Measurement of the Cable-Pulley Coulomb and Viscous Friction for aCable-Driven Surgical Robotic System

Muneaki Miyasaka1, Joseph Matheson1, Andrew Lewis2 and Blake Hannaford3

Abstract— In this paper we present experimentally obtainedcable-pulley Coulomb and viscous friction for cable-drivensurgical robotic systems including the RAVEN II surgicalrobotic research platform. In the study of controlling cable-driven systems a simple mathematical model which does notcapture physical behavior well is often employed. Even thoughcontrol of such systems is achievable without an accurate model,fully understanding the behavior of the system will potentiallyrealize more robust control. A surgical robot is one of thesystems that often relies on cables as an actuation method aswell as pulleys to guide them. Systems with such structureencounter frictional force related to conditions of cable andpulley such as cable velocity, tension, type and number of pulley,and angle of cable wrapping around pulley. Using a coupleof test platforms that incorporate cable, pulleys, and otherexperimental conditions corresponding to the RAVEN II system,it is shown that cable-pulley friction is function of tension, wrapangle, and number of pulleys and not of magnitude of cablevelocity.

I. INTRODUCTION

Robotic surgery has become an increasingly popular al-ternative to traditional minimally-invasive surgery. FDA-approved systems such as the da Vinci surgical system (In-tuitive Surgical, Inc.) have been used for well over a decade,and provide advantages such as the ability to scale motions,eliminate physiologic tremors, and improve dexterity [1].

Since minimally-invasive surgery requires a long and thintool to be inserted into the patient’s body, surgical robots areprimarily cable driven systems. Generally, cables used forsurgical robots have a diameter of less than a few millimetersand consist of multiple strands helically surrounding a centercore strand. When the cable is under stress, it exhibits uniqueand complex behavior due to its structure. Townsend [2]investigated input and output relations of tension elementdrives (e.g. cable, belt) and analyzed efficiencies basedon thermodynamic principles. Also, autonomous surgicalrobotics has recently been the focus of research [3] [4], and[5]. However, control systems for cable driven robots withsufficient accuracy for medical automation have not yet beendeveloped.

The eventual goal of our research is to improve theperformance of control by accurately defining a dynamicmodel of the system that takes into account cable stretch

1Muneaki Miyasaka (muneaki@uw.edu) and Joseph Matheson(m.joseph.matheson@gmail.com) Department of Mechanical Engi-neering

2Andrew Lewis (alewi@uw.edu) UW BioRobotics Laboratory andApplied Dexterity Inc.

3Blake Hannaford (blake@u.washington.edu) Departments ofElectrical Engineering, Mechanical Engineering, and Surgery

University of Washington, Seattle, WA 98195, USA

and friction. Our application is to achieve better positioncontrol of the RAVEN II Surgical Robotic Research Platform(Fig. 1) [6]. Developed by the University of Washington’sBioRobotics Lab, the RAVEN II has a workspace with threejoints (two rotational and one prismatic) and a four Degree ofFreedom (DOF) tool head, totaling seven cable drives. Thecable to transmit motor torque to each joint traverses multiplepulleys of three different diameters. Currently, control ofeach joint is based on an encoder and motor fixed on theproximal end of the cable. Therefore, accurate position andforce control of the robot is susceptible to position errorsbetween the motor and joint due to cable stretch, and forceerrors due to cable-pulley friction.

The contribution of this paper is the development ofa cable-pulley interaction model of Coulomb and viscousfriction based on experimental measurements. The model andthe experimental procedures can be applied to cable-drivensystems with structure, tension, load, and cable and pulleyparameters similar to the RAVEN II.

Fig. 1. The RAVEN II Surgical Robot is the target system for the cableresearch in this paper.

II. CABLE RELATED FORCE MODELRamadurai et al. [7] developed a dynamic model of the

RAVEN II for the purpose of simulation. The model associ-ated with cable stretch consists of an exponential spring andlinear dashpot system with constant parameters. The pulleyfriction is combined with motor friction and treated as aforce with constant parameters. In [8], Kosari et al. used asimilar cable force model for a single DOF RAVEN mock-up system to identify model parameters. Each parameter wasdynamically adjusted and estimated using position feedbackfrom an encoder mounted on a joint with the use of anUnscented Kalman Filter.

When a cable is in motion under tension, ideal pulleysrotate without slip at the cable and pulley interface. As

pulleys rotate, the frictional force at the pulley bearingdisturbs the motion, causing frictional loads on the cable.Because tension varies when linkages are actuated and itchanges the load on the pulley bearing, friction due to cable-pulley interaction should be a function of tension. Also,internal motion of cables exhibits internal damping withhysteresis behavior. There are a number of studies dedicatedto modeling the behavior of cables under load, both staticand dynamic, but most focus on models of bending, lateralvibration, or large-diameter cables and they are not directlyapplicable for surgical robot systems. Spak et al. summarizesestablished and recently developed modeling methods ofhelically twisted cable and categorized the internal dampingmodels based on their mechanisms, the inter-wire friction,variable bending stiffness, and material internal friction [9].One of the related studies discussed is [10] in which Sauteruses a Masing model [11] to capture dynamic behavior ofstockbridge dampers. Alternate hysteritic models include theBouc-Wen hysteresis model [12] which is used to describea wide range of systems that show hysteresis characteristicsand is investigated for modeling a wire rope spring in [13].A detailed model of hysteretic force is out of the scope ofthis paper but in general, hysteretic forces are modeled as afunction of displacement, velocity, and a hysteretic variable.

Based on the existing literature [7], [8], [14] and ananalysis of the structure of the robot, the force on thepulling side of a cable (FPulling) can be modeled with cablehysteretic force (Fh), cable damping (Fd), and cable-pulleyfriction (Fp). Hysteretic force is defined as a function ofthe difference in the joint side and motor side displacement(∆x) and velocity (∆x) and the hysteretic variable (z), whilecable-pulley friction is defined as a function of tension (T ) aswell as three possible variables: cable velocity (x), numberof pulleys (np), and wrap angle (θw).

FPulling = Fh[∆x,∆x, z] +Fd[∆x] +Fp[T, x, np, θw] (1)

In general, friction is modeled with Coulomb, viscous,stiction, and Stribeck terms [15]. Coulomb friction is theterm dependent on the direction of velocity but not on themagnitude of velocity and the viscous friction is representedas a friction linear to velocity. Stiction is the frictional forceat rest and its highest value is typically greater than themagnitude of Coulomb friction. Stribeck friction is the non-linear drop from the highest stiction toward the Coulombfriction as velocity increases that occurs at low velocity dueto presence of lubrication film. The pulleys we use consist ofrolling-element bearings which can be considered to have nostiction [16] and it consequently indicates Stribeck frictioncan also be omitted from the model. Hence, the cable-pulleyfriction (Fp) is written as

Fp = fc[T, np, θw]sign(x) + fv[T, np, θw]x (2)

where fc and fv are Coulomb and viscous friction fromcable-pulley interaction, respectively, and x is the velocityof the running cable. In this paper, we focus on determiningthe model of the cable-pulley friction (2) based on experi-mentally measured data.

III. IDENTIFICATION OF CABLE-PULLEYFRICTION MODEL

The range of cable tension and cable velocity for theexperiment was determined based on measurements of theactual tensions and velocities observed in use of the RAVENII. Cable tension ranges up to 33 N and velocities peakat 12 m/s. While the absolute maximum value of observedvelocity is 12 m/s, this is very rare and thus the upper limitof cable velocity for the experiments was set to the empiricalRMS value of 1.3 m/s. The cable and pulleys used in thisexperiment are the same as those used in the construction ofthe RAVEN II. The cable is pre-stretched and it is assumedcreep does not occur. Also, it is assumed there is no slipat the cable and pulley interface because slippage generallyoccurs when operation velocity is high. The cable is madeout of type 304 stainless steel and has a construction of 7twisted strands of 19 individual wires, a diameter of 0.61mm, and breaking strength of 311 N. Material of the pulleyis 6061 aluminum with hard anodized finish and the sheavediameters are 7.6 mm (small), 15.64 mm (medium), and 22.9mm (large) (Fig. 2). The pulley bearings have dynamic andstatic load ratings of 29.03 N and 9.07 N for the small pulleysand 36.74 N and 13.15 N for medium and large pulleys.

The RAVEN II uses up to 13 small, 10 medium, and 2large pulleys per cable run. The individual wrap angle (θw,i)is 30◦-180◦ for small and medium pulleys with the averageindividual wrap angles (θw,i,av) of 70◦ and 75◦ respectivelyand for large pulley, both θw,i and θw,i,av are 180◦. However,during the experiment we used up to 19 pulleys and extendeduniform θw,i and θw,i,av ranges of 15◦-180◦ and 45◦-150◦

for all pulleys in order to compare them side by side andinvestigate their behavior in depth.

Fig. 2. Small, medium, and large pulleys (from left to right) used for theRAVEN II.

A. Motor Driven System

1) Setup: We constructed the test setup shown in Fig. 3 totest how cable velocity correlates to pulley friction. In orderto eliminate the dynamic force and exclusively investigatethe viscous friction of the test pulleys, the cable was drivenat constant speed. The cable running through an array ofpulleys was fixed at a capstan mounted on a motor and thecable tension was kept with a mass hung at the other end ofthe cable. The system uses a PID controller to run the cableat a targeted constant velocity. The motor and servo amplifierused are a Maxon RE 30 brushed motor and an AMC Z12A8,

the same as the ones used for the RAVEN II system. For I/O,an Arduino Mega 2560 chip was employed. The Arduinoprovides control signals to the motor via servo amplifierand acquires motor current for analysis. Motor current wasconverted to torque based on the motor specification whichwas assumed to be accurate. Motor friction at each constantspeed was measured and subtracted to obtain just the cable-pulley friction. The software runs roughly at 5 kHz and iscapable of generating and measuring cable velocity of 7 m/sand cable tension of 48 N.

Fig. 3. Motor driven pulley board. The motor (right side) moves cablethrough the pulleys and tension is set by weight (left side).

2) Measurements and Results: In order to test dependencyof each variable on the cable-pulley viscous friction at atime, we plotted the cable-pulley friction as a function ofone variable while holding other variables as constants.

We first collected data for 9 medium-size test pulleys,keeping θw,i,av of 85◦, while applying various constanttensions. Since tension is not consistent throughout the cabledue to the friction at each pulley, we defined a nominaltension as the average of capstan side tension and massside tension. The results are summarized in Fig. 4. Thedata contain some noise when reading the motor current anddetermining the cable-pulley friction and it is amplified forthe larger tension. The plot shows that data are scatteredaround the mean without trend of friction force being a linearfunction of cable velocity.

Fig. 4. Cable-pulley friction against cable velocity for a setup with 9medium pulleys and θw,i,av = 85◦ at five different tensions.

Likewise, the data for the small and large pulleys werecollected for three different tensions each and results areplotted in Fig. 5. All data points are around the mean anddo not exhibit definite increasing trend as tension varies for

both small and large pulleys. Hence, it can be concluded thatviscous friction is not a function of cable tension.

Fig. 5. Cable-pulley friction against cable velocity for setups with 9 smalland large pulleys at three different tensions. θw,i,av is roughly 85◦ for bothpulleys.

Next, θw,i,av was changed to 50◦ and 135◦ while keepingthe number of the pulley the same (Fig. 6). To keep thetension constant, 1.5 kg mass was applied and the nominaltensions are calculated to be 15.8, 15.1, and 15.0 N for small,medium, and large pulley respectively. These tensions arealso used in Fig. 4 and 5 so that the results for θw,i,av =85◦ can be comparable. The results indicate that there is nodependency of the wrap angle on the viscous friction.

(a) θw,i,av = 50◦ (b) θw,i,av = 135◦

Fig. 6. Cable-pulley friction against cable velocity for setups with 9 small,medium and large pulleys respectively when θw,i,av = (a)50◦ and (b)135◦.

We then employed 5 and 15 pulleys with θw,i,av of 85◦

and took measurements for the same tension used in theprevious plot (Fig. 7). Including the results for 9 pulleys fromFig. 4 and 5, change in the frictional force as a functionof velocity is not observed from the results with differentnumber of pulley. Now that we are not able to observe thelinear increase of cable-pulley friction as the cable velocityincreases in all the experiments, it is concluded that theviscous term can be excluded from cable-pulley friction forall pulley sizes.

B. Gravity Driven System

1) Setup: Coulomb friction due to cable-pulley interactionwas measured using a gravity driven setup (Fig. 8) witha model fitting method. Due to absence of the motor and

(a) np = 5 (b) np = 15

Fig. 7. Cable-pulley friction against cable velocity for setups with (a)5and (b)15 small, medium and large pulleys.

noise associated with it, the measured value was consideredto be more accurate than the motor driven system. Thesystem measures linear displacement of cable from angulardisplacement measured by magnets placed radially on apulley which pass by a Hall Effect sensor. Tension wasdictated by identical loads on the ends of the cable. Motionwas induced by a weight added to one side of the cable.An Arduino Uno chip was used for I/O, which enablesufficiently accurate measurement of the displacement ofeach mass by means of the Hall effect sensor on either end.The effective distance between the magnets was 9.8 mm. Atime differential of the Hall effect data was used to estimatecable velocity at either end. To ensure the assumption of highstiffness in the cable, Hall Effect sensors were embeddedon both ends of the setup and only the trials where thedisplacement graphs were identical were used.

Fig. 8. Gravity driven pulley board. Pulleys instrumented with magnetsand Hall Effect sensors which are connected to Arduino microcontroller atthe back of the board.

2) Model: The system can be modeled as shown in Fig.9 and with a mathematical expression in state space form in(3), which includes the cable stiffness, damping, and cable-pulley friction terms. Since the cable translates only in onedirection and force applied on the cable is consistent, there isno effect of hysteresis, and the stiffness and damping forcesused here are simply linear to the displacement and velocity,respectively, and are represented in the model by k and b.The gravitational constant is represented by g.

Fig. 9. Schematic model of the gravity driven pulley board.

x1x1x2x2

=

0 1 0 0−km1

−b− fv4

m1

km1

b− fv4

m1

0 0 0 1km2

b− fv4

m2

−km2

−b− fv4

m2

x1x1x2x2

+

0

m2g − fc2

0

−m2g − fc2

(3)

The model is numerically solved using 4th order RungeKutta method. In order to search for parameters that leadto the best correlation, a least square method is used andfrom the average of several trials, parameters which resultin the highest correlations are calculated. For most of thetrials, data show very high correlation, often with an R2

value of more than 0.95. The model can be simplified asin Fig. 10 and (4) due to the fact that the tension is heldnearly constant because the masses applied are very similarand are given initial conditions so that the cable does notvibrate and the effect of cable stiffness and damping becomesnegligible. This was confirmed by fitting the models in (3)and (4) to the data obtained from the pulley setup with 9medium pulleys and θw,i,av of 85◦ at nominal cable tensionof 15.5 N. The value for fc for the two different modelformulations of equations (3) and (4) are identified to be0.816 and 0.818 N respectively. The plots of those modelsas well as the experimental data are shown in Fig. 11.

Fig. 10. Simplified schematic model of the gravity driven pulley board.

Fig. 11. Comparison of original and simplified models of the gravity drivenpulley board with experimental data.

[xx

]=

[0 1

0 − fvm1+m2

] [xx

]+

[0

(m1−m2)g−fcm1+m2

](4)

Furthermore, since the pulley friction is not a function ofcable velocity per the previous section, the pulley frictionsimply becomes Coulomb friction, and the model can bereduced to (5). This series of simplifications is just for thepurpose of Coulomb friction identification and is not forcharacterizing the dynamics of cable driven systems.[

xx

]=

[0 10 0

] [xx

]+

[0

(m1−m2)g−fcm1+m2

](5)

3) Measurements and results: First, θw,i,av was set to bea constant value. Here, θw,i,av can be any constant as long asit is within the range of interest. We selected a value of 85◦

and tested the effect of tension for the fc value by conductingtrials with system configurations consisting of 5, 9, 15, and19 pulleys to simultaneously test if fc could be a functionof number of pulleys in the system.

For each pulley size fc is plotted individually as a functionof tension (Fig. 12). Each line in the plot represents data fora board setup with a different number of pulleys. Withoutexception, fc linearly correlates with cable tension with R2

more than 0.99 and the slope and y-intercept become largeras the number of pulleys increases.

Fig. 12. fc (θw,i,av = 85◦) against tension for small, medium, and largepulley (from left to right). Each line indicates the linear fit to the data for5, 9, 15, and 19 pulleys respectively.

From this observation, the slope or ∂fc/∂T when θw,i,av

= 85◦ is plotted with respect to number of pulley in Fig. 13as well as the y-intercepts (values of fc when θw,i,av = 85◦

and T = 0). As a result, we identified

∂fc(θw,i,av = 85◦)

∂T= c1np (6)

fc(θw,i,av = 85◦, T = 0) = c2np (7)

Integrating (6) with respect to T yields

fc(θw,i,av = 85◦) = np(c1T + c2) (8)

The values of c1 and c2 are summarized in Table I.We then investigated how θw,i,av affects to fc. To begin

with, fc is plotted against θw,i,av with combinations of three

(a) (b)

Fig. 13. (a) Partial derivative of fc (θw,i,av = 85◦) with respect to tensionand (b) fc (θw,i,av = 85◦, T = 0) against number of pulley. Linear trendlines for each pulley size are shown.

different tensions and three number of pulleys, showingtotal of 9 data sets for each pulley size (Fig. 14). Mostof the data sets demonstrate high linearity and the slopes∂fc/∂θw,i,av appear to increase for higher tension and morepulleys. Acting on that trend, ∂fc/∂θw,i,av was plotted withrespect to np (Fig. 15). Then, as linear functions representthe trends of data points with high correlation, the slopes∂2fc/(∂θw,i,av∂np) are plotted as a function of tension (Fig.16).

From the trend lines in Fig. 16, we obtained

∂2fc∂θw,i,av∂np

= c3T (9)

and the trend lines in Fig. 15 give

∂fc(np = 0)

∂θw,i,av= 0 (10)

Then, integration of (9) with respect to np results

∂fc∂θw,i,av

= c3Tnp (11)

By integrating with respect to θw,i,av , (11) becomes

fc = c3Tnpθw,i,av + f(np, T ) (12)

where f(np, T ) is the constant of integration which is afunction of np and T . When θw,i,av = 85◦, this equationbecomes

fc(θw,i,av = 85◦) = c3npT85◦ + f(np, T ) (13)

Since (13) and (8) are equivalent, we get

f(np, T ) = np(c1T + c2) − c3npT85◦ (14)

By substituting (14) into (12), we obtain the expression offc as a function of T , θw,i,av , and np as follows.

fc = np((c1 + c3(θw,i,av − 85◦))T + c2) (15)

(a) Small Pulley

(b) Medium Pulley

(c) Large Pulley

Fig. 14. fc when np = 5, 9, and 15 (from left to right) against θw,i,av

for (a) small, (b) medium, and (c) large pulley. Each line shows the linearfit to the data for different constant tension.

In section III. A, we concluded there is no viscous frictionand hence (2) is written as

Fp = fc[T, np, θw,i,av]sign(x)

= np((c1+c3(θw,i,av−85◦))T+c2)sign(x) (16)

The values of c1, c2, and c3 for each pulley are summa-rized in Table I. Note that (16) with coefficients in TableI is valid under the range of variables we examined in theexperiments which is shown in Table II with similar cable

Fig. 15. Partial derivative of fc with respect to θw,i,av against numberof pulley for small, medium, and large pulley (from left to right). Lineartrend lines for different tensions are shown.

Fig. 16. ∂2fc/(∂θw,i,av∂np) against tension. Linear trend lines fordifferent pulley size are shown.

and pulleys employed for the RAVEN II system.

TABLE ISUMMARY OF COEFFICIENTS

Pulley size Radius c1 c2 c3[mm] [unitless] [N ] [1/degree]

Small 3.80 0.0128 0.0261 2.79×10−5

Medium 7.82 0.00416 0.0165 1.02×10−5

Large 11.45 0.00239 0.0129 0.737×10−5

TABLE IIRANGE OF VARIABLES IN WHICH THE MODEL IS VALID

Variable Rangex up to 1.3 [m/s]T up to 33 [N]np up to 19θw,i 15◦ to 180◦

θw,i,av 45◦ to 150◦

IV. DISCUSSIONS

Viscous friction, the dependency of the cable-pulley fric-tion force on magnitude of cable velocity, was not observedfor any of the pulleys and tensions tested. The fact that notonly section III.A showed the velocity independence, but alsothe data from section III.B demonstrated high correlationwithout viscous friction indicated the viscous term can beexcluded from the cable-pulley friction model. The resultmight vary if the cable was running at a much higher velocity,but the range of velocities we studied was in the range ofactual robot operation.

In the case of Coulomb friction, we confirmed that it wasa function of T , θw,i,av , and np. When the friction modelis solely Coulomb friction, the mean values in the plots ofthe motor driven experiment in section III.A represents fcand those values should be equivalent to the results fromthe gravity driven experiments in section III.B. Indeed, fcvalues calculated based on (16) were very close to thosefound in Fig. 4 to 7 with most within 5% and a maximumerror of about 10%. The error can mainly be associated withthe motor current reading since the signal needs to be filteredout due to presence of high noise which affects more forthe larger load or tension. However, since we derived themodel based on the gravity driven pulley board, the modelwas not affected by that uncertainty. One related sourceof error would be the approximation of tension using thenominal value. Furthermore, friction can vary dependingon the degree of lubrication, wear and oxidation of eachpulley and surrounding conditions such as temperature andhumidity.

V. CONCLUSIONS AND FUTURE WORK

We have determined the cable-pulley friction model as afunction of cable tension, average individual wrap angle, andtype and number of pulleys employed. The derived equationand coefficients can be applied to any cable-driven systemsas long as they are operated within the presented conditionsand parameters and use cable and pulleys similar to the onesemployed for the RAVEN II system (e.g. cable: type 304stainless steel, 7 strand and 19 wire construction, 0.61 mmdiameter. pulley: 6061 Aluminum with hard anodized finish,rolling element bearing).

If the presented cable-pulley friction model is used with anaccurate cable stretch model, the accuracy and robustness ofthe control can be potentially improved, which is necessaryfor automation of surgical tasks. This is under active pursuit.

The proposed model requires the knowledge of the realtime cable tension. Although one can integrate force sensorsinto systems, it would be costly and complicate the systems.Another method is to calculate tension from a cable stretchmodel, but the initial tension needs to be known. Eventhough it is possible to measure initial tension using a tensiongauge, measurement has to be taken every time the system isstarted. However, this step can be simplified with the cabletension estimation method presented in [8]. Hence, we planon implementing the proposed cable pulley friction modelwith a cable stretch model and the initial tension estimationtechnique for the RAVEN II.

The characteristic of stretch and the bending radius arenot the same for different cable types. How cables areconstructed (number and diameter of fiber composition) andhow materials (e.g. alloy and grade of steel) behave undercertain stretch and bending conditions makes a difference inthe interaction between the cable and pulley. Therefore, thecable-pulley friction model may depend on the type of thecable and further inspection is required.

The cables used in these experiments were new. As thecable experiences wear and creep over time and the condition

of cable-pulley interaction changes, this could also affect thefriction values. Knowing how the cables react to repeateduse and aging could help better predict how they respond tofuture operation, and one could adjust the friction model toaccount for this.

ACKNOWLEDGMENT

The authors would like to thank support from the Ko-rean Institute of Advanced Technology (KIST, Dr. HujoonProject), and National Science Foundation via Stanford Uni-versity (grant number 1227406), and the members of the UWBioRobotics lab for their support and contribution throughoutthe entire project.

REFERENCES

[1] A. R. Lanfranco, A. E. Castellanos, J. P. Desai, and W. C. Meyers,“Robotic surgery: a current perspective,” Annals of surgery, vol. 239,no. 1, p. 14, 2004.

[2] W. T. Townsend, “The effect of transmission design on force-controlled manipulator performance,” 1988.

[3] B. Kehoe, G. Kahn, J. Mahler, J. Kim, A. Lee, A. Lee, K. Nakagawa,S. Patil, W. D. Boyd, P. Abbeel, et al., “Autonomous multilateral de-bridement with the raven surgical robot,” in International Conferenceon Robotics and Automation, 2014.

[4] J. Mahler, S. Krishnan, M. Laskey, S. Sen, A. Murali, B. Kehoe,S. Patil, J. Wang, M. Franklin, P. Abbeel, et al., “Learning accuratekinematic control of cable-driven surgical robots using data cleaningand gaussian process regression.”

[5] D. Hu, Y. Gong, B. Hannaford, and E. J. Seibel, “Semi-autonomoussimulated brain tumor ablation with raven ii robot using behaviortree,” in Robotics and Automation (ICRA), 2015 IEEE InternationalConference on. IEEE, 2015 in Press.

[6] B. Hannaford, J. Rosen, D. W. Friedman, H. King, P. Roan, L. Cheng,D. Glozman, J. Ma, S. N. Kosari, and L. White, “Raven-ii: anopen platform for surgical robotics research,” Biomedical Engineering,IEEE Transactions on, vol. 60, no. 4, pp. 954–959, 2013.

[7] S. Ramadurai, S. N. Kosari, H. H. King, H. J. Chizeck, and B. Han-naford, “Application of unscented kalman filter to a cable drivensurgical robot: A simulation study,” in Robotics and Automation(ICRA), 2012 IEEE International Conference on. IEEE, 2012, pp.1495–1500.

[8] S. N. Kosari, S. Ramadurai, H. J. Chizeck, and B. Hannaford,“Control and tension estimation of a cable driven mechanism underdifferent tensions,” in ASME 2013 International Design EngineeringTechnical Conferences and Computers and Information in EngineeringConference. American Society of Mechanical Engineers, 2013, pp.V06AT07A077–V06AT07A077.

[9] K. Spak, G. Agnes, and D. Inman, “Cable modeling and internaldamping developments,” Applied Mechanics Reviews, vol. 65, no. 1,p. 010801, 2013.

[10] D. Sauter, “Modeling the dynamic characteristics of slack wire cablesin stockbridge dampers,” Ph.D. dissertation, TU Darmstadt, 2004.

[11] G. Masing, “Zur heynschen theorie der verfestigung der met-alle durch verborgen elastische spannungen,” in WissenschaftlicheVeroffentlichungen aus dem Siemens-Konzern. Springer, 1923, pp.231–239.

[12] Y.-K. Wen, “Method for random vibration of hysteretic systems,”Journal of the Engineering Mechanics Division, vol. 102, no. 2, pp.249–263, 1976.

[13] H. Nijmeijer and I. R. H. F. TUE, “Modelling and identification ofthe dynamic behavior of a wire rope spring.”

[14] J. Song and A. Der Kiureghian, “Generalized bouc–wen model forhighly asymmetric hysteresis,” Journal of engineering mechanics, vol.132, no. 6, pp. 610–618, 2006.

[15] H. Olsson, K. J. Astrom, C. C. De Wit, M. Gafvert, and P. Lischinsky,“Friction models and friction compensation,” European journal ofcontrol, vol. 4, no. 3, pp. 176–195, 1998.

[16] P. Dahl, “A solid friction model,” DTIC Document, Tech. Rep., 1968.