From Measurement to Interpretation - Kinematic Analysis of Human Movement - page 46

and

⎛ ⎞

− δ ⎛ −− (cos .sin) βγ ⎞ − 1 − 1

γ = tg

31

= tg

⎜ ⎟

⎜ ⎟

δ cos β .cos γ

⎝ ⎠

⎝ ⎠

33

This uncertainty, due to division by zero, corresponds to
what is often called the gimbal-lock. In the case of the knee,
there is no risk of this problem occurring as the varus-
valgus amplitude is low, but this sequence will not be
suitable for all joints.

A direct translation of Euler's theorem, which also
corresponds to the particular case of screwing (or helical
movement [WOL 85]) where translation is negligible,
involves expressing the 3D rotation between Ri-1 and Ri by
a single rotation about a spatial axis [CHE 00], called a
rotation vector :

The rotation axis is defined by its unit vector (unique to
a sign), which remains unchanged by rotation, and the value
of the rotation angle is also unique (its sign is determined

by the sign of the unit vector). As well as the uniqueness of
the solution, another advantage is that the rotation vector is
entirely represented by three scalar components in a given
coordinate system. With axis being unitary, its three
components correspond to two degrees of freedom, the thi rd
being given by the value of the rotation angle about this axis.
It is also easy to form the rotation operator (matrix 3

θ

3)
associated with this rotation vector, by the following
relation:

×

cos1cos. sin