Notes_11_04
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Three-Dimensional Constraints
General
0
0
q 0
q
t
q
t
q
0
q q q 2 q 0
q
q
q
q
qt
q
tt
q q q q 2 q t q t t
0
q
3 q q q q q q q q q q 3 q t q 3 q t q q q 3 q t t q t t t
q
0
q
q
4 q qq q 3 q qq q 6 q q q q q q q q q q q q q
4 q
4 q t q 12 q t q q q 4 q t q q q q 6 q t t q 6 q t t
qttt
tttt
q
q
q q q
Notes_11_04
Scleronomic constraints
independent of time such as mechanical joints
q q q q
q qq q
q qq q
Spherical
SPH rj P ri P 03x1
I 3
I 3
r i SPH
' i SPH
p i SPH
r j SPH
' j SPH
p j SPH
A i ~si 'P
2 A ~s 'P G
i
i
i
A j ~sj 'P
2 A j ~sj 'P G j
SPH 03x1
SPH Ai ~ i ' ~ i ' si 'P A j ~ j ' ~ j ' s j 'P
using
~ ]'[
~ ]'[
~ ]'[
~ ]'[
~ ]'[
~ ]'[
~ ]'
[H i ] 2[
i
i
i
i
i
i
i
{}SPH [Ai ][Hi ]{si }'P [A j ][H j ]{s j}'P
Double spherical
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Notes_11_04
SPH _ SPH d ij d ij L2 0
L cons tan t length
T
for
d r r
P
ij
P
j
2d ij
2d ij
d r r
P
and
i
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ij
P
j
i
T
r i SPH _ SPH
' i SPH _ SPH
p i SPH _ SPH
T
2d ij A i ~si 'P
T
4d ij A i ~si 'P G i
T
r j SPH _ SPH
' j SPH _ SPH
4d A ~s ' G
T
2d ij A j ~sj 'P
T
p j SPH _ SPH
P
ij
j
j
j
SPH _ SPH 0
d 2d
SPH _ SPH 2 d ij
T
T
ij
ij
SPH
d 2d {}
SPH _ SPH 6 d ij
T
T
ij
ij
SPH
Dot-1
DOT _1 a i a j 0
T
for
a i ri Q ri P
01x 3
01x 3
r i DOT _ 1
' i DOT _ 1
p i DOT _ 1
r j DOT _ 1
' j DOT _ 1
p j DOT _ 1
a r r
Q
and
j
P
j
j
A ~a ' a A ~a '
2a ' A A ~
a ' G 2a A ~
a ' G
a j 'T A j
T
i
j
i
i
T
T
j
T
i
j
T
i
i
i
j
i
i
i
2a ' A A ~
a ' G 2a A ~a ' G
T
T
a i 'T A i A j ~
a j ' a i A j ~
aj '
DOT _ 1 0
T
T
i
i
T
j
j
j
i
j
j
j
Notes_11_04
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A ~ '~ '2~ ' A A ~ '~ '~ 'A A a '
DOT _ 1 a j 'T A j
T
T
T
i
i
i
j
T
j
i
i
j
j
j
i
DOT _ 1 {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 ( [
j
j
j
j
j
i
i
i
~ ]'[
~ ]'[
~ ]' )[A ]T [A ][
~ ]'{a }'
3{a i }'T ( [
i
i
i
i
j
j
j
Dot-2
DOT _ 2 a i d ij 0
T
for
d r r
P
ij
P
j
d r r
P
and
i
ij
a i 'T A i a i
a i ' T A i a i
T
r i DOT _ 2
' i DOT _ 2
p i DOT _ 2
P
j
a i ri Q ri P
and
i
T
T
a i 'T ~si 'P d ij A i ~
ai '
T
2 a i 'T ~si 'P d ij A i ~
ai ' G i
T
r j DOT _ 2
' j DOT _ 2
T
2a ' A A ~s ' G 2a A ~s ' G
T
T
a i ' T A i A j ~s j ' P a i A j ~s j ' P
T
T
p j DOT _ 2
i
i
T
P
j
j
j
i
P
j
j
j
DOT _ 2 0
A ~ 'a 'd A ~ '~ 'a 'a ' A
DOT _ 2 2 d ij
T
T
i
i
i
ij
T
T
i
i
i
i
i
i
~ ]'{a }'3{d }T [A ] [
~ ]'[
~ ]'[
~ ]' {a }'
DOT _ 2 3{dij}T [A i ][
i
i
ij
i
i
i
i
i
{d ij}T [A i ][ Hi ]{a i }'{a i }'T [A i ]T {}SPH
Euler parameters
EUL pi pi 1 0
T
SPH
i
Notes_11_04
r i EUL
01x 3
p i EUL
2pi
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T
EUL 0
T
EUL 2p i p i
EUL 6pi p i
T
Fixed revolute rotation driver (angle about fixed axis [u v w ]T)
u
angle about fixed axis û v
w
e 0 C 2
e
uS
p 1 2
e 2 vS 2
e 3 wS
2
e1
e e 2
e
3
FRRD f t 0
1
may use dot products or 2 cos e 0 or 2 sin 1 e12 e 22 e 32
r i FRRD
01x 3
' i FRRD
u v w
p i FRRD
2 1 e02
2e0 u 2e0 v 2e0 w 2 sin
2e0 u 2e0 v 2e0 w
2
FRRD f t
FRRD f tt
3
FRRD f ttt
4
alternative
FRRD f t 0
e2 e2 e2
1
2
3
e0
or 2 tan 1
for
2 cos 1 e 0
problem for 5 5
Notes_11_04
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01x 3
r i FRRD
' i FRRD
u v w
p i FRRD
2(1 e02 ) 1/ 2 0 0 0
FRRD f t
FRRD 2e 0 e 02 (1 e 02 ) 3 / 2 f tt
3
2
3
FRRD 2 e 0 6 e0 e 0 e0 (1 e02 )3 / 2 6 e0 e 0 (1 e02 )5 / 2 f ttt
alternative
FRRD f t 0
for
e3 e 2
r i FRRD
only for û 1
2 sin 1 e1
e 0 e 0
p i FRRD
0 2e 0
1
T
1
0 0
problem for 175 and 175
FRRD f t
FRRD 2e1 e 12 e 03 f tt
3D relative distance driver (see double spherical)
3DRDD d ij d ij f t 0
T
0
01x 3
' i FRRD
for
0
d r r
P
ij
q i 3DRDD
q j 3DRDD
f (t) 0
2
j
P
i
and
q i
SPH _ SPH
q j SPH _ SPH
3DRDD 2 f f t
3DRDD SPH _ SPH 2 f t 2 f f tt
2
d r r
P
ij
j
P
i
Notes_11_04
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3DRDD SP H _ SP H 6 f t f tt 2 f f ttt
Dot-2 distance driver (see dot-2)
D 2 DD a i d ij/ L f t 0
L ai cons tan t length
T
for
d r r
P
ij
j
q j D 2 DD
and
d r r
P
ij
j
P
i
a i ri Q ri P
and
q i D 2 DD
P
i
q i DOT _ 2
L
q j DOT _ 2
L
D 2 DD f t
DOT _ 2
D 2 DD
f tt
L
DOT _ 2
D 2 DD
f ttt
L
General revolute rotation driver
GRRD f t 0
Screw distance-angle (see dot-2 and revolute rotation driver)
SDA a i d ij/ L 0
L ai cons tan t length , screw pitch , screw angle
T
for
d r r
P
ij
r i SDA
' i SDA
p i SDA
j
P
and
i
d r r
P
ij
j
P
i
a i ri Q ri P
and
r i DOT _ 2
L
' i DOT _ 2
L
p i DOT _ 2
L
u v w
2 1 e 02
2e 0 u 2e0 v 2e0 w
Notes_11_04
Local Joint Definition Frames
local joint definition frame at P
s i ' A s i ' P C i P s i ' ' A
F ' ' C A F
P T
on i
i
T
i
T ' ' C T '
P T
on i
i
on i
on i
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Notes_11_04
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Notes_11_04
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Notes_11_04
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Notes_11_04
Three-Dimensional Joint Constraints
Revolute
Prismatic
Helical (Screw)
Cylindrical
Universal (Hooke)
Globe (Spherical)
Flat (Planar)
R
P
H(S)
C
U(H)
G(S)
F(P)
Restricts
5
5
5
4
4
3
3
Allows
1
1
1
2
2
3
3
3D mobility
M = 6(nL-1) - 5 nJ1 - 4 nJ2 - 3 nJ3
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Notes_11_04
Spherical S (Globe G)
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Double Spherical SS
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Notes_11_04
Universal U (Hooke H)
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Revolute R
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Notes_11_04
Cylindrical C
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Prismatic P (Translational T)
Notes_11_04
Screw S (Helical H)
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Notes_11_04
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