03 IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems (IROS) November 3-7, 03. Tokyo, Japan Model of cyclotorson n a Tendon Drven Eyeball: theoretcal model and qualtatve evaluaton on a robotc platform Francesco Nor, Gulo Sandn, Gorgo Metta Abstract In ths paper we descrbe the mathematcal model of a tendon drven eye. We focus on descrbng ts movements, posng a specfc attenton on cyclotorson, that s the rotaton around the eye optcal axs. Ths study ams at understandng the cause of a cyclotorson effect that has been notced on a real setup. The paper starts from a qualtatve analyss of the effect. Then, t proposes two dfferent models for ts moton. Fnally both models are valdated by comparng ther predctons wth the outcomes on the real robot. Gven the complexty of the system and of ts moton results are for the moment just qualtatve and quanttatve comparson wll be the goal of our future works. I. INTRODUCTION In ths paper we descrbe the tendon drven eye whose pcture s gven n Fgure. The tendons are actuated by two motors, one for the vertcal tendon and one for the horzontal tendon. The tenson on the tendon s mantaned by a sprng whch pushes on the cable. Fgure shows also a scheme of the horzontal secton. The vertcal secton s smlar. Fg.. The left pcture shows the tendon drven eye. The two tendons are actuated by two motors. The frst motor moves the vertcal tendon. The second motor moves the horzontal tendon. The rght fgure sketches the actuaton scheme. II. CYCLOTORSION The present paper focuses on a cyclotorson effect that was notced on the real setup. Wth the term cyclotorson we ndcate a rotaton around the optcal axs of the camera (see Fgure ). Ths work was supported by the followng EC projects: CoDyCo, (FP7- ICT-0.. number 60076) and Xperence (FP7-ICT-009.. number 7073). Francesco Nor and Gulo Sandn are wth Robotcs, Bran and Cogntve Scences at the Isttuto Italano d Tecnologa. Gorgo Metta s wth the Cub Faclty at the Isttuto Italano d Tecnologa. name.surname(at)t.t Fg.. In order to measure the eye cyclotorson, a calbraton panel was placed n front of the camera. The fgure shows a typcal pcture taken by the camera. Let us brefly descrbe the undesred effect. In the current mplementaton the two motors are commanded n poston. It was notced that the same poston of the motors does not correspond a unque confguraton of the eyeball. In partcular, startng n a partcular confguraton of the eyeball and followng a cyclc path wth the motors, the systems does not return to the ntal confguraton and presents an evdent rotaton around the optcal axs. The cause of the experenced phenomena s clearly the lack of a drect control on the system cyclotorson. Clearly, ths phenomena s undesred snce t doesn t allow to reconstruct the poston of the eyeball gven the poston of the motors. In order to gve a frst characterzaton of the problem, we desgned an expermental setup based on the use of the Hough transform [4]. The dea conssts n placng, n front of the camera, a panel where a vertcal lne s panted (see Fgure ). In order to measure the cyclotorson angle, we used the mages taken from the camera whch s mounted on the eyeball. In ts ntal confguraton, the camera ponts towards a panel. Let 0 be the slope of the panted lne. Motors are then moved on cyclc path (.e. the motor postons s exactly the same after each path executon). After each cycle, the cyclotorson of the eyeball changes and consequently the slope of the lne changes (see Fgure 3). The amount of cyclotorson corresponds to the dfference n the lne orentaton,.e. = 0. The slope of the lne n the mage has been computed usng the Hough transform whch s practcally, a probablty densty of lnes n the mage plane. A. Prevous works Many eye-head systems have been desgned and bult n the last year but very few mmcs the human musculoskeletal Practcally, a cyclc path of the motors correspond to a seres of movements that start and fnsh n a unque confguraton. 97--4673-6357-0/3/$3.00 03 IEEE 59
cyclotorson effect smlar to that observed n the real setup. Fnally, Secton VII gves a qualtatve comparson between results obtaned n smulaton and wth the real setup. 0 =0 = = 0.5 = = 0.5 =0.75 = 0.5 =0.5 =0.75 = 0.5 =0.5 Fg. 3. Ths pcture shows the lne that has been used to measure the cyclotorson; s the slope of the lne n the mage and = 0 s the correspondng cyclotorson. In order to see the changes of orentaton (whch are very small and less than ), the pctures have been zoomed ten tmes horzontally. The rghtmost pcture was taken at the begnnng of the experment. The followng pctures, from left to rght, have been obtaned after some cyclc paths of the motors (.e. motors are n the same poston n all the pctures shown here). apparatus lke the one descrbed and analyzed n the present paper. Typcally, the goal has been the one of achevng two prncpal movements (typcally named a pan and a tlt movement) but very few systems were free to move along the cyclotorson axs. On the contrary a lot of work has been done on the analyss, modelng and control of human-lke eye movements [7], [6], [] Some recent works have proposed mechancal desgn wth strong smlartes wth humans. Among ths desgns t s worth ctng the work conducted by Cannata et al., [], [3] on the development of a complex tendon drven robotc eye whose moton dynamcs and relatons wth the Lstng s Law have been analyzed n deep. The system conssts of a sphere, actuated by four ndependent tendons drven by four motors wth the eye-ball hold by a low frcton support whch allows three rotatonal degrees of freedom. III. MOTIVATING THE ANALYSIS OF CYCLOTORSION As scentsts, we are not only nterested n measurng the cyclotorson of the eyeball but also n understandng ts causes. Clearly, n our specfc case, the causes of cyclotorson are not so evdent. Apparently, the two tendons do not provoke any cyclotorson on the system. Therefore the queston s: what does provoke the observed cyclotorson? Answerng ths queston wll gve us nsght n the knematcs of the systems suggestng elegant solutons to the encountered problem. In the rest of the paper we wll try to demonstrate that the observed cyclotorson s a consequence of the system knematcs. The paper s dvded nto four sectons. Secton IV descrbes a mathematcal model of the path descrbed by the tendons. Secton V descrbes two dfferent knematc models of the tendon drven eyeball; the frst model s unrealstc and s ntroduced only to smplfy the presentaton of the followng, more realstc model. Secton VI shows the numercal results obtaned by ntegratng the two knematc models; t s observed that the gven models present a IV. MATHEMATICAL MODEL OF THE TENDON GEOMETRY Snce the tendons are responsble for the movements of the eyeball, t s fundamental to descrbe the confguraton assumed by gven the confguraton of the eyeball. In partcular, n the present secton we gve the tools for computng the poston of the tendons gven the poston of the eyeball. We used MATLAB to create a model of the eye. The assumpton s that the tendons are attached to the eyeball n the ponts E, E, E 3, and E 4. At the same tme the tendons are forced to pass through the ponts C, C, C 3 and C 4 whose poston s fxed wth respect to an nertal reference frame (see Fgure 4). Consder an nertal reference frame, and let e be a reference frame attached to the eyeball. The orgn of the two reference frames are assumed to overlap. Practcally, the ponts E, E, E 3, E 4 are fxed n e and C, C, C 3, C 4 are fxed n. We wll use the followng notaton: Ej are the coordnates of E j n ; Ej e are the coordnates of E j n e. Let s now compute the poston of the tendon passng through C j to E j. The assumpton s that the tendon les on the geodesc path connectng C j to E j. The geodesc path on the eyeball can be easly computed snce ts surface s sphercal. Let Cj to E j be the coordnated of C j to E j n the nertal reference frame. The generc pont G on the geodesc satsfes the followng : (C j Ej ) G =0 G () = r where r s the eyeball radus, s the cross product, s the dot product and G are the coordnates of G n the reference frame. The length of the geodesc connectng Cj to E j s gven by: cos C j E j r! r. (3) V. KINEMATIC MODEL OF THE TENDON DRIVEN EYE We here descrbe a knematc model of the tendon drven eye. The secton s dvded nto two subsectons. We frst consder a smplfed model where a sngle tendon s used to push and to pull. Then we consder a more realstc model where tendons are only allowed to pull. The above condtons guarantees that G les on the great crcle that passes through C j and E j. To guarantee that the G s on the geodesc, we need also to mpose the followng condtons: C j G E j G Cj E j Cj E j whch guarantee that the (geodesc) dstance between G and E j and the dstance between G and C j do not exceed the dstance between C j and E j. () 59
where k R can be determned mposng a constant length of the tendon. Specfcally, let d be the tangent to the tendon n E (see rght Fgure 5). Moreover, let v E be the velocty of E,.e.: v E =! s E. (5) Then, n order to have a constant length of the tendon the projecton of v E along d must be equal to v,.e.: Substtutng (5) nto (6) we obtan: v E d = v. (6) Fg. 4. The pcture (left) and the sketch (rght) show the ponts E, E, E 3, and E 4 whch are attached to the eyeball. The ponts C, C, C 3 and C 4 are nstead n fxed wth respect to an nertal reference frame.! s E d = v, (7) whch can be used to determne k. Smlarly, f v =0we have:! s =(E C ) k (v ). () and agan k can be determned from:! s E d = v. (9) Fg. 5. The left pcture shows the two tendons consdered n the smplfed knematc model. Each tendon pushes and pulls the correspondng E j. The rght pcture shows the stuaton v = 0. Notce that n ths stuaton the angular velocty! s s orthogonal to the plane generated by E and C. Moreover, the norm of the vector! s can be determned mposng the constant length of the cable,.e v E d = v. A. Push-pull tendons In the real setup one tendon connects E, C, C 3 and E 3. The other tendon connects E, C, C 4 and E 4. When the frst motor rotates n one drecton, the tendon pulls E. When t rotates n the opposte drecton, the tendon pulls E 3. We here consder a smplfed model, assumng that the motor pushes and pulls E (see left Fgure 5). The other sde of the tendon (the one connected to E 3 ) s neglgble n ths framework and wll be therefore omtted; the tendon termnates wth the pont P whose velocty wll be denoted v (see rght Fgure 5). Smlarly, we assume that the second motor pushes and pulls E. Agan, the tendon termnates n the pont P whose velocty wll be denoted v. How does v and v constran the nstantaneous spatal angular velocty! s (see [5] for a defnton) of the eyeball? Let s frst consder a smplfed stuaton mposng a null velocty of the second tendon,.e. v =0. In ths smplfed stuaton t can be easly seen that the angular velocty! s s orthogonal to the plane generated by C and E (see rght Fgure 5). Therefore:! s = E C k (v ) (4) Fg. 6. Fnally, n the more general case v 6=0and v 6=0. In ths case the velocty! s belongs to the plane generated by E C and E C snce (5) and () both holds. We have:! s = E C k + E C k. (0) Equvalently, we can wrte:! s E C E C =0. () Once agan we need to mpose the two condtons on the length of the tendons. Specfcally we have:! s E d = v,! s E d = v, () whch can be rewrtten:! s E d = v,! s E d = v. (3) Condtons () and (3) fully specfy the angular velocty whch turns out to be the unque soluton of the followng system n three unknowns and three constrants: <! s E C E C =0! s E d = v (4) :! s E d = v. where t can be proven that: d = C E E C E E d = C E E C E E, (5). (6) The above system can be rearranged wth some trval computatons. In partcular we have: <! s E C E C =0! s E C =ṽ (7) :! s E C =ṽ. 593
wth: ṽ = C E E r v, () ṽ = C E E r v. (9) A. Normal form of the knematc model The models (4) and () are practcally dfferental equatons that descrbe the knematcs of the two consdered models. We here explctly derve the normal form of the dfferental equaton (4); the dervaton of the normal form of () s very smlar and therefore omtted. Let R(t) SO(3) be the rotaton matrx that maps ponts n the eye reference frame e nto ponts the nertal reference frame. Then the model (4) can be wrtten as follows: < :! s (t) R(t)E e C R(t)E e C! s (t) (R(t)E e d (t)) = v (t)! s (t) (R(t)E e d (t)) = v (t), =0 (3) Fg. 7. A more realstc model takes nto account the fact that tendons are only allowed to pull. The pcture shows ths stuaton. When the motor rotates n one drecton, E s pulled. When the motor rotates n the opposte drecton E 3 s pulled. B. Pull-only tendons In a more realstc stuaton the tendons are only allowed to pull (see Fgure 7). Therefore the above model needs to be changed. Let v > 0 ndcate the stuaton n whch E s pulled; smlarly v > 0 ndcates that E s pulled. Clearly, f v > 0 and v > 0 then Eq. (0) stll holds. However, f v < 0 and v > 0 then E 3 and E are pulled and the model must be changed. The left pcture n Fgure 7 shows that Eq. (0) must be modfed as follows:! s = E 3 C 3 k + E C k, (0) whch corresponds to:! s E 3 C 3 E C =0. () More generally, the model of the system s the followng:! s E C E C =0 f v 0,v 0! s E3 C 3 E C =0 f v < 0,v 0! s E C E 4 C 4 =0 f v 0,v < 0! s E3 C 3 E 4 C 4 =0 f v () < 0,v < 0! s E d = v! s E d = v. Ths model s more complcated but more realstc. It can be used to unquely determne! s gven the veloctes of the motors. VI. SIMULATING THE KINEMATIC MODEL In ths secton we show the results obtaned by ntegratng the knematc models (4) and (). The secton s dvded nto two parts. The frst subsecton descrbes the procedure to wrte the knematc models n the normal form. Ths step s fundamental f we want to ntegrate the models usng standard softwares, such as MATLAB. The second subsecton shows the numercal results obtaned by ntegratng the two proposed knematc models (4) and (). It s observed that both models present a cyclotorson effect smlar to the one observed on the real setup. where: Ej(t) =R(t)Ej e, d j (t) = C j E j (t) E j (t) Cj E j (t). (4) E j (t) The vector! s s related to the tme dervatve of R(t) by the followng relaton (see [5]): ˆ! s (t) =Ṙ(t)R> (t). (5) Therefore, (3) can be rearranged as a dfferental equaton n the normal form: Ṙ = f(r, v,v ). (6) for a sutably defned functon f( ). The dfferental equaton above can be ntegrated from the ntal condton R(0) = R 0 SO(3) gven the nputs v (t) and v (t). However, due to numercal errors we have that after few steps of ntegraton R(t) s no longer a rotaton matrx,.e. R(t) / SO(3). To avod these problems we can represent R(t) n exponental coordnates (t) R 3 such that R(t) =e ˆ(t). We have []: = I + ˆ +( (k k)) ˆ k k!! b (7) where! b s the nstantaneous body angular velocty [5], and (y) =(y/) cot(y/). The body angular velocty can be obtaned modfyng (4) as follows: h! b E e e ˆC E e e ˆC =0! b E e e ˆd = v ()! b E e e ˆd = v, Solvng explctly n the unknown! b : E e e ˆd 3! b 6 = 4 E e e ˆd 7 E e e ˆC E e e ˆC 5 4 v 3 v 5, (9) 0 > where A > ndcates A >. Substtutng (9) nto (7) we obtan the followng normal form: 594
! ˆ = I + ˆ +( (k k)) k k E e e ˆd 6 4 E e e ˆd E e e ˆC E e e ˆC 3 7 5 > 4 v 3 v 0 5. Alternatvely, startng from (7) we can obtan a relatvely smpler dfferental equaton that descrbes the system knematcs:! ˆ = I + ˆ +( (k k)) k k E e e ˆC 6 4 E e e ˆC E e e ˆC E e e ˆC 3 7 5 > 3 4ṽ ṽ 0 5. Fg. 9. Please zoom the.pdf for detals. Ths pcture s the analogous of Fgure but n ths case the underlyng knematc model () assumes that the tendons only push. Fnally, a more realstc model can be derved from (). Ths alternatve model s here omtted snce t can be trvally derved. B. Integraton of the knematc model Fg.. Please zoom the.pdf for detals. The fgure shows four frontal snapshots of the eyeball. The rghtmost pcture corresponds to the ntal condton R(0) = I. The other pctures, from rght to left, corresponds to the confguratons reached after one, two and three repettons of closed path (30). The underlyng knematc model (4) assumes that the tendons push and pull. We used MATLAB to ntegrate the normal forms of the knematc models (4) and (). The purpose of the smulaton s to see f movng the eyeball generates a cyclotorson movement,.e. a rotaton of the eyeball around the optcal axs of the camera. In a frst smulaton, we have moved the tendons on the followng closed path,.e.: v (t) = v (t) = v f t [0, ] 0 f t [, ] v f t [, 3 ] 0 f t [3, 4 ] 0 f t [0, ] v f t [, ] 0 f t [, 3 ] v f t [3, 4 ], (30). (3) where > 0 and v s a constant velocty. Repeatng the same closed path a three tmes, we obtaned the results shown n Fgure and Fgure 9. Fgure corresponds to the smplfed model (4), whch assumes that tendons can push and pull. Clearly, the systems present an evdent cyclotorson whch ncreases after each closed path. Fgure 9 shows the results obtaned wth the more realstc model (). In ths case the cyclotorson s stll present but t s less evdent and lmted. Heurstcally, smulatons reveal that wth ths second model the cyclotorson does not ncrease after each cycle: t ntally ncreases but does not apparently ncrease after some cycles. Ths s not the case of the frst model where cyclotorson ncreases after each closed path. In a second smulaton we tested the possblty of producng a reverse cyclotorson, n order to brng the eyeball back to the orgnal confguraton. In order to do ths test we frst moved the eyeball on the path (30) n tmes and then we moved the eyeball n tmes on the followng nverse path: 0 f t [0, ] v f t [, ] v (t) =, (3) 0 f t [, 3 ] v f t [3, 4 ] v (t) = v f t [0, ] 0 f t [, ] v f t [, 3 ] 0 f t [3, 4 ]. (33) Obtaned results are llustrated n Fgure 0 and Fgure. Fgure 0 shows that the frst model (4) s easly reversble snce the nverse path drves the system back to the ntal confguraton. Fgure shows that also n the second model () the nverse path produces an opposte cyclotorson. However, the confguraton reached by the system dffers from the ntal confguraton. VII. QUALITATIVE COMPARISON In ths secton we report the results obtaned n tryng to replcate on the real system the experments descrbed 595
Fg. 0. Please zoom the.pdf for detals. The fgure shows eght frontal snapshots of the eyeball. The pcture on the top rght corner corresponds to the ntal condton R(0) = I. The frst row, from rght to left, are the confguratons reached after 3 repettons of the drect path. The second row, s obtaned by 3 repettons of the nverse path. The underlyng knematc model (4) assumes that the tendons push and pull. Fg.. Please zoom the.pdf for detals. Ths pcture s the analogous of Fgure 0 but n ths case the underlyng knematc model () assumes that the tendons only push. 0 =0 = 0.5 = 0.5 = 0.5 = 0 = 0.5 = 0 = 0.5 = 0 = 0.5 =0.5 =0.75 = = = 0 =0.5 = 0 =0.75 = 0 = = 0 = Fg.. Expermental results reproducng the experments descrbed n Secton VI. The top rght pcture corresponds to one take wth the eye n ts ntal confguraton (zero cyclotorson, =0 ). The pctures on ts rght have been obtaned after one, two and three cyclc paths (34). The pctures on the second row have been obtaned after one, two, three and four nverse cyclc paths (35). 596
n Secton VI. Notce that on the real setup we can only control motors n poston whle smulatons assume tendons controlled n velocty. For our purposes ths, s not a real problem and the cyclc paths (30) and (3) can be easly translated nto paths for the poston of the motors. Specfcally, let be the poston of the frst motor and be the poston of the second motor. We consdered the followng cyclc path, whch s ntended to replcate what has been smulated n MATLAB: Intal confguraton =,0 and =,0 (34) Move to :,0 Move to :,0 Move to,0 :!,0 Move to,0 :!,0 Fnal confguraton =,0 and =,0. After each cyclc path, we took a pcture of the panel n order to measure the cyclotorson of the system. In order to measure the cyclotorson angle, we used the same procedure descrbed n Secton II. The dfference between the lne orentaton after and before the executon of the complete movement gves the cyclotorson angle = 0. The lne orentaton has been computed usng the Hough transform [4]. The obtaned results are shown n Fgure 3. Qualtatvely, results are smlar to those obtaned n smulaton. The cyclotorson ntally ncreases and then reaches a maxmum. In fact, after some cycles the cyclotorson does not ncrease sgnfcantly. Ths s qualtatvely n perfect agreement wth the smulatons whch has been done wth pull-only tendons model. As a second experment, we consder the problem of reversng the prevous path to see f t s possble to acheve an opposte the cyclotorson as observed n the smulatons. After havng executed the path (34) three consecutve tmes, we have executed three tmes also the nverse path: Intal confguraton =,0 and =,0 (35) Move to :,0 Move to :,0 The results of the experment are shown n Fgure. Remarkably, the obtaned results are qualtatvely n lne wth those obtaned by smulatng the knematc model wth push-only tendons. In partcular, the cyclotorson turns out to be reversble, n the sense that followng the nverse path produces an opposte cyclotorson. In Fgure t s shown that followng the path (34) produces a negatve cyclotorson ( < 0) whle followng the nverse path (35) produces a postve cyclotorson ( >0). The maxmum and mnmum cyclotorson that was experenced on the real setup are ndcated n the followng table 3 : Mnmum Maxmum Range cyclotorson cyclotorson cyclotorson 0.75.75 VIII. CONCLUSIONS AND FUTURE WORKS We have presented a knematc study of a tendon drven eyeball. The study was motvated by the necessty of understandng the cause of a cyclotorson effect that was experenced on the real setup. The frst part of the study was devoted to the development of a knematc model of the tendon drven eye. The second part was focused on a qualtatve comparson between data taken from the real setup and data obtaned ntegratng the knematc model. The comparson gave good results and leaves open the possblty of usng the model for obtanng some quanttatve data. Future works wll nvestgate how useful the gven model s for obtanng some quanttatve data. REFERENCES [] D Bamno, G Cannata, M Maggal, and A Pazza, MAC-EYE: a tendon drven fully embedded robot eye, 005, pp. 6 67. [] F. Bullo and R. M. Murray, Proportonal dervatve (PD) control on the Eucldean group, vol., June 995, pp. 09 097. [3] G Cannata and M Maggal, Implementaton of Lstng s Law for a Tendon Drven Robot Eye, 006, pp. 3940 3945. [4] R. O. Duda and P. E. Hart, Use of the hough transformaton to detect lnes and curves n pctures, Comm. ACM 5 (97), 5. [5] R. M. Murray, Z. L, and S. S. Sastry, A mathematcal ntroducton to robotc manpulaton, CRC Press, 994. [6] A D Polptaya and B K Ghosh, Modelng the dynamcs of oculomotor system n three dmensons, 003. [7] A D Polptya, B K Ghosh, C F Martn, and W P Dayawansa, Mechancs of the eye movement: geometry of the lstng space, 004. [] Ashoka D Polptya and Bjoy K Ghosh, Modellng and control of eyemovement wth musculotendon dynamcs, Proc of the 00 Amercan Control Conference ACC 00, vol. 3, 00, pp. 33 3. Move to,0 :!,0 Move to,0 :!,0 Fnal confguraton =,0 and =,0. 3 In smulaton, the maxmum and mnmum cyclotorson depend on the ampltude of the cyclc path. The bgger s the ampltude the bgger s the observed cyclotorson. Ths feature has been observed also on the real setup. 597