Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 174

To explain the Jacobian from a strictly mathematical point of view, consider the six arbitrary func-
tions of Equation 5.6 , each of which is a function of six independent variables. Given specific values for
the input variables, x i , each of the output variables, y i , can be computed by its respective function.

y 1 ¼ f 1 ðx 1 ; x 2 ; x 3 ; x 4 ; x 5 ; x 6 Þ
y 2 ¼ f 2 ðx 1 ; x 2 ; x 3 ; x 4 ; x 5 ; x 6 Þ
y 3 ¼ f 3 ðx 1 ; x 2 ; x 3 ; x 4 ; x 5 ; x 6 Þ
y 4 ¼ f 4 ðx 1 ; x 2 ; x 3 ; x 4 ; x 5 ; x 6 Þ
y 5 ¼ f 5 ðx 1 ; x 2 ; x 3 ; x 4 ; x 5 ; x 6 Þ
y 6 ¼ f 6 ðx 1 ; x 2 ; x 3 ; x 4 ; x 5 ; x 6 Þ

(5.6)

These equations can also be used to describe the change in the output variables relative to the
change in the input variables. The differentials of y i can be written in terms of the differentials of x i
using the chain rule. This generates Equation 5.7 .

dy i ¼ @f i

@x 1 dx 1 þ @f i @x 2 dx 2 þ @f i @x 3 dx 3 þ @f i @x 4 dx 4 þ @f i @x 5 dx 5 þ @f i

@x 6 dx 6

(5.7)

Equations 5.6 and 5.7 can be put in vector notation, producing Equations 5.8 and 5.9 , respectively.

dY ¼ @F

@X dX

(5.8)

dY ¼ @F

@X dX

(5.9)

Amatrix of partial derivatives, @F
@X

, is called the Jacobian and is a function of the current values of x i .
The Jacobian can be thought of as mapping the velocities of X to the velocities of Y (Eq. 5.10) .

Y ¼ JðXÞ X

(5.10)

At any point in time, the Jacobian is a function of the x i . At the next instant of time, X has changed
and so has the transformation represented by the Jacobian.
When one applies the Jacobian to a linked appendage, the input variables, x i , become the joint
values and the output variables, y i , become the end effector position and orientation (in some suitable
representation such as x - y - z fixed angles).

T

Y ¼½p x p y p z

a z

(5.11)

a x a y

y , to the velocities of the end

In this case, the Jacobian relates the velocities of the joint angles,

Y (Eq. 5.12) .
V ¼ Y ¼ JðyÞ y

effector position and orientation,

(5.12)
V is the vector of linear and rotational velocities and represents the desired change in the end effec-
tor. The desired change will be based on the difference between its current position/orientation to that
specified by the goal configuration. These velocities are vectors in three-space, so each has an x , y , and z
component (Eq. 5.13) .

T

V ¼ v x v y v z o x o y o z (5.13)