Decoupling Analysis of a Sliding Structure Six-axis Force/Torque Sensor

Transcription

1 10.478/msr MEASUREMEN SCIENCE REVIEW, Volume 13, No. 4, 013 Decoupling Analsis of a Sliding Structure Six-axis orce/orque Sensor Bo Wu, Ping Cai Department of Instrument Science and Engineering, Shanghai Jiao ong Uniersit, 0040, Shanghai, China, wb_9941@163.com his paper analzes the decoupling of a sliding structure six-axis force/torque sensor, which is used to measure the interactie force between surgical tools and soft tissue for the establishment of soft-tissue force model. Because this decoupling structure requires accurate sliding clearance and smmetric grooes, the influence of contact force between the elastic bod and the grooe sidewall on decoupling is analzed. he analsis results indicate that the contact force will produce additional coupling error. he robust design method of elastic bod size optimization is used to eliminate the influence of contact force. In the calibration test, the expanded uncertaint of the calibration deice is ealuated and the calibration results alidate the good decoupling. Kewords: ool-tissue interactie force, force/torque sensor, sliding structure, decoupling, robust design, uncertaint. 1. INRODUCION O IMPROVE the operators immersion sense in irtual surger, the haptic information acquisition is carried out to measure the tool-tissue interactie force for the establishment of the soft tissue force model. Man acquisition approaches are based on tensile or compressie test of non-destruction tissue samples [1] [], which cannot reeal the ariation of tool-tissue interactie force for tissue fracture comprehensiel [3]-[6]. Man kinds of sensors can be used to measure the force or torque [7]-[10]. pical surgical operations such as clamping, cutting, puncture and suture feature motion and application of force in a multidegree of freedom, so multi-axis force/torque sensors are used. Dimensional coupling elimination is a ke issue for multiaxis force/torque sensors. A. Gaillet [11] deeloped a multiaxis force/torque sensor based on the Stewart parallel structure. But owing to the coupling signal output of the sensor, the calculation of decoupling is complex. or a more direct acquisition of the force signal, the design of some sensors is based on mechanical structure decoupling of the elastic bod. loat beam structure and sliding structure are tpical mechanical decoupling approaches. A. G. Song [15] deeloped a kind of float beam structure based multidimensional force sensor. When force is applied, the non-sensing beams of the sensor become floating because of the compliance of non-sensing beams. he dimensional coupling is therefore eliminated. Due to the contradiction between compliance of non-sensing beams and sensor oerall stiffness, the decoupling is limited. In the sliding structure based multidimensional force sensor, the nonsensing beams become floating ia sliding along the guiding grooe, then the decoupling can be more effectie. Howeer, this decoupling structure requires accurate sliding clearance and smmetric grooes, especiall for the sensor with small measurement range. Otherwise, contact force between the elastic bod and the grooe sidewall will influence the decoupling. his research analzed the mechanical decoupling mechanism of a designed sliding structure six-axis force/torque sensor and the influence of contact force between elastic bod and grooe on decoupling. he contact force determined b structure size, machining error and assembl error will produce additional coupling error. Owing to the fact that the relationship between contact force and structure size and machining error is non-linear, higher machining and assembl accurac ma not be able to achiee the desired target. hus, a robust design method [17] of sliding structure for structure size optimization is adopted to eliminate the influence of contact force on decoupling.. DECOUPLING ANALYSIS O HE SLIDING SRUCURE SIX-AXIS ORCE/ORQUE SENSOR ig.1. Structure components. (1) elastic bod 1, () elastic bod, (3) top coer, (4) outer shell, (5) bottom coer. he structure components of the six-axis force/torque sensor are shown in ig.1. wo elastic bodies are composed of four identical parallel beams with double holes, respectiel. Elastic bod 1 is designed to measure force x, and torque M z, and elastic bod is designed to measure force z, torque M x and M. he clearance fit between the elastic bod and the grooe of the outer shell makes the elastic bod slide when the force is applied. he measurement range of the sensor for force is 0~0 N and for torque is 0~800 Nmm. he resolution of the sensor is 0 mn. he dimension parameters of the sensor are shown in able

2 able 1. Dimension parameters of the sensor height of the sensor diameter of the sensor height of elastic bod 1 height of elastic bod thickness of the thin wall distance between the two holes h = 8 mm D = 5 mm a = 4 mm b = 8 mm d = 0.5 mm l = 8 mm our strain gauges bonded to the thin wall of each parallel beam of the elastic bod compose a full Wheatstone bridge, thus 3 strain gauges and eight full Wheatstone bridges are needed in total. Each parallel beam can measure a single dimensional force, so two or four beams can be combined to measure the same force/torque for increased sensor sensitiit. (a) Sliding mechanism of elastic bod 1 When bending deformation occurs on beams, there are smmetrical tensile strain and compressie strain on two thin walls on both sides of the beam, respectiel. he sensitiit of output oltage can be increased 4 times with the Wheatstone bridge. When lateral bending deformation occurs on beams, the oerall strain of the strain gauge is zero due to the equal tension and compression on both sides of the strain gauge neutral laer, so beams will not sense lateral bending deformation. When torsion deformation occurs on beams, both normal stress and shear stress are produced in the cross section simultaneousl. Because of the great torsional stiffness of beams, the normal strain is er small and can be neglected. he strain gauges do not sense shear deformation, thus beams will not sense torsional deformation. hus, beams of the elastic bod can sense bending deformation but are insensitie to lateral bending deformation and torsional deformation. he deformation of each beam under a single dimensional force or torque is listed in able. able. shows that beam 11 and beam 13 sense the deformation under the force x ; beam 1 and beam 14 sense the deformation under the force ; beam 11, beam 1, beam 13 and beam 14 sense the deformation under the torque M z ; beam 1 and beam 3 sense the deformation under the torque M x ; beam and beam 4 sense the deformation under the torque M ; beam 1, beam, beam 3 and beam 4 sense the deformation under the force z. he output can be calculated b Uo K = ε (1) U 4 where ε = ( ε1, ε, ε3, ε4, ε5, ε6, ε7, ε8) U = ( U, U, U, U, U, U, U, U ) are o o1 o o3 o4 o5 o6 o7 o8 i and strain and the bridge output oltage of beam 11, beam 1, beam 13, beam 14, beam 1, beam, beam 3 and beam 4, respectiel, K is the gauge factor, and U i is the input oltage of the Wheatstone bridge. Considering the equal deformation of some beams under a single dimensional force or torque, U = ( U, U, U, U, U, U ) can be obtained as x z Mx M Mz the (b) Sliding mechanism of elastic bod ig.. Sliding structure of the six-axis force/torque sensor he sliding structure of elastic bod 1 and elastic bod is shown in ig..(a) and ig..(b), respectiel. When a single dimensional force or torque is applied, sensing beams produce bending deformation and non-sensing beams are floating owing to sliding, or produce lateral bending deformation, or torsional deformation U = U () O () indicates that the sensor is structure decoupling and the sensor sensitiit is doubled or quadrupled. 188

3 able. Deformation under a single dimensional force or torque Deformation mode ( B, -B, LB, denote tensile bending deformation, compressie bending deformation, lateral bending deformation and torsional deformation and 0 means no deformation) beam 11 beam 1 beam 13 beam 14 beam 1 beam beam 3 beam 4 x z M x M B 0 B 0 LB 0 LB 0 0 B 0 B 0 LB 0 LB LB LB LB LB B B B B LB LB B 0 -B 0 LB LB 0 B 0 -B M z B B -B -B LB LB LB LB 3. INLUENCE O CONAC ORCE ON DECOUPLING Contact force between the elastic bod and the grooe sidewall will emerge as a coupling force during the operation. Considering the small measurement range (0~0 N and 0~800 Nmm) of the sensor, it is necessar to analze the influence of the clearance and the asmmetric grooe. Y ig.3.(a) shows the ideal contact mode between the elastic bod and the grooe. ig.3.(b) shows the contact mode with angle θ and deiation distance h of grooe, where the deiation distance of the grooe is h= asinθ. In ig.3.(b), the branch force might deform the non-sensing beam to bring additional coupling. he structure of ig.3.(b) can be equialent to a first-order hperstatic structure of sliding structure in order to calculate the bending moment distribution of each beam, which can produce the bending deformation, as depicted in ig.4. X (a) Ideal contact (a) Mechanical model of elastic bod 1 (b) Contact with an angle ig.3. pical contact modes between the elastic bod and the grooe. (b) Mechanical model of elastic bod ig.4. irst-order hperstatic structure of the elastic bod in the sliding structure. 189

4 As shown in ig.4.(a), four unknown restraint reaction forces and a horizontal force are applied on the elastic bod 1. A redundant ertical reaction force B emerges after remoing the restraint at endpoint B. he force deformation compatibilit condition can be gien as δ 11 B δ1 0 + = (3) where δ11 and δ1 denote the deformation alue of the endpoint B in B direction when force B and are applied to the staticall determinate base, respectiel. δ 11 and δ1 can be obtained according to Mohr s theorem, and the ertical reaction force B is deried from (3) M sin θ B = K 1 = 3 3 Nsin( θ + α) + ( a + b ) / 3 + a b+ ab / (4) where M a b a b ab b = (( ) / ) + / 4, 3 tan (( ab ab)/ b /1)/( ab ab /) α = +, N a b ab a b ab b 3 = ( + /) + (( )/ /1). or M > N, it is clear that K 1 and B increase with the increase of θ. he bending moment of all beams, which is produced b the horizontal force and the ertical reaction force B, can be obtained based on the superposition principle. he strain of elastic bod 1 is gien as L ε5 = ε7 = (cosθ + K1sin θ) (5a) EW L ε 6 = ε8 = (sinθ + K1cos θ) (5b) EW ε1 = ε = ε3 = ε4 = 0 (5c) Assuming is applied in x direction, calculated in accordance with (1) and () U (cosθ + K1 sin θ ) (sinθ K1 cos θ ) + LKU 0 i = 4EW U can be (6) It follows that force x and emerge when force is applied in x direction, and force is considered as a coupling force. he coupling error E r ( x ) can be gien as (sinθ + K1 cos θ ) Er( x) = (cosθ + K sin θ ) rom (4) 0 K1 < 1 is obtained, then Δ sin( θ ) > K1Δsin( θ ) and Δ cos( θ ) > K1Δ cos( θ ) is deried. (6) and (7) indicate that the contact angle produces additional coupling. he coupling error E r ( x ) increases with the increase of θ. When moment M is applied on elastic bod, the mechanical model is shown in ig.4.(b). Similar method to that of elastic bod 1 can be used to obtain the bending moment distribution. Assuming M is applied in x direction, U can be obtained as U 0 0 LKU 0 i = 4EW (sinθ + K sin θ ) (sinθ + K cos θ ) 0 7absinθ where K =. 1ab + 10a he coupling error E r (M x ) produced b as 1 ε M (sinθ + K sin θ ) Er( Mx) = (cosθ + K sin θ ) (7) (8) can be gien (9) shows that the coupling error E r (M x ) increases with the increase of θ. 4. ROBUS OPIMIZAION DESIGN O SENSOR SLIDING SRUCURE Robust optimization design approach is used to ensure good qualit performance when controllable factors and uncontrollable factors deiate from the design alue. his approach aims to minimize the performance sensitiit to noise b selecting the appropriate design parameters instead of eliminating the noise factor. If the coupling error change caused b controllable factors and uncontrollable factors is allowed, the decoupling of sliding structure is robust. or sliding structure of the sensor, the size of the elastic bod a, b and the grooe width r are gien as undetermined random ariables and feature Gaussian distribution. he design ariables (controllable factors) are shown as (9) 190

5 (,, ) x = abr = ( x, x, x). 1 3 Δ a, Δb and Δr are machining errors, and noise factors (uncontrollable factors) are shown as (,, ) z = Δa Δb Δ r = ( z, z, z ). ( x,( /3) ) 1 3 x μ Δx can be obtained, where μ x expectation of x. he coupling error is determined b both controllable factors and uncontrollable factors and can be obtained as = E ( ) = ( xz, ) (10) r is design target function and the ideal alue is 0. he statistical mean can approach the target alue ia expectation μ and Δ can be minimized ia x min σ min. he target function with smaller-the- ariance better can be established as min (, ) = μ + βσ xz (11) where β is weighted coefficient for coordination of expectation and ariance. Adding constraint function g( x ), the robust optimization model can be gien as is x = ( x1, x, x3) min ( xz, ) = μ + βσ stg.. ( x) = x1 + x x3 0 x1 (45,55), x, x3 (3,5) (1) he initial alue, discrete increments and upper and lower bounds of design ariables are listed in able 3. β = 0.97 and ( Δx1, Δx, Δ x3) = (0.046,0.0,0.0) (GB/ I8) are gien. μ andσ can be calculated based on best square approximation method after sampling. When = Er( Mx) = ( xz, ), the same method can be used. he robust optimization design solution can be obtained b using one-dimensional traersal optimization. he optimized sizes of elastic bod are shown in able 4., and the corresponding mean and ariance of random function E r ( x ) and E r (M x ) is 0.34 % and 0.48 % for elastic bod 1, and 0.65 % and 0.7 % for elastic bod, respectiel. able 3. he initial alue, discrete increments, upper and lower bounds of design ariables Variables (mm) Discrete increments Upper bound Lower bound x ( ) 1 a x ( b ) x ( r ) able 4. Robust result of optimized design ariables a b r Δ a Δ b Δ r elastic bod 1 (mm) elastic bod (mm) CALIBRAION ES ig.5. Calibration deice of sensor (torque M x calibration) he uncertaint is a parameter that reasonabl shows the dissolution characteristic of a measurement alue [3]. ig.5. shows the calibration test of the fabricated six-axis force/torque sensor. In the calibration of the six-axis force/torque sensor, the combined standard uncertaint is obtained b combining the A tpe standard uncertaint due to lack of reproducibilit, the B tpe standard uncertaint from the calibration deice. he B tpe standard uncertaint due to the resolution of the data acquisition card. In order to calculate the uncertaint due to lack of reproducibilit, the force x, and z from 1 N to 0 N at 1 N steps, and torque M x, M and M z from 40 Nmm to 800 Nmm at 40 Nmm steps is measured, respectiel, fie times. he full-scale output oltage is 10 V. he aerage measured alues at each step are obtained and the standard 191

6 deiation S of the force x, and z, and torque M x, M and M z can be calculated with the maximum aerage measured alue from Bessel formula. he uncertaint u r can be expressed as S u r = (13) 5 he calculated uncertaint u r of the force x, and z, and torque M x, M and M z is V, V, V, V, V and V, respectiel. he freedom r = 5-1 = 4. he uncertaint from the calibration deice means the uncertaint due to the transmission of force and torque. he maximum force direction deiation angle α during the calibration can be measured. Multipling the coerage factor k = 3, the uncertaint u f from the calibration deice can be expressed as u f Uo(1 cos α) = (14) 3 Where U o is the measured output oltage of x,, z, M x, M or M z. he calculated uncertaint u f of the force x, and z, and torque M x, M and M z is V, V, V, V, V and V. he freedom f =. he reolution of the data acquisition card is V. Multipling the coerage factor k = 3, the uncertaint u d due to the resolution of data acquisition card can be calculated as V/ 3 = V. he freedom d =. he combined standard uncertaint u c can be written as u = u + u + u (15) c r f d he calculated uncertaint u c of the force x, and z, and torque M x, M and M z is V, V, V, V, V and V. he freedoms of x,, z, M x, M and M z are 4. he expanded uncertaint U is calculated b multipling the combined uncertaint u c b the coerage factor k = 1.9 (confidence leel 95%, t 0.95 (4) = 1.9). he equation for it can be written as U = ku c (16) he calculated expanded uncertaint U of x,, z, M x, M and M z is V, V, V, V, V and 0.01 V, respectiel. he following relationship between force and output oltage can be gien = [ DU ] (17) where = ( x,, z, Mx,, z), [D] is the static calibration matrix. During the calibration procedure of the force/torque sensor, the force x, and z are applied to the sensor from 1 N to 0 N at 1 N steps, respectiel, and the torque M x, M and M z are applied to the sensor from 40 Nmm to 800 Nmm at 40 Nmm steps, respectiel. In the meantime, the oltage output alues of the bridge from eight beams are recorded and 480 calibration data are obtained in total. U can be calculated from () and the static calibration matrix [D] can be calculated from (17) with least squares fitting as [ D ] = (18) he static calibration matrix indicates that the coupling errors of x are between 0.01 % and 0.54 %; the coupling errors of are between 0. % and 0.86 %; the coupling errors of z are between 0.16 % and 0.75 %; the coupling errors of M x are between 0.8 % and 0.74 %; the coupling errors of M are between 0.5 % and 0.89 %; the coupling errors of M z are between 0.18 % and 0.34 %. he maximum coupling error of the deeloped six-axis force/torque sensor is 0.89 % and the minimum coupling error is 0.01 %. It can be said that the measurement sensitiit is 5.4 mv/n for x and, 4. mv/n for M x and M, 4.8 mv/nmm for z and 5.3 mv/nmm for M z. 6. CONCLUSIONS In this paper, the decoupling mechanism of a designed sliding structure six-axis force/torque sensor is analzed. his sensor can decouple more thoroughl, for each beam of elastic bod can onl sense a single dimensional force/toque independentl. Because of sliding clearance and asmmetric grooe, elastic bod will come into contact with the grooe at a certain angle. hus, contact force will be applied to the non-sensing beams of elastic bod and produce additional coupling error. he contact angle is related to structure size of the sensor, machining error and assembl error. Robust optimization method was emploed b optimizing the elastic bod size a, b, and the grooe width r to eliminate the coupling error but not decrease the machining error. he calculated alues a, b and r are mm, 4.73 mm and mm for elastic bod 1, and mm, 4.60 mm and mm for elastic bod, respectiel. or calibration deice, the calculated expanded uncertaint of x,, z, M x, M and M z is V, V, V, V, V and 0.01 V, respectiel, with 10 V full-scale output oltage. he corresponding mean and ariance of coupling error is 0.34 % and 0.65 % for elastic bod 1, and 0.48 % and 0.7 % for elastic bod, respectiel. he calibration test was carried out and the results show that the coupling errors are between 0.01 % and 0.89 %. 19

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