1110kine.fodt

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        <annotation encoding="StarMath 5.0">left [{bold nitalic A}right]_ap
=
left [matrix{E^2_1 - E^2_2 - E^2_3 + E^2_4 # 2 left(E_1E_2 - E_3E_4 right) # 2 left(E_1E_3 + E_2E_4 right) ## 2 left(E_1E_2 + E_3E_4 right) # -E^2_1 + E^2_2 - E^2_3 + E^2_4 # 2 left(E_2E_3 - E_1E_4 right) ## 2 left(E_1E_3 - E_2E_4 right) # 2 left(E_2E_3 + E_1E_4 right) # -E^2_1- E^2_2 + E^2_3 + E^2_4}right]_ap</annotation>
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        <annotation encoding="StarMath 5.0">{partial left[{bold nitalic A} right]_ap} over {partial {bold nitalic q}_E} =
left[
matrix{
left[matrix{2E_1 ## -2E_2 ##- 2E_3 ## 2E_4}right] # left[matrix{2E_2 ## 2E_1 ## -2E_4 ## -2E_3}right] # left[matrix{2E_3 ## 2E_4 ## 2E_1 ## 2E_2}right] ##
left[matrix{2E_2 ## 2E_1 ## 2E_4 ## 2E_3}right] # left[matrix{-2E_1 ## 2E_2 ## -2E_3 ## 2E_4}right] # left[matrix{-2E_4 ## 2E_3 ## 2E_2 ## -2E_1}right] ##
left[matrix{2E_3 ## -2E_4 ## 2E_1 ## -2E_2}right] # left[matrix{2E_4 ## 2E_3 ## 2E_2 ## 2E_1}right] # left[matrix{-2E_1 ## -2E_2 ## 2E_3 ## 2E_4}right] 
}
right]</annotation>
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   <text:p text:style-name="equation">Using the convention that physical indices come first followed by variable indices, the following form is chosen. The physical indices are placed first because they are used most often in multiplications.</text:p>
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left[matrix{0 # 2 # 0 # 0 ## 2 # 0 # 0 # 0 ## 0 # 0 # 0 # -2 ## 0 # 0 # -2 # 0}right] # 
left[matrix{0 # 0 # 2 # 0 ## 0 # 0 # 0 # 2 ## 2 # 0 # 0 # 0 ## 0 # 2 # 0 # 0}right] ##
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left[matrix{-2 # 0 # 0 # 0 ## 0 # 2 # 0 # 0 ## 0 # 0 #- 2 # 0 ## 0 # 0 # 0 # 2}right] # 
left[matrix{0 # 0 # 0 # -2 ## 0 # 0 # 2 # 0 ## 0 # 2 # 0 # 0 ## -2 # 0 # 0 # 0}right] ##
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left[matrix{0 # 0 # 0 # 2 ## 0 # 0 # 2 # 0 ## 0 # 2 # 0 # 0 ## 2 # 0 # 0 # 0}right] # 
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matrix{
left[matrix{2 # 0 # 0 ## 0 # -2 # 0 ## 0 # 0 #- 2}right] # 
left[matrix{0 # 2 # 0 ## 2 # 0 # 0 ## 0 # 0 # 0}right] # 
left[matrix{0 # 0 # 2 ## 0 # 0 # 0 ## 2 # 0 # 0}right] # 
left[matrix{0 # 0 # 0 ## 0 # 0 # -2 ## 0 # 2 # 0}right] ##
left[matrix{0 # 2 # 0 ## 2 # 0 # 0 ## 0 # 0 # 0}right] # 
left[matrix{-2 # 0 # 0 ## 0 # 2 # 0 ## 0 # 0 # -2}right] # 
left[matrix{0 # 0 # 0 ## 0 # 0 # 2 ## 0 # 2 # 0}right] # 
left[matrix{0 # 0 # 2 ## 0 # 0 # 0 ## -2 # 0 # 0}right] ##
left[matrix{0 # 0 # 2 ## 0 # 0 # 0 ## 2 # 0 # 0}right] # 
left[matrix{0 # 0 # 0 ## 0 # 0 # 2 ## 0 # 2 # 0}right] # 
left[matrix{-2 # 0 # 0 ## 0 # -2 # 0 ## 0 # 0 # 2}right] # 
left[matrix{0 # -2 # 0 ## 2 # 0 # 0 ## 0 # 0 # 0}right] ##
left[matrix{0 # 0 # 0 ## 0 # 0 # -2 ## 0 # 2 # 0}right] # 
left[matrix{0 # 0 # 2 ## 0 # 0 # 0 ## -2 # 0 # 0}right] # 
left[matrix{0 # -2 # 0 ## 2 # 0 # 0 ## 0 # 0 # 0}right] # 
left[matrix{2 # 0 # 0 ## 0 # 2 # 0 ## 0 # 0 # 2}right] 
}
right]</annotation>
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=
{d left[{bold nitalic A} right]_ap} over {d t} =
left[{bold nitalic dot B} right]_ap
left[{bold nitalic C} right]^T_ap
+
left[{bold nitalic B} right]_ap
left[{bold nitalic dot C} right]^T_ap
=
2 left[{bold nitalic B} right]_ap
left[{bold nitalic dot C} right]^T_ap
=
2 left[{bold nitalic B} right]_ap
{partial left[{bold nitalic dot C} right]^T_ap} over {partial {bold nitalic dot q}_E}{bold nitalic dot q}_E
=
2 left[{bold nitalic B} right]_ap
{partial left[{bold nitalic C} right]^T_ap} over {partial {bold nitalic q}_E}{bold nitalic dot q}_E
newline
~~~
=
2 left[{bold nitalic B} right]_ap
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1 over 2 left[{bold nitalic B}right]^T_ap
{bold nitalic v}_%omega
=
left[{bold nitalic B} right]_ap
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left[{bold nitalic B}right]^T_ap
{bold nitalic v}_%omega</annotation>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Orientation Velocity from Assembly Frame,<draw:frame draw:style-name="fr1" draw:name="Object15" text:anchor-type="as-char" svg:width="0.9874in" svg:height="0.2543in" draw:z-index="61"><draw:object>
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   <text:p text:style-name="equation">From Haug pp 330. Define skew-symmetrix matrix</text:p>
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