Graphics Reference
In-Depth Information
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FIGURE 5.21
Example comparing the pseudoinverse of the Jacobian solution to inverse kinematics (a) without damped least-
squares and (b) with damped least squares for a two-dimensional three-link armature of lengths 15, 10, and 5.
The initial pose is {p/8, p/4, p/4} and goal is {
35, 5}. Panels show frames 0, 10, 18, 19, and 20 of a 21-frame
sequence in which the end effector tracks a linearly interpolated path to goal.
without, and then with, damping for the linkage used in the previous example. However, in this case,
the goal is at {
35, 5}—out of reach of the end effector. This example demonstrates better behavior
when damped as the linkage approaches the limits of its reach.
Adding more control
The pseudoinverse computes one of many possible solutions. It minimizes joint angle rates. The config-
urations produced, however, do not necessarily correspond to what might be considered natural poses. To
better control the kinematic model such as encouraging joint angle constraints, a control expression can be
added to the pseudoinverse Jacobian solution. The control expression is used to solve for control angle rates
with certain attributes. The added control expression, because of its form, contributes nothing to the desired
y ¼ðJ
þ
J IÞz
(5.25)
add anything to the velocities. As a consequence, the control expression can be added to the pseudoin-
verse Jacobian solution without changing the given velocities to be satisfied [
4
].
V ¼ J y
V ¼ JðJ
þ
J IÞz
V ¼ðJJ
þ
J IÞz
V ¼
(5.26)
0
z
V ¼
0
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