IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY METHOD FOR SIMPLIFYING MAGNETIC HAND MOTION CAPTURE: POSITION AND POSTURE ESTIMATION METHOD FOR FINGER SEGMENTS OF INDEX WITH TWO RECEIVERS Rong Tang *1,3, Masachka Sato 2, Kazutaka Mtobe 2, Noboru Yoshmura 3 1 Graduate School of Engneerng and Resource Scence, Akta Unversty 2 Graduate School of Engneerng Scence, Akta Unversty 3 Tohoku Unversty of Communty Servce and Scence DOI: 10.5281/zenodo.1407684 ABSTRACT Ths study presents a smplfyng method for magnetc hand moton capture, whch s mportant for capturng hand moton when usng scssors-lke surgcal tools, such as laparoscopc nstruments. In order to reduce physcal collson between nstruments and recevers, the recevers can only be placed on the dorsum of the hand and fngertps. Therefore, t s necessary to have a poston and posture measurement method that does not requre the placement of recevers on fnger segments wthout recevers. Thus, we propose a method that conssts of a jont center estmaton method from moton capture data, a constructon method for a skeletal fnger model, a calbraton method for recevers placed on fngertps, and an nverse knematcs method. An evaluaton was done wth the moton capture data from ndex fngers, and mean segment lengths durng a graspng moton were calculated by determnng the poston of jont centers. The standard devatons of all estmated segment lengths were under 0.4 mm. In addton, we compared the postons of our method wth a full set of four recevers. The mean dstance between jont center postons estmated by our methods and a full set of recevers were under 3.4 mm, whch was an mprovement over past research. Lastly, the Pearson correlaton coeffcents between the jont center poston estmated by our method and full set of recevers were calculated. The results found a strong correlaton between the jont center postons estmated by our method utlzng two recevers and a full set of four recevers, thus confrmng the robustness of our method. KEYWORDS: Moton Capture, Fnger Dexterous Moton, Jont Center Estmaton, Inverse Knematcs. I. INTRODUCTION Moton capture s an mportant technology for tranng and ergonomc assessment of hand motons n laparoscopc surgery. In a prevous research, one 6 Degree-Of-Freedom (DOF) magnetc trackng sensor was mounted onto each needle holder, and the moton data of sensors were recorded as the movement of laparoscopc nstruments [1]. By analyzng the movements of laparoscopc nstruments, t was possble to observe the dfferences between expert and novce surgeons. Other research has also nvestgated wrst angle and fnger jont angles that were recorded drectly wth glove-based moton capture (CyberGlove TM ) [2]. Wth surface electrodes placed on the upper arm and shoulders, muscle actvty durng coordnaton, peg transfer, precson cuttng and suturng were analyzed whle usng nstruments. Meanwhle, the movements of nstruments wthout sensors for capture has not been dscussed. The CyberGlove TM tself does not provde a 3D postonng mechansm for fnger jonts, and as such could not be used n laparoscopc tranng. Thus, addtonal postonng devces are needed for fnger jont poston measurement [3]. Addtonally, hand moton along wth fnger jont postons durng laparoscopc surgery has not been dscussed n prevous laparoscopc surgery related research because of the lack of a fnger poston measurng method for laparoscopc tranng. Hand moton capture methods whle usng tools, lke laparoscopc tranng, are rare. However, there s some research about hand moton capture methods for dexterous movements [4], [5], applyng methods such as optcal moton capture. Because a sngle optcal marker can only obtan one poston, at least three markers have to be placed on one fnger segment to calculate the posture. In a stuaton where the hand s not holdng anythng, three optcal markers placed on each fnger segment and dorsum of hand, can calculate fnger jont postons by usng http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [20]
Least Squares Optmzaton methods [4]. Furthermore, by solvng knematc problems wth a mathematcal hand model, t s possble to estmate fnger poston wth moton capture data from markers on the fngertps and the dorsum of the hand [5]. The mathematcal hand model s also called a skeletal hand model or lnk structure model n 3D CG modelng. A mathematcal hand model conssts of varous vectors that represent fnger bones and rotaton axs n a skeletal hand model. Flexon and extenson are represented by the rotaton axs of a fnger jont, whle the jont postons are represented by the rotaton of fnger bones. Jonts wth 1 DOF such as Dstal Interphalangeal Jont (DIP jont) and Proxmal Interphalangeal Jont (PIP jont) can be represented by one vector. For ergonomc evaluaton, some researchers made a hand model that s a combnaton of a CT scanned mesh hand model and a skeletal hand model for an ndvdual at reference posture [6]. Whle graspng or grppng, however, a self-occluson problem occurs when usng optcal moton capture devces, and re-labellng markers hdden from the camera s tme consumng. In some stuatons, these problems can be solved by optmzatons technques and knematc calculatons wth a skeletal hand model. One research study appled a combnaton of optcal moton capture and Mcrosoft Knect, and solved self-occluson problems, and markers could be labeled automatcally, when holdng nothng n the hands [7]. As such, prevous research ndcates that the self-occluson problem can be solved and fnger jont poston can be measured when capturng moton of the bare hand. To measure a sequence of surgcal hand motons whle usng scssor-lke tools, such as laparoscopc nstruments, some cases of the self-occluson problem wth optcal moton capture cannot be avoded, as optcal markers may be occluded by ether the hands or surgcal nstruments. To address ths, t may be possble to apply a magnetc moton capture devce that s capable of fnger moton measurement wthout a self-occluson problem, as the magnetc felds pass thru the hand. In prevous research, an Automatc Jont Parameter Estmaton (AJPE) method for the full body wth magnetc moton capture, whch s an optmzaton method, has been proposed [8]. However, the method lmted for a full body cannot determne sngle-axs jont centers lke the DIP jont of a fnger. The last problem that needs to be addressed s physcal collson between nstruments and magnetc recevers. When usng scssor-lke tools, the handler may collde wth magnetc recevers. In order to reduce physcal collson between nstruments and recevers, recevers can only be placed on the fngertps and dorsum of the hand. Postons and posture fnger segments wthout recevers can be calculated wth Inverse Knematcs (IK) methods. There are several IK solutons for computng the poston or posture va estmaton of each segment of a lnk structure. Some solutons of IK problems are tasked as a problem of fndng a local mnmum of a set of non-lnear equatons [9]. One famous IK method n a bomechancally constraned stuaton s the Cyclc Coordnate Descent (CCD) algorthm [10], whch s a heurstc teratve method wth low computatonal cost for each jont per teraton wthout matrx manpulatons. One IK soluton that has a very low computatonal cost s the trangulaton method [11]. Although the method has been mproved, especally for unnatural lookng problems for n-lnk IK problems, t s hard to apply to jont lmts, whch means that s hard to control the path n a 3D space. Because the trangulaton can reach the target va one teraton, t has greater performance benefts over CCD. Our proposed method s based on the trangulaton method. As our IK problem s focused on the hand, the soluton to ths typcal problem can be mproved by addng constrants and estmaton methods for specal condtons. One of the specal condtons we want to ntroduce s fnger extenson, whch usually s dscussed n moton capture [12], but has rarely been consdered n nverse knematcs. Ths basc research paper nvestgates hand moton measurement whle usng scssors-lke surgcal tools such as laparoscopc nstruments, by utlzng an estmaton method for poston and posture of a fnger segment wth magnetc moton capture recevers placed on the fngertps and dorsum of the hand. Fgure 1 shows the placement of magnetc hand moton capture recevers. The hand moton examned n our method s graspng of a crcular cylnder, whch s used n buldng a knematc hand model [13]. Fgure 1. Placement of magnetc hand moton capture recevers. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [21]
Fgure 2. A sample of placement of recevers and block dagram of expermental system. II. EXPERIMENTAL SYSTEM AND METHODS System overvew Our experment system conssts of a PC, Polhemus Lberty system, one transmtter, and sxteen magnetc recevers. Fgure 2 shows a sample of the placement of recevers and a block dagram of the expermental system. The magnetc feld from a transmtter generates the nduced electromotve force to the cols n the recevers. By analyzng the nduced electromotve force, the relatve poston and posture of recevers from the transmtter can be calculated, and automatcally recorded. There are ffteen recevers placed on hand and one attached on a stylus, a self-made poston measurng devce, whch wll be dscussed later. In order to ft the shape of fnger segments, the recevers (RX1-D) were processed nto a teardrop shape, wth a specally thnned cable. For each recever on the fnger bones, a specal knd of toupee tape that s thnner, but wth more strength was appled. In addton, knesology tape was used to wrap the recevers onto the fnger bones. As skn on the dorsum of hand extends and contracts whle movng, we used a specal fngerless glove to reduce skn movement, nstead of usng toupee tape and knesology tape. The cylnder used n the graspng moton s a PVC ppe 26 cm hgh and 3.8 mm n dameter. In addton, the cylnder s fxed on the table, so there s nether sgnfcant movement nor shape changes. One specal self-made postonng devce that wll be ntroduced s the stylus. The stylus s a devce for measurng poston, wth one recever fxed on the thcker end of a pencl that s lke wood. In order to get the calbraton data of the stylus, the sharper end pont s fxed on the test bench, and movement of the other end pont s of a sphere surface. Fgure 3 shows the moton for the stylus and the end pont calculaton method. Fgure 3(a) shows the moton for calbraton and Fgure 3(b) shows the calculaton method for the end pont of the stylus. From the stylus calbraton data, the sharper pont of the other end could be calculated by sphere fttng [14]. Once the vector (v stylus) between the recever and the sharper end pont s determned n the local coordnate of a recever, the poston of the sharper end pont could be calculated from moton data. Because the dstrbuton of the magnetc feld generated by the transmtter s very mportant for a correct readng, anythng that nfluences the dstrbuton of the magnetc feld, lke other metals, should be removed from the test space. Therefore, we used a 1-centmeter-thck acrylc plate as a test bench, and our measurements were executed n a wooden house. (a) (b) Fgure 3. Moton for calbraton and end pont estmaton method for stylus. (a) Moton for calbratng stylus. (b) Calculaton method of end pont. Constructon method of skeletal fnger model from moton capture data We used the AJPE calculaton method, whch cannot determne the poston of sngle-axs jonts [5]. Therefore, we took three steps to calculate the poston of jont centers and to construct a skeletal fnger model. Fgure 4 shows the jont center estmaton method from moton capture data. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [22]
Rotaton axs estmated from graspng moton Fgure 4(a) shows the estmaton method for rotaton axs from a graspng moton. Ths step used AJPE to calculate ponts from the data of a number of graspng motons. Ths allows us to obtan a set of ponts for each jont, whch are dfferent ponts on the rotaton axs of a fnger jont. Then, the axs of the graspng moton was calculated by lnear fttng of the ponts calculated above. Jont center estmaton at reference posture Fgure 4(b) shows the jont center estmaton method at reference posture. In ths paper, postures are represented by Euler angle (yaw ptch roll). The reference posture s the posture for defnng the Euler angle of fnger segments. Wth the stylus, the poston of the mdpont on the skn could be measured. The normal vector of the sagttal plane s the same as the X axs of coordnates of the transmtter. Therefore, wth the mdpont on the DIP and PIP jont, t was possble to determne the sagttal plane of a fnger. We defned the ntersecton of the jont axs and sagttal plane as the jont center for the DIP and PIP jont. To reduce calculaton costs, we also used the same method as appled to the DIP and PIP jont on the metatarsophalangeal jont (MP jont). Although the MP jont s not bometrcally a sngle-axs jont, there s a rotaton axs n the graspng moton. Constructon of skeletal fnger model Fgure 4(c) shows the constructon of a skeletal fnger model. The fnger skeletal model s constructed by v drm, v trd, l 1 and l 2. l 1 and l 2, whch was calculated from the DIP, PIP, and MP jont estmates. Meanwhle, v drm and v trd were calculated from poston and posture data of a recever placed on a fngertp and the dorsum of the hand. Calbraton method for magnetc recever placed on fngertp After placng recevers on fnger segments, the postures of the magnetc moton recevers are dfferent from the posture of fnger segments. For ordnary CG anmaton, the dfference between reference posture and moton capture data are gnorable. However, n surgcal tranng or practce, ths dfference may be sgnfcant. Therefore, t s necessary to have an accurate calbraton of recever posture. In ths paper, two postures were used for calbraton. Fgure 5 shows the posture calbraton method for a recever placed on a fngertp. Fgure 5(a) shows the hand posture for calculatng the Y axs of a recever placed on the fngertp, whch s calculated from vectors v pd and v pm. In order to determne the posture of a fngertp, one more axs s needed. Fgure 5(b) shows the posture for determnng the X axs of the recever. We used the vector from the DIP jont to the PIP jonts at the reference posture as the X axs of the recever. Wth the posture of a fngertp, the relatonshps wth data from the recevers were calculated, and the postures of the recever were calbrated. (a) (b) (c) Fgure 4. Constructon method for skeletal fnger model. (a) Calculaton method for sagttal plane and axs of rotaton. (b) Jont center estmaton method. (c) Constructon of skeletal fnger model. (a) (b) Fgure 5. Posture calbraton method for recever placed on fngertp. (a) Posture for calculatng Y axs. (b) Posture for calculatng X axs. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [23]
Inverse knematc method for estmatng fnger segments wthout recevers Fgure 6 shows the nverse knematc method for estmatng fnger segments wthout recevers. In our method, there are two knds of condtons. Fgure 6(a) shows the method for flexon, whch s the man movement of a graspng moton. Fgure 6(b) shows the method for over extenson, whch commonly happens when stretchng the fngers, but s usually not taken nto consderaton. We made a condton judgng vector v co, whch s the cross product of v dt and v dm. By comparng the y local coordnate of the recever placed on the fngertp wth v co, the fnger pose condton s determned. At flexon posture, v co s n the same drecton of y, and would be n a reversed drecton wth an over extenson posture. Wth the correspondng calculaton method, the jont center could be calculated. By mportng a skeletal hand model, the poston of the DIP and MP ( p dp, p mp ) jonts can be calculated from the moton data of the recevers placed on the fngertps and dorsum of a hand. Our method s shown n pseudo-code n algorthm 1. Algorthm 1. Jont center estmaton method Input moton data (#poston (x,y,z) and posture (yaw, ptch, roll) of recevers placed on fngertps and dorsum of the hand ) Input v drm, v trd, l 1, l 2 (#skeletal hand model) for all frame do calculate p mp, p dp, v dt, v dm f v dm l 1 + l 2 then (#Hand at stretchng posture. Sometmes, l 1, l 2 and v dm dd not form a trangle. θ = 180 Ths stuaton s nether flexon nor over extenson). else β = arccos ( l 1 2 + v dm 2 +l 2 2l 1 v ) dm v co = v dt v dm (#Left hand coordnate) calculate y (#The y-axs vector of local coordnate of recever placed on fngertp) f v co and y are n the same drecton then θ = α + β (#Flexon) else θ = (α + β ) (#Over extenson) end f end f calculate n dp (#Unt vector of v dp from rotatng v dt by θ about y ) calculate v dp (#Change the length of n dp to l 1 ) calculate p pp (#The poston of PIP by translatng p dp wth v dp end for ) (a) (b) Fgure 6. Fnger segment poston estmaton method for flexon and over extenson posture. (a) Flexon. (b) Over extenson. III. Results In order to evaluate the relablty of the proposed jont center estmaton method, the dstance between two jonts (segment length) were calculated from the data of a graspng moton. Fgure 7 shows fnger segment length and jont angle durng a graspng moton. The horzontal axs represents tme, and the frst vertcal axs represents http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [24]
jont angle and the second vertcal axs represents the length of segments. Although the angle of the PIP jont s changng (from about -20 degrees to 80 degrees) durng graspng, the lengths of ntermedate and proxmal phalange reman stable. Table 1 shows the mean and standard devaton (SD) of segment lengths for each subject. The mean lengths of ntermedate phalanges accordng to subjects are from 23.3 mm to 27.4 mm, and the mean lengths of proxmal phalanges are from 34.8 mm to 44.0 mm. All standard devatons of estmated segment lengths are under 0.4 mm, whch means almost no varaton on segment lengths durng graspng. Fgure 8 shows the trajectores of jont centers of an ndex generated by the method usng a full set of four recevers (4R method) and the proposed method wth only two recevers (2R method) durng one graspng moton that s descrbed n a 3D space. The orgn of ths space s the local coordnate of the recever placed on the dorsal sde. The red trajectory represents the center of a PIP jont, and yellow represents a MP jont. Meanwhle, the green and blue trajectory represents the center of a PIP and MP jont generated by the 4R method. The trajectory of a PIP jont generated by the 2R method (red) are concdent wth the trajectory generated by the 4R method (green). On the one hand, the trajectory of the MP jont generated by the 2R method are gathered as a pont, whle the trajectory of the MP jont generated by the 4R method are n an arc shape. Fgure 9 shows each component of poston (X, Y, Z) durng ten graspng motons that were analyzed. The horzontal axs represents tme and the vertcal axs represents the poston of center of a PIP jont. The red lnes represent the components that are generated by the 2R methods, and postons generated by the 4R method are n green. The red lnes are vsually well polymerzed to the correspondng green lnes, ndcatng that the 2R methods are vsually relable. For a statstcal evaluaton, we also calculated the Pearson correlaton coeffcent of the lnes n green and red. Table 2 shows the Pearson correlaton coeffcents between the center of the PIP jonts generated by the 2R method and 4R method. For the Y component, the lowest correlaton coeffcent s 0.82, whch showed a strong correlaton. For the X and Z components, the lowest correlaton was 0.97, and both showed a very strong correlaton. Table 3 shows the mean and standard devaton of the dstance between jont centers generated by the 2R method and 4R method. The mean dstance of the PIP and MP jont are under 3.4 mm, and the correspondng standard devatons are under 2.6 mm. In the nverse knematcs method, the calculaton of PIP jonts uses the poston of a MP jont, and the standard devaton of PIP jonts was greater than for MP jonts n most cases. The standard devaton of MP jonts are under 1.1 mm. Angle[degree] PIP Jont Angle Intermedate phalange Length Proxmal phalange Length Segment Length[mm] Tme[s] Fgure 7: Segment length and jont angle durng graspng moton. Table 1: Mean and standard devaton of segment length Subjects Mean (SD) of segment length [mm] Intermedate Phalange Proxmal Phalange A 25.2 (0.2) 38.5 (0.4) B 25.2 (0.2) 34.8 (0.2) C 26.5 (0.2) 43.4 (0.3) D 27.4 (0.4) 44.0 (0.4) E 23.3 (0.2) 40.3 (0.2) F 24.7 (0.2) 42.3 (0.4) http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [25]
Fgure 8: One sample of the trajectores generated by 2R method and 4R method durng one graspng moton. Fgure 9: Each component of poston durng graspng moton. Table 2: Pearson correlaton coeffcents between the center of PIP jonts generated by 2R method and 4R method Pearson correlaton coeffcents Subjects x y z A 0.99 0.99 0.99 B 0.98 0.99 0.99 C 0.99 0.82 0.97 D 0.97 0.93 0.97 E 0.98 0.97 0.99 F 0.99 0.98 0.99 Table 3: Mean dstance between jont center of generated by 2R method and 4R method Subjects Mean (SD) dstance [mm] PIP jont MP jont A 2.6 (1.1) 2.9 (1.0) B 1.6 (0.6) 1.8 (0.6) C 3.4 (2.6) 2.1 (0.6) D 2.8 (2.0) 1.9 (1.1) E 2.3 (1.0) 2.3 (0.8) F 1.4 (0.9) 1.6 (0.6) IV. DISCUSSION Ths paper proposed a method for estmatng the jont center from the ntersecton of the jont axs and sagttal plane. Each fnger jont that was estmated was algned n a straght lne when a fnger was n a stretched posture, whch means the fnger moton that s generated s more natural. For segment length estmaton made by Zong- Mng L, the standard devaton of ntermedate phalanges and proxmal phalanges were 1.2 mm and 1.6 mm [5], whle the standard devaton of fnger segment lengths n our results were under 0.4 mm for both ntermedate phalanges and proxmal phalanges. In the future, ths segment length estmaton method can also be used n musculoskeletal systems for moment predcton [15]. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [26]
The mean dstance between our 2R method and a 4R method jont was smaller compared to past research. For the PIP jont, the mean dstance of our method was about 2.4 mm whle past research [5] acheved about 5 mm (2.6 mm n X, 4.8 mm n Y, 3.6 mm n Z). In addton, our method can also be appled when a fnger s over extended, whch untl now has not been consdered by other methods. V. CONCLUSION By constructng a skeletal fnger model from magnetc moton capture data, we were able to estmate the fnger segment poston of the ndex fnger wth two recevers. We proposed, ntroduced, and valdated a calbraton method for a magnetc recever placed on the fngertp and an nverse knematc method for estmaton. In the evaluaton part, whch ncluded moton capture data from sx subjects, we compared the fnger segment poston estmated by our method wth a full set of four recevers. The standard devaton on length n our results were under 0.4 mm for ntermedate phalanges and proxmal phalanges. The Pearson correlaton coeffcents between the center of PIP jonts generated by our method wth two recevers and a method usng a full set of four recevers were all strong (over 0.82). Our future research wll focus on three steps. Frstly, we wll apply our method to other fngers, especally the thumb. Secondly, we wll make some laparoscopc nstruments wth non-metal handlers for experments and tranng. Lastly, we wll develop a measurement method for the contact surface of the hand and laparoscopc nstruments. REFERENCES [1] M. Uemura, M. Tomkawa, R. Kumashro, et al, Analyss of hand moton dfferentates expert and novce surgeons, Journal of surgcal research.,vol88,no.1,pp.8-13, 2014. [2] AE. Tapa-Araya, J. Uson-Gargallo, JA. Sanchez-Margallo, et al, Muscle actvty and hand moton n veternarans performng laparoscopc tranng tasks wth a box traner, Amercan Journal of Veternary Research.,Vol.77, No.2, PP.186-193, Feb. 2016. [3] J.Zhou, F.Malrc and S.Shrmohammad, A New Hand-Measurement Method to Smplfy Calbraton n CyberGlove-based Vrtual Rehabltaton, IEEE Transactons on Instrumentaton and Measurement, Nov. 2010. [4] L.Y. Chang and N.S. Pollard, Constraned Least-Squares Optmzaton for Robust Estmaton of Center of Rotaton, Journal of Bomechancs., Vol.40, No.6, pp.1392-1400, 2007. [5] R. Nataraj, Z.M. L, Robust dentfcaton of three-dmensonal thumb and ndex fnger knematcs wth a mnmal set of markers, Journal of Bomechancal Engneerng., Vol.135, 91002-91009, 2013. [6] Endo, Y. Tada M and Mochmaru M, Reconstructon of Dgtal Hand Models for Indvduals by Usng Moton Capture System, Journal of the Japan Socety for precson engneerng, Vol.79 No.9, pp.860-867, September, 2013.(n Japanese) [7] W. Zhao, J. Cha, Y.Q. Xu, Combnng Marker-based Mocap and RGB-D Camera for Acqurng Hghfdelty Hand Moton Data, In: Symposum on Computer Anmaton., pp.33-42, 2012. [8] J.F. O Bren, R.E. Bodenhemer, G.J. Brostow, J.K. Hodgns : Automatc jont parameter estmaton from magnetc moton capture data, n: Graphcs Interface Conference., pp. 53-60, 2000. [9] Janmn Zhao, Norman I. Badler, Inverse knematcs postonng usng nonlnear programmng for hghly artculated fgures,, ACM Transactons on Graphcs Vol.13, No.4, pp.313-336, 1994. [10] Chrs Welman, Inverse Knematcs and Geometrc Constrants for Artculated Fgure Manpulaton, Master Dssertaton, Smon Fraser Unversty, Department of Computer Scence, 1993. [11] R. Mu ller-cajar, R. Mukundan, Trangulaton: a new algorthm for nverse knematcs, n: Proc. of the Image and Vson Computng New Zealand 2007, New Zealand, pp.181-186, 2007. [12] P.Brado, X. Zhang, Quanttatve analyss of fnger moton coordnaton n hand manpulatve and gestc acts, Human Movement Sc., Vol.22, pp.661-678, 2004. [13] B. Buchholz and T. Armstrong, A knematc model of the human hand to evaluate ts prehensle capabltes, Journal of Bomechancs, Vol.25, No.2, pp.149 162, 1992. [14] V. Pratt, Drect least-squares fttng of algebrac surfaces, ComputerGraphcs Vol.21, pp.145-152, 1987. [15] J.Z. Wu, KN. An, R.G. Cutlp, M.E. Andrew, R.G. Dong, Modelng of the muscle/tendon excursons and moment arms n the thumb usng the commercal software anybody, Journal of Bomechancs, Vol.42, pp.383-388, 2009. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [27]