Graphics Reference
In-Depth Information
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FIGURE 5.24
Transpose of the Jacobian solution 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 {
20, 5}. Panels show frames 0, 5, 10, 15, and 20 of a 21-frame sequence in
which the end effector tracks a linearly interpolated path to goal. The scale term used in this example is 0.1.
Using the transpose of the Jacobian avoids the expense of computing the inverse, or pseudoinverse,
of the Jacobian. The main drawback is that even though a given instantaneous change vector might
contribute a small component to the velocity vector, the perpendicular component may take the end
effector well away from the desired direction. Also, there is an additional parameter that must be sup-
plied by the user.
Procedurally determining the angles: cyclic coordinate descent
Instead of relying on numerical machinery to produce desired joint velocities, a more flexible, proce-
dural approach can be taken. Cyclic coordinate descent considers each joint one at a time, sequentially
from the outermost inward [
5
]
. At each joint, an angle is chosen that best gets the end effector to the
goal position.
Figure 5.25
shows frames of a sequence using cyclic coordinate descent. While the interior joint
angles are different from the previous examples, they still look reasonable and the end effector tracks
the solution path well.
In a variation of this approach, each instantaneous change vector is compared to the desired
change vector and the joints are rank ordered in terms of their usefulness. The highest ranked joint
is then changed and the process repeats. The rank ordering can be based on how recently that joint
was last modified, joint limits, how close the joint gets the end effector to the goal, or any number of
other measures.
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FIGURE 5.25
Cyclic coordinate descent solution 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 {
20, 5}. Panels show frames 0, 5, 10, 15, and 20 of a 21-frame sequence in
which the end effector tracks a linearly interpolated path to goal.
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