Analytic Two-Bone IK in 2D - The solvable domain

Written by Ryan Juckett

Monday, 29 December 2008 07:46

Article Index
Analytic Two-Bone IK in 2D
Defining the problem
The solvable domain
Math review
Describing the problem with math
Solving for angle 2
Solving for angle 1
Writing the code
All Pages

Page 3 of 8

The solvable domain

There will be cases in which our bone chain cannot reach the desired point. If we allow our bones to rotate without any constraints, we can only find a valid solution within a an explicit range from the origin (i.e. the base of bone₁). The minimum distance we can reach is the absolute value of the difference of our bone lengths, $|d_1 - d_2|$ . The maximum distance we can reach is the sum of our bone lengths, $d_1 + d_2$ . If the distance to the target, $\sqrt { x^2 + y^2}$ , is within this range, we can solve for $\theta_1$ and $\theta_2$ . See figure 2 for a visual representation of the solvable area (a ring about the origin).