Notes_11_16
1 of 11
RSPU Generalized Coordinates
A = revolute R
B = spherical S
C = prismatic P, universal U
y

B
a
O
c
b
A
C
d
z
a = OA = ground (20.43 cm)
b = AB = input crank (4.00 cm)
in y-z plane
c = OC = ground (19.97 cm)
d = BC = variable
lenBD = square rod (35 cm)
 = crank angle in y-z plane
x
y1’
O
z4’
y4’
C
A
z1’

y2’
B
2
A
z3’ B
4
x4’
C
x2’
z2’
x4’’
x1’
3
y3’
D
x3’
Notes_11_16
2 of 11
CONSTRAINTS


revA
A, r1  r2 , i  A 2, j  A1
{r1 }A  {r2 }A


T
y 2 '  x1 '
g 2 f 1 , i  2, j  1
{ĝ 2 } {f̂ 1 }


T


z 2 '  x1 '
h 2 f 1 , i  2, j  1
{ĥ 2 } {f̂ 1 }




B
B


sphB
B, r2  r3 , i  B3, j  B2
{r2 }  {r3 }




T


x 4 '  z3 '
f 4 h 3 , i  4, j  3
{f̂ 4 } {ĥ 3 }


T
y4 '  z3 '
g 4 h 3 , i  4, j  3


{ĝ 4 } {ĥ 3 }


B
C
T
x 4 '  d ij
f 4 d ij , i  C4, j  B3
{d ij }  {r3 }  {r4 } {f̂ 4 } {d ij } 
B
C
T
{d }  {r3 }  {r4 } {ĝ 4 } {d ij }
y 4 '  d ij
g 4 d ij , i  C4, j  B3
   ij

x 4 '  y3 '
f 4 g 3 , i  4, j  3
21x1
{f̂ 4 }T {ĝ 3 }






univC
C, r1  r4 , i  C4, j  C1
{r1 }C  {r4 }C


T


x 4 '  y1 '
f 4 g 1 , i  4, j  1
{f̂ 4 } {ĝ 1 }





 Euler parameters
T
p2
{p 2 } {p 2 }  1



 Euler parameters
p3
{p 3 }T {p 3 }  1


T
p4
{p } {p 4 }  1

 Euler parameters




driver
driver


  C  f (t)
 r2 
p 
 2 
 r  
q   3 
21x1
p 3 
 r4 


p 4 
CONSTANTS
0
c 


 
{r1}A  0 {r1}C  0
a 
0
 
 
0
0
 0 
0
0
 
 


 
 
B
B
D
C
{s 2 }'  0 {s 2 }'   0  {s 3 }'   0  {s 3 }'  0 {s 4 }'  0
0
 b
lenBD 
0
0
 
 


 
 
A
Notes_11_16
3 of 11
INITIAL ESTIMATES
0
 
{r2 }  0
a 
 
 u  1
û 2    v   0
w  0
   
 2    5 deg
a  b


f̂ 3  unit  0 
 c 


 c 


ĥ 3  unit  0  {r3 }  {r1}C   lenBD 

a  b


0
ĝ 3   unit 1
0
 

e 0   C 2  1
 e     0
uS
p 2    1    2    
e 2   vS 2  0
e 3  wS   0
2

 
a  b2  c 2 ĥ 3 

 e1 
a 32  a 23 
1 
 

e 2  
 a 13  a 31 
e  4e 0 a  a 
 3
 21 12 
e  tr[A]  1 / 4
2
0
c 
 
{r4 }  0
0
 
p 4   p 3 
FIXED REVOLUTE DRIVER
 = 2 = 2 acos(e0) for link 2
y2’
{ĥ 1 }T {ĥ 2 }  C

z1’
{ĥ 1 }T {ĝ 2 }  S

 {ĥ }T {ĝ } 
  a tan 2 1 T 2 
 {ĥ } {ĥ } 
2 
 1
z2’
OUTPUT
d  norm d ij
 
~ 
~ d / d
d  d  d  
~ 
~ d  
~ 
~ d  
~ 
~ 
~ d  3
~ 
~ d / d
d  d  d  2
T
d  d ij d ij / d
T
ij
ij
3
3
ij
T
ij
ij
3
3
ij
3
3
ij
3
3
3
ij
3
3
ij
Notes_11_16
2[A 2 ][ ~s2 ]'A [G 2 ]
 2{f1}'T [A1 ]T [A 2 ][ ~
g 2 ]'[G 2 ]
~
T
T
 2{f1}' [A1 ] [A 2 ][ h 2 ]'[G 2 ]
 [I3 ]
03 x 3

01x 3
 01x 3
 01x 3
01x 3


 [I ]
 2[A 2 ][ ~s2 ]'B [G 2 ]
 [I3 ]
3


0
01x 4
01x 3
 1x 3
 01x 3
01x 4
01x 3

01x 4
{f 4 }'T [A 4 ]T
 01x 3
0
01x 4
{g 4 }'T [A 4 ]T
[  q ]   1x 3
01x 4
01x 3
 01x 3


03 x 4
03 x 3
 03 x 3
0
01x 4
01x 3
 1x 3


2{p 2 }T
03 x 3
 01x 3
 01x 3
01x 4
01x 3

01x 4
01x 3
 01x 3


2
01x 3
 01x 3  2 1  e 0 2e 0 u 2e 0 v 2e 0 w
4 of 11
JACOBIAN
03 x 4
03 x 3
01x 4
01x 3
01x 4
01x 3
2[A 3 ][ ~s3 ]'B [G 3 ]
01x 3
~
 2{f 4 }'T [A 4 ]T [A 3 ][ h 3 ]'[G 3 ]
~
 2{g 4 }'T [A 4 ]T [A 3 ][ h 3 ]'[G 3 ]
 2{f }'T [A ]T [A ][ ~s ]'B [G ]
4
4
3
3
3
01x 3
01x 3
 {f 4 }'T [A 4 ]T
 2{g 4 }'T [A 4 ]T [A 3 ][ ~s3 ]'B [G 3 ]  {g 4 }'T [A 4 ]T
 2{f }'T [A ]T [A ][ ~
g ]'[G ]
0
4
4
3
3
3
1x 3
03 x 4
 [I3 ]
01x 4
01x 3
01x 4
01x 3
T
2{p 3}
01x 3
01x 4
01x 3
01x 4
01x 3


01x 4


01x 4



01x 4


~
T
T

 2{h 3}' [A 3 ] [A 4 ][ f4 ]'[G 4 ]


 2{h 3}'T [A 3 ]T [A 4 ][ ~
g 4 ]'[G 4 ]
~

T ~ C
T
2{f 4 }' [ s4 ]' [G 4 ]  2{d ij} [A 4 ][ f4 ]'[G 4 ] 
2{g 4 }'T [~s4 ]'C [G 4 ]  2{d ij}T [A 4 ][ ~
g 4 ]'[G 4 ]

~
T
T
 2{g 3}' [A 3 ] [A 4 ][ f4 ]'[G 4 ]



2[A 4 ][ ~s4 ]'C [G 4 ]

~
T
T

 2{g1}' [A1 ] [A 4 ][ f4 ]'[G 4 ]



01x 4


01x 4

2{p 4 }T



01x 4

03 x 4
Notes_11_16
5 of 11
A, r1  r2 , i  A 2, j  A1
g 2 f1 , i  2, j  1
h 2 f1 , i  2, j  1
B, r2  r3 , i  B3, j  B2
f 4 h 3 , i  4, j  3
g 4 h 3 , i  4, j  3
f 4 d ij , i  C4, j  B3
g 4 d ij , i  C4, j  B3
f 4 g 3 , i  4, j  3
C, r1  r4 , i  C4, j  C1
f 4 g1 , i  4, j  1
p2
p3
p4
driver
  

 [G i ]
 2 
{p i }

{

}'
i 

Notes_11_16
6 of 11
VELOCITY
 0  A, r1  r2 , i  A 2, j  A1
 0 
g 2 f 1 , i  2, j  1


 0 
h 2 f 1 , i  2, j  1




 0  B, r2  r3 , i  B3, j  B2




 0 
f 4 h 3 , i  4, j  3


g 4 h 3 , i  4, j  3
 0 
 0 
f 4 d ij , i  C4, j  B3


 0 
g 4 d ij , i  C4, j  B3
   
f 4 g 3 , i  4, j  3
 0 




 0  C, r1  r4 , i  C4, j  C1
 0 
f 4 g 1 , i  4, j  1




 0 
p2


p3
 0 


p4
 0 
 0 


driver
 f t 
Notes_11_16
7 of 11
ACCELERATION
~ ]'[
~ ]'{s }'A

 A, r1  r2 , i  A 2, j  A1
[A 2 ][
2
2
2


T
T
~
~
 {f1}' ([A 1 ] [A 2 ][2 ]'[2 ]' ){g 2 }'
g 2 f1 , i  2, j  1


T
T
~
~


 {f1}' ([A 1 ] [A 2 ][2 ]'[2 ]' ){h 2 }'
h 2f1 , i  2, j  1




B
B
~
~
~
~

 B, r  r , i  B3, j  B2
[A 3 ][3 ]'[3 ]'{s 3}' [A 2 ][2 ]'[2 ]'{s 2 }'
2
3




  {h }'T ( [A ]T [A ][
~ ]'[
~ ]'2[
~ ]'T [A ]T [A ][
~ ]'[
~ ]'[
~ ]'[A ]T [A ] ) {f }' 
f 4 h 3 , i  4, j  3
3
3
4
4
4
3
3
4
4
3
3
3
4
4 

T
T
T
T
T
~
~
~
~
~
~
g 4 h 3 , i  4, j  3
 {h 3}' ( [A 3 ] [A 4 ][4 ]'[4 ]'2[3 ]' [A 3 ] [A 4 ][4 ]'[3 ]'[3 ]'[A 3 ] [A 4 ] ) {g 4 }'


s
C
B
~ ]'[
~ ]'{s }' -[A ][
~ ]'[
~ ]'{s }'
{ }  [A 4 ][
4
4
4
3
3
3
3


T
T
~ ]'{f }'{d } [A ][
~ ]'[
~ ]'{f }'{f }'T [A ]T { s }
f 4 d ij , i  C4, j  B3
 2{d ij} [A 4 ][


4
4
ij
4
4
4
4
4
4

~ ]'{g }'{d }T [A ][
~ ]'[
~ ]'{g }'{g }'T [A ]T { s }
g 4d ij , i  C4, j  B3
 2{d ij}T [A 4 ][
  

4
4
ij
4
4
4
4
4
4
T
T
T
T
T
~ ]'[
~ ]'2[
~ ]' [A ] [A ][
~ ]'[
~ ]'[
~ ]'[A ] [A ] ) {f }' 
  {g }' ( [A ] [A ][
f 4g 3 , i  4, j  3
3
3
4
4
4
3
3
4
4
3
3
3
4
4





 C, r1  r4 , i  C4, j  C1
C
~
~
[A 4 ][4 ]'[4 ]'{s 4 }'


~ ]'[
~ ]' ) {f }'
f 4g1 , i  4, j  1


 {g1}'T ( [A1 ]T [A 4 ][
4
4
4




T
p2


 2p 2  p 2 


T
p3
 2p 3  p 3 


T


p4
 2p 4  p 4 






driver
 f tt


dot  1
{}  {a i }T {a j }
~ ]'[
~ ]'2[
~ ]' T [A ]T [A ][
~ ]'[
~ ]'[
~ ]'[A ]T [A ] ) {a }'
  {a j }'T ( [A j ]T [A i ][
i
i
j
j
i
i
j
j
j
i
i
Notes_11_16
sphere
dot  2
{}  {rj }P  {rj }P
8 of 11
~ ]'[
~ ]'{s }'P [A ][
~ ]'[
~ ]'{s }'P
{}  [A i ][
i
i
i
j
j
j
j
~ ]'{a }'{d }T [A ][
~ ]' [
~ ]'{a }'{a }'T [A ]T { s }
{γ}  2{d ij}T [A i ][
i
i
ij
i
i
i
i
i
i
Notes_11_16
9 of 11
JERK


[A 2 ][ H 2 ]{s 2 }'

 A, r1  r2 , i  A 2, j  A1
 {f1}'T [A1 ]T [A 2 ][ H 2 ]{g 2 }'


g 2f1 , i  2, j  1
T
T


 {f1}' [A1 ] [A 2 ][ H 2 ]{h 2 }'
h 2f1 , i  2, j  1






[A 3 ][ H 3 ]{s 3}'B [A 2 ][ H 2 ]{s 2 }'B

 B, r2  r3 , i  B3, j  B2


   {h }'T [A ]T [A ][ H ]{f }'{f }'T [A ]T [A ][ H ]{h }'


3
3
4
4
4
4
4
3
3
3
 f 4 h 3 , i  4, j  3 




~ ]'[
~ ]'[
~ ]' )[A ]T [A ][
~ ]'{f }'3{f }'T ( [
~ ]'[
~ ]'[
~ ]' )[A ]T [A ][
~ ]'{h }'  
   3{h 3}'T ( [
3
3
3
3
4
4
4
4
4
4
4
4
3
3
3 




T
T
T
T



{
h
}'
[A
]
[A
][
H
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Notes_11_16
jerk product
dot  1
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10 of 11
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Notes_11_16
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11 of 11