Analytic Two-Bone IK in 2D - Describing the problem with math

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 5 of 8

Describing the problem with math

To solve the problem, we need to write it out in equation form. Our unknown variables are $\theta_1$ and $\theta_2$ . Our known variables are $d_1$ , $d_2$ , $x$ and $y$ . We are going to create two equations that define $x$ and $y$ in terms of the two bones.

Based on the right triangle ratios for $\sin\theta$ and $cos\theta$ , we can use the following equations to extract the adjacent and opposite sides of a right triangle.

$\mathrm{opposite} = (\sin \theta) ( \mathrm{hypotenuse} )$

$\mathrm{adjacent} = (\cos \theta) ( \mathrm{hypotenuse} )$

fig_3

figure 3

By breaking the problem into a couple right triangles as seen in figure 3, we can now define $x$ as a sum of the adjacent sides and $y$ as a sum of opposite sides.

$x = d_1 \cos \theta_1 + d_2 \cos(\theta_1 + \theta_2)$

$y = d_1 \sin \theta_1 + d_2 \sin(\theta_1 + \theta_2)$

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