Biomedical Engineering Reference
In-Depth Information
Table 6.2 Control laws for different values of the weight matrix
V
V
Minimize
Control law,
τ
Reference
d
{
||
2
I
||
τ
WBCF (Sentis and Khatib
2005
)
x
J q
M
1
J
T
JM
2
J
T
JM
1
h
€
þ
{
T
M
1
M
τ
τ
x
J
J
T
JM
1
J
T
JM
1
h
€
q
_
þ
JJ
T
{
M
2
||
M
1
||
2
τ
x
J q þ JM
1
h
MJ
T
τ
d
ﾼ
M
€
q
1
þ
h
,
,
ð
6
:
18
Þ
N
pðÞW
J
iW
€
x
i
J
i
_
€
q
i
ﾼ €
q
iþ
1
þ
q
i
∈
ﾽ
1
;
N
q
i
are projected into the nullspace of the higher
priority tasks; this guarantees that the framework is sound. However, this approach
is not optimal, because each task is solved independently and then projected into the
nullspace of the higher-priority tasks. This does not ensure the minimization of the
error of each task. In the case of the WBCF as reported in Sentis and Khatib (
2005
),
the hierarchical extension differs considerably. The reader is referred to the above
mentioned paper for a complete presentation of the WBCF.
The accelerations of each task
€
6.4.2 Hybrid Control
Hybrid position/force control can be realized by setting the joint space control
torques to
J
c
f
τ
0
ﾼ
h
ð
6
:
19
Þ
k
n
is the contact Jacobian,
f
*
are the desired contact forces, and
where
J
c
(
q
)
∈
ℝ
k
is the number of independent directions in which the robot can apply force.
Substituting
∈
ℝ
τ
0
into the desired control torques (
6.16
), we get
þ
x
J q
MJ
{
W
€
MN
W
M
1
J
c
f
τ
d
ﾼ
h
ð
6
:
20
Þ
where the applied forces act in the nullspace of the tracking task.
6.4.3 Task Space Inverse Dynamics
The development of a new method is compelling since none of the existing methods
is jointly sound, efficient, and optimal. For example, we would like to shove off the
computation of the inertia matrices pseudoinverses which have a cost of
O
(
N
3
). On
the contrary for specific choices of
V
and
τ
0
in (
6.15
), the solution takes the form of
M
€
q
1
þ
τ
d
ﾼ
h
which we can calculate without explicitly computing M through
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