1. Parametric Form of Curves Estimating beam deflection using strain information gained from sensors.

2. Parametric Form Where Condensed:

3. Given Boundary Conditions Given specific boundary conditions, such as p(0), p(1), p u (0), and p u (1) (where p u (u) is the first derivate of p(u) with respect to u), by solving for the coefficients, a form for p(u) can be found which includes the blending functions. Ex.) Is the first derivative form. Is the second derivative form. An example in beam theory: Where y = deflection along length x L = length of the beam and

4. Approaches Different approaches include: 1.) With more intermediate points, a single higher order polynomial can be estimated. 2.) Successive 3 rd order polynomials could be estimated between points and joined with continuity constraints. 3.) Interpolation using equations solved between sensors and using estimated information at u i as intermediate points.

5. Model (2 nd Approach) Image sensors at all positions u = [0 u 2 u 3 u 4… u N ]. We want to find the deflection at u = 1. x L y(0) = 0 θ(0) = 0 u = 0 u = 1 u2u2 u3u3 u4u4 uNuN

6. 2 Sensor Example From sensor readings (p uu ) at known locations (u = 0, u i, u j ), and the known boundary conditions, three 3 rd order equations can be solved for, and the position at u = 1 can be extrapolated. x L y(0) = 0 θ(0) = 0 u = 0 u = 1 uiui ujuj

7. 2 Sensor Example The deflection, slope, and curvature of the beam can be modeled as follows: where

8. 0 ≤ u ≤ u i Known: p(0), p u (0), and p uu (0), p uu (u i ).

9. u i ≤ u ≤ u j Using the coefficients found for 0 ≤ u ≤ u i, solve for p u (u i ) and p(u i ). Then, knowing p(u i ), p u (u i ), p uu (u i ), and p uu (u j ), solve for coefficients as above.

10. u j ≤ u ≤ 1 Using the second set of coefficients, solve for p u (u j ) and p(u j ). Now, we know p(u j ),p u (u j ), p uu (u j ), and p uu (1). After the last set of coefficients are found, p(1) can be calculated. In the end, the three part profile will be known for u along the length of the needle.

11. For Optimization In order to determine the best position for the sensors (u values used to construct the estimated profile): – Along a known curve (or set of curves), – Vary u, in given ranges such that u 1 < u 2 <…u N – Calculate sensitivity of positions u to error, abs(p- p_est). – Vary strains. Change known p uu values. (To give different curves.)