The International Journal of Robotics Researchrobotics.snu.ac.kr/fcp/files/_pdf_files_publications/3... · 2016-01-20 · The International Journal of Robotics Research Vol. 21, No

http://ijr.sagepub.comRobotics Research

The International Journal of

DOI: 10.1177/027836402321261922 2002; 21; 409 The International Journal of Robotics Research

Jungyun Kim, F C Park and Yeongil Park Design, Analysis and Control of a Wheeled Mobile Robot with a Nonholonomic Spherical CVT

http://ijr.sagepub.com/cgi/content/abstract/21/5-6/409 The online version of this article can be found at:

Published by:

http://www.sagepublications.com

On behalf of:

Multimedia Archives

can be found at:The International Journal of Robotics Research Additional services and information for

http://ijr.sagepub.com/cgi/alerts Email Alerts:

http://ijr.sagepub.com/subscriptions Subscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

© 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution. at SEOUL NATIONAL UNIV on February 1, 2007 http://ijr.sagepub.comDownloaded from

http://www.ijrr.org/multimedia.html

http://ijr.sagepub.com/cgi/alerts

http://ijr.sagepub.com/subscriptions

http://www.sagepub.com/journalsReprints.nav

http://www.sagepub.com/journalsPermissions.nav

http://ijr.sagepub.com

Jungyun KimF. C. ParkSchool of Mechanical and Aerospace EngineeringSeoul National UniversitySeoul 151-742, [email protected]@plaza.snu.ac.kr

Yeongil ParkInstitute of Precision Machinery TechnologySeoul National University of TechnologySeoul 139-743, [email protected]

Design, Analysis andControl of aWheeled MobileRobot with aNonholonomicSpherical CVT

Abstract

This article reports on the design, analysis and control of a newtype of wheeled mobile robot based on a nonholonomic sphericalcontinuously variable transmission (S-CVT). Our S-CVT based mo-bile robot is designed to increase the run time (i.e., the length oftime in which the robot can be operated), and to achieve full planaraccessibility with the design of a novel pivoting device that takesadvantage of the flexibility of the S-CVT. We examine the sources ofpower loss in the S-CVT, in particular spin loss. For a quantitativeanalysis of spin loss of the S-CVT, we develop a friction model for theS-CVT, and perform an in-depth contact analysis based on the rela-tive velocity field and normal pressure distribution. We also presenta nonlinear shifting controller based on feedback linearization thattakes into account the dynamics of the S-CVT. To evaluate the energyefficiency of our mobile robot and the performance of the S-CVT asa machine element, we perform experiments with a hardware pro-totype. The results are benchmarked numerically with a differentialdrive type mobile robot equipped with a reduction gear.

KEY WORDS—continuously variable transmission, spinloss, feedback linearization, mobile robot

1. Introduction

1.1. Continuously Variable Transmissions andNonholonomy

Depending on one’s perspective, the feature of nonholon-omy in input-output systems, which can be characterized bya reachable space whose dimension is larger than that of

The International Journal of Robotics ResearchVol. 21, No. 5-6, May-June 2002, pp. 409-426,©2002 Sage Publications

its input space, can be both a curse and an advantage. Al-though nonholonomy enables one to maneuver freely in ann-dimensional state space with fewer than n actuators, theassociated mathematical complexity makes control and plan-ning of such systems substantially more challenging. Exam-ples of robotic systems with nonholonomy abound in the lit-erature, e.g., systems with rolling constraints like a wheeledmobile robot, a car pulling multiple trailers, multibody sys-tems subject to angular momentum conservation laws like afree-flying space robot, or simply a ball rolling on a plane. Inmost of these systems nonholonomy occurs in a natural wayrather than by any intentional design.

More recently, robots that deliberately exploit nonholon-omy to advantage have been developed. The Cobot (Mooreet al. 1999) is a physically passive robotic device thatprovides virtual gliding surfaces for redirecting human-powered motions. The nonholonomic manipulator developedby Søerdalen et al. (1994) is an n-joint serial manipulatorthat can be controlled using only two independent actuators.What lies at the heart of both these robotic devices is a non-holonomic mechanical transmission device, involving rollingwheels in contact with a sphere, that effectively acts as a con-tinuously variable transmission (CVT).

CVTs have been the object of considerable research inter-est within the mechanical design community, driven primar-ily by automotive applications. Unlike conventional steppedtransmissions, a CVT allows for a continuous range of gearratios that can, up to certain device-dependent physical limits,be selected independently of the applied torque. This featureof the CVT allows for engine operation at the optimum fuelconsumption point, improving overall vehicle efficiency.

Existing CVTs can be classified into four types: beltdrive, variable stroke drive, hydrostatic/dynamic drive, and

409


www.sagepublications.com


410 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / May-June 2002

traction/friction drive (see report of U.S. Department of En-ergy (1982) for a description of the operating principles ofeach type). The last type of CVT transmits power through acontact between two rotating elements, with the drive ratiovaried by controlling the effective onset radius of the contactpoint. Only traction/friction drives directly exploit the non-holonomy inherent in rolling contacts, and combine them withshifting mechanism designs that are based on angular momen-tum conservation (see Figure 1, by courtesy of DOE report1982). Traction/friction drives also tend to be more compact,offering more flexibility in terms of their design and range ofapplications, and offer the possibility of precise positioningwithout backlash (Moronuki and Furukawa 1988; Kato 1990;Lee and Tomizuka 1996; Chang and Toumi 1998; Kurosawa,Takahashi, and Higuchi 1998).

While the kinematics of nonholonomic CVTs of the typeused in the Cobot (Moore et al. 1999) and the nonholonomicmanipulator (Søerdalen et al. 1994) are well-understood, bycontrast very little attention has been given to the understand-ing of their dynamic and elastic behaviors, e.g., spin and othersources of power loss, as well as elasto-dynamic modeling.Clearly such an understanding is essential to making nonholo-nomic CVTs a practical, reliable, and widely used machineelement for robotic systems.

1.2. Mobile Robot Design

In recent years there has been an explosion of research activ-ity in mobile robots, driven in part by the proliferation of ser-vice robots designed for, e.g., patient transportation, securitymonitoring, mobile manipulator platforms, etc. In contrast tothe extensive literature on mobile robot motion planning andcontrol—traceable in large part to the mathematical richnessof the nonholonomy problem—relatively little attention hasbeen given to hardware platforms and other mechanical as-pects of mobile robots (Jones and Flynn 1993; Borenstein,Everett, and Feng 1996).

While mobile robots vary widely in design, in general awheeled mobile robot requires a minimum of two actuators formoving about in the plane, each with a dedicated controller;this is true both for wheel drives employing differential gearsand differential drives. Mobile robots also typically rely onelectric motors for actuation, in particular dc motors, becauseof their relatively simple control, and the fact that power can besupplied from battery sources. The latest dc motors possessmotor drivers efficient enough to be used as variable speeddrives (Leonhard 1996; Kassakian, Schlecht, and Verghese1991). Despite these performance improvements, current mo-bile robots are still limited by battery lifetimes. It is also stilltrue that dc motors are most efficient in the low torque, highspeed operating regime (see Figure 2).

Reduction gears offer a means of operating a dc motor in itsregion of maximum efficiency, while simultaneously reduc-ing the load torque and increasing motor speed according to

the chosen gear ratio. Unlike automobiles, however, it is stillimpractical to equip mobile robots with conventional trans-mission devices given the manufacturing costs, size, and otherpractical limitations of current transmissions.

An increasingly practical alternative to reduction gears isthe CVT. By allowing for an infinite range of gear ratios, aCVT allows the motor to deliver a range of torques while con-tinuously operating at its most efficient speed. CVTs, providedthey are sufficiently compact and have only minimal powerloss, hold the potential of a mobile robot with improved ef-ficiency, capable of complete planar accessibility with justa minimal number of actuators and mechanical components.Developing a mobile robot around a CVT, however, intro-duces a number of new design challenges. For example, thereis the issue of how to design a simple pivoting device thateliminates the need for an actively controlled steering mech-anism. Designing a stable controller that takes into accountthe dynamics of the complete system, and maximizes powerefficiency, is also an issue.

1.3. Objectives of the Paper

In Kim et al. (2002), a novel type of nonholonomic CVT, theS-CVT, was proposed, together with a kinematic and dynamicanalysis of the device. This paper reports on the developmentof a new type of wheeled mobile robot, referred to as MOSTS(Mobile rObot with a Spherical Transmission System), basedon the S-CVT. Typical mobile robots achieve planar mobil-ity by employing an additional controlled actuator, such as asteering wheel or a motor for differentiating each wheel ve-locity. MOSTS is a minimal design in the sense that, withthe design of a novel pivoting device that takes advantageof the flexibility of the S-CVT, it can turn about its centerand change its direction of movement without the need for asteering actuator and controller. We also present a nonlinearshifting controller based on feedback linearization that takesinto account the dynamics of the S-CVT.

A second objective of this paper is to examine the sourcesof power loss in the S-CVT, in particular spin loss. Clearly spinloss strongly impacts the performance of all the nonholonomicCVT types mentioned above. However, an accurate analysis ofthe mechanism of slippage, particularly for spherical CVTs, isstill an open research topic. In this paper we develop a frictionmodel for the S-CVT, and perform a quantitative analysis ofspin loss. Our approach generalizes in a straightforward wayto more general spherical CVT designs such as the Cobot andnonholonomic manipulator, and serves as a theoretical tool insearching for ways to reduce spin loss.

To evaluate the energy efficiency of MOSTS and the per-formance of the S-CVT as a machine element, we performexperiments with a hardware prototype of MOSTS. The re-sults are benchmarked numerically with a differential drivetype mobile robot equipped with a reduction gear. Taken to-gether, we hope our results contribute to narrowing the gap



Kim, Park and Park / Design, Analysis, and Control of a Wheeled Mobile Robot 411

Fig. 1. Structures for traction/friction drive CVTs.

Fig. 2. Power efficiency of an armature-controlled dc motor(regions of highest efficiency are indicated in black).

between idealized nonholonomic CVTs and their use as prac-tical, reliable devices for robotic systems.

2. Nonholonomic Spherical CVT

In this section a new type of spherical continuously variabletransmission (S-CVT) is described. The S-CVT is marked byits simple kinematic design and IVT characteristics, i.e., theability to transition smoothly between the forward, neutral,and reverse states without the need for any brakes or clutches.Because the S-CVT transmits power via rolling resistance be-tween metal on metal, it has limitations on the overall trans-mitted torque, which is effectively determined by the staticcoefficient of friction and the magnitude of the normal forcesapplied to the sphere. Due to this torque limitation, the S-CVT is not intended for automobiles and other large capacitypower transmission applications. Target applications for theS-CVT include mobile robots, household electric appliances,small-scale machine tools, and other applications with mod-erate power transmission requirements. Although the currentdesign of the S-CVT is based on friction drive designs, it isour expectation that the power capacity of the S-CVT can beincreased by the use of traction oil, an issue which we do notpursue further in this paper.




Other spherical CVT structures have been proposed for usein passive mobile robots and for use as nonholonomic jointsin robot manipulators. The Cobot (Moore et al. 1999) adoptsa rotational CVT to provide smooth, hard virtual surfaces forpassive haptic devices in place of conventional motors. Its ro-tational CVT consists of a sphere caged by four rollers, andadopts the joint speeds and task space speeds along with thesteering angles as control inputs. Another application can befound in underactuated manipulators, designed by Søerdalenet al. (2001). This work proposes a new type of manipulatorarchitecture using a CVT-type robot joint that takes advantageof the inherent nonholonomy of the CVT. Although these sys-tems are designed to manipulate the speed ratio using a CVTmechanism, their main purpose is not for power transmissionto improve the energy efficiency. Furthermore, the shiftingmechanism of the S-CVT is quite different from these previ-ous designs, as will be described below.

2.1. Structure

The S-CVT is composed of three pairs of input and outputdiscs, variators, and a sphere (see Figure 3). The input discsare connected to the power source, e.g., an engine or an elec-tric motor, while the output discs are connected to the outputshafts. The sphere transmits power from the input discs tothe output discs via rolling resistance between the discs andthe sphere. The variators, which are connected to the shiftingcontroller, are in contact with the sphere like the discs, andconstrain the direction of rotation of the sphere to be tangentto the rotational axis of the variator. To transmit power fromthe discs to the sphere or from the sphere to the discs, a devicethat supplies a normal force to the sphere, such as a spring orhydraulic actuator, must be installed on each shaft. As can beseen in Figure 3, the structure and components of the S-CVTare simple enough to allow for a considerable reduction insize and weight compared to conventional transmissions. Theorientations of the input and output shafts can also be locatedfreely using rollers at arbitrary positions rather than discs.

2.2. Kinematic and Dynamic Analysis of the S-CVT

When the input device is actuated by a power source, the inputdisc rotates about the input shaft. This rotation in turn causesa rotation of the sphere, due to the condition of rolling contactwithout slip between the input discs and the sphere. Rotationof the sphere in turn causes a rotation of the output discs, andsubsequently of the output shaft. In the absence of any contactbetween the sphere and the variator, the axis of rotation of thesphere will largely be determined by an equilibrium conditionamong the various contact and load forces being applied tothe sphere.

By varying the axis of rotation of the sphere, it is in turnpossible to vary the radius of rotation of the contact point be-tween the input disc and the sphere, Ri , as well as the radius

of rotation of the contact point between the output disc andthe sphere, Ro (see Figure 4). In this way the speed-torqueratio of the S-CVT can be adjusted. Figure 4 shows the var-ious alignments of the variator for the forward, neutral, andreverse states of the output shaft of the S-CVT. The neutralstate, which corresponds to zero rotation of the output disc,is achieved when Ro becomes zero. As apparent from the fig-ure, the forward, neutral, and reverse states can all be achievedby smoothly manipulating the variator alignment, without theneed for any additional clutches or brakes.

Assuming roll contact without slip, the speed and torqueratio between the input and output discs is related to the vari-ator angle by the following relations:

ωout

ωin

= ri

rotan θ (1)

Tout

Tin

= ro

ricot θ (2)

where θ is the angular displacement of the variator, ωin andωout are the respective angular velocities of the input and out-put shafts, Tin and Tout are the respective input and outputtorques, and ri and ro are the respective radii of the contactpoints of the input and output discs (see Figure 4).

Although ideally an infinite torque ratio is possible withthe S-CVT, as seen in eq. (2), in practice there is a limit tothe torque that can be transmitted because power transmissionoccurs from rolling resistance of metal on metal. The limitingtorque Tmax is determined by the static coefficient of frictionµs and the normal force N exerted by the output disc springmechanism on the sphere according to the relation Tmax =rµsN , where r is the contact radius of the disc. When eitherthe input or output torque applied at the disc-sphere contactexceeds this limit, slippage can occur.

Using the free body diagrams of the sphere in Figure 5, thedynamic equations of the S-CVT can be derived as follows(Kim et al. 2002):[

2 (Ia+Iv+mε2)

ε2RIv

ε2

2 Iv

ε

Is

R+ 2RIv

ε2

] [θ

ω

]

=[

Fs

Fi cos θ − Fo sin θ

](3)

whereIa = moment of inertia of the shifting actuator shaft and

connecting rod [kg · m2],Iv = moment of inertia of the variator [kg · m2],Is = moment of inertia of the sphere [kg · m2],m = mass of the variator [kg],ε = eccentric distance between the centers of the shifting

actuator shaft and variator [m],R = sphere radius [m],Fs = shifting force [N],




(a) Standard structure of S-CVT. (b) 3-dimensional view.

Fig. 3. Spherical CVT.

ri

ro

RRi

Ro

ri

ro

RRi

ri

ro

R

Ri

Ro

variator angle

Fig. 4. Operating principles of S-CVT.




(a) Coordinate system and forces on S-CVT. (b) Forces on variator.

Fig. 5. Free body diagrams of the Spherical CVT.

Fi = driving force delivered from the input discs [N],Fo = Reaction force caused by the output discs connected

to the load torque [N],ω = spinning rate of the sphere [rad/s].

3. Spin Loss Analysis of the Spherical CVT

Spin loss and slip loss are the two main sources of power lossin traction and friction drives, excluding the losses due to theseals, thrust bearings, normal force loading device, and shift-ing actuator (particularly in traction drives, shifting actuatoris composed of hydraulics). Slip loss is caused by rotationalslippage at the contact points, resulting mainly from changesin the transmitted forces. When the exerted force from thedriving element is larger than the static friction force, a rota-tional slippage occurs at the contact points, and the magnitudeof the transmitted force is determined from the underlyingfriction mechanism. In this case the kinematic constraints ofthe S-CVT are no longer satisfied, and the dynamics of theoverall system needs to be rederived taking into account theextra degrees of freedom that now become available. Pre-dicting slip loss therefore requires information on velocities,accelerations, and forces at the contact points together with amuch more complicated dynamic model of the system. Sliploss analysis is at best difficult and unreliable for the S-CVT,and we instead focus on spin loss in this paper.

The other source of loss, spin loss, is also one of the maindesign issues in traction drives. Spin loss results from the elas-tic contact deformation of rotating bodies that have different

Pump

Thrust bearing

Creep and Spin

� � � � � � � � � � � � � � � � � � � � ! � � � � � � % & � � ) �

, - / 0 2

9.7%

67%

7.5% Seals

Fig. 6. Spin loss in traction drives.

rotational velocities. Relative velocities resulting from elasticcontact between rotating bodies usually give rise to frictionmechanisms, in which case friction moments (spin loss) occurat the contact region. Spin loss is caused not by changes in theapplied or exerted forces, but by the difference in rotating ve-locities and geometric properties of the bodies in contact. Spinloss (see Figure 6, Choi et al. 1999) is of particular concernin traction drives. To reduce spin loss in traction and fric-tion drives, designers have investigated different approachesto optimal contact geometry design, applying normal loads,and controller design (Loewenthal et al. 1981; Wang and Fries1989; Tanaka et al. 1989).




For nonholonomic spherical elements like S-CVT, spinloss is present whenever the rotational axis of the sphere is notparallel to the roller’s axis (Moore 1997). Although many ef-forts have been dedicated to compensate for slip in a variety ofmechanical systems, an accurate analysis of the mechanism ofslippage is still an open topic of research. Furthermore thereis at present no theoretical framework for the analysis andquantification of spin loss for devices like the spherical CVT.

3.1. Modified Friction Model for the S-CVT

Contrary to the predictions derived from the classical stick-slip friction model, it has been experimentally found thatsmall relative displacements between two bodies in contactoccur when the applied tangential force is less than that pro-duced by static friction (Courtney-Pratt and Eisner 1957).Dahl (1977) provided a model of this pre-sliding displacementphenomenon, known as the Dahl model, that assumes frictionforce is a function of displacement x and time t ; however,it cannot be applied when the velocity x � 0. In addition,several researchers have found a source for this discrepancyin the Stribeck effect, and experimentally derived a model offriction variation with velocity.

In order to predict the spin loss of friction drives includ-ing the S-CVT, one must consider the pre-sliding effect inthe vicinity of zero relative velocity. Friction models basedon Dahl’s have difficulties in numerical integration due to thehigh stiffness and damping coefficients; moreover these co-efficients must be obtained through a careful experimentalanalysis. For our purposes we develop a modified version ofthe classical stick-slip friction model that includes a Stribeckeffect-like term:

Ff =[(µs − µk) exp

(−(�V

Vstr

)2)

+ µk

]Psgn(�V ) (4)

whereVstr is the critical Stribeck velocity,µs, µk are static andkinetic coefficients of friction, P the normal load, and �V

the relative velocity, respectively. Here we neglect viscousfriction as there is no lubricant layer in the S-CVT.

The friction force on the contact surface is determined bythe normal force and friction coefficients. Considering that thekinetic coefficient of friction is related to the relative veloc-ity �V between two rotating bodies, we first investigate therelative velocity field on the contact surface. In the followingsubsection, Hertzian results (Timoshenko and Goodier 1951;Chung and Han 1996) for elastic deflection are employed toconstruct the geometric parameters of the contact surface.

3.2. Disc-Sphere Contact

Figure 7(a) shows the contact surface between the sphere andupper variator. We set at the center of the contact surfaceS a local coordinate frame {ξyη} along the xyz directions,with ξ in the rolling direction. Supposing there is no bending

deformation of the variator along the x and z axes, the normaldeflection δ and contact surface radius c can be calculatedfrom Hertzian theory to be

c = 1.109 × 3

√P

ER , δ = 2.64 × 3

√P 2

E2

1

R, (5)

where P is the normal force, R the sphere radius, and E

represents the equivalent Young’s modulus.To obtain each velocity field, we first recall that the sphere

has a pure rotational speed of ω in the z direction and thevariator a rotational speed ofωv in they direction. The velocityfield of the sphere V1(ξ, η) and that of variator V2(ξ, η) onthe contact surface can be obtained as follows:

V1(ξ, η) = [ (R − δ)ω, 0 ] ,V2(ξ, η) = [ (ε + η)ωv, − ξωv ] ,

(6)

where ε is the distance between the contact surface centerand variator center. Consequently, the relative velocity field�Vv(ξ, η) can be derived as

�Vv(ξ, η) = [ (R − δ)ω − (ε + η)ωv, ξωv ] . (7)

The relative velocity fields at the other contact points (inputand output discs) can be similarly obtained.

As shown in eq. (7), there is a relative velocity componentof �Vv in the η direction, �Vη, on the contact surface. How-ever, �Vη does not accumulate total relative velocity in the η

direction, because it is symmetric along the η axis. Therefore,�Vη contributes to spin along the direction normal to the ξη

plane. In the case of �Vξ , the relative speed of −(δω + ηωv)

occurs in the rolling direction (recall the rotational speed re-lation of Rω = εωv). Note that δ becomes small enough tobe neglected compared R (for example, δ = 2.4 × 10−3 mm,c = 0.59 mm, and R = 30 mm for the case of the S-CVTprototype); therefore �Vξ can be approximated to be −ηωv.The contribution of �Vξ is also a spin, similar to �Vη.

The vector diagram of relative velocity is obtained usingtypical values of ω, ωv, ε, R, P , and E that correspond to theS-CVT prototype specification (see Figure 7(b)). The spinvelocity field can be found straightforwardly, although thereare no excessive forces that cause slippage. From this result,we can be assured that there must be spin in the contact surfacearound the origin of the local coordinate frame of the contactpoint in the S-CVT regardless of the existence of shear forceresulting slippage. This relative velocity field in turn givesrise to spin moments. Moreover the resulted spin moment haslittle effect on the kinematic constraints of S-CVT from thefacts that the relative velocity �Vv(ξ, η) does not accumulatein the rolling direction and the amount of normal deflection δ

is small enough to neglect compared to R.Now we consider the normal pressure distribution on the

contact patch to obtain the friction force in eq. (4). A Hertzianpressure distribution develops in the circular shaped contact




(a) Velocity vector field.

-0.0006 -0.0004 -0.0002 0.0000 0.0002 0.0004 0.0006

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

(b) Typical relative velocity.

Fig. 7. Relative velocity on S-CVT.

patch (with radius c) between the sphere and disc. The pres-sure p at each point in the contact surface is known to be ofthe form

p (ξ, η) = 3

2

P

πc3

√c2 − ξ 2 − η2 . (8)

The maximal normal pressure pmax is located at the center ofthe contact surface; at the boundaries, the normal pressure p

becomes zero. In the following subsection, we now perform aqualitative analysis of spin loss using the obtained results onrelative velocity field and normal pressure distribution.

3.3. Quantitative Analysis of Spin Loss

Consider the infinitesimal area at the contact surface S, withthe friction force of the ith area in the rolling direction (ξ )denoted Fξi , and Fηi the force in the η direction. The totalfriction forces Fξ and Fη can be obtained using the followingequations:

Fξ =c∫

−c

c∫−c

Fξi(ξ, η) dξdη , Fη =c∫

−c

c∫−c

Fηi(ξ, η) dξdη .

Recall that the normal pressure distribution has symmetriesalong the ξ and η axes, and that �Vξ in eq. (7) varies alongthe η direction (neglecting δ), and �Vη along the ξ direction;there are no total relative velocities in the ξ and η directions.Therefore one can conclude that Fξ and Fη become zero, andthat these forces do not cause any rolling directional slippageresulting in slip loss.

Using the proposed friction model in eq. (4), the spin mo-ment Tspin for the variator can be calculated as

Tspin =c∫

−c

c∫−c

(ηFξi + ξFηi) dη dξ (9)

where

Fξi =[(µs − µk) exp

(−(

�Vξ

Vstr

)2)

+ µk

]p(ξ, η)sgn(�Vξ) ,

Fηi =[(µs − µk) exp

(−(

�Vη

Vstr

)2)

+ µk

]p(ξ, η)sgn(�Vη) ,

�Vξ = −ηωv , �Vη = ξωv .

Rearranging and integrating by parts, eq. (9) becomes

Tspin = 3Pc

4π

(µk + (µs − µk)

2

c4

Vstr2

ωv2

{c2 − Vstr

2

ωv2[

1 − exp

(−(cωv

Vstr

)2)]})

(10)

where c is the radius of the contact surface, which can becalculated using eq. (5).

To investigate the amount of spin loss at the contact pointsof the S-CVT, we calculate the respective spin losses usingeq. (10) for the input and output discs and variators with typi-cal values forµk, µs, Vstr , P . Figure 8 shows the numerical re-sults for spin loss and the rotational speeds of the input/outputdiscs and variator at a constant input speed of 3000 rpm with




0 10 20 30 40 50 60 70

0

2000

4000

6000

8000

decreasing as theinput speed increases

0.020

0.025

0.07

0.08

Fig. 8. Spin losses on S-CVT at input speed of 3000 rpm.

respect to changes in the variator angle. Spin moments occurat six contact regions in the S-CVT, and the total amount ofspin loss reaches almost 0.076 Nm. Spin moments decrease asthe variator angle increases, except for the input discs whosespin moments remain constant (because there are no speedchanges in the input discs). This is due to the characteristicsof our friction models (Fξi and Fηi) in eq. (9). Raising theinput speed increases the rotational speed of variator, caus-ing the relative velocity �V to become large. The increasedrelative velocity in turn reduces the friction force from thatcorresponding to static friction to kinetic friction.

The average value of spin loss obtained from our numeri-cal results is almost 0.072 Nm. Considering the input torqueis limited under a static friction torque of 1.962 Nm, the ratioof spin loss to static friction torque is almost 3.67%. Consid-ering normal operating conditions, at which the input torqueis smaller than the limiting torque, one can note that the ra-tio of spin loss becomes much greater. To reduce this loss, itis helpful to operate the S-CVT with high input speeds; theincreased relative velocity reduces the relevant friction force.

4. Shifting Controller for the S-CVT

The shifting controller realizes the target gear ratio, whichin turn determines the input/output power ratio. When theshifting command for a certain gear ratio is given, the shiftingsystem must be stabilized so as to realize the demanded gearratio with the desired performance (e.g., minimum shiftingeffort, short settling time, etc.). The shifting command for aCVT can be either a final value or a trajectory of the targetgear ratio. The shifting controller is denoted a stabilizer (orregulator) in the former case and tracker (or servo) in the lattercase.

Because the shifting system of the S-CVT has second-order nonlinear dynamics, and the original open-loop system

reveals unstable characteristics, in this section, we developa feedback controller based on exact feedback linearization.The central idea is to algebraically transform a nonlinear sys-tem dynamics into a (fully or partly) linear one, so that linearcontrol techniques can be applied. We first present a stabilityanalysis of the shifting system, followed by the tracker designand a numerical analysis of its performance.

4.1. Stability Analysis of the S-CVT Shifting System

Lyapunov’s (indirect) linearization method is involved withthe local stability of a nonlinear system. It is a formalizationof the intuition that a nonlinear system should behave simi-larly to its linearized approximation for small range motions.Because all physical systems are inherently nonlinear, Lya-punov’s linearization method serves as the fundamental jus-tification of using linear control techniques in practice, i.e.,that stable design by linear control guarantees the stability ofthe original physical system locally.

We first investigate the local stability of the S-CVT shiftingsystem using Lyapunov’s linearization method. We first recastthe shifting dynamics of eq. (3) into state-space form. Lettingx1 = θ, x2 = θ be the states, the corresponding state-spaceequation assumes the following form:

x1 = x2

x2 = 1

D[a22Fs − a12(Fi cos x1 − Fo sin x1)] ,

(11)

where [a11 a12

a21 a22

]=[

2 (Ia+Iv+mε2)

ε2RIv

ε2

2 Iv

ε

Is

R+ 2RIv

ε2

],

D = a11a22 − a12a21 .

Using the trigonometric transformation

a sin x + b cos x = √a2 + b2 sin(x + φ), φ = tan−1

(b

a

),

eq. (11) can be written

x1 = x2

x2 = 1

D{a12

√F 2

i + F 2o

sin(x1 − φ) + a22Fs} (12)

where

φ = tan−1

(Fi

Fo

).

Noting that D = 2Is(Ia + Iv + mε2)

εR+ 4RIv(Ia + mε2)

ε3

is always larger than zero, the equilibrium point is given by

x∗1 = φ = tan−1 Fi

Fo

, x∗2 = 0, F ∗

s= 0 . (13)




The equilibrium point of interest is x∗ = (φ, 0). Physically,this point corresponds to steady state of the shifting system inwhich no shifting occurs.

The Jacobian matrix J of the shifting system (12) linearizedabout the equilibrium point becomes

J =[

0 1a12

D

√F 2

i + F 2o

0

]. (14)

The eigenvalues of J are λi = ±√

a12

D

√F 2

i + F 2o

, indicat-

ing that the linearized system is unstable at this equilibriumpoint. Physically this implies that when the input-output forcerelation (Fi cos θ = Fo sin θ ) is broken (i.e., steady state is de-stroyed) by some disturbances from the input or output force,maintaining stability requires a change in variator angle (gearratio) from the shifting actuator, or by a change in input forcefrom the power source controller. In order to make the shift-ing system stable, one can conclude that an appropriate feed-back controller is necessary. We now discuss the design of afeedback controller based on the exact feedback linearizationmethod.

4.2. Input-State Linearization

Consider an affine single-input nonlinear system

x = f(x) + g(x) · u, (15)

where x ∈ Rn, u ∈ R. The objective of input-state lineariza-

tion is to find a transformation z = T(x) and a control of theform u = α(x) + β(x) · ν such that

z1 = z2,

z2 = z3,...

zn = ν.

(16)

We refer the reader to, e.g., Khalil (1996) and Slotine and Li(1991) for a more in-depth treatment of feedback lineariza-tion, including the conditions for the existence of a solution.For our purposes the vector fields f and g of the shifting systemare given by

f =[x2

a12

D

√F 2

i + F 2o

sin(x1 − φ)

]T

,

g =[0

a22

D

]T. (17)

The Lie bracket adf g of two vector fields f and g is defined tobe

adf g = ∇g · f − ∇f · g,

which is also often denoted[f, g

]. Repeated Lie brackets can

be defined recursively according to

ad0f g = g,

adif g = [

f, adi−1f g

].

With these preliminaries, input-state linearization of theshifting system can be performed via the following steps:

1. Construct the vector fields g, adf g, . . . , adn−1f g for our

system.

2. Check the controllability and involutivity conditions.

3. Find the first new state z1 from

∇z1 · adkf g = 0 k = 0, 1, . . . , n − 2 ,

∇z1 · adn−1f g �= 0 .

(18)

4. Compute the diffeomorphism that transforms the state

x into the new state z, T(x) = [z1 Lfz1 . . . Ln−1

f z1

]T,

and the input transformation using

α(x) = − Lnf z1

LgLn−1f z1

, β(x) = 1

LgLn−1f z1

. (19)

Knowing that the system order n = 2 and ∇g = 0, thecorresponding Lie bracket becomes

adf g = ∇g · f − ∇f · g

= 0 −[

0 1a12

D

√F 2

i + F 2o

cos(x1 − φ) 0

][0a22

D

]

=[−a22

D0]T

. (20)

A simple check of the rank of the controllability matrix

[g adf g ad2

f g . . . adn−1f g

] = 0 −a22

Da22

D0

, (21)

which is full rank, confirms that the shifting system is input-state linearizable.

We now find the diffeomorphism T(x) that transforms theoriginal shifting dynamics into a linear system. Using the re-sults of eq. (18), the necessary conditions for the first state z1

are

∂z1

∂x1

�= 0,∂z1

∂x2

= 0 .

Thus z1 must be a function of x1 only. Among the variouscandidates for z1, the simplest solution is z1 = x1 − φ. Theother state can be obtained from z1 as

z2 = ∇z1f = x2 .




The corresponding diffeomorphism T(x) can be obtained as

z = T(x) =[

x1 − φ

x2

]. (22)

Accordingly the input transformation in eq. (19) is

u = ν − ∇z2f∇z2g

which can be written explicitly as

u = D

a22

{ν − a12

D

√F 2

i + F 2o

sin(x1 − φ)} . (23)

As a result of the above state and input transformations, weend up with the following set of linear equations:

z1 = z2 , z2 = ν (24)

ν = a12

D

√F 2

i + F 2o

sin(x1 − φ) + a22

Du , (25)

thus completing the input-state linearization.

4.3. Shifting Controller Design

Using the above input-state linearization results, we now de-sign a shifting controller for tracking. In this case, it is desiredto have the variator angle θ track a prescribed trajectory θd .The input ν is chosen to be of the form

ν = ¨z1d − k1e − k2e, (26)

where e = z1 − z1d and z1d = θd − φ. The associated errordynamics is of the form with the gain values k1 and k2 chosenappropriately to ensure stability.

e + k2e + k1e = 0 . (27)

The resulting closed-loop dynamics of the shifting system(27) can be viewed as the canonical form of a general second-order oscillation problem:

s2 + 2 ζωns + ωn2 = 0 .

Hence one can give physical meaning to the feedback gains asthe respective damping ratio ζ and the natural frequency ωn.The relation between the feedback gains and ζ, ωn are simply

k1 = ωn2, k2 = 2 ζωn . (28)

In this study, we desire our shifting controller to providethe most rapid response according to the shifting commandwithout overshoot; we designate the settling time of the shift-ing system (the time in reaching the new equilibrium state)to be less than 1 s. Hence, we select the system damping

ratio ζ to 1, which corresponds to the case of critical damp-ing. For a given initial excitation, a critically damped systemtends to approach the equilibrium position the fastest withoutany overshoot. Moreover, these feedback gains guarantee theasymptotic stability and tracking performance of the S-CVTshifting system. We consider two cases of k1, k2 (see Table 1).

Simulation results for the tracking controller are given fortwo sets of gain values. For the reference trajectory of the

variator angle, we consider a sinusoidal functionπ

3sin(

π

2t −

φ) (see Figure 9(a)), with the initial states of the system chosenas

θ = 42◦, θ = 0, Fi = 1, Fo = 1 .

Maintaining the input-output force as the initial values, thecalculated variator angle changes are depicted in Figure 9(b).As expected, both cases of feedback gains show asymptoticconvergence of tracking error. The relevant tracking error andcorresponding shifting effort are shown in Figure 10. Notethat the time to reach the zero tracking error for case A isalmost 0.7 s while for case B is almost 1.2 s. Thus we selectthe feedback gains for case A.

5. An S-CVT based Mobile Robot: MOSTS

In this section we present the design and prototype construc-tion of an S-CVT based mobile robot, MOSTS (Mobile rObotwith a Spherical Transmission System), including numericaland experimental results on its performance.

5.1. Pivot Device for Planar Accessibility

In typical mobile robot designs, an additional controlled ac-tuator, such as a steering wheel or a motor for differentiatingeach wheel velocity, is necessary in order to move a mobilerobot in the plane. Employing a novel pivot device, however,we can eliminate the need for an additional steering actuatorand controller. To change its heading direction, MOSTS turnsabout its center (or pivots) by rotating one of the wheels in thereverse direction. For this to occur, we have been inspired bythe fact that the S-CVT can locate arbitrarily the orientationof an output shaft. To achieve this, it is necessary to locateone of the output shafts on the opposite side of the sphere (seeFigure 11(a)).

To achieve this operation we have adopted an internalgear driven by a simple actuator (see Figure 11(b)), e.g., alimit switch used in automated windows, and an uncontrolled

Table 1. Candidates for k1, k2

Case A k1 = 100, k2 = 20Case B k1 = 50, k2 = 10

√2




0 2 4 6 8 10

-60

-40

-20

0

20

40

60

(a) Reference shifting command θd.

0.0 0.3 0.6 0.9 1.2 1.5 1.8

-60

-40

-20

0

20

40

60

0.0 0.2 0.4 0.6 0.840

45

50

55

60

(b) Variator angle changes.

Fig. 9. Tracking performance of the S-CVT shifting system.

0.0 0.2 0.4 0.6 0.8 1.0 1.2

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

(a) Tracking error.

0 2 4 6 8 10

-0.2

0.0

0.2

0.4

0.6

Fs

(b) Control.

Fig. 10. Tracking error and corresponding control.




(a) Pivot inspiration. (b) Realization by use of an internal gear.

Fig. 11. Pivot device for planar accessibility of MOSTS.

motor. Using simple analog devices, we build a pivot switchthat can be turned off according to a pre-set current limit.

During pivot motion, each wheel rotates in opposite direc-tions with the same magnitude, while the driving motor ro-tates continuously without any changes of state. The amountof pivot angle is determined by the amount of angular dis-placement of each wheel, which is controlled by the shiftingactuator, or variator. Moreover, if a controlled actuator is usedto rotate the movable output shaft, steering motion can be ob-tained. Designed in this fashion, MOSTS has the capabilityto move in the plane with one drive motor, one controller forthe S-CVT, and one switching actuator.

5.2. Prototype Design

For the construction of the mobile robot platform, we have setthe following performance targets:

1. A top speed of 5 m/s;

2. A maximum ascending angle of 10◦;

3. A combined vehicle-payload mass of 50 kg.

From the above hardware specifications and material prop-erties of the S-CVT, we choose a specific dc motor that pro-duces a power of 150 W with 12 V under nominal operatingconditions as the driving motor (see the details provided inTable 2).

The body of the mobile robot is designed to have a cylin-drical shape, and a caster wheel is added to provide stable

Table 2. DC Motor Characteristic Coefficients of MOSTSRated voltage 12 VRated power 150 W

Motor-torque constant 0.0164 Nm/ABack emf constant 0.0164 Volt/s/rad

Rotor winding resistance 0.1174Stall torque 2.03 Nm

support. The internal body consists of three layers: a mechan-ical base for the transmission system, an intermediate layerfor the battery pack and controller, and a top layer for periph-erals and accessories, e.g., navigation sensors, manipulators.Rotary encoders sensing the speeds of the input and outputshafts are also included. The overall size of the platform is260 mm in radius, and 500 mm in height (see Figure 12).

5.3. Numerical and Experimental Results

In this subsection, we present numerical and experimentalresults that demonstrate the operation of MOSTS, and theenergy savings possible from the use of the S-CVT mechanismover standard reduction gears. The reference path is shown inFigure 13(a); there are three linear movements and two pivotmotions during 22 s. The distance traversed by the robot is20 m. During the pivot motion, there is an auxiliary 2 s periodfor actuating the pivot switch, which is necessary to move oneof output shafts of the S-CVT to the opposite direction. Withthis reference trajectory, we calculate the necessary wheelvelocity profile satisfying the time constraints by using a sine




Fig. 12. Photograph of MOSTS and S-CVT prototype.

N

S

EW

(a) Designed moving path.

0 5 10 15 20 25-100

-50

0

50

100

150

200

250

300

(b) Calculated wheel velocity profile.

Fig. 13. Motion simulation of MOSTS.




0 5 10 15 20 25

-2

0

2

4

6

8

Fig. 14. Trajectory of variator angle.

function (see Figure 13(b)). The pivot motions in the path arespecified as a 90◦ counter-clockwise rotation, followed by a90◦ clockwise rotation.

First, we calculate the value of the output speed acceler-ation from the driving pattern under the assumption that theinput voltage is held constant at 12 V. The exerted load torqueis set to 2.5215 Nm, and the equivalent inertia with respect tothe motor shaft is set to 0.01 km2. With these values and theoutput speed, we extract the necessary variator angle θ usinga computed torque control algorithm. Finally, the trajectoryof the variator angle is presented in Figure 14.

5.3.1. Numerical Results

Using the equations derived in the previous section, we havedeveloped a simulation program that computes the motorspeed, produced torque, and the power consumption. We usethe Runge-Kutta fourth-order algorithm for numerical inte-gration in the simulation program.

In Figure 15(a), the initial motor speed is about 7000 rpm,which is obtained from the no-load condition of the dc mo-tor considered here. During the whole operation period, themotor speed varies freely between 6500 rpm and 7000 rpmthrough the entire stop, start, and pivot motions; this is al-most the nominal speed with a no-load condition. Besides thespeed aspect, the necessary motor torque is calculated in Fig-ure 15(b). Generally, maximal torques are exerted when themotor of general mobile robots starts rotating. However, in thecase of our robot, almost zero torque is exerted at the start,and the torque variations are quite small during the whole pe-riod. This fact of reducing the peak current is one of the mainadvantages of S-CVT equipped mobile robot; the maximumavailable peak current is the critical problem of commercial-

ized batteries in practical robot designs. Moreover, it can bepossible to design the driving motor of a mobile robot in smallcapacity, also.

To investigate the increase in energy savings, we calculatethe energy consumption rate of our mobile robot for the ref-erence trajectory. As a benchmark, we consider a differentialdrive type mobile robot having a reduction gear unit with agear ratio of six under the same load condition. The differ-ential drive type robot considered has two driving motors of150 W at each wheel shaft and follows the same reference tra-jectory. Consequently, we calculate the energy consumptionrates for each case using the following equation:

Energy =∫

| ea(t) × ia(t) | dt.

The calculated total energy consumption is 1389.61 J for thedifferential drive with reduction gear unit and 727.86 J forour CVT-based mobile robot. Simulation results indicate thatour mobile robot with an S-CVT consumes less than 47.6%of the energy consumed by the differential drive, a significantimprovement in energy efficiency.

5.3.2. Experimental Results

Using the sequential manipulation of the variator angle ac-cording to calculated values of Figure 14, we experimentallydetermined the actual energy consumption of MOSTS underthe same reference trajectory mentioned above. The actual en-ergy consumption is 1294.92 J, which is larger than the idealcase by 567.06 J. However, this is still smaller than the cal-culated energy consumption of 1389.61 J for the differentialdrive case (the actual energy consumption for this case willmost likely be significantly higher than the calculated idealrate). To investigate the reason behind this difference in to-tal energy consumption, the induced motor current and theactual motor speed for the first two seconds are depicted inFigure 16. As the reference motion trajectory considered herehas five repetitive sequences, it is sufficient to investigate thefirst two second period experimental results.

Observe that the initial motor current is almost 4 A,whereas the ideal value is almost zero. This initial inducedmotor current is mainly due to the power loss resulting frommanufacturing errors including bearing friction, gear back-lash, etc. Consequently, this power loss makes the drivingmotor run at lower speeds, causing the overall power effi-ciency to decrease. Actually, adopting the S-CVT into a mo-bile robot may increase the robot’s weight adversely. Howeverour mobile robot still holds the possibility of energy savingstaking into account that the repeated start and stop motionsof a mobile robot, in which large torques (requiring the largemotor current) are applied from the driving motor, happenfrequently.




0 5 10 15 20 25

6600

6700

6800

6900

7000

(a) Motor speed.

0 5 10 15 20 25-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

(b) Motor torque.

Fig. 15. Motor behaviors of MOSTS.

Table 3. Energy Consumption; MOSTS vs. Differential Drive

MOSTS Simulation result 727.86 JExperimental result 1294.92 J

Differential drive Simulation result 1389.61 Jwith reduction gear Experimental result ??

0.0 0.5 1.0 1.5 2.06200

6300

6400

6500

6600

6700

6800

6900

7000

(a) Motor speed.

0.0 0.5 1.0 1.5 2.0-1

0

1

2

3

4

5

6

7

8

(b) Motor induced current.

Fig. 16. Experimental results.




6. Conclusion

In this paper, we have presented the design of a CVT-basedmobile robot using a minimal number of actuator and controlcomponents, by taking advantage of the performance featuresof the S-CVT. Such a CVT-based mobile robot has the advan-tage of being able to operate the motors in their regions ofmaximum efficiency, thereby prolonging the total run time ofthe robot. The addition of a novel pivot device also enablesthe mobile robot to achieve steering (more precisely, chang-ing its heading direction) by using only a single drive motorand controller, unlike most existing mobile robot platforms.

We examine spin loss in the S-CVT by first developinga modified friction model based on Hertzian elasticity the-ory that includes Stribeck effects, and perform a quantitativeanalysis that is generalizable to other nonholonomic CVT de-vices. A shifting controller is developed for the S-CVT basedon input-state linearization of the dynamic equations. We alsoperform an in-depth analysis of the energy efficiency of ourmobile robot taking into account features of the dc motors,the S-CVT, and the mobile robot dynamics. The results arebenchmarked numerically with a differential drive type mo-bile robot equipped with a reduction gear. Furthermore, weperform an experiment using the prototype robot to verify therobot’s operation and the CVT characteristics. The numeri-cal and experimental results show that our mobile robot withan S-CVT consumes less power than differential drive typerobots, and suggests the S-CVT can be a useful nonholonomicmachine element for a wide variety of robotics systems.

Acknowledgement

This research was supported in part by the BK21 program inmechanical engineering, and the National Research Labora-tory for CAD/CAM, both at Seoul National University.

References

Borenstein J., Everett H. R., and Feng L. 1996. Where am I?Sensors and Methods for Mobile Robot Positioning, TheUniversity of Michigan.

Chang, W. S., and Toumi, K. Y. 1998. Modeling of an omni-directional high precision friction drive positioning stage.In Proc. of Inter. Conf. on Robotics and Automation,pp. 175–180, Leuven, Belgium.

Choi, Y., Yeo, I., and Kim, J. 1999. Technical Report: Trendsof Continuously Variable Transmission, Technical Center,Daewoo Motor Company.

Chung, S., and Han, D. C. 1996. Standards of MechanicalDesign, Dongmyung-Sa, Korea.

Courtney-Pratt, J., and Eisner, E. 1957. The effect of a tan-gential force on the contact of metallic bodies. Proc. of theRoyal Society, vol. A238, pp. 529–550.

Dahl, P. R. 1977. Measurement of solid friction parametersof ball bearing. Proc. of the 6th Annual Symposium onIncremental Motion, Control Systems and Devices, pp. 49–60, University of Illinois. 89-0148 B.

Jones, J. L., and Flynn, A. M. 1993. Mobile Robots: Inspira-tion to Implementation. Massachusetts: A. K. Peters.

Kassakian, J. G., Schlecht, M. F., Verghese, G. C. 1991. Prin-ciples of Power Electronics, chap. 1-8. Cambridge, MA:Addison-Wesley.

Kato, K. 1990. Fundamentals and applications of frictiondrive. In Journal of the Japan Society for Precision En-gineering, vol. 9, pp. 1602–1606.

Khalil H. K. 1996. Nonlinear Systems, 2nd Ed., Prentice Hall.Kim, J., Park, F. C., Park, Y., and Shizou, M. 2002. Design

and analysis of a spherical continuously variable trans-mission. In Journal of Mechanical Design 124(1):21–29.ASME.

Kim, J., Yeom, H., Park, F. C. 1999. MOSTS: A mobilerobot with a spherical continuously variable transmission.In Proc. IEEE/RSJ Inter. Conf. on Intelligent Robots andSystems, vol. 3, pp. 1751–1756.

Kobayashi, N., Nonami, K., and Yasuzumi, I. 1996. Attitudecontrol of free-flying space robot with discrete time slidingmode control algorithm. In Transactions of the Society ofInstrument and Control Engineers 32(8):1206–1211.

Kurosawa, M. K., Takahashi, M., and Higuchi, T. 1998. Elas-tic contact conditions to optimize friction drive of surfaceacoustic wave motor. In Transactions on Ultrasonics, Fer-roelectrics, and Frequency Control 45(5):1229–1237.

Lee, H. S., and Tomizuka, M. 1996. Robust motion controllerdesign for high-accuracy positioning systems. In Transac-tions on Industrial Electronics 43(1):48–55.

Leonhard, W. 1996. Control of Electrical Drives, chap. 1-9,2nd ed. Braunschweig, Germany: Springer.

Loewenthal, S. H. et al. 1981. Evaluation of a high perfor-mance fixed-ratio traction drive. In Transactions of theASME, vol. 103.

Moore, C. A. 1997. Continuously Variable Transmission forSerial Link Cobot Architectures, M.Sc. Thesis, Northwest-ern University.

Moore, C. A., Peshikin, M. A., Colgate, J. E. 1999. Designof a 3R cobot using continuously variable transmission. InIEEE International Conference on Robotics and Automa-tion, Detroit, Michigan, pp. 3249–3254.

Moronuki, N., and Furukawa, Y. 1988. On the design of pre-cise feed mechanism by friction drive. In Journal of theJapan Society for Precision Engineering, vol. 11, 2113–2118.

Nakamura, Y., Chung, W., and Sørdalen, O. J. 2001 Designand control of the nonholonomic manipulator. In IEEETransactions on Robotics and Automation 17(1):48–59.

Neimark, N. I., and Fufaev, N. A. 1972. Dynamics of nonholo-nomic systems. In Translations of Mathematical Mono-graphs, vol. 33, American Mathematical Society.




Singh, T., and Nair, S. S. 1992. A mathematical reviewand comparison of continuously variable transmissions.In Transactions of SAE, No. 922107.

Slotine, J. E., and Li, W. 1991. Applied Nonlinear Control,Prentice Hall.

Tanaka, H. et al. 1989. Spin moment of a thrust ball bearingin traction fluid. In Journal of JSME.

Timoshenko, S., and Goodier, J. N. 1951. Theory of Elasticity,2nd Ed., McGraw-Hill Book Co.

U. S. Department of Energy. 1982. Advanced AutomotiveTransmission Development Status and Research Needs,DOE/CS/50286-1.

Wang, B. T., and Fries, R. H. 1989. Determination of creepforce, moment, and work distribution in rolling contactwith slip. In Transactions of the ASME, vol. 111.

Weast, R. C., Astle, M. J., and Beyer, W. H. 1987. CRCHandbook of Chemistry and Physics, 68th ed. Boca Ra-ton, Florida: CRC Press, Inc.



Documents

The International Journal of Robotics Researchrobotics.snu.ac.kr/fcp/files/_pdf_files_publications/3... · 2016-01-20 · The International Journal of Robotics Research Vol. 21, No