Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 173

L 2

2

L 1

180 2

( X, Y )

X 2 Y 2

Y

1

T

(0, 0) X

X

acos( T ) =

X 2 Y 2

X
X 2 Y 2

T = acos

L 2 X 2 L 2 Y 2

cos( 1 - T ) = (cosine rule)

X 2 Y 2

2 L 1

L 2 X 2 Y 2 L 2

1 = acos T

X 2 Y 2 2 L 1

L 2 L 2 ( X 2 Y 2 )

(cosine rule)

cos(180 - 2 ) = -cos( 2 ) =

2 L 1 L 2

L 2 L 2 X 2 Y 2

2 = acos

2 L 1 L 2

FIGURE 5.15
Analytic solution to a simple IK problem.

Figure 5.15 ; the inverse cosine is two-valued in both plus andminus theta ( y ). However, for more com-

plicated cases, there may be infinitely many solutions that will give the desired end effector location.
The joint values for relatively simple linkages can be solved by algebraicmanipulation of the equations
that describe the relationship of the end effector to the base frame. Most linkages used in robotic applica-
tions are designed to be simple enough for this analysis. However, for many cases that arise in computer
animation, analytic solutions are not tractable. In such cases, iterative numeric solutions must be relied on.
5.3.2 The Jacobian
Many mechanisms of interest to computer animation are too complex to allow an analytic solution. For
these, the motion can be incrementally constructed. At each time step, a computation is performed that
determines a way to change each joint angle in order to direct the current position and orientation of the
end effector toward the desired configuration. There are several methods used to compute the change in
joint angle, but most involve forming the matrix of partial derivatives called the Jacobian .