Graphics Reference
In-Depth Information
orientation. Thismeans that if too big a step is taken in joint angle space, the end effector may not appear to
travel in the direction of the goal. If this appears to happen during an animation sequence, then taking smal-
ler steps in joint angle space and thus recalculating the Jacobian more often may be in order.
As an example, consider a two-dimensional three-joint linkage with link lengths of 15, 10, and 5.
Using an initial pose vector of {
p
/8,
p
/4,
p
/4} and a goal position of {
20, 5}, a 21-frame sequence is
calculated
2
for linearly interpolated intermediate goal positions for the end effector.
Figure 5.20
shows
frames 0, 5, 10, 15, and 20 of the sequence. Notice the path of the end effector (the end point of the third
link) travels in approximately a straight line to the goal position.
Even the underconstrained case still has problems with singularities. A proposed solution to such
bad behavior is the damped least-squares approach [
1
][
2
]. Referring to Equation
5.24
, a user-supplied
parameter is used to add in a term that reduces the sensitivity of the pseudoinverse.
y ¼ J
T
ðJJ
T
2
IÞ
1
þ l
V
(5.24)
It is argued that this form behaves better in the neighborhood of singularities at the expense of rate
of convergence to a solution.
Figure 5.21
shows solutions using the pseudoinverse of the Jacobian
30
30
30
20
20
20
10
10
10
30
20
10
10
20
30
30
20
10
10
20
30
30
20
10
10
20
30
10
10
10
20
20
20
0
5
10
30
30
20
20
10
10
30
20
10
10
20
30
30
20
10
10
20
30
10
10
20
20
15
20
FIGURE 5.20
IK 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. (a) The pseudoinverse of Jacobian solution without damped least-squares.
(b) The pseudoinverse of Jacobian solution with damped least-squares.
2
The examples showing solutions of numeric IK were generated using the
LinearSolve
function of Mathematica.
Search WWH ::
Custom Search