Notes_11_12
1 of 10
RSUR Generalized Coordinates
A = revolute R
B = spherical S
C = universal U
D = revolute R
y
a

B
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’
z3’
B
x1’
3

D
z4’
C
y4’
y3’
x3’
x4’ C
y4’’
Notes_11_12
2 of 10
CONSTRAINTS
 r2 
p 
 2 
 r  
q   3 
21x1
p 3 
 r4 



p 4 

revA
 {r1 }A  {r2 }A 


T
y 2 '  x1 '
 {ĝ 2 } {f̂ 1 } 
T
 {ĥ 2 } {f̂ 1 } 
z 2 '  x1 '




 {r }B  {r }B 
sphB
3
 2



 {r }C  {r }C 
univC
4
 3

T
x4 '  x3'
 {f̂ 4 } {f̂ 3 } 



   D
D 
revD
21x1
 {r1 }  {r4 } 
T
 {f̂ 4 } {ĥ 1 } 
x 4 '  z1 '


T
y 4 '  z1 '
 {ĝ 4 } {ĥ 1 } 




{p 2 }T {p 2 }  1 Euler parameters
{p }T {p }  1 Euler parameters
3
 3

T
{p 4 } {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
B, r2  r3 , i  3, j  2
C, r3  r4 , i  4, j  3
f 4 f 3 , i  4, j  3
D, r1  r4 , i  4, j  1
f 4 h 1 , i  4, j  1
g 4 h 1 , i  4, j  1
p2
p3
p4
driver
CONSTANTS
0
d 
 
 
D
{r1 }  0 {r1 }  0
a 
0
 
 
A
0
0
0
0
c 
0
 
 
 
 
 
 
B
B
C
C
D
{s 2 }'  0 {s 2 }'   0  {s 3 }'  0 {s 3 }'  0 {s 4 }'  0 {s 4 }'  0
0
 b
e 
0
0
0
 
 
 
 
 
 
A
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   0
 2
Notes_11_12
 d 


ĥ 3  unit  c 
a  b


d
 
{r3 }   c
0
 
a  b


f̂ 3  unit  0 
 d 


 
 4    270 deg
A 3   f̂ 3 

e 02  tr[A]  1 / 4
d 
 
{r4 }  0 
0 
 
3 of 10
ĥ  f̂  ĥ 
3
0  u 
û 4   0   v 
1 w 
   
e 0   C 2   0.7071
e      0 
uS

p 4    1    2   

e 2   vS 2   0 
e 3  wS    0.7071 
 2
 = 2 = 2 acos(e0) for link 2

z1’
{ĥ 1 }T {ĥ 2 }  C
{ĥ 1 }T {ĝ 2 }  S
 {ĥ }T {ĝ } 
  a tan 2 1 T 2 
 {ĥ } {ĥ } 
2 
 1

z2’
OUTPUT LINK
y4’
{f̂ 1 }T {f̂ 4 }  C
x4’
{f̂ 1 }T {ĝ 4 }  C(  90  )  S

x1’
3
 e1 
a 32  a 23 
1 
 

e 2  
 a 13  a 31 
e  4e 0 a  a 
 3
 21 12 
FIXED REVOLUTE DRIVER
y2’
3
  {f̂ 1 }T {ĝ 4 } 

  a tan 2
 {f̂ }T {f̂ } 
4
 1

Notes_11_12
4 of 10
JACOBIAN
 [I3 ]
 A, r1  r2 , i  2, j  1
2[A 2 ][ ~s2 ]'A [G 2 ]
03 x 3
03 x 4
03 x 3
03 x 4


T
T
~
 2{f1}' [A1 ] [A 2 ][ g 2 ]'[G 2 ]
01x 3
01x 4
01x 3
01x 4
 01x 3
 g 2f1 , i  2, j  1
~
T
T
 01x 3
 h 2f1 , i  2, j  1
 2{f1}' [A1 ] [A 2 ][ h 2 ]'[G 2 ]
01x 3
01x 4
01x 3
01x 4




B
B
~
~
 [I ]
 B, r  r , i  3, j  2
 2[A 2 ][ s2 ]' [G 2 ]
 [I3 ]
2[A 3 ][ s3 ]' [G 3 ]
03 x 3
03 x 4
 3
 2 3


C
C
0
 C, r  r , i  4, j  3
~
~
03 x 4
[I3 ]
 2[A 3 ][ s3 ]' [G 3 ]
 [I3 ]
2[A 4 ][ s4 ]' [G 4 ]
 3x 3
 3 4
~
~
T
T
T
T
 01x 3
01x 4
01x 3  2{f 4 }' [A 4 ] [A 3 ][ f3 ]'[G 3 ] 01x 3  2{f 3}' [A 3 ] [A 4 ][ f4 ]'[G 4 ]  f 4f 3 , i  4, j  3




[ q ] 
 03 x 3
 D, r1  r4 , i  4, j  1
03 x 4
03 x 3
03 x 4
 [I3 ]
2[A 4 ][ ~s4 ]'D [G 4 ]


~
T
T
01x 4
01x 3
01x 4
01x 3  2{h1}' [A1 ] [A 4 ][ f4 ]'[G 4 ]  f 4 h1 , i  4, j  1
 01x 3
0
01x 4
01x 3
01x 4
01x 3  2{h1}'T [A1 ]T [A 4 ][ ~
g 4 ]'[G 4 ] g 4 h1 , i  4, j  1
 1x 3



T
0

p2
2{p 2 }
01x 3
01x 4
01x 3
01x 4
 1x 3

T
p3
01x 4
01x 3
2{p 3}
01x 3
01x 4
 01x 3



T
p4
01x 4
01x 3
01x 4
01x 3
2{p 4 }
 01x 3





2
driver
01x 4
01x 3
01x 4
 01x 3  2 1  e 0 2e 0 u 2e 0 v 2e 0 w 01x 3

  

 [G i ]
 2 
{p i }
 { i }' 
Notes_11_12
5 of 10
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  B, r2  r3 , i  3, j  2




 0  C, r  r , i  4, j  3
3
4


f 4 f 3 , i  4, j  3
 0 


   
 0  D, r1  r4 , i  4, j  1
 0 
f 4 h 1 , i  4, j  1


g 4 h 1 , i  4, j  1
 0 




p2
 0 
 0 
p3


 0 
p4




 f t 
driver
Notes_11_12
6 of 10
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}' ([A 1 ] [A 2 ][ 2 ]'[ 2 ]' ){g 2 }'


T
T
~ ]'[
~ ]' ){h }'


h 2 f1 , i  2, j  1
 {f1}' ([A 1 ] [A 2 ][
2
2
2




B
B
~
~
~
~

 B, r  r , i  3, j  2
[A 3 ][3 ]'[3 ]'{s 3 }' [A 2 ][ 2 ]'[ 2 ]'{s 2 }'
2
3




C
C

 C, r  r , i  4, j  3
~
~
~
~
[A 4 ][ 4 ]'[ 4 ]'{s 4 }' [A 3 ][3 ]'[3 ]'{s 3 }'
3
4


T
T
T
T
T
~
~
~
~
~
~
f 4 f 3 , i  4, j  3
 {f 3 }' ( [A 3 ] [A i ][4 ]'[ 4 ]'2[3 ]' [A 3 ] [A 4 ][ 4 ]'[3 ]'[3 ]'[A 3 ] [A 4 ] ) {f 4 }'



  

D
~
~
[A 4 ][ 4 ]'[ 4 ]'{s 4 }'

 D, r1  r4 , i  4, j  1
T
T
~
~


f 4 h 1 , i  4, j  1
 {h 1}' ( [A 1 ] [A 4 ][ 4 ]'[ 4 ]' ) {f 4 }'


T
T
~
~
g 4 h 1 , i  4, j  1
 {h 1}' ( [A 1 ] [A 4 ][ 4 ]'[4 ]' ) {g 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
  {â i }T {â j }
sphere
~ ]'[
~ ]'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
{}  {rj }P  {rj }P
~ ]'[
~ ]'{s }'P [A ][
~ ]'[
~ ]'{s }'P
  [A i ][
i
i
i
j
j
j
j
Notes_11_12
7 of 10
JERK


[A 2 ][ H 2 ]{s 2 }'A

 A, r1  r2 , i  2, j  1
T
T

{
f
}'
[A
]
[A
][
H
]{
g
}'
1
1
2
2
2


g 2f1 , i  2, j  1


 {f1}'T [A1 ]T [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  3, j  2


C
C


[A 4 ][ H 4 ]{s 4 }' -[A3 ][ H 3 ]{s 3}'

 C, r3  r4 , i  4, j  3
T
T
T
T

  {f 3}' [A 3 ] [A 4 ][ H 4 ]{f 4 }'{f 4 }' [A 4 ] [A 3 ][ H 3 ]{f 3}'

f 4 f 3 , i  4, j  3

T
T
T
T
~
~
~
~
~
~
~
~



  3{f 3}' ( [3 ]'[3 ]'[3 ]' )[A 3 ] [A 4 ][4 ]'{f 4 }'3{f 4 }' ( [4 ]'[4 ]'[4 ]' )[A 4 ] [A 3 ][3 ]'{f 3}' 
  

D

 D, r1  r4 , i  4, j  1
[A 4 ][ H 4 ]{s 4 }'


f 4 h1 , i  4, j  1
 {h1}'T [A1 ]T [A 4 ][ H 4 ]{f 4 }'


g 4 h1 , i  4, j  1


 {h1}'T [A1 ]T [A 4 ][ H 4 ]{g 4 }'




p2


T
 6p2  p 2 


p3
T


 6p3  p 3 


p4
T
 6p4  p 4 






driver


  3 / 4  f ttt for link 2
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
  [A i ][Hi ]{s i }'P [A j ][H j ]{s j }'P
Notes_11_12
8 of 10
RSUR Coordinates - Nori Photogrammetry
A = revolute R
B = spherical S
C = universal U
D = revolute R
z
a

B
O
d
b
e
A

x
D
a = OA = ground (20.43 cm)
b = AB = input crank (4.00 cm)
in x-z plane
c = CD = follower (10.00 cm)
in y-z plane
d = OD = ground (19.97 cm)
e = BC = coupler (30.42 cm)
 = crank angle in x-z plane
= follower angle in y-z plane
c
C
y
y1’
O

y2’
A
z1’
B
2
D
A
x2’
4
z2’
z3’
B
x1’
3

D
z4’
C
y4’
y3’
x3’
x4’ C
y4’’
Notes_11_12
markers on coupler body 5 for DH notation
planar cluster parallel to x-z plane
% markers on coupler body 5 measured in local coordinates
x5_markers = [ -7 -11 -19 -23 -19 -11 ;
-3
-3
-3
-3
-3
-3 ;
0
7
7
0
-7
-7 ];
3 cm
stem
B
5
4
5
8 cm
arms
6
3
1
2
C
15 cm
x5
y5
z5
markers on coupler body 3 for Haug notation
planar cluster parallel to x-z plane
% markers on coupler body 3 measured in local coordinates
s3pP = [ 0
6.93
6.93
0
-6.93 -6.93 ;
10.0 10.0
10.0
10.0 10.0
10.0 ;
15.5 19.5
27.5
31.5 27.5
19.5 ];
10 cm
stem
5
4
6
z3’
B
3
8 cm
arms
1
2
23.5 cm
3
C
y3’
x3’
9 of 10
Notes_11_12
10 of 10
RSUR Coordinates - Inertial Properties
link 2
density  = 1.18 g/cm3 plexiglass
diameter D = 5 inch = 12.7 cm
t = 0.5 inch = 1.27 cm
m2  tD2 / 4  189.84 g = 0.18984 kg

y2’
B
2
J XX _ 2  mR 2 / 2  mD 2 / 8  3.8274 kg.cm2
A
J YY _ 2  mR 2 / 4  mt 2 /12  1.9177 kg.cm2
z2’
J ZZ _ 2  mR 2 / 4  mt 2 /12  1.9177 kg.cm2
link 3
density  = 1.18 g/cm3 plexiglass
diameter D = 0.5 inch = 1.27 cm
length L = 30.42 cm
m3  LD 2 / 4  45.47 g = 0.04547 kg
x2’
z3’
B
3
y3’
J XX _ 3  mL2 /12  3.5064 kg.cm2
J YY _ 2  mL2 /12  3.5064 kg.cm2
shift centroid
initial estimate
 0 
s3'B   0 
 e / 2


 0 
s3'C   0 
 e / 2


4

D
z4’
J YY _ 2  mR 2 / 4  mt 2 /12  24.9534 kg.cm2
J ZZ _ 4  mR / 2  mD / 8  49.879 kg.cm
2
2
y3’ old
x3’ old
 d/2 


{r3}    c / 2 
a  b  / 2


link 4
density  = 1.18 g/cm3 plexiglass
diameter D = 9.5 inch = 24.13 cm
t = 0.5 inch = 1.27 cm
m4  tD2 / 4  685.32 g = 0.68532 kg
J XX _ 2  mR 2 / 4  mt 2 /12  24.9534 kg.cm2
2
C
x3’
J ZZ _ 2  mR 2 / 2  mD 2 / 8  0.009167 kg.cm2
y4’
x4’ C
y4’’