Free-Form Tactile Sensor Using 3-Dimensional Shape Capture Sheet

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1 Free-Form Tactle Sensor Usng 3-Dmensonal Shape Capture Sheet Takayuk Hosh and Hroyuk Shnoda The Unversty of Tokyo {star, Abstract We are developng a novel sensng devce named 3-dmensonal capture sheet (3DCS). The cloth-lke sheet measures ts own 3D confguraton. The sheet enables us to make a soft tactle sensor by wrappng a compressble materal n t. The number, shapes, and postons of obects are obtaned from the surface deformaton of the materal. The contact forces are also estmated. There s a lattce structure nsde the sheet, and each lnk of the structure has a traxal accelerometer. The roll and ptch angles of the lnk are derved from the gravty vector measured by the accelerometer. Furthermore, the relatve yaw angle s also determned owng to the lattce structure. The posture of each lnk s fully descrbed by these three angles. Then, the whole shape of the sheet s estmated by combnng all of the lnks n computaton. 1. Introducton Measurement of the 3D shape of an obect s usually conducted by the optcal methods (e.g. stereo vson, structured lght proecton, shape from focus/defocus and so on). They need external equpments and suffer from the occluson problem. We propose another shape measurement approach, n whch a cloth-lke sensng devce named 3- dmensonal capture sheet (3DCS) s used. The sheet has the ablty to acqure ts own 3D confguraton, and measures the shape of an obect by wrappng t drectly. Ths method needs only the foldable sheet and no other external equpments. Wth the sheet, you can measure the shape and sze of an obect wth ease. The human posture can also be measured by the sheet worn. The sheet also can be utlzed as a soft tactle sensor. Softness s one of the most mportant factors for tactle sensor skns on robots [1]. The tactle sensor s a compressble materal, such as urethane foam, wrapped Fgure 1. Structure of 3DCS. by the sheet. The number, shapes, and postons of contact obects are obtaned because the whole surface profle of the materal s observed by the sheet. The contact forces are also estmated from the surface deformaton and Young s modulus of the materal. The structure of the tactle sensor s so smple that t enables us to attach t to a free-form surface of a robot, or make a stuffed toy whch has tactle, haptc, and somatc sensaton. The 3DCS, the key component of the proposed tactle sensor, s the sheet n whch there s a lattce structure (Fg. 1). A traxal accelerometer s attached on each lnk, and measures the gravty vector. The posture of the lnk s estmated from the measured gravty. The gravty measurement has been used n moton capture n the precedng reports [, 3]. They use the gravty to derve only the roll and ptch angles of a human arm because the output of a sngle traxal accelerometer contans only the nformaton about the two angles. They use other types of sensors such as an electromagnetc compass or a gyro sensor to know the yaw angle. Unlke them, we use mult accelerometers and thereby estmate also the yaw angles wthout any other sensors. The 3D confguraton of the sheet can be

2 Fgure. 7 7 lattce model made of.5-cm tubes combned wth strngs. Fgure 3. Defnton of rotatonal angles. measured wthout beng affected by any electrc or magnetc noses because the sensng theory of the sheet s based on the gravty. Besdes, the theory s based on geometry wthout temporal operaton. Therefore the measurement s less subect to the offset of the sensor output than the temporal ntegraton method. Ths devce s motvated by the recent trend of MEMS and sensor networkng technologes that enable us to mplant a large number of low-cost mnaturzed sensors n elastc cloth-lke materals [4]. The rest of ths paper s organzed as follows. Frstly, Secton II descrbes the basc structure and the theory of 3D shape reconstructon of the 3DCS. Next, the feasblty of the sheet s examned by smulaton n Secton III. Then, the prototype s presented n Secton IV. Fnally, we conclude ths paper n Secton V.. Three-dmensonal capture sheet.1. Structure The llustraton of the 3DCS s shown n Fg. 1. There s a lattce structure nsde of t, and each lnk of the structure has a traxal accelerometer. The accelerometers measure the gravty vector, and the measured data are sent to the host computer. The x-axs of the traxal accelerometer s algned along the lnk drecton, and the z-axs s algned along the drecton Fgure 4. Sde vew. The sensor coordnate s rotated accordng to α and β. of the normal vector on the sheet. The all lnk lengths are the same. The stretchng property of the lattce structure s ansotropc as s the case wth a textle cloth,.e. t s stretchable only along the dagonal drectons). Assumng a smooth curved shape, we can cover ts surface wth the structure. Fg. shows the lattce model made of tubes combned wth strngs for demonstraton... Sensng theory In the "shape from gravty" theory, we frstly ntroduce the followng assumpton. [Assumpton 1] The acceleraton by moton of the lnk s neglgble compared wth the gravty acceleraton. Then the shape estmaton n dynamc moton s out of consderaton at least n ths stage. The posture of each lnk s descrbed by three angles based on the world coordnate; roll α [rad], ptch β [rad], and yaw γ [rad] (Fg. 3). Whereas only the roll and ptch angles are derved from the measured sensor data, the relatve yaw angle s determned owng to the lattce structure. After the angles of the all lnks are obtaned, the whole shape of the 3DCS s estmated by combnng the lnks n a computatonal 3D space. Note that the estmated shape stll has the uncertanty of orentaton n the world coordnate. The angles of the lnk are obtaned as follows. Here we assume that each axs of the traxal accelerometer s algned along the correspondng axs of the world coordnate (.e. the x-axs of the accelerometer to the x- axs of the world coordnate) at the ntal condton Roll and ptch angles. These angles are derved drectly from the gravty vector measured by the accelerometer on each lnk. The rotaton matrx G βα, from the world coordnate to the sensor coordnate (Fg. 4), s descrbed as

3 Fgure 5. Assumpton, about drectonal vectors. G βα snβ = snβ snβ 1 1 snα snα snβ snα snβ α α. (1) cos sn snα Each column of G βα means the axes of the sensor coordnate drecton n the world coordnate after rotated, and hence the output vector of the accelerometer s=[s x, s y, s z ] T s represented as a product of the transposed matrx of G βα and the gravty vector g=[,, -g] T (where g [m/s ] s the gravty acceleraton); s= G T βα g snβ. () g= g snα g Then we can derve α and β by solvng ()..1.. Yaw angles. Ths angle cannot be derved from the output of the sngle accelerometer. Then we take note of one unt of the lattce structure composed of four lnks formng a quadrangle to derve the yaw angles, makng two assumptons. The frst assumpton s about the drectonal vector d ( s the lnk dentfcaton) whch s represented as d snγ snγ cosγ G 1 βα 1 =. (3) snγ snβ Because the lattce unt s a closed loop (Fg. 5), the two routes reach the same pont n the 3D space. That results n the assumpton of the lnk rgdness as Fgure 6. Assumpton 3, about normal vectors. Symmetrc transformatons, (a) squashng and (b) foldng. The other assumpton s about the normal vector n whch s represented as snγ n snγ cosγ Gβ α 1 1 snβ + snγ snα =. (5) snγ snβ cosγ snα We assume the 3DCS s mounted on a smooth curved shape, and the lattce unt transforms n symmetrc manners (Fg. 6). In other words, one average of the normal vectors of an opposed lnk par must be equal to the other, that s [Assumpton 3] n + n = n +. (6) 1 n3 We can get γ by solvng (4) and (6). Note that at least one of γ should be fxed at a certan value prelmnarly to determne the four parameters γ n prncple. In other words, all we can do s to determne the relatve yaw angels of γ 1, γ and γ 3 to γ. Estmatng the confguraton of the mult-unt lattce structure s straghtforward. After the same algorthm s appled to each lattce unt, the whole confguraton s estmated by combnng the estmated unt shapes..3. Unsolvablty analyss The unsolvable condtons of (4) and (6) are clarfed by the sngular value decomposton [5]. Now (4) and (6) are rewrtten as a matrx form, that s, [Assumpton ] d + d = d +. (4) 1 3 d

4 Fgure 7. Procedure of lattce model generaton. (7) where s and c stand for sn and cos, respectvely. The z-components of (4) and (6) are excluded because they do not contan any γ. Here γ s a fxed number to be the root of the orentaton, and the other γ (=1,, 3) are obtaned as relatve values to γ. If the 4 6 coeffcent matrx has less than three nonzero sngular values, (7) s underdetermned and the confguraton of the lattce unt cannot be estmated unquely. The state of the lattce unt s characterzed by four angle parameters; corner θ c [rad], fold θ f [rad], roll θ r [rad], and ptch θ p [rad] (Fg. 7). In the followng calculaton, we quantze each angle s dvded nto dscrete values at even ntervals. Fg. 8 reveals the unsolvable stuatons of (θ c, θ f, θ r, θ p ). The ponts plotted n Fg. 8 stand for the condtons on whch the rato of the thrd sngular value λ 3 to the frst largest sngular value λ 1 s less than.1 (.e. λ 3 /λ 1 <.1). From Fg. 8, t turns out that all the unsolvable condtons are roughly ncluded n the followng two cases: [Case 1] The all lnks of the lattce unt are lad on a horzontal plane,.e. (θ f, θ r, θ p )=(,, ), (, π, ), (π,, ), or (π, π, ). [Case ] The lattce unt s fully squashed,.e. θ c =, π, or θ f =π. Fgure 8. Results of unsolvablty analyss. The (θ c, θ f, θ r, θ p )s satsfyng λ 3 /λ 1 <.1are plotted n the parameter space. Note that Case s avodable by settng lmts to the lnk onts n real devces. As a result, the possble unsolvable condton s Case 1, the horzontal case.

5 Fgure 9. Smulaton result. The far and near plots are the model and estmated shapes, respectvely. 3. Smulatons and results 3.1. Methods In the data acquston process, we frstly desgn a sprng-mass model of the 3DCS. In the model, the lnk s modeled as a very hard sprng so that the lnk length does not change, and the ont as a mass. After the model s lad over a target shape by teratve calculaton, the drecton of the sensor axes are smulated, and then the sensor outputs are calculated. In the shape estmaton process, the roll and ptch angles are derved analytcally from (). Because t s hard to derve the yaw angles analytcally from (4) and (6), the followng numercal calculaton s conducted. Frstly, we modfy (4) and (6) nto a mnmzaton problem, that s P { x, y, z} {( d + d1 d d3 ) ( n n + n n ) } mn. (8) where s the coordnate dentfcaton. Now, α and β are gven. If the mnmum value of P s equal to zero, the soluton for (8) are also the soluton for (4) and (6). Secondly, we solve (8) by the steepest descent method. Although (8) has several local mnmums, we can overcome them by tryng multple ntal values. 3.. Shape estmaton result Frstly, we show an example of general stuatons usng a Gaussan as a target shape. Fg. 9 shows the model shape and the estmated shape. The Gaussan shape s successfully reproduced. Fgure 1. Smulaton results on effect of nose. The mean values (dots) and the standard devatons (error bars) of the estmaton error Effect of nose on sensor data The prevous result (Fg. 9) s obtaned wthout nose. The possble causes of dsturbance are the nose on the sensor data (.e. angle error), the change of the lnk length, and the volaton of the normal vector assumpton (6). Snce the maor factor seems the nose on the sensor data, the smulaton nvestgates the stablty of the 3DCS under varous S/N ratos of the accelerometers. The nose s generated usng the Mersenne Twster algorthm [6] and added to each component of the sensor data. Here, the nose level s represented as the percentage of the nose compared to the gravty g. Fg. 1 shows the mean values and the standard devatons of the estmaton errors, the dstances between the correspondng onts of the model and estmated shapes. The dstances are calculated after overlappng the estmated shape to the model shape by the least-squares method to match ther orentatons and postons. 1,96 samples (196 onts 1 trals) are used n statstcal calculaton. Accordng to our observaton, the estmaton seems to fal dscernbly when the estmaton error goes beyond 5 % of the sde length of the 3DCS. Then, Fg. 1 suggests that the 3DCS decently works wth the nose level up to 5 % (.e. about.5 m/s n acceleraton). We thnk that the accuracy s practcally achevable. 4. Prototype of lattce unt 4.1. System Specfcatons The developed prototype s one lattce unt (Fg. 11). It conssts of four acrylc lnks ( cm n length) and four traxal accelerometers (5 5 mm ) mounted on

6 Fgure 11. (a) Prototype of one lattce unt. (b) The traxal accelerometer. small substrates (1 1 cm ). The lnks are onted by metal rngs. The PC mports the analog sensor readouts va 5- Hz-cutoff LPFs and 1-bt A/D converters. The effectve samplng rate of the system s 145 Hz. Here, the conugate gradent method [5] s appled n the estmaton process for real-tme performance. 4.. Demonstraton Fg. 1 shows the demonstraton of the shape estmaton of the one-unt prototype. The shape of the prototype s successfully estmated n real tme. The quanttatve evaluaton of the prototype s conducted. Wth the shape of the prototype set to certan angle parameters, (θ c, θ f, θ r, θ p )=(π/6, 3π/1, π, 3π/1), the coordnate values of the estmated ont postons are recorded. The mean value and the standard devaton of the estmaton errors s 11.3 % and.8 % of the lnk length (.e. about. cm and 1.6 mm n length for the -cm lnk), respectvely. Ths estmaton error s due to the algnment error of the accelerometer, the characterstcs of the ont, the computatonal error n the estmaton process, and the nose n the analog sgnal lnes. 5. Concluson A new tactle sensor s proposed n ths paper. The sensor s based on the sheet whch measures ts own 3- dmensonal confguraton usng dstrbuted traxal accelerometers. The structure and sensng theory of the sheet are descrbed. The theory clarfes that the proposed estmaton method works well outsde the trval cases. A one-unt prototype s developed and the 3D shape estmaton s demonstrated wth 11.3 % estmaton error. Fg. 1. Demonstraton of shape estmaton. After examnng and mprovng the accuracy of the prototype, n the next stage, we wll develop the smallszed sensor chp on whch an LSI s mounted wth the traxal accelerometer. The LSI measures the sensor readouts by A/D converters and sends dgtal data to the host computer va the two-dmensonal communcaton sheet [4]. Then the practcal 3DCS s realzed wthout complcated, long, and/or analogsgnal wres. The proposed tactle sensor s developed as one applcaton of the sheet. References [1] M. H. Lee and H. R. Ncholls, Tactle sensng for mechatroncs - a state of the art survey, Mechatroncs, vol. 9, pp. 1-31, [] J. Lee and I. Ha, Real-tme moton capture for a human body usng accelerometers, Robotca, vol. 19, pp , 1. [3] N. Mller, O. C. Jenkns, M. Kallmann, and M. J. Matarc, Moton capture from nertal sensng for untethered humanod teleoperaton, Proc. 4th IEEE/RAS Internatonal Conference on Humanod Robots (Humanods 4), vol., pp , 4. [4] Y. Makno, H. Chgusa and H. Shnoda, Twodmensonal sensor ntegraton usng resonant proxmty connector Basc technology and applcaton to elastc nterface devce, Proc. 3rd Internatonal Conference on Networked Sensng Systems (INSS 6), pp. 196-, 6. [5] W. H. Press, S. A. Teukolsky, W. T. Vetterlng, and B. P. Flannery, Numercal Recpes n C: The Art of Scentfc Computng Second Edton, Cambrdge Unversty Press, 199. [6] M. Matsumoto and T. Nshmura, Mersenne Twster: A 63-dmensonally equdstrbuted unform pseudorandom number generator, ACM Trans. on Modelng and Computer Smulaton, vol. 8, no. 1, pp.3-3, 1998.