Notes_04_04
1 of 8
Two-Dimensional Constraints
General
  0
   0
    q    0
q
t
 q   
   t 
q
   0
    q   q  q   2 q      0
q
q
qt
q
tt
   q q q q   2 q t q    t t 
 q  
q
  0
 q  
  3 q  q  q  q  q  3 q 3 q  q  3 q   
q
q
q
q
q
qt
q
qt
qtt
q
ttt
 0
 q 
  4 q q   3 q q  6 q  q  q    q  q  q  q 


 4 q  12 q  q  4 q  q  q   6 q  6 q  q 
 4 q    
q
q
qt
qttt
q
q
qt
tttt
q
q
q
qt
q
q
q
q
q
qtt
q
q
qtt
q
q
Notes_04_04
2 of 8
Scleronomic constraints
independent of time such as mechanical joints
   q q q q 
     q qq q 
      q qq q 
Revolute
REV  rj P  ri P  0 2 x1 
 
 
qi REV
qj REV

  I 2 

Note: Haug uses ri   r j 
P
Bi s i ' P
 I 
B s ' 

P
2
j
j
REV  0 2x1 
REV   j 2 A j s j ' P  i 2 A i s i ' P
REV   j 3 B j s j 'P 3  j j A j s j ' P  i 3 Bi s i 'P 3  i i A i s i ' P
REV  6  j 2 j B j s j ' P 4  jj  3 j 2   j 4 A j s j ' P
2
2
4
 6  i  i Bi s i ' P 4  ii  3 i   i A i s i ' P
Double revolute
 REV _ REV  d ij  d ij  L2  0
T
for
and
and
and
 
 
d   r   r 
d   r   r 
d   r   r 
d   r  r
P
ij
P
j
i
P
ij
j
ij
j
ij
j
P
i
P
P
i
P
qi REV _ REV
qj REV _ REV
P
i
 
 2d   
 2d ij   qi
T
REV
T
ij
qj REV
L  cons tan t length
P
Notes_04_04
 REV _ REV  0
  d 
T
 REV _ REV  2d ij  REV  2 d ij
T
ij
  d 
T
REV _ REV  2d ij  REV  6 d ij
T
ij
  d  6d  d 
T
 REV _ REV  2d ij  REV  8 d ij
T
T
ij
ij
ij
Parallel vectors (planar parallel-1)
a i  parallel
to a j 
 PARALLEL  a i  R  a j   0
T
for
T
a i   ri Q  ri P
 
 
qi PARALLEL
qj PARALLEL

 0
Q
j
j
P
j

 a  a 
 a i  a j 
 01x 2 
1x 2
a   r   r 
and
T

T
i
j
 PARALLEL  0
 PARALLEL  0
PARALLEL  0
 PARALLEL  0
Pin-in-slot (planar parallel-2)
a i  parallel
to d ij 
 PIN _ SLOT  a i  R  d ij   0
T
for
and
T
d   r   r 
P
ij
j
P
i
d  d  d 
ij
ij
ij
and
a i   ri Q  ri P
from above
3 of 8
Notes_04_04
 
 
 
 a  R   
 a i  R   qi
T
qi PIN _ SLOT
T
qj PIN _ SLOT
T
REV

 01x 2 
4 of 8
a i T d ij 
T
i
qj REV
 PIN _ SLOT  0
  

  
 
T
T
2
 PIN _ SLOT  a i  2  i d ij  R   i d ij  REV


 
T
3
T
2
PIN _ SLOT  a i  3  i d ij  3 i d ij   i d ij  R  3  i d ij  3  i i d ij  REV
 PIN _ SLOT  a i 
T
 
  
 
 
  
 4  i dij  6  i d ij  4 i   i 3 d ij  6  i 2  i d ij 

  R T 6  2 d  12   d  4   3 2   4 d  
i
ij
i i
ij
i i
i
i
ij
REV


Relative angle driver
 ANGLE   j   i  C  f ( t )  0
 
 
qi ANGLE
ji ANGLE
C  cons tan t
 0 0  1
 0 0 1
 ANGLE  f t
 ANGLE  f tt
ANGLE  f ttt
 ANGLE  f tttt
Gear pair driver (chain/sprockets, belt/pulleys)
 GEAR   j  Ki  C  0
K  cons tan t, C  cons tan t
external gears K  i /  j , int ernal gears K  i /  j
 
 
qi GEAR
qj GEAR
 0 0  K
 0 0 1







Notes_04_04
 GEAR  0
 GEAR  0
GEAR  0
 GEAR  0
Gear pair on rotating link k
 GEAR _ ON _ K   j  k   Ki  k   C  0
 
 
 
qi GEAR _ ON _ K
qj GEAR _ ON _ K
K  cons tan t , C  cons tan t from above
 0 0  K 
 0 0 1
qk GEAR _ ON _ K
 0 0
K  1
 GEAR _ ON _ K  0
 GEAR _ ON _ K  0
GEAR _ ON _ K  0
 GEAR _ ON _ K  0
Relative coordinate driver (translation, rotation, gears, pure rolling)
 RCD  q j  Kq i  C  f ( t )  0
 
 
 
  K 0 0
 
 
 
 1 0 0
qi RCD
qi RCD
qi RCD
qj RCD
qj RCD
qj RCD
 0  K 0
 0 0  K
 0 1 0
 0 0 1
qi  xi
q i  yi
q i  i
qj  xj
qj  yj
q j  j
K  cons tan t. C  cons tan t
5 of 8
Notes_04_04
6 of 8
 RCD  f t
 RCD  f tt
RCD  f ttt
 RCD  f tttt
Planar parallel-2 distance driver (see pin-in-slot)
 PP2 DD  a i  d ij / L  f (t )  0
L  a i   cons tan t length
T
 
 
qi PP2 DD
qj PP2 DD


 
 a   
 a i   qi
REV
T
i
a i T R T dij/ L
 01x 2 
T
qj REV
/L
 PP2 DD  f t

 


  
 
  
  
  
T
T
2
 PP2 DD  a i   2  i R  d ij   i d ij  REV / L  f tt


 
T
T
3
2
PP2 DD  a i   R  3  i d ij  3 i d ij   i d ij   3  i d ij  3  i i d ij  REV / L  f ttt
 
 PP2 DD  a i 
 

Pure rolling along planar parallel-2 distance
 ROLL  a i  d ij/ L   j  i   C  0
T
L  a i   cons tan t length ,   rolling radius , C  cons tan t
 
 
qi ROLL
qj ROLL

  R T 4  i dij  6  i d ij  4 i   i 3 d ij  6  i 2  i d ij  

/L  f
tttt
  6  2 d  12   d  4   3 2   4 d   
i
ij
i
i
ij
i
i
i
i
ij
REV


T
  
 a   
 a i   qi
T
REV
T
i
qj REV
 ROLL   PP2 DD
for

 01x 2 
 01x 2 
 ROLL  0
f tt  0
a i T R T d ij  L/ L
L/ L
Notes_04_04
ROLL  PP2 DD
for
f ttt  0
 ROLL   PP2 DD
for
f tttt  0
Planar relative distance driver (see double revolute)
 PRDD  d ij  d ij  f t   0
T
 
 
qi PRDD
qj PRDD
2
f t   0
 
  
  qi
REV _ REV
qj REV _ REV
 PRDD  2 f f t
 PRDD   REV _ REV  2 f t  2 f f tt
2
 PRDD   REV _ REV  6 f t f tt  2 f f ttt
 PRDD   REV _ REV  6 f tt  8 f t f ttt  2 f f tttt
2
7 of 8
Notes_04_04
8 of 8
Acceleration Right-hand Side for Revolute
   q q q q   2 q t q    tt 
REV  rj P  ri P  0 2 x1 
r 
q i    i 
r 
q i    i 
 i 
 i 
  I 2 
Bi s i ' P

 q    I 
Bi s i 'P
r   r   B s '
 
qi REV
qi

i
2
 q 
qi
i
qi

 0 2 x 2 
 
i
qi REV
qi

i
 I 
2
   0 
qi t
2x3
 t   0 2x1 

i
i
P
i
Bi i  A i 

Bi s i 'P

  q   0 
qi t
i
2 x1
 tt   0 2x1 
   q q q q   2 q t q    tt 
REV   j2 A j s j'P

i
r 
2
 i A i s i ' P  i    i A i s i ' P
 i 
2x 2
REV   i 2 Ai si 'P
i

 i A i s i ' P
 q  q    0 
qi
i
for body i
for body j
REV   j 2 A j s j ' P  i 2 A i s i ' P