anova aplicado a tesis de maestria en ingenieria mecanica

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AN EXPERIMENTAL INVESTIGATION

OF THE

COBOT WHEEL CONTACT PATCH

Submitted By

Gregory W. Bachman

In Partial Fulfillment of the Requirements

for the Degree of Masters in Science in

Mechanical Engineering

Northwestern University

Department of Mechanical Engineering

December 4, 1997



ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS.......................................................................................................................... VI

ABSTRACT ...............................................................................................................................................VII

1. INTRODUCTION .....................................................................................................................................1

1.1 COBOT RESEARCH ...............................................................................................................................1

1.2 NONHOLONOMIC JOINTS.........................................................................................................................1

1.3 THE PROBLEM OF SLIDING .....................................................................................................................2

1.4 AN ELASTIC CONTACT MODEL...............................................................................................................2

1.5 CHAPTER OVERVIEWS ............................................................................................................................2

2. WHEELS....................................................................................................................................................3

2.1 WHEEL CHARACTERISTICS .....................................................................................................................3

2.1.1 Durometer ......................................................................................................................................3

2.1.2 Profile.............................................................................................................................................4

2.1.3 Core................................................................................................................................................4

2.1.4 Diameter.........................................................................................................................................42.2 URETHANE COMPOSITION ......................................................................................................................4

2.3 IN-LINE SKATING WHEEL TYPES............................................................................................................4

2.3.1 All-Around Wheels..........................................................................................................................5

2.3.2 Racing Wheels ................................................................................................................................5

2.3.3 Aggressive Wheels..........................................................................................................................5

2.3.4 Hockey Wheels................................................................................................................................5

3. WHEEL TESTING ...................................................................................................................................6

3.1 TESTING GOALS .....................................................................................................................................6

3.2 TAGUCHI METHODS ...............................................................................................................................6

3.2.1 Factorial Experiments....................................................................................................................6

3.2.2 Orthogonal Arrays .........................................................................................................................7

3.2.2.1 Full Factorial Experiment Column Allocation.........................................................................................83.2.2.2 FFE Factor Allocation.............................................................................................................................. 9

3.2.2.3 Confounding Data .................................................................................................................................. 10

3.3 PARAMETER DETERMINATION ..............................................................................................................11

3.3.1 Parameter Level Selection............................................................................................................11

3.3.2 Orthogonal Array Selection .........................................................................................................12

3.4 SETTING UP THE EXPERIMENTS ............................................................................................................12

3.4.1 The Basic Testing Apparatus........................................................................................................13

3.4.2 The Modified Apparatus for Friction Testing...............................................................................153.4.2.1 Applying the Force.................................................................................................................................15

3.4.2.2 Measuring the Force...............................................................................................................................16

3.4.3 Testing Strategy............................................................................................................................16

4. TEST RESULTS......................................................................................................................................17

4.1 METHODS OF ANALYSIS .......................................................................................................................17

4.1.1 Column Effects Method ................................................................................................................17

4.1.2 Plotting Methods ..........................................................................................................................18 4.1.2.1 Plotting Levels .......................................................................................................................................18

4.1.2.2 Plotting Interactions...............................................................................................................................19

4.1.3 Analysis of Variance (ANOVA) ....................................................................................................214.1.3.1 The Mean and Deviation........................................................................................................................ 21

4.1.3.2 Sum of Squares ......................................................................................................................................21

4.1.3.3 Variance and Degrees of Freedom......................................................................................................... 22



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4.1.4 ANOVA with Controlled Factors..................................................................................................234.1.4.1 Sum of the Squares and Degrees of Freedom for Factors ...................................................................... 24

4.1.4.2 Sum of the Squares and Degrees of Freedom for Interactions ...............................................................25

4.1.4.3 Sum of the Squares and Degrees of Freedom for the Error ....................................................................26

4.1.4.4 The Variance for the Factors, Interactions, and Error ............................................................................ 26

4.1.4.5 The F-Test.............................................................................................................................................. 27

4.1.4.6 Percent Contribution..............................................................................................................................27

4.1.4.7 ANOVA Example ..................................................................................................................................284.2 TEST RESULTS ......................................................................................................................................31

4.2.1 Wheel Compression......................................................................................................................33

4.2.2 Friction Tests................................................................................................................................35

4.2.3 Contact Patch Size........................................................................................................................38

4.2.4 Analysis Summary ........................................................................................................................40

5. CONTACT MODELS.............................................................................................................................41

5.1 HERTZ THEORY FOR CONTACT OF ELASTIC SOLIDS..............................................................................41

5.2 ELASTIC FOUNDATION MODEL .............................................................................................................42

5.3 ELASTIC MODEL ANALYSIS AND CONCLUSIONS ...................................................................................46

5.4 ELASTIC MODEL SUMMARY .................................................................................................................48

6. SUMMARY..............................................................................................................................................50

APPENDICES..............................................................................................................................................51

A. Elastic Foundation Depth Estimate ............................................................................................................................ 51

B. Parameters ..................................................................................................................................................................52

C. Standard L16 OA Used for the Experiments .............................................................................................................. 53

D. Raw Test Results - Wheel Compression, Contact Patch Size, Friction ...................................................................... 54

E. Transverse Load Data .................................................................................................................................................55

BIBLIOGRAPHY........................................................................................................................................57



v

TABLE OF TABLES

Table 3.1: L16 Standard Two-Level Orthogonal Array ........... ............ ............ ............ ............ ............ .......... 8

Table 3.2: Assignment of Factors/Interactions for Columns 9-15 for a Four Factor, Two Level L16 OA.....9

Table 3.3: L8 Standard Two Level Orthogonal Array................. ............ ............ ............ ............ ............ .......9

Table 3.4: Interaction Table for an L8 Standard Two Level Orthogonal Array ............. ............ ............. .....10

Table 3.5: Allocation of Factors/Interactions for an L8 OA with Three, Two Level Factors.................... ...10Table 3.6: The Confounding of Factors and Interactions for an L8 OA with Four Two Level Factors........11

Table 3.7: Final Allocation of Factors and Interactions for an L8 OA with Four Two Level Factors..........11

Table 3.8: Selection of Factor Levels.......... ............ ........... ........... ........... ............ ........... ........... ......... ......... 12

Table 3.9: Factors and Interactions Column Allocation ........... ............ ............ ............ ............. ............ ....... 12

Table 3.10: Testing Apparatus Loading Schemes ............ ............ ............ ............ ............ ........... .......... .......15

Table 4.1: Sample Data Set for an L8 Standard OA Experiment ........... ............ ............ ............. ............ .....17

Table 4.2: ANOVA Summary ......................................................................................................................31

Table 4.3: Initial Test Results....... ........... ........... ........... ........... ........... ........... ............ .......... .......... .......... ....32

Table 4.4: Wheel Compression ANOVA Summary............. ............. ............ ............. ............. ............ ......... 33

Table 4.5: Column Effects Analysis for Wheel Compression ............ ............. ............. ............. .............. .....34

Table 4.6: Testing Accuracy Experiment for the GT Comp 76mm, 81A Wheel Under a 20 kg Load.........34

Table 4.7: Friction Test ANOVA Summary................. ............ ............ ............ ............ ............ ........... ......... 35

Table 4.8: Lateral Load (Friction Force/Gravity) Column Effects Analysis ............. .............. .............. .......35Table 4.9: Coefficients of Static Friction................ ........... ............ ............ ........... ............ ........... .......... .......37

Table 4.10: Contact Patch Size ANOVA .....................................................................................................39

Table 4.11: Contact Patch Column Effects Analysis ............ ............. ............ ............. ............. ............ ......... 39

Table 5.1: Test Results and Parameters for Analyzing the Elasticity Model.............. ............. ............. ........47

Table 5.2: Elastic Foundation Model Load Prediction Test Results............. ............ ........... ............ ........... ..48

Table 5.3: % Error in the Predicted Semi-Axis Lengths for the Full Radius Wheel................ .............. .......48



vii

ABSTRACT

Current research at the Laboratory for Intelligent Mechanical Systems is focused on developing a new class

of passive robots called cobots, which use nonholonomic joints rather than actuated joints. The

nonholonomic joint takes advantage of the nonholonomic velocity constraint imposed by a wheel to relate

both linear and angular velocities. As a result, the performance of the cobot is in part dependent on how

well the wheels function. This paper looks at what effect varying different wheel parameters has on the

performance of the cobots. Using Taguchi methods we were able to examine how the compression of the

wheel, the contact patch size, and coefficient of friction, were affected by the diameter, durometer, and

profile of the urethane wheels used on many cobots. Additionally, by modifying Hertz theory for the

contact of elastic bodies using an elastic foundation simplification it was possible to develop an equation for

the load which was a function of measured parameters. By comparing the theoretical results with the actual

results, it was then possible to determine the validity of modeling the contact using the assumptions found in

Hertz theory.



1

1. INTRODUCTION

1.1 COBOT Research

In 1995 a new type of robot called a cobot was developed at Northwestern University [2]. A cobot is a

physically passive device which assists workers in performing a task by defining virtual surfaces within

which an operator can maneuver. Traditionally, for safety reasons, a robot operator would remain outsideof the workspace of the robot while the robot is in operation. However, in recent years more and more

applications are arising in which it would be propitious for an operator and robot to interact through a

haptic interface. Some of these applications include part installation on automotive assembly lines and

robot assisted surgery. In these situations the actuated joints used on most haptic robots are a cause for

safety concerns. Cobots differentiate themselves from other robots because they use nonholonomic joints

rather than actuated joints.

Actuated joints resist motion by applying a force to oppose that motion. An example of an actuated joint is

the human elbow. As shown by Colgate and Brown [1] passivity is important for ensuring system stability

and human operator safety. Consider the following example which you may remember from your

childhood. A friend tells you to sit at a table with your elbows on the table, your hands clasped together,

and your forearms extended at about a 45 degree angle with respect to the table, effectively making you the

system. That friend (the operator) then bets you that he can force your hands to the table. You of courseaccept the bet and struggle to prevent your hands from moving as your friend pushed down on them. Your

friend then removes his hands from yours, your fists go flying up towards your face (the system becomes

unstable), and everyone laughs as you punch yourself in the head (a loss of safety). Though childish, this

example exemplifies the safety concerns of using actuated joints for haptic interfaces, especially in cases

where large actuators (automotive assembly lines) or precise movements (robot assisted surgery) are

involved.

1.2 Nonholonomic Joints

The nonholonomic cobot joint is much safer because it does not apply a direct force itself, but rather

redirects the force that is being applied [5]. In other words it can not provide any motive force of its own,

so it will not hit you in the face. The basis mechanism behind the nonholonomic cobot joint is the wheel.Thus far there are two basic types of joints which have been developed. The first is the simple wheel. As

shown by Colgate, Peshkin, and Wannasuphoprasit [2] it is possible to develop a unicycle device with a

steerable wheel that:

1. emulates a caster and

2. enforces a virtual boundary through which the wheel can not be pushed by aligning the wheel

along the boundary when it is reached.

An extension of the unicycle cobot is the freestanding tricycle cobot developed by Wannasuphoprasit,

Colgate, and Peshkin, [8].

The other recently developed nonholonomic cobot joint for use in serial joints is called the continuously

variable transmission or CVT [5]. Where the unicycle and tricycle relate linear velocity ratios according tothe angle the wheel makes relative to the coordinate axis,

tanθ rad

y

x

v

v= , (1.1)

the CVT relates angular velocity ratios. As shown in Figure 1.1, the CVT consists of a central transmission

sphere around which two drive rollers and two steering rollers are placed.



2

Figure 1.1: CVT Schematic [5]

One of the drive rollers supplies an input angular velocity that results in output angular velocity in the other

drive roller based on the axis of rotation of the sphere. The rotation of the sphere is controlled by the two

steering rollers [5].

1.3 The Problem of Sliding

In both of these nonholonomic joints it is fundamentally important that the wheels do not slide. Take the

unicycle for example. To prevent the wheel from crossing a virtual boundary, the wheel aligns itself with

that boundary. The coulomb friction between the wheel and the surface it is riding on prevents the wheel

from sliding laterally. However, once the friction force is exceeded the wheel will slip. A similar

phenomenon occurs with the CVT, where it is possible to supply a torque to the drive roller that results in a

force that exceeds the allowable coulomb friction force causing slippage. In order to improve the

performance of cobots this paper examines how the choice of wheel parameters effects the coulomb friction

and attempts to determine design criteria based on these results.

1.4 An Elastic Contact Model

Previous work on modeling the contact patch for both the wheel and the CVT has been done by Gillespie

[3] and Moore [5]. Gillespie’s work focused on developing a model of the contact patch which assumed

rigid body contact. Moore’s work used Hertz theory for the normal contact of elastic solids to determine a

pressure distribution over the contact patch which was then used to develop an equation for the slipping

torque. This paper will attempt to look at the feasibility of modeling the contact patch using Hertz theory

by examining results obtained using an elastic foundation simplification of Hertz theory.

1.5 Chapter Overviews

The rest of this paper is structured as follows. Chapter 2 focuses on the characteristics and parameters of

in-line skating wheels being used in cobots. Chapter 3 explains the testing goals, testing methods, and

testing apparatus. This leads into Chapter 4 which deals with the analysis of the data gathered in the tests

defined in Chapter 3. Chapter 5 uses the elastic foundation model to test the validity of using Hertz theory

to model the contact. The results from Chapters 4 and 5 are then summarized in Chapter 6.



3

2. WHEELS

2.1 Wheel Characteristics

To ensure that the cobot stays within its workspace, the wheels on a tricycle cobot must be able to withstand

relatively large forces without slipping or sliding. This requires that a large friction force, Ff = µN, existbetween the wheel and riding surface. Conversely, the wheels must also be able to be rotated quickly. This

rotation requirement means that the size and pressure distribution of the contact patch plays a direct role in

determining how much torque will need to be applied to a wheel in order to rotate it. Ideally the best wheel

for the application will have a large coefficient of friction and a small contact patch. These two goals are

somewhat antithetical. Softer materials which deform more and thus have larger contact patches often have

larger coefficients of friction. Previous work in the LIMS lab has found that the best wheels on the market

for use in wheeled cobot applications are standard polyurethane in-line skating wheels. It will be these

wheels that will be examined in this paper.

Most in-line skating wheels are very similar. There are three basic components to the wheel: the shell, the

core, and the bearings. The customization of the wheel for various skating applications revolves around

changing four basic parameters of the wheel: the durometer, the diameter, the profile, and the core type.

Diameter

Core

Bearings

Urethane Shell

Profile}

Depth

Figure 2.1: Basic Components of an In-Line Skating Wheel

2.1.1 Durometer

The hardness, or resistance to permanent indentation, of urethane is measured using an instrument called a

durometer. The indenter on the durometer is pressed into the material using a rapidly applied constant load

for one second. The depth of penetration of the indenter is inversely related to the hardness of the material.

For in-line skating wheels the most commonly used hardness scale is the durometer “A” scale, though the

durometer “D” scale is also used. Both the “A” and “D” scales range from 0 to 100. A 100 score indicates

that no indentation has occurred. The “A” scale uses a blunt indenter with a 1 kg applied load and is used

on softer materials. For use on harder materials, the “D” scale uses a sharper indenter with a 5 kg applied

load. Comparing scales a 97A ≈ 60D ≈ 50R. Occasionally the Rockwell “R” scale is also used. For harder

materials the “D” and “R” scales provide greater sensitivity to differences in hardness. For example, a 97A

≈ 60D while a 95A ≈ 45D. Harder wheels deform less and general provide a smaller friction force, though

this is somewhat dependent upon the intended use of the wheel and the composition of the urethane shell.



4

2.1.2 Profile

The profile of the wheel plays an important role in determining the contact patch size of the wheel. Profiles

vary in shape from a large nearly flat radius to a small very tapered radius. Variations in profile

significantly alter the performance of the wheel. Narrower profiles generally are more maneuverable.

Figure 2.2 shows the range of profiles found on in-line skating wheels.

Large

Radius

Small, Tapered

Radius

Full

Radius

Tapered

Radius

Figure 2.2: In-Line Skating Wheel Profiles

2.1.3 Core

The core of the in-line skating wheel is usually made of nylon, but is occasionally made out of aluminum.

In order to reduce the weight of the wheel, most cores have spokes. A spokes trade off strength for weight.

The outer surface of the hub connects directly to the urethane surface of the wheel and the inner surface of

the hub houses the bearings and bearing spacer through which the axle passes. The urethane shell is

molded to the core to prevent the wheel from failing at that interface.

2.1.4 Diameter

The diameter of an in-line skating wheel is measured from the outer most diameter of the wheel. Diameters

range in value from ~45mm to ~80mm depending on the designed application of the wheel. The diameter

of the wheel greatly effects the speed and response of the wheel. Larger wheels are generally faster though

less stable and less responsive.

2.2 Urethane Composition

A fifth parameter as mentioned in the section on durometer is the urethane composition of the wheel. This

is also an important parameter, but it can not be precisely controlled by the consumer. There is no standard

measure of urethane composition within the industry and wheel manufacturers do not like to reveal their

compositions. For these reasons urethane composition is more important to the manufacturer than the

consumer.

2.3 In-Line Skating Wheel Types

Ideally one would like to be able to chose any combination of diameter, durometer, profile, and core.

Wheel manufacturers have found that certain parameter combinations are desired according to the type of

skating for which they will be used. As a result there are limitations set on the parameter combinations.

The four basic types of wheels are: all-around, aggressive, racing, and hockey. The following portions of this section summarize the general characteristics of these wheel types.



5

2.3.1 All-Around Wheels

All-around wheels are the most general purpose wheels available. They come in

average diameters, average durometer, and average profile. They provide a benchmark

with which to compare other types of wheels.

Diameter: ~ 70 - 76 mm

Durometer: ~ 78A - 88A

Profile: Full radius

2.3.2 Racing Wheels

Racing wheels are designed for speed. They are taller, thinner, and lighter than all-

around wheels. To reduce the weight of the wheel, the core is ultra light weight and

oversized. The light weight core is not intended for use under heavy loads and may

break under these conditions. The oversized core also means that the depth of the

urethane coating is substantially reduced as is the life span of the wheel.



Profile: Larger taper

2.3.3 Aggressive Wheels

Aggressive wheels are generally harder but smaller wheels. To take the abuse that

aggressive skaters impart on them, the core is usually solid, and is often made of

urethane. Aggressive skaters like to be able to do controlled slides. The urethane

composition is also usually slicker. For these reasons the coefficient of friction is also

generally lower on these wheels.

Diameter: ~48 - 67 mm


Profile: Wide radius

2.3.4 Hockey Wheels

Hockey wheels are designed to enable rapid changes of direction and good

maneuverability and the urethane composition is generally stickier. Manufactures

further breakdown hockey wheels depending on whether they are for indoor or outdoor

use. Indoor hockey wheels usually have a thinner profile and small contact patch than

their outdoor counterpart. Outdoor wheels are also usually softer to absorb the

irregularities in the skating surface and as a result have a shorter life span than indoorwheels. Both indoor and outdoor hockey wheels have high coefficients of friction.



Profile: Medium radius



6

3. WHEEL TESTING

3.1 Testing Goals

Based on the wheel type generalities presented in Section 2.2 one might conclude that the most applicable

in-line skating wheel for use on wheeled cobots is the indoor hockey wheel. Regardless of whether or not

this is the case, it is still important to optimize the performance of each wheel for every situation.Additionally, the wheel requirements for CVTs are not exactly the same as the requirements for wheeled

cobots. Knowledge of the effects of each parameter on the behavior of the wheel will enable the cobot

designer to select the appropriate wheel for a particular application. In this paper the effects of each

parameter will be examined in an attempt to better understand

1. How much force can be applied under a certain condition before the wheels slip,

2. How far the wheels compress under a certain load, and

3. How much torque is required to rotate the wheels.

Of the previously discussed wheel parameters, only diameter, durometer, and profile will effect the behavior

of the wheel in these regards. The core of the wheel is for all intensive purposes rigid and will not effect the

design of the wheel as long as it does not break. The effects of diameter, durometer, and profile on

performance, can be determined using Taguchi methods of experimentation.

3.2 Taguchi Methods

The goal of this section of Chapter 3 is to familiarize the reader with Taguchi methods and not to instruct

the reader in everything there is to know. There are many well written books focusing entirely on Taguchi

methods which can provide a reader with a thorough examination of the topic.

3.2.1 Factorial Experiments

Taguchi methods of experimentation provide techniques for organizing experiments which optimize the

number of conclusions that can drawn from a data set. These methods are used to simultaneously evaluate

the effects of multiple factors on the quality of a product [7]. Traditional, full-factorial testing says that tofully explore the effects of all factors, all possible combinations of factors must be tested. For small

experiments this works well, but as the number of variables and the number of levels of those variables

increase, the number of experiments required to examine all the possible combinations increases

exponentially:

# of experiments = ab, (3.1)

where a is the number of levels and b is the number of factors. A full-factorial experiment containing four

factors (A, B, C, and D) with each factor having two levels (1 and 2) is schematically represented in Figure

3.1. There are sixteen, 24, different combinations of factors and levels for this experiment..

A1 A2

B1 B2 B1 B2C1 D1 A1B1

C1D1A1B2C1D1

A2B1C1D1

A2B2C1D1

D2 A1B1C1D2

A1B2C1D2

A2B1C1D2

A2B2C1D2

C2 D1 A1B1C2D1

A1B2C2D1

A2B1C2D1

A2B2C2D1

D2 A1B1C2D2

A1B2C2D2

A2B1C2D2

A2B2C2D2

Figure 3.1: Full-Factorial Experiment with Four Factors and Two Levels



7

The benefit of using a full-factorial experiment is that it provides the greatest level of resolution possible in

an experiment. It enables one to evaluate the effects of individual factors as well as the effects of the

interactions of factors: AxB, AxC, AxD, BxC, BxD, CxD, AxBxC, AxBxD, AxCxD, BxCxD, and

AxBxCxD. A full-factorial experiment is said to be orthogonal because “the factors can be evaluated

independently of one another; the effect of one factor does not bother the estimation of the effect of another

factor [7].” In practice this means that each factor has the same number of occurrences at each level. Forexample, each factor in Figure 3.1 has eight occurrences at level one and eight occurrences at level two.

Orthogonality, however, is not limited to full-factorial experiments. It is possible to construct fractionally

factorial experiments (FFEs) based on full-factorial experiments which are also orthogonal and thus enable

a fair comparison of factors. Figures 3.2 and 3.3 show two examples of FFEs based on the previously

discussed four factor, two level full factorial experiment. In each case the number of trials is a fraction of

the number of trials in a full factorial experiment. They are not unique.

A1 A2

B1 B2 B1 B2

C1 D1 A1B1C1D1

A2B2C1D1

D2 A1B2

C1D2

A2B1

C1D2

C2 D1 A1B2C2D1

A2B1C2D1

D2 A1B1C2D2

A2B2C2D2

Figure 3.2: A ½FFE for a Four Factor, Two Level Experiment

A1 A2

B1 B2 B1 B2

C1 D1 A1B1C1D1

A2B2C1D1

D2

C2 D1

D2 A1B1C2D2

A2B2C2D2

Figure 3.3: A ¼ FFE for a Four Factor, Two Level Experiment

The obvious advantage to FFEs is the reduction in the number of required trials. For experiments with large

numbers of factors and levels, FFEs can be very useful as screening experiments to reduce the number of

factors to be examined in later more detailed experiments. While they can be very cost and time efficient,

FFEs do not provide the resolution of a full factorial experiment and will not be able to determine the

effects of all of the interactions.

3.2.2 Orthogonal Arrays

To make it easier for an experimenter to set-up an experiment, Taguchi developed sets of arrays calledOrthogonal Arrays (OAs). OAs make use of the orthogonal properties of full factorial and fractionally

factorial experiments in an easily to understand format. They enable an experimenter who is not well

versed in statistics to take full advantage of statistical analysis without having to know a considerable

amount about statistics. By knowing the number of factors, the number of levels or each factor, and the

resolution required an experimenter can use Taguchi’s methods to design and execute an efficient

experiment.



8

As indicated in the previous section a full-factorial experiment with four, two level factors requires sixteen

different trials to determine the effects of the factors as well as the interactions of factors. The OA used to

perform such an experiment is the L16 Standard Two-Level Orthogonal Array as shown in Table 3.1.

Column Number

Trial # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

3 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2

4 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1

5 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2

6 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1

7 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1

8 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2

9 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

10 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1

11 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1

12 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2

13 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1

14 2 2 1 1 2 2 1 2 1 1 2 2 1 1 215 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2

16 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1

Table 3.1: L16 Standard Two-Level Orthogonal Array

The structure of OAs are such that each column of the array represents a different factor or interaction of

factors and each row represents a trial to be conducted in which the factors are set at the levels indicated by

the columns. OAs are also designed to make the allocation of factors and interactions to the appropriate

columns as logical as possible.

3.2.2.1 Full Factorial Experiment Column Allocation

For a full factorial experiment the assignment of factors to columns proceeds as follows. Factors A and B

are assigned to the first two columns since there cannot be an interaction without at least two factors. The

interaction of A and B is then assigned to the next available column, column 3. Not coincidentally, the

mathematical sum of the column numbers of the contributing factors in the interaction equals the column

number of the interaction.

Factors: A and B = AxB

Columns: 1 + 2 = 3

Having exhausted all the interactions between factors A and B, factor C is assigned to column 4. This

additional factor adds three additional interactions to the experiment: AxC, BxC, and AxBxC. The

assignment of columns for these interactions follows the same mathematical pattern as before:

Factors: A and C = AxC

Columns: 1 + 4 = 5

Factors: B and C = BxC

Columns: 2 + 4 = 6

Factors: A and B and C = AxBxC

Columns: 1 + 2 + 4 = 7



9

The mathematical relationship between factors and assigned columns holds for all combinations of factors

and interactions as can be seen in the following redundant column assignment expressions for the

interaction AxBxC:

Factors & Interactions: AxB and C = AxBxC

Columns: 3 + 4 = 7

Factors & Interactions: A and BxC = AxBxCColumns: 1 + 6 = 7

Factors & Interactions: B and AxC = AxBxC

Columns: 2 + 5 = 7

The assignment of the columns with the addition of factor D proceeds in the same manner:

Column Factor/Interaction

9 AxD

10 BxD

11 AxBxD

12 CxD

13 AxCxD

14 BxCxD15 AxBxCxD

Table 3.2: Assignment of Factors/Interactions for Columns 9-15 for a Four Factor, Two Level L16 OA

3.2.2.2 FFE Factor Allocation

The experiment just discussed has the highest resolution possible, all factors and interactions are able to be

examined, but Taguchi methods also allow one to reduce the number of different trials in an experiments if

not all the interaction data is deemed important. In many cases the most complicated interactions have little

to no effect on the performance of the design. Because they are less likely to be important some

interactions are ignored when there are time and expense issues. Screening experiments to reduce the

number of factors being examined also may not require a very high resolution. In the end the resolution of

the experiment will dictate the size of the OA and the number of experiments performed. The ½ FFE

represents the first standard reduction in resolution of a full factorial experiment. The OA representing the

½ FFE for the example from Section 3.2.2.1 is the eight trial OA in Table 3.3.

Column Number

Trial # 1 2 3 4 5 6 7

1 1 1 1 1 1 1 1

2 1 1 1 2 2 2 2

3 1 2 2 1 1 2 2

4 1 2 2 2 2 1 1

5 2 1 2 1 2 1 2

6 2 1 2 2 1 2 1

7 2 2 1 1 2 2 1

8 2 2 1 2 1 1 2

Table 3.3: L8 Standard Two Level Orthogonal Array

The assignment of columns in this case is not as straight forward as it is in a full factorial experiment. As a

result, two well established resources have been created for assigning columns: interaction tables or linear

graphs. Interaction tables and linear graphs are available for most situations. Table 3.4 and Figure 3.4

show the interaction table and linear graphs for the L8 OA in Table 3.3. In the interaction table, the

intersection of the rows and columns of the table denote the columns of the OA which represents the

interaction of other columns of the OA. For example, if factor A is assigned to column 1 of the OA and



10

factor B is assigned to column 2 of the OA then the interaction of those two factors will be found in column

3 as shown by the shaded block in Table 3.4.

Column #

Column # 1 (A) 2 (B) 3 4 5 6 7

1 (A) 3 (AxB) 2 5 4 7 6

2 1 6 7 4 53 7 6 5 4

4 1 2 3

5 3 2

6 1

7

Table 3.4: Interaction Table for an L8 Standard Two Level Orthogonal Array

Linear graphs are based on the same principles as the interaction tables, however they present the data in a

slightly different format. The nodes of the linear graphs represent the columns of the OA and the lines

connecting the nodes represent the interactions of those columns. Again in Figure 3.4 the interaction of

columns 1 and 2 occurs in column 3.

1

2 4

3 5

6

7

1

23

45

6

7Figure 3.4: Two Linear Graphs for an L8 Standard Two Level Orthogonal Array

3.2.2.3 Confounding Data

To examine how the data is confounded in an FFE an L8 OA is used to examine four, two level factors.

Start by noting that the L8 OA represents a full factorial experiment for three, two level factors. For three

factors the column assignments are as would be expected (see Table 3.5).


1 A

2 B

3 AxB

4 C

5 AxC

6 BxC

7 AxBxCTable 3.5: Allocation of Factors/Interactions for an L8 OA with Three, Two Level Factors

Adding a fourth factor, D, without increasing the size of the OA automatically confounds and confuses data

regardless of the column to which the factor is assigned. Since all the factors must be assigned to different

columns, factor D must be assigned to column 3, 5, 6, or 7. Assigning factor D to column 3,5, or 6

automatically confounds it with a two factor interaction (AxB, AxC, or BxC). Assigning factor D to

column 7 confounds it with a three factor interaction. Regardless of the column to which factor D is

assigned, it will be impossible to differentiate between factor D and the interaction already assigned to that



11

column based on full factorial experiment with three factors. However, since the three factor interaction is

statistically less likely to be important, it is logical to assign factor D to is column 7. The interaction table

in Table 3.5 shows that placing factor D in column 7 results in the confounding of factors and interactions

shown in Table 3.6.


1 A & BxCxD2 B & AxCxD

3 AxB & CxD

4 C & AxBxD

5 AxC & BxD

6 BxC & AxD

7 D & AxBxC

Table 3.6: The Confounding of Factors and Interactions for an L8 OA with Four Two Level Factors

There are three things to notice about Table 3.6. First, there is no information available about the

interaction AxBxCxD. Second, the three factor interactions in columns 1, 2, 4, and 7 are confounded by the

factors already assigned to those columns. Since factors are much more likely to be important it is often

just assumed that the data in these columns relates to the factors rather than the interactions. Third, in

columns 3, 5, and 6 two factor interactions are confounded with two factor interactions such that eitherinteraction is equally likely to be the important interaction. If the data shows one of the interactions in these

columns to be important, further tests would need to be conducted to differentiate between them. The final

assignment of factors/interactions is usually assumed to be as shown in Table 3.7.


1 A

2 B

3 AxB & CxD

4 C

5 AxC & BxD

6 BxC & AxD

7 D

Table 3.7: Final Allocation of Factors and Interactions for an L8 OA with Four Two Level Factors

As shown by this example the allocation of factors and interactions to columns can be somewhat confusing,

especially as the resolution of the experiment decreases. Usually column allocation techniques follow the

previous example in which the factors are confounded with the highest level interaction possible.

Sometimes however, decisions need to be made as to factors/interactions to confound. In these cases the

allocation of factors is somewhat subjective.

3.3 Parameter Determination

For the cobot wheels three important factors have already been determined: diameter, durometer, and

profile. A fourth factor which is not a function of the wheels but is important in determining the frictional

characteristics of the wheels is the normal load. Since the frictional force is proportional to the normal load,

the effect of the load on the frictional force will provide a baseline from which to judge the importance of

the other factors and interactions.

3.3.1 Parameter Level Selection

By comparing the results of the tests on each parameter for two different levels it is possible to determine

the effect that parameter has on the system. The effects of further variations in the levels can be

extrapolated from these results. In order to better judge the relative effects of each factor the levels of the



12

factors were chosen such that each represents a similar portion of the total range over which the factor can

be varied. Vagueness in the ranges of the factors and limitations in available wheel parameter combinations

made it impossible to do this exactly. Limitations in the available wheels also meant that only a small

percentage of the total range could be covered in the tests. Ideally a larger range would have been

preferred. Table 3.8 shows the total range of values for each factor, the chosen levels for each factor, and

the percentage of the total range covered by the chosen levels.

Factor Total Range Factor Levels (1 - 2) Est. % of Total Range

Load 0kgs - 45kgs 20kgs - 25kgs 11%

Diameter 50mm - 80mm 72mm - 76mm 13%

Durometer 74A - 94A 81A - 84A 15%

Profile Large Radius - Small, Tapered

Radius

Full Radius - Tapered Radius 20%

Table 3.8: Selection of Factor Levels

3.3.2 Orthogonal Array Selection

Having defined the factors and levels of the factors, the next thing to determine in is the resolution of the

experiment. Because there are only four, two level factors it is not necessary to perform a screening

experiment. Also, the savings in time and expense in performing a lower resolution experiment do not

outweigh the advantages of performing a full resolution experiment. The likelihood of an interaction being

important mandates that a high resolution, full factorial experiment be performed. Otherwise additional

experiments would need to be performed to determine which interaction of the confounded interactions is

important. For these reasons the L16 Standard Two-Level OA shown in Table 3.1 was selected. The

allocation of factors to columns is as described in Table 3.9.

Column # Factor/Interaction

1 Diameter (mm)

2 Durometer

3 Diameter x Durometer

4 Profile

5 Diameter x Profile6 Durometer x Profile

7 Diameter x Durometer x Profile

8 Load

9 Diameter x Load

10 Durometer x Load

11 Diameter x Durometer x Load

12 Profile x Load

13 Diameter x Profile x Load

14 Durometer x Profile x Load

15 Diameter x Durometer x Profile x Load

Table 3.9: Factors and Interactions Column Allocation

3.4 Setting up the Experiments

With the factors and the levels of the factors determined it is necessary to design tests to answer the

questions posed in Section 3.1.1. It was determined that the tests on the wheels should be as simple as

possible. A trade-off was made between the accuracy of the measurements and the ease and simplicity of

the experiments. The goal of the experiments was not to have exact numbers for the limited number of

wheels that could be efficiently tested but rather to gain an understanding of how various properties of in-

line skating wheels affect performance in situations likely to arise in cobot applications. So the decision



14

C.M.

176mm

228mm

232mm

278mm

379mm

481mm

549mm

43mm

105mm70mm

Figure 3.6: Testing Apparatus Dimensions

The Figure 3.7, the equation for the sum of the moments is :

0 1 1 2 2 3 3 4 4 5 6= + + + + −F r F r F r F r W r N r * * * * * * . (3.2)

Solving for the normal force, N, yields:

( ) N F r F r F r F r W r r = + + + +1 1 2 2 3 3 4 4 5 6* * * * * . (3.3)

F1 F2 F3 F4

N

W

r4

r3

r2

r5

r1

r6

Figure 3.7: Forces Contributing to the Moment

Since all the distances from the hinge to the forces are known the loads (F1-F4) can be varied to come up

with different normal forces on the wheel. Note that W is a constant in the preceding equations. Table 3.10

shows the loading schemes used to provide different normal forces for the experiments.



15

N

(kg)

F1

(kg)

F2

(kg)

F3

(kg)

F4

(kg)

3 3 0 0 0

5 4 1 0 0

10 4 2 1 1

15 4 3 2 220 4 4 3 3

25 4 5 4 4

30 4 6 5 5

35 4 7 6 6

Table 3.10: Testing Apparatus Loading Schemes

The contact patch of the wheels is measured by placing the apparatus on a 1cm thick sheet of glass and

looking directly up from underneath. The compression of the wheel is measured at the same time.

3.4.2 The Modified Apparatus for Friction Testing

3.4.2.1 Applying the Force

In order to test how the wheel properties affect the friction force, some modifications were made to the

device. Initially, it was thought that the friction force could be determined by applying a force along the

axle. Experiments revealed that the moment caused by the 36mm to 38mm offset between the applied force

at the axle and the friction force at the contact point between the wheel and the surface caused the wheel

and the entire apparatus including the weights on the arm of the apparatus to tilt (see Figure 3.8). Neither

the U-bracket nor the wheel fixture was strong enough to keep the wheel from tilting as the axial force was

increased. Both eventually began to bend and had to be replaced. The tilt was determined to be

unacceptable and a new strategy was developed.

F

Figure 3.8: The Tilting of the Apparatus when Applying a Force Along the Axle

To remedy the situation a metal link was fixed to the end of the axle so that the applied force acts along the

same line as the friction force. This effectively removed the offending moment created in the other

configuration and eliminated the tilt of the device.



16

3.4.2.2 Measuring the Force

The final piece of the puzzle that needed to be solved was how to effectively apply a measurable force to

the apparatus. Under normal operating conditions each wheel may be subjected to loads in excess of 30kg

and the coefficient of static friction for various plastics can be somewhat greater than unity depending on

the polyurethane and the surface on which it is riding. Not enough weights were available to provide therequired force under these conditions and the largest spring scale available had a 50lb maximum. As a

result a simply pulley system was incorporated into the apparatus, halving the input force required to

determine the friction force. The force was applied manually and its magnitude was measured using a 50lb

spring scale (see Figure 3.9). Again several reasonable assumptions were made to simplify the calculations.

The wire was assumed to be inextensible, the pulley was assumed to be frictionless, the mass of the pulley

and the wires were ignored, and all forces are assumed to be parallel so that force applied to the wheel

through the pulley, F, was double the force applied to the spring scale, F/2.

FFixed

End

Spring

ScaleF/2

Figure 3.9: Pulley System for Measuring the Friction Force

For all the experiments the base of the testing apparatus was fixed to the testing surface using C-clamps to

stabilize the apparatus and prevent it from tipping over.

3.4.3 Testing Strategy

Having determined the test set up it was necessary to determine the sample size required to make the

experiment statistically valid and sufficiently sensitive. The test data is variable data rather than attribute

data, so a student’s t distribution was assumed for the data set. Based on the student’s t distribution a

confidence level for the data was obtained. The confidence level provided a statistical measure of the

certainty that differences between levels will be noticed. A standard testing guideline is to achieve 95%

confidence that a variation of one standard deviation will be noticed. Using a table from Ross [7] it was

determined that repeating each trial 3 times (48 total data points) would provide this confidence level (95%

with 1.07 standard deviations).

Additionally, the order of testing was randomized to prevent any unknown or uncontrolled factors from

influencing the experiment. The order in which the wheels were tested was randomly chosen from among

the 24 test wheels (3 wheels x 23 combinations of diameter, durometer, and profile). To save time each

wheel was then loaded at both load levels, though the order in which the loads were applied was randomly

selected. A complete randomization of all four factors, including load, would have doubled the time needed

to perform the test as there would need to have been twice as many setup changes.



17

4. TEST RESULTS

An analysis of the initial tests, the results of which are contained in Table 4.3, will provide a good idea of

the effects of the different parameters on the behavior of the wheel. There are many methods available for

analyzing the data obtained in the Taguchi method experiment. Since a goal of the analysis is to determine

if any of the factors or interactions of factors have an effect on the results of the experiments column effects

analysis, plotting methods, and analysis of variance will be used.

4.1 Methods of Analysis

For the purposes of discussing the various methods of analyzing experimental data gathered from an

orthogonal array it will be useful to generate a small sample set of data. This allows for the full exploration

of the analysis methods using fewer calculations than would be required if the actual data were to be used.

The sample data set is an L8 Standard OA with four two level factors. With four factors there exists some

confounding of interactions in columns three, five, and six. The presence of this confounding will provide

the reader a better idea of why the L16 Standard OA was chosen for the actual data. As with the actual

data, three measurements were taken for each trial in the experiment. The sample data is presented in Table

4.1.

Factors and Interactions Data

Trial A B AxBCxD

C AxCBxD

BxCAxD

D Test#1

Test#2

Test#3

1 1 1 1 1 1 1 1 23 24 27

2 1 1 1 2 2 2 2 14 13 14

3 1 2 2 1 1 2 2 15 17 17

4 1 2 2 2 2 1 1 30 29 32

5 2 1 2 1 2 1 2 21 21 20

6 2 1 2 2 1 2 1 9 10 9

7 2 2 1 1 2 2 1 13 12 13

8 2 2 1 2 1 1 2 24 25 25

Sum1 255 205 227 223 225 301 231Sum2 202 252 230 234 232 156 226

Difference -53 47 3 11 7 -145 -5

Table 4.1: Sample Data Set for an L8 Standard OA Experiment

4.1.1 Column Effects Method

The column effects method is a relatively quick and easy method of examining the results of an experiment

to find out which factors are important, the relative importance of each of these factors, and at which level

the improvement occurs [7]. The method simply involves summing the test results by level and then

comparing the magnitudes of the differences between the levels. Those factors and interactions for which

the difference between the sums is substantial are the ones which are important.

An examination of factor A, shows that the sum of the test results at level one (Sum1) is somewhat larger

than the sum at level two (Sum2). These summations were obtained by adding together all the test data for

factor A at each specified level:

Sum1 = 23+24+27+14+13+14+15+17+17+30+29+32 = 255 (4.1)

Sum2 = 21+21+20+9+10+9+13+12+13+24+25+25 = 202 (4.2)

Difference = Sum2 - Sum1 = 202 - 255 = -53. (4.3)



18

The summations for the remaining factors and interactions are contained in Table 4.2. The greater the

difference in the magnitudes of these sums, the more likely that a factor/interaction has an effect. Also, the

larger the magnitude of the difference the more important the effect. The sample data shows that interaction

BxC/AxD has the strongest effect on the experiment. (The slash between the interactions is meant to

indicate the confounding of data such that the effect of an individual interaction is indistinguishable from

the group of interactions. The effect of interaction BxC can not be distinguished from the effect of

interaction AxD.) To a lesser degree factors A and B also seem to have an effect on the experiment whilstfactors C and D and interactions AxB/CxD and AxC/BxD do not have a noticeable effect on the

experiment.

The sign (+/-) of the difference between the two levels tells what kind of effect a factor has on the

experiment. For example, if the goal of the experiment was to determine a way to reduce the output value,

the analysis suggests using level 1 of factor B and level 2 of factor A. Assume for example, factor A is a

continuous variable representing a length which can be changed from 1 to 100 meters and in the experiment

level one corresponds to a length of 75 meters and level two corresponds to a length of 50 meters. The

experiment indicates that further reducing that factor would continue to improve the desired result. A visual

representation of this same information can be obtained by plotting the results.

4.1.2 Plotting Methods

4.1.2.1 Plotting Levels

The first method of plotting is to plot the levels of the factors. This is strictly a visual interpretation of the

results obtained in the column effects analysis. When plotting all the factors by their levels an idea of the

influence of a factor can be obtained by examining the slope of the line.

It is important to note that one of the problems with all these analysis methods which compare the relative

effects of different factors, resides in accurately determining the levels of the factors. If the two levels of a

factor are not chosen well they may not adequately reflect the impact of that factor on the experiment

relative to the other factors. If the two levels of the factor are too close together they may under estimate

the factor’s influence, while if the levels are too far apart they may overestimate the factor’s influence. In a

visual representation where the levels are all viewed on the same scale, such as is the case with plottinglevels, it is easy to incorrectly estimate the effect of a factor on the results. Assuming that the levels of all

the factors have been appropriately determined, the slope of the lines shows that interaction BxC/AxD and

factors A and B are the most influential factors/interactions and the relative importance of each. For the

factors, the slope of the line also shows how to vary the factors to obtain the desired results.



19

150

170

190

210

230

250

270

290

Sum1 Sum2

A

B

AxB/CxDC

AxC/BxD

BxC/AxD

D

Figure 4.1: Plotting Interaction Levels

4.1.2.2 Plotting Interactions

A second plotting method which can be helpful for examining interactions is to plot the combinations of

levels for the interactions. In this method the average of each possible combination is plotted. When a

basic, unmodified OA is used for the experiment there will automatically be equal numbers of trials for each

combination of levels, resulting in an unbiased estimate of the effects of the combinations. The number of

possible combinations of factors and levels is equal to the product of levels of each factor:

# of combinations = (levels in factor 1)*(levels in factor 2)*…*(levels in factor n) (4.4)

where n is the number of factors in the interaction. The AxB interaction for example has the following four

combinations A1B1, A2B2, A1B2, A2B1. From the example the values of these combinations are 19.17,

23.33, 16.67, and 19.00 respectively.

This information can be plotted in two different ways to show the same information. The difference is in

the semantics. Either one plots the effect of factor B on factor A as shown in the first of the two graphs in

Figure 4.2 or one plots the effect of factor A on factor B as shown in the second of the two graphs in Figure

4.2. In either case the results will be the same. If the two lines are nearly parallel as they are in Figure 4.2

then there is little or no interaction between the factors. If the two lines cross the interaction between the

two factors is important.

10

15

20

25

B1 B2

A1

A2

10

15

20

25

A1 A2

B1

B2

Figure 4.2: Plots for the Interaction AxB



20

Note that in this example interaction AxB is confounded with interaction CxD. Plotting interaction CxD

gives similar results.

15

20

C1 C2

D1

D2

Figure 4.3: Interaction Plot for CxD

The interaction plot for BxC/AxD provides an idea of what a plot looks like for a meaningful interaction.

Regardless of how the interactions are plotted, the lines for this interaction cross. However, once againbecause of the confounding of data it is not possible to distinguish between interactions BxC and AxD. As

a result, further experiments need to be conducted to determine which of the interactions is important.

0

5

10

15

20

25

30

B1 B2

C1

C2

0

5

10

15

20

25

30

C1 C2

B1

B2

Figure 4.4: Interaction Plots for BxC

0

5

10

15

20

25

30

A1 A2

D1

D2

0

5

10

15

20

25

30

D1 D2

A1

A2

Figure 4.5: Interaction Plots for AxD



21

4.1.3 Analysis of Variance (ANOVA)

4.1.3.1 The Mean and Deviation

Analysis of variance (ANOVA) is “a statistically based, objective decision-making tool for detecting any

differences in average performance of groups of items tested [7].” One of the most basic statisticalcomputations that can be applied to a set of data is the computation of the mean. The mean , x , of a data set

is the average value of that data set. It is defined as:

x N

xi

i

N

==∑1

1

(4.5)

where

N = number of data points

xi = the value of data point i.

To get an idea of how much variation exists in a data set it is often helpful to separate each data point into

two components, the mean and the deviation from the mean.

Data Point = Mean + Deviation (4.6)

For example, if a data set has a mean of 25 a data point of 30 could be broken down as follows:

30 = 25 + 5. (4.7)

4.1.3.2 Sum of Squares

It can also be helpful to know the total amount of variation that exits in a data set. The sum of the set of the

values in the deviation term of equation 4.6 always equals zero, which does not make it very helpful

measure of the total variation in the system. A better measure of variation can be obtained by summing thesquares of the data. Mathematically one can prove that the total sum of the squares of the data points, SSt,

equals the sum of the squares of the means, SSm, plus the sum of the squares of the error or deviation, SSd.

SS x SS SS t i

i

N

m d = = +=∑ 2

1

(4.8)

Using equation 4.5 the sum of the squares of the mean can be rewritten as:

SS N x N N

X X

N m = = =( ) ( )

2 22

1(4.9)

where

X xi

i

N

==∑

1

. (4.10)

Solving equation 4.8 for the sum of the squares of the deviation yields

SS SS SS d t m= − (4.11)



22

SS x X

N d i

i

N

= −=∑ 2

2

1

(4.12)

SS x X

N

X

N d i

i

N

= − +=∑ 2

2 2

1

2 (4.13)

( )SS x x X Nxd i

i

N

= − +=∑ 2 2

1

2 (4.14)

SS x x x xd i i

i

N

i

N

i

N

= − += ==∑ ∑∑ 2

1

2

11

2 (4.15)

SS x x x xd i i

i i

N

= − +=

∑ [ ( ) ]2 22 (4.16)

SS x xd i

i i

N

= −=∑ ( )

2(4.17)

where x xi − defines the deviation of a data point from the mean.

4.1.3.3 Variance and Degrees of Freedom

The sample variance, S, is defined as:

S

x x

N

i

i

N

=

−

−=∑ ( )2

1

1 (4.18)

where, N-1, is a measure of the degrees of freedom, v, of the measurand. The degrees of freedom provide a

measure of how restricted a statistic is from taking any value. The total degrees of freedom in a data set

equals the number of data points, N in that data set. A loss of one degree of freedom is associated with each

estimate of the data. In this case the only estimate on the data used in determining the sample variance is

the mean, x . Therefore, there are N - 1 degrees of freedom associated with the sample variance.

The degrees of freedom for a system can be broken down in the same way the data is broke down,

according to the total, mean, and deviation from the mean:

vt = vm + ve. (4.19)

As previously discussed, one degree of freedom is associated with the mean, vm. Therefore, there are N-1

degrees of freedom associated with the deviation, ve. The same result can be obtained by remembering that

a degree of freedom is associated with each independent comparison that can be drawn from the data. As

the following example shows, N-1 independent comparisons can be made for N data points. Consider the

ages of Martha, Bill, and Mike (N = 3). If Martha is four years younger than Bill (comparison 1) and Bill

is thirty-two years older than Mike (comparison 2) then Martha is necessarily twenty-eight years older than

Mike (comparison 3). Hence, only two (N-1) independent comparisons that can be made from the data [6].



23

Having defined degrees of freedom associated with the deviation, ve, it is now possible to write an equation

for the variance of the deviation, Sd, in terms of the sum of the squares of the deviation and the degrees of

freedom associated with the error

S SS

vd

d

e

= . (4.20)

Similarly, the variance of the mean can be written in terms of the sum of the squares of the mean and the

degrees of freedom associated with the mean, which is always one

S SS

m

m=1

. (4.21)

The variance of the deviation in equation 4.20 gives a good idea of the variation contained in the system

since it is identical to the sample variance as defined in equation 4.18. By adding controlled factors to the

system it is possible to further breakdown the variance of the deviation into components to obtain a better

idea of what is causing a variation in the data. This is traditionally associated with ANOVA.

4.1.4 ANOVA with Controlled Factors

Taguchi methods involve examining the effects of controlled factors on a product or process. As a result,

the total variation can be decomposed into:

1. The variation due to each factor

2. The variation due to the interactions of factors

3. The variation due to error

4. The variation due to the mean.

This provide a way of allocating the variation otherwise associated with the deviation from the mean to

factors and interactions. For example, the sum of the squares of the deviation for two controlled factors is

decomposed as follows:

SSd = SSA + SSB + SSAxB + SSe (4.22)

where e is the error which hasn’t been allocated to any of the factors or the interaction. Since the variation

of the mean does not affect the variation of the factors, interactions, or error the analysis can be simplified

by excluding the mean:

SS SS x x x X

N t d i

i i

N

i

i

N

= = − = −= =∑ ∑( )2 2

2

1

. (4.23)

The equation for the total sum of the squares would be

SS SS SS SS t factors interactio e= + +∑ ∑ ns . (4.24)

For a two factor experiment equation 4.24 equals:

SS SS +SS +SS +SS t A B AxB e= . (4.25)



24

Since the mean is being excluded from the analysis, the degrees of freedom of the mean must also be

subtracted from the equation for the degrees of freedom:

v v N v v vt d factors interactio e= = − = + +∑ ∑1 ns (4.26)

which for the example yields

v v +v +v +vt A B AxB e= (4.27)

This makes sense if one recalls from equation 4.23 that SSt = SSd. The total degrees of freedom in the

system now equals the degrees of freedom associated with the deviation from the mean, N-1.

4.1.4.1 Sum of the Squares and Degrees of Freedom for Factors

Previously the deviation of data points from the mean was determined through equation 4.17. Using

Taguchi methods data points can be categorized by factor as well as the level of the factor/interaction. By

finding the average value for every factor and level combination and comparing that to the mean of the

entire data set it is possible to determine a factor’s influence on the variation. Calculating the sum of the

squares for a factor, such as A, is similar to calculating the sum of the squares of the deviation with theindividual data points in the SSd calculation replaced by the average value of all data points at that level.

SS n A x n A x n A x A A A A k K = − + − + + −

1 212

22 2

( ) ( ) ... ( ) (4.28)

SS n A x A A i

i

k

i= −

=∑ ( )

2

1

(4.29)

where,

Ai = average of all observations of factor A at level i

n Ak = total number of observations of factor A at level i

k

= number of levels of factor A.

By noting that

X x n Ai

i

N

A i

i i

k

i

A

= == −∑ ∑

1

(4.30)

N n A

i i

k

i

A

=−∑ (4.31)

x X

N

= (4.32)

and expanding the square in equation 4.29, the sum of the squares of a factor can be written as

SS n A A x x A A i i

i

k

i

a

= − +=∑ ( )2 2

1

2 (4.33)



25

SS n A x n A x n A A i

i

k

A i

i

k

A

i

k

i

a

i

a

i

a

= − += = =∑ ∑ ∑2

1 1

2

1

2 (4.34)

SS n A xX NX

N A A i

i

k

i

a

= − +

=

∑ 2

1

2

22 (4.35)

SS n A X

N

X

N A A i

i

k

i

a

= − +=∑ 2

1

2 2

2 (4.36)

SS n A

n

X

N A A

i

Ai

k

i

i

a

= −=∑

2

21

2

(4.37)

SS A

n

X

N A

i

Ai

k

i

a

= −=∑

2

1

2

. (4.38)

Equation 4.38 becomes increasingly easier to use than equation 4.29 as the number of levels of a factor

increase.

The degrees of freedom associated with each factor is determined in a similar manner as before. The

number of independent comparison that can be made between data points at a specified level is equal to the

number of data points at that level minus one. For example,

vA = k A - 1. (4.39)

4.1.4.2 Sum of the Squares and Degrees of Freedom for Interactions

Having calculated the sum of the squares for the individual factors it is now necessary to calculate the sum

of the squares for the interactions, AxB for example. This requires a knowledge of all the interactioncombinations that exist. Two levels for each of factors A and B results in four possible interaction

combinations; A1B1, A1B2, A2B1, and A2B2. As done previously with the factors, the sum of the squares for

an interaction, SS α , is found by taking the square of the difference between the average of each combination

and the mean:

SS n

X

N

i

i

k

i

α α

α α

= −=∑

2

1

2

(4.40)

where the interaction, α , has k α interaction combinations. This equation however, includes the effects of

the main factors as well as of any lower order interactions. AxB, AxC, and BxC are lower order

interactions of AxBxC. Therefore, to find the effect of an interaction the effects of its constituent factors

and lower order interactions must be subtracted out of the answer:

SS n

X

N SS SS

i

i

k

i

α α

α α

= − − −=∑ ∑ ∑

2

1

2

constituent factors lower order interactions . (4.41)



26

When determining the first order interaction for two factors with the same number of trials for each possible

combination of levels equation 4.41 reduces to the following:

[ ]

SS AxB AxB

N AxB( )

( ) ( )=

−1 2

2

(4.42)

where,

( ) ( ) ( ) AxB A xB A xB1 1 1 2 2= +∑ ∑ (4.43)

( ) ( ) ( ) AxB A xB A xB2 1 2 2 1= +∑ ∑ . (4.44)

Equations 4.42-4.44 can be very helpful when using ANOVA on a experiment designed using orthogonal

arrays. The column in an OA that represents the interaction of two, two level factors automatically allocates

level one status to the combinations of factors in which the factors are both at the same level, such as is the

case in (AxB)1. A level two designation is given to those cases where the factors are at different levels,

(AxB)2. By definition an orthogonal array has en equal number of trials at each level.

The degrees of freedom for an interaction is found by multiplying together the degrees of freedom of the

constituent factors.

v v v v lα = ( )( )... ( )1 2 (4.45)

where there are l factors in the interaction.

4.1.4.3 Sum of the Squares and Degrees of Freedom for the Error

After determining the sum of the squares and the degrees of freedom associated with all the factors and

interactions it is easy to determine the sum of the squares and degrees of freedom of the error by

rearranging equations 4.24 and 4.26:

SS SS SS SS e t factors= − −∑ ∑ interactions (4.46)

and

v N v ve factors= − − − ∑∑1 interactions . (4.47)

4.1.4.4 The Variance for the Factors, Interactions, and Error

The variance for each factor and interaction is obtained by dividing its sums of the squares by its degrees of

freedom:

V SS

v= . (4.48)



27

4.1.4.5 The F-Test

The importance of the variance calculations still needs to be considered. One way of measuring the

importance of a factor/interaction variance is by comparing it to the error variance. This can be done using

the F-test. The F-test, named after British statistician Sir Ronald Fisher, is a ratio of variances:

F V

V

factor

e

= / interaction . (4.49)

The F-test predicts with a certain confidence level whether or not the two variances can be considered

equal. If the two variances are not equal then it can be assumed that the variance of the factor/interaction is

derived from more than one population and the factor/interaction is important. By using the degrees of

freedom associated with each variance estimate it is possible to look up in tables the value of F necessary to

determine at a specified confidence level that two variances are different.

4.1.4.6 Percent Contribution

After using the F-test to determine which factors and interactions are important, the relative importance of

each factor and interaction needs to be determined. This can be done by determining the percent

contribution of a specific factor/interaction to the total variation. It is important to note that previously

determined variances, V , actually contain some amount of variation that is actually due to the error. As a

result, the actual variance, V’, in a factor/interaction is

V V V e' = − . (4.50)

Since the degrees of freedom associated with each variance is known equation 4.50 can be rewritten in

terms of sums of squares:

SS

v

SS

v

V e'

= − . (4.51)

Solving this equation for SS’ yields

SS SS vV e' = − . (4.52)

The percentage of the total sum of squares represented by SS’ provides a good measure of a

factor/interaction ’s importance:

P SS

SS t %

'*=

100 . (4.53)

Since a portion of the effect of the error within each factor/interaction has been removed, the total percentcontribution will be less than 100% unless the portion of the error which has been removed from the factors

and interactions is added to the error. After determining the percent contribution of all the factors and

interactions the remaining unused percentage is the percent contribution of the error.

The percent contribution of the error provides a good idea as to whether or not all the important factors

have been identified. Convention suggests that if the percent contribution due to the error is less than 15%

it can be reasonably assumed that all important factors have been covered. On the other hand if the percent

contribution due to the error is greater than 50%, it can be reasonably assumed that not all the important



28

factors have been addressed, the conditions under which the experiment were run were not properly

controlled, or excessive measurement error existed.

4.1.4.7 ANOVA Example

For the purpose of example, an analysis of variance will be done to the sample test data which has served as

an example throughout Chapter 4. For convenience the test data from Table 4.1 has been reproduced again.

Factors and Interactions Data

Trial A B AxBCxD

C AxCBxD

BxCAxD

D Test#1

Test#2

Test#3

1 1 1 1 1 1 1 1 23 24 27

2 1 1 1 2 2 2 2 14 13 14

3 1 2 2 1 1 2 2 15 17 17

4 1 2 2 2 2 1 1 30 29 32

5 2 1 2 1 2 1 2 21 21 20

6 2 1 2 2 1 2 1 9 10 9

7 2 2 1 1 2 2 1 13 12 13

8 2 2 1 2 1 1 2 24 25 25Sum1 255 205 227 223 225 301 231

Sum2 202 252 230 234 232 156 226

Difference -53 47 3 11 7 -145 -5

Table 4.1: Example Data set for an L8 Standard OA Experiment

First, the relevant sums of squares are determined: SS t , SS A, SS B, SS C , SS D, SS AxB, SS AxC , and SS BxC . Because

of the confounding of columns the sums of squares for the various confounded columns will be the same.

For example, SS AxB = SS CxD. It does not matter which sum of squares is calculated, though both may be

calculated as a check.

First, the total sum of the squares, SS t , is determined,

X xi

i

N

= ==∑

1

457 (4.54)

N n A

i i

k

i

A

= =−

∑ 24 (4.55)

x X

N = (4.56)

SS SS x x x X

N

t d i

i i

N

i

i

N

= = − = − = − == =

∑ ∑( ) . .2 2

2

1

9815 8702 04 1112 96 . (4.57)

Next the sum of the squares of the factors are determined:

SS A

n

X

N A

i

Ai

k

i

a

= − = − ==∑

2

1

2

8819 08 8702 04 117 04. . . . (4.58)



29

SS B

n

X

N B

i

Bi

k

i

B

= − ==∑

2

1

2

8794.08 -8702.04 = 92.04 (4.59)

SS C

n

X

N C

i

C i

k

i

C

= − = − =

=

∑2

1

2

8707 08 8702 04 5 04. . . (4.60)

SS D

n

X

N D

i

Di

k

i

D

= − = − ==∑

2

1

2

8703 08 8702 04 1 04. . . (4.61)

Since all the sum of squares of the factors are now known, the sums of squares of the interactions can be

determined. Confounding of data makes it impossible to determine anything about the lower order

interactions. However, since lower order interactions do not usually have much of an effect this may not be

important. The sums of squares for the first order interaction AxB is calculated as follows:

( ) AxB 1 23 24 27 14 13 14 13 12 13 24 25 25 227= + + + + + + + + + + + = (4.62)

( ) AxB 2 15 17 17 30 29 32 21 21 20 9 10 9 230= + + + + + + + + + + + = (4.63)

[ ]SS

AxB AxB

N AxB( )

( ) ( ) ( ).=

−=

−=1 2

2 2227 230

240 375 . (4.64)

The remaining interactions are calculated in the same manner,

[ ]SS

AxC AxC

N AxC ( )

( ) ( ) ( ).=

−=

−=1 2

2 2225 232

242 04 (4.65)

[ ]SS

BxC BxC

N BxC ( )

( ) ( ) ( )

.=

−

=

−

=1 2

22

301 156

24 876 04 . (4.66)

Finally, the sum of squares of the error is calculated,

SS SS SS SS e t factors= − −∑ ∑ interactions (4.46)

SS e = − − − − − − − =1112 96 117 04 92 04 504 104 0 38 2 04 876 04 19 34. . . . . . . . . . (4.67)

The next step in ANOVA is to determine the degrees of freedom associated with each sum of squares:

vt = N-1 = 24 - 1 = 23 (4.68)

v A = n A - 1 = 1 (4.69)

v B = n B - 1 = 1 (4.70)

vC = nC - 1 = 1 (4.71)

v D = n D - 1 = 1 (4.72)

v(AxB) = (v A)(v B) = (1)(1) = 1 (4.73)



30

v(AxC) = (v A)(vC ) = (1)(1) = 1 (4.74)

v(BxC) = (v B)(vC ) = (1)(1) = 1 (4.75)

ve = vt - v A - v B - vC - v D - v(AxB) - v(AxC) - v(BxC) = 23 - 1 - 1 - 1 - 1 - 1 - 1 - 1 = 16. (4.76)

Since all the factors have two levels all of the factors/interactions have only one degree of freedom, making

the variance calculations easy. For all but the error, the variance is equal to the sum of the squares:

V SS

SS factor n

factor n

factor n /

/

/ interactio

interactio

interactio= =1

. (4.77)

As a result,

V A = 117 04. , V B = 9204. , V C = 504. , V D = 104. , (4.78)

V AxB( ) .= 038, V AxC ( ) .= 2 04 , and V BxC ( ) .= 867 04

The variance of the error is,

V SS

ve

e

e

= = =1934

16120

.. . (4.79)

From here it is easy to determine the F ratio for all the factors/interactions using equation 4.49,

F V

V

factor n

e

= / interactio. (4.49)

This yields

F A = 9886. , F B = 7617. , F C = 417. , F D = 086. (4.80)

F AxB( ) .= 031, F AxC ( ) .= 168 , and F BxC ( ) = 725 .

Now the experimental values of F can be compared with the tabulated values of F for a one degree of

freedom variance in the numerator and a sixteen degree of freedom variance in the denominator. For a 99%

confidence level the F value from the table is 8.53. For 95% and 90% confidence levels the respective F

values are 4.49 and 3.05. Comparing these values with the calculated values shows with 99% confidence

that factors A and B and interaction BxC/AxD are important. At a 90% confidence level factor C also

appears to be important

The relative importance of each factor/interaction is determined by finding the percent contribution of each

factor and interaction. For factor A,

SS SS v V A A A e' . ( )( . ) .= − = − =117 04 1 120 11584 (4.81)

PSS

SS A

A

t

=

=

='

*.

.* .100

11584

1112 96100 10 41% . (4.82)

The percent contribution for the rest of the factors and interactions are determined the same way. It should

be noted that the sum of the squares for factors D and interaction AxB are significantly small and can not be



31

distinguished from the error. The portion of the sum of the squares allocated to the error is larger that the

sum of the squares. For factor D,

SS SS v V D D D e' . ( )( . ) .= − = − = −086 1 120 017 . (4.83)

Consequently, these factors have zero percent contribution and can lumped together with the error. After

determining the percent contribution of all the factors and interactions the percent contribution of the erroris determined,

P P P P P P P Pe A B C D AxB AxC BxC = − − − − − − − =100% 2 41%( ) ( ) ( ) . . (4.84)

The ANOVA analysis for this example is summarized in Table 4.2 It agrees with both the column effects

method of analysis and the plotting method of analysis in identifying factors A, B, and C and interaction

BxC/AxD as important. The advantage of using ANOVA is that ANOVA provides a statistically based

reason for drawing these conclusions. The F ratio shows that the impact of factor C is not as certain as the

other factors and interactions. In conjunction with the percent contribution portion of the analysis, which

shows only a small contribution for factor C, the decision might be made to ignore factor C. The percent

contribution column also show that the interaction BxC/AxD at 79% is by far the most significant factor or

interaction. Yet, due to the confounding of the experiment it can not be determined if BxC or AxD is the

important interaction. Another experiment should be run to make that determination. Lastly the low

percent contribution for the error, 2.41% show that the experiment was well designed and conducted and

that no important factors were left out.

Axial Deflection

SS D. of F. Variance F Ratio SS' P

T 1112.96 23

A 117.04 1 117.04 96.86 99% 115.83 10.41%

B 92.04 1 92.04 76.17 99% 90.83 8.16%

C 5.04 1 5.04 4.17 90% 3.83 0.34%

D 1.04 1 1.04 0.86 -0.17 0.00%

AxB/CxD 0.38 1 0.38 0.31 -0.83 0.00%

AxC/BxD 2.04 1 2.04 1.68 0.83 0.07%

BxC/AxD 876.04 1 876.04 725.00 99% 874.83 78.60%

E 19.34 16 1.20 2.41%

Table 4.2: ANOVA Summary

4.2 Test Results

The initial test results on wheel compression, contact patch size, and friction levels are shown in Table 4.3.

The data on wheel compression and contact patch size used the same experimental setup and thus were

taken at the same time. The friction force data was gathered under the same conditions but in a second

experiment. The primary method of analysis was ANOVA, since ANOVA provides a means of determining

which factors are important and the relevant importance of each factor. Column effects analysis and

plotting methods were used to provide details as to the specific effect of an interaction. The integrated useof all these analysis methods will become more apparent as the analysis of the data progresses.



3 2

W h e e l C o m p r e s s i o n ( m m )

C o n t a c t P a t c h S i z e ( m m

2 )

T r a n s v e r s e L o a d o n G l a s s ( k g )

T r i a l

1

2

3

A v e r a g e

1

2

3

A v e r a g e

1

2

3

A v e r a g e

1

1 . 1

9

1 . 0

7

1 . 2

3

1 . 1

6

1 4 3 .

7 1

1 3 5 . 0

0

1 3 1 .

0 3

1 3 6 . 5

8

2 6 . 4

0

2 2 .

7 0

2 6 . 4

0

2 5 . 1 7

2

1 . 5

8

1 . 4

7

1 . 4

5

1 . 5

0

1 6 3 .

1 0

1 5 0 . 3

1

1 5 0 .

3 5

1 5 4 . 5

9

3 2 . 7

0

3 0 .

9 0

3 0 . 0

0

3 1 . 2 0

3

1 . 4

1 . 3

5

1 . 3

8

1 . 3

8

1 2 1 .

5 5

1 2 2 . 4

7

1 2 6 .

0 4

1 2 3 . 3

5

2 2 . 7

0

2 0 .

0 0

2 1 . 8

0

2 1 . 5 0

4

1 . 7

2

1 .

6

1 . 6

8

1 . 6

7

1 4 6 .

0 0

1 4 2 . 8

7

1 4 2 .

6 8

1 4 3 . 8

5

2 3 . 6

0

2 1 .

8 0

2 4 . 5

0

2 3 . 3 0

5

1 . 4

2

1 . 0

3

0 . 7

8

1 . 0

8

1 2 4 .

6 5

1 2 0 . 0

5

1 1 8 .

5 6

1 2 1 . 0

9

2 5 . 5

0

2 5 .

5 0

2 3 . 6

0

2 4 . 8 7

6

1 . 5

4

1 . 1

9

1 . 0

6

1 . 2

6

1 3 4 .

7 2

1 4 1 . 6

9

1 3 8 .

2 5

1 3 8 . 2

2

3 0 . 0

0

2 9 .

1 0

2 9 . 1

0

2 9 . 4 0

7

1 . 5

3

1 . 1

7

1 . 1

6

1 . 2

9

1 1 3 .

9 8

1 0 7 . 4

4

1 1 5 .

3 8

1 1 2 . 2

7

2 0 . 0

0

1 9 .

1 0

2 0 . 0

0

1 9 . 7 0

8

1 . 8

7

1 . 3

2

1 . 3

1 . 5

0

1 2 7 .

6 8

1 3 7 . 9

3

1 3 9 .

5 7

1 3 5 . 0

6

2 4 . 5

0

2 3 .

6 0

2 2 . 7

0

2 3 . 6 0

9

1 . 2

8

1 . 1

9

1 . 0

4

1 . 1

7

1 3 7 .

9 7

1 4 0 . 6

3

1 3 7 .

0 2

1 3 8 . 5

4

2 4 . 5

0

2 6 .

4 0

2 7 . 3

0

2 6 . 0 7

1 0

1 . 5

4

1 .

6

1 . 4

8

1 . 5

4

1 6 5 .

0 2

1 5 9 . 7

4

1 6 0 .

6 7

1 6 1 . 8

1

3 0 . 0

0

3 3 .

6 0

3 3 . 6

0

3 2 . 4 0

1 1

1 . 2

0 . 9

5

1 . 2

3

1 . 1

3

1 3 1 .

7 5

1 3 3 . 8

0

1 2 4 .

4 0

1 2 9 . 9

8

2 1 . 8

0

2 2 .

7 0

2 5 . 5

0

2 3 . 3 3

1 2

1 . 4

8

1 . 3

3

1 . 5

9

1 . 4

7

1 5 2 .

9 2

1 4 4 . 7

1

1 4 6 .

2 6

1 4 7 . 9

6

2 9 . 1

0

2 7 .

3 0

2 8 . 2

0

2 8 . 2 0

1 3

0 . 7

8

1 . 0

7

1 . 0

3

0 . 9

6

1 2 9 .

0 0

1 3 2 . 6

2

1 3 2 .

0 0

1 3 1 . 2

1

2 6 . 4

0

2 6 .

4 0

2 5 . 5

0

2 6 . 1 0

1 4

1 . 2

1

1 . 1

8

1 . 1

1

1 . 1

7

1 5 5 .

4 3

1 5 0 . 9

3

1 5 5 .

8 2

1 5 4 . 0

6

3 1 . 8

0

3 4 .

5 0

2 8 . 2

0

3 1 . 5 0

1 5

1 . 0

5

1 . 3

3

1 . 3

1 . 2

3

1 2 7 .

0 0

1 1 8 . 6

8

1 2 0 .

2 2

1 2 1 . 9

7

2 4 . 5

0

2 1 .

8 0

2 2 . 7

0

2 3 . 0 0

1 6

1 . 5

1

1 .

7

1 . 6

1

1 . 6

1

1 3 9 .

7 6

1 3 7 . 8

7

1 4 0 .

6 0

1 3 9 . 4

1

2 8 . 2

0

2 5 .

5 0

2 7 . 3

0

2 7 . 0 0

T a b l e 4 . 3 : I n i t i a l T e s t R e s u l t s



33

In order to keep with the convention used in the examples each factor was accorded a letter:

factor A = diameter,

factor B = durometer,

factor C = profile, and

factor D = load

and the levels for each factor were assigned a number, 1 or 2. Table 3.8 in section 3.3.1 shows how the

levels were allocated. These number and letter designations condensed the analysis tables by enabling

factors, interactions, and levels to be abbreviated.

4.2.1 Wheel Compression

Since the wheel compression data was the easiest data to measure it was an obvious place atwhich to start the analysis. The ANOVA analysis for the wheel compression is summarized inTable 4.4.


T 2.8463 47

A 0.0602 1 0.0602 2.1412 0.0321 1.13%

B 0.1610 1 0.1610 5.7260 95% 0.1329 4.67%

C 0.3745 1 0.3745 13.3197 99% 0.3464 12.17%

D 1.0092 1 1.0092 35.8906 99% 0.9811 34.47%

AxB 0.0108 1 0.0108 0.3841 -0.0173 0.00%

AxC 0.0102 1 0.0102 0.3630 -0.0179 0.00%

AxD 0.0140 1 0.0140 0.4982 -0.0141 0.00%

BxC 0.1474 1 0.1474 5.2424 95% 0.1193 4.19%

BxD 0.0234 1 0.0234 0.8325 -0.0047 0.00%

CxD 0.0027 1 0.0027 0.0960 -0.0254 0.00%

AxBxC 0.1083 1 0.1083 3.8515 90% 0.0802 2.82%AxBxD 0.0021 1 0.0021 0.0759 -0.0260 0.00%

AxCxD 0.0052 1 0.0052 0.1852 -0.0229 0.00%

BxCxD 0.0140 1 0.0140 0.4982 -0.0141 0.00%

AxBxCxD 0.0033 1 0.0033 0.1185 -0.0248 0.00%

E 0.8998 32 0.0281 40.56%

Table 4.4: Wheel Compression ANOVA Summary

The analysis of the F Ratio and Percent contribution shows that the most important factors affecting the

wheel compression are load followed by profile. Both factors have 99% confidence ratings and a percent

contribution of an order of magnitude greater that any other factor except the error. Durometer and the

interaction of durometer and profile both have 95% confidence ratings which suggests that they have an

effect on the wheel compression. Their percentage contributions in the 4% range suggest, however, that

these effects are small.

The actual effects of the important factors/interactions were determined from the column effects analysis in

Table 4.5 and the interaction plot in Figure 4.6. According to Table 4.5:

1. Increasing the load increases the compression,

2. Decreasing the radius of the profile increases the compression, and

3. Decreasing the durometer increases the compression.



34

Factor Level 2 - Level 1(mm)

D 0.29

C 0.18

A -0.07

B -0.12

Table 4.5: Column Effects Analysis for Wheel Compression

The interaction plot in Figure 4.6 reveals that:

4. Decreasing the durometer and the radius of the profile in combination increases the

compression.

1

1.5

B1 B2

C1

C2

Figure 4.6: Interaction Plot for Durometer and Profile

Of the four conclusions which were drawn from the analysis of the data it should again be noted that the

first two will most likely have the largest effect on the wheel compression since they had the largest percent

contributions.

Also, it should be noted that percent contribution of the error is fairly large. This suggests that perhaps

either an important factor has been excluded from the analysis or there is some problem with measurement

strategy. Tests on the measurement strategy revealed that this large error may have been caused by

limitations in the measuring system. The test involved taking five measurements of the compressed wheel

height for the tapered radius, 76mm, 81A wheel under a 20 kg load. The wheel compression is equal to the

uncompressed wheel height minus the compressed wheel height, where the uncompressed wheel height is a

constant. Prior to each measurement of the wheel the calipers used to take the measurements were zeroed at

a different unknown value by a third party, enabling unbiased measurements to be taken to determine the

testing accuracy. Table 4.6 contains the results of these tests. Considering that the for the entire experiment

the range of values varied by only 1.09mm and the standard deviation for the testing accuracy experiment is

0.03mm it seems likely that the large error contribution is a result of a lack of measurement precision.

Compressed Wheel Height (mm)

Test # Reading Zero Value Actual Value

1 3.89 21.98 25.872 -6.63 32.46 25.83

3 6.09 19.78 25.87

4 13.19 12.66 25.85

5 -14.91 40.71 25.8

Table 4.6: Testing Accuracy Experiment for the GT Comp 76mm, 81A Wheel Under a 20 kg Load



35

4.2.2 Friction Tests

The friction tests provide an idea of how large a friction force can be applied to the wheels before they slip

by measuring the lateral load applied to the wheel. For practical purposes failure occurs at the first instance

slip in the wheel is noticed. However, tests revealed that the wheels had a tendency to slip slightly prior to

total failure, total failure being defined as the applied lateral load under which the wheel continues to slide.

Friction in the pulley system also caused an occasional jerk in the spring scale, which was difficult todifferentiate from the initial wheel slippage. Consequently, for testing purposes the recorded values for the

lateral loads were the values where continuous slip was observed. To ensure that each wheel was tested

under identical conditions and to minimize slip both the wheels and the contact surface were cleaned with

acetone prior to each test. The ANOVA for the initial tests on glass are recorded in Table 4.7.


T 695.65917 47

A 66.740833 1 66.7408 28.3075 99% 64.38 9.25%

B 6.75 1 6.7500 2.8629 90% 4.39 0.63%

C 257.61333 1 257.6133 109.2643 99% 255.26 36.69%

D 254.84083 1 254.8408 108.0884 99% 252.48 36.29%

AxB 0.27 1 0.2700 0.1145 -2.09 0.00%AxC 12 1 12.0000 5.0897 95% 9.64 1.39%

AxD 3.5208333 1 3.5208 1.4933 1.16 0.17%

BxC 0.0008333 1 0.0008 0.0004 -2.36 0.00%

BxD 0.27 1 0.2700 0.1145 -2.09 0.00%

CxD 11.213333 1 11.2133 4.7560 95% 8.86 1.27%

AxBxC 0.3008333 1 0.3008 0.1276 -2.06 0.00%

AxBxD 1.08 1 1.0800 0.4581 -1.28 0.00%

AxCxD 0.75 1 0.7500 0.3181 -1.61 0.00%

BxCxD 2.5208333 1 2.5208 1.0692 0.16 0.02%

AxBxCxD 2.3408333 1 2.3408 0.9928 -0.02 0.00%

E 75.4467 32 2.3577 14.28%

Table 4.7: Friction Test ANOVA Summary

An examination of the F-ratio and percent contribution in Table 4.7 shows that diameter, profile, and load

have the greatest affect on the friction force. Profile and load have by far the largest percent contribution.

The interaction of diameter and profile and the interaction of profile and load also seem important. Lastly

there is a 90% certainty that durometer is important.

The column effects analysis of the data in Table 4.8 shows that:

1. Increasing the load increases the friction force,

2. Increasing the diameter increases the friction force,

3. Increasing the radius of the profile increases the friction force, and

4. Increasing the durometer decreases the friction force.

Factor Level 2 - Level 1(kg)

D 4.61

A 2.36

B -0.75

C -4.63

Table 4.8: Lateral Load (Friction Force/Gravity) Column Effects Analysis



36

The interaction plots for diameter and profile and for profile and load in Figure 4.7 reveal two more

conclusions:

5. Increasing the diameter and the radius of the profile in combination increases the friction force

and

6. Increasing the radius of the profile and the load in combination increases the friction force.

20

22

24

26

28

30

A1 A2

C1

C2

20

22

24

26

28

30

32

C1 C2

D1

D2

Figure 4.7: Interaction Plot for Diameter and Profile (AxC) and for Profile and Load (CxD)

The percent contribution of the error is small enough at 14.28% not to be a concern so the test is valid.

According to the six conclusions that were drawn from analysis of the Taguchi experiment, the full radius,

76mm, 81A wheel should have the largest friction force, while the tapered radius, 72mm, 84A wheel should

have the smallest friction force. Using these two wheels additional tests were conducted to determine the

response of the wheels on different materials which were thought to be potential surfaces for cobots. These

materials were glass, acrylic, aluminum, and anodized aluminum. In these tests the tangential load that

could be applied to the wheels was determined for normal loads of 10kg, 15kg, 20kg, and 25kg. The results

are plotted in Figure 4.8 and Figure 4.9.

GlassAluminumAcrylicAnodized Al

5 10 15 20 25 305

10

15

20

25

30

Normal Load (kg)

L a t e r a l L o a d (

k g )

Figure 4.8: Plot of Normal vs. Lateral Load on Several Materials, Full Radius 76mm 81A Wheel



37

GlassAluminum

AcrylicAnodized Al

5 10 15 20 25 305

10

15

20

25

30

Normal Load (kg)

L a t e r a l L o a d (

k g )

Figure 4.9: Plot of Normal vs. Lateral Load on Several Materials, Tapered Radius 72mm 84A Wheel

Using the polyfit function of Matlab® to fit these points to a linear function of the form y=ax+b in a least

squares sense, an estimate of the coulomb friction,

µ sF

N = =

Lateral Load

Normal Load (4.85)

for each material was obtained.

Coefficient of Friction ( s

Surface Material Tapered Radius 72mm 84A Full Radius 76mm 81A

Acrylic 0.6382 0.9436

Glass 0.5973 0.8555

Aluminum 0.5873 0.9591

Anodized Aluminum 0.5936 0.7882

Table 4.9: Coefficients of Static Friction

As shown in both the plot and the estimate for the coefficient of friction the choice of wheels makes a

considerable difference in the friction force. To a lesser extent the surface material also plays a role in

determining the friction force.



38

4.2.3 Contact Patch Size

The last set of data analyzed was the contact patch size, the shape of which was assumed to be elliptical. To

test the validity of this assumption photographs of the contact patches were scanned into a computer and fit

to an ellipse. As Figure 4.10 shows, the elliptical contact patch assumption is valid.

Tapered Radius 72mm 84A @ 20 kg

Figure 4.10: Sample Contact Patch Photograph w/ Ellipse Overlay

The contact patch area data in Table 4.2, was determined by plugging measured values of the major and

minor axis lengths into the equation for the area of an ellipse:

area ab= π

44.86

where,

a

b

The ANOVA calculations in Table 4.10 show that the load with a 54.22% contribution is the most

important factor. To a lesser degree diameter, durometer, and profile also effect the contact patch. The

interaction of diameter and durometer also effects the contact patch size though the interaction’s 0.66%

contribution is quite small.



39


T 8818.0928 47

A 673.4887 1 673.4887 39.2104 99% 656.3124 7.44%

B 1303.5295 1 1303.5295 75.8912 99% 1286.3532 14.59%

C 1267.8432 1 1267.8432 73.8136 99% 1250.6669 14.18%

D 4798.6149 1 4798.6149 279.3743 99% 4781.4387 54.22%

AxB 75.5966 1 75.5966 4.4012 99% 58.4203 0.66%

AxC 20.0888 1 20.0888 1.1696 2.9125 0.03%

AxD 1.8193 1 1.8193 0.1059 -15.3570 0.00%

BxC 20.6749 1 20.6749 1.2037 3.4986 0.04%

BxD 0.0394 1 0.0394 0.0023 -17.1369 0.00%

CxD 1.2160 1 1.2160 0.0708 -15.9603 0.00%

AxBxC 34.0628 1 34.0628 1.9831 16.8865 0.19%

AxBxD 1.0588 1 1.0588 0.0616 -16.1175 0.00%

AxCxD 66.6502 1 66.6502 3.8804 90% 49.4739 0.56%

BxCxD 1.7341 1 1.7341 0.1010 -15.4422 0.00%

AxBxCxD 2.0344 1 2.0344 0.1184 -15.1419 0.00%

E 549.6413 32 17.1763 8.08%Table 4.10: Contact Patch Size ANOVA

As was done in the previous two sections, column effect analysis (Table 4.11) and interaction plot analysis

(Figure 4.10) were used in conjunction with ANOVA to draw several conclusions:

1. Increasing the load increases the area of the contact patch,

2. Increasing the diameter increases the area of the contact patch,

3. Increasing the durometer decreases the area of the contact patch,

4. Increasing the radius of the profile increases the area of the contact patch, and

5. Increasing the diameter and durometer in combination increases the area of the contact patch.

Factor Level 2 - Level 1

(kg)D 20.00

A 7.49

C -10.28

B -10.42

Table 4.11: Contact Patch Column Effects Analysis

125

130

135

140

145

A1 A2

B1

B2

Figure 4.11: Contact Patch Size Interaction Plot for AxB



40

4.2.4 Analysis Summary

The conclusions from the previous three sections can be summarized as follows:.

1. Increasing the load increases the wheel compression, friction force, and contact patch size;

2. Increasing the diameter increases the friction force and contact patch size;

3. Increasing the durometer decreases the wheel compression, friction force, and contact patchsize;

4. Increasing the radius of the profile increases the friction force and contact patch size, but

decreases the compression;

5. Increasing the durometer and the radius of the profile in combination decreases the

compression;

6. Increasing the diameter and the radius of the profile in combination increases the friction

force;

7. Increasing the load and the radius of the profile in combination increases the friction force;

and

8. Increasing the diameter and the durometer in combination increases the area of the contact

patch.



41

5. CONTACT MODELS

A concern with cobots is the amount of torque generated in rotating the wheel relative to the contact

surface. A good estimate of the torque provides a guideline for sizing the steering motors which rotate the

wheels. In the CVT, the torque also dictates the maximum obtainable transmission ratio for the device. In

previous cobot work, Moore [5] derived a slipping torque equation using a pressure distribution derived

from Hertz theory for the normal contact of elastic solids. Hertz’s pressure distribution is [4]

p x y P

ab

x

a

y

b( , ) = −

−

3

21

2 21

2

π (5.1)

where P is the normal load and a and b are the semi-axis of the contact patch. The total slipping torque

caused by friction, µs, is determined by integrating equation 5.1 over the contact patch:

τ µ s s p x y x y dxdy= +∫∫ ( , )( )2 2 12 . (5.2)

Equation 5.2 is elliptical and can not be solved analytically. More importantly, the accuracy with whichequation 5.2 predicts the torque rests on the ability of the elastic model to predict the behavior of the

system. This can be checked by using Hertz theory to develop a relationship between load, wheel

compression, and major and minor axes.

5.1 Hertz Theory for Contact of Elastic Solids

In order to analyze the stresses at the contact of two elastic solids, Hertz uses the following assumptions in

defining the surfaces [4]:

1. Each body is assumed to be an elastic half-space, loaded over a elliptical contact patch, so that

the highly localized contact stresses can be disassociated from other general stress in the body.

In order to use the elastic half-space simplification two conditions must be satisfied. The

dimensions of the contact patch must be small relative to the dimensions of the body (i.e.boundary effects can be neglected) and the semi-axes of the contact patch ellipse must be

small relative to the respective radii of curvature (i.e. strains are small and within the linear

theory of elasticity).

2. Each surface is assumed to be frictionless so that the pressure is only transmitted in the normal

direction.

Hertz theory says that an applied normal load will result in a deformation of the system as shown in Figure

5.1. Assuming that point Q is a distant point outside of the deforming region, the application of force P

causes point Q to move a distance δ towards the origin O.



42

P

.δ

O

az

• Q

P

z

Figure 5.1: Cross Section for a Wheel Contact Assuming Hertz Contact Theory

Within the contact region the elastic deformation, z , is

z x y z x y( , ) ( , )= −δ (5.3)

where z(x, y) is the profile of the wheel. The elastic deformation is also related to the pressure distribution.

However, because of the elastic half-space assumption, it is very difficult to analytically determine this

relationship and eventually requires solving an elliptical integral. It is much easier to relate the elastic

deformation and pressure distribution if the elastic half-space assumption is replaced by an elastic

foundation assumption. Once the relationship between the elastic deformation and the pressure distribution

is know it becomes possible to relate the applied load, the wheel compression, and contact patch size, all of

which were measured in previous experiments.

5.2 Elastic Foundation Model

Unlike elastic half-space where the pressure at any point on the contact surface is dependent on the

distribution of pressure throughout the entire contact surface, the pressure in an elastic foundation is only a

function of the displacement at that point. As will be seen later in this section, the elastic foundation

assumption simplifies the mathematical relationship between the elastic deformation and the pressure

distribution. An elastic foundation is like a mattress with individual coils. As a load is applied to the

mattress, the compression of each spring is dependent only upon the load applied to that spring (see Figure

5.2) [4].

P

h

δ

Figure 5.2: The Mattress Representation of the Elastic Foundation Model



43

The standard elastic foundation model assumes that an elastic foundation of depth, h, is loaded under a

force, P, using a rigid indenter such that the indenter moves a distance δ into the foundation at the point of

contact. However, for the cobot wheel the indenter (the wheel) is the elastic foundation and the mattress

(the riding surface) is rigid. This reversed representation of the elastic foundation model is pictured in

Figure 5.3. As long as the contact patch is small, the depth of the elastic foundation for the wheel can be

assumed to be the thickness polyurethane shell on the wheel (see Appendix A).

O.δ

az

• Q

P

h z

Figure 5.3: Elastic Foundation Representation for the Cobot wheel

As shown in Figure 5.3, the elastic displacement within the contact patch can once again be represented by

equation 5.3. Additionally, the elastic foundation assumption requires the elastic displacement outside the

contact patch equal zero. Thus, equation 5.3 can be rewritten as follows

z x y z x y

( , )( , ),

,=

− >≤

δ δ

δ

0

0 0. (5.3a)

Since there is no interaction between adjoining elements all shear stresses can be ignored. As shown by

Johnson [4] the stress throughout the contact patch is therefore related to the displacement at each point

p x y K

h z x y( , ) ( , )=

(5.4)

where K is a representation of the elasticity of the materials in contact. Combining equations 5.3a and 5.4

yields the following relationship:

{ } p x y K

h z x y( , ) ( , )=

−δ . (5.5)

Assuming that the contact patch is small, the profile of the wheel near the contact patch can be

approximated as a quadratic:

z x y Ax By( , ) = +2 2. (5.6)



44

Constants A and B in equation 5.6 are related to the principle radii of curvature of the wheel at the origin

such that along the x and y axes,

z x z x Ax x( , ) ( )02= = (5.7)

z y z y By y( , ) ( )0 2= = . (5.8)

By noting that the curvature,1

R, is the second derivative of equations (5.7) and (5.8) A and B can be

related to the principle radii of curvature, R1 and R2 , of the wheel at the origin:

z A R

A R

x″ = = ⇒ =( )0 2

1 1

21 1

(5.9)

z B R

B R y

″ = = ⇒ =( )0 21 1

22 2

. (5.10)

The result is an approximation for the profile of an ellipsoidal wheel near the origin,

( ) z x y x

R

y

R, = +

2

1

2

22 2(5.11)

which can then be plugged into equation 5.5 so that

p x y K

h

x

R

y

R( , ) =

−

−

δ

2

1

2

22 2. (5.12)

Additionally, since the elastic displacement at the boundary of the contact is zero, the principle radii of curvature are related to the lengths of the semi-axes of the contact ellipse through the following equations:

R a

1

2

2=

δ (5.13)

R b

2

2

2=

δ . (5.14)

Plugging these equations into equation (5.12) yields the following equation for the pressure distribution

p x y K

h

x

a

y

b( , ) = − −

δ 1

2 2

(5.15)

which can be analytically integrated to reveal an equation for the load,

P K ab

h=

π δ

2. (5.16)



45

Because of simplifications in the elastic foundation model K will not be exactly equal to E. However

according to Johnson [4] the values for K should fall within the following ranges depending on the

eccentricity of the elliptical contact patch

118 170. . E h

aK E

h

a

≤ ≤

(5.17)

where

1 1 1 12 2 2

E E E E

w

w

s

s

w

w

= −

+ −

≈ −υ υ υ

(5.18)

E w = Young’s Modulus for the wheel

E s = Young’s Modulus for the rolling surface

ν s = Poisson’s Ratio for the wheel

ν w = Poisson’s Ration for the rolling surface

Often Young’s modulus for the rolling surface is orders of magnitude larger than Young’s modulus for the

wheel. When this is the case, the contribution of the rolling surface to equation 5.18 is small and can beignored.

The upper bound for K represents the case where two spheres are contacting each other while the lower

bound represents the case where two cylinders are contacting each other. For all the wheels tested the

major axis was much longer than the minor, meaning that the value for K should be close to the lower

bound. For the wheels with a full radius, K was chosen as

K E h

a=

125. (5.19)

and for the wheels with the tapered radius K was chosen as

K E h

a=

120. . (5.20)

Values for E for the 81A and 84A durometer wheels were based on averages from strain-strain data

obtained from Kryptonics® for 82A and 85A durometer polyurethane respectively (see Appendix B).

These values are

E (81A) = 4.7MPa (5.21)

and

E (84A) = 5.7MPa. (5.22)

The final result of all this work is that two equations have been developed which relate parameters that have

all previously been determined or measured:

P E b

= 1252

. π δ

(Full Radius Wheels) (5.22)

and

P E b

= 1202

. π δ

(Tapered Radius Wheels) (5.23)



46

5.3 Elastic Model Analysis and Conclusions

Table 5.1 provides a review of the data obtained from the previous chapter’s Taguchi experiment. It also

shows the corresponding values of the other parameters required to evaluate equations (5.22) and (5.23).

Table 5.2 goes on to show the actual load, the predicted load, and the percentage error between the actual

and predicted. The was on average a 30% difference between the predicted load and the actual load. A

closer examination of the results shows that the model is slightly skewed. The average predicted value foran actual 20kg load was 24.9kg with a standard deviation of 4.1kg. Similarly for a 25kg load the predicted

load was 33.4kg with a 3.8kg standard deviation. Given a better estimate of the modulus of elasticity of the

material and the eccentricity of the elliptical contact a better estimate of K may be able to be made, resulting

in small decrease in the error. However, considering the number of approximations used in the model a

30% deviation between the modeled and actual load is reasonable. Hertz theory, with its elastic half-space

material assumption, should more accurately model the system. However, neither model is going to fit

exactly.

The problem lies in the fact that both the elastic half-space and the elastic foundation assume dimensions for

the contact patch which are small relative to the dimensions of the body. While the length of the major axis

is considerably less than the radius of the wheel, the length of the minor axis is not significantly smaller than

the radius of the profile. This can be seen by examining how well equations (5.13) and (5.14) predict the

major and minor semi-axis lengths. Table 5.3 shows that while the average error in the major semi-axisprediction is only 5. 7%, the error in the minor semi-axis prediction is double that at 12.0%.



4 7

T r i a l

L o a d

M a j o r a n d M i n o r A x i s L e n g t h s ( m m )

E

K

h

#

( k g )

( m m )

1

2

3

( M P a )

( M P a )

( m m )

1

2

3

M a j o r M i n o r M a j o r

M i n o r M a j o r

M i n o r

1

2

3

1

2 0

1 . 1 9

1 . 0 7

1 . 2 3

1 9 . 1 6

9 . 5

5

1 8 . 1 7

9 . 4 6

1 7 . 9

9 . 3 2

4 . 7

1

3 . 1 7

3 . 2 7

1 2 . 7 5

2

2 5

1 . 5 8

1 . 4 7

1 . 4 5

2 0 . 5

1 0 . 1

3

1 9 . 3 9

9 . 8 7

1 9 . 2 2

9 . 9 6

4 . 7

2 . 4 7

2 . 6 8

2 . 6 8

1 2 . 7 5

3

2 0

1 . 4

1 . 3 5

1 . 3 8

1 9 . 5 9

7 . 9

1 9 . 5 4

7 . 9 8

1 9 . 6 9

8 . 1 5

4 . 7

3 . 0 0

2 . 9 7

2 . 8 9

1 2 . 7 5

4

2 5

1 . 7 2

1 . 6

1 . 6 8

2 1 . 2 2

8 . 7

6

2 1 . 0 3

8 . 6 5

2 1 . 1

8 . 6 1

4 . 7

2 . 5 4

2 . 5 9

2 . 6 0

1 2 . 7 5

5

2 0

1 . 4 2

1 . 0 3

0 . 7 8

1 6 . 8 3

9 . 4

3

1 7 . 0 6

8 . 9 6

1 6 . 8 1

8 . 9 8

5 . 7

2 . 8 8

2 . 9 9

3 . 0 3

1 2 . 7 5

6

2 5

1 . 5 4

1 . 1 9

1 . 0 6

1 7 . 8 3

9 . 6

2

1 8 . 3 9

9 . 8 1

1 8 . 2 6

9 . 6 4

5 . 7

3 . 0 7

2 . 9 2

2 . 9 9

1 2 . 7 5

7

2 0

1 . 5 3

1 . 1 7

1 . 1 6

1 9 . 0 7

7 . 6

1

1 9 . 1 6

7 . 1 4

1 9 . 0 3

7 . 7 2

5 . 7

2 . 9 2

3 . 1 0

2 . 8 9

1 2 . 7 5

8

2 5

1 . 8 7

1 . 3 2

1 . 3

2 0 . 0 2

8 . 1

2

1 9 . 9 8

8 . 7 9

2 0 . 4 5

8 . 6 9

5 . 7

2 . 6 7

2 . 4 7

2 . 4 4

1 2 . 7 5

9

2 0

1 . 2 8

1 . 1 9

1 . 0 4

1 9 . 2 2

9 . 1

4

1 9 . 1 5

9 . 3 5

1 8 . 8 6

9 . 2 5

4 . 7

3 . 3 4

3 . 2 8

3 . 3 6

1 4 . 7 5

1 0

2 5

1 . 5 4

1 . 6

1 . 4 8

2 0 . 6 8

1 0 . 1

6

2 0 . 4

9 . 9 7

2 0 . 5 8

9 . 9 4

4 . 7

2 . 9 0

3 . 0 0

2 . 9 8

1 4 . 7 5

1 1

2 0

1 . 2

0 . 9 5

1 . 2 3

2 0 . 2 1

8 . 3

2 0 . 5

8 . 3 1

2 0 . 0 5

7 . 9

4 . 7

3 . 7 3

3 . 6 7

3 . 9 5

1 4 . 7 5

1 2

2 5

1 . 4 8

1 . 3 3

1 . 5 9

2 2

8 . 8

5

2 1 . 4 5

8 . 5 9

2 1 . 4 3

8 . 6 9

4 . 7

3 . 2 5

3 . 4 4

3 . 4 1

1 4 . 7 5

1 3

2 0

0 . 7 8

1 . 0 7

1 . 0 3

1 8 . 2 5

9

1 8 . 0 4

9 . 3 6

1 8 . 5 3

9 . 0 7

5 . 7

5 . 8 6

5 . 7 0

5 . 7 3

1 4 . 7 5

1 4

2 5

1 . 2 1

1 . 1 8

1 . 1 1

1 9 . 9 7

9 . 9

1

1 9 . 4 7

9 . 8 7

2 0 . 0 6

9 . 8 9

5 . 7

3 . 9 2

4 . 0 4

3 . 9 1

1 4 . 7 5

1 5

2 0

1 . 0 5

1 . 3 3

1 . 3

1 9 . 8 9

8 . 1

3

1 9 . 2 5

7 . 8 5

1 9 . 4

7 . 8 9

5 . 7

4 . 4 2

4 . 7 3

4 . 6 7

1 4 . 7 5

1 6

2 5

1 . 5 1

1 . 7

1 . 6 1

2 1 . 2 1

8 . 3

9

2 0 . 7

8 . 4 8

2 1 . 2 1

8 . 4 4

5 . 7

3 . 4 9

3 . 5 4

3 . 4 7

1 4 . 7 5

T a b l e 5 . 1 : T e s t R e s u l t s a n d P a r a m e t e r s f o r A n a l y z i n g t h e E l a s t i c i t y M o d e l



48

Trial Load (kg) % Error Between Actual

# Actual Predicted and Predicted Load

1 2 3 1 2 3

1 20 26.22 23.35 26.45 31.10% 16.76% 32.24%

2 25 36.93 33.47 33.32 47.70% 33.89% 33.28%

3 20 24.50 23.86 24.91 22.48% 19.30% 24.55%

4 25 33.37 30.65 32.04 33.48% 22.61% 28.15%

5 20 37.47 25.82 19.60 87.33% 29.11% -2.01%

6 25 41.45 32.66 28.59 65.81% 30.65% 14.36%

7 20 31.27 22.44 24.05 56.37% 12.19% 20.27%

8 25 40.79 31.17 30.34 63.14% 24.66% 21.38%

9 20 26.99 25.67 22.19 34.96% 28.35% 10.97%

10 25 36.10 36.80 33.94 44.39% 47.21% 35.76%

11 20 22.06 17.48 21.52 10.30% -12.58% 7.61%

12 25 29.01 25.30 30.60 16.04% 1.21% 22.41%

13 20 19.64 28.02 26.14 -1.79% 40.11% 30.70%

14 25 33.55 32.59 30.72 34.20% 30.35% 22.86%

15 20 22.93 28.04 27.55 14.65% 40.22% 37.75%

16 25 34.03 38.72 36.50 36.12% 54.89% 46.00%

Table 5.2: Elastic Foundation Model Load Prediction Test Results

Trial % Error

# 1 2 3

Major Minor Major Minor Major Minor

1 3.38% 10.75% 3.39% 6.01% 5.15% 15.37%

2 4.06% 20.30% 6.12% 19.10% 6.32% 17.22%

5 20.16% 22.52% 0.96% 9.82% 10.84% 4.65%

6 18.12% 25.07% 0.67% 7.81% 4.31% 3.55%

9 2.63% 20.01% 0.68% 13.12% 5.72% 6.89%

10 4.63% 18.42% 8.11% 23.01% 3.07% 18.66%13 15.62% 4.86% 0.02% 7.15% 4.51% 8.49%

14 3.96% 7.62% 2.72% 6.71% 8.43% 3.28%

Table 5.3: % Error in the Predicted Semi-Axis Lengths for the Full Radius Wheel

5.4 Elastic Model Summary

In summary, the analysis of the contact showed that while the elastic foundation model and Hertz theory can

both be used to model the interface, neither is going to be extremely accurate and should not be heavily

relied upon to provide more than order of magnitude estimate. A major concern with both models is that

they assume the contact area is small relative to the wheel dimensions. An examination of equations (5.13)

and (5.14) revealed that the size of minor axis can not be considered small relative to the radius of curvature

in that direction, thus violating that assumption.

As stated in the beginning of the chapter, one of the reasons for modeling the contact patch is to determine a

slipping torque which can then be used to size the steering motors. A schematic of the relationship between

the slipping and motor torque is shown in Figure 5.4.



49

Jm Jw

α

τs

τm

Figure 5.4: Schematic of the Slipping and Motor Torque Relationship

From Figure 5.4, the equation of motion relating the motor torque, τ m, to the slipping torque, τ s, is

τ α τ

mw m s J n J

n=

+ +( )2

(5.24)

where,

Jm = mass moment of inertia of the motor

Jw = mass moment of inertia of the wheel assembly

n = transmission ratio

a = angular acceleration.

Solving equation 5.24 using the slipping torque estimate in equation 5.2 based on Hertz theory would in the

provides at best an order of magnitude estimate of the motor torque. An order of magnitude estimate will

probably not be accurate enough to use when sizing motors.



50

6. SUMMARY

This paper has examined several aspects of the contact patch of the cobot wheel and the riding surface in an

attempt to get a better understanding of

1. How much force can be applied under a certain condition before the wheels slip,

2. How far the wheels compress under a certain load, and3. How much torque is required to rotate the wheels.

In Chapter 4 the first two of these questions were examined. Using Taguchi methods of experimentation

along with several methods of analysis including ANOVA, rules were developed to show how varying the

diameter, durometer, profile, and load affect the compression of the wheel, the size of the contact patch, and

the friction force. These rules are summarized as follows:

1. Increasing the load increases the wheel compression, friction force, and contact patch size;

2. Increasing the diameter increases the friction force and contact patch size;

3. Increasing the durometer decreases the wheel compression, friction force, and contact patch

size;

4. Increasing the radius of the profile increases the friction force and contact patch size, butdecreases the compression;

5. Increasing the durometer and the radius of the profile in combination decreases the

compression;

6. Increasing the diameter and the radius of the profile in combination increases the friction

force;

7. Increasing the load and the radius of the profile in combination increases the friction force;

and

8. Increasing the diameter and the durometer in combination increases the area of the contact

patch.

Chapter 4 also showed how the friction force varied with the surface material. Table 4.9 shows the

coefficients of static friction for these materials using the best (full radius 76mm 81A) and worst (tapered

radius 72mm 84A) wheels as determined by the Taguchi experiment.

Coefficient of Friction ( s

Surface Material Tapered Radius 72mm 84A Full Radius 76mm 81A

Acrylic 0.6382 0.9436

Glass 0.5973 0.8555

Aluminum 0.5873 0.9591

Anodized Aluminum 0.5936 0.7882

Table 4.9 Coefficients of Static Friction

Chapter 5 looked at how well the elastic foundation simplification of Hertz theory modeled the contact

between the wheel and the rolling surface. This was done to get an idea of whether or not a Hertz pressure

distribution could be used to estimate the slipping torque. The elastic foundation model load prediction was

on average 30% greater than the actual load. This suggests that Hertz theory should only be used to provide

order of magnitude estimates. One of the problems with using Hertz theory is that the assumption of a small

contact area relative to the wheel dimensions is violated with regard to the minor axis. An order of

magnitude estimate of the torque will most likely not be accurate enough to reasonably size the motors

required to steer the wheels.



51

APPENDICES

A. ELASTIC FOUNDATION DEPTH ESTIMATE

If zw(x,y) and zc(x,y) represent the profiles of the wheel and the core of the wheel respectively, thedifference in profiles can be represented as:

z x y z x y z x y h x y x yd c w( , ) ( , ) ( , ) ( , ) ( , )= − = + ∆ . (A.1)

where

h x y h( , ) ≈ . (A.2)

As seen in Figure A.1, for a contact patch that is small relative to the radii of curvature of the core, ∆ will be

small and the thickness of the polyurethane shell on the wheel can be used to approximate the depth of the

elastic foundation, h, along the entire contact region.

∆

Figure A.1: Approximation of the Change in Foundation Model Depth, ∆, for a Wheel



52

B. PARAMETERS

Uncompressed polyurethane shell thickness - 12.75mm (for 72mm wheels)

- 14.75mm (for 76mm wheels)

Poisson’s Ratio = 0.45 (estimated from tables)

Elasticity estimates are based on stress/strain information obtained from Kryptonics® engineers for wheels

of 82A and 85A hardness. Data was not available for the 81A and 84A wheels used in the experiment.

82A

Strain Stress Young's Modulus

% elongation PSI MPa PSI MPa

100 600 4.14 600 4.14

300 1200 8.27 400 2.76

500 2400 1.65 480 3.31

650 4500 3.10 692.3077 4.77

E (Ave.) = 3.74 MPa

85A

Strain Stress Young's Modulus

% elongation PSI Pa PSI MPa

100 700 4826360 700 4826360

300 1500 10342200 500 3447400

500 3000 20684400 600 4136880

550 4500 31026600 818.1818 5641200

E (Ave.) = 4.51 MPa



5 3

C . S T A N D A R D L 1 6

O A U S E D F O R T H E E X P E R I M E N T S

1

2

3

4

5

6

7

8

9

1 0

1 1

1 2

1 3

1 4

1 5

T r i a l #

D i a .

( m m )

D u r .

D i a . ,

D u r .

P r o f i l e

( R a d i u s )

D i a . ,

P r o f i l e

D u r . ,

P r o f i l e

D i a . ,

D u r . ,

P r o f i l e

L o a d

( k g )

D i a . ,

L o a d

D u r . ,

L o a

d

D i a . ,

D u r . ,

L o a d

P r o f i l e ,

L o a d

D i a . ,

P r o f i l e ,

L o a d

D u r . ,

P r o f i l e ,

L o a d

D i a . ,

D u r . ,

P r o f i l e ,

L o a d

1

7 2

8 1 A

1

F u l l

1

1

1

2 0

1

1

1

1

1

1

1

2

7 2

8 1 A

1

F u l l

1

1

1

2 5

2

2

2

2

2

2

2

3

7 2

8 1 A

1

T a p e r e d

2

2

2

2 0

1

1

1

2

2

2

2

4

7 2

8 1 A

1

T a p e r e d

2

2

2

2 5

2

2

2

1

1

1

1

5

7 2

8 4 A

2

F u l l

1

2

2

2 0

1

2

2

1

1

2

2

6

7 2

8 4 A

2

F u l l

1

2

2

2 5

2

1

1

2

2

1

1

7

7 2

8 4 A

2

T a p e r e d

2

1

1

2 0

1

2

2

2

2

1

1

8

7 2

8 4 A

2

T a p e r e d

2

1

1

2 5

2

1

1

1

1

2

2

9

7 6

8 1 A

2

F u l l

2

1

2

2 0

2

1

2

1

2

1

2

1 0

7 6

8 1 A

2

F u l l

2

1

2

2 5

1

2

1

2

1

2

1

1 1

7 6

8 1 A

2

T a p e r e d

1

2

1

2 0

2

1

2

2

1

2

1

1 2

7 6

8 1 A

2

T a p e r e d

1

2

1

2 5

1

2

1

1

2

1

2

1 3

7 6

8 4 A

1

F u l l

2

2

1

2 0

2

2

1

1

2

2

1

1 4

7 6

8 4 A

1

F u l l

2

2

1

2 5

1

1

2

2

1

1

2

1 5

7 6

8 4 A

1

T a p e r e d

1

1

2

2 0

2

2

1

2

1

1

2

1 6

7 6

8 4 A

1

T a p e r e d

1

1

2

2 5

1

1

2

1

2

2

1

D i a . = D i a m e t e r

D u r . = D u r o m e t e r

T a p e r e d W h e e l - K r y p t o n i c s ® G

T C o m p W h e e l

F u l l R a d i u s W h e e l - K r y p t o n i c s ® D - s t r o y e r W h e e l



5 4

D . R A W T

E S T R E S

U L T S - W H E E L C O M P R E S S I O N ,

C O N T A C T P A T C H S I Z E , F R I C T I

O N

T r i a l

L o a

d e d S h e l l T h i c k n e s s

M a

j o r & M i n o r A x i s L e n g t h s ( m m ) *

T r a n s v e r s e L o a d o n G l a s s

#

( m m )

1

2

3

( k g )

1

2

3

M a j o r

M i n o r

M a j o r

M i n o r

M a j o r M

i n o r

1

2

3

1

2 3 . 8

1

2 3 . 9 3

2 3 . 7 7

1 9 . 1 6

9 . 5 5

1 8 . 1 7

9 . 4 6

1 7 . 9

9 . 3 2

2 6 . 4

0

2 2 . 7

0

2 6 . 4

0

2

2 3 . 4

2

2 3 . 5 3

2 3 . 5 5

2 0 . 5

1

0 . 1 3

1 9 . 3 9

9 . 8 7

1 9 . 2 2

9 . 9 6

3 2 . 7

0

3 0 . 9

0

3 0 . 0

0

3

2 3 . 6

2 3 . 6 5

2 3 . 6 2

1 9 . 5 9

7 . 9

1 9 . 5 4

7 . 9 8

1 9 . 6 9

8 . 1 5

2 2 . 7

0

2 0 . 0

0

2 1 . 8

0

4

2 3 . 2

8

2 3 . 4

2 3 . 3 2

2 1 . 2 2

8 . 7 6

2 1 . 0 3

8 . 6 5

2 1 . 1

8 . 6 1

2 3 . 6

0

2 1 . 8

0

2 4 . 5

0

5

2 3 . 5

8

2 3 . 9 7

2 4 . 2 2

1 6 . 8 3

9 . 4 3

1 7 . 0 6

8 . 9 6

1 6 . 8 1

8 . 9 8

2 5 . 5

0

2 5 . 5

0

2 3 . 6

0

6

2 3 . 4

6

2 3 . 8 1

2 3 . 9 4

1 7 . 8 3

9 . 6 2

1 8 . 3 9

9 . 8 1

1 8 . 2 6

9 . 6 4

3 0 . 0

0

2 9 . 1

0

2 9 . 1

0

7

2 3 . 4

7

2 3 . 8 3

2 3 . 8 4

1 9 . 0 7

7 . 6 1

1 9 . 1 6

7 . 1 4

1 9 . 0 3

7 . 7 2

2 0 . 0

0

1 9 . 1

0

2 0 . 0

0

8

2 3 . 1

3

2 3 . 6 8

2 3 . 7

2 0 . 0 2

8 . 1 2

1 9 . 9 8

8 . 7 9

2 0 . 4 5

8 . 6 9

2 4 . 5

0

2 3 . 6

0

2 2 . 7

0

9

2 5 . 7

2

2 5 . 8 1

2 5 . 9 6

1 9 . 2 2

9 . 1 4

1 9 . 1 5

9 . 3 5

1 8 . 8 6

9 . 2 5

2 4 . 5

0

2 6 . 4

0

2 7 . 3

0

1 0

2 5 . 4

6

2 5 . 4

2 5 . 5 2

2 0 . 6 8

1

0 . 1 6

2 0 . 4

9 . 9 7

2 0 . 5 8

9 . 9 4

3 0 . 0

0

3 3 . 6

0

3 3 . 6

0

1 1

2 5 . 8

2 6 . 0 5

2 5 . 7 7

2 0 . 2 1

8 . 3

2 0 . 5

8 . 3 1

2 0 . 0 5

7 . 9

2 1 . 8

0

2 2 . 7

0

2 5 . 5

0

1 2

2 5 . 5

2

2 5 . 6 7

2 5 . 4 1

2 2

8 . 8 5

2 1 . 4 5

8 . 5 9

2 1 . 4 3

8 . 6 9

2 9 . 1

0

2 7 . 3

0

2 8 . 2

0

1 3

2 6 . 2

2

2 5 . 9 3

2 5 . 9 7

1 8 . 2 5

9

1 8 . 0 4

9 . 3 6

1 8 . 5 3

9 . 0 7

2 6 . 4

0

2 6 . 4

0

2 5 . 5

0

1 4

2 5 . 7

9

2 5 . 8 2

2 5 . 8 9

1 9 . 9 7

9 . 9 1

1 9 . 4 7

9 . 8 7

2 0 . 0 6

9 . 8 9

3 1 . 8

0

3 4 . 5

0

2 8 . 2

0

1 5

2 5 . 9

5

2 5 . 6 7

2 5 . 7

1 9 . 8 9

8 . 1 3

1 9 . 2 5

7 . 8 5

1 9 . 4

7 . 8 9

2 4 . 5

0

2 1 . 8

0

2 2 . 7

0

1 6

2 5 . 4

9

2 5 . 3

2 5 . 3 9

2 1 . 2 1

8 . 3 9

2 0 . 7

8 . 4 8

2 1 . 2 1

8 . 4 4

2 8 . 2

0

2 5 . 5

0

2 7 . 3

0

* M a j o r a n d m i n o r a x e s r e p r e s e n t t h e f u l l l e n g t h a n d w i d t h o f t h e c o n t a c t p a t c h , t h e a c t u a l m a j o r a n d m i n o r s e m i - a x e s a r e ½ o

f t h e s e v a l u e s .



5 5

E . T R A N S V E R S E L O A D D A T A

T r a n s v e r s e

L o a d ( l b . ) f o r A l u m i n u m c l e a n e d

w i t h A c e t o n e

W h e e l #

1

2

3

T r i a l #

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1 0 k g n o r m a l l o a d

D 7 6 8 1 A

2 6

2 8

2 6

2 8

2 8

2 8

2 8

3 2

3 0

3 0

3 2

3 0

3 2

3 0

3 0

3 0

3 0

3 0

3 0

3 0

3 2

G T 7 2 8 4 A

1 6

1 8

1 6

1 8

1 6

1 6

1 8

1 6

1 6

1 6

1 6

1 6

1 6

1 6

1 8

1 8

1 8

2 0

2 0

2 0

2 0


D 7 6 8 1 A

3 8

3 8

3 8

3 8

4 0

4 0

4 0

4 0

4 2

4 4

4 4

4 4

4 6

4 6

4 6

4 0

4 4

4 2

4 4

4 2

4 2

G T 7 2 8 4 A

2 4

2 4

2 4

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

3 0

2 8

2 6

3 0

3 0

3 0

3 0


D 7 6 8 1 A

5 6

5 4

5 6

5 2

5 2

5 2

5 2

5 2

5 4

5 2

5 0

5 4

5 4

5 0

5 4

5 4

5 4

5 4

5 0

5 2

5 0

G T 7 2 8 4 A

3 6

3 6

3 2

3 2

2 8

3 0

3 0

3 2

3 6

3 6

3 6

3 8

3 6

3 6

3 0

3 0

3 2

3 2

3 0

3 2

3 0


D 7 6 8 1 A

6 2

6 0

5 8

6 2

6 4

6 0

6 0

6 6

6 4

6 4

6 2

6 2

6 2

6 2

5 8

5 8

5 8

5 8

6 0

6 2

6 0

G T 7 2 8 4 A

3 4

3 4

3 2

3 4

3 4

3 2

3 4

3 6

3 6

3 4

3 4

3 4

3 4

3 4

4 0

4 2

4 0

4 0

4 0

4 0

4 0

T r a n s v e r

s e L o a d ( l b . ) f o r G l a s s c l e a n e d w

i t h A c e t o n e

W h e e l #

1

2

3

T r i a l #

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

7


D 7 6 8 1 A

3 0

3 0

3 0

2 8

2 8

3 0

2 8

3 2

3 2

3 0

2 8

2 8

2 8

3 0

3 0

3 0

2 8

2 8

3 0

3 0

3 0

G T 7 2 8 4 A

2 4

2 2

2 2

2 6

2 6

2 2

2 2

2 0

1 6

1 6

1 6

1 6

1 8

1 8

1 6

1 8

1 8

1 8

2 0

1 8

1 8


D 7 6 8 1 A

4 0

3 8

3 8

3 6

3 8

3 8

3 6

3 8

4 0

3 8

3 8

4 0

4 0

4 2

4 0

4 0

4 0

4 0

3 8

4 0

3 8

G T 7 2 8 4 A

3 0

3 2

3 0

3 2

3 2

3 2

3 4

2 4

2 4

2 2

2 2

2 2

2 2

2 2

2 4

2 2

2 2

2 4

2 2

2 4

2 4


D 7 6 8 1 A

4 8

4 6

4 8

5 4

5 0

5 2

4 8

4 6

4 6

4 4

4 2

4 6

5 0

5 0

5 6

5 8

5 8

6 0

5 8

6 0

6 0

G T 7 2 8 4 A

3 6

3 6

3 2

3 4

3 4

3 2

3 6

3 2

3 4

3 6

3 6

3 6

3 8

3 4

3 4

3 0

3 0

3 2

3 0

3 2

3 2


D 7 6 8 1 A

5 6

5 6

5 6

5 4

5 4

5 0

5 0

5 6

5 4

5 4

6 2

6 0

6 0

5 8

6 0

5 4

6 0

5 8

5 8

6 0

5 8

G T 7 2 8 4 A

4 0

4 8

4 8

4 8

4 6

4 6

4 6

3 6

3 4

3 6

3 6

3 6

3 6

3 4

3 4

3 4

3 6

3 6

3 6

3 4

3 4



5 6

T r a n s v e r s e L o a

d ( l b . ) f o r A c r y l i c ( P l e x i g l a s ) c l e a n e d w i t h A c e t o n e

W h e e l #

1

2

3

T r i a l #

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

7


D 7 6 8 1 A

3 0

3 0

3 0

3 0

3 0

3 0

2 8

3 6

3 0

3 0

3 2

3 2

3 2

3 0

3 4

3 4

3 2

3 0

3 0

3 0

3 0

G T 7 2 8 4 A

2 2

2 4

2 4

2 4

2 4

2 4

2 6

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 0

2 0

2 0

2 0

2 0

2 0

2 0


D 7 6 8 1 A

4 2

4 0

3 8

4 0

4 2

4 2

4 2

4 4

4 4

4 2

4 2

4 2

4 2

4 2

4 4

4 4

4 4

4 4

4 6

4 4

4 4

G T 7 2 8 4 A

3 4

3 4

3 4

3 4

3 6

3 4

3 4

2 8

2 8

2 8

2 8

2 8

2 8

2 8

3 0

2 6

2 6

2 6

2 6

2 6

2 6


D 7 6 8 1 A

5 6

5 0

5 2

5 0

5 4

5 0

4 8

5 0

5 2

5 2

5 4

5 0

5 2

5 4

4 8

4 8

4 8

5 0

5 0

4 8

4 8

G T 7 2 8 4 A

3 4

3 2

3 6

3 8

3 4

3 8

4 0

4 0

4 2

4 0

3 8

3 8

3 8

4 2

3 6

3 6

3 4

3 4

3 4

3 2

3 6


D 7 6 8 1 A

6 6

6 4

6 4

6 4

6 4

6 2

6 6

6 0

6 2

6 4

6 4

6 2

6 2

6 2

6 2

6 2

6 2

6 2

6 2

6 2

6 2

G T 7 2 8 4 A

4 8

5 2

5 0

4 8

4 8

5 0

4 8

4 6

4 4

4 4

4 2

4 2

4 2

4 0

3 6

3 6

3 8

3 8

3 8

3 8

3 6

T r a n s v e r s e L o a d ( l b . ) f o r A n o d i z e d A l u m i n u m c l e a n e d w i t h A c e t o n e

W h e e l #

1

2

3

T r i a l #

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

7


D 7 6 8 1 A

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 0

2 2

2 2

2 2

2 0

2 2

2 2

2 2

2 2

G T 7 2 8 4 A

1 8

1 6

1 6

1 6

1 8

1 6

1 6

1 6

1 6

1 6

1 6

1 6

1 6

1 6

1 8

1 8

1 8

1 8

1 8

1 8

1 8


D 7 6 8 1 A

3 2

3 2

3 2

3 2

3 0

3 0

3 2

3 2

3 2

3 2

3 2

3 0

3 0

3 0

3 4

3 2

3 2

3 2

3 2

3 2

3 2

G T 7 2 8 4 A

2 6

2 4

2 4

2 4

2 4

2 4

2 4

2 4

2 6

2 6

2 4

2 4

2 4

2 4

2 6

2 8

2 6

2 6

2 6

2 6

2 6


D 7 6 8 1 A

3 6

4 0

4 0

4 0

4 2

4 2

3 8

3 8

3 8

3 8

3 6

3 8

3 8

3 8

3 4

3 6

3 6

3 6

3 4

3 6

3 6

G T 7 2 8 4 A

3 2

3 2

3 0

3 0

3 0

2 8

3 0

3 0

3 0

3 2

3 2

3 0

3 0

3 0

3 2

3 2

3 2

3 2

3 0

3 0

3 4


D 7 6 8 1 A

5 4

5 2

5 2

5 0

5 0

5 0

5 0

4 6

4 6

4 6

4 6

4 6

4 6

4 6

5 0

5 0

5 0

4 8

4 8

4 8

4 8

G T 7 2 8 4 A

3 8

3 8

3 6

3 4

3 4

3 4

3 4

3 4

3 6

3 6

3 6

3 6

3 6

3 6

3 6

3 8

4 0

4 0

3 8

4 0

4 0

D r e f e r s t o t h e D - s t r o y e

r w h e e l w i t h t h e f u l l r a d i u s , G T r e f e r s

t o t h e G T C o m p w h e e l w i t h t h e t a p e r e

d r a d i u s



BIBLIOGRAPHY

1. Colgate, J. E. and Brown, J. M., "Factors Affecting the Z-width of a Haptic Display ," Proceedings of

the IEEE International Conference on Robotics and Automation, IEEE, San Diego, Vol. 4, pp. 3205-

10, 1994.

2. Colgate, J. E., Wannasuphoprasit W., and Peshkin, M. A., “Cobots: Robots for Collaboration with

Human Operators”, Proceedings of the ASME Dynamic Systems and Control Division, ASME, Atlanta,

Vol. 58, pp. 433-439, 1996.

3. Gillespie, R. B., Moore, C. A., Peshkin, M. A., and Colgate, J. E., “Kinematic Creep in Continuously

Variable Transmissions: Traction Drive Mechanics for Cobots”, to be submitted: IEEE Transactions

on Robotics & Automation, Dec. 1997.

4. Johnson, K. L., Contact Mechanics, Cambridge University Press, Cambridge, England, 1985.

5. Moore, C., Continuously Variable Transmission for Serial Link Cobot Architectures, M. S.,

Northwestern University, 1997.

6. Peace, Glen Stuart, Taguchi Methods: A Hands-On Approach, Addison-Wesley Publishing Company,

Inc., Reading, Massachusetts, 1993.

7. Ross, Phillip J., Taguchi Techniques for Quality Engineering, 2nd

Edition, McGraw-Hill, New York,

1996.

8. Wannasuphoprasit, W., Gillespie, R. B., Colgate, J. E., and Peshkin, M., “Cobot Control”, Proceedings

of the IEEE 1997 International Conference on Robotics and Automation, IEEE, Albuquerque, pp.

3571-3576, 1997.

anova aplicado a tesis de maestria en ingenieria mecanica

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