Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 178

g 1

Motions induced
by joint
articulation

g 2

g 3

Joint 1 Joint 2 Joint 3

End effector

Goal

desired motion

FIGURE 5.19
Simple example of a singular configuration.

(i.e., J has full column rank), then ( J T J) 1 exists and instead, the pseudoinverse , J þ , can be used as in
Equation 5.20 . This approach is viable because a matrix multiplied by its own transpose will be a square
matrix whose inverse may exist.

V ¼ J y
J T V ¼ J T J y

(5.20)

ðJ T JÞ 1 J T V ¼ðJ T JÞ 1 J T y

J þ V ¼ y

If the rows of J are linearly independent, then JJ T is invertible and the equation for the pseudoinverse
is J þ ¼J T ( JJ T ) 1 . The pseudoinverse maps the desired velocities of the end effector to the required

changes of the joint angles. After making the substitutions shown in Equation 5.21 ,LUdecomposition
can be used to solve Equation 5.22 for b . This can then be substituted into Equation 5.23 to solve for

y .

J þ V ¼ y

(5.21)

Þ 1 V ¼ y J T ðJJ T

Þ 1 b ¼ðJJ T V

(5.22)

ðJJ T Þb ¼ V

J T b ¼ y

(5.23)

Simple Euler integration can be used at this point to update the joint angles. The Jacobian has chan-
ged at the next time step, so the computation must be performed again and another step taken. This
process repeats until the end effector reaches the goal configuration within some acceptable (i.e.,
user-defined) tolerance.
It is important to remember that the Jacobian is only valid for the instantaneous configuration for which
it is formed. That is, as soon as the configuration of the linkage changes, the Jacobian ceases to accurately
describe the relationship between changes in joint angles and changes in end effector position and