Parametric%20Form%20of%20Curves

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Parametric Form of Curves

• Estimating beam deflection using strain
  information gained from sensors.

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Parametric Form
Where Condensed
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Given Boundary Conditions
Given specific boundary conditions, such as p(0),
p(1), pu(0), and pu (1) (where pu (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
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Approaches

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

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Model (2nd Approach)

• Image sensors at all positions u 0 u2 u3 u4
  uN. We want to find the deflection at u 1.

u2
u3
u4
u 0
uN
u 1
y(0) 0 ?(0) 0
x
L
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2 Sensor Example

• From sensor readings (puu) at known locations (u
  0, ui, uj) , and the known boundary conditions,
  three 3rd order equations can be solved for, and
  the position at u 1 can be extrapolated.

ui
uj
u 0
u 1
y(0) 0 ?(0) 0
x
L
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2 Sensor Example

• The deflection, slope, and curvature of the beam
  can be modeled as follows
• where

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0 u ui

• Known p(0), pu(0), and puu(0), puu(ui).

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ui u uj

• Using the coefficients found for 0 u ui,
  solve for pu(ui) and p(ui).
• Then, knowing p(ui), pu(ui), puu(ui), and
  puu(uj), solve for coefficients as above.

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uj u 1

• Using the second set of coefficients, solve for
  pu(uj) and p(uj).
• Now, we know p(uj),pu(uj), puu(uj), and puu(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.

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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 u1 lt u2 ltuN
• Calculate sensitivity of positions u to error,
  abs(p-p_est).
• Vary strains. Change known puu values. (To give
  different curves.)