Tactile whisker sensor using flexible 3D printed transducers

Tactile Whisker Sensor using flexible 3D printed transducers

B. (Bram) Eijking

BSc Report

C e

Prof.dr.ir. G.J.M. Krijnen Ing. R.G.P. Sanders Dr.ir. M. Odijk

July 2017 022RAM2017 Robotics and Mechatronics

EE-Math-CS University of Twente

P.O. Box 217

7500 AE Enschede

The Netherlands

Contents

1 Introduction 4

1.1 Biomimetic background . . . . 4

1.1.1 Animal Whiskers . . . . 4

1.1.2 Previous Research . . . . 4

1.2 Content of this report . . . . 5

2 Design 6 2.1 Whisker design . . . . 6

2.2 Mechanical model . . . . 7

2.3 Resistive sensing . . . . 11

3 Fabrication 12 3.1 Materials used . . . . 12

3.1.1 Flexible materials . . . . 12

3.1.2 Conductive material . . . . 12

3.2 3D printing . . . . 12

3.3 Linear samples . . . . 12

3.4 3D model . . . . 13

4 Experimental setup 15 4.1 Measurement setup . . . . 15

4.2 Resistive Measurements . . . . 15

4.3 Whisker measurements . . . . 15

5 Results 17 5.1 Resistive measurements . . . . 17

5.2 Whisker measurements . . . . 18

5.3 Model validation . . . . 19

6 Discussion 25 7 Conclusion 27 7.1 Recommendation . . . . 27

A Model derivations 28

2

CONTENTS 3

A.1 Beam displacement due to moment . . . . 28 A.2 Beam displacement due to horizontal force . . . . 30 A.3 Whisker . . . . 31

B 3D prints 33

Bibliography 35

Chapter 1

Introduction

1.1 Biomimetic background

1.1.1 Animal Whiskers

Tactile sense is one of the main senses that is utilized by humans for proximal sensing. The tactile sense of humans is provided by receptors on the skin. These create impulses upon touching an object which are then processed in the brain. Many other mammals use whiskers to provide information for tactile sensing, this is done by means of whiskers. These whiskers are hairs with specific mechanical properties. The whiskers are used for various different applications by different species. Rodents use their whiskers for applications such as localizing objects, orientation, tracking as well as texture discrimination [1]. The rats whisker system is used for active sensing in order to perform these tasks. By moving the whiskers along the surface rats can explore and discriminate textures. Micro as well as macro vibrissae help to sense textures at different levels of detail [2].

It has been shown that these senses are very useful in situations where other senses are unable to give enough information such as in darkness. A different type of species that uses their whiskers in underwater applications are seals. They use their whiskers to sense flows, currents and vibrations in water. Using this information it is shown that seals are able to track the wake of artificial fish enabling them to hunt down prey even in dark conditions [3]. The accuracy of the tactile senses of animals has been shown to be very high and in some cases animals have shown to be able to distinguish object using their whiskers with the same accuracy as with vision by using their whiskers [4].

1.1.2 Previous Research

As stated above the tactile senses of animals by means of whisking can be used in environments such as darkness where other types of sense are not usable, as well as the potential for highly accurate sensing. This concept shows great potential for artificial sensors based on this principle and inspiration of this way of sensing is taken from rats [5] to apply in sensing techniques. Several implementations have already been made using whiskers, whisker arrays and active whisking. D. Jung and A. Zelinsky have made an implementation of a proportional whisker [6], where the whisker information is used to navigate an autonomous robot. J.H. Solomon and M.J. Hartmann have shown an implementation where a whisker array is used to sense complex environmental features [7]. O. Bebek and M. Cenk Cavusoglu have shown an implementation of a whisker sensor for surgery tools. The whisker sensor is used to determine the position of the surface of a beating heart in three-dimensions. This information is used for positioning surgical tools and account for the movement of the heart [8]. Read out is achieved using

4

CHAPTER 1. INTRODUCTION 5

various techniques. Some of the techniques that are used to provide a read out of the whisker sensors are [5]:

• Potentiometers [9]

• Load-cells [10]

• Electret microphones [11]

• Resistive arrays [12]

• Strain gauges [8]

• Piezoelectricity [13]

• Magnetic hall-effect sensors [14]

• Capacitive sensors [15]

Each having certain advantages and disadvantages for certain applications.

1.2 Content of this report

In this report a first step towards a fully 3D-printed flexible tactile whisker sensor is shown. The goal is to make

a passive whisker that can be used to measure in 2 degrees of freedom. The whisker will be used to measure both

force and moment acted upon the whisker. By measuring this the point of action can be determined by evaluating

the arm at which the force is applied. The system will have two degrees of freedom therefore two strain gauges

will be used to detect these quantities. A first design is proposed and printed using a 3D printer. The materials

used are evaluated and the conductive material used for the strain gauges is characterized. The measurement

setup is shown and obtained results are compared to the mechanical model that describes the mechanics of the

system.

Chapter 2

Design

2.1 Whisker design

The function of the whisker inspired tactile sensor is to determine both the force F _ext acting on the whisker as well as its point of action s. This can be done by using the horizontal displacement along the x-direction to measure the force by determining the translational stiffness of the base on which the whisker is place. By using the lever of the whisker the moment can be measured from the rotation of the base of the whisker. This can be done by determining the rotational stiffness of the base on which the whisker is placed and measuring the angle of rotation. After the moment and force are calculated the point of action s can be determined. A schematic of this principle is shown in figure 2.1.

F ext

s F r t ^r

F ext

s x r q ^r

Figure 2.1: Schematic of the proposed whisker inspired tactile sensor with rotational stiffness (green) and translational stiffness (blue).

The way this rotational as well as translational stiffness is obtained is by placing the whisker on a horizontal beam. This beam is clamped at the two endpoints such that the moment and force can be obtained by making a model of the deflection of the beam. This deflection is measured by using two strain gauges that are placed on the two sides of the horizontal beam. A model of the design is shown in 2.2.

6

CHAPTER 2. DESIGN 7

Figure 2.2: Schematic of the proposed design.

The horizontal force applied on will apply a moment on the center of the horizontal beam. This moment will bend the horizontal beam symmetrically around the center point. The deflection will give a strain in the two sections of the beam where the gauges are placed. The horizontal force will displace the center of the whisker base therefore stretching one side and compressing the other side of the horizontal beam. This will cause a positive strain on one end and a negative strain on the other end. By measuring the sum as well as the difference between the two gauges the desired quantities may be obtained.

2.2 Mechanical model

The Whisker sensor is modeled as two separate beams that are connected at the center point of the whisker.

Figure 2.3: Free body diagram of the vertical beam(left) and horizontal beam(right).

The horizontal base of the whisker is modeled as a clamped-clamped beam with cross-section A, Young’s

modulus E, second moment of area I base and length L base . The beam is loaded at position x = a by a

CHAPTER 2. DESIGN 8

moment M 0 . The left and right support have reaction forces F A , F _B respectively and reaction moments M A , M _B respectively. Since the horizontal forces are evaluated separately a second free body diagram was made where only the horizontal forces are taken into account. The two results for the deflection are combined by using superposition. The free-body diagram of the horizontal beam is shown in figure 2.3. The moment and force balance in the vertical direction are taken. This generates 2 equations with 4 unknowns therefore the deflection integrals are taken in order to solve this problem.

The force balance in the y-direction is taken:

F A + F B = 0 (2.1)

The moment balance taken around the center point gives the following:

M _A + M ₀ − F _A · a + F _B · b − M _B = 0 (2.2)

By making a cut along the beam the internal moments can be evaluated, this yields the following 2 equations:

M (x) + M A − F _A · x = 0

M (x) = F A · x − M _A f or (x < a) (2.3)

M (x) − M _B + F _B · (L − x) = 0

M (x) = −F _B · (L − x) + M _B f or (x > a) (2.4)

The internal moment along the beam can be expressed as M (x) = EI · w00(x), with w(x) as the vertical deflection of the beam.

w00(x) = 1

EI _base (F _A · x − M _A ) f or(x < a) (2.5) w00(x) = 1

EI _base (−F B · (L − x) + M _B ) f or(x > a) (2.6) This result is integrated twice to obtain w(x):

w(x) = 1 2EI _base ( 1

3 F A · x ³ − M _A · x ² + C 1 · x + C ₂ ) f or(x < a) (2.7)

= 1

2EI _base (− 1

3 F _B · (L − x) ³ + M _B · x ² + C ₃ · x + C ₄ ) f or(x > a) (2.8) To determine the integration constants the boundary conditions are applied. The beam is clamped at x = 0 and x = L therefore w(0) = w0(0) = 0 and w(L) = w0(L) = 0. Applying these boundary conditions gives the following result:

w(x) = 1 2EI _base ( 1

3 F A · x ³ − M _A · x ² ) f or(x < a) (2.9)

= 1

2EI _base (− 1

3 F _B · (L − x) ³ + M _B · (L − x) ² ) f or(x > a) (2.10) In order to solve the equations two more conditions are needed, which are the continuity conditions at the center point of the beam, therefore setting w l (L/2) = w r (L/2) = 0 and w l 0(L/2) = w _r 0(L/2) = 0, where {l,r}

indicate left and right of the center of the beam. This gives the following equations:

w  L 2



= 1

2EI _base 1

3 F A ·  L 2

 3

− M _A ·  L 2

 2 !

= 1

2EI _base − 1

3 F B ·  L 2

 3

+ M B

 L 2

 2 !

= 0

(2.11)

CHAPTER 2. DESIGN 9

w0  L 2



= 1

2EI _base F A ∗  L 2

 2

− 2M _A ∗  L 2

 !

= 1

2EI _base F B ∗  L 2

 2

− 2M _B ∗  L 2

 !

= 0 (2.12) By combining the result with equations 2.1 and 2.2 the reaction forces and moments can be expressed in M 0 . Combining this result with equations 2.9 and 2.10 the final equation describing the deflection of the beam is obtained.

w(x) = M 0

EI _base

 1

4L x ³ − 1 8 x ²



f or

 x ≤ L

2



(2.13)

= M ₀ EI _base



− 1

4L (L − x) ³ + 1

8 (L − x) ²



f or

 x ≥ L

2



(2.14) From this the strain in the beam is calculated using an integral to calculate the curve-length S of the neutral axis.

S = Z L

0

s

1 +  dw dx

 2

dx (2.15)

This integration can in most cases only be done numerically therefore it is calculated by using the approximation:

√

1 + a ² ≈ 1 + ^a ₂

which is valid for small displacements. Now the integration becomes for the left-hand side of the beam:

S L = Z L/2

0

1 + 1 2

 M 0

EI _base

 2  3

4L x ² − 1 4 x

 2

dx (2.16)

Which will be equal to the right-hand side of the beam due to the symmetry of the problem. This integration yields:

S _L = L

2 + 1.302 × 10 ⁻⁴

 M ₀ EI _base

 2

· L ³ (2.17)

This curve length is then equated to the strain in the following way:

ε _L,R = ∆L

L = S − ^L ₂

L 2

= 2.604 × 10 ⁻⁴

 M ₀ EI _base

 2

· L ² (2.18)

Where ∆L is the change in length of the beam which is obtained by taking the curve-length and subtracting the original length of the beam. {L,R} indicate the strain on the left and right strain gauges, an elaboration on the derivation of this model can be found in appendix A.1.

In order to model the horizontal deflection of the whisker superposition is used. The beam is clamped at both ends and a horizontal force is applied at the center of the beam. Two separate equations for the deflection are determined one describing the deflection of a beam that is clamped at one end and a force applied in the center. The second equation is found by calculating the deflection of the beam under a compressing load at the end of the free end of the beam, this compressive force is then set equal to the reaction force. The sum of forces in the x direction along the beam are:

F _ext + R _A − R _B = 0 (2.19)

The deflection of the beam under an applied load F _ext that is applied at the center of the beam is:

u(x) = F _ext x

EA f or

 x < L

2



(2.20)

= F ext L

2EA f or

 x > L

2



(2.21)

CHAPTER 2. DESIGN 10

Next the displacement of the beam under a compressive force that is set equal to R B due to the support on the right side is:

u(x) = − R _B x

EA f or (x < L) (2.22)

Now these contributions are added to obtain the final equation for the horizontal deflection.

u(x) = F ext x

EA − R B x

EA f or

 x < L

2



(2.23)

= F _ext L

2EA − R _B x

EA f or

 x > L

2



(2.24) Now applying the boundary conditions that the total deflection at u(L) = 0 it is found that the reaction force R B

is equal to ^F ₂

. The reaction force R _A can then be calculated and is equal to − ^F ₂

. The final equation describing the deflection in the horizontal direction is therefore:

u(x) = F _ext x

2EA f or

 x < L

2



(2.25)

= F _ext (L − x)

2EA f or

 x > L

2



(2.26) From this it follows that the displacement at the center of the whisker due to an external load is equal to:

u  L 2



= F _ext L

4EA (2.27)

From this it follows that the force related to a horizontal displacement can be seen as a translational spring with spring constant:

F _ext = k · u  L 2



= 4EA

L · u (2.28)

Therefore the strain on the left side of the beam is:

ε L = ∆L

L =

F

L 4EA L 2

= F ext

2EA (2.29)

and on the right hand side it is equal but opposite since the beam is compressed in stead of stretched:

ε R = − ^F _4EA

^L

L 2

= − F ext

2EA (2.30)

The final step was to combine equations 2.18, 2.29 and 2.30 to obtain the total strain in the two strain gauges due to the applied force and moment:

ε _L = 2.604 × 10 ⁻⁴

 M ₀ EI base

 2

· L ² + F _ext

2EA (2.31)

ε R = 2.604 × 10 ⁻⁴

 M 0

EI _base

 2

· L ² − F ext

2EA (2.32)

The whisker itself is modeled as a cantilever beam suspended by a torsional spring and a translational spring.

A free-body diagram is shown in figure 2.3. These springs represent the stiffness of the horizontal beam the whisker is placed on. The rotational stiffness was found to be:

θ = − M ₀ · L

EI base · 16 (2.33)

CHAPTER 2. DESIGN 11

And the stiffness in the horizontal direction was found to be:

u = F _ext · L

4EA (2.34)

Using these values the mechanical model for the whisker was derived.

w(x) = 3F ext · s · x ²

2EI _whisker + F ext · x ³

6EI _whisker − F ext · s · L _base · x

16EI _base · EI _whisker + F ext · L _base

4EA _base · EI _whisker f or (x < s) (2.35)

=  3F _ext · s ²

2EI _whisker − F _ext · s · L _base 16EI _base · EI _whisker



· x − 5F _ext · s ³

6EI _whisker + F _ext · L _base

4EA _base · EI _whisker f or(x > s) (2.36) This completes the model. From the strain gauges the force and moment are determined, from this the arm length can be determined and using the equation describing the whisker the point of action can be determined in two dimensions.

2.3 Resistive sensing

The sensing technique used in the whisker sensor described in the previous section is resistive sensing. The resistance of a conducting channel that is subjected to strain changes. This change can be caused by geometrical effects or piezoresitive effects. For strain gauges the relation between strain and change in resistance is governed by the gauge factor(GF) in the following way [16]:

∆R

R = GF · ε (2.37)

Applying the model derived in section 2.2, the relative change in resistance can be related to the applied force and moment on the whisker:

∆R _L R L

= GF _L · ε _L = GF _L 2.604 × 10 ⁻ 4

 M ₀ EI base

 2

· L ² + F _ext 2EA

!

(2.38)

∆R _R R R

= GF R · ε _R = GF R 2.604 × 10 ⁻ 4

 M ₀ EI base

 2

· L ² − F _ext 2EA

!

(2.39) This gives 2 equations with two unkowns M ₀ and F _ext which can be calculated form the two resistance changes in the following way:

M 0 = ± v u u u t

∆R

GF

·R

+ _GF ^∆R

R

·R

5.208 × 10 ⁻⁴

 L EI

 F =

 ∆R L

GF L · R _L − ∆R R

GF R · R _R



· EA (2.40)

Chapter 3

Fabrication

3.1 Materials used

3.1.1 Flexible materials

The materials used for the whisker are flexible TPU-based materials. The first material that was used is Ninjaflex.

This is a very flexible non-conductive TPU that is used for the whisker and base of the design. This material has a tensile modulus of 12 MPa and has an elongation at break of 660% [17].

3.1.2 Conductive material

The conductive material that is used is a flexible PI-ETPU called Carbon black. This material is used for the strain gauges. The material has a tensile modulus of 12 MPa and an elongation at break of 250% [18]. The volume resistivity of the material is <300 Ω cm, the conductivity is provided by carbon black fillers in the filament.

3.2 3D printing

The sensors are printed on a Flashforge Creater Pro printer [19]. The used materials are very flexible therefore the printer was equipped with a flexion extruder from Diabase Engineering [20], which allows for better performance when printing with flexible filaments. This extruder is equipped with two nozzles which allows for printing of 2 different materials. For the Ninjaflex a nozzle of 600 µm is used, the PI-ETPU is printed with a nozzle of 800 µm.

The sensors are printed with a print speed of 2000 mm min ⁻¹ (33.3 mm s ⁻¹ ). After making several test prints the best results were obtained with a print temperature of 220 ^◦ C for the Ninjaflex and 230 ^◦ C for the PI-ETPU, the hot-bed temperature was kept at 50 ^◦ C. Both materials are printed at a layer thickness of 200 µm. Since the printer only allowed for 2 materials to be printed at a time, no support material could be used. The bridging capabilities of the material is very low therefore the decision was made to print the sensor flat on the bed.

3.3 Linear samples

A linear sample was designed which was used to characterize the electrical behavior of the conductive material.

This sample was printed at different thicknesses. First a small layer of ninjaflex was printed and on top of this

12

CHAPTER 3. FABRICATION 13

layers of ETPU were printed. The samples were printed at thicknesses of several layers up to 3 mm. A model of this sample is shown in figure 3.1.

Figure 3.1: Design of the samples printed for electrical characterization.

3.4 3D model

The design shown in section 2.1 was modeled using Fusion 360 by Autodesk [21]. The 3D model of the design is shown in figure 3.2. The sensor has a thickness of 3 mm. The dimensions of the conductive channels are shown in figure 3.3. These conductive channels will be used as the strain gauges and have a thickness of 0.8 mm.

To allow for contacts to be soldered into the conductive material rectangular plates were included at the end of

the strain gauges.

CHAPTER 3. FABRICATION 14

Figure 3.2: Design showing the dimensions in mm of the body of the whisker sensor.

Figure 3.3: Design showing the dimensions in mm of the strain gauges.

Chapter 4

Experimental setup

4.1 Measurement setup

After the samples were printed a first set of measurements were conducted to characterize the resistive behavior of the conductive ETPU. For this 4-point measurements were used to minimize the influence of the resistance of the contacts. Measurements were done at different lengths along the sample, which had different thicknesses to see how the resistance related to the dimensions of the samples and to determine the conductivity of the material.

The measurements were done by measuring the voltage over the samples at different supplied currents to see if the material shows an ohmic relationship between current and voltage. After this measurements were done on the whisker that is shown in section 3.4. In order to validate the derived model a setup was needed where a force could be applied to the whisker which could be controlled and measured. Besides the force measurements the resistance of the two strain gauges had to be measured. These resistance measurements were done in a similar manner as the first samples by using a 4-point measurement. The measurements were compared to the model derived in section 2.2, which relates the applied force to the relative change in resistance of the strain gauges.

4.2 Resistive Measurements

The first set of measurements were done using a Keithley 2410 SourceMeter [22], which was set programmed to do 4-point measurements at several predefined currents. This Sourcemeter was connected through a serial connection and read-out was directly stored into Matlab.

4.3 Whisker measurements

For the measurements that were done on the whisker a linear actuator(Smac LCA25-050-15F [23]) was used to apply a force on the whisker, this force was measured using a Model 1004 load cell from VPG [24]. The load cell has a rated capacity of 0.3 kg and was mounted on the shaft of the actuator. The load cell was read-out by a HP 34401A multimeter [25] connected via GPIB. The resistance was measured simultaneously using the same setup as in the first set of measurements.

The whisker was clamped using a plastic vise to ensure that the base of the whisker stays in the correct position. The source-meter only allowed for 1 measurement at a time therefore the strain gauges were measured in 2 separate measurements while applying the same force on the whisker. After this set of measurements the

15

CHAPTER 4. EXPERIMENTAL SETUP 16

setup was altered to support the use of two SourceMeters simultaneously. This was done by connecting the two

SourceMeters as well as the multimeter through a GPIB connection. This allowed for the measurements of the

two resistances and the load-cell to be done at the same time.

Chapter 5

Results

5.1 Resistive measurements

The results of the resistance measurements of the first set of samples is shown in figure 5.1. For this measurement several samples at thicknesses ranging from 1 mm to 3 mm were used. The applied current was kept constant for this measurements. During these measurements it was seen that there was no clear relation between thickness and resistance was found. This gave the impression that the resistance had a large proportion of surface conduction since the length over which the measurement was done showed a clear relation to the resistance measurement.

Therefore a second set of samples was printed where the conductive ETPU was printed at a thickness of only several layers. The results of these measurements are shown in figure 5.2. During these measurements the current was stepped from −10 µA to 10 µA to see if any nonlinearities in the resistance occurred. During these measurements it was clearly seen that the resistance was in the same order of magnitude as the previous measurements, and similarly did not show a clear resemblance between thickness and resistance. Secondly it was seen that most samples showed a clear ohmic behavior, except for the sample that had only 1 layer. The reason for the resistance to be determined for a large portion by the surface conduction was not found and due to time restrictions this was not further investigated.

Figure 5.1: Measurements showing the measured resistance for various different thicknesses.

17

CHAPTER 5. RESULTS 18

Figure 5.2: Measurements showing the measured resistance for various number of layers of ETPU.

5.2 Whisker measurements

The parameters needed for the calculations of the model were determined. These parameters are shown in table 5.1.

Table 5.1: Table of parameters Parameter Value

L base 29.5 mm

s 33 mm

I base 8.1 × 10 ⁻¹² m ⁴

A 1.08 × 10 ⁻⁵ mm

E 12 MPa

For the whisker measurements the actuator was programmed to excite the whisker in a sine-wave manner. The

measurement results are shown in figure 5.3 and figure 5.4. The measurements were done separately and the

time-axis were synchronized afterwards. The first observation of the measured data was that a large amount of

drift could be seen in the resistance measurement. The force measurement showed very clear results. A plot was

made to show the relation between force and resistance, since this is what is going to be related to the derived

model. From this plot it can be seen that the drift has a too large influence to determine a clear relation between

force and resistance.

CHAPTER 5. RESULTS 19

Figure 5.3: Measurements showing the measured resistance and force over time as well as the resistance set out over force for the first strain gauge.

Figure 5.4: Measurements showing the measured resistance and force over time as well as the resistance set out over force for the second strain gauge.

5.3 Model validation

The drift was filtered out by designing a digital high-pass filter in Matlab with a cut-off frequency of 0.1 Hz.

The filter was applied using a zero-phase digital filtering function in Matlab [26] to ensure the measurements

were in phase after filtering. This resulted in the change in resistance which was divided by the mean value of

CHAPTER 5. RESULTS 20

the resistance measurement to obtain the relative change in resistance. After filtering the resistance was plotted against the measured force, this is shown in figure 5.5. A second degree polynomial fit was made to the data points and from this the coefficients were found for the quadratic term as well as the linear term. The fit is also shown in figure 5.5.

Figure 5.5: Measurements showing the measured resistance set out against measured force, for the 2 strain gauges respectively.

It was found that the quadratic term caused by the moment had a much larger influence than was first expected from the model. Therefore the model was altered and the terms were separated. By doing this two gauge factors were obtained, one for the strain caused by the moment(GF _M ) and one for the strain caused by the horizontal force(GF F ).

∆R _L R L

= GF _L · ε _L = GF _LM · 2.604 × 10 ⁻ 4

 M ₀ EI base

 2

· L ² + GF _LF · F _ext

2EA + C _offset1 (5.1)

∆R _R R R

= GF R · ε _R = GF RM · 2.604 × 10 ⁻ 4

 M ₀ EI base

 2

· L ² − GF _RF · F _ext

2EA + C offset2 (5.2) The obtained values for the gauge factors, offsets and mean values of resistances of the gauges are:

GF _LF = 1.8835 GF _RF = 0.2279 GF _LM = 16.916 GF RM = 40.315

C offset1 = −4.4698 × 10 ⁻⁴ C _offset2 = −1.1 × 10 ⁻³

R _L = 1.4569 × 10 ⁶ Ω

R _R = 9.3843 × 10 ⁵ Ω

CHAPTER 5. RESULTS 21

Using these corrections to calculate the resistance of the two measurements from the measured force. The result of this calculation is shown in figures 5.6 and 5.7. From this figure it can be seen that the model is capable of calculation the change in resistance caused by an applied force after the corrected gauge factors are applied to the model.

Figure 5.6: Measurements showing the measured resistance and calculated resistance from the measured force, for the 2 strain gauges respectively.

Figure 5.7: Zoomed in measurements showing the measured resistance and calculated resistance from the measured force, for the 2 strain gauges respectively.

By applying the calculation shown in 2.40 the force was calculated from the measured resistances after applying

the corrections described above. This is shown in figure 5.8. The same was done for the moment, this result is

CHAPTER 5. RESULTS 22

shown in figure 5.9.

Figure 5.8: Force calculated from resistance measurements using the model.

Figure 5.9: Moments calculated from resistance measurements using the model.

After it was found that the measurements had such a large deviation of the model and that there was a large

deviation between the two gauges. The measurement was repeated with the setup were the resistances and force

were measured at the same time. This was done to make sure that no other factors could make a difference

between the two measurements. The measurements were filtered using the same filter and the relative change in

resistance was plotted against the force, this is shown in figure 5.10.

CHAPTER 5. RESULTS 23

Figure 5.10: Measurements showing the measured resistance set out against measured force, for the 2 strain gauges respectively.

The same method was applied to calculate the gauge factors and determine the mean values of the resistances, they were found to be:

GF LF = 0.6166 GF RF = 1.3251 GF _LM = 11.6395 GF _RM = 21.1434 C _offset1 = −4.593 × 10 ⁻⁴ C _offset2 = −8.3056 × 10 ⁻⁴

R _L = 8.7950 × 10 ⁵ Ω R _R = 4.9178 × 10 ⁵ Ω

Similarly to the first measurement a much larger influence of the strain due to the moment term was obtained.

Also it was seen that the resistance of the two strain gauges decreased dramatically. This second measurement

was done 2 weeks after the first measurement. This slow decrease over time of the resistance was also observed

during the measurements in the form of drift. Using these values the resistance was calculated from the measured

force, this result is shown in figure 5.11.

CHAPTER 5. RESULTS 24

Figure 5.11: Measurements showing the measured resistance and calculated resistance from the measured force, for the 2 strain gauges respectively.

This second measurement showed very different values for the gauge-factors. However the offsets were in the

same order of magnitude. After these values were determined and the force was calculated from the measured

resistances it was found that no good results could be obtained to calculate the force.

Chapter 6

Discussion

The first resistive measurements showed that the resistance of the samples that were made had a resistance which did not depend on the thickness of the samples. A second set of samples which were only a few layers thick showed similar results. This could be caused by a surface conduction that determines the resistance of the samples. Nonlinearities were observed in the sample that featured only 1 layer of ETPU. The electrical characterization was not worked out into more detail.

During the measurements that were done on the whisker it was found that a large deviation of the mechanical model was obtained. The influence of the moment on the relative change in resistance was much larger than expected. The model was based on several assumptions:

• Euler Bernoulli beam theory.

• Small angle approximations.

• Approximation of curve length integral.

• Assumption of constant gauge factor.

• Assumption that the supports were stiff.

• Superposition was used to derive the model.

• Force in the vertical direction is neglected.

The two parts of the whisker were modeled by applying the Euler Bernoulli beam theory. This theory however is only accurate for small deflections, it was assumed that the deflections of the horizontal beam were small enough such that this theorem could be applied. Looking back at the results it is clearly seen that the model shows resemblance with the measurements, however for larger forces a large deviation of the model can be seen.

This can be caused by the fact that the deflections in the beam are too large for the beam theory to be valid.

Secondly it was found that the squared moment term in the model had a much larger influence than was expected from the model. The derivation of the moment relies on an approximation that was done where it was assumed that a small-angle approximation could be used. The angles which are created in the horizontal beam might however be too large for this approximation to hold, this could explain why the model has a deviation with the obtained results.

25

CHAPTER 6. DISCUSSION 26

Another observation was that a very large hysteresis effect can be seen in the measured data. This was not taken into account when the model was derived for the sensor. Due to this large influence of hysteresis the calculated force from the resistance shows large deviations with the measured force.

Despite these large deviations of the model it was seen that the relation between the applied force and moment on the whisker can be related to the change in resistance by a quadratic model. However the model needed to be altered to ensure a good fit to the measurements was obtained. This shows that in a first approximation the behavior of the strain-gauges can be related to the applied force, however a more accurate derivation is needed.

In order to do this the assumptions that are made need to be investigated into more detail.

Another factor that could potentially play a role in the mechanics of the whisker is the fact that the sen- sor is 3D printed. 3D printed objects are anisotropic due to the fact that layers are stacked on top of each other.

The influence this anisotropy might have on the mechanics as well as electrical properties are not investigated but might play a role.

Due to the large deviations from the mechanical model, the model that describes the deflection of the whisker

was not studied into more detail. This model could potentially be used to accurately determine the point of

action in two dimensions. For this an accurate way of calculating the arm at which the force acts is needed which

was not possible with the results that were obtained.

Chapter 7

Conclusion

In this report a first step was shown towards a fully 3D printed tactile whisker sensor. A design was proposed that uses strain gauges to measure the desired quantities. A model was derived to describe the mechanics of the design. From the derived model a relation was made between the measured resistances of the strain gauges and the applied force and moment on the whisker. The obtained model showed a quadratic dependence on the moment as well as a linear dependence on the horizontal force that was applied. The measurements showed a quadratic model could indeed be used to relate the applied force and moment to the resistance change in the strain gauges. However some adjustments had to be made to the model to ensure a good fit was obtained with the measured data. The force calculated from the two resistances showed that the principle can be used as a first approximation. However the measurements that were used to obtain the applied force and moment deviated too much from the real values that it was not possible to accurately determine the arm at which the force was applied. Based on the observations that were done from the measurements the principle of the design shows potential. However more detailed research into the material properties as well as a more detailed mechanical model will be needed to make an accurate whisker sensor that is able to measure the two degrees of freedom.

7.1 Recommendation

In order to develop an accurate sensor the model needs to be further elaborated and the assumptions need to be investigated into more detail to ensure that they are valid assumptions. The material shows behavior that was unexpected, this needs to be investigated into further detail to get a good understanding of how the material behaves. This is necessary because it might also give more insight in why the material behaves the way it does in the strain gauges that were designed. Another factor that was not investigated is the influence of the print orientation that is used. This could have large influences the mechanic properties of the design and needs to be investigated into more detail.

27

Appendix A

Model derivations

A.1 Beam displacement due to moment

The base of the whisker is modeled as a beam clamped at the two endpoints of the beam with cross-section A, Young’s modulus E, second moment of area I base and length L base . The beam is loaded in the center by a moment M 0 such that x = a where a < L base and L base − a = b. The left and right support have reaction forces F _A , F _B respectively and reaction moments M _A , M _B respectively. Since the horizontal forces are evaluated separately the moment and force balance in the vertical direction are taken. This generates 2 equations with 4 unknowns therefore the deflection integrals are taken in order to solve this problem. The force balance in the y-direction is taken:

F _A + F _B = 0 (A.1)

The moment balance taken around the center point gives the following:

M _A + M ₀ − F _A · a + F _B · b − M _B = 0 (A.2)

Now by making a cut along the beam at a position x < a the internal moment can be evaluated.

M (x) + M _A − F _A · x = 0 (A.3)

M (x) = F _A · x − M _A (A.4)

and for x > a.

M (x) − M B + F B · (L − x) = 0 (A.5)

M (x) = −F B · (L − x) + M _B (A.6)

now the internal moment can be expressed as M (x) = EI _base · w00(x) with the deflection w(x) in the vertical direction.

w00(x) = 1

EI _base (F A · x − M _A ) f or(x < a) (A.7) w00(x) = 1

EI _base (−F _B · (L − x) + M _B ) f or(x > a) (A.8)

28

APPENDIX A. MODEL DERIVATIONS 29

These equations are integrated twice to obtain the equation describing w(x) for the left and right part of the beam.

w _l (x) = 1 2EI base

( 1

3 F _A · x ³ − M _A · x ² + C ₁ · x + C ₂ ) f or(x < a) (A.9) w r (x) = 1

2EI _base (− 1

3 F _B · (L − x) ³ + M _B · x ² + C 3 · x + C ₄ ) f or(x > a) (A.10) Now applying the boundary conditions that the deflection as well as the rotation at x = 0 and x = L are zero the integration constants are determined. C 1 = C 2 = 0 and C 3 = −2M B L, C 4 = −M B L ² + 2M B L ² = M B L ² . Applying these into the equations gives:

w l (x) = 1 2EI _base ( 1

3 F A · x ³ − M _A · x ² ) f or(x < a) (A.11) w _r (x) = 1

2EI _base (− 1

3 F _B · (L − x) ³ + M _B · (L − x) ² ) f or(x > a) (A.12) In order to solve the equations two more conditions are needed, which are the continuity conditions at the center point of the beam, therefore setting w _l (L/2) = w r (L/2) and w _l 0(L/2) = w _r 0(L/2).

w  L 2



= 1

2EI _base 1

3 F _A ·  L 2

 3

− M _A ·  L 2

 2 !

= 1

2EI _base − 1

3 F _B ·  L 2

 3

+ M _B  L 2

 2 !

(A.13)

w0  L 2



= 1

2EI _base F _A ∗  L 2

 2

− 2M _A ∗  L 2

 !

= 1

2EI _base F _B ∗  L 2

 2

− 2M _B ∗  L 2

 !

(A.14) From this the reaction forces can be determined by combining these equations with A.1 and A.2.

F A = − 2M 0

3L (A.15)

F _B = 2M ₀

3L (A.16)

M A = − M 0

4 (A.17)

M _B = M ₀

4 (A.18)

Now by combining this result with equations A.11 and A.12 the following is obtained for the deflection of the base of the whisker:

w(x) = M ₀ EI base

 1

4L x ³ − 1 8 x ²



f or



x <  L 2



(A.19) w(x) = M 0

EI _base



− 1

4L (L − x) ³ + 1

8 (L − x) ²



f or



x >  L 2



(A.20) From this the angle under which the beam rotates at the centre can be obtained by taking ^dw _dx at the centre this yields:

θ = − M 0

EI _base L

16 (A.21)

In order to obtain the strain in the whisker from this deflection curve the following integral is taken to calculate the curve-length:

S = Z L

0

s

1 +  dw dx

 2

dx (A.22)

APPENDIX A. MODEL DERIVATIONS 30

This integration can in most cases only be done numerically therefore it is calculated by using the approximation:

√

1 + a ² ≈ 1 + ^a ₂

which is valid for small displacements. Now the integration becomes for the left-hand side of the beam:

S L = Z L/2

0

1 + 1 2

 M ₀ EI _base

 2  3

4L x ² − 1 4 x

 2

dx (A.23)

Which will be equal to the right-hand side of the beam due to the symmetry of the problem.

S _L = Z L/2

0

1 + 1 2

 M ₀ EI base

 2  9

16L ² x ⁴ − 3

8L x ³ + 1 16 x ²

 2

dx (A.24)

S L =

"

x + 1 2

 M 0

EI _base

 2  9

80L ² x ⁵ − 3

32L x ⁴ + 1 48 x ³

 # L/2 0

(A.25)

S L = L

2 + 1.302 × 10 ⁻⁴

 M 0

EI _base

 2

· L ³ (A.26)

From this the strain along the neutral axis can be determined in the following way:

ε = ∆L

L = S − ^L ₂

L 2

= 2.604 × 10 ⁻⁴

 M 0

EI _base

 2

· L ² (A.27)

A.2 Beam displacement due to horizontal force

In order to model the horizontal deflection of the whisker superposition is used. The beam is clamped at both ends and a horizontal force is applied at the center of the beam. Two separate equations for the deflection are determined one describing the deflection of a beam that is clamped at one end and a force applied in the center.

The second equation is found by calculating the deflection of the beam under a compressing load at the end of the free end of the beam, this compressive force is then set equal to the reaction force. The sum of forces in the x direction along the beam are:

F _ext + R _A − R _B = 0 (A.28)

The deflection of the beam under an applied load F _ext that is applied at the center of the beam is:

u(x) = F _ext x

EA f or

 x < L

2



(A.29)

= F ext L

2EA f or

 x > L

2



(A.30) Next the displacement of the beam under a compressive force that is set equal to R B due to the support on the right side is:

u(x) = − R _B x

EA f or (x < L) (A.31)

Now these contributions are added to obtain the final equation for the horizontal deflection.

u(x) = F _ext x

EA − R _B x

EA f or

 x < L

2



(A.32)

= F ext L

2EA − R B x

EA f or

 x > L

2



(A.33)

APPENDIX A. MODEL DERIVATIONS 31

Now applying the boundary conditions that the total deflection at u(L) = 0 it is found that the reaction force R B

is equal to ^F ₂

. The reaction force R _A can then be calculated and is equal to − ^F ₂

. The final equation describing the deflection in the horizontal direction is therefore:

u(x) = F _ext x

2EA f or

 x < L

2



(A.34)

= F _ext (L − x)

2EA f or

 x > L

2



(A.35) From this it follows that the displacement at the center of the whisker due to an external load is equal to:

u  L 2



= F _ext L

4EA (A.36)

From this it follows that the force related to a horizontal displacement can be seen as a translational spring with spring constant:

F _ext = k · u  L 2



= 4EA

L · u (A.37)

Therefore the strain on the left side of the beam is:

ε L = ∆L

L =

F

L 4EA L 2

= F ext

2EA (A.38)

and on the right hand side it is equal but opposite since the beam is compressed in stead of stretched:

ε R = − ^F _4EA

^L

L 2

= − F ext

2EA (A.39)

A.3 Whisker

The whisker is modeled as a beam suspended on a torsional spring and a translational spring. The rotation of the torsional spring has been related to the applied moment on the base of the whisker. The rotation of the base as a result of the moment is taken as torsional spring constant.

θ = − M _spring EI base

L _base

16 (A.40)

and horizontal displacement due to applied load F ext is taken as translational spring constant:

u = F _spring L _base 4EA base

(A.41) The sum of moments in the beam is taken around the point of rotation of the spring, and the sum of forces in the x direction is taken.

X M = M _spring − F _ext · s = 0 (A.42)

X F = F ext − F _spring = 0 (A.43)

The internal moments and forces are taken by making a cut in the beam. The moments are taken around the point of rotation in the spring:

M (y) + M spring − V (y) · y = 0 (A.44)

F spring − V (y) = 0 (A.45)

APPENDIX A. MODEL DERIVATIONS 32

Combining the equations for the internal moment at point y is found

M (y) = −M _spring + F _spring · y (A.46)

Integrating twice gives the deflection curve of the whisker:

EI _whisker · w _whisker (y) = M spring · y ²

2 + F spring · y ³

6 + C 1 · y + C ₂ (A.47)

The boundary conditions used are w(0) = u and w ⁰ (0) = θ, according to small angle approximations. Applying these boundary conditions give C 1 = θ and C 2 = u.

EI _whisker · w(y) = M spring · y ²

2 + F spring · y ³

6 + θ · y + u (A.48)

Now applying equation (1), θ can be eliminated and the final deflection of the whisker is as follows:

EI _whisker · w(y) = M _spring · y ²

2 + F _ext · y ³

6 − M _spring EI _base

L _base

16 · y + F _spring L _base

4EA _base (A.49)

Now by applying equation (2) the Moment of the spring can be expressed in the applied force M _spring = F _ext · s and the force of the translational spring can be expressed as F spring = F _ext .

EI _whisker · w(y) = F _ext · s · y ²

2 + F _ext · y ³

6 − F _ext · s EI _base

L _base

16 · y + F _ext · L _base

4EA _base (A.50)

The equations above only hold for y ≤ s. For y > s no internal moments are bending the beam therefore w00 = 0. The beam is continuous in y = s, from this follows that the displacement of the beam for y > s is:

EI _whisker · w(y) =  3F _ext · s ²

2 − F ext · sL _base 16EI _base



· y − 5F ext · s ³

6 + F ext · L _base

4EA _base (A.51)

such that w(s) and w0(s) are continuous.

Appendix B

3D prints

Figure B.1: Sample used for electrical characterization.

33

APPENDIX B. 3D PRINTS 34

Figure B.2: 3D printed sensor, numbers 1 and 2 indicate the two strain gauges.

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