Biomedical Engineering Reference
In-Depth Information
2.7
Determining the position of a multi-segmental link:
forward kinematics
The forward kinematics problem is characterized by determining the final position
and orientation of a link (e.g., anatomical landmark on the hand) with knowledge
of the joint variables. One can think of the forward kinematics (sometimes called
direct kinematics) as a black box that contains the necessary calculations for
accepting joint coordinates as input, and producing position and orientation para-
meters as output, for any of the segmental links in the chain. This black box
approach is depicted in
Figure 2.13
.
Consider the arm of a person constrained to move on the surface of a table.
Assume that this arm is represented by only two joints, characterized by two variables,
q
1
and
q
2
. The question that will be addressed throughout this chapter is as follows:
Given a displacement of
q
1
5
15
and
q
2
5
30
, what is the final position of
the hand?
In order to answer this question, it is necessary to formulate an equation that
contains the two independent variables
q
1
and
q
2
as its parameters, and that evalu-
ates to a position. Because two parameters are involved, we would also need two
independent coordinate systems, which we shall denote by frame 0 and frame 1
shown in
Figure 2.14
.
For this simple example, it can be seen from geometry that the
x
- and
y
-values
of the hand with respect to the first coordinate system
x
0
and
y
0
are
x
4 cos
q
1
1
6 cos
ð
q
1
1
q
2
Þ
(2.45)
5
and
y
4 sin
q
1
1
6 sin
ð
q
1
1
q
2
Þ
(2.46)
5
Joint
variables
Forward
kinematics
Position and
orientation
FIGURE 2.13
Understanding forward kinematics.
5.28
4
6
q
2
y
0
y
1
8.10
x
0
x
1
q
1
FIGURE 2.14
The upper limb with two embedded coordinate systems.
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