Notes_11_03
1 of 6
Three-Dimensional Kinematics
Position
x i 
x i   a i 11 a i 12
 
  
 y i    y i   a i 21 a i 22
z 
 z  a
 i
 i   i 31 a i 32
P
ri P  ri   A i s i ' P
a i 13  x i 
 
a i 23   y i 
a i 33   z i 
'P
Velocity
ri P  ri   A i s i ' P
i x 
i   i y 
 
 iz
ri P  ri  i  si P
 0
~ i    i z
 i y

 i z
0
i x
i y 

 i x 
0 
~  s P
i  s i P  
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~ s P  r   
~ A s  ' P  r   A 
~ ' s ' P
ri P  ri   
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s i P  A i s i ' P
s i ' P  A i T s i  P
i   Ai i '
i '  A i T i 
A   ~ A   A ~ '
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~ i   A i A i T
~ i '  A i T A i   A i T ~ i A i 
Notes_11_03
Acceleration
 s ' P
ri P  ri   A
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 i   A i  i 
 i   A i T  i 
~  A s ' P 
~  A
~  A s ' P 
~ 
~ A s ' P
 s ' P  r   
ri P  ri   
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 i   ~ i   ~ i ~ i 
ri P  ri    i A i s i ' P
~ ' s ' P A 
~ ' s ' P  r   A 
~ ' 
~ ' s ' P A 
~ ' s ' P
ri P  ri   A i 
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 i '  ~ i '~ i ' ~ i '
ri P  ri   A i  i ' s i ' P
A   ~  ~ ~  A    A 
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A   ~  ~ ~  A   A  '
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~   A  A 
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~ 
~ 
 
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~ '  A  A  ~ ' ~ '
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2 of 6
Notes_11_03
3 of 6
Jerk
riP  ri  A i s i ' P
~  
  A 
  A 
 

~ 
  A T 
  A T 


~
~  
~  
~  
~   
~ 
~ 
~   A s ' P
   2
riP  ri  
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i  ~ i   2~ i  ~ i  ~ i  ~ i   ~ i ~ i ~ i 
riP  ri  i A i s i ' P
~ ' 
~ '  2
~ ' 
~ '
~ ' 
~ ' 
~ '  s ' P
~ '
riP  ri  A i 
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i '  ~ i '~ i ' ~ i '  2~ i ' ~ i '~ i ' ~ i ' ~ i   ~ i 'Hi '
riP  ri Ai i 'si ' P
~ ]'[
~ ]'[
~ ]'[
~ ]'[
~ ]'[
~ ]'[
~ ]'
[H i ] '  2[
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jerk product
A   ~   2~  ~  ~  ~  ~ ~ ~   A 
~ ' 
~ '  2
~ ' 
~ '
~ ' 
~ ' 
~ ' 
~ '
 A 
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~   A A 
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 
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 
~ 
~   
~ 
~  
~ 
~ 
~ 
2
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~ '  A  A  ~ ' ~ '2~ ' ~ '~ ' ~ ' ~ '
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NOT angular momentum
Notes_11_03
Snap
r  r A s '
P
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P
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  A  2A~    A ~   ~  ~   



~ 
~   
~ 
~ 
  A T 

  AT   2AT 
   
 
r  
P
r  r  A s '
P
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  
 
 
~  3 
~  3 
~ 
~  
~ 
~ 
~
 

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 i

~ 
~ 
~   2
~ 
~ 
~   
~  
~ 
~  A s ' P
 ri    3 
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 ~ i ~ i ~ i~

  i i i i 



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 
P
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 i  ~i   3~ i  ~ i   3~ i ~ i  ~ i  ~ i  3~ i  ~ i ~ i   2~ i  ~ i ~ i   ~ i  ~ i ~ i 
~ 
~ 
~ 
~ 
 
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++++++++++++++++++++++++++++++++++
~ ' 
~ '  2
~ ' 
~ '
~ ' 
~ ' 
~ '  s ' P
~ '
riP  ri  A i 
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i '  ~ i '~ i ' ~ i '  2~ i ' ~ i '~ i ' ~ i ' ~ i 
riP  ri  A i i ' s i ' P
A   ~   2~  ~  ~  ~  ~ ~ ~   A 
~ ' 
~ '  2
~ ' 
~ '
~ ' 
~ ' 
~ ' 
~ '
 A 
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~   A A 
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 
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 
~ 
~   
~ 
~  
~ 
~ 
~ 
2
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~ '  A  A  ~ ' ~ '2~ ' ~ '~ ' ~ ' ~ '
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4 of 6
Notes_11_03
5 of 6
2D partial derivatives
ri P  ri  Ai si 'P
by inspection
r  
P
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 I 2 
r  
P
i
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 Bi si 'P
ri P  ri   i Bi si 'P
ri P  I2 ri   Bi si 'P  i  0t
chain rule
ri P  ri P r i ri   ri P i  i  ri P t
compare to terms in chain rule
r  
P
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r  
 I 2 
P
i
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 Bi si 'P
r    0
P
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t
3D partial derivatives
ri P  ri  Ai si 'P
~ 's  ' P
ri P  ri   Ai 
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ri P  ri   A i  i '  s i ' P   ri   A i  s i ' P  i ' 
ri P  ri   A i ~si ' Pi '
i '  2G i p i 
ri P  ri   2 A i ~si ' P G i p i 
ri P  I3 ri   2Ai ~si ' PGi p i  0t
chain rule
ri P  ri P r i ri   ri P  p i p i   ri P t
compare to terms in chain rule
r  
P
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partial derivative wrt Euler parameters
ri P  ri   A i ~si ' Pi '
ri P  I3 ri   Ai ~si ' P i '0t
ri
r  
 I3 
P
i
r  
P
i
pi
pi
 2 A i ~si ' PG i 
 2 A i ~si ' PG i 
r    0
P
i
t
Notes_11_03
ri P  ri P r i ri   ri P  ' i i 'ri P t
chain rule
compare to terms in chain rule
r  
P
i
partial derivative wrt  ' directions
*   2 * G 
pi
6 of 6
' i
i
ri
 I3 
r  
P
i
*   * G 
' i
1
2
' i
T
pi
i
r  
P
i
' i
  A i ~si '
P
  A i ~si '
P
r    0
P
i
t