9/16/2015
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Nguyen Q. Hoang, Nguyen V. Khang
45
nonlinear equations
more unknowns than equations
Given
( )
n
t
q
( , )
0
f x q
Application of pseudo-inverse: Inverse kinematics
( ) [ ,
]
m
E
E
t
x
r O
determined
Number of solutions
Finite number of solutions
Infinite number of solutions
Methods: analytical and numerical
Problems
Nguyen Q. Hoang, Nguyen V. Khang
46
Method based on the pseudo-inverse of jacobian matrix
( )
( )
d
or
dt
f q
q
x
J q q
x
q
1
1
1
1
2
1
2
...
( )
...
...
...
...
...
n
m
m
m
n
f
f
f
q
q
q
f
f
f
q
q
q
f
J q
q
system of m linear equations, with n unknown
1
2
min
T
J
q Wq
introduction the functional
W : positive weighting matrix
Solution
1
1
1
( )[ ( )
( )]
T
T
q W J q J q W J q
x
1
( )[ ( )
( )]
T
T
q
J q J q J q
x
J x
if W is identical matrix
1
( )[ ( )
( )]
T
T
J
J q J q J q
Application of pseudo-inverse: Inverse kinematics
0
( )
(0)
t
t
dt
q
q
q
Nguyen Q. Hoang, Nguyen V. Khang
47
Application of pseudo-inverse: Inverse kinematics
Blockdiagram of Inverse kinematics
x
†
( )
W
J q
1/s
Solving for q
0
(0)
x
q(t)
(0)
q
q
Drawback of this diagram :
accumulated errors due to rounding and integral method makes
don’t satisfy the constraint equations, so the end-effector moves out the
given trajectory.
( ), ( )
t
t
x
q
( ( ), ( ))
0
t
t
f x
q
Nguyen Q. Hoang, Nguyen V. Khang
48
Application of pseudo-inverse: Inverse kinematics
Method of Error feedback
,
( )
( )
e
Ke
e
x
f q
e
x
J q q
Ke
( )
J q q
x
Ke
(
)
q
J x
Ke
t
( )
0
e
Null space of jacobian matrix
( )
J q
o
(
) (
)
q
J x
Ke
E
J J z
By putting
o
z
the redundancy of manipulator can be exploited:
- advoiding obstacles
- advoiding impact with the joint limitations
- advoiding singular configurations
