TABLE II.

Nomenclature, pt. 2

Symbol	Description
ϕi	Relative extension of the ith tube or section
Φi	Maximum relative extension of the ith tube or section
ϕp, Φp	Extension variables for proximal section
ϕd, Φd	Extension variables for distal section
kix(s)	Bending stiffness of ith tube about x as function of arc length, s
kiy(s)	Bending stiffness of ith tube about y as function of arc length, s
kB	Bending stiffness of distal manipulation section
kA	Bending stiffness of distal navigation section
D	Stiffness ratio of a robot section with respect to proximal section
θi	Rotation of the ith tube
αi	Relative rotation of the ith tube with respect to tube 1
${\mathcal{T}}_u$	Unconstrained robot tube set
$\mathcal{T}$	Robot tube set satisfying design rules of Section II-A
q	Set of kinematic input variables
RoC	Radius of curvature, equivalent to $\frac{1}{k}$
r	Radius of a tube
Bi, $\mathcal{B}$	The set of surgical task frames, $\mathcal{B}=\{B_i,i=1,\dots ,t\}$
E	Frame of entry into the anatomy
Eg	Initial guess for frame of entry into the anatomy
A	Frame of the manipulator base
Ag	Initial guess for manipulator base frame
$\mathcal{R}(E,A)$	Clinician selected regions for frames E and A
$e_x^B$	x-axis vector for frame B
Γ, Γm, Γn	Representation of the anatomy
Δ, Δn	Penalty function for the anatomy
v	Poisson’s ratio
Rz(θ)	Rotation matrix of θ around the z-axis
Ω	Occupancy volume for anatomical model
Ωr	Occupancy volume eroded by radius r
Ur	Distance map corresponding to Ωr
Sr	Spherical structural element of radius r
Pi	Point on robot centerline, i=1,…, o