Graphics Reference
In-Depth Information
To explain the Jacobian from a strictly mathematical point of view, consider the six arbitrary func-
tions of Equation
5.6
, each of which is a function of six independent variables. Given specific values for
the input variables,
x
i
, each of the output variables,
y
i
, can be computed by its respective function.
y
1
¼ f
1
ðx
1
; x
2
; x
3
; x
4
; x
5
; x
6
Þ
y
2
¼ f
2
ðx
1
; x
2
; x
3
; x
4
; x
5
; x
6
Þ
y
3
¼ f
3
ðx
1
; x
2
; x
3
; x
4
; x
5
; x
6
Þ
y
4
¼ f
4
ðx
1
; x
2
; x
3
; x
4
; x
5
; x
6
Þ
y
5
¼ f
5
ðx
1
; x
2
; x
3
; x
4
; x
5
; x
6
Þ
y
6
¼ f
6
ðx
1
; x
2
; x
3
; x
4
; x
5
; x
6
Þ
(5.6)
These equations can also be used to describe the change in the output variables relative to the
change in the input variables. The differentials of
y
i
can be written in terms of the differentials of
x
i
using the chain rule. This generates Equation
5.7
.
dy
i
¼
@f
i
@x
1
dx
1
þ
@f
i
@x
2
dx
2
þ
@f
i
@x
3
dx
3
þ
@f
i
@x
4
dx
4
þ
@f
i
@x
5
dx
5
þ
@f
i
@x
6
dx
6
(5.7)
dY ¼
@F
@X
dX
(5.8)
dY ¼
@F
@X
dX
(5.9)
Amatrix of partial derivatives,
@F
@X
, is called the
Jacobian
and is a function of the current values of
x
i
.
Y ¼ JðXÞ X
(5.10)
At any point in time, the Jacobian is a function of the
x
i
. At the next instant of time,
X
has changed
and so has the transformation represented by the Jacobian.
When one applies the Jacobian to a linked appendage, the input variables,
x
i
, become the joint
values and the output variables,
y
i
, become the end effector position and orientation (in some suitable
representation such as
x
-
y
-
z
fixed angles).
T
Y ¼½p
x
p
y
p
z
a
z
(5.11)
a
x
a
y
y
, to the velocities of the end
In this case, the Jacobian relates the velocities of the joint angles,
Y
(Eq. 5.12)
.
V ¼ Y ¼ JðyÞ y
effector position and orientation,
(5.12)
V
is the vector of linear and rotational velocities and represents the desired change in the end effec-
tor. The desired change will be based on the difference between its current position/orientation to that
specified by the goal configuration. These velocities are vectors in three-space, so each has an
x
,
y
, and
z
component
(Eq. 5.13)
.
T
V ¼ v
x
v
y
v
z
o
x
o
y
o
z
(5.13)
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