Kinematic Linkages - Computer Animation: Algorithms and Techniques - page 172

orientations) can first be calculated and each one of these then used as input to an IK problem. In this
way, the path the end effector takes is prescribed by the animator.
If the mechanism is simple enough, then the joint values (the pose vector) required to produce the
final desired configuration can be calculated analytically. Given an initial pose vector and the final pose
vector, intermediate configurations can be formed by interpolation of the values in the pose vectors,
thus animating the mechanism from its initial configuration to the final one. However, if the mecha-
nism is too complicated for analytic solutions, then an incremental approach can be used that employs a
matrix of values (the Jacobian ) that relates changes in the values of the joint parameters to changes in
the end effector position and orientation. The end effector is iteratively nudged until the final config-
uration is attained within a given tolerance. In addition to the Jacobian, there are other incremental
formulations that can be used to effect inverse kinematic solutions.

5.3.1 Solving a simple system by analysis
For sufficiently simple mechanisms, the joint values of a final desired position can be determined ana-
lytically by inspecting the geometry of the linkage. Consider a simple two-link arm in two-dimensional
space with two rotational DOFs. Link lengths are L 1 and L 2 for the first and second link, respectively. If
a position is fixed for the base of the arm at the first joint, any position beyond | L 1 L 2 | units from the
base of the link and within L 1 þL 2 of the base can be reached (see Figure 5.14 ) .
Assume for now (without loss of generality) that the base is at the origin. In a simple IK problem, the
user gives the ( X , Y ) coordinate of the desired position for the end effector. The joint angles, y 1 and y 2 ,
can be determined by computing the distance from the base to the goal and using the law of cosines to
compute the interior angles. Once the interior angles are computed, the rotation angles for the two links
can be computed (see Figure 5.15 ) . Of course, the first step is to make sure that the position of the goal
is within the reach of the end effector; that is, | L 1 L 2 |

p

X L 1 þL 2 .

In this simple scenario, there are only two solutions that will give the correct answer; the configura-
tions are symmetric with respect to the line from (0, 0) to ( X , Y ). This is reflected in the equation in

2

þ Y

2

L 1

L 1

L 2

Configuration Reachable workspace

FIGURE 5.14
Simple linkage.