2009

Deriving the default normal to the zero plane
Obviously, this method won’t extend to the north or south poles. They are
the singular points. When the EE axis moves across the poles, the normal will
suddenly change direction: it flips from the users’ viewpoint.
Normally, the preferred pose is the one when the solver is first assigned. So,
the plane on which one lays the joints corresponds to the horizontal plane
here. Rule A makes sure that the chain will stay on the plane if one moves the
goal on the plane.
Rule B means that, when you move the goal along the great circle vertical to
the equator, the chain will stay vertical, except when it passes through the
poles, which are the singular points of this map.
Parent Space
So far, we have described things as if the whole world comprises only IK
elements. In practice, the IK chain and goal might sit at points of separate
transformation hierarchies. Ultimately, we need to map the position of the
end effector that is described in the world to a point on the sphere. Depending
how the sphere is mounted relative to the end effector position, the readings
3402 | Chapter 15 Animation