Algorithm 1 Numerical estimation of the arc curvature ($\kappa$)
Inputs:
${\mathbf{r}}_t^0=[r_x{r}_{y}0{]}^T$   ⊳ Robotically actuated delivery sheath tip position
${\epsilon }_{\mathrm{th}}$   ⊳ Specified tolerance
Parameters:
$\ell$   ⊳ Arc length
$l_t$   ⊳ Rigid link length
Outputs:
$\hat{\kappa }$   ⊳ Estimated arc curvature
Method:
1: $\varphi =\arctan \left(\frac{r_y}{r_x}\right)$   ⊳ Arc plane angle
2: $\tan (\hat{\theta })=\frac{|r_t^0|}{\ell +l_t}$
3: $\hat{\kappa }=\frac{2}{\ell +l_t}\tan (\hat{\theta })$   ⊳ Initial estimate $\hat{\kappa }$
4: while $e>{\epsilon }_{\mathrm{th}}$ do
5: $\left[\begin{array}{c}{\hat{\mathbf{r}}}_t^0\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]={\mathbf{H}}_t^0(\hat{\kappa })\left[\begin{array}{c}{\mathbf{o}}_t\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$   ⊳ Compute tip position with $\hat{\kappa }$
6: $e=|{\hat{\mathbf{r}}}_t^0|-|{\mathbf{r}}_t^0|$   ⊳ Compute error (x- and y-axes)
7: $\hat{\kappa }=\hat{\kappa }+G·e$   ⊳ Re-estimate $\hat{\kappa }$, with $G=4450$ (empirically determined)
8: end while