914 IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, OCTOBER 2010

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1 914 IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, OCTOBER 2010 Modelng of Transmsson Characterstcs Across a Cable-Condut System Varun Agrawal, Student Member, IEEE, Wllam J. Pene, Member, IEEE, andbnyao, Senor Member, IEEE Abstract Many robotc systems, lke surgcal robots, robotc hands, and exoskeleton robots, use cable passng through conduts to actuate remote nstruments. Cable actuaton smplfes the desgn and allows the actuator to be located at a convenent locaton, away from the end effector. However, nonlnear frctons between the cable and the condut account for major losses n tenson transmsson across the cable, and a model s needed to characterze ther effects n order to analyze and compensate for them. Although some models have been proposed n the lterature, they are lumped parameter based and restrcted to the very specal case of a sngle cable wth constant condut curvature and constant pretenson across the cable only. Ths paper proposes a mathematcally rgorous dstrbuted parameter model for cable-condut actuaton wth any curvature and ntal tenson profle across the cable. The model, whch s descrbed by a set of partal dfferental equatons n the contnuous tme-doman, s also dscretzed for the effectve numercal smulaton of the cable moton and tenson transmsson across the cable. Unlke the exstng lumped-parameter-based models, the resultant dscretzed model enables one to accurately smulate the partal-movng/partal-stckng cable moton of the cable-condut actuaton wth any curvature and ntal tenson profle. The model s further extended to cable-condut actuaton n pull pull confguraton usng a par of cables. Varous smulatons results are presented to reveal the unque phenomena lke backlash, cable slackng, nteracton between the two cables, and other nonlnear behavors assocated wth the cable conduts n pull pull confguraton. These results are verfed by experments usng two dc motors coupled wth a cable-condut par. The expermental setup has been prepared to emulate a typcal cable-actuated robotc system. Expermental results are compared wth the smulatons and varous mplcatons are dscussed. Index Terms Cable-condut actuaton, cable complance, frcton, pull pull confguraton, surgcal robot. I. INTRODUCTION SURGICAL robots often utlze cable-condut pars n a pull pull confguraton to actuate the patent-sde manpulators and slave nstruments [2], [3]. Cable transmssons are preferred because they can provde adequate power through narrow Manuscrpt receved December 10, 2009; revsed Aprl 27, 2010; accepted July 28, Date of publcaton September 7, 2010; date of current verson September 27, Ths paper was recommended for publcaton by Assocate Edtor A. Albu-Schäffer and Edtor W.K.Chunguponevaluaton of therevewers comments. Ths work was supported by Meere Company, South Korea and n part by the Natonal Scence Foundaton under Grant CMMI Ths paper was presented n part at the IEEE Internatonal Conference on Robotcs and Automaton, Pasadena, CA, 2008 [1]. The authors are wth the School of Mechancal Engneerng, Purdue Unversty, West Lafayette, IN USA (e-mal: vagrawal@purdue.edu, pene@purdue.edu, byao@purdue.edu). Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /TRO tortuous pathways and allow the actuators to be located safely away from the patent. Cables are lght weght and cost effectve and greatly smplfy the transmsson. Cable-condut actuaton, whch s also sometmes known as tendon sheath, or Bowden cable actuaton, s also used n many robotc hands [4] [6], as well as colonoscopy devces [7], [8]. To develop power dense yet ergonomc actuaton for wearable nterfaces, cable actuaton s also used n exoskeleton robots [9] [11]. The control of these systems, however, s challengng due to cable complance and frcton wthn the condut. These nonlneartes ntroduce sgnfcant tenson losses across the cable and gve rse to moton backlash, cable slack, and nput-dependent stablty of the servo system [12], [13]. In the absence of a transmsson model for the cable-condut system, these nonlnear behavors are not accounted for [9] [11], leadng to poor system performance. Although varous physcal measures are adopted ncludng usng Polytetrafluoroethylene (PTFE)-coated steel cables sldng nslghtly preloaded Kevlar-renforced housngs [10] and keepng the cable-wrappng angles and pretenson to low levels, they can only mprove the system performance to a certan degree.beyond that, one has to rely on effectve control algorthms to mprove the performance, whch stresses the mportance of developng a model for the transmsson characterstcs. Ths paper develops a model for the transmsson characterstcs n cable-condut mechansms to effectvely analyze such a system. Kaneko et al. [12] performed experments on torque transmsson from the actuator to the fnger jont usng a par of cables passng through conduts. However, no analytcal model was developed. These experments assumed a large value of pretenson n the cable to avod any slackng. Snce frcton forces are drectly dependent on cable pretenson, t leads to a tradeoff between tenson losses and cable slackng. Thus, cable slackng s an mportant phenomenon that should be addressed. Later on, the authors developed a model for a sngle cable passng through the condut of fxed constant curvature wth a gven constant pretenson throughout the cable [14], [15]. Based on the model, they analytcally calculated the equvalent cable stffness for a sngle cable. Furthermore, the authors proposed a lumpedmass numercal model for tenson transmsson across the cable. Through ther model, they demonstrated the cable-condut system dsplay drecton-dependent behavor and, hence, cannot be treated as a smple sprng. However, ther model essentally assumes that all ponts on the cable have the same ntal pretenson of a constant value and, as such, cannot consder the general behavor of a cable-condut transmsson, where the ntal tenson depends on the spatal postons. The calculaton of last movng pont when usng multple-lumped elements also assumes the same constant pretenson across all elements, whch essentally /$ IEEE

2 AGRAWAL et al.: MODELING OF TRANSMISSION CHARACTERISTICS ACROSS A CABLE-CONDUIT SYSTEM 915 gnores the spatal dependence of the tenson transmsson across the cable and prevents accurate study of some of the unque phenomena assocated wth the cable-condut actuaton mechansms, such as partal movng/partal stckng. In practce, due to the presence of frcton, the resdual tenson or ntal tenson profle depends on the tme hstory of past appled forces and cannot be assumed to be unform across the cable. Moreover, n many applcatons, lke surgcal robots and exoskeletons, the condut curvatures are path dependent, and thus, the model cannot be appled drectly for these applcatons. Because of these ssues, the assumpton of constant curvature and a predetermned constant pretenson across the entre cable severely lmts the usefulness of the model. Pall and Melchorr [16], [17] further refned the model usng a dynamc Dahl s frcton model nstead of the smple Coulomb frcton model but made the same assumptons of constant pretenson and curvature for the lumped-element models. Furthermore, all these exstng models only focus on power transmsson usng a sngle cable condut and, therefore, cannot address the unque phenomena of cable slackng and cable nteracton assocated wth the systems usng a par of cables for power transmsson, lke the ones studed n ths paper. Instead of usng the lumped-mass analyss, ths paper frst develops an exact, contnuous tme-doman model descrbed by a set of partal dfferental equatons (PDEs), whch s applcable to cable-condut systems wth any pretenson and curvature profles. In addton, ths paper consders the complex nteracton between a par of cables n pull pull. The exact nfntedmensonal model s then dscretzed to generate effectve numercal smulaton algorthms for moton and power transmssons. Ths can be used to solve the nonlnear system response to predct cable slackng and overall transmsson characterstcs of the system. The model s valdated through experments. Unlke the sngle-cable system dscussed n detal n the earler research, the use of par of cables nduces cable nteracton leadng to behavor that s completely absent n the pror cases. Whle the prevous models have been developed usng lumpedmass analyss wth nerta, our work uses the exact dstrbuted system dynamcs to generate the dscretzed model for analyss and smulaton, although the cable nerta s neglected. Furthermore, the phenomenon of partal cable segment, whch causes the cable nteracton, can be explaned. The approxmaton errors n the dscretzaton process have been clearly lad out as well. Moreover, whle only force transmsson has been analyzed n prevous research, moton transmsson has also been presented here, whch s partcularly mportant for surgcal devces, whch are usually operated n poston control mode. In the followng sectons, the setup of the problem s dscussed followed by detals of the proposed model and the smulaton results. The methodology of experments s outlned, and expermental results are presented and compared wth the smulaton results. II. MOTIVATION AND EXPERIMENTAL SETUP In cable-drven robots, the slave manpulators are mechancally actuated usng cable passng through a thn tube or condut. Fg. 1. Fg. 2. Expermental setup. Model of the expermental setup. Nonlneartes are ntroduced n moton transmsson due to the frcton forces between the cable and the condut. Moreover, tenson losses across the cable necesstate much hgher actuatng forces for relatvely small loads. Whle hgh pretenson s desred to avod cable slackng, t comes wth a drawback of hgher frcton forces. However, lower pretenson leads to cable slackng. Thus, a tradeoff s requred between cable slackng and large actuaton forces. Snce t s dffcult to place sensors at the dstal ends of the hghly mnaturzed nstrument, such as n a surgcal robot, the poston and appled forces of the tool tp are dffcult to estmate and control. Hence, the resultant accuracy of the system s extremely poor, as compared wth ndustral robots. In surgcal robots, ths results n contnuous adjustment of the actuatng nput by the human n the loop, thereby potentally affectng the performance of the surgeon. The objectve of ths research s to model cable actuaton n such a system and characterze the force and moton transmsson from the actuator to the load. Ultmately, these models can be used to mprove the control strategy of the system. A typcal load actuaton system of a cable actuated robot has been emulated n the expermental setup shown n Fg. 1. A schematc of the setup s shown n Fg. 2. A two-cable pull pull transmsson s used, actuated wth two brushed dc motors

3 916 IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, OCTOBER 2010 Fg. 4. Forces balance dagram of cable element. Fg. 3. Moton of the cable element. mounted on lnear sldes. The frst motor s controlled as the nput or the drve motor, whle the second motor smulates a passve load or envronment. Each cable passes through a flexble condut and s wrapped around pulleys attached to each of the dc motors. The tghtly wound sprng wre conduts are fxed at each end usng two plates attached to the same platforms on whch the lnear sldes are mounted. Ths way, the platforms holdng the plates are free to move n space, and applyng a tenson n the cable s counteracted by a compresson n the condut wth no forces beng transmtted through the ground. The cable and the condut, therefore, act as sprngs n parallel. The actuator or the drve motor s run n poston control mode, whle the follower motor s run n torque control mode. The load s smulated as a torsonal sprng such that the restorng torque appled by the motor s proportonal to the angle of rotaton. Encoders are used to measure the angular rotaton of the two motors. The current flowng across the two motors s used to estmate the torque appled by the pulley, whch s proportonal to the dfference n the two cable tensons on each sde. The sum of the tensons beng appled by the two cables on each motor s measured usng load cells mounted between the lnear slde and condut-termnaton plate. Usng the torque values and the load cell measurements, tenson at the two ends of each cable can be calculated. III. DYNAMIC MODEL A. System Governng Equatons Consder the setup shown n Fg. 3, where two flexble cables,.e., cable A and cable B, pass through fxed conduts of predefned curvature R(x), where x denotes the poston along the condut. At t = 0, actuator starts to move the cables. To model the moton of the cable, we make the followng assumptons. 1) Inerta effects n the cable can be neglected. 2) The cable s restrcted to move along condut (no transverse moton). 3) Interacton between the cable and the condut s through a normal force and frcton (Coulomb frcton). 4) Consttutve behavor of the cable s assumed to be elastc, defned by the standard Hooke s law. When at relaxng state (.e., no tenson) wthout slackng, ponts on the cable can be unquely ndexed by the condut poston x. For the pont on the cable ndexed by x, let u(x,t) denote ts axal dsplacement at tme t, and let T(x,t) be the correspondng axal tenson of the cable. For notatonal smplcty, n the followng, the partal dervatve of a functon T(x,t) wth respect to the spatal varable x wll be denoted by T (x,t) and the partal dervatve wth respect to the tme varable t by T (x, t). To obtan the dynamc model of the moton and force transmtted through the cable, consder the movement of an nfntesmal cable segment [x, x + dx], as shown n cable A of Fg. 3 wth an enlarged vew shown n Fg. 4. In Fg. 4, N(x,t) denote the normal force between the cable and the condut. f(x,t) s the frctonal force actng on the cable. For ths nfntesmal segment, the radus of curvature can be assumed to be constant, gven by R(x), and the nfntesmal angle dθ(x), shown n Fg. 4, s related to dx by dx = R(x)dθ(x). As there s no cable movement along the radal drecton of the condut, through the force balancng equaton, the normal reacton force actng on ths nfntesmal cable element s related to the tensons at the two ends by ( ) ( ) dθ(x) dθ(x) N(x, t) =T (x, t)sn + T (x + dx, t)sn 2 2 T (x, t)dθ(x). (1) Thus, from the Coulomb frcton model, we know that f (x, t) µn (x, t) = µt (x, t) dθ (x) =µt (x, t) dx R (x). As the nerta of the cable s neglected, the force balance equaton apples to the axal drecton of the condut (.e., the cable movement drecton) as well. Therefore, when the net axal tenson force T(x+dx,t) T(x,t) = T (x,t)dx s less than the rght-hand sde of (2),.e., T (x, t) <µt(x, t)/r(x), the cable segment wll not move, and the actual frcton f(x,t) has the same magntude as the net axal tenson force,.e., (2) u(x, t) =0 and f(x, t) =T (x, t)dx. (3)

4 AGRAWAL et al.: MODELING OF TRANSMISSION CHARACTERISTICS ACROSS A CABLE-CONDUIT SYSTEM 917 On the other hand, when the cable segment moves due to the net axal tenson forces, frcton wll be at ts maxmum value, as gven by (2),.e., f(x, t) =(µt (x, t)/r(x))sgn ( u(x, t)) dx. Thus, from the force balance equaton T µt (x, t) (x, t)dx = sgn ( u(x, t))dx when u 0. R(x) (4) To calculate the cable stran along the condut path u (x, t),t s assumed that, when stretchng, Hooke s law of elastcty can be used by modelng the cable as a lnear sprng wth stffness K, and when compressng or cable slackng, no force s transmtted through the cable. Snce the cable and the condut act n parallel, K s the combned stffness of the system. Thus T (x, t) =Ku (x, t) when u (x, t) > 0 T (x, t) =0 when u (x, t) 0 1 where K = (5) K cable K condut Combng (3) (5), the overall dstrbuted dynamc model of the moton and tenson across the cable s descrbed by the followng set of PDEs: when u (x, t) > 0, T(x, t) =Ku (x, t) and ) u (x, t) µ R(x) u (x, t)sgn ( u(x, t)) = 0 f u (x, t) = µ R(x) u (x, t) ) u(x, t) = 0, otherwse when u (x, t) 0 ) T (x, t) =0. (6) Asde from the aforementoned governng equatons, to calculate the moton and tenson transmsson across the cable, one needs the ntal condtons and the boundary condtons as well. Thus, to be able to precsely descrbe the cable dynamc behavor, t s absolutely necessary to specfy the ntal cable dsplacement profle u 0 (x),.e., u(x, 0) = u 0 (x) (7) and the boundary condtons of the cable at the two ends, dependng on the envronment to whch the cable s attached. For example, consder a cable wth the end at x =0connected to an envronment havng a predefned movement of g d (t) and the other end at x = L fxed to a stff envronment havng a stffness of K e ; then, notng (7), the boundary condtons for solvng the cable movement would be u (0,t)=g d (t) u (L, t) = 1 K e T (L, t) = K K e u (L, t). (8) Remark 1: In the earler development of the mathematcally rgorous dstrbuted parameter model for cable-condut actuaton, no restrcton s put on the curvature of the condut,.e., R(x) could be any functon. Ths s n contrast wth the prevous work n [14] [17], where constant curvature s assumed across each cable segment. Furthermore, no restrcton s put on the ntal cable dsplacement profle u 0 (x), and thus, the ntal tenson profle of T (x, 0) = Ku (x, 0) = Ku 0 (x) (assume that u 0 (x) 0). It s noted that all the prevous work [14] [17] assume a constant ntal tenson profle of T (x, 0) = T 0, whch s hardly true n realty due to the dstrbuted frcton effect across the cable. Thus, although some of the dscretzed equatons on the tenson transmsson for a partcular segment ntroduced n the followng secton may look somewhat smlar to those n [14] [17], the overall modelng process s fundamentally dfferent from the prevous work. In addton, cable slackng s explctly taken nto account n the proposed model, whch cannot be addressed usng prevous work. B. Dscretzed Model Snce t s mpossble to analytcally solve the PDEs (6), n practce, dscretzed element models based on these governng equatons are obtaned for realstc computer smulatons, whch s the subject of ths secton. As shown by cable B n Fg. 3, the each cable s dvded nto n cable segments, wth nodes at x 1 = 0, x 2 = x 1, x 3 = x 2 + x 2,...,and x n = x n 1 + x n = n =1 x = L. The dsplacement and tenson of the two ends of each segment wll be calculated at dscrete tme nstants usng the dscretzed elemental equatons as follows. Consder the th cable segment between nodes and + 1. Let T(x,t j ) and u(x,t j ) be the tenson and the dsplacement of the th node, respectvely, at tme t j. We neglect small varatons n radus of curvature over the cable segment and denote t by R(x ). It should be noted that such a standard dscretzaton approxmaton s dfferent from the assumpton of constant curvature across the entre cable n the prevous work [14] [17] as one can always choose the dscretzaton segment length x small enough to make the approxmaton error arbtrarly small n the proposed approach. Wthout consderng the segments, whch are completely slackng (.e., the segment of u (x, t) < 0, whch could happen at the two ends of the cable due to certan mposed boundary condtons) as they are not the normal workng modes for cable actuated devces, all cable segments can be dvded nto three dfferent categores as follows. Case 1: The entre segment s movng. Snce u (x, t) 0for the cable segment (x,x +1 ], the frst case of (6) or, equvalently, (4) apples. Thus, notng the dscretzaton approxmaton assumpton that R(x) =R(x ) and sgn ( u(x, t)) = sgn ( u(x,t)) x (x,x +1 ], one can ntegrate (4) over the segment as follows: x T (x, t) x T (x, t) dx = x x µ R(x ) sgn ( u(x,t)) dx x (x,x +1 ]. (9) On ntegratng over the segment (x,x], we get ( ) µ (x x ) T (x, t) =T (x,t)exp sgn ( u(x,t)). (10) R (x ) From (5), wth the tenson dstrbuton of (10) over the segment, the dsplacement or the stretch n the cable segment can be

5 918 IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, OCTOBER 2010 analytcally calculated by x u (x, t)dx x = 1 x ( ) µ (x x ) T (x,t)exp sgn ( u(x,t)) dx (11) K x R (x ) whch, upon ntegraton, gves us the followng equaton: u(x, t) u(x,t)= R (x ) Kµ sgn ( u(x,t)) T (x,t) [ ( ) ] µ (x x ) exp sgn ( u(x,t)) 1. R (x ) (12) Therefore, at the dscrete tme nstant t j, from (10) and (12), tenson and dsplacement of the two ends of the th cable segment can be approxmated as gven by the followng equatons: ( ) T j +1 = T j µ (x+1 x ) exp S j (13) R (x ) [ ( µ (x+1 x ) exp ) ] u j +1 uj = R (x ) Sj Kµ T j S j 1 (14) R (x ) where for compactness, the notaton T j = T (x,t j ), u j = u (x,t j ), and S j = sgn (u(x,t j ) u(x,t j 1 )) have been used. Case 2: The entre cable segment s statonary. Snce the cable segment s statonary, the stran of the segment wll not change. Subsequently, the tenson of the segment does not change ether. Thus, dsplacement and tenson of the two nodes reman unchanged from the prevous values,.e., T j = T j 1 u j = uj 1 and T j +1 = T j 1 +1 and u j +1 = uj (15) Case 3: A part of the cable segment s movng, whle the rest of t s statonary. Wthout loss of generalty, assume that node s movng, whle node +1 s statonary. Let ξ be the last movng pont on the cable segment,.e., ξ = max {x [x,x +1 ] : u (x, t j ) u (x, t j 1 )}. Therefore, Case 1 apples for the secton (x, ξ], whle Case 2 for the rest of the segment. Thus, the tenson and dsplacement reman unchanged over the secton (ξ,x +1 ],.e., T (x, t j )=T (x, t j 1 ) and u(x, t j )=u(x, t j 1 ) x (ξ,x +1 ]. (16) Snce only node nformaton s preserved n the dscrete tme model, the actual tenson and dsplacement of T (x, t j 1 ) and u(x, t j 1 ) for the secton x (ξ,x +1 ) n the prevous tme nstance are lost at the current tme nstance. Thus, one cannot explctly calculate the exact tenson and dsplacement varaton across the secton (ξ,x +1 ), and some sorts of further approxmatons have to be made. Fortunately, snce a small cable segment s assumed n the dscretzaton process, we may approxmate the stran over the segment by the stran calculated based on the average tenson of the two ends of the cable segment,.e., assumng u (x, t) (1/K)((T j + T j +1 )/2), x (x,x +1 ]. Wth ths approxmaton, the cable dsplacement over the segment can be calculated as follows: u(x, t) u j = T j + T j +1 (x x ) x (x,x +1 ]. (17) 2K Note that cable stretch, as calculated n (17), s accurate n case of constant tenson across the segment. In the case of the entre segment beng n moton as n Case 1, t represents a secondorder approxmaton of the tenson profle gven by (12), whch can be proven by notng (e y 1)/y = (1 + e y )/2 + O(y 3 ). Combnng (16) and (17), the tenson and dsplacement transmssons over the segment are calculated by u j +1 = uj 1 +1, Tj +1 = T j 1 +1 (18) u j = uj +1 T j + T j +1 (x +1 x ). (19) 2K These three cases cover all the possble scenaros of moton durng the normal operatng condtons. The analyss has been carred out based on the system dynamcs, makng no assumpton related to the dscretzaton for all the movng nodes. Spatal dscretzaton assumpton s only made for the partally movng cable segment. Thus, ts effects are mnmal. Snce no assumpton has been made related to the temporal dscretzaton, t does not drectly affect the smulatons. However, t affects the last movng pont n the partally movng node and, thus, ndrectly affects the accuracy of the smulatons. Although no analyss has been presented here to determne the number of elements nto whch the cable should be dvded, smulatons can be used to fnd the optmal number of elements. We can now use these equatons to smulate the moton and torque transmsson characterstcs of a cable-condut system. IV. SIMULATION RESULTS To smulate the moton and torque transmsson, we start by defnng condut shape, ntal condton, and boundary condtons, as dscussed n (7) and (8). Although the ntal pretenson may not be constant across the entre cable n practce, n order to compare the proposed method wth prevous work, a constant pretenson T 0 s assumed n the followng smulatons, whch translates to the ntal condton: u (x, 0) = T 0 /K or u(x, 0) = T 0 x/k + u(0, 0). (20) The number of cable segments for smulatons can be determned based on the accuracy levels desred, ether by analyss or by teratons. Wth ntal condton n (20) and boundary condton n (8), we smulate the system for requred number of segments for each cable, assumng that the system starts from rest (or some pre-specfed state). Because of the presence of frcton, moton transmsson s not nstantaneous. Thus, all the segments do not start movng mmedately when an nput moton s appled. Therefore, at any tme nstant, for each cable segment, t s dentfed whether the segment s movng, partally movng, or statonary. Based on ths nformaton, the last movng node of the cable s estmated.

6 AGRAWAL et al.: MODELING OF TRANSMISSION CHARACTERISTICS ACROSS A CABLE-CONDUIT SYSTEM 919 Consder the moton of cable A. For smplcty, we assume that at t 0 = 0, all the nodes are statonary. Wthout loss of generalty, consder the case when the cable s beng pulled at x 1 = 0 wth an enforced boundary gven by (8), as shown n Fg. 3. Therefore, at t = t 1, u(x 1,t)=ġ d (t) < 0 and the moton starts propagatng along the cable. By the tme t p, let node k be the last movng node,.e., the moton has been propagated over the segments from 1 to k 1 but has not reached node k+1. Thus, Case 1 n prevous secton apples to the frst k 1 segments, Case 3 for the segment k, and Case 2 for the rest of the segments, whch leads to the followng set of equatons: T p 1 exp [ µ x1 R (x 1 ) Sp 1 ] T p 2 = 0.. [ ] T p k 1 exp µ xk 1 R (x k ) Sp k 1 T p k = 0...k 1 eqns. u p 2 R (x [ ( 1) Kµ Sp 1 T p µ x1 1 exp u p 3 up 2 R (x 2) Kµ Sp 2 T p 2 R (x 1 ) Sp 1 [ exp ( µ x2 R (x 2 ) Sp 2 u p k up k 1 R (x [ k 1) S p k Kµ T p k 1 exp u p k + T p k 2 x k K k+1 T p 1 k+1 2 = up 1 x k K ) ] 1 = g d (t p )... ) ] 1 =0 ( µ xk 1 R (x k 1 ) Sp k (21) ) ] 1 =0...k eqns. (22) Wth the boundary condtons mposed by the actuatng moton of the end of the cable, u(x 1,t p ) = g d(t p ), and statonary node k+1, u(x k+1,t p ) = u(x k+1,t p 1), the moton of all the ntermedate nodes u(x 2,t p ), u(x 2,t p ),..., u(x k,t p ) are unknown (k 1 unknowns). In addton, tenson of the frst k nodes T(x 1,t p ), T(x 2,t p ),...,T(x k,t p ) are unknown (k unknowns), the tenson of node k+1 beng known from prevous tme step. Therefore, these 2k 1 unknown dsplacement and tenson varables can be calculated by smultaneously solvng the 2k 1 equatons n (21) and (22). The kth cable segment starts to move, and node k + 1 becomes the last movng node at tme t q when the followng condton s satsfed: T (x k+1,t q 1 ) S p k T (x k,t q ) S p k exp [ µ xk R (x k ) Sp k ]. (23) As the nodes at x 1 and x n+1 keep on movng, tenson at the nput ends keep on changng, and moton propagates along the two cables A and B. Eventually, the last movng nodes of both the cables concde and the cables start to move en masse. Apart from computatonal perspectve, the concept of last movng node also helps us n physcally analyzng the partal moton transmsson across the cable. Ths becomes partcularly useful for understandng the coupled moton transmsson n a twocable system, where one cable starts pullng another cable, as elaborated upon later n ths secton. If, at any pont n tme, tenson on one of the cables becomes zero (cable goes slack), e.g., cable B, only the nodes of the other cable, as well as the dstal node of the slack cable (.e., node 1, 2,...,n,2n n ths case), need to be solved for moton transmsson. Snce the moton of two cables are constraned by the pulleys connectng them, assumng no slackng at the two pulleys u(x n,t) u(x 2n,t)=2R o θ o u(x n,t)+u(x 2n,t)=2L. (24) Smulatons are carred out assumng neglgble frcton losses at the pulleys as compared wth the losses n the conduts. Thus, the boundary condton n (8) can be smplfed as follows: τ o = K e θ o =(T 2 T 4 )R o (25) where τ o s the external torque beng appled by the load, θ o s ts angular rotaton at the output, R o s the radus of the pulley attached to the load motor, K e s the smulated envronment stffness, 2L s the sum of two condut lengths correspondng to cables A and B, and T k (k = 1, 2, 3, 4) denote the tenson at the two ends of each cable,.e., T 1 (t) = T(x 1,t), T 2 (t) = T(x n,t), T 3 (t) = T(x n+1,t), and T 4 (t) = T (x 2n,t). The nput motor s smulated to follow a snusodal oscllatory moton profle. At each tme step, based on the aforementoned dscusson, the last movng node of each cable s estmated for the gven nput moton. The tenson and dsplacements of all the nodes up to the last movng node are calculated accordngly. If one of the cables goes slack, the parameters for the other cable are calculated as an ndependent cable wth the correspondng load, snce the slack cable no longer affects ts moton. Based on the earler analyss, the moton of the system can be dvded n two categores: 1) both cables taut or 2) ether cable slack. Smulatons are carred out for the two-cable system, where each cable s of length 2 m, looped thrce, wth a pretenson of T 0 = 10 N; the equvalent stffness of the cable-condut system s 1 kn/m, and the envronment stffness s 0.4 kn m. The cables are dvded n 16 sectons each, and a tme step of 0.25 ms was used for the smulatons. A snusodal moton of ampltude 1 rad and frequency of 1 Hz s appled as the nput. Smulaton results are shown n Fg. 5. Varous tme nstants have been marked by numbers to facltate the comparson of varous parameters as well as correlatng the trends n dfferent plots. For smplcty, tme nstant 1 has been chosen when the drecton of moton changes for the frst tme. The tenson transmsson across the two cables,.e., cable A and cable B, s shown n Fg. 5(a) and (b), respectvely. Although the tenson transmsson profles are largely smlar to the case of sngle cable transmsson, dfferences due to cable couplng are vsble as peaks n the transmsson profle (hghlghted by the dotted crcle), whch are dscussed n detal later n the secton as phase II (one cable pullng another cable). The overall tenson profles are smlar for the two cables, as shown n Fg. 5(a) and (b),

7 920 IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, OCTOBER 2010 Fg. 5. Transmsson profle wth varous tme nstants. (a) Tenson transmsson across cable A (T 1 versus T 2 ) and (b) across cable B (T 3 versus T 4 ). (c) Torque transmsson from the actuator (τ n ) to the load (τ out ). (d) Angular rotaton transmsson from the drve pulley (θ n ) to the follower pulley (θ out ). however, wth dfferent states at any tme nstant. Apart from comparng the tenson varaton, n the case of the two-cable system, we can also compare torque and angular moton transmsson from the actuator to the load. The torque transmsson s shown n Fg. 5(c), and the angular moton propagaton s shown n Fg. 5(d). Both the torque and moton transmsson follow a backlash type of profle, however, wth dfferent slopes and wdths. To understand the mechansm of moton propagaton across the cable, consder Fg. 6, showng the varaton of output torque versus nput torque (τ n versus τ out ). The transmsson profle can be dvded n varous phases, as marked n the fgure. These phases can be brefly descrbed as follows. 1) Output pulley not movng: When the moton has not propagated to the dstal end of ether of the two cables (.e., the output load), both the cables move ndependently, and as a result, no torque s transmtted to the output, and the output pulley does not move. Ths corresponds to the wdth of the backlash, as represented by the flat sectons (tme ntervals 1 2 and 5 6 n Fg. 5). 2) One cable pullng another cable: Because of dfference n tenson across the two cables, frcton levels are also dfferent, and as a result, the rate of moton propagaton varyng across the two cables s also dfferent. Therefore, there are tme nstances when moton propagates to the end of only one cable (wthout loss of generalty assume cable A), whle the other cable (cable B) s partally mov- Fg. 6. Torque transmsson profle. ng. Thus, cable A s causng the output pulley to move, as well as the dstal end of cable B, whle the (partal) moton of the drve end of cable B does not nfluence the moton of the load pulley. Ths gves rse to the secton wth small slope n the backlash profle (tme ntervals 2 3 and 6 7), snce only one cable s actve n moton transmsson to the load, whch s also referred to as soft sprng [12]. In the tenson transmsson profle, ths gves rse to the phase

8 AGRAWAL et al.: MODELING OF TRANSMISSION CHARACTERISTICS ACROSS A CABLE-CONDUIT SYSTEM 921 Fg. 7. Tenson varaton across the length of the two cables. Fg. 8. Expermental and smulaton results for T 0 = 7.3 N, usng the orgnal parameter estmates. when (n cable B) although the tenson at the drve end of the cable s decreasng, the tenson at the follower end s ncreasng (due to pullng by cable A), as shown by the sold brown lne n Fg. 7 and dotted crcle n Fg. 5(a). Thus, the nteracton between the two cables gves rse to these counter-ntutve peaks n tenson transmsson profle, whch s not vsble n the case of sngle cable. 3) Both cables movng: When the last movng nodes of both the cables concde, both the cables collectvely move, transmttng torque to the output, whch corresponds to the slope of the backlash n the torque as well as moton transmsson (tme ntervals 3 4 and 7 8). 4) One cable slack, whle other cable s movng: Dependng on the nput moton profle and the pretenson, large tenson drop across one of the cable can lead to cable slackng, whle the tenson across the other cable ncreases, and t keeps movng. Ths decreases the slope of the backlash, snce only one cable s effectvely transmttng moton, and hence, the apparent stffness of the system reduces (tme ntervals 8 1 and 4 5). Ths phase can be further subdvded nto two cases: a) Moton of the drve pulley contnues n the same drecton, and b) drve pulley changes ts drecton of moton. These four phases defne the cable moton. Whle phases I and II are generally of short tme duraton, phases III and IV govern the moton durng most of the operaton. Note that the occurrence and strength of all these phases are dependent on cable pretenson as well as the ampltude of the nput moton, apart from physcal parameters of the system lke cable length, stffness, frcton level, and envronmental stffness. From these plots, t s evdent that frcton not only causes a backlash type of transmsson profle but also leads to other phenomenon, such as changes n the slope of the transmsson due to cable slackng, ntroducton of small slopes n the torque transmsson, as well as opposte tenson varatons at two ends of the cable due to partal moton propagaton. V. EXPERIMENTAL RESULTS To valdate our smulaton, experments were performed usng the setup descrbed n Secton II. The actuaton cables were 0.52 mm n dameter, uncoated stanless steel 7 19 wre rope that s approxmately 1.6 m n length and wrapped around 12-mm-dameter motor pulleys. The stanless steel conduts were 1.2 m n length, made from 0.49-mm-dameter wre wrapped nto a close-packed sprng wth an nner dameter of 1.29 mm. The two motors were controlled usng the dspace control board. For ths study, the pretenson n the cables and shape of the condut could be vared and controlled. Pulley rotaton was measured wth a resoluton of Pulley torque was measured wth an accuracy of 0.1 mn m over a range of 100 mn m, whle the combned cable tenson measured by the load cell had an accuracy of 0.1 N over a range of 45 N. To correlate the expermental results wth the smulaton, several system propertes were expermentally estmated. In partcular, the stffness of the cable and conduts were measured to be and kn/m, respectvely. Snce the cable and the conduts act as sprngs n parallel, the equvalent cable stffness was calculated to be kn/m. Force relaxaton or the creep of the cable was measured to be approxmately 10.5%, wth a tme constant of approxmately 30 s and, therefore, deemed neglgble for these ntal experments. The frcton coeffcent between the cable and the condut was measured to be 0.147, and the vscous frcton was neglgble and could be gnored as expermental error. Fg. 8 shows the expermental and smulaton results for a half loop n the condut. The pretenson n the experments was approxmately set at T 0 = 7.3 N, applyng a unform pretenson across the cable not beng possble due to the frcton effects. The smulatons capture all the major trends observed n experments and match the numbers closely. In the expermental results, for the tenson transmsson profle, we observe a hump or a peak as predcted by the smulatons. In addton, the

9 922 IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, OCTOBER 2010 Fg. 9. Fttng of expermental results and smulaton results for the recalculated cable parameter k = 7.5 kn/m, and µ = TABLE I NORMALIZED ERROR PERCENTAGE backlash n the torque and moton transmsson s smlar to what s predcted n the smulaton results. To analyze, goodness of ft between smulaton and expermental results, the model was ftted on the expermental data to back calculate the cable stffness and frcton coeffcents as 7.75 kn/m and 0.156, respectvely. Usng these parameters, the smulatons were carred out agan and compared wth expermental results, as shown n Fg. 9. Table I shows the normalzed rms error percentage over one cycle for varous parameters. In the followng sectons, the effect of varaton n cable pretenson and condut path has been studed, and the change of behavor as captured by smulatons and experments are compared. A. Varaton of Condut Path Frcton s exponentally correlated to the angle of curvature of the condut. Thus, ncreasng the curvature angle should ncrease the frcton and, hence, larger backlash wdth. Larger frcton also leads to longer tme perods when one of the cables s slack, whle the other cable s movng (phase IV), as well as smaller slope durng ths phase. To verfy ths, the curvature angle of the condut s vared, whle all the other parameters are kept constant, and ts effect on the transmsson profle s observed. For smplcty, the condut shape was changed by addng addtonal half-loops or 180 of bend. For ntroducng loops, the entre length of the conduts was used such that the radus of curvature remaned constant throughout the condut. To mantan unformty n pretenson, cables were loaded after changng the number of loops. The smulaton results n Fg. 10(b) match well wth the expermental results n Fg. 10(a). An ncrease n Fg. 10. Varaton n torque transmsson wth number of loops. (a) Expermental results for dfferent number of loops n condut. (b) Smulaton results for correspondng number of loops. the backlash wdth, as well as change n the slope, as seen n expermental results, have been well predcted by the smulatons. B. Varaton of Pretenson Snce, accordng to (2), the tenson loss s drectly proportonal to pretenson, ncreasng the pretenson ncreases the backlash. For a large pretenson of 7 N, cables do not slack, and therefore, phase IV s not present. For a pretenson of 3.5 N, phase IV s present, when one cable goes slack, whle the other s stll movng. Ths s also vsble for the pretenson of 0.7 N, but n ths case, an addtonal trend s present, showng the second case of phase IV when one cable remans slack, whle the other cable s movng, and the nput swtches ts drecton of moton. Expermental results n Fg. 11(a) show the change n the backlash wdth as well as cable slackng (both phase IVa and IVb), whch can also be observed n smulaton results n Fg. 11(b). C. Varaton of Loop Radus Accordng to (4), tenson varaton across the cable s related to x/r. = θ. Therefore, for a constant angle of curvature θ, the model predcts that there s no effect on system behavor wth a change n the radus of curvature. To verfy ths, experments were carred out for three dfferent loop rad of 4.57, 6.35, and 7.62 cm (1.8, 2.5, and 3 n), whle keepng the curvature angle

10 AGRAWAL et al.: MODELING OF TRANSMISSION CHARACTERISTICS ACROSS A CABLE-CONDUIT SYSTEM 923 Fg. 12. Varaton n torque transmsson wth loop radus. Expermental results for loop radus (a) 4.57 cm, (b) 5.35 cm, (c) 7.62 cm, and (d) correspondng smulaton result. Fg. 11. Varaton n torque transmsson wth pretenson n the cables. (a) Expermental results for dfferent cable pretenson. (b) Smulaton results for correspondng varaton n pretenson. same. Constant radus was mplemented by wrappng the condut around a crcular object. Smlar to Secton V-A, pretenson was appled after changng the number of loops. Correspondng expermental torque profles are shown n Fg. 12(a) (c), whle the smulaton result s shown n Fg. 12(d). From the plots, t can be nferred that the varaton wth the loop radus s mnmal, as predcted by the model. VI. CONCLUSION Although usng a par of cables n pull pull confguraton provdes smple and cost-effectve power transmsson n a surgcal robot as well as other robotc devces, ts use has been lmted due to the nonlneartes generated due to frcton and complance present n the system. A system model was needed to analyze the nonlneartes n the system and to understand the tradeoffs nvolved arsng from tenson losses and cable slackng due to frcton. Whle transmsson models have been developed earler, ther applcaton scope was qute restrcted due to the assumptons of sngle-cable transmsson, constant curvature, and cable pretenson. In ths paper, startng from the system dynamcs, we have developed a dscretzed model of the transmsson characterstcs of the system. The model has been valdated on the expermental setup developed, whch emulates a typcal robot actuaton. Smulatons were successful not only n predctng the trends of the transmsson characterstcs n the ex- permental setup but n the magntudes of varous parameters wth hgh accuracy as well. The dfferences n expermental and smulaton results can be attrbuted to varous approxmatons nherent n the system modelng, expermental errors, as well as errors n system parameter dentfcaton. Although due care was taken, knks may have been nadvertently ntroduced n the cables, whch also deterorate the system performance. For the modelng cable nerta that has been neglected, typcal cable mass s less than 2 gm/m for the steel cables used. However, at extremely low-tenson levels and nflecton or sngularty ponts n moton trajectory, nerta may not be neglgble. The smulatons use statc Coulomb frcton model. Usng a dynamc frcton model, lke Dahl s frcton model, may provde better results. The use of Coulomb frcton model, together wth neglgble nerta, mght be the reason of sharp transton n smulatons, whch are not present n the expermental results. The model also neglects the effects of force relaxaton and frcton effects at the two pulleys. Although placement of the loop along the condut has not been explctly dscussed, the model captures ts effects by approprately defnng the condut curvature. Loop placement does not drectly change the capstan effect; however, t changes the cable elongaton and, therefore, also changes the overall transmsson profle. Although the envronmental load has been assumed as a torsonal sprng, t can be convenently modfed for a generc load n (25). Although correspondng smulatons have not been carred out, the results are expected to follow a smlar frcton dependent backlash-type behavor, snce the deadband s dependent on frcton due to cable pretenson and not on the external load. For the experments, a constant radus has been used. However, n practce, varous sensors, e.g., fber optc sensors can be placed along the condut, whch can be used to estmate the condut curvature. The model can be further mproved by ncorporatng the load dynamcs. The current model assumes hgh-condut bendng stffness and does not model the changes

11 924 IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, OCTOBER 2010 n pretenson, whch occur due to the changes n the path of condut due to lateral forces exerted by the cable. Ths system model, whle renforcng the results obtaned n other publcatons, also brng up some key phenomena not observed earler, partcularly due to cable couplng. An attempt has been made to duly hghlght all these aspects usng the smulaton results. Apart from torque transmsson, moton transmsson, whch s necessary for poston control, has also been presented, whch was completely gnored n prevous work. Furthermore, due analyss has been done to hghlght all the physcal parameters, as well as expermental condtons, whch affect the moton transmsson. Snce the model has been valdated, n the future, ths model can be used to develop new control strateges for both force control as well as poston control. An effectve controller mplementaton can lead to actve usage of cable drves n robotc systems, partcularly n surgcal robots, where the cable conduts can brng dexterty and flexblty n laparoscopc surgcal robots, whch the current systems lack. REFERENCES [1] V. Agrawal, W. J. Pene, and B. Yao, Modelng of closed loop cablecondut transmsson system, n Proc. IEEE Int. Conf. Robot. Autom., Pasadena, CA, May 2008, pp [2] G. S. Guthart and J. K. Salsbury, The ntutve telesurgery system: Overvew and applcatons, n Proc. IEEE Int. Conf. Robot. Autom., San Francsco, CA, Apr. 2000, pp [3] R. J. Fanzno, The Laprotek surgcal system and the next generaton of robotcs, Surg. Cln. N. Amer., vol. 83,no.6,pp ,Dec [4] L. Bagott, F. Lott, C. Melchorr, G. Pall, P. Tezz, and G. Vassura, Development of UB hand 3: Early results, n Proc. IEEE Int. Conf. Robot. Autom., Barcelona, Span, Apr. 2005, pp [5] F. Lott and G. Vassura, A novel approach to mechancal desgn of artculated fnger for robotc hands, n Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., Lausanne, Swtzerland, Oct. 2002, pp [6] A. Nahv, J. M. Hollerbach, Y. Xu, and I. W. Hunter, Investgaton of the transmsson system of a tendon drven robot hand, n Proc. IEEE Int. Conf. Intell. Robots Syst., Munch, Germany, Sep. 1994, vol. 1, pp [7] I. Kassm, W. S. Ng, G. Feng, S. J. Phee, P. Daro, and C. A. Mosse, Revew of locomoton technques for robotc colonoscopy, n Proc. IEEE lnt. Conf. Robot. Autom., Tape, Tawan, Sep. 2003, pp [8] S. J. Phee, W. S. Ng, I. M. Chen, F. Seow-Choen, and B. L. Daves, Locomoton and steerng aspects n automaton of colonoscopy, Eng. Med. Bol. Mag., IEEE, vol. 16, no. 6, pp , Nov./Dec [9] J. F. Veneman, R. Ekkelenkam, R. Krudhof, F. C. T. van der Helm, and H. Van Der Kooj, Desgn of a seres elastc- and Bowden-cable-based Actuaton system for use as torque actuator n exoskeleton-type robots, Int. J. Robot. Res., vol. 25, no. 3, pp , [10] A. Schele, P. Leter, R. van der Lnde, and F. Van Der Helm, Bowden Cable actuator for force-feedback exoskeletons, n Proc. IEEE Int. Conf. Intell. Robot. Syst., Bejng, Chna, Oct. 2006, pp [11] A. Schele, Performance dfference of Bowden Cable relocated and nonrelocated master actuators n vrtual envronment applcatons, n Proc. IEEE Int. Conf. Intell. Robots Syst., Nce, France, Sep. 2008, pp [12] M. Kaneko, W. Paetsch, and H. Tolle, Input-dependent stablty of jont torque control of tendon-drven robot hands, IEEE Trans. Ind. Electron., vol. 39, no. 2, pp , Apr [13] W. T. Townsend and J. K. Salsbury, Jr., The effect of coulomb frcton and stcton on force control, n Proc. IEEE Int. Conf. Robot. Autom., Mar. 1987, vol. 4, pp [14] M. Kaneko, T. Yamashta, and K. Tane, Basc consderatons on transmsson characterstcs for tendon drve robots, n Proc. Int. Conf. Adv. Robot., 1991, vol. 1, pp [15] M. Kaneko, M. Wada, H. Maekawa, and K. Tane, A new consderaton on tendon tenson control system of robot hands, n Proc. IEEE Int. Conf. Robot. Autom., Sacramento, CA, 1991, vol. 2, pp [16] G. Pall and C. Melchorr, Model and control of tendon-sheath transmsson systems, n Proc. IEEE Int. Conf. Robot. Autom., Orlando, FL, May 2006, pp [17] G. Pall and C. Melchorr, Optmal control of tendon-sheath transmsson systems, n Proc. 8th Int. IFAC Symp. Robot Control, Bologna, Italy, Sep. 2006, vol. 8, no. 2, pp Authors photographs and bographes not avalable at the tme of publcaton.