cylindric joint Ai translates along its fixed axis zi, the
subworkspace with respect to the system AixiYizi is the
volume between two concentric cylinders without considering
the two ends of the subworkspace. The radii of the external
and the internal cylinders are respectively the maximum and
the minimum radii of the torus described by Ci without
translation. Taking the derivative of r expressed by Eq.
(3.9) with respect to 8bi and making the resulting
expression equal zero, we can obtain the maximum and minimum
radii of the torus from the following equation
dr
r ---- -biSebi(biCSbiS2abi + ai) = 0 (3.10)
d8bi
Then we obtain the following two equations
SBbi = 0 (3.11)
and
ai
Cebi = a(3.12)
biS2abi
When Sabi = 0 (abi = 0 or n) (the circles described by
Bi and Ci are coplanar), or ai > biS2obi, no real roots of
8bi can be found in Eq. (3.12). Hence we only have two
roots of 8bi from Eq. (3.11), i.e. 9 i = 0 and 8 i = n. In
this case, the values of r corresponding to 48i and 6ji are
respectively the maximum and minimum values of r.
Therefore, substituting the value of e8i and e8i into Eq.
(3.9) yields the maximum and minimum values of r as follows: