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Due to their complexity, inverse kinematics (IK) problems are often solved with iterative solutions. While iterative solutions can handle different IK poblems with a single algorithm, they can also become computationally expensive. If a specific IK problem needs to be computed often, it is worth considering an analytic solution.

If we limit the problem to a two bone chain in a two-dimensions, we can derive the anaytic solution without much complexity. This will generally be more efficient than its iterative alternatives. While this specific case lends itself to a 2D world, it can also be used in 3D as long as the kinematic chain's motion is limited to a single plane.

The goal of is to create a robust function with the following requirements.

1. Let the user specify which direction to bend the bone chain (there can be up to two valid solutions, each bending in an opposite direction).
2. Notify the user when there was no valid solution based on the input.
3. When there is no valid solution, calcualte the closest alternative.
4. Produce the expected results for every target position.  Most implementations found online or in books only function properly when the target is in a certain quadrant or when it remains in a specific range.

If you aren't interested in the exciting math ahead, jump to the end of the article for sample code. You can also download the RJ_Demo_IK application and source code to see a running implementation. If you're the type that likes to know how things work (like me), we will walk through the derivation in detail. It might look extensive at first, but it's only because I've broken it down step by step. Hopefully you'll be able to follow along without much difficulty.

You might also be interested in the following articles on different IK solvers:

• Constraint Relaxation IK in 2D
• Cyclic Coordinate Descent in 2D