sensors Article

A Novel Low-Cost, Large Curvature Bend Sensor Based on a Bowden-Cable Useok Jeong and Kyu-Jin Cho * School of Mechanical and Aerospace Engineering/IAMD, Seoul National University, Seoul 08826, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-880-1663 Academic Editor: Vittorio M. N. Passaro Received: 12 May 2016; Accepted: 20 June 2016; Published: 24 June 2016

Abstract: Bend sensors have been developed based on conductive ink, optical fiber, and electronic textiles. Each type has advantages and disadvantages in terms of performance, ease of use, and cost. This study proposes a new and low-cost bend sensor that can measure a wide range of accumulated bend angles with large curvatures. This bend sensor utilizes a Bowden-cable, which consists of a coil sheath and an inner wire. Displacement changes of the Bowden-cable’s inner wire, when the shape of the sheath changes, have been considered to be a position error in previous studies. However, this study takes advantage of this position error to detect the bend angle of the sheath. The bend angle of the sensor can be calculated from the displacement measurement of the sensing wire using a Hall-effect sensor or a potentiometer. Simulations and experiments have shown that the accumulated bend angle of the sensor is linearly related to the sensor signal, with an R-square value up to 0.9969 and a root mean square error of 2% of the full sensing range. The proposed sensor is not affected by a bend curvature of up to 80.0 m´1 , unlike previous bend sensors. The proposed sensor is expected to be useful for various applications, including motion capture devices, wearable robots, surgical devices, or generally any device that requires an affordable and low-cost bend sensor. Keywords: bend sensor; flexible sensor; shape sensor; bend measurement; low-cost sensor; Bowden-cable; cable conduit

1. Introduction Bend sensors have been widely used in various applications, from simple angle measurements to flexion and extension measurements of the body joints [1–11] and shape-sensing in mechanical structures [12–14]. Flex sensors and optical fiber sensors have proved useful in these applications. A flex sensor is a thin film printed with conductive ink whose resistance changes depending on the bend curvature and angle of the sensor [1,15]. It is the most affordable bend sensor on the market, and several companies sell off-the-shelf versions of various dimensions [16,17]. Some research [12–14,18] and do-it-yourself projects [2] use flex sensors because they are cheap, easy to integrate, and do not require complicated signal processing. Motion capture gloves measuring the posture of the fingers using flex sensors have been developed [1,3–5] and commercialized [6] because they can be easily attached to fabric without adding volume. However, flex sensors tend to have large nonlinearities, including time-dependent creep and a nonlinear relationship between curvature and resistance change [3,4,19,20]. The nonlinearity of flex sensors prevents them from measuring large curvatures and the absolute angles of objects. They are also limited in length (maximum, 95.25 mm for the Spectra Symbol flex sensor) and so cannot measure a bend change for a large area [4]. Optical fiber-based sensors detect bending by measuring changes of intensity, phase, wavelength, or polarization of the light inside a bending optical fiber [3,4]. The elaborate and intensive techniques used to manufacture the optical fiber and sophisticated signal processing of the light spectrum enable Sensors 2016, 16, 961; doi:10.3390/s16070961

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highly sensitive measurements of fiber shape changes [21]. Also, these sensors take advantage of Sensors 2016, 16, 961 2 of 20 the physical properties of optical fibers, including small diameter and flexibility. Optical fiber-based sensors can be divided into two types depending on whether they measure the spectrum or the overall intensive techniques used to manufacture the optical fiber and sophisticated signal processing of the intensity of light [22]. Grating-based sensors including fiber Bragg gratings (FBGs) and long-period light spectrum enable highly sensitive measurements of fiber shape changes [21]. Also, these sensors fiber take gratings (LPFGs) measure the intensityofvariation of the fiber’s resonant dip wavelength [23–26]. advantage of the physical properties optical fibers, including small diameter and flexibility. An expensive device—an interrogator—is used measure and analyze spectrum of the light, Optical fiber-based sensors can be divided intototwo types depending on the whether they measure the and complicated signal processing is required to convert the spectrum data into shape information [3,4,21]. spectrum or the overall intensity of light [22]. Grating-based sensors including fiber Bragg gratings It is possible butlong-period difficult to fiber minimize the(LPFGs) effect ofmeasure temperature and nonlinearities viafiber’s special fabrication (FBGs) and gratings the intensity variation of the resonant dip wavelength [23–26]. fiber-based An expensivesensors device—an used to measure and analyze the as techniques [24,27]. Optical thatinterrogator—is measure light intensity were proposed [7,8,22] spectrum of the light, and complicated signalsensors. processing is required to intensity convert the spectrum a low-cost alternative to spectrum-analyzing These measure changes in data the light into information [3,4,21]. It is possible but difficult toas minimize effect of temperature source ofshape light-emitting diodes (LEDs) with photodiodes the fiberthebends. However, thisand type of nonlinearities via special fabrication techniques [24,27]. Optical fiber-based sensors that measure light sensor is used only to measure small curvatures [22]. intensity were proposed [7,8,22] as a low-cost alternative to spectrum-analyzing sensors. These Electronic textiles (e-textiles) are an emerging technology that can be used for bend sensing [9–11]. measure intensity changes in the light source of light-emitting diodes (LEDs) with photodiodes as The fabric or thread used to make these itsto electric properties depending the fiber bends. However, this type oftextiles sensor ischanges used only measure small curvatures [22].on the stretch of the material. They are advantageous for wearable devices because the sensing material itself can Electronic textiles (e-textiles) are an emerging technology that can be used for bend sensing [9–11]. be anThe integral of theused device. However, textiles nonlinearities [11]onand fabricpart or thread to make these electronic textiles changes its show electrichigh properties depending the have stretch of the material. Theyfor are advantageous for wearable devices because the sensing material limited application other than wearable devices. itself paper can be proposes an integralapart of the device. However, electronic textiles showthat highoperates nonlinearities [11] This novel bend sensor based on the Bowden-cable on principles and have limited application other than for wearable devices. completely different from those at work in flex sensors and optical fiber-based sensors. A Bowden-cable This paper proposes athat novelconsists bend sensor onwire the Bowden-cable operates principlesforce is a wire-driven mechanism of anbased inner and an outerthat sheath thatontransmits completely different from those at work in flex sensors and optical fiber-based sensors. A Bowden-cable through the inner wire. It provides high degrees of freedom in the routing path because the is a wire-driven mechanism that consists of an inner wire and an outer sheath that transmits force transmission consists of flexible components. It has been used in bicycle brake systems for decades and through the inner wire. It provides high degrees of freedom in the routing path because the in many robotic applications, fromcomponents. exoskeletonItrobots [28–31] tobicycle surgical devices [32–35]. One issue transmission consists of flexible has been used in brake systems for decades with and Bowden-cables is property changes [36], especially error of the inner in many robotic applications, from exoskeleton robotsrelated [28–31]to to position surgical devices [32–35]. One wire originating from the varying shape of the coil sheath [32–35]. related to position error of the inner issue with Bowden-cables is property changes [36], especially wire originating from the varying of the coil in sheath This study takes advantage of shape position error the [32–35]. Bowden-cable to detect the bend angle of This study takes advantage of position error in the Bowden-cable to detect the bend angle of theof the the sheath. When one end of the inner wire is fixed to the sheath, the displacement change sheath. When one end of the inner wire is fixed to the sheath, the displacement change of the inner the inner wire can be measured on the other end of the sheath. There is a linear relationship between wire can be measured on the other end of the sheath. There is a linear relationship between displacement change of the inner wire and the accumulated bend angle of the sheath such the that the displacement change of the inner wire and the accumulated bend angle of the sheath such that the sheath can be used as the sensor body of a bend sensor. The design and material of the proposed sensor sheath can be used as the sensor body of a bend sensor. The design and material of the proposed is simple, low cost, and does not require any special fabrication techniques or complicated signal sensor is simple, low cost, and does not require any special fabrication techniques or complicated processing. Based onBased theseonattributes, we used a Bowden-cable low-cost, easy-to-use signal processing. these attributes, we used a Bowden-cableto to develop develop a alow-cost, easy-to-use bendbend sensor (Figure 1) capable of sensing over a wide range of accumulative bend angle with a large sensor (Figure 1) capable of sensing over a wide range of accumulative bend angle with a large ˝ bend angle and 80.0 m´−11 curvature for the first prototype). curvature (up to 450 curvature (up to 450° bend angle and 80.0 m curvature for the first prototype).

Figure 1. A Bowden-cable–based bend sensor consists of an inner wire and an outer sheath.

Figure 1. A Bowden-cable–based bend sensor consists of an inner wire and an outer sheath.

This paper will discuss the basic concept, modeling, and preliminary evaluation of the proposed

This paper will discuss the basic modeling, and preliminary evaluation of theof proposed sensor. The concept and basic designconcept, of the sensor will be described in Section 2. The modeling the proposed bend sensor’s bend angle, bend curvature, sensor output willin beSection discussed Section 3. sensor. The concept and basic design of the sensorand will be described 2.inThe modeling of the proposed bend sensor’s bend angle, bend curvature, and sensor output will be discussed in

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2016, 16, 961 3 of 20 SectionSensors 3. Characteristics and considerations of the proposed sensor will be covered in Section 4. Sensors 2016, 16,5961 3 of 20 Finally, Sections and 6 will describe development of a prototype and our preliminary experimental Characteristics and considerations of the proposed sensor will be covered in Section 4. Finally, Characteristics and considerations of the proposed sensor will be covered in Section 4. Finally, Sections 6 will describe development ofof a prototype and our preliminary experimental results results for it. 5 and Sections 5 and 6 will describe development a prototype and our preliminary experimental results

Characteristics and considerations of the proposed sensor will be covered in Section 4. Finally, forfor it.it. Sections 5 and 6 will describe development of a prototype and our preliminary experimental results 2. Concept and Design for it. 2. 2. Concept and Design Concept and Design

2.1. Principle

2. Concept and Design 2.1. Principle

2.1.working Principle principle of the proposed bend sensor starts from the idea that the displacement The The working principle ofof the proposed sensor from the that the displacement ofof 2.1.line Principle The working principle the proposed bend sensor starts from the idea that the displacement of the passing through the center of abend helical coilstarts increases asidea the coil bends, as shown in the line passing through the center of a helical coil increases as the coil bends, as shown inin line passing through the center of a helical coil increases as the coil bends, as shown Figuresthe 2 The and 3. This occurs because each coil pitch remains in contact inside the curvature but principle ofbecause the proposed bend sensor startsin from the idea thatthe thecurvature displacement of Figures 2working and 3. 3. This occurs each coil pitch remains contact inside but the Figures 2 and This occurs because each coil pitch remains in contact inside theshown curvature but the gap between each coil pitch on the outside of the curvature increases, as in Figure the line passing through the center of a helical coil increases as the coil bends, as shown inthe 3b. gap between each coil pitch onon the outside ofof the curvature increases, asas shown inin Figure 3b.3b. This gap between each coil pitch the outside the curvature increases, shown Figure This Figures 2 and 3. This occurs because each coil pitch remains contact inside the but the This results in aa displacement change thecenter center line (this isinlinearly related to curvature the bend angle of the results in displacement change ofof the line (this is is linearly related toto the bend angle ofof the results in a each displacement change of the center line (this linearly related the bend angle the gap between coil pitch on the outside of the curvature increases, as shown in Figure 3b. This helical coil, which will be addressed in more detail in the next section). The displacement change can helical coil, which will bebe addressed inin more detail inin the next section). The displacement change can helical coil, which will addressed more detail the next section). The displacement change can results in a displacement change of the center line (this is linearly related to the bend angle of the be measured with a wire that is placed through the center of the coil and fixed at one end of the coil bebe measured with a wire that is is placed through the center of the coil and fixed atat one end ofof the coil measured with a wire that placed through center the coil and fixed one end the coil coil, which will be addressed in more detailthe in the nextofsection). The displacement change can and a displacement sensor such as a potentiometer or Hall-effect sensor. and ahelical displacement sensor such as a potentiometer or a Hall-effect sensor. and a displacement sensor such as a potentiometer or a Hall-effect sensor. be measured with a wire that is placed through the center of the coil and fixed at one end of the coil and a displacement sensor such as a potentiometer or a Hall-effect sensor.

Figure 2. Structure of the bending partofofthe theproposed proposed bend sensor, which consists of aof helical coil, coil, Figure 2. Structure of the part consists a helical Figure 2. Structure ofbending the bending part of the proposedbend bendsensor, sensor, which which consists of a helical coil, ® lining and an end-cap. ® lining an inner wire, a Teflon ® an inner wire, a Teflon and an end-cap. an inner wire, a Teflon lining and an end-cap. Figure 2. Structure of the bending part of the proposed bend sensor, which consists of a helical coil, an inner wire, a Teflon® lining and an end-cap.

(a)(a)

(b)(b)

Figure 3. Basic working(a) principle of the proposed bend sensor using (a) When the (b)a Bowden-cable. Figure 3. Basic working principle of the proposed bend sensor using a Bowden-cable. (a) When the Figure 3. Basic working principle of theisproposed bend sensor using a Bowden-cable. (a) When the sensor is straight; (b) When the sensor bent. sensor is straight; (b) When the sensor is bent. Figure 3. Basic(b) working of the proposed bend sensor using a Bowden-cable. (a) When the sensor is straight; Whenprinciple the sensor is bent. sensor is straight; (b) When the sensor is bent. This idea can bebe extended toto the whole body ofof the sheath: Central displacement changes ofof the This idea can extended the whole body the sheath: Central displacement changes the sensor are related to the accumulated bend angle of the sensor (depicted in Figure 4). sensor are related to the accumulated bend angle of the sensor (depicted in Figure 4). This idea can be extended to the whole body of the sheath: Central displacement changes This idea can be extended to the whole body of the sheath: Central displacement changes of theof the sensor are related to the accumulated sensor(depicted (depicted Figure sensor are related to the accumulatedbend bendangle angle of of the the sensor in in Figure 4). 4).

Figure 4. Sensor bending with multiple bending points. The displacement change of the center line is Figure 4. Sensor bending with multiple bending points. The displacement change of the center line is proportional to the accumulated bend angle throughout the helical coil, = + + + . proportional to the accumulated bend angle throughout the helical coil, = + + + . Figure 4. Sensor bending with multiple bending points. The displacement change of the center line isline is Figure 4. Sensor bending with multiple bending points. The displacement change of the center proportional to the accumulated bend angle throughout the helical coil, = + + + .

proportional to the accumulated bend angle throughout the helical coil, φ = φ1 + φ2 + φ3 + φ4 .

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2.2. Maintaining Internal Tension 2.2. Maintaining Internal Tension The tension of the sensing wire inside the helical coil should be maintained over a certain range to prevent it from slack, prevent measurement of thea central The tension of becoming the sensing wirewhich inside would the helical coil accurate should be maintained over certain displacement change. There are two possible to prevent keep theaccurate sensingmeasurement wire taut. One use a range to prevent it from becoming slack, whichways would of is thetocentral passive element such as a spring to provide an elastic force to the sensing wire. Figure 5 shows displacement change. There are two possible ways to keep the sensing wire taut. One is to use examples this method a linear and a torsional spring. free end ofwire. the sensing can a passive of element such asusing a spring to provide an elastic force The to the sensing Figure wire 5 shows be connected tomethod a linearusing spring (Figure or wound around a spool a torsional examples of this a linear and 5a) a torsional spring. The free end ofwith the sensing wire spring can be (Figure 5b).toThe spring should have5a) some initial around stretch atospool keep with the sensing wire taut, (Figure even for a connected a linear spring (Figure or wound a torsional spring 5b). straight-shaped sensor. increases as the bends, displacement of the The spring should have Inner some tension initial stretch to keep thesensor sensing wire and taut,central even for a straight-shaped coil increases as well. increases In this case, the sensor operating range ofcentral the spring should cover thecoil travel range of sensor. Inner tension as the bends, and displacement of the increases as the which is determined the bendshould sensingcover rangethe and the range designofparameters the well.sensing In this wire, case, the operating range ofby the spring travel the sensingofwire, sensor. other method is bend to maintain internal tension using an actuator such as sensor. an electric which isThe determined by the sensing range and the design parameters of the Themotor other or piezoismotor. This method can provide constant tension sensing method to maintain internal tension using either an actuator suchorasvariable an electric motorto or the piezo motor.wire, This which enables the sensor have variable bending stiffness orsensing optimizes hysteresis properties. In all method can provide eithertoconstant or variable tension to the wire, which enables the sensor cases, should be maintained within a range that induces onlyInnegligible elongation of the to havetension variable bending stiffness or optimizes hysteresis properties. all cases, tension should be wire. The wire stretch should considered calculating the bend elongation of the wire maintained within a range thatbeinduces only when negligible elongation of theangle wire.ifThe wire stretch should is than negligible. bemore considered when calculating the bend angle if elongation of the wire is more than negligible.

(a)

(b)

Figure ways of of maintaining maintaining internal Figure 5. 5. Two Two ways internal tension tension and and measuring measuring the the displacement displacement change change of of the the sensing wire. (a) Using a linear spring and a linear transducer; (b) Using a spool with a torsional sensing wire. (a) Using a linear spring and a linear transducer; (b) Using a spool with a torsional spring spring and a rotational transducer. and a rotational transducer.

2.3. Displacement Change Measurement 2.3. Displacement Change Measurement One end of the sensing wire is attached to the end of the helical coil, and the other free end of One end of the sensing wire is attached to the end of the helical coil, and the other free end of the sensing wire is connected to a displacement sensing device to measure the central displacement the sensing wire is connected to a displacement sensing device to measure the central displacement change of the helical coil. There are two possible ways to measure the position of the free end of the change of the helical coil. There are two possible ways to measure the position of the free end of the sensing wire. The first method merely measures the linear position of the free end of the sensing wire sensing wire. The first method merely measures the linear position of the free end of the sensing using a linear position transducer (Figure 5a). Available linear position transducers include wire using a linear position transducer (Figure 5a). Available linear position transducers include potentiometers, linear variable differential transformers (LVDTs), and Hall-effect sensors. The second potentiometers, linear variable differential transformers (LVDTs), and Hall-effect sensors. The second method measures the rotation angle of a spool wound by a sensing wire (Figure 5b). In both cases, method measures the rotation angle of a spool wound by a sensing wire (Figure 5b). In both cases, the the operating range and resolution of the displacement transducer influence the performance of the operating range and resolution of the displacement transducer influence the performance of the bend bend sensor. For example, it is preferable to choose a linear potentiometer that has a travel length as sensor. For example, it is preferable to choose a linear potentiometer that has a travel length as close as close as possible to the maximum range of the central displacement change of the sensor. possible to the maximum range of the central displacement change of the sensor.

within an allowable curvature range. 3.1. Basic Model Figure 6 delivers the basic modeling concept for the relationship between bend angle and the of 20 change of the coil. Each coil pitch is represented as a rectangle with coil5wire diameter, b, and outer diameter, d. The vertexes of the rectangles on the inside of the bend keep in contact as the helical coil bends, and the angle between each coil pitch uniformly increases. Total 3. Modeling bend angle, ϕ, and length of the sensor, L, can be represented by the number of coil pitches, n: This section provides a sensor model that describes the relationship between bend angle and n curvature of the sensor, the displacement changes of the sensing wire, and sensor output. The resultant   i (1) model shows that the bend angle of the sensor and i 1 the sensor output are linearly related within an allowable curvature range.

Sensors 16, 961 central2016, displacement



L

3.1. Basic Model

n

b

(2)

i 1

Figure 6 delivers the basic modeling concept for the relationship between bend angle and the The small changechange in the of central displacement between two pitchesasisarepresented as xcoil andwire can central displacement the coil. Each coil pitch is represented rectangle with be calculated using a cosine rule:d. The vertexes of the rectangles on the inside of the bend keep in diameter, b, and outer diameter, contact as the helical coil bends, and the angle between each coil pitch uniformly increases. Total bend d d b coil pitches, n:  number angle, φ, and length of the sensor, x  L, can 2 be 1 represented cos    by 2the  (3)  1  cos of

2

2



L



n ÿ αi (1) Then the central displacement changes φ of“the helical coil along the sensor, ∆l, can be derived from the summation of the displacement changes:i“1 n

n

l 

x i 1

i

dL ÿ   b   L “ 2 b1  cos     2 b i “1   L 

(2) (4)

Figure 6. 6. Schematic Schematic of of aa bent the displacement change of of the the line line passing passing through through Figure bent helical helical coil coil showing showing the displacement change the center of the coil. Each coil pitch uniformly bends by α, resulting in a central displacement the center of the coil. Each coil pitch uniformly bends by α, resulting in a central displacement increment of of x. x. increment

Figure 7 illustrates the simulation result of the derived Equation (3), showing the relationship The small change in the central displacement between two pitches is represented as x and can be between the central displacement change of the helical coil and the bend angle of the sensor. The calculated using a cosine rule: simulation result indicates that it is possible to simplify the mathematical model. d ˆ ˙ da d b x“ 2 p1 ´ cos αq “ 2 1 ´ cos φ (3) 2 2 L Then the central displacement changes of the helical coil along the sensor, ∆l, can be derived from the summation of the displacement changes: ∆l “

n ÿ i “1

dL xi “ 2b

d " ˆ ˙* b 2 1 ´ cos φ L

(4)

Figure 7 illustrates the simulation result of the derived Equation (3), showing the relationship between the central displacement change of the helical coil and the bend angle of the sensor. The simulation result indicates that it is possible to simplify the mathematical model.

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Figure 7. Simulation result of derived Equation (3) when coil wire diameter b = 0.5 mm. It shows the Figure 7. Simulation result of derived Equation (3) when coil wire diameter b = 0.5 mm. It shows linear relation between bend angle and (3) the linear relation between bend angle andthe thecentral centraldisplacement displacementchange, change, although although Equation Equation (3) contains nonlinear functions. The curves with different length, L, are overlapped with the curves for contains nonlinear functions. The curves with different length, L, are overlapped with the curves for the same same coil coil diameter, diameter, d. d. the

3.2. Linearity 3.2. Linearity To further analyze the linearity of the bend angle and the central displacement change, To further analyze the linearity of the bend angle and the central displacement change, Equation (3) Equation (3) can be expressed as the curvature of the sensor bend, κ: can be expressed as the curvature of the sensor bend, κ:

 φ

κ“ ∆l l “ 



LL

dd a 22t11´cos cospbκqu A ¨φ    b ¨φ “ A pκq 2bκ 2 b a 2 t1 ´ cos pbκqu d A pκq “ ¨ 2d 2 1 bκ  cos  b 

(5) (5) (6) (6)

(7) (7) A the  displacement change ratio in regard to the bend  central Equations (6) and (7) indicate that 2 b angle, A(κ), is a function of the design parameters (helical coil diameter, d, and coil wire diameter, b) Equations (6) and (7) thatisthe central displacement change ratio in regard to the bend and the bend curvature, κ. indicate Here, there a possibility that the bend curvature could change during angle, A(κ), is a function of the design parameters (helical coil diameter, d, and coil wire diameter, b) sensor operation and affect the measurement of the bend angle. and the bend curvature, κ. Here, there is a possibility that the the normalized bend curvature could change during Equation (7) can be non-dimensionalized by defining length change, Λ (length sensor operation and affect the measurement of the bend angle. change, ∆l, divided by the radius of the coil, d/2) and the normalized curvature, K (curvature, κ Equation (7) coil can wire be non-dimensionalized by defining the normalized length change, Λ (length multiplied by the diameter, b): change, ∆l, divided by the radius of the coil, d/2) and the normalized curvature, Κ (curvature, κ a multiplied by the coil wire diameter, b):2 t1 ´ cos pKqu (8) ¨ φ “ X pKq ¨ φ Λ“ K 2 1  cos    (8)  X(K),    with   aTaylor    series (see Appendix A for The normalized change rate, can be expressed  more detail): 8 ÿ The normalized change rate, Χ(Κ), with a Taylor series (see Appendix A for 1can 1 be expressed 1 X pKq “ K2n “ 1 ` O pKq2 « 1 (9) n more detail): p n q 16 n! p3{2q n “0

1to 1 1if the approximation 1 22 Equation (9) can be 1 enough.  approximated  2 n  1error  OO(K)       issmall (9) n n  16 n !  3 2change Figures 8 and 9 show how the rate is affected by varying curvature with n  0displacement  different design parameters. The displacement change rate is linearly proportional to coil diameter, d, 2 is small enough. Equation (9) can (7), be approximated to 1 if the approximation error as shown in Equation and it has a nonlinear relation to curvature andO(Κ) coil wire diameter, b. Figure 8 Figures and 9 show how therate, displacement rate when is affected bywire varying curvature with shows that the8 displacement change A, does notchange vary much the coil diameter, b, is small. different designthe parameters. The displacement change rate is linearly proportional coil diameter, This is because helical coil is approximated to the continuum structure with anto incompressible d, as shown in Equation (7), and it has a nonlinear relation to curvature and coil wire diameter, b. wall inside the curve if the coil wire diameter is small enough relative to the coil diameter. In this Figure 8 shows that the displacement change rate, A, does not vary much when the coil wire diameter, case, the displacement change rate can be considered constant regardless of the bend curvature along b, is small. This is because the helical coil is approximated to the continuum structure with an incompressible wall inside the curve if the coil wire diameter is small enough relative to the coil 



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diameter. diameter. In In this this case, case, the the displacement displacement change change rate rate can can be be considered considered constant constant regardless regardless of of the the bend bend curvature curvature along along the the sensor. sensor. Figure Figure 99 shows shows that that even even in in an an extreme extreme curvature curvature condition, condition, Κ Κ == 0.5 0.5 (e.g., (e.g., −1 1, −1 κ = 1000 m , b = 0.50 mm), the change rate only varies by 1.0%. The ratio of the coil diameter and the κ = sensor. 1000 m Figure , b = 0.50 mm), that the change only varies by 1.0%.condition, The ratio K of =the diameter and the the 9 shows even inrate an extreme curvature 0.5coil (e.g., κ = 1000 m´ diameter illustrated here is the proper of practical design parameters wire diameter illustrated here is within within the1.0%. proper range ofthe practical design and parameters because bwire = 0.50 mm), the change rate only varies by Therange ratio of coil diameter the wirebecause diameteraa high coil wire diameter increases the stiffness of the sensor, which is not desirable. high coil wire therange stiffness of the sensor, which is not desirable. illustrated herediameter is withinincreases the proper of practical design parameters because a high coil wire diameter increases the stiffness of the sensor, which is not desirable.

Figure Simulation result showing the rate varies with the Figure result showing how how the displacement changechange rate varies the bend Figure8.8. 8.Simulation Simulation result showing how the displacement displacement change ratewith varies with curvature, the bend bend curvature, κ, coil diameter, when == 33 mm. displacement rate be κ, and the coil wire diameter, b, when d = 3 b, mm. Thedddisplacement rate canchange be assumed to be curvature, κ, and and the the coil wire wire diameter, b, when mm. The Thechange displacement change rate can can be assumed to be constant within a certain range of design parameters and curvatures. constant a certain within range of design range parameters and parameters curvatures. and curvatures. assumedwithin to be constant a certain of design

Figure Simulation result of the approximation error depending on the normalized curvature. Large Figure9.9. 9.Simulation Simulationresult resultof ofthe theapproximation approximationerror errordepending dependingon onthe thenormalized normalizedcurvature. curvature.Large Large Figure variation of the normalized curvature during sensor operation can lead to measurement error, but variationofofthe thenormalized normalizedcurvature curvature during sensor operation to measurement error, but variation during sensor operation cancan leadlead to measurement error, but not −1 not by much—the change rate varies within 1.0% with extreme conditions when Κ = 0.5 (κ = 1000 m ´1−1 notmuch—the by much—the change varies within 1.0% with extreme conditions when Κ 0.5 = 0.5 = 1000 by change raterate varies within 1.0% with extreme conditions when K= (κ (κ = 1000 mm when 0.50 mm). This can be approximated constant value during actual conditions of when bbb===0.50 0.50mm). mm).This Thiscan canbe beapproximated approximatedtoto toaaaconstant constantvalue valueduring during actual actual conditions conditions of of when sensor operation. sensor operation. sensor operation.

The The effective effective displacement displacement change change rate, rate, A Aeft eft,, which which can can be be used used as as aa sensor sensor gain, gain, can can be be derived derived The effective displacement change rate, Aeft , which can be used as a sensor gain, can be derived with the approximation in Equation (9): with the approximation in Equation (9): with the approximation in Equation (9):

d Aeft  d      d eft Ae f t “ 2¨ X pKq «

d 2

(10) (10) (10)

2 2 Equation Equation (10) (10) indicates indicates that that geometric geometric model model in in Figure Figure 66 can can be be simply simply approximated approximated with with aa Equation (10) indicates that× geometric model in Figure 6 can be simply approximated with perimeter calculation ∆l = d/2 ϕ with the small normalized curvature. It can be used for perimeter calculation ∆l = d/2 × ϕ with the small normalized curvature. It can be used for the the a perimeter calculation ∆l = d/2 ˆ φ with the small normalized curvature. It can be used for the

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model-based sensor sensor gain gain before before precise precise calibration calibration of of the the bend bend sensor. sensor. The The accuracy accuracy of of the the derived derived model-based model will be compared with experimental results in Section 6. model will be compared with experimental results in Section 6. 3.3. Advanced Advanced Model Model 3.3. The gap gap between between the the sensing sensing wire, wire, the the Teflon Teflon®® tube, tube, and and the the helical helical coil coil together together with with the the The cross-sectional shape shape of of the the coil coil wire wire should should be be considered considered for for aa more more accurate accurate description description of of the the cross-sectional relationship between the bend angle of the sensor and the central displacement change. relationship between the bend angle of the sensor and the central displacement change. 3.3.1. 3.3.1. Gap Gap between between Parts Parts Gaps Gaps between between the the sensing sensing wire, wire, the the Teflon Teflon tube, tube, and and the the helical helical coil coil are are necessary necessary to to prevent prevent excessive excessive friction friction between between these these parts parts and and to to ensure ensure that that the the sensing sensing wire wire slides slides well well with with the the angle angle change change of of the the sensor. sensor. Thus Thus the the sensing sensing wire wire does does not not retain retain its its central central position position inside inside the the coil, coil, and and itit contacts contacts the the inner inner wall wall as as the the helical helical coil coil bends bends when when the the sensing sensing wire wire isistensioned. tensioned. As As aa result, result, the the effective effective distance distance between betweenthe thesensing sensingwire wireand andthe theouter outerwall wallof ofthe thehelical helicalcoil, coil,ddeft1 eft1,, is is smaller smaller than than the the radius radius of of the the helical helical coil. coil. This This is is depicted depicted in in Figure Figure10 10and anddescribed describedin inthe thefollowing followingequation: equation:

1 btube`1 dwire d w ire de df t1eft“ `btube 1 b b 22

(11) (11)

whereb,b,bbtube tube, and dwire where arethe thewall wall thickness thickness of of the the helical helical coil, coil, the the wall wall thickness thickness of the Teflon tube, wireare and the the diameter diameter of ofthe thesensing sensingwire, wire,respectively. respectively. and

Figure 10. The sensing wire does not maintain its position in the center of the coil because there are Figure 10. The sensing wire does not maintain its position in the center of the coil because there are gaps between the sensor’s parts. The effective distance between the center of the sensing wire and the gaps between the sensor’s parts. The effective distance between the center of the sensing wire and the outside wall of the helical coil, deft , should be considered when calculating the sensor gain derived in outside wall of the helical coil, deft, should be considered when calculating the sensor gain derived in Section 3.1. Section 3.1.

3.3.2. Shape of the Cross Section 3.3.2. Shape of the Cross Section In the helical coil model in Figure 6, the coil wire was assumed to have a square cross-sectional In the helical coil model in Figure 6, the coil wire was assumed to have a square cross-sectional shape for purposes of simplification. However, the actual distance between the contact point of each coil shape for purposes of simplification. However, the actual distance between the contact point of each pitch and the sensing wire decreases if the cross section of the coil wire is circular in shape. Figure 11b coil pitch and the sensing wire decreases if the cross section of the coil wire is circular in shape. shows a close up of the contact point between two pitches with rounded ends. The modified distance, Figure 11b shows a close up of the contact point between two pitches with rounded ends. The deft2 , can be approximated to a constant value if the angle between two pitches, α, is small enough: modified distance, deft2, can be approximated to a constant value if the angle between two pitches, α, is small enough: d b lim de f t2 “ ´ (12) 2d 2 b α Ñ0 lim d eft 2   (12)  0

2

2

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(a)

(b)

Figure 11. The model should be modified for a coil wire with a circular cross-sectional area for more

Figure 11. The model should area for more (a) be modified for a coil wire with a circular cross-sectional (b) accurate modeling. (a) Model modification for a circular cross section; (b) Close view of the contact accurate modeling. (a) Model modification for a circular cross section; (b) Close view of the contact point11. between the two coilbe pitches. Figure The model should modified for a coil wire with a circular cross-sectional area for more point between the two coil pitches. accurate modeling. (a) Model modification for a circular cross section; (b) Close view of the contact

3.4. Sensor Gain the two coil pitches. point between

3.4. Sensor The Gainresultant sensor gain, which transforms the measured position of the sensing wire to the 3.4. Sensor Gain bend angle ofsensor the sensor, be derived from the described in Sections The resultant gain,can which transforms theconsiderations measured position of the sensing 3.1–3.3: wire to the bend The resultant sensor gain, which transforms the measured position of the sensing wire angle of the sensor, can be derived from the described in Sections 3.1–3.3: to the  considerations 1 bend angle of the sensor, can be derived fromthe considerations described in Sections 3.1–3.3: d w ire  l  b (13) φ  2  b11tube  2  ¯ “ (13)  ´  d ∆l  l  b2b ` btube `d wwire  (13) ire 2  btube    2 r 2   φ   b r d (14) ¯  (14) “ ´   btube  w ire  r dwire2  b 2 ∆θ  2 ` btube ` 2 d ire   b (14)  btubewith  aw linear Equations (13) and (14) describe a sensor  2 gain  position transducer and a rotational Equations (13) and (14) describe a sensor  gain with a2linear  position transducer and a rotational angle transducer, respectively. The operating range of each position transducer should be determined angle transducer, respectively. The operating range of each position transducerand should be determined considering of motion bend sensor increase the resolution theasensitivity Equations the (13)range and (14) describeofa the sensor gain withtoa linear position transducer and rotational of considering the range of motion of the bend sensor to increase the resolution and the sensitivity of the thetransducer, bend sensor. A small-diameter spool increases of spool rotationshould per sensor bend, which angle respectively. The operating range of the eachratio position transducer be determined bend sensor. A small-diameter spool increases the ratio of spool rotation per sensor bend, which will increase and sensitivity but sensor decrease sensing considering the resolution range of motion of the bend to the increase therange. resolution and the sensitivity of will increase resolution and sensitivity but decrease the the bend sensor. A small-diameter spool increases thesensing ratio of range. spool rotation per sensor bend, which Axial Force Robustness will3.5. increase resolution and sensitivity but decrease the sensing range.

3.5. Axial Force Robustness

The helical coil can stretch, causing a sensing error, if an axial force is exerted on the sensor, as 3.5. Axial Force Robustness shown in Figure 12. stretch, Stretching of the helical coil along axial axisforce is prevented by on the the tension of as The helical coil can causing a sensing error,the if an axial is exerted sensor, the sensing wire and the initial extension force of the helical coil. However, when the external axial The helical coil can stretch, causing a sensing error, if an axial force is exerted on the sensor, as shown in Figure 12. Stretching of the helical coil along the axial axis is prevented by the tension of the force higher than the summation of theof wire tension and the initial tension, sensing error axial results shown inis Figure 12. Stretching of the force helical coil the axial axis is prevented by the tension of force sensing wire and the initial extension thealong helical coil. However, whena the external the bend The robustness of the axial can becoil. increased by choosing helical coil with thefor sensing wiresensor. and the initial extension force of force the helical However, when thea external axial is higher than the summation of the wire tension and the initial tension, a sensing error results for the a higher initial tension (generallyofproportional to wire diameter and stiffness) and byerror increasing force is higher than the summation the wire tension and the initial tension, a sensing resultsthe bend sensor. The robustness of the axial force can be increased by choosing a helical coil with a higher of theThe sensing wire. of the axial force can be increased by choosing a helical coil with forpre-tension the bend sensor. robustness initial tension (generally to wire diameter and stiffness) and byand increasing the pre-tension a higher initial tension proportional (generally proportional to wire diameter and stiffness) by increasing the of the sensing wire. pre-tension of the sensing wire.

(a)

(b)

Figure 12. Axial force robustness. (a) Safe region; (b) A sensing error occurs when the external axial (a) (b) force is higher than the wire tension and the initial tension. Figure 12. Axial force robustness. (a) Safe region; (b) A sensing error occurs when the external axial

Figure 12. Axial force robustness. (a) Safe region; (b) A sensing error occurs when the external axial force is higher than the wire tension and the initial tension. force is higher than the wire tension and the initial tension.

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3.6.Axial Axial Rotation Robustness 3.6. Rotation Robustness 3.6. Axial Rotation Robustness Axial rotation of the helical coil can cause geometrical changes in the The sheath. The Axial Axialrotation rotationofofthe thehelical helicalcoil coilcan cancause causegeometrical geometricalchanges changesininthe thesheath. sheath. Thecountercountercounter-clockwise rotation shown in Figure 13a further coils the sheath, which increases the total clockwise rotation shown inin Figure further coils the sheath, which increases the total length ofof clockwise rotation shown Figure13a 13a further coils the sheath, which increases the total length length of the sheath, as if the sensing wire were being pulled by the sensor’s bending: the sheath, asas if if the sensing wire were being pulled byby the sensor’s bending: the sheath, the sensing wire were being pulled the sensor’s bending: b ∆L “ (15) L L    bϕ axial (15)  axial (15) 22π  axial

2

Thelength lengthchange changeofofthe thesheath sheathcan canbeberepresented representedasasshown shownininEquation Equation(15), (15),where whereϕϕaxial the is is the The axial is the The length change of the sheath can be represented as shown in Equation (15), where ϕaxial ˝ axialrotation rotationangle angleofofthe thesheath sheathalong alongthe thesensor sensorininthe theradian. radian.AsAsananexample, example,90° 90rotation rotationofofthe the axial axial rotation angle of the sheath along the sensor in the radian. As an example, 90° rotation of the ˝ . However, sheath clockwise will increase sheath length 0.125 mm, causing a measurement error of 4.9 sheath sheathclockwise clockwisewill willincrease increasesheath sheathlength length0.125 0.125mm, mm,causing causinga ameasurement measurementerror errorofof4.9°. 4.9°. this effectthis is much smaller in the real world because there is there friction between each coil pitch and the However, effect is much smaller in the real world because is friction between each coil pitch However, this effect is much smaller in the real world because there is friction between each coil pitch diameter of the helical coil does not change uniformly along the sheath. and the diameter of the helical coil does not change uniformly along the sheath. and the diameter of the helical coil does not change uniformly along the sheath.

(a)(a)

(b)(b)

Figure 13.13. Axial rotation along thethe helical coil causes only a small length change ofof thethe helical coil. Figure Axial rotation along helical coil causes Figure 13. only a small length change helical coil. The sheath (a) contracts with counter-clockwise rotation and; (b) extends with clockwise rotation. The The sheath sheath (a) (a) contracts contracts with with counter-clockwise counter-clockwise rotation rotationand; and;(b) (b)extends extendswith withclockwise clockwiserotation. rotation.

4. 4. Characteristics ofof the Proposed Bend Sensor Characteristics the Proposed Bend Sensor Characteristics of the Proposed Bend Sensor The proposed sensor has several characteristics The proposed sensor has several characteristicsthat thatdifferentiate differentiateit itfrom fromflex flexsensors sensorsand and sensors and optical-fiber based sensors. The properties of these sensors are compared in Table 1 and Figure 14. optical-fiber based based sensors. sensors. The properties properties of of these these sensors sensors are are compared compared in in Table Table11and andFigure Figure14. 14. optical-fiber

Figure 14.14. Comparison of of thethe representative characteristics between the proposed sensor, thethe flex Figure Comparison representative characteristics between proposed sensor, flex Figure 14. Comparison of the representative characteristics between the the proposed sensor, the flex sensor, and the optical fiber-based sensor. sensor, and the optical fiber-based sensor. sensor, and the optical fiber-based sensor.

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Table 1. Comparison of sensor characteristics. Property

Proposed Sensor

Flex Sensor

Optical Fiber-Based Sensor

Size Length Measurand Module size 3 Signal processing Fabrication Cost 4 Directionality Curvature range Stiffness 5 Nonlinearity Resolution 6

Thin Unlimited 1 Position Medium Simple Simple Cheap Any-direction Wide Adjustable Low High

Wide Fixed Electrical resistance Compact Simple Off the shelf Cheap Uni-/bi-direction Narrow Fixed High Low

Very thin Unlimited 1 Light 2 Large Complicated Complicated Expensive Any-direction Restricted Depends on fiber High High

1

Theoretically, the length can be increased infinitely, but unmodeled properties can affect the characteristics of sensors if they are lengthened a great deal; 2 Measures intensity, phase, wavelength, and polarization of light. An interrogator is necessary for spectrum analysis; 3 Size of the device that detects the measurand. It can be a Hall-effect sensor or a potentiometer (as in the proposed bend sensor) or a photodiode (as in an optical fiber-based sensor); 4 Costs for the proposed sensor in US dollars: $3.60 for the potentiometer version and $52.60 for the Hall-effect sensor version. Flex sensor: approximately $10 depending on size. Optical fiber-based sensor: Approximately $12,000, the manufacturer's suggested retail price for the interrogator (PXIe-4844; National Instruments, Corp., Austin, TX, USA); 5 Bending stiffness of the optical fiber depends on the material and the inner and outer diameter of the fiber; 6 Resolution of the proposed sensor depends on the performance of the position measurement of the sensing wire and is bounded by mechanical properties (friction and backlash). On the other hand, the resolution of the optical fiber-based sensor is closely related to its fabrication process and sophisticated signal processing.

1.

Low cost The proposed sensor is made of low-cost materials and can be manufactured easily without specialized fabrication techniques. The sensor can be assembled with off-the-shelf parts within 15 min.

2.

High degrees of freedom for the sensor’s dimensions The dimensions of the sensor are determined by the diameter and length of the helical coil. Thus the size of the sensor can be easily and finely adjusted because there is a wide range of choices for the diameter of the extension spring, and the spring can be cut to the desired size.

3.

Intuitive and easy to use The principle of the proposed sensor is the mechanical conversion of information about bending to information about displacement of the sensing wire. Thus the measurand of the proposed sensor is the displacement of the sensing wire. Because there are various options for the displacement transducer, the user can select one based on their preferences. No complex signal processing or specialized data acquisition device is required because the measurement from the displacement transducer is proportional to the bend angle of the sensor.

4.

No directionality This sensor can detect bending in any direction because the helical coil’s shape and properties are symmetrical along the axial line. And because the sensing wire is placed in the center of the helical coil, the axial displacement of the sensing wire changes with bending in any direction along the helical coil.

5.

Measures the accumulated bend angle This sensor cannot detect the direction of bending or the specific location where bending occurs because it only measures one-degree-of-freedom information, that is, central displacement

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changes of the coil. However, it can detect the accumulated bend angle along the sensor, as depicted in Figure 4, because any local bending along the helical coil causes a central displacement change along the sheath. 6.

Large sensing range The sensor’s physical sensing range is restricted only by the maximum bend curvature of the helical coil that does not cause permanent strain on the sheath. Widely used extension springs made of spring steel or stainless steel have high elasticity and can enable large curvature of the sensor.

7.

Theoretical performance The theoretical performance of the sensor depends on the design parameters and performance of the sensor that is used to measure the displacement change of the sensing wire. The Hall-effect sensor provides high performance for both a linear and a rotary configuration. An incremental Hall-effect encoder with 8192 counts per resolution can provide 0.2˝ resolution for a bend sensor consisting of a 3-mm outer diameter helical coil and a 6-mm diameter spool. A potentiometer is an economical option that provides sufficient performance.

8.

Bend stiffness The bend stiffness of the sensor is determined by the stiffness of the helical coil and the tension of the sensing wire. A large spring constant and a large-diameter coil lead to high bend stiffness of the sensor because energy is required to elongate the spring coil, as shown in Figure 6. Also, greater tension in the sensing wire and a larger helical coil diameter contribute to the restoring force from the sensing wire, which increases the bend stiffness. Bend stiffness increases as the sensor bends if a spring is used to provide tension on the sensing wire. Because bend stiffness can be modeled as a function of design parameters, a bend stiffness profile can be adjusted by changing the radius of the spool and the constant of the linear spring. An actuator can be used as a tensioning device to provide more degrees of freedom for the stiffness profile or to allow variable sensor stiffness.

9.

Environmental robustness The absence of electronic components on the bending part ensures good robustness to environmental conditions. Stainless steel, Teflon, and aramid, which are used in the helical coil, liner, and sensing wires, respectively, have high chemical resistance and thermal resistance (the melting point of Teflon and aramid is 327 ˝ C and 500 ˝ C, respectively). Because the sensor is mechanically operated, it is immune to electromagnetic interference and moisture, allowing it to be used even when submerged in boiling water or high-temperature oil.

5. Prototype Development A prototype of the proposed bend sensor was developed using a spool to provide internal tension and measure the position change of the sensing wire (Figures 15 and 16). A miniature extension spring was attached between the spool and the frame to provide a restoring torque to the spool. A Hall-effect angle sensor and a magnet were placed to measure the rotational angle change of the spool. The Hall-effect sensor provides higher resolution than a rotary potentiometer, but it is more expensive. The frame of the sensing part was fabricated using a three-dimensional (3D) printer. A continuous length extension spring (Φ3.0 mm ˆ Φ2.0 mm ˆ 500 mm—Φ indicates diameter) whose coil wire had a circular cross section was used as the helical coil of the bending part. An extension spring can be cut to the desired length of the sensor, and it is easy to find an off-the-shelf continuous length spring up to 1 m long on the market.

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A Teflon was used inside the coil to reduce friction along the sensing wire. Two 13 Teflon Sensors 2016,tube 16, 961 of 20 tubes Sensors 2016,diameters 16, 961 13 oftogether 20 with different (Φ1.38 mm ˆ Φ1.88 mm and Φ1.40 mm ˆ Φ0.90 mm) were fitted A Teflon tube was used inside the coil to reduce friction along the sensing wire. Two Teflon to increase the wall thickness and place the sensing wire as close as possible to the center of the coil, tubes Awith different diameters (Φ1.38the mm Φ1.88 mmfriction and Φ1.40 Φ0.90 mm) fitted Teflon tube was used inside coil× to reduce alongmm the ×sensing wire. were Two Teflon which increases the ratio of the central displacement change to bending of the sensor. The sensing wire together to increase thediameters wall thickness place the sensing wire as close to the center of tubes with different (Φ1.38and mm × Φ1.88 mm and Φ1.40 mmas×possible Φ0.90 mm) were fitted and Teflon tube should not be tightly fit because that causes too much friction, resulting in hysteresis the coil, which increases the ratio of the central displacement change to bending of the sensor. The together to increase the wall thickness and place the sensing wire as close as possible to the center of of thesensing sensor. Using aTeflon 0.9 mm Teflon tube with acauses 0.45tomm diameter andincreases tube should tightly fit because that too much friction, resulting the coil,wire which theinner ratio diameter ofnot thebecentral displacement change bending of theDyneema sensor. The(Royal DSM,insensing Heerlen, Netherlands) wire ensures that the wire moves smoothly. A 3D-printed end-cap hysteresis of the sensor. Using a 0.9 mm inner diameter Teflon tube with a 0.45 mm diameter wire and Teflon tube should not be tightly fit because that causes too much friction, resulting was Dyneema (Royal DSM, Netherlands) wire ensures that wire smoothly. A and in hysteresis of the Usingcoil a 0.9tomm inner diameter Teflon tube withmoves a 0.45 diameter attached to both ends ofsensor. theHeerlen, helical provide rigidity and to the connect the coilmm to the frame 3D-printed end-cap was attached to both ends of the helical coil to provide rigidity and to connect Dyneema (Royal DSM, Heerlen, Netherlands) wire ensures that the wire moves smoothly. A mm) fix the end of the sensing wire. The sensing wire was built with braided Dyneema wire (Φ0.45 the coil to the frame and fix the end of the sensing wire. The sensing wire was built with braided 3D-printed end-cap was attached to both ends of the helical coil to provide rigidity and to connect because this material has high modulus (109–132 GPa tensile modulus [37]), ensuring that the wire did Dyneema this of material has high modulus (109–132 GPawas tensile [37]), the coil towire the(Φ0.45 frame mm) and because fix the end the sensing wire. The sensing wire builtmodulus with braided not elongate and violate the assumptions of the derived model. Dyneema wire is flexible and easily ensuring the wire mm) did not elongate violate assumptions of the derived model. Dyneema Dyneemathat wire (Φ0.45 because this and material hasthe high modulus (109–132 GPa tensile modulus [37]), wound around a small-diameter spool (6and and it does not contact with spool and the wire is flexible andwire easily around amm), small-diameter spool (6 lose mm), and it does notthe lose contact ensuring that the didwound not elongate violate the assumptions of the derived model. Dyneema innerwith wall of the Teflon tube. Metal wires can lose contact with the surface of a spool because spool and inner wall of the Teflon tube. Metal wires lose contact withnot thelose surface of their wirethe is flexible andthe easily wound around a small-diameter spool can (6 mm), and it does contact bending stiffness is and higher than of polymer Also, Dyneema wire is reported haveisofa small awith spool because theirthe bending stiffness isTeflon higherwires. thanMetal that ofwires polymer wires. Also, Dyneema wire the spool inner that wall of the tube. can lose contact with the to surface reported to havewhen atheir small friction coefficient when used with tubeswires. (aof friction coefficient 0.055 a spool because bending stiffness is tubes higher than that Teflon ofcoefficient polymer Also, Dyneemaofwire is friction coefficient used with Teflon (a friction 0.055 compared to 0.092 for compared to 0.092 for steel [36,38]). reported to have a small friction coefficient when used with Teflon tubes (a friction coefficient of 0.055 steel [36,38]). compared to 0.092 for steel [36,38]).

(a) (b) (a) (b) Figure 15. Computer-assisted design (CAD) model of the bend sensor prototype. (a) Overall view of Figure 15. Computer-assisted design (CAD) model of the bend sensor prototype. (a) Overall view of the sensor of a sensing part(CAD) and a bending (b) Close view of the sensing part, view which Figure 15.consisting Computer-assisted design model ofpart; the bend sensor prototype. (a) Overall of the sensor consisting of a sensingwire partand and a bending part; (b) Close view of the sensing part, which provides tension to theofsensing measures its position. the sensor consisting a sensing part and a bending part; (b) Close view of the sensing part, which provides tension to the wire and position. provides tension to sensing the sensing wire andmeasures measures its its position.

Figure 16. Developed prototype of the bend sensor. Figure 16. Developed prototype of the bend sensor.

Figure Developed prototype of the bendenabling sensor. manufacture of a lowAll of the materials used to 16. fabricate the sensor are inexpensive, cost bend sensor. The total price of the components used, exclusive of the displacement sensing All of the materials used to fabricate the sensor are inexpensive, enabling manufacture of a lowelement, was $2.60 US dollars. This study used a Hall-effect sensor module with an embedded analog costofbend The used total price of the components used, of the enabling displacement sensing All the sensor. materials to fabricate the sensor are exclusive inexpensive, manufacture of to digital converter (ADC), which isofsomewhat expensiveused, for fast prototyping. Hall-effect element, was $2.60 US dollars. This used a Hall-effect sensor module with an Using embedded analog a low-cost bend sensor. The total pricestudy the components exclusive of the displacement sensing sensor ICs with their own electronics further reduce the cost.Using The total cost ofanalog to digital converter (ADC), which iswould somewhat expensive formanufacturing fast module prototyping. Hall-effect element, was $2.60 US dollars. This study used a Hall-effect sensor with an embedded the sensor reduced to $3.60 USwould dollarsfurther if a rotary potentiometer is used instead of atotal Hall-effect sensor ICscould with be their own electronics reduce the manufacturing cost. The cost of to digital converter (ADC), which is somewhat expensive for fast prototyping. Using Hall-effect sensor sensor. Thecould specifications and of the components listed in Table 2. instead of a Hall-effect the sensor be reduced toprices $3.60 US dollars if a rotaryare potentiometer is used ICs with their own electronics would further reduce the manufacturing cost. The total cost of the sensor. The specifications and prices of the components are listed in Table 2.

sensor could be reduced to $3.60 US dollars if a rotary potentiometer is used instead of a Hall-effect sensor. The specifications and prices of the components are listed in Table 2.

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Table 2. Specification and price list of the prototype components (US dollars). Sensors 2016, 16, 961 Component

Specification

14 of 20 Price

Helical coil

Extension spring: Φ3.0 mm × Φ2.0 mm × 500 mm

$1

Lining

Outer Teflon tube: mm components × Φ1.88 mm;(US dollars). Table 2. Specification and price list of the Φ1.38 prototype Inner Teflon tube: Φ1.40 mm × Φ0.90 mm

Component

Wire

Specification

Braided Dyneema wire: Φ0.45 mm

Extension spring: Φ3.0 mm ˆ Φ2.0 mm ˆ 500 mm 3D printed with VeroWhitePlus™ (Objet Connex; Stratasys Ltd., Eden Prairie, Outer Teflon tube: Φ1.38 mm ˆ Φ1.88 mm; Structure Lining Inner Teflon tube: MN,Φ1.40 USA)mm ˆ Φ0.90 mm Wire Braided Dyneema wire: Φ0.45 mm Spring Miniature spring Ltd., Eden Prairie, MN, USA) Structure 3D printed with VeroWhitePlus™ (Objetextension Connex; Stratasys Spring Miniature extension Hall-effect sensor: RMB20IC13BC10 (RLS; 8192spring pulses per turn, ±0.5° accuracy, Displacement Hall-effect sensor: RMB20IC13BC10 (RLS; 8192 pulses per turn, ˘0.5˝ accuracy, 0.18˝ hysteresis, onboard filter 720 Hz cut-offfor frequency for sine andof Displacement sensor 0.18° hysteresis, onboard RC filterRC with 720with Hz cut-off frequency sine and cosine signal sensor magnetic flux); Magnet: Φ4 mm ˆ 4 mm cosine signal of magnetic flux); Magnet: Φ4 mm × 4 mm Helical coil

$0.2 Price

$0.1 $1

$1 $0.2 $0.1 $0.3 $1 $0.3

$50 $50

6. Experiments Experiments Two Two experiments experiments were were conducted conducted to to determine determine the the output output response response of of the sensor in terms of the reference bend angle and curvature. curvature. First, a quasi-static experiment was conducted to compare the ˝ to 120˝ using an external reference angle continuous response of the sensor at a bend angle from 00° 120° sensor. quasi-static experimental setup encouraged a second sensor. Limitation Limitation of the bend angle range in the quasi-static experiment that permitted large deflections. Finally, a static experiment was conducted to determine the response of the sensor to large deflections (0 (0°˝ to 450 450°˝ bend angle) with different bend curvatures 1 , and 80.0 −1 1 ) using bending brackets. −1,1 ,40.0 (κ 40.0mm−1´ (κ == 26.7 m´ , and 80.0 m m)´using bending brackets. Hall-effect sensors with an incremental quadrature pulse output (RMB20IC; RLS, Komenda, Komenda, Slovenia, accuracy accuracy ±0.7°, ˘0.7˝ hysteresis , hysteresis0.18°) 0.18˝and ) and a magnet (Φ4 mm ˆ 4 mm) were used to measure a magnet (Φ4 mm × 4 mm) were used to measure the the spool angle of the bend sensor and referenceangle. angle.The Theoutput outputsignal signalofofthe the sensor sensor and the spool angle of the bend sensor and thethe reference reference acquisition device device (cRIO-9082, (cRIO-9082, National National Instruments Instruments Corp.) Corp.) reference angle were acquired with a data acquisition with a differential digital input input module module (NI (NI 9411, 9411,National NationalInstruments InstrumentsCorp.). Corp.).Data Datawas wassaved savedonona ahost hostPC PCusing usingan anEthernet Ethernetinterface interfacewith withaa1-kHz 1-kHzsampling samplingfrequency. frequency. No No filter filter was applied after data acquisition embedded ADC with filters. AllAll of acquisition because becausethe theHall-effect Hall-effectsensor sensormodule moduleincluded includedanan embedded ADC with filters. the brackets used in in both experiments were made with a a3D of the brackets used both experiments were made with 3Dprinter printer(Object (ObjectConnex, Connex,Stratasys StratasysLtd.). Ltd.). 6.1. Quasi-Static Experiment Experiment 6.1. Quasi-Static Figure continuous bend Figure 17 17 shows shows the the experimental experimental setup setup for for measuring measuring the the sensor sensor signal signal with with aa continuous bend ˝ ˝ angle change from 0 to 120 . To set the desired rotation angle, a rotating bracket with a reference angle change from 0° to 120°. To set the desired rotation angle, a rotating bracket with a reference ˝ ˝ ˝ ˝ angle angle sensor sensor was was moved moved by by hand. hand. The The moving moving bracket bracket was was rotated rotated from from 00° to to 120 120° and and 120 120° to to 00° three three times times sequentially. sequentially.

Figure 17. 17. Setup Setup for for the the quasi-static quasi-static experiment. experiment. The constrained to the desired desired Figure The sensor sensor sheath sheath was was constrained to the bend angle, and the setup was moved by hand. bend angle, and the setup was moved by hand.

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Error (deg) Error (deg)

Bend angle (deg) Bend angle (deg)

Sensor raw signal (cnt) Sensor raw signal (cnt)

of 20 Figures 18 and 19 show the experimental results of the quasi-static experiment in the time 15 domain 18 angle and 19 show the experimental results of the quasi-static in the time and theFigures reference domain, respectively. Data was plotted to compareexperiment the reference angle, the Figures 18 and 19 show the experimental results of the quasi-static experiment in the time domain and the reference angle domain, respectively. Data was plotted to compare the reference model-based sensor output, and the calibrated sensor output. The model-based sensor output was domain and the reference angle domain, respectively. Data was plotted to compare the reference angle, the model-based output, calibrated sensor The model-based sensor calculated from the derivedsensor sensor model,and thethe design parameter of output. the prototype, and the raw signal angle, the model-based sensor output, and the calibrated sensor output. The model-based sensor output was calculated from the derived sensor model, the design parameter of thederived prototype, and the 3. from the Hall-effect sensor. It shows the modeling accuracy of the sensor model in Section output was calculated from the derived sensor model, the design parameter of the prototype, and the signal fromthat the Hall-effect sensor.sensor It shows the modeling accuracy the sensor model derived Theraw result shows model-based output was lower than theof reference angle, with raw signal from thethe Hall-effect sensor. It shows the modeling accuracy ofactual the sensor model derived in Section 3. The result shows that the model-based sensor output was lower than the actual reference ˝ a root mean square error (RMSE) of 8.42 . This means that the actual displacement change of the in Section 3. The result shows that the model-based sensor output was lower than the actual reference angle, with a root mean square error (RMSE) of 8.42°. This means that the actual displacement change sensing thansquare for the derived model because of the uncertainty ofdisplacement the model. However, angle,wire withisa lower root mean error (RMSE) of 8.42°. This means that the actual change of the sensing wire is lower than for the derived model because of the uncertainty of the model. Figure 19asensing shows wire that the sensor output signal has high linearity canuncertainty be calibrated a linear of the is lower than for the derived model becauseand of the of with the model. However, Figure 19a shows that the sensor output signal has high linearity and can be calibrated function to increase accuracy. (ordinary squares) without the intercept term However, Figure 19a shows Linear that theregression sensor output signalleast has high linearity and can be calibrated with a linear function to increase accuracy. Linear regression (ordinary least squares) without the a linear on function to increase Linear (ordinary squares)the without the waswith conducted the raw signal ofaccuracy. the sensor and regression the reference angle least to calibrate developed intercept term was conducted on the raw signal of the sensor and the reference angle to calibrate the intercept term was conducted onplotted the rawwith signala of the sensorline andin the reference angle to calibrate theof sensor. The calibrated signal was red dotted Figures 18 and 19. The RMSE developed sensor. The calibrated signal was plotted with a red dotted line in Figures 18 and 19. The ˝was ˝ ). The18 developed sensor. The calibrated plotted with of a red dotted line(120 in Figures and 19. value The theRMSE calibrated signal was reduced tosignal 5.15 , which 4.29% scale R-square of the calibrated signal was reduced to is 5.15°, whichthe is full 4.29% of the full scale (120°). The RMSE of the calibrated signal was reduced to 5.15°, which is 4.29% of the full scale (120°). The (coefficient determination) ofdetermination) the calibrated of sensor output issensor 0.9887,output whichisensures the sensor to be R-squareof value (coefficient of the calibrated 0.9887, which ensures R-square value (coefficient of determination) of the calibrated sensor output is 0.9887, which ensures used with linear calibration function. However, the sensor signal can be calibrated with polynomials the sensor to be used with linear calibration function. However, the sensor signal can be calibrated the sensor to be sensing used with linear calibration function. However, the sensor signal can be calibrated to further increase accuracy. with polynomials to further increase sensing accuracy. with polynomials to further increase sensing accuracy.

(a) (a)

(b) (b)

Error (deg) Error (deg)

Sensor output (deg) Sensor output (deg)

Figure 18. Results for the quasi-static experiment in the time domain. (a) Reference angle (gray thick Figure 18.18. Results forfor thethe quasi-static (a)Reference Referenceangle angle(gray (graythick thick Figure Results quasi-staticexperiment experimentin inthe the time time domain. domain. (a) line), model-based sensor output (black solid line), and calibrated sensor output (red dotted line) in line), model-based sensor output (black solid line), and calibrated sensor output (red dotted line) inin the line), model-based sensor output (black solid line), and calibrated sensor output (red dotted line) the time domain; (b) Sensing error of model-based sensor output (black solid line) and calibrated time domain; (b) Sensing error oferror model-based sensorsensor outputoutput (black(black solid line) sensor the time domain; (b) Sensing of model-based solid and line)calibrated and calibrated sensor output (red dotted line) in the time domain. sensor output (red dotted line) in the time domain. output (red dotted line) in the time domain.

(a) (a)

(b) (b)

Figure 19. Results for the quasi-static experiment in the reference angle domain. (a) Reference angle Figure 19. Results for quasi-staticexperiment experimentin inthe the reference reference angle angle domain. (a) angle Figure Results thethe quasi-static domain.sensor (a)Reference Reference angle (gray19.thick line),for model-based sensor output (black solid line), and calibrated output (red (gray thick line), model-based sensor output (black solid line), and calibrated sensor output (red (gray thickline) line), sensordomain; output(b) (black soliderror line),ofand calibratedsensor sensoroutput output (red solid dotted dotted in model-based the reference angle Sensing model-based (black dotted line) in the reference angle domain; (b) Sensing error of model-based sensor output (black solid line) in the (b) Sensing error of model-based sensor output (black solid line) line) andreference calibratedangle sensordomain; output (red dotted line). line) and calibrated sensor output (red dotted line). and calibrated sensor output (red dotted line).

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The results show low levels of hysteresis with a maximum offset of about 3˝ . The sensor’s 16 of 20 hysteresis may have originated in wire friction. It can be compensated for with hysteresis Sensors 2016, 16, 961 16 of 20 modeling minimized analysis and optimization of the design parameter The[32–35] results or show low levelsby of hysteresis hysteresis with a maximum offset of about 3°. The sensor’s hysteresis may have show originated in wireoffriction. It can be compensated for with of the sensor. The results low levels hysteresis with a maximum offset of hysteresis about 3°. modeling The sensor’s Sensors 2016, 16, 961

[32–35] or minimized byoriginated hysteresis in analysis and optimization of the designfor parameter of the sensor. hysteresis may have wire friction. It can be compensated with hysteresis modeling

6.2. Static Large Deflection Experiment [32–35] or minimized by hysteresis analysis and optimization of the design parameter of the sensor. 6.2. Static Large Deflection Experiment

A static large deflection experiment was conducted to determine the sensor’s response when 6.2. Static Large A static large Deflection deflectionExperiment was conducted to determine the sensor’s response subjected to large bend anglesexperiment of different curvatures. Five brackets having a 90˝ bendwhen angle were subjected to large bend angles of different curvatures. Five brackets having a 90° bendresponse angle´were 1 A static large deflection experiment was conducted to determine the sensor’s whenm´1 , placed onto breadboard for experiments with three different bend curvatures (κ = 26.7 m , 40.0 placed onto breadboard for angles experiments with three different bend curvatures (κ =a 26.7 m−1, 40.0 m−1were , ´ 1 subjected to large bend of different curvatures. Five brackets having 90° bend angle andand 80.0 m m−1),),as shown in Figure Figure20. 20.Sensor Sensor output was measured with 11 different bend angles by 80.0 as shown in output was measured with 11 different bend angles by placed onto breadboard for experiments different curvatures (κ = 26.7 m−1, 40.0 m−1, ˝ to with ˝three ˝ to 0bend ˝. adding and subtracting brackets from 0 450 and 450 adding and subtracting brackets from 0° to 450° and 450° to 0°. and 80.0 m−1), as shown in Figure 20. Sensor output was measured with 11 different bend angles by adding and subtracting brackets from 0° to 450° and 450° to 0°.

(a)

(b)

(a)deflection experiment using a 90° bending (b) Figure 20. Setup for the static large bracket with three Figure 20. Setup for the static large deflection experiment using a 90˝ bending bracket with three different curvatures. (a) κ = 80.0 m−1´1 ; (b) κ = 26.7 m−1. ´1 Figure 20. Setup(a) forκ the static deflection using a 90° bending bracket with three different curvatures. = 80.0 m large ; (b) κ = 26.7 experiment m . different curvatures. (a) κ = 80.0 m−1; (b) κ = 26.7 m−1.

Figure 21 shows the result of the static large deflection experiment. Data was plotted with different curvatures, andresult the compared to the reference angle. it was found FigureFigure 21 shows the of thesignal static large deflection experiment. DataOverall, was with different 21 shows thesensor result of thewas static large deflection experiment. Dataplotted was plotted with that the sensor output shows a linear increment with an increasing bend angle up to 450°, with curvatures, and the sensor signal was compared to the reference angle. Overall, it wasitfound that the different curvatures, and the sensor signal was compared to the reference angle. Overall, was found ˝ , with R-square values of 0.9959, and 0.9969 forincreasing the calibrated sensor output the three different sensor output shows a linear0.9968, increment with an bend angle up tofor 450 R-square values that the sensor output shows a linear increment with an increasing bend angle up to 450°, with curvatures. Also, theof tendency of thecalibrated sensor output wasoutput not affected bythree the bend curvature, of 0.9959, 0.9968, and 0.9969 the sensor forsensor the different R-square values 0.9959,for 0.9968, and 0.9969 for the calibrated output for the curvatures. threewhich differentAlso, wascurvatures. the expected result the modeling and simulation. Also, thefrom tendency of the sensor output was not affected by the bend curvature, which

(a)

Error (deg)

Error (deg)

Sensor output (deg)

Sensor output (deg)

the tendency of the sensor output was not affected by the bend curvature, which was the expected expected result from the modeling and simulation. result was fromthe the modeling and simulation.

(b) (a)

(b)

Figure 21. Experimental results showing the output signal with various bending angles and curvatures. (a) Sensor output with difference curvatures showing model-based output (black) and 21. Experimental showing the output signalvarious with various angles and FigureFigure 21. Experimental resultsresults showing the output signal with bendingbending angles and curvatures. calibrated output (red); (b) Sensing error with different curvatures, showing model-based output curvatures. (a) Sensor output with difference curvatures showing model-based output (black) (a) Sensor output with difference curvatures showing model-based output (black) and calibratedand output (black) and calibrated output (red). calibrated (b) Sensing error withshowing different model-based curvatures, showing (red); (b) Sensingoutput error (red); with different curvatures, outputmodel-based (black) and output calibrated (black) and calibrated output (red).

output (red).

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The sensor output based on the model is still lower than the reference angle as in the quasi-static experiment, with RMSEs for the model-based sensor output of 38.05˝ , 37.90˝ , and 49.39˝ for the three curvatures. Additionally, the RMSEs of the calibrated sensor output are 9.05˝ , 8.10˝ , and 8.29˝ for the three curvatures, which represents a 2% maximum error of the full sensing range. The sensing range of this experiment was restricted by the length of the sensor. A longer bend sensor would have allowed a larger bend angle to be measured. 7. Discussion The experimental results in Section 7 show some unmodeled measurement error. The expected error of the proposed sensor can be considered in terms of thermal properties and mechanical precision. The helical stainless steel coil had a positive thermal expansion ´ coefficient (αss = 10.1 ˆ 10 6 ´ 17.3 ˆ 10´6 K´1 ), while Dyneema wire had a negative coefficient (αdyneema = ´12 ˆ 10´6 K´1 ) [37]. This resulted in a 0.011–0.015 mm displacement error and a 0.65–0.86˝ measurement error per unit temperature (K´1 ) for the 500 mm length of the sensor. However, the huge temperature change that occurred during sensor measurement is not typical, and the temperature effect described above is much smaller than the temperature effect of flex sensors (˘30% resistance tolerance) [1]. The precision of parts and the assembly process is another factor affecting the measurement error because the principle of the sensor is the mechanical conversion of the bend angle to the displacement change. Improper finishing of the sensing wire can lead to a dead signal zone or a zero offset change of the sensor. Also, the tilt axis of the spool can affect sensor gain depending on the bend angle, as shown in the tendency of an error in Figure 19. In addition, friction between the sensing wire and Teflon tube can affect the nonlinearities of the sensor signal, although the tension of the sensing wire is kept as low as possible to reduce the friction effect. Furthermore, during long-term use of the sensor, sensor contamination can affect its friction and hysteresis characteristics. 8. Conclusions This study proposes a new type of bend sensor that uses a Bowden-cable and a displacement sensing element. The basic principle of the sensor is based on the displacement change of the inner wire when the shape of the sheath changes, which previous studies have considered to be an error. It has been shown that the output signal of the sensor has high linearity with the bend angle and can measure a wide range of accumulated bending angles with large curvatures. The proposed bend sensor has three main advantages over previous bend sensors, including flex sensors and optical fiber-based sensors. First, it is low cost. The design and manufacturing process for the proposed sensor are very simple. It can be easily made in laboratories with cheap and affordable parts that do not require any special fabrication technique. Second, it is easy to use. It does not need complicated signal processing because the bend angle of the sensor is proportional to the output signal of the displacement sensing module. Third, it can measure a wide range of bend angles and curvatures (up to 450˝ bend angle and 80.0 m´1 curvature for the first prototype). The sensor signal is linear with large curvatures (R-square value up to 0.9969 with a curvature 80.0 m´1 ), and the length of the sensor can be easily extended to measure large angles. Nevertheless, this study has some limitations. Some non-linearity, which is shown in the initial bend of the sensor, can accumulate if the sensor is subjected to multiple bend points. However, many applications that use other types of bend sensors do not involve multiple bend points. Also, some sensor nonlinearities can be compensated for with more precise modeling and calibration, because the signal is repeatable. Despite these limitations, the proposed sensor is expected to overcome the usability and accessibility deficiencies of flex sensors and optical fiber-based sensors. Many existing studies can be further developed by using the proposed bend sensor for more precise control of a Bowden-cable [28–36] or to obtain shape feedback from soft-bodied structures like soft robots [39,40].

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Acknowledgments: The authors would like to thank Jeong-Ryul Song for his assistance with drawing figures; Figure 1 and an abstract graphic. This research was supported by the convergence technology development program for the bionic arm through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2015M3C1B2052817). Author Contributions: Useok Jeong contributed the original idea, developed the prototype, designed and performed the experiment, analyzed the data, and wrote the paper; Kyu-Jin Cho supervised the research, revised the manuscript, and approved the final version to be published. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A The Taylor series for Equation (8) can be derived as follows: a 8 ÿ 2 t1 ´ cos pKqu Xpnq px0 q px ´ x0 qn X pKq “ “ K n!

(A1)

n “0

a 0

X p0q “ lim

KÑ 0

« a 1

X p0q “ ´ „ ? 2

X p0q “

2

2p1´cos Kq K3

2 p1 ´ cos Kq K2

` ? K

X pKq “ 1 ´

cos K 2p1´cos Kq

2 p1 ´ cos Kq “1 K sin K ` a K 2 p1 ´ cos Kq

´

? ?2sin K 2 K 1´cos K

(A2) ff “0

(A3)

K“ 0



´

2 ? sin K 2 2p1´cos Kq3{2 K“0

K2 K4 K6 K8 ` ´ ` `¨¨¨ 24 1920 322560 92897280

“

1 12

(A4) (A5)

8 ÿ 1 1 1 K2n n 16 n! p3{2qpnq n “0

(A6)

Γ px ` nq “ x px ` 1q ¨ ¨ ¨ px ` n ´ 1q Γ pxq

(A7)

X pKq “ where x(n) is the Pochhammer symbol [41], x pnq ”

References 1. 2. 3. 4.

5.

6. 7. 8.

Jurgens, J.; Patterson, P.E. Development and evaluation of an inexpensive sensor system for use in measuring relative finger positions. Med. Eng. Phys. 1997, 19, 1–6. [CrossRef] Mulder, A. How to build an instrumented glove based on the Powerglove flex sensors. PCVR Mag. 1994, 16, 10–14. Simone, L.K.; Kamper, D.G. Design considerations for a wearable monitor to measure finger posture. J. NeuroEng. Rehabil. 2005, 2. [CrossRef] [PubMed] Dunne, L.E.; Smyth, B.; Caulfield, B. A Comparative Evaluation of Bend Sensors for Wearable Applications. In Proceedings of the 11th IEEE International Symposium on Wearable Computers, Boston, MA, USA, 11–13 October 2007. Gentner, R.; Classen, J. Development and evaluation of a low-cost sensor glove for assessment of human finger movements in neurophysiological settings. J. Neurosci. Methods 2009, 178, 138–147. [CrossRef] [PubMed] Sturman, D.J.; Zeltzer, D. A survey of glove-based input. IEEE Comput. Graph. Appl. 1994, 14, 30–39. [CrossRef] Zimmerman, T.G. Optical Flex Sensor. U.S. Patent 4 542 291, 17 September 1985. Harvill, Y.L.; Zimmerman, T.G.; Grimaud, J.J.G. Motion Sensor which Produces an Asymmetrical Signal in Response to Symmetrical Movement. U.S. Patent 5 097 252, 17 March 1992.

Sensors 2016, 16, 961

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

31.

32. 33.

19 of 20

Tognetti, A.; Lorussi, F.; Carbonaro, N.; de Rossi, D. Wearable Goniometer and Accelerometer Sensory Fusion for Knee Joint Angle Measurement in Daily Life. Sensors 2015, 15, 28435–28455. [CrossRef] [PubMed] Stoppa, M.; Chiolerio, A. Wearable Electronics and Smart Textiles: A Critical Review. Sensors 2014, 14, 11957–11992. [CrossRef] [PubMed] Huang, C.T.; Shen, C.L.; Tang, C.F.; Chang, S.H. A wearable yarn-based piezo-resistive sensor. Sens. Actuators A Phys. 2008, 141, 396–403. [CrossRef] Ponticelli, R.; Gonzalez de Santos, P. Full perimeter obstacle contact sensor based on flex sensors. Sens. Actuators A Phys. 2008, 147, 441–448. [CrossRef] Peng, W.; Tianhuai, D.; Feng, X. A novel flexible sensor for compression stress relaxation. In Proceedings of IEEE Sensors, Toronto, ON, Canada, 22–24 October 2003. Starck, J.R.; Murray, G.; Lawford, P.V.; Hose, D.R. An inexpensive sensor for measuring surface geometry. Med. Eng. Phys. 1999, 21, 725–729. [CrossRef] Gentile, C.T.; Wallace, M.; Avalon, T.D.; Goodman, S.; Fuller, R.; Hall, T. Angular displacement sensors. U.S. Patent 5 086 785, 11 Feburay 1992. Flex Sensor Manufacturers | Spectra Symbol. Available online: http://www.spectrasymbol.com/flex-sensor (accessed on 12 April 2016). Flexpoint Sensor Systems. Available online: http://www.flexpoint.com/ (accessed on 12 April 2016). Kure, K.; Kanda, T.; Suzumori, K.; Wakimoto, S. Flexible displacement sensor using injected conductive paste. Sens. Actuators A Phys. 2008, 143, 272–278. [CrossRef] Saggio, G. Electrical Resistance Profiling of Bend Sensors Adopted to Measure Spatial Arrangement of the Human Body. In Proceedings of the 4th International Symposium on Applied Sciences in Biomedical and Communication Technologies, Barcelona, Spain, 26–29 October 2011. Saggio, G. Mechanical model of flex sensors used to sense finger movements. Sens. Actuators A Phys. 2012, 185, 53–58. [CrossRef] Qi, Y.; Ma, L.; Sun, J.; Kang, Z.; Bai, Y.; Jian, S. Highly sensitive bending sensor based on multimodemultimode-coreoffset fiber structure. Opt. Laser Technol. 2015, 75, 52–56. [CrossRef] Kuang, K.S.C.; Cantwell, W.J.; Scully, P.J. An evaluation of a novel plastic optical fibre sensor for axial strain and bend measurements. Meas. Sci. Technol. 2002, 13, 1523–1534. [CrossRef] Liu, Y.; Williams, J.A.R.; Bennion, I. Optical bend sensor based on measurement of resonance mode splitting of long-period fiber grating. IEEE Photonics Technol. Lett. 2000, 12, 531–533. [CrossRef] Chen, S.; Tong, Z.; Zhao, Q.; Liu, Z.; Dong, X. A smart bending sensor with a novel temperature- and strain-insensitive long-period grating. Sens. Actuators A Phys. 2004, 116, 103–106. [CrossRef] Dong, Y.; Lu, Y.; Shen, C.; Chu, J.; Zou, X.; Zhong, C.; Dong, X. Novel bending sensor based on a meniscus shaped beam with LPG. Opt. Int. J. Light Electron. Opt. 2013, 124, 6737–6739. [CrossRef] Taghipour, A.; Rostami, A.; Bahrami, M.; Baghban, H.; Dolatyari, M. Comparative study between LPFG- and FBG-based bending sensors. Opt. Commun. 2014, 312, 99–105. [CrossRef] Miao, Y.; Zhang, K.; Liu, B.; Lin, W.; Yao, J. Characteristics of bend sensor based on two-notch Mach-Zehnder fiber interferometer. Opt. Fiber Technol. 2012, 18, 509–512. [CrossRef] In, H.; Kang, B.B.; Sin, M.; Cho, K.J. Exo-Glove: A Wearable Robot for the Hand with a Soft Tendon Routing System. IEEE Robot. Autom. Mag. 2015, 22, 97–105. [CrossRef] Jeong, U.; In, H.-K.; Cho, K.-J. Implementation of various control algorithms for hand rehabilitation exercise using wearable robotic hand. Intell. Serv. Robot. 2013, 6, 181–189. [CrossRef] Veneman, J.F.; Kruidhof, R.; Hekman, E.E.G.; Ekkelenkamp, R.; Asseldonk, E.H.F.V.; Van Der Kooij, H. Design and Evaluation of the LOPES Exoskeleton Robot for Interactive Gait Rehabilitation. IEEE Trans. Neural Syst. Rehabil. Eng. 2007, 15, 379–386. [CrossRef] [PubMed] Ding, Y.; Galiana, I.; Asbeck, A.; Quinlivan, B.; De Rossi, S.M.M.; Walsh, C. Multi-joint actuation platform for lower extremity soft exosuits. In Proceedings of the 2014 IEEE International Conference on Robotics and Automation (ICRA), Hongkong, China, 31 May–7 June 2014. Phee, S.J.; Low, S.C.; Dario, P.; Menciassi, A. Tendon sheath analysis for estimation of distal end force and elongation for sensorless distal end. Robotica 2010, 28, 1073–1082. [CrossRef] Bardou, B.; Nageotte, F.; Zanne, P.; Mathelin, M. Improvements in the control of a flexible endoscopic system. In Proceedings of the 2012 IEEE International Conference on Robotics and Automation (ICRA), Saint Paul, MN, USA, 14–18 May 2012; pp. 3725–3732.

Sensors 2016, 16, 961

34.

35. 36.

37. 38. 39. 40. 41.

20 of 20

Do, T.N.; Tjahjowidodo, T.; Lau, M.W.S.; Phee, S.J. Adaptive control of position compensation for Cable-Conduit Mechanisms used in flexible surgical robots. In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Vienna, Austria, 1–3 September 2014; pp. 110–117. Kesner, S.B.; Howe, R.D. Robotic catheter cardiac ablation combining ultrasound guidance and force control. Int. J. Robot. Res. 2014, 33, 631–644. [CrossRef] Jeong, U.; Cho, K.J. Feedforward friction compensation of Bowden-cable transmission via loop routing. In Proceedings of the 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Hamburg, Germany, 28 September–2 October 2015; pp. 5948–5953. Dyneema Comprehensive Factsheet. Available online: http://www.pelicanrope.com/pdfs/DyneemaComprehensive-factsheet-UHMWPE.pdf (accessed on 9 June 2016). Carlson, L.E.; Veatch, B.D.; Frey, D.D. Efficiency of Prosthetic Cable and Housing. J. Prosthet. Orthot. 1995, 7, 96–99. Rus, D.; Tolley, M.T. Design, fabrication and control of soft robots. Nature 2015, 521, 467–475. [CrossRef] [PubMed] Lee, J.Y.; Kim, W.B.; Choi, W.Y.; Cho, K.J. Development of a Modularized Design Concept for Soft Robotics: Introducing SoBL, a Soft Robotic Block. IEEE Robot. Autom. Mag. 2016, 23, in press. Knuth, D.E. Two notes on notation. Am. Math. Mon. 1992, 99, 403–422. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).