Inverse Kinematics

A harder task is to get the skeleton to perform a particular task,
i.e., to place one of its end effectors at a particular location in space.

==>  This is the problem that  "Inverse Kinematics"  is trying to solve.

Let's look at a simple 2-link skeleton in 2D space:

The task is to find the two joint angles that will move the end effector to loction P.
For this simple case we can find a direct closed form solution.
The first equation below sets the angle q2 so that the length of the combined arm matches the distance from the root to P;
and the second equation then turns the combined arm into the right direction.

But in general, this task is much harder!

These are some of the problems:

There may be no solution at all!  (What should be the end-configuration that the system returns ?)

If there is one solution, there may be others as well.  (Which one should we pick ?)

Already for 3-link arms there may be infinitely many solutions.  (Which is the "best" one ?).