Sensor Placement Estimation

1
Sensor Placement Estimation

• Force Based and Deflection Based Models

2
Deflection Estimation using Beam Theory
ex strain measured by FBG sensor ? radius of
curvature d distance from neutral axis
Sensor 1
Sensor 2
x1
x2
Curvature (1/?)
x
1
Curvature

?
f(x) ax2bxc
Slope
Slope ? f(x) dx
x
Deflection ?? f(x) dx
Deflection
x
y
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Force-Based Displacement

• 2 locations of force concentration defined.
• Moment of bending calculations determine
  curvature.
• True deflection model calculated from curvature.
• Estimate curvature determined from x1, x2, two
  sensor locations.
• Estimated deflection ??curvature.

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Model Construction
L 15cm
2 / L
F2
F1
Sensor 2
Sensor 1
Tip Deflection
x1
x2
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Sensitivity

• The sensitivity of the positions x1 and x2 where
  found as the deflection error at the tip of the
  needle with respect to x1 and x2.

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Results Force-Based Model

• Magnitudes of forces had no effects on
  distribution of deflection error and
  sensitivities.
• Whether the forces where applied in opposing
  directions also had no effect.
• Only location of the applied forces changed the
  distribution of deflection error and
  sensitivities.

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Error Distribution
2 Cases of varying loads applied at the same two
locations. One case also had opposing forces.
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Error Distribution
2 Cases of loads applied at differing locations.
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Error Sensitivities Distributions
Force of 1 gram at half the needle length (75mm)
and 2 grams at the tip (150mm).
Deflections Sensitivity to
x1 Sensitivity to x2
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Deflection Curvature Estimate
Force of 1 gram at half the needle length (75mm)
and 2 grams at the tip (150mm). The sensors are
placed at 25mm and 82mm (based on sensitivity and
deflection error) results.
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Deflection Based Model

• Given a known curve, 3rd order equations for each
  deflected segment (2) are found.

(xd, yd)
(xd2, yd2)
y(0) 0 ?(0) 0
L
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BCs and System of Equations
The system of equations to solve is
Because the deflection and slope at x 0 is
zero, the first equation has no first order or
y-intercept terms. The continuity constraints
are
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Finding Coefficients
With at least the point of deflection and a point
in the y2 region, the system can be solved. A
closer solution is found with more (x,y) points.
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Deflection Based Method

• Equations for Y1 and Y2 are found.
• A true curvature model is found by taking the
  second derivative of the deflection.
• An estimate curvature is calculated from the
  actual curvatures at two sensor positions.
• Estimated deflection ??curvature.