Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 175

y is a vector of joint value velocities, or the changes to the joint parameters, which are the unknowns
of the equation (Eq. 5.14) .

h i T

y ¼ y 1 y 2 y 3 y n

(5.14)

...

J , the Jacobian, is a matrix that relates the two and is a function of the current pose (Eq. 5.15) .

2
4

3
5

@p x
@ y 1

@p x
@ y 2

@p x
@ y n

...

@p y
@ y 1

@p y
@ y 2

@p y
@ y n

...

J ¼ (5.15)

... ... ... ...

@a z
@ y 1

@a z
@ y 2

@a z
@ y n

...

Each term of the Jacobian relates the change of a specific joint to a specific change in the end
effector. For a revolute joint, the rotational change in the end effector, o , is merely the velocity of
the joint angle about the axis of revolution at the joint under consideration. For a prismatic joint, the
end effector orientation is unaffected by the joint articulation. For a rotational joint, the linear change
in the end effector is the cross-product of the axis of revolution and a vector from the joint to the end
effector. The rotation at a rotational joint induces an instantaneous linear direction of travel at the end
effector. For a prismatic joint, the linear change is identical to the change at the joint (see Figure 5.16 ) .
The desired angular and linear velocities are computed by finding the difference between the cur-
rent configuration of the end effector and the desired configuration. The angular and linear velocities of
the end effector induced by the rotation at a specific joint are determined by the computations shown in
Figure 5.16 . The problem is to determine the best linear combination of velocities induced by the var-
ious joints that would result in the desired velocities of the end effector. The Jacobian is formed by
posing the problem in matrix form.
In assembling the Jacobian, it is important to make sure that all of the coordinate values are in the
same coordinate system. It is often the case that joint-specific information is given in the coordinate

Z i ( E J i )

i

i

Z i Z i

Z i

E

E

E J i

J i

J i

a) Angular velocity, i b) Linear velocity, Z i ( E J i )

E — end effector
J i — i th joint
Z i — i th joint axis
i — angular velocity of i th joint

FIGURE 5.16
Angular and linear velocities induced by joint axis rotation.