Abstract: A five-axes machine is capable of rotating its tool spindle in a high
angle, and therefore largely increases efficiency by machining multiple
faces of a workpiece with only one setup. The rotation motion type of
five-axes machining is a two dimensional submanifold of the special
Orthogonal group SO(3). For decades, the Hooke joint model is taken for
granted to describe such a motion type. However, the Hooke joint model
has a topology of a torus, which contradicts the fact that the spindle
rotation has a topology of a sphere. The Hooke joint model is not an
ideal model to type synthesize parallel kinematics machine with
omni-directional high angle capability. Inspired by a recent survey
study, we discovered that there exists a unified model for the motion
types ranging from kinesiology of Da Vinci and Listing?s law of eye
movement to Rzeppa?s constant velocity joint, Zlatanov?s multi-mode
parallel mechanism and five-axes machining. This newly discovered model,
known as an ?exponential submanifold?, has the correct sphere topology
and successfully explains the inconsistency of the Hooke joint model. It
is shown that exponential submanifold has its unique topological and
geometrical properties, which naturally leads to an omni-directional
high angle capability. Therefore, exponential submanifold is not only
the correct kinematics model for five-axes machines, rehabilitation
robotics, eye movement, constant velocity joints etc., but also an
indispensable mathematics tool for type synthesis.
- This event has passed.