mi>
           <mi mathvariant="italic">JeIe</mi>
          </msub>
          <mo stretchy="false">+</mo>
          <mfrac>
           <msub>
            <mi>r</mi>
            <mi>I</mi>
           </msub>
           <msub>
            <mi>r</mi>
            <mi>J</mi>
           </msub>
          </mfrac>
         </mrow>
         <mrow>
          <mrow>
           <msub>
            <mi>ϕ</mi>
            <mi mathvariant="italic">IeJe</mi>
           </msub>
           <mo stretchy="false">−</mo>
           <mi>π</mi>
          </mrow>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">%phi_JeIe + {r_I} over {r_J} %phi_IeJe - %pi = 0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:p>
   <text:h text:style-name="Heading_20_3" text:outline-level="3">Pulley Constraint<draw:frame draw:style-name="fr1" draw:name="Object52" text:anchor-type="as-char" svg:width="1.0071in" svg:height="0.2201in" draw:z-index="87"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>G</mi>
          <mi mathvariant="italic">pulley</mi>
         </msub>
         <mrow>
          <mo fence="true" form="prefix" stretchy="false">(</mo>
          <mrow>
           <mrow>
            <mi>t</mi>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">q</mi>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">s</mi>
            </mstyle>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="false">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">G_{pulley}
(
t,
{bold nitalic q},
{bold nitalic s}
)</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:h>
   <text:p text:style-name="Standard">This is similar to gears.</text:p>
   <text:p text:style-name="equation"/>
   <text:p text:style-name="equation"/>
   <text:p text:style-name="equation"/>
   <text:p text:style-name="equation"/>
   <text:p text:style-name="equation"/>
  </office:text>
 </office:body>
</office:document>