Design Analysis and Control

7/27/2019 Design Analysis and Control

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Design, Analysis and Control of a

Spherical Continuously Variable Transmission

By

Jungyun Kim

Submitted in Partial Fulfillment of The

Requirements for The Degree of

Doctor of Philosophy

in theSchool of Mechanical and Aerospace Engineering

atSeoul National University

February 2001


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To My Parents

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Abstract

This dissertation is concerned with the design, analysis, and control of a novel con-

tinuously variable transmission, the spherical CVT (S-CVT). The S-CVT has a

simple kinematic structure, infinitely continuously variable transmission character-

istics, and transmits power via dry rolling friction on the contact points between

a sphere and discs. The S-CVT is intended to overcome some of the limitations

of existing CVT designs. Its compact and simple design and relatively simple con-

trol make it particularly effective for mechanical systems in which excessively large

torques are not required.

We describe the operating principles behind the S-CVT, including a kinematic

and dynamic analysis. A prototype is constructed based on a set of design spec-

ifications and results of theoretical performance analysis. In order to provide a

quantitative analysis of the spin loss of the S-CVT, which is one of the main sources

of power loss, we develop an explicit formulation using a modified classical friction

model, and an in-depth study of the velocity fields and normal pressure distribution

on the contact regions. The proposed friction model includes the pre-sliding effect,

i.e., Stribeck effects. Actual transmission ratios and power efficiency are obtained

from experiments with a prototype testbench.

The open-loop shifting system of the S-CVT reveals nonlinearity and unstable

characteristics. In order to cancel the nonlinearity of the shifting system and to

make it stable to shifting commands, we develop an input-state feedback controller

based on exact feedback linearization. We also investigate the power efficiency of

a generic dc motor, and present the results of a numerical investigation of the S-CVTs energy savings possibility benchmarked against a standard reduction gear.

Furthermore, we develop a minimum energy control law for the S-CVT driven by a

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dc motor, and present numerical simulation results that confirm the performance of

the controller.

Finally, we design and construct an S-CVT based mobile robot to realize the

various advantages of the S-CVT into practical use. One of the key features of

the mobile robot is the design of a novel pivot mechanism for planar accessibility.

Results of both numerical simulations and experiments are presented to validate the

robots performance advantages obtained as a result of using the S-CVT.

Keywords: Continuously variable transmission; infinitely variable transmission;

dry rolling friction; spin loss; feedback linearization; minimum energy control;

mobile robot.

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Contents

Dedication i

Abstract ii

List of Tables viii

List of Figures xi

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 CVTs for Passenger Cars . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Dynamic Analysis of the Spherical CVT 23

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Kinematics of S-CVT . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.2 Operating Principles . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Dynamics of S-CVT . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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2.3.1 Motion of Sphere . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.2 Shifting Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Reaction Forces of the S-CVT . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1 Normal Reaction Force Exerted on the Variator: Fn . . . . . 36

2.4.2 Shifting Reaction Force on the Sphere: D . . . . . . . . . . . 37

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Prototype Design and Experimental Results 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Issues in Mechanical Design . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Normal Force Loading Device . . . . . . . . . . . . . . . . . . 41

3.2.2 Capacity of Shifting Actuator . . . . . . . . . . . . . . . . . . 42

3.3 Prototype Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.1 Performance of S-CVT . . . . . . . . . . . . . . . . . . . . . . 47

3.4.2 Strength and Life Prediction of S-CVT . . . . . . . . . . . . 49

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Slip Analysis of the Spherical CVT 53

4.1 Friction Model Review . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Modified Friction Model for S-CVT . . . . . . . . . . . . . . . . . . . 59

4.3 Spin Loss of the S-CVT . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.1 Velocity Fields on the Contact Surface . . . . . . . . . . . . . 60

4.3.2 Normal Pressure Distribution . . . . . . . . . . . . . . . . . . 65

4.3.3 Quantitative Analysis of Spin Loss . . . . . . . . . . . . . . . 66

4.4 Slip Motion of the S-CVT . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4.1 Stick and Slip States . . . . . . . . . . . . . . . . . . . . . . . 70

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4.4.2 Slip Loss of the S-CVT . . . . . . . . . . . . . . . . . . . . . 70

4.4.3 Slip Involved Contact Analysis . . . . . . . . . . . . . . . . . 70

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Shifting Controller Design via Exact Feedback Linearization 74

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Stability Analysis of S-CVT Shifting System . . . . . . . . . . . . . . 76

5.3 Differential Geometric Preliminaries . . . . . . . . . . . . . . . . . . 78

5.4 Shifting Controller Design via Input-State Linearization . . . . . . . 81

5.4.1 Controllability and Linearizability . . . . . . . . . . . . . . . 82

5.4.2 Input-State Linearization . . . . . . . . . . . . . . . . . . . . 83

5.5 Shifting Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 84

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Optimal Control of an S-CVT equipped Power Transmission 91

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2 Power Efficiency of a DC Motor . . . . . . . . . . . . . . . . . . . . . 936.2.1 DC Motor Dynamics . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.2 Power Efficiency of a DC Motor . . . . . . . . . . . . . . . . 95

6.3 Investigation of S-CVT Energy Savings . . . . . . . . . . . . . . . . 96

6.3.1 Control Design based on the Computed Torque Method . . . 98

6.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 99

6.4 Minimum Energy Control via a B-Spline Parameterization . . . . . . 101

6.4.1 B-Spline Parameterization . . . . . . . . . . . . . . . . . . . . 102

6.4.2 Gradients of the Objective Function and Constraint . . . . . 103


6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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7 Case Study: An S-CVT based Mobile Robot 109

7.1 Motivation for Mobile Robot Applications . . . . . . . . . . . . . . . 110

7.2 MOSTS: An S-CVT Mobile Robot . . . . . . . . . . . . . . . . . . . 112

7.2.1 Pivot Device for Planar Accessibility . . . . . . . . . . . . . . 112

7.2.2 Prototype Design . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . 115


7.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 120

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8 Conclusion 123

References 125

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List of Tables

3.1 Specifications of prototype. . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Endurance test condition. . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1 Maximal normal pressure comparison. . . . . . . . . . . . . . . . . . 66

5.1 Candidates for k1, k2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.1 Characteristic coefficients of dc motor. . . . . . . . . . . . . . . . . . 97

6.2 Energy consumption; reduction gear vs. S-CVT. . . . . . . . . . . . 101

6.3 Energy consumption with the minimum energy control. . . . . . . . 108

7.1 Hardware specifications of general mobile robots. . . . . . . . . . . . 110

7.2 DC motor charateristic coefficients of MOSTS. . . . . . . . . . . . . 115

7.3 Energy consumption; MOSTS vs. differential drive. . . . . . . . . . . 121

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List of Figures

1.1 Classification of transmissions for vehicles. . . . . . . . . . . . . . . . 2

1.2 Fuel consumption reduction for an engine. . . . . . . . . . . . . . . . 4

1.3 Engine speed variation. . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Classification of CVTs. . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Belts for belt drive CVT. . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Belt drive CVTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7 Variable stroke drives. . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.8 Full toroidal CVT, by courtesy of Torotrak. . . . . . . . . . . . . . . 10

1.9 Structures for traction and friction drive CVT. . . . . . . . . . . . . 11

1.10 Geometries of toroidal CVT, by courtesy of Torotrak and NSK. . . . 13

1.11 Optimal operating line of an engine. . . . . . . . . . . . . . . . . . . 17

1.12 Typical variogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1 Standard structure of S-CVT. . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Velocity constraint diagram. . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Operating principles of S-CVT. . . . . . . . . . . . . . . . . . . . . . 28

2.4 Ideal speed ratio of S-CVT. . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Transmittable torque. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Coordinate system and forces on S-CVT. . . . . . . . . . . . . . . . 31

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2.7 Forces on variator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1 3-dimensional concept view. . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Normal force loading device using a spring. . . . . . . . . . . . . . . 42

3.3 Schematic diagram of S-CVT. . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Assembly drawing of S-CVT. . . . . . . . . . . . . . . . . . . . . . . 45

3.5 S-CVT prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.6 Testbench of S-CVT. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.7 Experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.8 Power efficiency of S-CVT. . . . . . . . . . . . . . . . . . . . . . . . 49

3.9 Endurance test result of input disc. . . . . . . . . . . . . . . . . . . . 51

4.1 Spin loss in traction drives. . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Classical model of static, kinetic, and viscous friction. . . . . . . . . 55

4.3 Pre-sliding displacement phenomenon. . . . . . . . . . . . . . . . . . 57

4.4 Proposed friction model. . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5 Contact of two bodies with different curvature. . . . . . . . . . . . . 61

4.6 Velocity vector field on contact point. . . . . . . . . . . . . . . . . . 63

4.7 Typical relative velocity vector diagram. . . . . . . . . . . . . . . . . 64

4.8 Friction forces at the infinitesimal area of the contact surface. . . . . 67

4.9 Spin losses on S-CVT at input speed of 3000 rpm. . . . . . . . . . . 69

4.10 Dislocation of contact center. . . . . . . . . . . . . . . . . . . . . . . 71

4.11 Change of normal pressure distribution in XZ plane. . . . . . . . . . 72

5.1 Stability of the S-CVT shifting system. . . . . . . . . . . . . . . . . . 875.2 Tracking performance of the S-CVT shifting system. . . . . . . . . . 88

5.3 Tracking error and corresponding control. . . . . . . . . . . . . . . . 88

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5.4 System behaviors of S-CVT during the gear ratio change. . . . . . . 89

6.1 Diagram of an armature-controlled dc motor. . . . . . . . . . . . . . 93

6.2 Efficiency of an armature-controlled dc motor. . . . . . . . . . . . . . 96

6.3 Target profile of output speed. . . . . . . . . . . . . . . . . . . . . . 97

6.4 Computed variator angle time profile. . . . . . . . . . . . . . . . . . 99

6.5 Motor behaviors; reduction gear vs. S-CVT. . . . . . . . . . . . . . . 100

6.6 Power consumption; reduction gear vs. S-CVT. . . . . . . . . . . . . 100

6.7 Interpretation oftilde07Eg(p). . . . . . . . . . . . . . . . . . . . . . 104

6.8 Optimal variator angle time profile. . . . . . . . . . . . . . . . . . . . 106

6.9 Motor behaviors with the minimum energy control. . . . . . . . . . . 107

6.10 Output behaviors with the minimum energy control. . . . . . . . . . 107

7.1 Pivot device for planar accessibility of MOSTS. . . . . . . . . . . . . 112

7.2 Electric circuit diagram of pivot switch and driving motor. . . . . . . 113

7.3 Hardware prototype of MOSTS. . . . . . . . . . . . . . . . . . . . . 116

7.4 The desired trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.5 Calculated wheel velocity profile. . . . . . . . . . . . . . . . . . . . . 117

7.6 Trajectory of variator angle. . . . . . . . . . . . . . . . . . . . . . . . 118

7.7 Motor behaviors of MOSTS. . . . . . . . . . . . . . . . . . . . . . . . 119

7.8 Power consumption; MOSTS vs. differential drive. . . . . . . . . . . 120

7.9 Experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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Chapter 1

Introduction

Power transmissions are a universal element in nearly all mechanical systems, from

a small-sized reduction gear in a compact disc drive, to a complex gear box (usually

referred to as a transmission) in a vehicle. Although their components, sizes, and

operating principles vary, their main objective is to effect changes in the sources

power in the manner that corresponds to the load condition by manipulating the

transmission ratio (or the gear ratio, i.e., the ratio of the input speed to output

speed). Well-designed power transmissions eliminate the need for oversized power

sources, and increase the power efficiency of overall the system. Even though power

transmissions are required in various engineering fields, research activities are driven

mainly by automobile manufacturers for their conventional transmissions. Thus, in

this dissertation, an overview of power transmissions will be focused on automobile

applications.

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1.1 Overview

The generated power from ordinary power sources (internal combustion engines,

electric motors, etc.) is much different from the necessary tractive force for driving

vehicles. Hence, it is necessary to transform the power adequately from the source

to the tractive force; the transmission of the vehicle takes this role. General trans-

missions for vehicles can be primarily classified into manual transmissions (MTs)

and automatic transmissions (ATs), according to its actuating mechanism for the

shifting action (decision of shifting time, engaging/disengaging of the power flow el-

ements, selecting the ratio, etc.). A detailed classification of transmissions is shown

in Figure 1.1.

A MT consists of dry clutch, which engages and/or disengages the power flow,

a pair of synchronizing devices and constant meshing gear train for each gear ratio,

and gear ratio selecting devices. Its structure and components are simple enough to

allow for a considerable reduction in size and weight compared to a conventional AT.

Transmission

ContinuouslyVariable

Transmission

Figure 1.1: Classification of transmissions for vehicles.

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Moreover, a MT is built up with pure mechanical components and has no external

power loss, such as a hydraulic system; thus the power efficiency is quite better than

that of an AT.

A conventional AT consists of wet clutches, planetary gear trains for each gear

ratio, a hydraulic system for shifting action, an electro-hydraulic servo system for

shifting control, and a torque converter. A torque converter is a unique device which

has the multiple roles of a torque multiplication device, starting device, and torsional

damper. Besides disadvantages in size and weight, a hydraulic system including a

torque converter shows significant power loss, reducing the the overall efficiency of

an AT (and ultimately the fuel economy of an AT equipped vehicle). However as

the driving comfort of vehicle becomes the main concern, and greater effort is made

toward improving the efficiency of ATs, the market share of AT equipped vehicles

is growing rapidly.

Power sources have complex efficiency characteristics according to driving con-

ditions. For example, an internal combustion engine has different fuel consumption

rates (or, brake specific fuel consumption: BSFC) according to its speed and torque

while producing the same amount of power (see Figure 1.2). In this figure, there

are two engine operating points which produce the same power for 120 km/hr with

regard to different gear ratios. In the case of gear ratio A, which is greater than B,

the BSFC value of this point is smaller than that of gear ratio B by 10%; hence one

can conclude that a wide-spread of gear ratios is helpful for improving a fuel econ-

omy. In addition, making more gear ratios can enhance the acceleration performance

for the same reason. Many transmission engineers therefore endeavor to develop a

transmission having more gear ratios. But making more gear ratios increases the

size and weight of a transmission.

The continuously variable transmission (CVT) has continued to be an object

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Engine

torque

Engine speed

100 %

110 %

120 %

130 %

140 %

150 %Constant power

at 120 kph

Driving resistancecurve B

Driving resistance

curve A

BSFC curve

Engine torquecurve at W.O.T.

Vehicle speed 120 kph with gear ratio A

Vehicle speed 120 kph with gear ratio B

10% FC

reduction

Figure 1.2: Fuel consumption reduction for an engine.

of considerable research interest within the mechanical design community, driven

primarily by the automotive industrys demands for more energy efficient and en-

vironmentally friendlier vehicles. Unlike conventional stepped transmissions (MTs

and ATs), in which the gear ratio cannot be varied continuously, a CVT has a con-

tinuous range of gear ratios that can, up to device-dependent physical limits, be

selected independently of the transmitted torque. This feature of the CVT allows

for engine operation at the optimum fuel consumption point consistent with the

given output power requirements, thereby improving the engines power efficiency.

Moreover, the CVT does not suffer from shifting shock (see Figure 1.3).

In 1886, a CVT with rubber belt and pulleys made by Daimler Benz company

was known as the first CVT to have been applied to a passenger car. About 1930,

General Motors acquired the patent of the toroidal drive, which will be mentioned

in a subsequent section in more detail, and tried to develop their own CVT. But

they failed to commercialize it, and finished the related research in 1935. A different

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time

speed

time

speed

time

speed

Stepped

Transmission

CVT

Vehicle

Power source

repeat accordingto the shifting

stay aroundsome point

regardless of shifting

Figure 1.3: Engine speed variation.

toroidal type CVT, known as a Heyes Self-Selector, was adopted in many Austin

cars, although its production ceased after two years.

The first commercially successful CVT for a passenger car was the rubber belt

Variomatic of DAF Co., developed in 1958. The Variomatic was not popular, be-

cause it failed to resolve the problems of rubber belt failure and the performance

degradation due to deformation and wear. In the 1960s, a CVT using a metal belt

and variable pulleys was developed by Hub Van Doorne, but did not make it to the

market due to its insufficient torque capacity.

In the 1970s, due to the worldwide oil-crisis and the raised environmental recogni-

tion, many countries strengthened the regulations of the fuel economy and exhausted

emissions of vehicles. Moreover by the advances of metallurgy and production tech-

nology, inherent restraints of CVT could overcome; the research and development

for CVT was much encouraged from the late of 1980s. Currently several automobile

manufacturers have developed various prototype CVTs that are soon expected to

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appear in commercial vehicles (see [1]-[5] and references therein).

1.2 CVTs for Passenger Cars

According to the power transmission element and shifting mechanism, existing CVTs

can be classified into belt drive, traction drive, variable stroke drive, and hydro-

static/dynamic drive (see Figure 1.4).

Belt Drive CVT

In a belt drive CVT, a rubber or steel belt running on conically shaped variable

diameter pulleys is used to transmit power at different drive ratios. According to

the belt material, belt drive CVT can be divided into rubber, chain, and metal belt

type. In Figure 1.5, the schematic diagrams of rubber, chain, and metal belt are

shown. Because of their small power capacity, rubber belt CVTs are adopted in

ContinuouslyVariable Transmission

Friction Drive

Figure 1.4: Classification of CVTs.

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(a) Rubber belt. (b) Chain belt. (c) Metal belt.

Figure 1.5: Belts for belt drive CVT.

compact cars and machine tools. Passenger cars equipped with a chain belt CVT

had previously appeared on the market, but their production halted before long

(a) ACVT of Aichi Co. (b) Multimatic of Honda Co.

Figure 1.6: Belt drive CVTs.

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owing to chain noise and vibration problems.

Currently almost conventional CVTs have push type metal belts of Van Doornes

Transmissie b.v. or a revised form. Although the metal belt still suffers from a

small torque capacity, the number of production units has been rising steadily in

the worldwide market. The torque capacity has recently increased with the aid of

advances in metallurgy and improvements in the hydraulic system. Figure 1.6 shows

the rubber belt drive CVT made by Aichi Co. and metal belt drive CVT by Honda

Co.

Hydrostatic/Dynamic and Variable Stroke Drive CVT

Hydrostatic/dynamic drives use an incompressible fluid as the transmission medium,

by connecting a hydraulic pump directly to a variable displacement hydraulic actu-

ator. It can realize neutral, forward, and reverse stages, but is not typically applied

to passenger cars, owing to its own low power efficiency, weight, and noise. It has

(a) Cylinder type. (b) Link type.

Figure 1.7: Variable stroke drives.

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seen limited applications to heavy equipment.

A variable stroke drive is made with one-way clutches and crank devices which

can adjust the crank arm length. The rotational motion of the drive shaft transforms

into translational motion, and the one-way clutch rectifies the motion into a uni-

directional motion. This type of CVT cannot manage properly the pulsative output

torques, and is therefore not adopted in vehicles (see Figure 1.7, by courtesy of DOE

report [6]).

Traction and Friction Drive CVT

Friction wheels of unequal diameter were one of the earliest speed changing mech-

anisms. It is speculated that their use even predates that of gearing toothed

wheels, whose beginnings date back to the time of Archimedes, circa 250 B.C. [7].

Even today, friction drives may be found in equipment where a simple and eco-

nomical solution to speed regulation is required: phonograph drives, self-propelled

lawnmowers, or even amusement park rides driven by a rubber tire are a few of the

more common examples. In these examples, simple dry contact is involved, and thetransmitted power levels are low. However, this same principle can be harnessed

in the construction of an oil-lubricated, all steel component transmission which can

carry hundreds of horsepower using todays technology. In fact, oil-lubricated trac-

tion drives have been in industrial service as speed regulators for more than 70 years

[8].

Great progress in tribology research since late 1960s, particularly research on

elasto-hydrodynamic lubrication (EHL) traction, has made it easier to understand

the traction drive mechanism. Traction drives transmit power through an increased

shear force, which results from elasto-hydraulic shear stress of the traction oil be-

tween two rotating solid bodies. A coefficient of traction is typically 0.1, and macro-

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scopic slip occurs at any time. Since there is no direct contact between the rotating

bodies, wear phenomenon does not occur. The drive ratio is varied by controlling

the effective onset radius of the contact point.

According to the geometries of rotating elements, there are various type of trac-

tion drive CVTs: nutating drive, half toroidal type, and full toroidal type. The half

toroidal CVT has semi-circular discs, while full toroidal CVTs have full-circular discs

as input/output rotating elements. According to the curvature radii of discs, they

have different attainable gear ratios, torque capacity, and spin loss. Figure 1.8 shows

the full toroidal drive and pertinent CVT made by Torotrak Ltd. in UK. Apart from

this, many automobile manufacturers have developed half toroidal CVTs with dif-

ferent torque capacities in Japan. Along with the traction oil developers (Santotrak,

Shell companies), they have presented various prototypes of traction drive CVTs in

the market [9].

Generally, a traction drive shows rapid shifting response compared to belt drives,

(a) Full toroidal traction drive. (b) Pertinent CVT.

Figure 1.8: Full toroidal CVT, by courtesy of Torotrak.

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and can be adopted to medium-sized (even to large-sized) vehicles, because the

highly-pressurized traction oil endures more shear stress than belt drives. The draw-

backs of a traction drive are known to be as follows: the need for careful temperature

control, sealing and supply of traction oil, and complicated shifting control due to

the three dimensional contact curvature of the rolling elements.

Finally, there exist friction drives where the power transmission mechanism is

via rolling resistance and friction force in direct contact, though its structure and

operating principle are much similar to traction drives (see Figure 1.9, by courtesy

of DOE report [5]). Friction drives have been also found in several types of wood-

working machinery dating back to before the 1870s. For example, [10] reports of a

frictional gearing being used to regulate the feed rate of wood on machines in which

one wheel was made of iron and the other, typically the driver, of wood (or iron

covered with wood).

Figure 1.9: Structures for traction and friction drive CVT.

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Friction drives have not been considered for passenger cars due to its low torque

capacity, wear, heat dissipation problems, etc.However, friction drives have received

significant attention from the perspective of tribology, because precise positioning

can be accomplished while avoiding backlash [11]-[15]. Furthermore, traction and

friction drives provide much design flexibility in terms of their structure and al-

lowance for compact-sized designs.

Although each type of CVT has its own particular set of advantages and disad-

vantages, common difficulties shared by current CVTs are the complicated shifting

controller design, and the need for a large-capacity, typically inefficient shifting ac-

tuator [5]. Also, these CVT designs do not have infinitely variable transmission

(IVT) capabilities, i.e., they do not include zero output speed among its available

ratios, and therefore require a clutch or other type of starting and engaging device

for initially driving the vehicle.

1.3 Literature Review

There is a vast amount of literature regarding the design, analysis, control, and

application of CVTs in engineering fields. This section focuses on the areas of

design and control of traction/friction drives, because the proposed spherical CVT

in this thesis shows similar characteristics with respect to operating principles, power

transmission and shifting mechanisms, and control laws.

Traction Drive Designs

It is well known that one of the earliest examples of the friction drive was the patent

of C. W. Hunt in 1877 [16]. Basically the mechanism of that drive was a toroidal

drive, which was developed for more than a decade thereafter. Applications of

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traction drives to automobiles have been studied since the beginning of this century.

Prior to 1935, cars were called Friction Drive Cars, experimentally installed with

such drives, had received some attention: it was widely believed that power was

transmitted by friction between the rolling metallic elements.

In the latter half of the 1960s, when elsto-hydraulic lubrication (EHL) became

better understood [17], it was recognized that power was transmitted by traction.

The performance of a traction drive depends to a large extent on the rheological

properties of the fluid in the EHL contact [18]-[29]. In the 1970s, synthetic traction

oil was developed which had a traction coefficient almost 50% higher than before,

and practical use of the traction drive CVT was thought to be close at hand. It was,

however, not realized, because the heat treatment of the rolling elements could not

be achieved. A new type of traction oil being developed for automotive use shows

some promise [30], [31].

(a) Full toroidal CVT. (b) Half toroidal CVT.

Figure 1.10: Geometries of toroidal CVT, by courtesy of Torotrak and NSK.

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As stated earlier, there are two main design streams in traction drives, for auto-

motive use full and half toroidal type CVTs. A full toroidal CVT [32] has full-circular

discs as input/output media and power rollers as shifting devices (see Figure 1.10

(a)). In a full toroidal CVT, power rollers are located at the center of the toroidal

shaped input and output discs. A hydraulic loading system has commonly been used

to supply the normal force, which is necessary to transmit power via traction. Its

shifting mechanism is based on the side-slip force generated by the velocity difference

of the contact point.

A half toroidal CVT uses semi-circular discs instead full-circular ones, though

the shifting mechanism is not different from full toroidal CVTs (Figure 1.10 (b)).

Many engineers including P. W. R. Stubbs (1980), Lubomyr O. Hewko (1986), M.

Nakano (1991, 1999), H. Kumura (1999), and H. Machida (1999) have presented the

trends and issues on half toroidal CVT designs for use in full-sized cars as a future

driveline technology [33]-[38]. Nakano (1991) reported that the main reasons for the

unsuccessful commercialization of toroidal CVTs were thought to be the inability to

obtain sufficient performance with respect to the traction and viscosity performance

of the traction fluid, the fatigue strength of the rolling elements, power transmission

efficiency, transient ratio change controllability, and the issue of synchronization

control in connection with the parallel arrangement of the traction elements [35].

A traction drive CVT changes its speed ratio by controlling the side-slip force

on the Hertzian contact area. Tanaka and Eguchi (1991) showed the principles of

the speed ratio changing mechanism of half-toroidal CVTs and highlighted a digital

compensation method for stabilization of the electro-hydraulically operated speed

ratio control mechanism [39]. Fellows and Greenwood (1991) reported that it might

not be possible to suppress hunting of the shifting control signal, depending on the

control system used [40]. In addition to these results, there are many materials

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related to toroidal CVT controller designs (for example [41], [42], and references

therein).

When the ratio changes in a half toroidal traction CVT, the necessary contact

force does not vary significantly compared to a full toroidal CVT [43]. Consequently,

a loading cam system of a half toroidal CVT that produces contact force in pro-

portion to the input torque can provide high efficiency over the entire speed ratio

range, contrary to the hydraulic loading system of a full toroidal type. Moreover

it has been reported that the full toroidal traction CVT suffers larger spin moment

at the contact points than the half toroidal type, which tends to reduce its power

capacity [44].

The current design issues on toroidal type traction drives can be summarized as

follows:

the material of rolling elements is not sufficiently reliable because of high pres-sure and high temperature on the traction contact point;

there is no affordable traction oil which satisfies all the conditions of automo-

biles, although it has been reported that an adequate traction oil has been

developed recently [9], [31];

there are no bearings which can support high speeds and a large axial load;

the normal force loading system (e.g., hydraulics or loading cam), which isnecessary to produce traction force, is inefficient and needs precise control for

equalizing the normal forces on the contact points;

there are difficulties on the control of transient ratio change and synchronizedprecise control in connection with the parallel arrangement of the traction

elements.

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CVT Controls

A CVT is originally intended to operate the power source in power efficient regimes

by means of manipulating its gear ratio. Many previous efforts are focused on finding

the power efficient regimes of sources and controlling the gear ratio of a CVT in order

to run the source within those regimes. Here we review the previous analysis results,

which describe ways of controlling a CVT for maximizing the fuel economy of a

CVT-equipped vehicle as well as how to establish the shift schedule (the so called

variogram, which describes the graphic relation between the engine and vehicle

speeds) of a CVT for the vehicles performance objectives.

Generally the power efficiency of a source is maximized at only one point over

its operating region. In an internal combustion engine (see Figure 1.2), the fuel

consumption is lowered for higher engine torque. On the other hand, it worsens for

high engine speeds as the mechanical loss is large at those speed points. The pumping

loss tends to be large for low engine speeds; hence, the fuel consumption also worsens

for low engine speeds. These characteristics are consistent with theory. If we operate

the engine only at the most efficient point, however, the driving performance may

not be satisfied, because the driving torque to be generated for each vehicle speed is

limited. Therefore, the control and optimization of automotive powertrain systems

with a CVT is achieved by cooperative control of the engine and CVT (see [2]-[4],

[45]). A drive-by-wire structure using an electric throttle control device is adopted

for this engine consolidated CVT control [46].

Figure 1.11 shows an optimal operating line (OOL) of a typical engine for max-

imum fuel economy. Generally, an OOL is constructed simply by connecting staticBSFC contours through optimization. For improving the fuel economy of a vehicle,

it is definitely helpful to control the CVT gear ratio so as to run the engine along

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100 %

110 %

120 %

130 %

140 %

150 %

Figure 1.11: Optimal operating line of an engine.

this operating line. Most CVT-equipped vehicles use shift schedules (or variograms)

in look-up table form, presetting the optimal gear ratios obtained from the static

performance data of the engine and road tests of the prototype vehicle (see Figure

1.12). However, this OOL does not involve the vehicle dynamics including accel-

eration, performance objectives, because there is no consideration for the engine

dynamics.

The classical way to control CVT cars is to use some information on the gear

ratio or on the transmitted torque which is then fed back by a PID controller [47]-[49].

Only when using gain-scheduled controllers with typically 100 different gain points

could the required performance be achieved. Kolmanovsky et al. (1999) explored the

use of a CVT for torque management during mode transitions in lean burn gasoline

engines [50]. They demonstrated that an intuitively sound CVT gear ratio control

strategy which attempts to completely cancel the engine torque disturbance may

result in unstable zero dynamics. They concluded the coordination of engine torque

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Figure 1.12: Typical variogram.

production and CVT gear ratio control during mode transitions was mandatory.

Takahashi (1998) proposed a scheme to minimize rate of fuel consumption by a

direct fuel injection engine used by combination with CVT [45]. Target values for

the engine and transmission which minimize fuel consumption ensuring driving per-

formance were calculated based on the nonlinear optimization method. As a result

of optimization, target values for air-fuel ratio and gear ratio were calculated and

controlled by tracking. Because the calculation of partial differential was impossible

at some operating points, he used a simplex method that did not require calculat-

ing differential values. For minimization of fuel consumption function under various

restrictions, penalty functions were also introduced.

The non-minimum phase behavior of the CVT based powertrain system (without

a torque converter) was mentioned in [51]. Considering this phase behavior of CVT,

Guzzella and Schmid (1995) addressed an exact feedback linearization approach for

a controller of CVT equipped vehicle [52]. In their works, the plant dynamics were

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exactly linearized over the complete operating range using feedback linearization.

And as an application of the exact linearization approach, a kick-down controller

was designed.

1.4 Outline and Contributions

This dissertation deals with the design, analysis, and control of a spherical CVT. A

conceptual design of a particular spherical CVT (S-CVT) was proposed by Joukou

Mitsusida in [53]. The S-CVT consists of a sphere, input and output discs, and

variators. The rotating input and output discs are connected to the power source

and output shafts, respectively, while the sphere is situated between the input and

output discs. The transmission ratio is controlled by adjusting the location of the

variator on the sphere, which in turn controls the axis of rotation of the sphere. It

transmits power via dry rolling friction on the contact points of sphere and discs;

therefore, there exists a torque limitation decided by the static friction force.

The S-CVT, intended to overcome some of the aforementioned limitations of

existing CVT designs, is marked by its simple kinematic design and IVT charac-

teristics, i.e., the ability to transition smoothly between the forward, neutral, and

reverse states without the need for any brakes or clutches. Moreover its relatively

simple control makes it particularly effective for mechanical systems in which ex-

cessively large torques are not required (e.g., mobile robots, household appliances,

small-scale machining centers, etc.).

In order to put the S-CVT to practical use, an analysis of its operating principles,

power transmission and shifting mechanisms, and power capacity together with the

consideration for issues of hardware design, needs to be performed. This dissertation

is aimed at providing theoretical and practical solutions for these concerns, through

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an in-depth study of the design, dynamics, and control of the S-CVT. This work can

be categorized into four parts:

analysis of theoperating principles, kinematics, and dynamics in Chapter 2;

hardware design issues including a slip loss analysis in Chapter 3, 4;

shifting controller and minimum energy control law design in Chapter 5, 6;

application for a wheeled mobile robot as a case study in Chapter 7.

The subsequent achievements in this work can be exploited to the design and analysis

of traction or friction drives having similar structure.

Currently, we are carrying out the development of other S-CVT applications

for small-capacity speed changers, e.g., bicycles, laundry machines, wind-propelled

generating systems, potters spinning wheels, etc.There still remain several practical

problems, such as realizing precise shaft alignments and increasing the torque capac-

ity. Currently research efforts are being directed toward the application of traction

fluid for the purpose of adopting the S-CVT for large torque capacity applications,e.g., hybrid vehicles, compact cars, etc.

The detailed outlines and contributions of each chapter can be stated as follows.

Operating Principles, Kinematics, and Dynamics

Chapter 2 describes the conceptual design and operating principles of the S-CVT

together with a detailed kinematic and dynamic analysis of its performance. In

addition, there shows analytic interpretation for the reaction forces of S-CVT which

are normally exerted on variator and discs, along with the definitions of their physical

meanings.

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Hardware Design Issues and Slip Loss

Chapter 3 presents the prototype specifications and a discussion of the main design

issues, focusing on the normal force loading device and the shifting actuator capacity.

Some experimental results are given on the actual transmission ratios and power

efficiency obtained from a prototype testbench, to validate the operating principles

and performance of S-CVT. We briefly address the strength and life estimation for

the S-CVT, based on the well-known ball-bearing life theory.

Spin loss of S-CVT, which is one of the main power losses of the S-CVT (and

more generally for friction and traction drives) due to slippage, is formulated using

a modified classical friction model in Chapter 4. The proposed friction model can

involve pre-sliding effect i.e., Stribeck effects. For this, we perform an in-depth

study of velocity fields and the normal pressure distribution generated on the contact

regions. We also provide a quantitative analysis of the spin loss of the S-CVT. In

addition, we discuss contact analysis involving slip, in which a shear force resulting

from friction occurs on the contact surface.

Shifting Controller Design and Minimum Energy Control

The shifting system of the S-CVT has second-order nonlinear dynamics, for which

typical open-loop control systems are likely to develop unstable characteristics. In

order to cancel the nonlinearity of the S-CVT shifting system and to make it stable

and responsive to shifting commands, we develop a feedback controller based on

the exact feedback linearization method in Chapter 5. We first investigate the

instability of the S-CVT shifting system using the Lyapunovs indirect method. Wethen present the input-state feedback controller design of the S-CVT shifting system,

and investigate the stabilizing and tracking performance of the dedicated shifting

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controller by numerical simulation.

Chapter 6 deals with a minimum energy control law for the S-CVT connected

to a dc motor. We first investigate the general power efficiency of a dc motor. We

then present the results of a numerical investigation of the S-CVT energy saving

possibility benchmarked with a standard reduction gear. For this investigation, a

computed torque control algorithm for the S-CVT is proposed. In addition, we

describe a minimum energy control law of S-CVT connected to a dc motor. To do

this, we describe the general power efficiency characteristics of a dc motor. Then

the minimum energy control design is carried out via B-spline parameterization.

Numerical results obtained from simulations illustrate the validity of our minimum

energy control design.

An S-CVT based Mobile Robot

Finally, we propose an S-CVT based mobile robot (denoted as MOSTS for a Mobile

rObot with a Spherical Transmission System) to realize the various advantages of

the S-CVT, including the originally intended CVT characteristic of energy efficiency,into practical use in Chapter 7. In this chapter, we first address the motivation

for applying the S-CVT to a wheeled mobile robot by first reviewing the current

hardware designs of mobile robots and their power efficiency. We then present the

hardware design of our S-CVT based mobile robot in accordance with the target

performance. In addition, we propose a novel pivot mechanism which is necessary

for planar accessibility using an internal gear and an uncontrolled dc motor. We

perform both numerical simulations and experiments for various motion plans, in

order to validate the realization of the robots operation, the CVT characteristics,

and its energy saving possibility.

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Chapter 2

Dynamic Analysis of the

Spherical CVT

2.1 Introduction

In this chapter a new type of spherical continuously variable transmission (S-CVT)

is described. The S-CVT, intended to overcome some of the aforementioned limi-

tations of existing CVT designs, is marked by its simple kinematic design and IVT

characteristics, i.e., the ability 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 between metal on

metal, it has limitations on the overall transmitted torque, which is effectively de-

termined by the static coefficient of friction and the magnitude of the normal forces

applied to the sphere. Due to this torque limitation, the S-CVT is not intendedfor automobiles and other large capacity power transmission applications. Target

applications for the S-CVT include mobile robots, household electric appliances,

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small-scale machine tools, and other applications with moderate power transmis-

sion requirements. Although the current design of the S-CVT is based on friction

drive designs, it is our expectation that the power capacity of the S-CVT can be

increased by the use of traction oil, an issue which we do not pursue further in this

dissertation.

Other spherical CVT structures have been proposed for use in passive mobile

robots and for use as nonholonomic joints in robot manipulators. Carl A. Moore et

al. (1999) have reported a 3R passive robot, called the Cobot. The Cobot adopts a

rotational CVT to provide smooth, hard virtual surfaces for passive haptic devices in

place of conventional motors. Its rotational CVT consists of a sphere caged by four

rollers, and adopts the joint speeds and task space speeds along with the steering

angles as control inputs [54]. Another application can be found in underactuated

manipulators, designed by Serdalen et al. (1994). This work proposes a new type

of manipulator architecture using a CVT-type robot joint that takes advantage of

the inherent nonholonomy of the CVT [55]. Although these systems are designed

to manipulate the speed ratio using a CVT mechanism, their main purpose is not

for power transmission to improve the energy efficiency. Furthermore, the shifting

mechanism of the S-CVT is quite different from these previous designs, as will be

described below.

In this chapter, the conceptual design and operating principles of the S-CVT

are described together with a detailed kinematic and dynamic analysis of its perfor-

mance. Section 2 describes the basic kinematic structure and operating principles

of the S-CVT. In Section 3, we examine the dynamics of the S-CVT by deriving

the equations of motion and its shifting mechanism. Finally in Section 4, we exam-

ine the reaction forces on the S-CVT, in particular those exerted normally on the

variator and discs.

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2.2 Kinematics of S-CVT

2.2.1 Structure

The S-CVT is composed of three pairs of input and output discs, variators, and a

sphere (see Figure 2.1). The input discs are connected to the power source, e.g., an

engine or an electric motor, while the output discs are connected to the output

shafts. The sphere, which is the main component of the S-CVT, transmits power

from the input discs to the output discs via rolling resistance between the discs and

the sphere. The variators, which are connected to the shifting controller, are incontact with the sphere like the discs, and constrain the direction of rotation of the

sphere to be tangent to the rotational axis of the variator.

Figure 2.1: Standard structure of S-CVT.

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The speed and torque transmission ratios of the S-CVT vary with the angular

displacements of the variators; this will be described in further detail in the following

subsection on the operating principles of the S-CVT. To transmit power from the

discs to the sphere or from the sphere to the discs, a device that supplies a normal

force to the sphere, such as a spring or hydraulic actuator, must be installed on each

shaft. As can be seen in Figure 2.1, the structure and components of the S-CVT

are simple enough to allow for a considerable reduction in size and weight compared

to conventional transmissions. The orientations of the input and output shafts can

also be located freely using rollers at arbitrary positions rather than discs.

2.2.2 Operating Principles

When the input device is actuated by a power source, the input disc rotates about the

input shaft. This rotation in turn causes a rotation of the sphere, due to the condition

of rolling contact without slip between the input discs and the sphere. Rotation of

the sphere in turn causes a rotation of the output discs, and subsequently of the

output shaft. In the absence of any contact between the sphere and the variator, theaxis of rotation of the sphere will largely be determined by an equilibrium condition

among the various contact and load forces being applied to the sphere.

The role of the variator is to control the axis of rotation of the sphere. Specifically,

referring to Figure 2.2, the variator contacts the sphere at a point (marked by P)

located directly above the sphere center. Since the variator rotates about an axis

normal to the variator disc and passing through the variator center (marked by C1,

C2), it follows that the contact point between the variator and the sphere undergoes

a linear velocity in a direction tangential to the variator disc (marked by V1, V2).

By adjusting the location of the variator (from C1 to C2) it is therefore possible to

control the axis of rotation of the sphere (from 1 to 2); the axis will be parallel

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V1

V2

V

1

2

P

C2

C1

Figure 2.2: Velocity constraint diagram.

to the line between the variator center and the sphere-variator contact point, and

passing through the sphere center.

By varying the axis of rotation of the sphere, it is in turn possible to vary the

radius of rotation of the contact point between the input disc and the sphere, Ri,

as well as the radius of rotation of the contact point between the output disc and

the sphere, Ro (see Figure 2.3). In this way the speed-torque ratio of the S-CVT

can be adjusted. Figure 2.3 shows the various alignments of the variator for the

forward, neutral, and reverse states of the output shaft of the S-CVT. The neutral

state, which corresponds to zero rotation of the output disc, is achieved when Ro

becomes zero. As apparent from the figure, the forward, neutral, and reverse states

can all be achieved by smoothly manipulating the variator alignment, without the

need for any additional clutches or brakes.

Assuming roll contact without slip, the speed and torque ratio between the input

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and output discs is related to the variator angle by the following relations:

outin

=riro

tan (2.1)

ToutTin

=rori

cot (2.2)

where is the angular displacement of the variator, in and out are the respective

angular velocities of the input and output shafts, Tin and Tout are the respective

input and output torques, and ri and ro are the respective radii of the contact

points of the input and output discs (see Figure 2.3). There are two design variables

that prescribe the transmission ratio: the ratio of the input and output contact

ri

ro

RRi

Ro

ri

ro

R

Ri

ri

ro

R

Ri

Ro

variator angle

Figure 2.3: Operating principles of S-CVT.

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-50-30

-1010

3050

0.10.3

0.50.7

0.91.1

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Figure 2.4: Ideal speed ratio of S-CVT.

radiiriro

, and the variator angle . From Equation (2.1) it is apparent that a large

range of available transmission ratios is possible even with a sphere of small radius.

Assuming that there is no slip or other physical effects, the ideal speed ratio of the

input speed to output speed is shown in Figure 2.4.

Although ideally an infinite torque ratio is possible with the S-CVT as seen in

Equation (2.2), in practice there is a limit to the torque that can be transmitted

because power transmission occurs from rolling resistance of metal on metal. Figure

2.5 shows a plot of the torque ratio as a function of the variator angle, for a given

fixed input torque. The actual torque ratio of the S-CVT will lie somewhere in the

operating region as indicated in the figure because of power loss due to friction, slip,

heat generation. The limiting torque Tmax is determined by the static coefficient of

friction s and the normal force N exerted by the output disc spring mechanism on

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the sphere according to the relation Tmax = rsN, where r is the contact radius of

the disc. When either the input or output torque applied at the disc-sphere contact

exceeds this limit, slippage can occur. Taking into account this limiting torque, the

output torque for a given input torque Tin is given as follows:

Tout = Tmax sat( TinTmax

rori

cot ) TLoss (2.3)

where the saturation function sat() is defined by

sat(x) =

sgn(x) if|x|

> 1

x if |x| 1,

and TLoss is the torque loss in S-CVT. Though assuming roll contact without slip

(i.e., the speed ratio can be realized as the ideal case), the torque loss cannot be

zero, because there exist some torque losses resulted from the spin moments and

internal loads, etc., which will be discussed in Chapter 3 and 4.

Figure 2.5: Transmittable torque.

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2.3 Dynamics of S-CVT

2.3.1 Motion of Sphere

To investigate the shifting mechanism of the S-CVT, we designate a reference frame

XYZ situated at the center of the sphere, and a moving reference frame xyz, with

z coinciding with the spin axis of the sphere (see Figure 2.6). The various external

forces acting on the sphere are also shown in this figure, neglecting the normal

forces exerted on the sphere-discs contact points to hold the sphere and the weights

of sphere and discs.We define the driving forces which are delivered from the input discs as FZi1 and

FZi2, and the reaction forces exerted by the load torque from the output discs as

Figure 2.6: Coordinate system and forces on S-CVT.

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FZo1 and FZo2. Ftv1 and Ftv2 denote the forces generated by the shifting actuator

acting at the sphere-variator contact points. The remaining reaction forces at the

input and output discs and variators are respectively denoted by FXi1, FXi2, FY o1,

FY o2, Fnv1, Fnv2.

Assuming that the sphere center does not move and the rotational axis of the

sphere lies on the xy plane, the force equilibrium conditions for each coordinate are

as follows:

FXo1 FXo2 + FXi1 FXi2 + (Fnv1 Fnv2)cos (Ftv1 Ftv2)sin

FY i1

FY i2 + FY o2

FY o1 + (Fnv1

Fnv2)sin + (Ftv1

Ftv2)cos FZo1 FZo2 + FZi1 FZi2 + FZv1 FZv2

= 0. (2.4)With respect to the specified coordinate frames, we can derive the dynamic equations

relating the angular momentum change with the resultant moment acting on the

sphere, i.e.,

d

dtHo =

Mo

where Ho is the angular momentum and Mo is the resultant moment. Theangular momentum of the sphere is given by:

Ho = Is = 25

msR21

where is the angular velocity of the sphere, Is is its mass moment of inertia, ms

the mass, and R the radius. Expressing the angular momentum of the sphere in

terms of the moving coordinate frames, we obtain the derivatives of this momentum

and the resultant moments, leading to the following set of second-order differential

equations:

Is

=

(Fnv1 + Fnv2)R (FZi1 + FZi2)R sin (FZo1 + FZo2)R cos

(FY o1 + FY o2 + FXi1 + FXi2)R

(Ftv1 + Ftv2)R (FZo1 + FZo2)R sin + (FZi1 + FZi2)R cos

(2.5)

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where is the spinning rate of the sphere.

Considering the attributes of the external forces, as stated earlier, we can restate

those forces in Equation (2.5) as follows:

FZi1 + FZi2 = Fi,

FZo1 + FZo2 = Fo,

Ftv1 + Ftv2 = Ft,

Fnv1 + Fnv2 = Fn.

In the above equations, Fn should not be regarded as an active force for shifting,

but rather as a loss-like force acting to resist any variator displacements. Examining

the reaction forces at the input and output discs caused by changes in the sphere

axis of rotation, we can also conclude that the magnitudes of these forces must be

equal, otherwise the sphere will be distorted:

FXi1 = FXi2 = FY o1 = FY o2 = D. (2.6)

The relevant forces can therefore be summarized as follows:

Fi = Driving force delivered from the input discs;

Fo = Reaction force caused by the output discs connected to the load torque;

Ft = Shifting force on sphere delivered from the variator in the tangential direction;

Fn = Loss-like reaction force exerted normally on variator;

D = Reaction force on sphere generated by the shifting.

2.3.2 Shifting Dynamics

To establish the dynamic relations between the sphere and variator, we first define

the forces on the upper sphere-variator contact point and the connected shifting

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Figure 2.7: Forces on variator.

actuator (see Figure 2.7). To permit spinning motion of the variator, bearings arelocated on the connecting rod, which connects the shifting actuator and variator.

In this figure, denotes the angular displacement of the shifting actuator, which

consists of the same number of variators, m is the mass of the variator, and the

eccentric distance between the centers of the shifting actuator shaft and variator. In

addition, Fsv1 is the shifting force delivered by the shifting actuator, a is the linear

acceleration of the variator center, and v1 the rotational speed of variator.

Using the velocity constraint on the sphere-variator contact point, one can obtain

the rotational speed of variator v1

v1 = +R

(2.7)

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where is the spinning rate of the sphere, and R the sphere radius as defined earlier.

Let the moment of inertia of the shifting actuator shaft and connecting rod be Ia,

and that of the variator be Iv. The shifting force delivered from the variator onto

the sphere in the tangential direction (Ftv1) can be written as

Ivv1 = Ftv1. (2.8)

By the force relation Fsv1 = ma+Ftv1, and using the fact that the linear acceleration

of the variator a = , as well as Equations (2.7), (2.8), we can express the shifting

torque Fsv1

,

Fsv1 = (Iv + m2 + Ia) + Iv

R

. (2.9)

We assume that the lower variator always runs synchronously with the upper

one; then the total shifting force Fs = 2Fsv1 and v1 = v2. Rearranging the

equations of the sphere and variator (2.5) and (2.9), we obtain the following set of

second-order differential equations for the S-CVT:

2(Ia+Iv+m2)

2RIv

2

2 Iv Is

R + 2 RIv

2

= Fs

Fi cos Fo sin . (2.10)The reaction forces are given by

Fn =IsR

+ Fi sin + Fo cos , (2.11)

D =1

4

IsR

, (2.12)

Ft = 2Iv

( +R

). (2.13)

2.4 Reaction Forces of the S-CVT

The two main reaction forces of the S-CVT are Fn and D, which are exerted respec-

tively at the contact points between the sphere and discs, and sphere and variators.

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In order to prevent slippage, they must be smaller than the maximal friction force.

In this section we derive analytic expressions for Fn and D, and examine their effect

on the performance of the S-CVT.

2.4.1 Normal Reaction Force Exerted on the Variator: Fn

Fn, the reaction force which is exerted normally on the variators, can be considered

as a loss force, and restricts the available gear ratios. Since Fn acts ultimately on

the bearings located within the connecting rod, which connects the shifting actuator

and variator (see Figure 2.7), it can therefore cause excessive bearing normal forces

and bending moments on the variator and connected shafts.

From Equation (2.11), Fn at steady state becomes

Fn = Fi sin + Fo cos . (2.14)

From the fact that the shifting effort Fs is zero at steady state, the relation between

Fi and Fo of Equation (2.10) becomes

Fi cos = Fo sin .

Substituting Fi into Equation (2.14), Fn becomes

Fn =Fo

cos . (2.15)

Beyond a certain variator angle, the magnitude of Fn becomes larger than the max-

imal friction force which is determined by the static coefficient of friction s and

the normal force N; slippage therefore occurs at the sphere-variator contact point

(similar to the limiting torque Tmax).

During transient states, the dynamics of sphere and variator influences the

magnitude of Fn additionally. More than any other reaction forces on the S-CVT,

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Fn varies considerably together with input/output force and the shifting dynamics;

thus it contributes the limit of available gear ratios of S-CVT. The allowable range

of Fn during transient states is

Fn =Fo

cos +

IsR

sN .

Rearranging this, we obtain a range for the gear ratio :

| | cos1 FosN IsR

. (2.16)

In order to increase the range of available gear ratios, one can reduce the internal

load and hence increase the output force Fo, or decrease the shifting response , as

well as improve material properties with respect to s, N.

2.4.2 Shifting Reaction Force on the Sphere: D

There are four contact points between the sphere and input/output discs in the S-

CVT (see Figure 2.1). When shifting (i.e., changes in gear ratio) occurs, the reaction

force D, which resists the angular momentum change of the sphere, appears at each

contact point. The reaction force D is normally exerted on the discs, and it acts

directly on the bearings located within the input/output shafts; thus it can be

considered as loss force like Fn.

Moreover from Equation (2.12), D is related with the shifting response , and acts

to restrict the available shifting response. As is the case for Fn, slippage resulting

from a reaction force larger than the maximal friction force makes the S-CVT unable

to transmit power; therefore the following inequalities must be hold:

D sN , 4RIs

sN . (2.17)

From this relation, we can conclude that the shifting response is constrained by the

material properties s, the sphere geometries R, Is, and the normal force N.

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2.5 Summary

In this chapter, we have presented the design and analysis of a newly developed

spherical continuously variable transmission (S-CVT) focusing on its basic structure

and operating principles, shifting mechanism, and its reaction forces. The S-CVT is

intended to overcome some of the limitations of existing CVTs, e.g., difficult shifting

controller design, and the necessity of a large-capacity and typically inefficient shift-

ing actuator. It is marked by its simple configuration, infinite variable transmission

(IVT) characteristics and realization of forward, neutral, and reverse states without

any brakes or clutches.

Because the S-CVT transmits power through rolling resistance between metal on

metal, torque limitations prevent current versions of the S-CVT from being applied

to large capacity power transmission systems like passenger cars. However, our

study suggests that it can be well-suited for applications involving small mechanical

systems such as mobile robots, household electric appliances, small-scale machining

centers, etc.

Finally, we have investigated the reaction forces which are exerted normally on

the variator and discs. Both Fn and D constitute sources of power loss for the

S-CVT; in particular, the magnitude variation of Fn along the variator angle is

steeper than any other forces on S-CVT. Moreover Fn can be a dominant factor in

determining the available range of gear ratios of the S-CVT. The shifting reaction

force D is related with the shifting response and acts to restrict the available

shifting response.

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Chapter 3

Prototype Design and

Experimental Results

3.1 Introduction

In designing a transmission, one must consider both the power capacity of the trans-

mission and power source as well as the load conditions. In this chapter, we first

define the design objectives of the S-CVT, taking into account its inherent charac-

teristics such as the power transmission mechanism based on friction force, shifting

mechanism, and operating principles.

The proposed S-CVT is intended for use in small capacity mechanical systems,

e.g., mobile robots, household electric appliances, small-scale machine tools, and

other applications with moderate power transmission requirements. In determining

the hardware specifications of the S-CVT, practical issues such as the amount ofnormal force required to assure rolling resistance at the contact points of the S-

CVT, and the capacity of the shifting actuator that can realize the desired shifting

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response, and the range of available gear ratios must all be considered. Based on the

kinematic and dynamic analysis results of the previous chapter, we have designed

and built the following S-CVT prototype.

In this chapter, we present the prototype hardware specifications for the S-CVT,

and an analysis of its performance. Using a prototype testbench, we obtain ex-

perimental results that serve to validate the operating principles and performance

of the S-CVT. In Section 2, we discuss various issues in the mechanical design of

the S-CVT, focusing on the normal force loading device and the shifting actuator

capacity. To assure rolling resistance force at the contact points of the S-CVT, we

adopt compressible springs because of their simple structure and the ease in ad-

justing the preset load. To determine the shifting actuator capacity, we derive the

Figure 3.1: 3-dimensional concept view.

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numerical relationship between the necessary power and the shifting demand using

the previous dynamic analysis results. Section 3 shows the hardware specifications

and schematic drawings of the prototype S-CVT. In Section 4, we present experi-

mental results on the actual transmission ratios and power efficiency obtained from

the prototype testbench. Finally, we briefly address the strength and life estimation

of the S-CVT, based on the well-known ball-bearing life theory.

3.2 Issues in Mechanical Design

Among the relevant issues in designing the S-CVT, we will focusing in particular

on the normal force loading device and the capacity of the shifting actuator. The

considered issues are mainly related to power capacity, namely the maximal trans-

mittable force and the shifting actuator design.

3.2.1 Normal Force Loading Device

In order to assure rolling resistant force at the contact points of the S-CVT, an appro-

priate normal force should be applied on the sphere and discs. Compressible springs

are employed at each shaft, which are connected to the variators and input/output

discs, to make the mechanical structure simple and to adjust the amount of normal

force easily (see Figure 3.2). Since the spring force is closely related with the limit

of transmittable force, we need to measure and adjust it. Using a set-screw, the

amount of normal force can be adjusted by fixing the preset displacement of the

spring. To set an accurate spring force, strain gauges are attached to each relevant

shaft.

Regarding the amount of normal force, the larger spring force increases the trans-

mittable force. However, applying too large normal force causes yielding and plastic

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Figure 3.2: Normal force loading device using a spring.

deformation of the sphere and discs; careful consideration for the normal stress on

the contact regions must be carried out. In this study, we have designated the nor-

mal force amount as 100 kgf using the corresponding finite element analysis results

obtained by ANSYS.

3.2.2 Capacity of Shifting Actuator

In order to determine the capacity of the shifting actuator, it is necessary to inves-

tigate the variation of shifting force Fs along with the desired performance. From

Equation (2.10), in steady state Fs is zero and the input-output force relation be-

comes Fi cos = Fo sin . To achieve shifting (i.e., gear ratio change), a non-zero Fs

must be induced by the shifting actuator in an appropriate manner.

For example, we consider the case when shifting occurs by the amount d at a

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certain steady state instant. At the beginning of shifting, we can assume that the

input-output force relation still holds. Rearranging Equation (2.10), the shifting

force Fs becomes

Fs = 2 (Ia + Iv + m2)

2 I

2v R

2

(Is2 + 2 IvR2)

d . (3.1)

As seen in Equation (3.1), Fs necessary for shifting is determined by R, d, and

the mass moments of inertia of the sphere Is, variator Iv, and connected elements

Ia + m2. Considering that shifting forces of other traction or belt drives must be

large enough to resist the traction or friction force, which is generated directly by

the transmitted torque, the overall magnitude of shifting force of the S-CVT will

likely be much smaller than that of other existing CVTs.

The necessary power Ps of the shifting actuator is calculated using Equation

(3.1):

Ps = Fsd = 2

(Ia + Iv + m2) 2I

2v R

2

(Is2 + 2 IvR2)

d d (3.2)

where d is the corresponding angular velocity to the required shifting demand d.

3.3 Prototype Specifications

Based on the numerical investigation results from the previous studies, we have des-

ignated the hardware specifications of the S-CVT prototype. The overall layout of

the power transmission is shown in Figure 3.3. Because of the maximum limiting

torque, a reduction gear with a ratio of three is added to the prototype; this ratio

also includes a safety factor. This additional reduction gear can be eliminated by

improving the material properties such as the static coefficient of friction and in-

creasing the normal force at the contact point. The final assembly drawing is shown

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Driving Motor

PivotMotor

z=36z=18

z=18z=18

z=36

z=

72

z=

36

z=

26

z=

52 z

=26

z=

52

z=

26

z=

39z

=26

z=

39

Variator

Sphere

Wheel

Wheel

10

VariatorMotor

z=18 z=66

z=66z=18

13

13

Input disc

Figure 3.3: Schematic diagram of S-CVT.

in Figure 3.4. In Figure 3.3 and 3.4, a dc motor referred to as the pivot motor, and

internal gears are included in the power-flow line of the S-CVT. These elements are

added in order to make each output shaft rotate in opposite directions. This novel

pivot mechanism is proposed for the application to the CVT-based mobile robot,

which is the main subject of Chapter 7.

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load spring

housing

Figure 3.4: Assembly drawing of S-CVT.

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Element Nomenclature and Specification Material

Spheremass (ms) = 0.882 kg

radius (R) = 30 mm

Steel ball

of ball bearing

Input/output disc

Variator

mass (m) = 0.095 kg

radius (r) = 16 mm

contact radius (ri/o, ) = 10 mm

SNCM 8 class

Input/output shafts SCM 4 class

Gears Refer to Figure 3.3 SCM 21 class

Mass moments

of inertia

Sphere (Is) = 3.1758 104 kg m2

Input parts (Iin) = 2.3581 105 kg m2

Output parts (Iout) = 3.8609 104 kg m2

Variator (Iv) = 1.0514 105

kg m2

Variator connected parts (Ia) = 1.0585 104 kg m2

Table 3.1: Specifications of prototype.

The detailed specifications for numerical studies and experiments are shown in

Table 3.1. The prototype S-CVT has been built and is shown in Figure 3.5.

Figure 3.5: S-CVT prototype.

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3.4 Experimental Results

In order to validate the operating principles and performance of the S-CVT, we have

built a testbench for it. Two eddy-current type AC servo motors (input: 3-phase

AC, 122 V, 9 A; output: 1500 Watts; rated speed: 2000 rpm) are used for a driving

power source and a driven load generator. In the testbench (see Figure 3.6), the

variator angle is controlled by a dc stepped motor with an angular resolution of

0.024/pulse. The rotational speeds of the input and output shafts are measured

through incremental optical encoders attached to the shafts.

3.4.1 Performance of S-CVT

Setting the external load torque to zero, we observe the output speed together with

the variator angle displacement while the input speed is set respectively to 748,

Figure 3.6: Testbench of S-CVT.

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1502, and 2001 rpm. A steady-state speed ratio curve of the S-CVT is extracted

for the no-load condition (see Figure 3.7 (a)). Note that the overdrive of the output

speed, which implies that the output speed is faster than the input speed, occurs

when the variator angle exceeds 50. In addition, there is a large deviation between

the ideal value and the test result beyond a variator angle of 65, which indicates

the onset of slippage. These less than ideal output speeds arise from the increase of

reaction force normally exerted on variator Fn, which is described in Section 2.4. In

the experimental result, moreover, there must be a certain amount of internal load

induced by manufacturing and other errors, which makes Fo large (see more details

in Section 2.4).

Using slip-ring type torque sensors, we have also observed the output torque

together with the variator angle displacement by adjusting input/output torque to

realize the pre-obtained steady state speed ratio (see Figure 3.7 (b)). The actual

torque ratio is limited to under 20, which is determined mainly by the static coeffi-

0 10 20 30 40 50 60 70

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

(a) Speed ratio of S-CVT.

0 10 20 30 40 50 60 70

-5

0

5

10

15

20

25

30

35

40

45

(b) Torque ratio of S-CVT.

Figure 3.7: Experimental results.

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0 10 20 30 40 50 60

0

10

20

30

40

50

60

70

80

90

100

Figure 3.8: Power efficiency of S-CVT.

cient of friction and the exerted normal force.

Finally, we calculate the power efficiency of the S-CVT using the obtained speed

and torque ratios (see Figure 3.8). The efficiency is almost 85% for variator an-

gles under 15, while the average efficiency beyond this angle is about 65%. The

power efficiency of the prototype S-CVT is somewhat low; this is mainly due to the

manufacturing errors including bearing friction loss, gear backlash, etc. From exper-

iments with the prototype S-CVT, we have also found that slight misalignments of

the shafts may cause bending moments in the shafts and discs, resulting in increased

bearing friction loss and slippage, although for applications accurate shaft alignment

will have to be separately addressed.

3.4.2 Strength and Life Prediction of S-CVT

Perhaps the most common form of mechanical failure in friction and traction drives

is by wear. The laws governing the overall friction and wear between two surfaces

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seem to depend primarily on the total force transmitted across the two surfaces

rather than on the local distributions of stress and strain, and one may assume that

the two bodies in contact are perfectly rigid. In studying the details of the actual

mechanism of wear and friction, however, one must take into account the extremely

small areas of actual load contact between two bodies and the elastic and plastic

deformations in these regions. Another factor which must be considered in studying

the detailed mechanism is the surface condition of the metal, since this condition

may be such that these local points of contact behave in a manner quite different

from that of the same material in bulk form. It has been generally accepted that

the addition of a reasonable tangential force to a rolling contact has no appreciable

effect on drive life. This is so only when spin is almost entirely absent.

Dawe and Lohr (1993) reported that application of a realistic tangential trac-

tion force at the contacts does not seem to cause dramatic reduction in life, and

circular contacts appear to offer the best

Design Analysis and Control

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