Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 176

system local to that joint. In forming the Jacobian matrix, this information must be converted into some
common coordinate system such as the global inertial (world) coordinate system or the end effector
coordinate system. Various methods have been developed for computing the Jacobian based on attain-
ing maximum computational efficiency given the required information in local coordinate systems, but
all methods produce the derivative matrix in a common coordinate system.

A simple example
Consider the simple three-revolute-joint, planar manipulator of Figure 5.17 . In this example, the objec-
tive is to move the end effector, E , to the goal position, G . The orientation of the end effector is of no
concern in this example. The axis of rotation of each joint is perpendicular to the figure, coming out of
the paper. The effect of an incremental rotation, g i , of each joint can be determined by the cross-product
of the joint axis and the vector from the joint to the end effector, V i ( Figure 5.18 ) , and form the columns
of the Jacobian. Notice that the magnitude of each g i is a function of the distance between the locations
of the joint and the end effector.
The desired change to the end effector is the difference between the current position of the end
effector and the goal position (Eq. 5.16) .

2 3

ðG EÞ x
ðG EÞ y
ðG EÞ z

4 5

V ¼ (5.16)

A vector of the desired change in values is set equal to the Jacobian matrix (Eq. 5.17) multiplied by a
vector of the unknown values, which are the changes to the joint angles.

2 3

ðð 0 ; 0 ; 1 ÞEÞ x ðð 0 ; 0 ; 1 ÞðE P 1 Þ x ðð 0 ; 0 ; 1 ÞðE P 2 Þ x

4 5

J ¼ ðð 0 ; 0 ; 1 ÞEÞ y ðð 0 ; 0 ; 1 ÞðE P 1 Þ y ðð 0 ; 0 ; 1 ÞðE P 2 Þ y (5.17)

ðð 0 ; 0 ; 1 ÞEÞ z ðð 0 ; 0 ; 1 ÞðE P 1 Þ z ðð 0 ; 0 ; 1 ÞðE P 2 Þ z

G

P 1

L 1

2

L 3

E

1

3

L 2

P 2

(0, 0)

FIGURE 5.17
Planar three-joint manipulator.