Notes_11_14
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RSSR Generalized Coordinates
A = revolute R
B = spherical S
C = spherical S
D = revolute R
y

B
a
O
d
b
e
A

z
D
a = OA = ground (20.43 cm)
b = AB = input crank (4.00 cm)
in y-z plane
c = CD = follower (10.00 cm)
in x-y plane
d = OD = ground (19.97 cm)
e = BC = coupler (30.42 cm)
 = crank angle in y-z plane
 = follower angle in x-y plane
c
C
x
y1’
O

y2’
A
z1’
B
2
D
A
x2’
4
z2’
B
x1’

use
doublespherical
D
z4’
y4’
C
x4’ C
y4’’
Notes_11_14
2 of 8
CONSTRAINTS
 r2 
p 
q   2 
14 x1
 r4 
p 4 
revA


{r1 }A  {r2 }A


T
y 2 '  x1 '
{ĝ 2 } {f̂ 1 }




z 2 '  x1 '
{ĥ 2 }T {f̂ 1 }




{d }  {r }C  {r }B {d }T {d }  L2  double  sph BC
4
2
ij
ij
 ij



D
D


revD
{r1 }  {r4 }

  

T
x 4 '  z1 '
14 x1
{f̂ 4 } {ĥ 1 }




y 4 '  z1 '
{ĝ 4 }T {ĥ 1 }




T

 Euler parameters
{p 2 } {p 2 }  1


{p 4 }T {p 4 }  1

 Euler parameters






driver
  C  f (t)
A, r1  r2 , i  2, j  1
g 2 f 1 , i  2, j  1
h 2 f 1 , i  2, j  1
i  B2, j  C4
D, r1  r4 , i  4, j  1
f 4 h 1 , i  4, j  1
g 4 h 1 , i  4, j  1
p2
p4
driver
CONSTANTS
0
d 


 
{r1 }A  0 {r1 }D  0
a 
0
 
 
0
0
c 
0






 
{s 2 }'A  0 {s 2 }'B   0  BC  e {s 4 }'C  0 {s 4 }'D  0
0
 b
0
0
 
 
 
 
INITIAL ESTIMATES
0
 
{r2 }  0
a 
 
 2    0 deg
 u  1
û 2    v   0
w  0
   
e 0   C 2  1
 e     0
uS
p 2    1    2    
e 2   vS 2  0


 
e 3 
 
wS 2  0
Notes_11_14
d 
 
{r4 }  0 
0 
 
 4    270 deg
0  u 
û 4   0   v 
1 w 
   
3 of 8
e 0   C 2   0.7071
e      0 
uS

p 4    1    2   

e 2   vS 2   0 




e 3 
 
wS 2   0.7071 
FIXED REVOLUTE DRIVER
 = 2 = 2 acos(e0) for link 2
y2’

z1’

{ĥ 1 }T {ĥ 2 }  C
{ĥ 1 }T {ĝ 2 }  S
 {ĥ 1 }T {ĝ 2 } 

  a tan 2
 {ĥ }T {ĥ } 
2 
 1
z2’
OUTPUT LINK
y4’
{f̂ 1 }T {f̂ 4 }  C
x4’
{f̂ 1 }T {ĝ 4 }  C(  90  )  S

x1’
  {f̂ 1 }T {ĝ 4 } 

  a tan 2
 {f̂ }T {f̂ } 
4
 1

Notes_11_14
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JACOBIAN
  [I3 ]

 01x 3
 01x 3


 2d T
ij


 0
 3x 3
 01x 3

 01x 3


 01x 3
 0
 1x 3

 0
 1x 3
2[A 2 ][ ~s2 ]'A [G 2 ]
 2{f1}'T [A1 ]T [A 2 ][ ~
g 2 ]'[G 2 ]
~
 2{f1}'T [A1 ]T [A 2 ][ h 2 ]'[G 2 ]
T
4d ij A 2 ~s2 'B G 2 
01x 3
01x 3
T
03 x 4
 [I3 ]
01x 4
01x 3
01x 4
01x 3
2p 2 
01x 3
01x 4
01x 3
2e 0 u 2e 0 v 2e 0 w
01x 3
 A, r1  r2 , i  2, j  1

g 2f1 , i  2, j  1
01x 4


h 2f1 , i  2, j  1
01x 4


T
C
~
i  B2, j  C4
 4d ij A 4 s4 ' G 4  


D
 D, r  r , i  4, j  1
~
2A 4 s4 ' G 4 
1
4

~
T
T

f
h
,
i
 4, j  1


 2{h1}' [A1 ] [A 4 ][ f4 ]' G 4
4 1

g 4 h1 , i  4, j  1
 2{h1}'T [A1 ]T [A 4 ][ ~
g 4 ]' G 4 


p2
01x 4

T

p4
2p 4 



driver
01x 4

03 x 4
2d ij
T
 2 1  e 02
03 x 3
  

 [G i ]
 2 
{p i }
 { i }' 
Notes_11_14
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VELOCITY
 0  A, r1  r2 , i  2, j  1
 0 
g 2 f 1 , i  2, j  1


 0 
h 2 f 1 , i  2, j  1




 0 
i  B2, j  C4




 0  D, r  r , i  4, j  1
1
4
   
f 4 h 1 , i  4, j  1
 0 
 0 
g 4 h 1 , i  4, j  1




 0 
p2


p4
 0 




 f t 
driver
Notes_11_14
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ACCELERATION
~ ]'[
~ ]'{s }'A

 A, r1  r2 , i  2, j  1
[A 2 ][
2
2
2


T
T
~
~
g 2 f1 , i  2, j  1
  {f1}' ([A1 ] [A 2 ][2 ]'[2 ]' ){g 2 }' 
~ ]'[
~ ]' ){h }' 
  {f1}'T ([A1 ]T [A 2 ][
h 2f1 , i  2, j  1
2
2
2




T 
T
s



i  B2, j  C4
 2{d ij} {d ij}  2{d ij} { }




D

 D, r  r , i  4, j  1
~
~
[A 4 ][4 ]'[4 ]'{s 4 }'

1
4
  

T
T
~
~
f 4 h1 , i  4, j  1
  {h1}' ( [A1 ] [A 4 ][4 ]'[4 ]' ) {f 4 }' 
~ ]'[
~ ]' ) {g }'
 {h1}'T ( [A1 ]T [A 4 ][
g 4 h1 , i  4, j  1
4
4
4




T


p2
 2p 2  p 2 


T
p4
 2p 4  p 4 








driver
 f tt
dot  1
{}  {â i }T {â 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
sphere
{}  {rj }P  {rj }P
double  spherical
{d ij}  {rj }P  {ri }P
~ ]'[
~ ]'{s }'P [A ][
~ ]'[
~ ]'{s }'P
  [A i ][
i
i
i
j
j
j
j
{}  {d ij}T {d ij}  L2
  2{d ij}T {d ij}  2{d ij}T { s }
Notes_11_14
7 of 8
JERK

 A, r1  r2 , i  2, j  1
[A 2 ][ H 2 ]{s 2 }'A


T
T
g 2f1 , i  2, j  1
  {f1}' [A1 ] [A 2 ][ H 2 ]{g 2 }' 
  {f1}'T [A1 ]T [A 2 ][ H 2 ]{h 2 }' 
h 2f1 , i  2, j  1




  6{d }T {d }  2{d }T {s } 
i  B2, j  C4
ij
ij
ij




D

 D, r  r , i  4, j  1
[A 4 ][ H 4 ]{s 4 }'

1
4
  

T
T
f 4 h1 , i  4, j  1
  {h1}' [A1 ] [A 4 ][ H 4 ]{f 4 }' 
 {h1}'T [A1 ]T [A 4 ][ H 4 ]{g 4 }'
g 4 h1 , i  4, j  1




T


p2
 6p2  p 2 


T
p4
 6p4  p 4 






   3 / 4  f ttt for link 2 
driver
jerk product
dot  1
~ ]' [
~ ]'[
~ ]' [
~ ]'[
~ ]' [
~ ]' [
~ ]'
[H i ]  2[
i
i
i
i
i
i
i
   {a j }'T [A j ]T [A i ][ H i ]{a i }'{a i }'T [A i ]T [A j ][ H j ]{a j }'



~ ]' [
~ ]'[
~ ]' )[A ]T [A ][
~ ]'{a }'3{a }'T ( [
~ ]' [
~ ]'[
~ ]' )[A ]T [A ][
~ ]'{a }' 
  3{a }'T ( [
j
j
j
j
j
i
i
i
i
i
i
i
i
j
j
j 

sphere
double  spherical
  [A i ][H i ]{s i }'P [A j ][H j ]{s j }'P
{d ij}  {rj }P  {ri }P
{}  {d ij}T {d ij}  L2
  6{d ij}T {d ij}  2{d ij}T {s }
Notes_11_14
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