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Super-structured D-shape fibre Bragg grating contact force sensor 1 2 3 4 5 6 7 8 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 34 35 36 37 38 39 40 41 42 43 44 45 46 47 Superstructured DShape sensor Christopher R. Dennison and Peter M. Wild Department of Mechanical Engineering, University of Victoria, British Columbia, Canada Corresponding Author: Chris R. Dennison University of Victoria Department of Mechanical Engineering P.O. Box 3055 Victoria, B.C. V8W 3P6 Ph: (250) 853-3198 Fax: (250) 721-6051 e-mail: [email protected] Abstract 1 Super-structured D-shape fibre Bragg grating contact force sensor 48 49 1. Introduction In-fibre Bragg gratings (FBGs) can be configured as sensors for various parameters 50 including displacement [1], strain [2], temperature [3], pressure [3], contact force [4], 51 humidity [5], and radiation dose [6] among others. FBGs are an attractive alternative to 52 other piezoelectric, resistive or other solid-state sensing technologies because they are: 53 small (125 μm diameter and smaller), biocompatible, mechanically compliant, chemically 54 inert, resistant to corrosive environments, immune to electromagnetic interference, and 55 are capable of simultaneous multi-parameter sensing when suitably configured [1, 7-11]. 56 Moreover, multiple FBG sensors can be multiplexed along a single optical fibre thereby 57 allowing spatially distributed measurements [12]. FBG-based contact force sensors 58 typically comprise Bragg gratings in birefringent optical fibre. 59 Birefringent fibre has different refractive indices along two orthogonal, or 60 principal, directions that are commonly referred to as the slow (higher index) and fast 61 (relatively lower index) axes. The difference in the refractive indices along the slow and 62 fast axis is termed the birefringence. A Bragg grating in a birefringent fibre reflects two 63 spectra, one polarized along the fast axis and one along the slow axis, with centre 64 wavelength of each spectrum given by the Bragg condition for reflection [12]. The centre 65 wavelength of each spectrum is a function of, among other parameters, the refractive 66 index of the fast and slow axis. 67 The physical mechanism causing sensitivity to contact force is stress-induced 68 changes in birefringence [13] that are governed by the stress-optic effect [14]. Contact 69 forces on the fibre induce stresses (or strains), the principal directions of which are 70 uniform in the region of the fibre core. However, the magnitudes of the principal stresses 71 are a function of the magnitude of the contact force [4]. Therefore, when the contact 2 Super-structured D-shape fibre Bragg grating contact force sensor 72 force magnitude changes, a predictable change in the principal stress magnitudes and, 73 therefore, birefringence is induced. These force-induced changes in birefringence cause 74 predictable changes in the spectrum reflected by the Bragg grating, including shifts in the 75 Bragg spectra corresponding to the fast and slow axes of the fibre. Conversely, applied 76 axial forces/strains and changes in fibre temperature cause relatively lower, in the case of 77 certain fibres insignificant, changes in fibre birefringence [12]. Therefore, one advantage 78 of Bragg grating sensors in birefringent fibres is that force measurements are not 79 confounded by changes in axial strain and temperature. More specifically, D-shaped 80 optical fibre has been shown to experience virtually no change in fibre birefringence with 81 changes in fibre temperature [4] or changes in axial strain that can result from, among 82 other influences, fibre bending [15]. 83 Bragg gratings in birefringent fibres possess higher sensitivity to contact force 84 than gratings in non-birefringent fibres because they are typically constructed with stress 85 concentrating features embedded in the fibre clad, or clad geometries, that increase force- 86 induced changes in birefringence [12]. These features are generally referred to as stress- 87 applying parts. Each commercially available birefringent fibre possesses unique stress- 88 applying parts or clad geometry. Therefore, the contact force sensitivity of each type of 89 birefringent fibre is unique. Contact force sensitivity is also a function of the orientation 90 of the stress-applying parts relative to the direction of load [4]. Therefore, one limitation 91 of birefringence-based contact force sensors is the necessity to control fibre orientation 92 with respect to the contacting surface applying the force. 93 A number of researchers have studied the relationship between contact force and 94 birefringence for several types of birefringent fibres. Udd et al. (1996) [16] applied Bragg 3 Super-structured D-shape fibre Bragg grating contact force sensor 95 gratings in 3M, Fujikura and Corning birefringent fibres to simultaneously measure 96 temperature and force-induced axial and transverse strain. Wierzba and Kosmowski 97 (2003) [17] applied Side-Hole birefringent fibre to force measurements. Chehura et al. 98 (2004) [4] investigated the force sensitivity of D-shape, Elliptical core, TruePhase, 99 Panda, Bow Tie and Elliptical clad birefringent fibres as a function of fibre orientation. 100 More recently, Abe et al. (2006) [18] measured force using chemically-etched Bow Tie 101 and Elliptical clad birefringent fibres to understand the influence that reduction in fibre 102 diameter has on fibre birefringence and force sensitivity. 103 In all of the studies outlined above, fibre orientation relative to the contact force 104 was controlled. However, there are applications for FBG-based contact force 105 measurements where fibre orientation is difficult or impossible to control. One example 106 application from the authors’ previous work is contact force measurements between the 107 articulating surfaces of cartilage within intact human joints such as the hip [19]. Contact 108 force measurements over the cartilage surfaces of the hip, and other articular joints, can 109 indicate the mechanics of the joint, which is closely linked to the etiology of joint 110 degeneration and osteoarthritis [20]. In an intact human hip, the contoured cartilage 111 surfaces of the femoral head and acetabulum are covered by a synovial fluid layer and 112 sealed by a fibrous joint capsule. The presence of synovial fluid within the joint prevents 113 permanent fixation of sensors to cartilage surfaces with conventional adhesives or 114 mechanical fixation. Because fixation is impossible, the orientation of fibre-based sensors 115 with respect to the cartilage surfaces cannot be controlled. Moreover, because the joint 116 capsule surrounds the entire joint and prevents optical access, it is impossible to orient 4 Super-structured D-shape fibre Bragg grating contact force sensor 117 sensors relative to cartilage surfaces using the techniques applied in the above 118 applications of birefringent fibre. 119 To avoid the challenge of controlling sensor orientation, a non-birefringent Bragg 120 grating contact force sensor, with orientation-independent force sensitivity, was 121 developed and applied ex vivo to intact cadaveric hips. This sensor addresses limitations 122 of the current standard film-based sensors [21, 22] for contact force measurements in 123 joints because it can be implanted while leaving the joint capsule and synovial fluid 124 layers intact. 125 The contact force sensor comprises a non-birefringent optical fibre containing a 1 126 mm FBG that is coaxially encased within a silicone annulus and polyimide sheath 127 (outside diameter 240 microns) [19]. Contact forces applied to the outside of the sheath 128 cause compressive strain in both the sheath and silicone annulus surrounding the optical 129 fibre. Because the silicone annulus is approximately incompressible, the annulus volume 130 is conserved by tensile Poisson strain along the axis of the optical fibre containing the 131 FBG. Furthermore, because the magnitude of Poisson strain is directly proportional to the 132 contact force, Bragg wavelength shifts of the FBG within the non-birefringent fibre are 133 directly proportional to contact force. 134 This sensor exhibits orientation independence of sensitivity because of its coaxial 135 configuration. However, like other sensors based on FBGs written in non-birefringent 136 fibre, this sensor is also sensitive to both applied axial strain and temperature changes, 137 which are two mechanical parameters that can confound force measurements on the 138 contoured surfaces of articular joints. More recently, contact force sensors based on tilted 139 Bragg gratings have been proposed that also exhibit orientation independence of 5 Super-structured D-shape fibre Bragg grating contact force sensor 140 sensitivity [23]. However, these approaches are also susceptible to confounding errors 141 associated with axial strain, bending and temperature. (mention limited in multiplexing 142 due to high spectral width?) 143 The objective of the work reported here is to develop a birefringent Bragg 144 grating-based contact force sensor hat addresses limitations of non-birefringent Bragg 145 grating sensors, namely co-sensitivity to axial strain and temperature, and one limitation 146 of Bragg grating sensors in birefringent fibres: the necessity to control sensor orientation. 147 D-shape birefringent fibre is used in the sensor presented because it possesses negligible 148 co-sensitivity to temperature and axial strain. 149 2. Materials and methods 150 151 2.1 Principles of FBGs, birefringence and D-shape optical fibre FBGs are formed in optical fibres by creating a periodic variation in the refractive 152 index of the fibre core [12, 24]. The length of the FBG and the magnitude and period of 153 the variation in the refractive index determine the optical spectrum that is reflected by the 154 FBG [12, 25]. When light spanning a broad range of wavelengths travels along the core 155 of a non-birefringent fibre and encounters a Bragg grating (Figure 1a), a single-peaked 156 spectrum of wavelengths is reflected. This spectrum is centered at the Bragg wavelength, 157 B , which is given by: 158 B 2n0 (1) 159 where, as shown in Figure 1a, is the spatial-period of the variation in the refractive 160 index and, n0 , is the effective refractive index of the fibre core [12]. However, when light 161 travels along the core of a birefringent optical fibre and encounters a grating, two single 162 peaked spectra are reflected. One spectrum corresponds to light polarized along the slow 163 axis (s) and the other corresponds to light polarized along the fast axis (f) (Figure 1b): 6 Super-structured D-shape fibre Bragg grating contact force sensor s 2ns f 2 n f 164 (2) 165 where the subscripts s and f denote the slow and fast axis, respectively. In general, 166 because ns and n f differ in a birefringent fibre, the slow and fast axis Bragg wavelengths 167 will be different, and the magnitude of the difference is termed the spectral separation. 168 The spectral separation between the slow and fast axis Bragg wavelengths, s f is 169 given by: 170 s f 2(ns n f ) 2B 171 where the refractive index difference between the slow and fast axis is referred to as the 172 birefringence, B . 173 174 175 (3) For any optical fibre, the birefringence in the fibre core can be expressed as the sum of three contributions [26]: B ns n f BG BIS BE (4) 176 where BG is the geometric contribution that, typically, is found only in optical fibres with 177 asymmetric or elliptical cores; BIS is the internal stress contribution that, typically, is 178 found in optical fibres with internal stress applying parts; and BE is the external 179 contribution to fibre birefringence that is typically caused by externally applied contact 180 forces. There are two main classes of birefringent fibre that are distinguished by the 181 predominant birefringence contribution that exists in the fibre core: either internal stress- 182 induced birefringence or geometric birefringence. 183 In fibres with internal stress-induced birefringence, the difference in refractive 184 index is created primarily by thermal residual stresses that are created within the fibre 7 Super-structured D-shape fibre Bragg grating contact force sensor 185 when the optical fibre cools from its drawing temperature to ambient temperature. 186 Typically, there is no geometric contribution to birefringence. The thermal residual 187 stresses are created by different thermal expansion coefficients of the fibre clad, stress 188 applying parts which are embedded in the clad, and core. When the fibre, which initially 189 has uniform temperature throughout its cross section, cools from its drawing temperature 190 to ambient temperature thermal residual stresses evolve because the clad, stress applying 191 parts and core undergo differing volume contractions. The configuration of the stress 192 applying parts results in a uniform principal stress field in the region of the core, which 193 through the stress-optic effect [14] results in a uniform difference in refractive index, or, 194 alternatively, uniform BIS throughout the core. In the context of contact force sensing, 195 applied contact forces cause changes in the birefringence contribution BE and, therefore, 196 the Bragg wavelengths associated with the fast and slow axis of the fibre. The 197 mechanisms underlying changes in Bragg wavelengths will be detailed after the 198 principles of D-shape fibre are discussed. Examples of fibres with stress-induced 199 birefringence include Side-hole, PANDA, and Bow Tie. (Need more refs for this 200 paragraph). 8 Super-structured D-shape fibre Bragg grating contact force sensor 201 202 203 204 205 206 207 208 Figure 1: a) schematic of D-shape fibre showing the clad, core and Bragg grating comprising regions of modified refractive index, n spaced with period, Λ. b) schematic of D-shape fibre showing variable names assigned to nominal major and minor outside diameters of fibre (Dmajor and Dminor) and minimum core offset from clad, r; major and minor diameters of the elliptical core (dmajor and dminor); and the directions of the fibre axis, slow axis and fast axis. 209 D-shape optical fibre (Figure 1a and 1b) is one example of the second class of 210 birefringent fibre that is based on geometric birefringence. While the physics and 211 mathematics behind geometric birefringence is beyond the scope of this work, a 212 qualitative discussion of birefringence in D-shaped fibres with elliptical cores will be 213 included. 214 The majority of birefringence in D-shaped fibres is caused by the geometry of the 215 fibre clad and elliptical core [27]. More specifically, the geometric birefringence of an 9 Super-structured D-shape fibre Bragg grating contact force sensor 216 optical fibre with an elliptical core is a function of the core location relative to the air 217 surrounding the fibre, r (Figure 1b); the major and minor diameters of the elliptical core, 218 dmajor and 219 through the core; and the refractive indices of the clad, core and air surrounding the fibre 220 [28]. The internal stress contribution to birefringence is negligible at the wavelengths 221 used for Bragg grating sensing applications [29]. When the major axis of the fibre core is 222 aligned as shown in Figure 1b, the fast axis of the fibre is normal to the flat-side of the D- 223 shape clad (Figure 1b). The slow axis is orthogonal to the fast axis and is aligned with the 224 major axis of the elliptical core (Figure 1b). In the context of force sensing, applied 225 forces cause changes in the birefringence contribution, BE and, therefore, the Bragg 226 wavelengths of the fast and slow axes. dminor, respectively (Figure 1b); the wavelength of the light propagating 227 There are two mathematical formulations that can be used to calculate changes in 228 the Bragg wavelengths of the fast and slow axes as a function of applied contact forces. 229 The first, or stress-optic, formulation relates changes in contact force-induced principal 230 stresses to changes in fibre birefringence. The changes in birefringence can then be used 231 to calculate spectral separation (Equation 3). 232 The second, or strain-optic, formulation can calculate both spectral separation and 233 the magnitudes of the Bragg wavelengths for the fast and slow axes. This formulation 234 relates changes in force-induced strains to changes in the Bragg wavelength of the fast 235 axis and slow axis, which can then be used to calculate spectral separation. Because 236 experimental data for the D-shape sensor will comprise fast axis, slow axis and spectral 237 separation data, we will apply the strain-optic formulation because it can be used to 238 calculate Bragg wavelengths for the fast and slow axes and spectral separation. 10 Super-structured D-shape fibre Bragg grating contact force sensor 239 240 2.2 Strain-optic principles As mentioned above, the centre wavelength of a Bragg grating changes as a function of 241 the mechanical strains in the core of the optical fibre. In the case of birefringent optical 242 fibre, the changes in the Bragg wavelength of the light polarized along the slow axis (x- 243 axis) and fast axis (y-axis) are given by: n2 s s z s pxz z pxx x pxy y 2 244 n 2f f f z pyz z pyx x pyy y 2 (5) 245 where denotes a change in Bragg wavelength; denotes mechanical strain with the 246 subscripts referencing the fibre coordinate system shown in Figure 1b; and 247 pxz pxy p yx p yz 0.252 and pxx p yy 0.113 are strain-optic constants [12]. Note 248 that the subscripts on the photoelastic constants are referred to the fibre coordinate 249 system. To calculate the Bragg wavelength of the fast and slow axis for any state of strain 250 the changes in Bragg wavelength (Equations 5) are added to the initial Bragg 251 wavelengths (i.e. the Bragg wavelengths when the fibre is unstrained). 252 To predict the contact force-induced changes in Bragg wavelength for the sensor 253 presented in this work the equations relating contact force and mechanical strain must be 254 established. To establish the equations, the sensor geometry and orientation to contact 255 force is presented. 256 257 2.3 Super-structured D-shape fibre sensor and force/strain models As shown in Figure 2a, the contact force sensor consists of a D-shaped birefringent fibre 258 around which there is a super-structure that both transmits contact forces to the fibre and 259 maintains the orientation of the sensor with respect to the contact force (Figure 2b). The 260 super-structured D-shape fibre sensor, hereafter referred to as the sensor, comprises an 11 Super-structured D-shape fibre Bragg grating contact force sensor 261 outside Polyimide© sheath (SmallParts Inc., Miami FL, Ds = 165 micron, tW = 18 micron 262 nominally) that contains a Nitinol© wire (Need the manuf. Details, Dw = 51 micron 263 nominally) and a D-shape birefringent fibre (KVH Industries Inc., Middletown RI, Dmajor 264 = 125 micron, Dminor = 76 micron, r = 14 micron, dmajor = 8 micron, dminor = 5 micron) 265 with a 5 mm Bragg grating photo-inscribed in the fibre core (TechnicaSA, Beijing China, 266 the grating details). The length, along the fibre axis (Figure 1b), of the Polyimide© 267 sheath and Nitinol© wire is 12 mm at the center of which is the 5 mm Bragg. The 268 clearance space (Figure 2a) between the sheath, wire and fibre is filled with a compliant 269 adhesive (Dow Corning®, Midland MI, silicone 3-1753) that serves to maintain the 270 alignment of the features comprising the super-structure. A Polyimide© alignment 271 feature (Figure 2a) is affixed to the outside of the Polyimide© sheath by submerging the 272 sheath in a heat curable Polyimide© solution (HD MicroSystems™, Parlin NJ, solution 273 PI-2525) and curing it into the geometry shown. The length of the alignment feature is 274 nominally the same as the length of the sheath (i.e. 12 mm). The width of the alignment 275 feature was XXX (I actually made a bunch of them, how can I convey this). 276 Contact forces, F, are applied to the outside of the sheath and are aligned with the 277 internal features (e.g. wire and fibre) as shown in Figure 2b (alignment feature omitted 278 for clarity). The upper force is transmitted through the sheath, wire, and onto the fibre at 279 contact 1. The lower force is transmitted through the sheath and onto the fibre at contact 280 2. 281 The key difference between the sensor shown in Figure 2 and force sensors based 282 on bare (i.e. not superstructured) D-shape fibre [4] is that the contact force transmitted to 283 the flat side of the D-shape fibre of the super-structured sensor (Figure 2b) is 12 Super-structured D-shape fibre Bragg grating contact force sensor 284 concentrated by the wire. Conversely, force sensors using bare D-shape fibre distribute 285 forces over the entire flat side of the D-shape clad (Figure 1). As will be shown, 286 concentrating the contact force onto the fibre through the wire will result in increased 287 force sensitivity of the super-structured D-shape sensor relative to force sensors based on 288 bare D-shape fibre. This increase in force sensitivity is predicted by the equations, 289 developed below, relating contact force to strain in the fibre core. 290 13 Super-structured D-shape fibre Bragg grating contact force sensor 291 292 293 294 295 296 297 Figure 2: a) schematic of super-structured D-shape sensor cross section showing Polyimide© sheath, Nitinol© wire, D-shape fibre and Polyimide© alignment feature. b) contact forces, F are applied to the outside of the sheath and are aligned with the internal features of the sensor as shown. The upper contact force is transmitted through the sheath, to the wire and to the fibre at contact 1. The lower force is transmitted through the sheath to the fibre at contact 2. 298 A plane elasticity model is applied to calculate the stresses and strains created in 299 the D-shape fibre core as a result of applied contact forces. To calculate the stresses and 300 strains, the D-shape fibre is approximated as an elastic half space [30] that is subjected to 301 contact forces of magnitude F (N/mm, normalized to sensor length along the fibre axis). 302 One of the key assumptions associated with elastic half space theory is that 303 stresses and strains vanish at distances infinitely far from the point of force application. 304 In the case of D-shape fibre, which is finite in size, this assumption cannot be satisfied. 305 To address this seemingly irreconcilable difference between the boundary conditions of 306 half space theory and the D-shape fibre sensor a composite half space model is proposed 307 that accounts for the shape and boundary conditions of the super-structured D-shape fibre 308 sensor. 309 As shown in Figure 3a, elastic half space formulations of contact consist of a 310 contact force applied at the origin of a large elastic continuum [30]. To satisfy boundary 311 conditions, when x and z are large all stresses and strains vanish and static equilibrium is 312 achieved by internal stresses that balance the external force. The stresses at any location 313 in the elastic continuum are: 14 Super-structured D-shape fibre Bragg grating contact force sensor x2y x (x 2 +y 2 ) 2 2F 314 y3 (x 2 +y 2 ) 2 y 2F xy 2F (6) xy 2 (x 2 +y 2 ) 2 315 where x , y , xy are the normal stresses aligned with the x and y directions and the 316 shear stress that shears the xy plane [30]. 317 To address the difference in boundary conditions between the elastic half space 318 (Figure 3a) and the D-shape fibre sensor (Figure 3b) Equations 6 are adapted to formulate 319 the stresses for the composite half space. 320 321 322 323 324 325 326 327 328 329 Figure 3: a) schematic showing contact force, F applied to elastic half space with zero stress boundary condition at regions far from co-ordinate system origin. b) schematic showing D-shape fibre cross section as a composite elastic half space. The core of the fibre is located relative to two co-ordinate systems with origins at contact 1 and contact 2. The co-ordinate of the core relative to contact 1 and contact 2 is x1, y1 and x’2, y’2, respectively. c) schematic representation of half space model for a bare D-shape fibre subjected to distributed contact force over the flat side of the clad (pressure, P) and concentrated contact force on the round side of the clad. 330 The sensor (Figure 2) is approximated using the boundary conditions shown in 331 Figure 3b. We assume that the contact forces applied to the top and bottom of the sheath 15 Super-structured D-shape fibre Bragg grating contact force sensor 332 (Figure 2b) are transmitted to the D-shape fibre. To calculate the stresses at any location 333 in the D-shape fibre, we assume that the state of stress in the fibre is due to a stress field, 334 given by Equations 6 due to the contact 1 and a stress field, also given by equations like 335 6, due to contact 2. Because we assume linear elasticity, we apply superposition to find 336 the total state of stress due to these contacts as: 337 x x 22 y'2 2 F x12 y1 + (x12 +y12 ) 2 (x 22 +y'22 ) 2 y y'32 2 F y13 + 2 2 2 2 2 2 (x1 +y1 ) (x 2 +y'2 ) xy x 2 y'22 2 F x1 y12 + (x12 +y12 ) 2 (x 22 +y'22 ) 2 (7) 338 In the region of the centre of the fibre core, where the majority of light is transmitted, the 339 dimensions x are small relative to the dimensions y and Equations 7 can be simplified as: x 0 y 340 2F 1 1 y1 y'2 (8) xy 0 341 Based on Equations 8, in the region of the centre of the fibre core there is negligible shear 342 and there is one principal stress. The simple state of stress can be used with Hooke’s Law 343 for plane strain [30] to obtain strains: (1 ) y 1 (1 2 ) x (1 ) y E E (1 2 ) y 1 y (1 2 ) y (1 ) x E E 2(1 ) xy xy 0 E x 344 (9) 16 Super-structured D-shape fibre Bragg grating contact force sensor 345 where E and are the Young’s modulus and Poisson ratio of the silica glass of the fibre, 346 77 GPa and 0.17, respectively [29] (check these values and the reference). When 347 Equations 7 are substituted into Equations 9, the resulting expressions for strain are 348 functions of contact force, F, and the co-ordinates of the core relative to contact 1 and 349 contact 2. For the sensor presented in Figure 2, y1 and y’2 are constant: 14 microns and 62 350 microns, respectively. 351 For the purpose of eventual comparison of the contact force sensor presented in 352 this work and contact force sensors based on bare D-shape fibre, we have also developed 353 a half-space model to estimate the stresses and strains in a bare D-shape fibre (Figure 3c). 354 In this model, contact force on the flat side of the clad is distributed, shown as contact 355 pressure, P, while contact force on the opposite side of the fibre remains concentrated. 356 We apply similar reasoning as that described for Figure 3b to obtain the stresses in the 357 region of the fibre core as: x 358 y P P xy 0 sin sin 2F 1 y'2 (10) 359 360 where P is the applied pressure (N/mm2) and α is the included angle (i.e. 2.73 radians for 361 D-shape fibre) that is subtended by the pressure. The boundary conditions shown in 362 Figure 3b do not completely satisfy those of half-space theory because the pressure 363 boundary condition extends to dimensions comparable to the fibre diameter. Nevertheless, Equations 10 can serve to approximate the state of stress and be used to 17 Super-structured D-shape fibre Bragg grating contact force sensor 364 365 gain insight into increases/decreases in sensitivity between the sensor presented here and 366 those based on bare D-shape fibre. The state of stress given by Equations 10 can be 367 substituted into Hooke’s Law for plane strain to obtain the strains in a similar form to 368 those presented as Equations 9. 369 The force-induced strains for the sensor presented here (Equations 9), and those 370 for a bare D-shape sensor, can then be substituted into Equations 5 to calculate the Bragg 371 wavelength shifts for the fast and slow axis for the sensor presented in this work and that 372 of a bare D-shape fibre subjected to contact force. 373 The process of Bragg wavelength shift calculation described above was 374 completed for contact forces ranging from 0 N/mm to 2 N/mm to predict force 375 sensitivity. Equations 5 were also used to calculate the axial strain sensitivity, for the 376 contact force sensor, as described below. 377 As will be described in detail below, the contact force sensor was tested 378 experimentally for sensitivity to axial strain. The experimental set-up for this test 379 consisted of the force sensor affixed to the top of an aluminum cantilever. Prescribed 380 deflections were applied to the cantilever, and the fast and slow axis Bragg wavelengths 381 were monitored. To estimate axial strain at the sensor, equations from classical beam 382 theory [31] were used to calculate the axial strain at the location along the beam of the 383 force sensor. To verify that results from this experiment agreed with strain-optic theory 384 and the known behavior of D-shape fibres subjected to axial strain [15], the axial strains, 385 and corresponding transverse Poisson strains, for the cantilever beam were used with 18 Super-structured D-shape fibre Bragg grating contact force sensor 386 Equations 5 to predict the Bragg wavelength shifts. Verification involved comparison of 387 the experimentally measured wavelength shifts to the strain-optic model predicted 388 wavelength shifts. 389 2.4 Sensor calibration protocols 390 A single force sensor was constructed and calibrated for sensitivity to contact-force, axial 391 strain and temperature in three separate calibration protocols. 392 The apparatus used for force calibration was identical to that used in a previous 393 study (Figure 4a) [19]. The force sensor and a support cylinder of identical outside 394 diameter were subjected to contact force between two metrology gauge blocks (Class 0, 395 24.1 mm × 24.1 mm, steel, Mitutoyo Can., Toronto, ON). The bottom gauge block 396 supported the force sensor and support cylinder and the top block applied contact forces. 397 Both gauge blocks were constrained by guide blocks allowing motion only in the 398 direction of load application. 399 Contact forces were applied by compressing a calibrated spring with a manual 400 screw-follower (not shown in Figure 4a). These forces were transmitted through a pre- 401 calibrated load cell (445 N capacity, ±0.1% FS non-repeatability, Futek Inc., Irvine, CA), 402 fixed to the top gauge block, connected to a data-acquisition system implemented in 403 LabView™ (version 8, National Instruments Inc., Austin, TX). The force was transmitted 404 through the top gauge block, through the sensor and support cylinder and finally to the 405 bottom gauge block. Applied contact forces in all calibrations ranged from 0 N mm−1 to, 406 nominally, 2 N mm−1. This range of forces was chosen because it brackets the range of 407 forces experienced in the hip during walking [57] 19 Super-structured D-shape fibre Bragg grating contact force sensor 408 409 410 411 412 413 414 Figure 4: a) schematic showing relevant features of contact force calibration apparatus. Force is applied by compressing a calibrated spring that is in contact with a load cell and top gauge block. Forces are transmitted through the force sensor and support cylinder, and to the bottom gauge block. b) schematic showing forces applied to force sensor without alignment feature, and c) with alignment feature. 415 To allow comparison of sensor sensitivity for a sensor without and with an 416 alignment feature, calibration was completed first for a sensor without an alignment 417 feature (Figure 4b), and subsequently with the alignment feature (Figure 4c). For the 418 sensor without the alignment feature, sensor orientation was controlled with an external 419 fibre rotator while contact forces were applied. The alignment feature was then added to 420 the sensor and calibration was repeated without external orientation control. One 421 calibration was performed for the sensor without an alignment feature, while three were 422 performed for the sensor with the alignment feature. Bragg wavelength shifts of the fast 20 Super-structured D-shape fibre Bragg grating contact force sensor 423 and slow axis were measured using an optical spectrum analyzer (ANDO AQ6331, 424 Tokyo, Japan) as described below. 425 Bragg wavelengths were demodulated by directing light from a broad C-band 426 light source (AFC-BBS1550, Milpitas, California) into a linear polarizer (PR 2000, JDS 427 Uniphase, Milpitas, California) and then into one of the input channels of a 3 dB optical 428 coupler (Blue Road Research). The light was then directed via the coupler to the FBG in 429 the test fiber, and the reflected spectrum was directed back through the optical coupler 430 and into the optical spectrum analyzer. The polarization of the light was adjusted to 431 illuminate either the fast or the slow axis of the fiber by using the tuning facilities of the 432 linear polarizer. The demodulation scheme described is similar to that presented by 433 numerous researchers [3, 32, 33]. 434 The sensitivity to force, for both the slow and the fast axes, was calculated by 435 using linear regression as the slope in the recorded data (slope ± standard deviation for 436 slope), in terms of wavelength shift versus applied force. Sensitivity was also calculated 437 based on the spectral separation of the slow and fast axes (i.e., the difference between the 438 slow and the fast axis sensitivities). 439 In the axial strain protocol, the force sensor was affixed to the top surface of an 440 aluminum cantilever (Figure 5, w = 25 mm, h = 0.25 mm, L = 45 mm, and x”s = 10 mm) 441 using a UV curable adhesive (Norland Products Inc., Cranbury NJ, Blocking adhesive 442 107). To create axial strains in the cantilever, prescribed deflections, δ, were created at 443 the end of the cantilever. The prescribed deflections were measured using a mechanical 444 indicator (Mitutoyo Corp., JP, Indicator 2046F, 10 mm range), while Bragg wavelengths 445 of the fast and slow axis were recorded. As described in Section 2.3, the axial strains 21 Super-structured D-shape fibre Bragg grating contact force sensor 446 transmitted to the sensor from the cantilever were estimated from the deflections using 447 classical beam equations [31] by assuming that the tensile strains at the sensor/beam 448 interface (Section A-A, Figure 5) were transmitted to the sensor. The estimated axial 449 strains ranged from 0 με to 70 με. The protocol was conducted twice. The sensitivity to 450 strain, for both the slow and the fast axes and based on spectral separation, was calculated 451 by using linear regression as the slope in the recorded data, in terms of wavelength shift 452 versus strain. 453 454 455 456 457 458 Figure 5: schematic showing relevant feature axial strain calibration apparatus. The force sensor was affixed to the top of the cantilever (Section A-A). Prescribed deflections were applied to the end of the cantilever (x” = L). 459 Controls Inc., Winona MN, Model CascadeTEK, controller series 982). The force sensor 460 and a pre-calibrated thermocouple (Omega Engineering Inc., Stamford CT, Type-T), 461 interrogated with a thermocouple amplifier (Omega Engineering, Super MCJ), were fixed 462 to a glass slide (VWR International™, 76 mm by 26 mm by 1 mm) and placed within the Temperature calibration was performed within a controllable oven (Watlow 22 Super-structured D-shape fibre Bragg grating contact force sensor 463 oven. The temperature, as reported by the thermocouple, of the force sensor was 464 increased from 30 ˚C to 60 ˚C, in 5 ˚C increments, while the Bragg wavelengths of the 465 fast and slow axis were recorded. This temperature range was chosen because it spans a 466 larger range than that experienced in most ex vivo experiments with cadaveric material. 467 temperature The protocol was repeated three times. The sensitivity to temperature, for 468 both the slow and the fast axes and based on spectral separation, was calculated by using 469 linear regression as the slope in the recorded data, in terms of wavelength shift versus 470 temperature. 471 472 473 3. Results Figure 6 shows spectra recorded using the optical spectrum analyzer for both the fast and 474 slow axis. As shown, the Bragg spectra of both the fast and slow axis are symmetric 475 about their initial Bragg wavelength when the sensor is not subjected to contact force. As 476 force is applied to the sensor, both the fast and slow axis Bragg wavelengths shift to 477 longer wavelengths while retaining symmetry. The symmetry exhibited by the fast and 478 slow axis spectrums, when the sensor is subjected to force, indicates that the strains 479 experienced along the Bragg grating are uniform along the grating length. Conversely, 480 non-uniform strains over the grating length would manifest in increased width, along the 481 wavelength axis, of the Bragg spectrums and decreased peak light intensity [25]. As 482 shown in Figure 6, the slow axis Bragg wavelength experiences greater shift than the fast 483 axis. Therefore, as described below, the slow axis Bragg wavelength exhibits greater 484 sensitivity to contact force than the fast axis (Figure 7 and Table 1). 485 23 Super-structured D-shape fibre Bragg grating contact force sensor 486 487 488 489 490 491 492 493 Figure 6: Bragg spectra for the fast and slow axis recorded using the optical spectrum analyzer. Contact forces applied to sensor cause shifts to longer Bragg wavelengths. The symmetry of the spectra indicate that strains along grating are uniform while the relative shifts of the slow and fast axis indicate that the slow axis has greater force sensitivity than the fast axis. 494 wavelengths obtained from the half-space/strain-optic model, while Figure 7b shows the 495 measured Bragg wavelength shifts for the contact force sensor without the alignment 496 feature. As shown, for the super-structured sensor, the model predicted (Figure 7a) and 497 measured (Figure 7b) slopes, or sensitivities to contact force, match to within 1.8 % and Figure 7a shows the predicted wavelength shifts of the fast and slow axis Bragg 24 Super-structured D-shape fibre Bragg grating contact force sensor 498 13.2 % (relative to experimental slope) for the slow and fast axis, respectively. Relative 499 to a bare D-shape sensor, the modeled slow and fast axis sensitivities for the super- 500 structured sensor are 125.4 % and 3.5% greater than those modeled for a bare D-shape 501 sensor. 502 25 Super-structured D-shape fibre Bragg grating contact force sensor 503 504 505 506 507 508 509 510 511 512 513 514 515 Figure 7: a) Predicted sensitivities of the fast and slow axis Bragg wavelengths obtained from the half-space/strain-optic model for both the super-structured sensor and a bare Dshape sensor. b) measured wavelength shifts of the fast and slow axis Bragg wavelengths for the sensor without the alignment feature. Sensitivities are reported as slope from regressions (dashed lines). Error bars shown are not clearly visible at given scale but convey non-repeatability of contact force measurements (±0.02 N/mm) and observed non-repeatability of wavelength shift measurements (mean of ±1 pm for all data points presented) from optical spectrum analyzer. Table 1: Summary of experimental results for slow and fast axis sensitivity to contact force, axial strain and temperature of contact force sensor. Sensitivities reported obtained using linear regression and are reported as the best-fit slope ± the standard deviation on slope. Correlation coefficients, r2, also obtained from regression calculations. Sensitivity (pm/measurand unit) Calibration protocol Measurand unit Trial Slow axis r† 2 Fast axis r† 2 Spectral separation force, without alignment feature (N/mm) 1 255.9±21.1 0.96 61.66±7.98 0.91 194.2 force, with alignment feature (N/mm) 1 2 3 mean 228.6±3.98 232.1±4.30 234.3±4.06 231.7 0.99 0.99 0.99 0.99 62.42±3.71 62.48±4.05 61.57±5.36 62.16 0.98 0.98 0.96 0.97 166.2 169.6 172.7 169.5 axial strain, with alignment feature (micro-strain) 1 2 mean 1.100±0.0786 1.130±0.0827 1.115 0.97 0.96 0.97 1.091±0.0996 1.129±0.0240 1.11 0.95 0.99 0.97 0.009‡ 0.001‡ 0.005‡ degree Celsius 1 2 3 mean 8.435±0.156 8.451±0.126 8.408±0.144 8.431 0.99 0.99 0.99 0.99 9.577±0.310 9.527±0.309 9.548±0.310 9.551 0.99 0.99 0.99 0.99 -1.142 -1.076 -1.140 -1.119 temperature, with alignment feature notes: 516 517 518 † r2 is abbreviated notation for linear correlation coefficient from least squares regression ‡ only one significant digit reported because of relatively low magnitude compared to slow and fast axis sensitivity Figure 8 shows typical measured Bragg wavelength shifts for the contact force 519 sensor with the alignment feature (trial 1, results also noted in Table 1). As shown by the 520 results in Table 1 (force, with alignment feature), both the slow and fast axis sensitivities 521 are nearly constant over the three trials in which the sensor aligns itself (experimental 522 configuration shown in Figure 4c. The lowest and highest sensitivities over the three 523 trials deviate from the tabulated mean sensitivities for the slow and fast axis by only 524 -1.3% and 1.1% for the slow axis and -0.9% and 0.5% for the fast axis (Table 1). The 26 Super-structured D-shape fibre Bragg grating contact force sensor 525 lowest and highest spectral separation sensitivities deviate from the mean (i.e. 169.5 526 pm/(N/mm)) by only -1.9% and 1.9% (Table 1). 527 528 529 530 531 532 533 534 Figure 8: measured wavelength shifts of fast and slow axis Bragg wavelengths for contact force sensor with alignment feature (trial 1 in Table 1). Error bars not clearly visible at given scale but convey non-repeatability of measurements conveyed in caption for Figure 7b. Figure 9a shows the strain/strain-optic model predicted wavelength shifts for the 535 fast and slow axis as a function of axial strain along the Bragg grating. As shown, the 536 model predicted sensitivities of the fast and slow axis differ by only 0.008% of the mean 537 sensitivity of the fast and slow axis (i.e.1.24475 pm/microstrain). The measured changes 538 in Bragg wavelength versus axial strain (Figure 9b) also show nearly identical sensitivity 539 of the fast and slow axis (also shown in Table 1, axial strain, with alignment feature). The 540 mean difference between the fast and slow axis sensitivities reported for trial 1 and 2, 541 shown in Table 1, is only 0.18%. 542 27 Super-structured D-shape fibre Bragg grating contact force sensor 543 544 545 546 547 548 549 Figure 9: a) Predicted sensitivities to axial strain of the fast and slow axis Bragg wavelengths obtained from the strain/strain-optic model. b) measured (trial 2) wavelength shifts of the fast and slow axis Bragg wavelengths for the sensor with the alignment feature. Sensitivities are reported as slope from regressions (dashed lines). Error bars shown are not clearly visible at given scale but convey non-repeatability of axial strain values (±1.69 micro-strain) and observed non-repeatability of wavelength shift 28 Super-structured D-shape fibre Bragg grating contact force sensor 550 551 552 553 measurements (mean of ±1 pm for all data points presented) from optical spectrum analyzer. 554 protocol (Table 1, temperature, with alignment feature). Like the results for axial strain 555 calibration, the fast and slow axis sensitivities to temperature were nearly identical. For 556 example, as shown by the results based on spectral separation (Table 1), the spectral 557 separation sensitivity was only 12.4 % of the mean of the fast and slow axis sensitivities 558 (i.e. 8.99 pm/degree Celsius). 559 560 561 562 563 564 565 566 567 568 569 Figure 10 shows typical results obtained during the temperature calibration Figure 10: measured (trial 1) fast and slow axis Bragg wavelengths versus temperature for the contact force sensor with the alignment feature. Sensitivities are reported as slope from regressions (dashed lines). Error bars shown are not clearly visible at given scale but convey non-repeatability of thermocouple measurements (±0.1 degrees Celsius) and observed non-repeatability of wavelength shift measurements (mean of ±1 pm for all data points presented) from optical spectrum analyzer. 29 Super-structured D-shape fibre Bragg grating contact force sensor 570 571 4. Discussion The contact force sensor presented in this work has desirable characteristics when 572 compared to other birefringence-based force sensors and the authors’ previous work [19] 573 including: increased sensitivity to contact force relative to bare-Dshape fibre and other 574 birefringent fibres; near-constant force sensitivity without external orientation control; 575 and negligible co-sensitivity to extraneous axial strain and temperature changes. Insights 576 gained from the half-space/strain-optic model, with reference to sensor design features, 577 can be used to elucidate the causes of these benefits. 578 As shown by the results plotted in Figure 7, the slow axis Bragg wavelength of 579 the super-structured sensor has sensitivity 125.4 % greater than the slow axis of a bare D- 580 shape fibre. Conversely, the fast axis sensitivities of the super-structured and bare D- 581 shape sensors are nearly identical, and differ by only 3.5%. The principal cause of the 582 increase in slow axis sensitivity is a 273% increase, relative to the case of bare D-shape 583 fibre, in compressive stresses, y , that result from concentrating contact forces at contact 584 1 with the Nitinol© wire. Conversely, fast axis sensitivities of the super-structured and 585 bare D-shape sensors match to within 3.5% because the state of stresses in the two 586 sensors (Equations 8 for super-structured and Equations 10 for bare fibre) lead to 587 mechanical strains in the core that cause nearly identical fast axis wavelength shifts for a 588 given contact force. 589 Experimental measurements reinforce the model-predicted increases in sensitivity 590 for the slow axis of the super-structured sensor. The half-space/strain-optic model 591 predicted slow axis sensitivity is 260.6 pm/(N/mm), which is 45% higher than previously 592 reported maximum slow axis sensitivity for bare D-shape fibre (i.e. 180 pm/(N/mm) [4]). 593 The model-predicted slow axis sensitivity of the bare D-shape fibre is 115.6 pm/(N/mm), 30 Super-structured D-shape fibre Bragg grating contact force sensor 594 which is much lower than the previously sensitivity (180 pm/(N/mm)). 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