Page 382 - Control Theory in Biomedical Engineering

348 Control theory in biomedical engineering

This unit triangle requires three parameters to be characterized, which
include the side a and the angles α and β. It follows that
L AB ¼ a, (8)
L BC ¼ a sin αðÞ=sin βðÞ, (9)

L AC ¼ a sin α + βÞ=sin β ðÞ: (10)
ð
When in its folded state, the position of the same triangle is characterized
by the height h and the angle of twist ϕ as well as the radius r of the cylinder.
Again, it follows that
π

l AB ¼ 2rsin , (11)
n
where n is the number of repetitions of the unit cell. Since there are different
folded states with different heights and twist angles, it is not possible to deter-
mine them using the three constants, namely a, α, and β. Thus, there is a
need to introduce strains E AB , E BC, E AC to link the variables to the constants,
where

l AB L AB
E AB ¼ (12)
L AB
and so on. The deformation energy stored in one strip of the paper is then
given by

nEA 2 2 2
U ¼ L AB E AB + L BC E BC + L AC E AC , (13)
2
where EA is the tensile rigidity. This deformation energy shows that there
are apparent bi-stable states where the fully extended as well as the fully col-
lapsed states have minimum energy. There exists an energy barrier between
these two states, which is evidence that once fully extended, the material is
stiff and requires a higher amount of energy to collapse again. A more
detailed derivation of the results can be found in Zhai et al. (2018). By vary-
ing the angles α and β, it is possible to select the energy barrier and hence the
stiffness change required for bespoke applications.
While this method of using collapsible structures showed promise as a
variable stiffness mechanism, it also had a major disadvantage with respect
to actuation. As mentioned earlier in the section objectives, we aimed to
create a combined actuation and variable stiffness method using a single
structure. When tested for actuation capabilities, it was observed that the
device was only able to collapse and extend and did not provide useful