TABLE I.

Nomenclature

a	Bold typeface for vectors
A	Bold, upright typeface for matrices
n	Number of tubes in concentric tube robot
pB	Position of backbone Bishop frame
RB	Rotation matrix of backbone Bishop frame
ψi	Angle between the material frame of tube i and RB
Rψ	Canonical rotation about z-axis of angle ψ
uB	Bishop frame curvature
·̂	The conversion from ℝ3 to the cross product matrix
(·)∨	Inverse function of ·̂, i.e. (x̂)∨ = x
(·)*	Denotes variable in undeformed state
${\mathit{u}}_i^{\ast }$	Precurvature of the ith tube in the local material frame
kib	Bending stiffness of tube i
kit	Torsional stiffness of tube i
Ki	Linear elastic constitutive map
βi	Arc length where tube i is held
αi	Absolute rotational actuation of tube i
Li	Total length of tube i
β	mini {βi}
L	maxi {βi + Li}
(·)′	Derivative with respect to arc length/dimensionless length
ei	ith standard basis vector
ui	Curvature of the ith tube in the local material frame
θi	Relative angle ψi − ψ1
h, h, hi	Allowable variation function(s)
𝓢	Linear second variation operator
σ	Dimensionless arc length
Fα	Partial derivative of a function F with respect to a
λ	Bifurcation parameter