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White Paper: Swivel Angle of the HI IK Solver 431
1. Start joint
2. End effec tor
3. EE axis
Solver Plane
Let’s call the plane passing all the joints the Solver
Plane. When joints do not lie on a plane, we
w ill define it to be the plane that (A) passes the
Start Joint and End Joint and (B) is closest to the
remaining joint in a certain sense.
The Swivel Ang le describes the degree of freedom
of the Solver Plane and it constrains only the Start
Joint.
Zero Plane Map
In order to describe the solver plane in terms of
a numerical quantity, we have to agree to what 0
means. Given the end-effector position, where is
the Zero (Solver) Plane? The Zero Plane Map takes
as the argument EE Axis and produces the norma l
to the zero plane.
The IK system a llows individual solver plug-ins
to define their own Zero Plane Maps. When not
defined, the IK system provides a default one.
The argument to the Zero Plane Map is a unit
vector to give the direction of the EE axis.
Equivalently, when the EE slides along the EE
axis, the solver plane should be fixed. Therefore,
the Z ero Plane Map defines a vector field on a
sphere. Given a point on the sphere, it produces
a tangential unit vector to be interpreted as the
normal to the zero plane.
1. Normal to the zero plane
Solver Plane Flipping
It is a mathematical fact that there does not exist
a continuous vector field on a sphere. No matter
howhardyoutry,therewillalwaysbeapoint
on the sphere where neighboring vectors change
dramatically. This is where the solver plane will
flip when the end effector axis approaches to it.
This is because, on one h and, the history
independent requirement demands us to assign
a fixed vector to the singular point. On the other
hand, no matter what vector is assigned, it will be
dramatically different from some vectors assigned
to the neighboring points.
Int r ins ic R ef erence Fr a me for the S pher e
InordertodefinetheZeroPlaneMap,weneed
to define a reference frame for the sphere. This
reference frame is intrinsic to the joint chain itself.
A sphere can be defined by the center, the
horizontal plane, and the meridian (zero
longitude). The center is assig ned to the start joint.