Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 181

To bias the solution toward specific joint angles, such as the middle angle between joint limits, z is
defined as in Equation 5.27 , where y i are the current joint angles, y ci are the desired joint angles, and a i
are the joint gains. This does not enforce joint limits as hard constraints, but the solution can be biased
toward the middle values so that violating the joint limits is less probable.

2

z ¼ a i ðy i y ci Þ (5.27)

The desired angles and gains are input parameters. The gain indicates the relative importance of the
associated desired angle; the higher the gain, the stiffer the joint. 4 If the gain for a particular joint is
high, then the solution will be such that the joint angle quickly approaches the desired joint angle. The
control expression is added to the conventional pseudoinverse of the Jacobian (Eq. 5.28) . If all gains are
zero, then the solution will reduce to the conventional pseudoinverse of the Jacobian. Equation 5.28 can
be solved by rearranging terms as shown in Equation 5.29 .

y ¼ J þ V þðJ þ J IÞz

(5.28)

y ¼ J þ V þðJ þ J IÞz
y ¼ J þ V þ J þ Jz Iz
y ¼ J þ ðV þ JzÞz
y ¼ J T ðJJ T Þ 1

(5.29)

ðV þ JzÞz

h i

z

y ¼ J T

ðJJ T Þ 1 ðV þ JzÞ

( JJ T ) 1 ( VþJz ) so that Equation 5.30 results. Use LU decompo-
sition to solve for b in Equation 5.31 . Substitute the solution for b in Equation 5.30 to solve for

To solve Equation 5.29 , set b¼

y .

y ¼ J T b z

(5.30)

V þ Jz ¼ðJJ T Þb (5.31)

In Figure 5.22 , the difference between using a larger gain for the second joint versus using a larger
gain for the third joint is shown. Notice how the joints with increased gain are kept straighter in the
corresponding sequence of frames.

Alternative Jacobian
Instead of trying to push the end effector toward the goal position, a formulation has been proposed that
pulls the goal to the end effector [ 1 ] . This is implemented by simply using the goal position in place of
the end effector in the pseudoinverse of the Jacobian method. Comparing Figure 5.19 to Figure 5.23 for
the simple example introduced earlier, the results of this approach are similar to the standard method of
using the end effector position in the calculations.

4

Stiffness refers to howmuch something reacts to being perturbed. A stiff spring is a strong spring. A stiff joint, as used here, is
a joint that has a higher resistance to being pulled away from its desired value.