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Design, Analysis and Control of a
Spherical Continuously Variable Transmission
By
Jungyun Kim
Submitted in Partial Fulfillment of The
Requirements for The Degree of
Doctor of Philosophy
in theSchool of Mechanical and Aerospace Engineering
atSeoul National University
February 2001
To My Parents
i
Abstract
This dissertation is concerned with the design, analysis, and control of a novel con-
tinuously variable transmission, the spherical CVT (S-CVT). The S-CVT has a
simple kinematic structure, infinitely continuously variable transmission character-
istics, and transmits power via dry rolling friction on the contact points between
a sphere and discs. The S-CVT is intended to overcome some of the limitations
of existing CVT designs. Its compact and simple design and relatively simple con-
trol make it particularly effective for mechanical systems in which excessively large
torques are not required.
We describe the operating principles behind the S-CVT, including a kinematic
and dynamic analysis. A prototype is constructed based on a set of design spec-
ifications and results of theoretical performance analysis. In order to provide a
quantitative analysis of the spin loss of the S-CVT, which is one of the main sources
of power loss, we develop an explicit formulation using a modified classical friction
model, and an in-depth study of the velocity fields and normal pressure distribution
on the contact regions. The proposed friction model includes the pre-sliding effect,
i.e., Stribeck effects. Actual transmission ratios and power efficiency are obtained
from experiments with a prototype testbench.
The open-loop shifting system of the S-CVT reveals nonlinearity and unstable
characteristics. In order to cancel the nonlinearity of the shifting system and to
make it stable to shifting commands, we develop an input-state feedback controller
based on exact feedback linearization. We also investigate the power efficiency of
a generic dc motor, and present the results of a numerical investigation of the S-
CVT’s energy savings possibility benchmarked against a standard reduction gear.
Furthermore, we develop a minimum energy control law for the S-CVT driven by a
ii
dc motor, and present numerical simulation results that confirm the performance of
the controller.
Finally, we design and construct an S-CVT based mobile robot to realize the
various advantages of the S-CVT into practical use. One of the key features of
the mobile robot is the design of a novel pivot mechanism for planar accessibility.
Results of both numerical simulations and experiments are presented to validate the
robot’s performance advantages obtained as a result of using the S-CVT.
Keywords: Continuously variable transmission; infinitely variable transmission;
dry rolling friction; spin loss; feedback linearization; minimum energy control;
mobile robot.
iii
Contents
Dedication i
Abstract ii
List of Tables viii
List of Figures xi
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 CVTs for Passenger Cars . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Dynamic Analysis of the Spherical CVT 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Kinematics of S-CVT . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Operating Principles . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Dynamics of S-CVT . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
iv
2.3.1 Motion of Sphere . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.2 Shifting Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Reaction Forces of the S-CVT . . . . . . . . . . . . . . . . . . . . . . 35
2.4.1 Normal Reaction Force Exerted on the Variator: Fn . . . . . 36
2.4.2 Shifting Reaction Force on the Sphere: D . . . . . . . . . . . 37
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Prototype Design and Experimental Results 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Issues in Mechanical Design . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.1 Normal Force Loading Device . . . . . . . . . . . . . . . . . . 41
3.2.2 Capacity of Shifting Actuator . . . . . . . . . . . . . . . . . . 42
3.3 Prototype Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Performance of S-CVT . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Strength and Life Prediction of S-CVT . . . . . . . . . . . . 49
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Slip Analysis of the Spherical CVT 53
4.1 Friction Model Review . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Modified Friction Model for S-CVT . . . . . . . . . . . . . . . . . . . 59
4.3 Spin Loss of the S-CVT . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Velocity Fields on the Contact Surface . . . . . . . . . . . . . 60
4.3.2 Normal Pressure Distribution . . . . . . . . . . . . . . . . . . 65
4.3.3 Quantitative Analysis of Spin Loss . . . . . . . . . . . . . . . 66
4.4 Slip Motion of the S-CVT . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4.1 Stick and Slip States . . . . . . . . . . . . . . . . . . . . . . . 70
v
4.4.2 Slip Loss of the S-CVT . . . . . . . . . . . . . . . . . . . . . 70
4.4.3 Slip Involved Contact Analysis . . . . . . . . . . . . . . . . . 70
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Shifting Controller Design via Exact Feedback Linearization 74
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Stability Analysis of S-CVT Shifting System . . . . . . . . . . . . . . 76
5.3 Differential Geometric Preliminaries . . . . . . . . . . . . . . . . . . 78
5.4 Shifting Controller Design via Input-State Linearization . . . . . . . 81
5.4.1 Controllability and Linearizability . . . . . . . . . . . . . . . 82
5.4.2 Input-State Linearization . . . . . . . . . . . . . . . . . . . . 83
5.5 Shifting Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Optimal Control of an S-CVT equipped Power Transmission 91
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Power Efficiency of a DC Motor . . . . . . . . . . . . . . . . . . . . . 93
6.2.1 DC Motor Dynamics . . . . . . . . . . . . . . . . . . . . . . . 93
6.2.2 Power Efficiency of a DC Motor . . . . . . . . . . . . . . . . 95
6.3 Investigation of S-CVT Energy Savings . . . . . . . . . . . . . . . . 96
6.3.1 Control Design based on the Computed Torque Method . . . 98
6.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 Minimum Energy Control via a B-Spline Parameterization . . . . . . 101
6.4.1 B-Spline Parameterization . . . . . . . . . . . . . . . . . . . . 102
6.4.2 Gradients of the Objective Function and Constraint . . . . . 103
6.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 105
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
vi
7 Case Study: An S-CVT based Mobile Robot 109
7.1 Motivation for Mobile Robot Applications . . . . . . . . . . . . . . . 110
7.2 MOSTS: An S-CVT Mobile Robot . . . . . . . . . . . . . . . . . . . 112
7.2.1 Pivot Device for Planar Accessibility . . . . . . . . . . . . . . 112
7.2.2 Prototype Design . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.3 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . 115
7.3.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 118
7.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 120
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8 Conclusion 123
References 125
vii
List of Tables
3.1 Specifications of prototype. . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Endurance test condition. . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Maximal normal pressure comparison. . . . . . . . . . . . . . . . . . 66
5.1 Candidates for k1, k2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.1 Characteristic coefficients of dc motor. . . . . . . . . . . . . . . . . . 97
6.2 Energy consumption; reduction gear vs. S-CVT. . . . . . . . . . . . 101
6.3 Energy consumption with the minimum energy control. . . . . . . . 108
7.1 Hardware specifications of general mobile robots. . . . . . . . . . . . 110
7.2 DC motor charateristic coefficients of MOSTS. . . . . . . . . . . . . 115
7.3 Energy consumption; MOSTS vs. differential drive. . . . . . . . . . . 121
viii
List of Figures
1.1 Classification of transmissions for vehicles. . . . . . . . . . . . . . . . 2
1.2 Fuel consumption reduction for an engine. . . . . . . . . . . . . . . . 4
1.3 Engine speed variation. . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Classification of CVTs. . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Belts for belt drive CVT. . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Belt drive CVTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Variable stroke drives. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Full toroidal CVT, by courtesy of Torotrak. . . . . . . . . . . . . . . 10
1.9 Structures for traction and friction drive CVT. . . . . . . . . . . . . 11
1.10 Geometries of toroidal CVT, by courtesy of Torotrak and NSK. . . . 13
1.11 Optimal operating line of an engine. . . . . . . . . . . . . . . . . . . 17
1.12 Typical variogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Standard structure of S-CVT. . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Velocity constraint diagram. . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Operating principles of S-CVT. . . . . . . . . . . . . . . . . . . . . . 28
2.4 Ideal speed ratio of S-CVT. . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Transmittable torque. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 Coordinate system and forces on S-CVT. . . . . . . . . . . . . . . . 31
ix
2.7 Forces on variator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 3-dimensional concept view. . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Normal force loading device using a spring. . . . . . . . . . . . . . . 42
3.3 Schematic diagram of S-CVT. . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Assembly drawing of S-CVT. . . . . . . . . . . . . . . . . . . . . . . 45
3.5 S-CVT prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Testbench of S-CVT. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7 Experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.8 Power efficiency of S-CVT. . . . . . . . . . . . . . . . . . . . . . . . 49
3.9 Endurance test result of input disc. . . . . . . . . . . . . . . . . . . . 51
4.1 Spin loss in traction drives. . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Classical model of static, kinetic, and viscous friction. . . . . . . . . 55
4.3 Pre-sliding displacement phenomenon. . . . . . . . . . . . . . . . . . 57
4.4 Proposed friction model. . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 Contact of two bodies with different curvature. . . . . . . . . . . . . 61
4.6 Velocity vector field on contact point. . . . . . . . . . . . . . . . . . 63
4.7 Typical relative velocity vector diagram. . . . . . . . . . . . . . . . . 64
4.8 Friction forces at the infinitesimal area of the contact surface. . . . . 67
4.9 Spin losses on S-CVT at input speed of 3000 rpm. . . . . . . . . . . 69
4.10 Dislocation of contact center. . . . . . . . . . . . . . . . . . . . . . . 71
4.11 Change of normal pressure distribution in XZ plane. . . . . . . . . . 72
5.1 Stability of the S-CVT shifting system. . . . . . . . . . . . . . . . . . 87
5.2 Tracking performance of the S-CVT shifting system. . . . . . . . . . 88
5.3 Tracking error and corresponding control. . . . . . . . . . . . . . . . 88
x
5.4 System behaviors of S-CVT during the gear ratio change. . . . . . . 89
6.1 Diagram of an armature-controlled dc motor. . . . . . . . . . . . . . 93
6.2 Efficiency of an armature-controlled dc motor. . . . . . . . . . . . . . 96
6.3 Target profile of output speed. . . . . . . . . . . . . . . . . . . . . . 97
6.4 Computed variator angle time profile. . . . . . . . . . . . . . . . . . 99
6.5 Motor behaviors; reduction gear vs. S-CVT. . . . . . . . . . . . . . . 100
6.6 Power consumption; reduction gear vs. S-CVT. . . . . . . . . . . . . 100
6.7 Interpretation of tilde07Eg(p). . . . . . . . . . . . . . . . . . . . . . 104
6.8 Optimal variator angle time profile. . . . . . . . . . . . . . . . . . . . 106
6.9 Motor behaviors with the minimum energy control. . . . . . . . . . . 107
6.10 Output behaviors with the minimum energy control. . . . . . . . . . 107
7.1 Pivot device for planar accessibility of MOSTS. . . . . . . . . . . . . 112
7.2 Electric circuit diagram of pivot switch and driving motor. . . . . . . 113
7.3 Hardware prototype of MOSTS. . . . . . . . . . . . . . . . . . . . . 116
7.4 The desired trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.5 Calculated wheel velocity profile. . . . . . . . . . . . . . . . . . . . . 117
7.6 Trajectory of variator angle. . . . . . . . . . . . . . . . . . . . . . . . 118
7.7 Motor behaviors of MOSTS. . . . . . . . . . . . . . . . . . . . . . . . 119
7.8 Power consumption; MOSTS vs. differential drive. . . . . . . . . . . 120
7.9 Experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
xi
Chapter 1
Introduction
Power transmissions are a universal element in nearly all mechanical systems, from
a small-sized reduction gear in a compact disc drive, to a complex gear box (usually
referred to as a transmission) in a vehicle. Although their components, sizes, and
operating principles vary, their main objective is to effect changes in the source’s
power in the manner that corresponds to the load condition by manipulating the
transmission ratio (or the gear ratio, i.e., the ratio of the input speed to output
speed). Well-designed power transmissions eliminate the need for oversized power
sources, and increase the power efficiency of overall the system. Even though power
transmissions are required in various engineering fields, research activities are driven
mainly by automobile manufacturers for their conventional transmissions. Thus, in
this dissertation, an overview of power transmissions will be focused on automobile
applications.
1
1.1 Overview
The generated power from ordinary power sources (internal combustion engines,
electric motors, etc.) is much different from the necessary tractive force for driving
vehicles. Hence, it is necessary to transform the power adequately from the source
to the tractive force; the transmission of the vehicle takes this role. General trans-
missions for vehicles can be primarily classified into manual transmissions (MTs)
and automatic transmissions (ATs), according to its actuating mechanism for the
shifting action (decision of shifting time, engaging/disengaging of the power flow el-
ements, selecting the ratio, etc.). A detailed classification of transmissions is shown
in Figure 1.1.
A MT consists of dry clutch, which engages and/or disengages the power flow,
a pair of synchronizing devices and constant meshing gear train for each gear ratio,
and gear ratio selecting devices. Its structure and components are simple enough to
allow for a considerable reduction in size and weight compared to a conventional AT.
Transmission
ContinuouslyVariable
Transmission
Figure 1.1: Classification of transmissions for vehicles.
2
Moreover, a MT is built up with pure mechanical components and has no external
power loss, such as a hydraulic system; thus the power efficiency is quite better than
that of an AT.
A conventional AT consists of wet clutches, planetary gear trains for each gear
ratio, a hydraulic system for shifting action, an electro-hydraulic servo system for
shifting control, and a torque converter. A torque converter is a unique device which
has the multiple roles of a torque multiplication device, starting device, and torsional
damper. Besides disadvantages in size and weight, a hydraulic system including a
torque converter shows significant power loss, reducing the the overall efficiency of
an AT (and ultimately the fuel economy of an AT equipped vehicle). However as
the driving comfort of vehicle becomes the main concern, and greater effort is made
toward improving the efficiency of ATs, the market share of AT equipped vehicles
is growing rapidly.
Power sources have complex efficiency characteristics according to driving con-
ditions. For example, an internal combustion engine has different fuel consumption
rates (or, brake specific fuel consumption: BSFC) according to its speed and torque
while producing the same amount of power (see Figure 1.2). In this figure, there
are two engine operating points which produce the same power for 120 km/hr with
regard to different gear ratios. In the case of gear ratio A, which is greater than B,
the BSFC value of this point is smaller than that of gear ratio B by 10%; hence one
can conclude that a wide-spread of gear ratios is helpful for improving a fuel econ-
omy. In addition, making more gear ratios can enhance the acceleration performance
for the same reason. Many transmission engineers therefore endeavor to develop a
transmission having more gear ratios. But making more gear ratios increases the
size and weight of a transmission.
The continuously variable transmission (CVT) has continued to be an object
3
Eng
ine
torq
ue
Engine speed
100 %
110 %
120 %
130 %
140 %
150 %Constant power
at 120 kph
Driving resistancecurve B
Driving resistancecurve A
BSFC curve
Engine torquecurve at W.O.T.
Vehicle speed 120 kph with gear ratio A
Vehicle speed 120 kph with gear ratio B
10% FCreduction
Figure 1.2: Fuel consumption reduction for an engine.
of considerable research interest within the mechanical design community, driven
primarily by the automotive industry’s demands for more energy efficient and en-
vironmentally friendlier vehicles. Unlike conventional stepped transmissions (MTs
and ATs), in which the gear ratio cannot be varied continuously, a CVT has a con-
tinuous range of gear ratios that can, up to device-dependent physical limits, be
selected independently of the transmitted torque. This feature of the CVT allows
for engine operation at the optimum fuel consumption point consistent with the
given output power requirements, thereby improving the engine’s power efficiency.
Moreover, the CVT does not suffer from “shifting shock” (see Figure 1.3).
In 1886, a CVT with rubber belt and pulleys made by Daimler Benz company
was known as the first CVT to have been applied to a passenger car. About 1930,
General Motors acquired the patent of the toroidal drive, which will be mentioned
in a subsequent section in more detail, and tried to develop their own CVT. But
they failed to commercialize it, and finished the related research in 1935. A different
4
time
speed
time
speed
time
speed
SteppedTransmission
CVT
Vehicle
Power source
repeat accordingto the shifting
stay aroundsome point
regardless of shifting
Figure 1.3: Engine speed variation.
toroidal type CVT, known as a Heyes Self-Selector, was adopted in many Austin
cars, although its production ceased after two years.
The first commercially successful CVT for a passenger car was the rubber belt
Variomatic of DAF Co., developed in 1958. The Variomatic was not popular, be-
cause it failed to resolve the problems of rubber belt failure and the performance
degradation due to deformation and wear. In the 1960s, a CVT using a metal belt
and variable pulleys was developed by Hub Van Doorne, but did not make it to the
market due to its insufficient torque capacity.
In the 1970s, due to the worldwide oil-crisis and the raised environmental recogni-
tion, many countries strengthened the regulations of the fuel economy and exhausted
emissions of vehicles. Moreover by the advances of metallurgy and production tech-
nology, inherent restraints of CVT could overcome; the research and development
for CVT was much encouraged from the late of 1980s. Currently several automobile
manufacturers have developed various prototype CVTs that are soon expected to
5
appear in commercial vehicles (see [1]-[5] and references therein).
1.2 CVTs for Passenger Cars
According to the power transmission element and shifting mechanism, existing CVTs
can be classified into belt drive, traction drive, variable stroke drive, and hydro-
static/dynamic drive (see Figure 1.4).
Belt Drive CVT
In a belt drive CVT, a rubber or steel belt running on conically shaped variable
diameter pulleys is used to transmit power at different drive ratios. According to
the belt material, belt drive CVT can be divided into rubber, chain, and metal belt
type. In Figure 1.5, the schematic diagrams of rubber, chain, and metal belt are
shown. Because of their small power capacity, rubber belt CVTs are adopted in
ContinuouslyVariable Transmission
Friction Drive
Figure 1.4: Classification of CVTs.
6
(a) Rubber belt. (b) Chain belt. (c) Metal belt.
Figure 1.5: Belts for belt drive CVT.
compact cars and machine tools. Passenger cars equipped with a chain belt CVT
had previously appeared on the market, but their production halted before long
(a) ACVT of Aichi Co. (b) Multimatic of Honda Co.
Figure 1.6: Belt drive CVTs.
7
owing to chain noise and vibration problems.
Currently almost conventional CVTs have push type metal belts of Van Doorne’s
Transmissie b.v. or a revised form. Although the metal belt still suffers from a
small torque capacity, the number of production units has been rising steadily in
the worldwide market. The torque capacity has recently increased with the aid of
advances in metallurgy and improvements in the hydraulic system. Figure 1.6 shows
the rubber belt drive CVT made by Aichi Co. and metal belt drive CVT by Honda
Co.
Hydrostatic/Dynamic and Variable Stroke Drive CVT
Hydrostatic/dynamic drives use an incompressible fluid as the transmission medium,
by connecting a hydraulic pump directly to a variable displacement hydraulic actu-
ator. It can realize neutral, forward, and reverse stages, but is not typically applied
to passenger cars, owing to its own low power efficiency, weight, and noise. It has
(a) Cylinder type. (b) Link type.
Figure 1.7: Variable stroke drives.
8
seen limited applications to heavy equipment.
A variable stroke drive is made with one-way clutches and crank devices which
can adjust the crank arm length. The rotational motion of the drive shaft transforms
into translational motion, and the one-way clutch rectifies the motion into a uni-
directional motion. This type of CVT cannot manage properly the pulsative output
torques, and is therefore not adopted in vehicles (see Figure 1.7, by courtesy of DOE
report [6]).
Traction and Friction Drive CVT
Friction wheels of unequal diameter were one of the earliest speed changing mech-
anisms. It is speculated that their use even predates that of gearing “toothed”
wheels, whose beginnings date back to the time of Archimedes, circa 250 B.C. [7].
Even today, friction drives may be found in equipment where a simple and eco-
nomical solution to speed regulation is required: phonograph drives, self-propelled
lawnmowers, or even amusement park rides driven by a rubber tire are a few of the
more common examples. In these examples, simple dry contact is involved, and the
transmitted power levels are low. However, this same principle can be harnessed
in the construction of an oil-lubricated, all steel component transmission which can
carry hundreds of horsepower using today’s technology. In fact, oil-lubricated trac-
tion drives have been in industrial service as speed regulators for more than 70 years
[8].
Great progress in tribology research since late 1960s, particularly research on
elasto-hydrodynamic lubrication (EHL) traction, has made it easier to understand
the traction drive mechanism. Traction drives transmit power through an increased
shear force, which results from elasto-hydraulic shear stress of the traction oil be-
tween two rotating solid bodies. A coefficient of traction is typically 0.1, and macro-
9
scopic slip occurs at any time. Since there is no direct contact between the rotating
bodies, wear phenomenon does not occur. The drive ratio is varied by controlling
the effective onset radius of the contact point.
According to the geometries of rotating elements, there are various type of trac-
tion drive CVTs: nutating drive, half toroidal type, and full toroidal type. The half
toroidal CVT has semi-circular discs, while full toroidal CVTs have full-circular discs
as input/output rotating elements. According to the curvature radii of discs, they
have different attainable gear ratios, torque capacity, and spin loss. Figure 1.8 shows
the full toroidal drive and pertinent CVT made by Torotrak Ltd. in UK. Apart from
this, many automobile manufacturers have developed half toroidal CVTs with dif-
ferent torque capacities in Japan. Along with the traction oil developers (Santotrak,
Shell companies), they have presented various prototypes of traction drive CVTs in
the market [9].
Generally, a traction drive shows rapid shifting response compared to belt drives,
(a) Full toroidal traction drive. (b) Pertinent CVT.
Figure 1.8: Full toroidal CVT, by courtesy of Torotrak.
10
and can be adopted to medium-sized (even to large-sized) vehicles, because the
highly-pressurized traction oil endures more shear stress than belt drives. The draw-
backs of a traction drive are known to be as follows: the need for careful temperature
control, sealing and supply of traction oil, and complicated shifting control due to
the three dimensional contact curvature of the rolling elements.
Finally, there exist friction drives where the power transmission mechanism is
via rolling resistance and friction force in direct contact, though its structure and
operating principle are much similar to traction drives (see Figure 1.9, by courtesy
of DOE report [5]). Friction drives have been also found in several types of wood-
working machinery dating back to before the 1870s. For example, [10] reports of a
frictional gearing being used to regulate the feed rate of wood on machines in which
one wheel was made of iron and the other, typically the driver, of wood (or iron
covered with wood).
Figure 1.9: Structures for traction and friction drive CVT.
11
Friction drives have not been considered for passenger cars due to its low torque
capacity, wear, heat dissipation problems, etc.However, friction drives have received
significant attention from the perspective of tribology, because precise positioning
can be accomplished while avoiding backlash [11]-[15]. Furthermore, traction and
friction drives provide much design flexibility in terms of their structure and al-
lowance for compact-sized designs.
Although each type of CVT has its own particular set of advantages and disad-
vantages, common difficulties shared by current CVTs are the complicated shifting
controller design, and the need for a large-capacity, typically inefficient shifting ac-
tuator [5]. Also, these CVT designs do not have infinitely variable transmission
(IVT) capabilities, i.e., they do not include zero output speed among its available
ratios, and therefore require a clutch or other type of starting and engaging device
for initially driving the vehicle.
1.3 Literature Review
There is a vast amount of literature regarding the design, analysis, control, and
application of CVTs in engineering fields. This section focuses on the areas of
design and control of traction/friction drives, because the proposed spherical CVT
in this thesis shows similar characteristics with respect to operating principles, power
transmission and shifting mechanisms, and control laws.
Traction Drive Designs
It is well known that one of the earliest examples of the friction drive was the patent
of C. W. Hunt in 1877 [16]. Basically the mechanism of that drive was a toroidal
drive, which was developed for more than a decade thereafter. Applications of
12
traction drives to automobiles have been studied since the beginning of this century.
Prior to 1935, cars were called “Friction Drive Cars”, experimentally installed with
such drives, had received some attention: it was widely believed that power was
transmitted by friction between the rolling metallic elements.
In the latter half of the 1960s, when elsto-hydraulic lubrication (EHL) became
better understood [17], it was recognized that power was transmitted by traction.
The performance of a traction drive depends to a large extent on the rheological
properties of the fluid in the EHL contact [18]-[29]. In the 1970s, synthetic traction
oil was developed which had a traction coefficient almost 50% higher than before,
and practical use of the traction drive CVT was thought to be close at hand. It was,
however, not realized, because the heat treatment of the rolling elements could not
be achieved. A new type of traction oil being developed for automotive use shows
some promise [30], [31].
(a) Full toroidal CVT. (b) Half toroidal CVT.
Figure 1.10: Geometries of toroidal CVT, by courtesy of Torotrak and NSK.
13
As stated earlier, there are two main design streams in traction drives, for auto-
motive use full and half toroidal type CVTs. A full toroidal CVT [32] has full-circular
discs as input/output media and power rollers as shifting devices (see Figure 1.10
(a)). In a full toroidal CVT, power rollers are located at the center of the toroidal
shaped input and output discs. A hydraulic loading system has commonly been used
to supply the normal force, which is necessary to transmit power via traction. Its
shifting mechanism is based on the side-slip force generated by the velocity difference
of the contact point.
A half toroidal CVT uses semi-circular discs instead full-circular ones, though
the shifting mechanism is not different from full toroidal CVTs (Figure 1.10 (b)).
Many engineers including P. W. R. Stubbs (1980), Lubomyr O. Hewko (1986), M.
Nakano (1991, 1999), H. Kumura (1999), and H. Machida (1999) have presented the
trends and issues on half toroidal CVT designs for use in full-sized cars as a future
driveline technology [33]-[38]. Nakano (1991) reported that the main reasons for the
unsuccessful commercialization of toroidal CVTs were thought to be the inability to
obtain sufficient performance with respect to the traction and viscosity performance
of the traction fluid, the fatigue strength of the rolling elements, power transmission
efficiency, transient ratio change controllability, and the issue of synchronization
control in connection with the parallel arrangement of the traction elements [35].
A traction drive CVT changes its speed ratio by controlling the side-slip force
on the Hertzian contact area. Tanaka and Eguchi (1991) showed the principles of
the speed ratio changing mechanism of half-toroidal CVTs and highlighted a digital
compensation method for stabilization of the electro-hydraulically operated speed
ratio control mechanism [39]. Fellows and Greenwood (1991) reported that it might
not be possible to suppress hunting of the shifting control signal, depending on the
control system used [40]. In addition to these results, there are many materials
14
related to toroidal CVT controller designs (for example [41], [42], and references
therein).
When the ratio changes in a half toroidal traction CVT, the necessary contact
force does not vary significantly compared to a full toroidal CVT [43]. Consequently,
a loading cam system of a half toroidal CVT that produces contact force in pro-
portion to the input torque can provide high efficiency over the entire speed ratio
range, contrary to the hydraulic loading system of a full toroidal type. Moreover
it has been reported that the full toroidal traction CVT suffers larger spin moment
at the contact points than the half toroidal type, which tends to reduce its power
capacity [44].
The current design issues on toroidal type traction drives can be summarized as
follows:
• the material of rolling elements is not sufficiently reliable because of high pres-
sure and high temperature on the traction contact point;
• there is no affordable traction oil which satisfies all the conditions of automo-
biles, although it has been reported that an adequate traction oil has been
developed recently [9], [31];
• there are no bearings which can support high speeds and a large axial load;
• the normal force loading system (e.g., hydraulics or loading cam), which is
necessary to produce traction force, is inefficient and needs precise control for
equalizing the normal forces on the contact points;
• there are difficulties on the control of transient ratio change and synchronized
precise control in connection with the parallel arrangement of the traction
elements.
15
CVT Controls
A CVT is originally intended to operate the power source in power efficient regimes
by means of manipulating its gear ratio. Many previous efforts are focused on finding
the power efficient regimes of sources and controlling the gear ratio of a CVT in order
to run the source within those regimes. Here we review the previous analysis results,
which describe ways of controlling a CVT for maximizing the fuel economy of a
CVT-equipped vehicle as well as how to establish the shift schedule (the so called
“variogram”, which describes the graphic relation between the engine and vehicle
speeds) of a CVT for the vehicle’s performance objectives.
Generally the power efficiency of a source is maximized at only one point over
its operating region. In an internal combustion engine (see Figure 1.2), the fuel
consumption is lowered for higher engine torque. On the other hand, it worsens for
high engine speeds as the mechanical loss is large at those speed points. The pumping
loss tends to be large for low engine speeds; hence, the fuel consumption also worsens
for low engine speeds. These characteristics are consistent with theory. If we operate
the engine only at the most efficient point, however, the driving performance may
not be satisfied, because the driving torque to be generated for each vehicle speed is
limited. Therefore, the control and optimization of automotive powertrain systems
with a CVT is achieved by cooperative control of the engine and CVT (see [2]-[4],
[45]). A ‘drive-by-wire’ structure using an electric throttle control device is adopted
for this engine consolidated CVT control [46].
Figure 1.11 shows an optimal operating line (OOL) of a typical engine for max-
imum fuel economy. Generally, an OOL is constructed simply by connecting static
BSFC contours through optimization. For improving the fuel economy of a vehicle,
it is definitely helpful to control the CVT gear ratio so as to run the engine along
16
100 %
110 %
120 %
130 %
140 %
150 %
Figure 1.11: Optimal operating line of an engine.
this operating line. Most CVT-equipped vehicles use shift schedules (or variograms)
in look-up table form, presetting the optimal gear ratios obtained from the static
performance data of the engine and road tests of the prototype vehicle (see Figure
1.12). However, this OOL does not involve the vehicle dynamics including accel-
eration, performance objectives, because there is no consideration for the engine
dynamics.
The “classical” way to control CVT cars is to use some information on the gear
ratio or on the transmitted torque which is then fed back by a PID controller [47]-[49].
Only when using gain-scheduled controllers with typically 100 different gain points
could the required performance be achieved. Kolmanovsky et al. (1999) explored the
use of a CVT for torque management during mode transitions in lean burn gasoline
engines [50]. They demonstrated that an intuitively sound CVT gear ratio control
strategy which attempts to completely cancel the engine torque disturbance may
result in unstable zero dynamics. They concluded the coordination of engine torque
17
Figure 1.12: Typical variogram.
production and CVT gear ratio control during mode transitions was mandatory.
Takahashi (1998) proposed a scheme to minimize rate of fuel consumption by a
direct fuel injection engine used by combination with CVT [45]. Target values for
the engine and transmission which minimize fuel consumption ensuring driving per-
formance were calculated based on the nonlinear optimization method. As a result
of optimization, target values for air-fuel ratio and gear ratio were calculated and
controlled by tracking. Because the calculation of partial differential was impossible
at some operating points, he used a simplex method that did not require calculat-
ing differential values. For minimization of fuel consumption function under various
restrictions, penalty functions were also introduced.
The non-minimum phase behavior of the CVT based powertrain system (without
a torque converter) was mentioned in [51]. Considering this phase behavior of CVT,
Guzzella and Schmid (1995) addressed an exact feedback linearization approach for
a controller of CVT equipped vehicle [52]. In their works, the plant dynamics were
18
exactly linearized over the complete operating range using feedback linearization.
And as an application of the exact linearization approach, a “kick-down” controller
was designed.
1.4 Outline and Contributions
This dissertation deals with the design, analysis, and control of a spherical CVT. A
conceptual design of a particular spherical CVT (S-CVT) was proposed by Joukou
Mitsusida in [53]. The S-CVT consists of a sphere, input and output discs, and
variators. The rotating input and output discs are connected to the power source
and output shafts, respectively, while the sphere is situated between the input and
output discs. The transmission ratio is controlled by adjusting the location of the
variator on the sphere, which in turn controls the axis of rotation of the sphere. It
transmits power via dry rolling friction on the contact points of sphere and discs;
therefore, there exists a torque limitation decided by the static friction force.
The S-CVT, intended to overcome some of the aforementioned limitations of
existing CVT designs, is marked by its simple kinematic design and IVT charac-
teristics, i.e., the ability to transition smoothly between the forward, neutral, and
reverse states without the need for any brakes or clutches. Moreover its relatively
simple control makes it particularly effective for mechanical systems in which ex-
cessively large torques are not required (e.g., mobile robots, household appliances,
small-scale machining centers, etc.).
In order to put the S-CVT to practical use, an analysis of its operating principles,
power transmission and shifting mechanisms, and power capacity together with the
consideration for issues of hardware design, needs to be performed. This dissertation
is aimed at providing theoretical and practical solutions for these concerns, through
19
an in-depth study of the design, dynamics, and control of the S-CVT . This work can
be categorized into four parts:
• analysis of theoperating principles, kinematics, and dynamics in Chapter 2;
• hardware design issues including a slip loss analysis in Chapter 3, 4;
• shifting controller and minimum energy control law design in Chapter 5, 6;
• application for a wheeled mobile robot as a case study in Chapter 7.
The subsequent achievements in this work can be exploited to the design and analysis
of traction or friction drives having similar structure.
Currently, we are carrying out the development of other S-CVT applications
for small-capacity speed changers, e.g., bicycles, laundry machines, wind-propelled
generating systems, potter’s spinning wheels, etc.There still remain several practical
problems, such as realizing precise shaft alignments and increasing the torque capac-
ity. Currently research efforts are being directed toward the application of traction
fluid for the purpose of adopting the S-CVT for large torque capacity applications,
e.g., hybrid vehicles, compact cars, etc.
The detailed outlines and contributions of each chapter can be stated as follows.
Operating Principles, Kinematics, and Dynamics
Chapter 2 describes the conceptual design and operating principles of the S-CVT
together with a detailed kinematic and dynamic analysis of its performance. In
addition, there shows analytic interpretation for the reaction forces of S-CVT which
are normally exerted on variator and discs, along with the definitions of their physical
meanings.
20
Hardware Design Issues and Slip Loss
Chapter 3 presents the prototype specifications and a discussion of the main design
issues, focusing on the normal force loading device and the shifting actuator capacity.
Some experimental results are given on the actual transmission ratios and power
efficiency obtained from a prototype testbench, to validate the operating principles
and performance of S-CVT. We briefly address the strength and life estimation for
the S-CVT, based on the well-known ball-bearing life theory.
Spin loss of S-CVT, which is one of the main power losses of the S-CVT (and
more generally for friction and traction drives) due to slippage, is formulated using
a modified classical friction model in Chapter 4. The proposed friction model can
involve pre-sliding effect i.e., Stribeck effects. For this, we perform an in-depth
study of velocity fields and the normal pressure distribution generated on the contact
regions. We also provide a quantitative analysis of the spin loss of the S-CVT. In
addition, we discuss contact analysis involving slip, in which a shear force resulting
from friction occurs on the contact surface.
Shifting Controller Design and Minimum Energy Control
The shifting system of the S-CVT has second-order nonlinear dynamics, for which
typical open-loop control systems are likely to develop unstable characteristics. In
order to cancel the nonlinearity of the S-CVT shifting system and to make it stable
and responsive to shifting commands, we develop a feedback controller based on
the exact feedback linearization method in Chapter 5. We first investigate the
instability of the S-CVT shifting system using the Lyapunov’s indirect method. We
then present the input-state feedback controller design of the S-CVT shifting system,
and investigate the stabilizing and tracking performance of the dedicated shifting
21
controller by numerical simulation.
Chapter 6 deals with a minimum energy control law for the S-CVT connected
to a dc motor. We first investigate the general power efficiency of a dc motor. We
then present the results of a numerical investigation of the S-CVT energy saving
possibility benchmarked with a standard reduction gear. For this investigation, a
computed torque control algorithm for the S-CVT is proposed. In addition, we
describe a minimum energy control law of S-CVT connected to a dc motor. To do
this, we describe the general power efficiency characteristics of a dc motor. Then
the minimum energy control design is carried out via B-spline parameterization.
Numerical results obtained from simulations illustrate the validity of our minimum
energy control design.
An S-CVT based Mobile Robot
Finally, we propose an S-CVT based mobile robot (denoted as MOSTS for a Mobile
rObot with a Spherical Transmission System) to realize the various advantages of
the S-CVT, including the originally intended CVT characteristic of energy efficiency,
into practical use in Chapter 7. In this chapter, we first address the motivation
for applying the S-CVT to a wheeled mobile robot by first reviewing the current
hardware designs of mobile robots and their power efficiency. We then present the
hardware design of our S-CVT based mobile robot in accordance with the target
performance. In addition, we propose a novel pivot mechanism which is necessary
for planar accessibility using an internal gear and an uncontrolled dc motor. We
perform both numerical simulations and experiments for various motion plans, in
order to validate the realization of the robot’s operation, the CVT characteristics,
and its energy saving possibility.
22
Chapter 2
Dynamic Analysis of the
Spherical CVT
2.1 Introduction
In this chapter a new type of spherical continuously variable transmission (S-CVT)
is described. The S-CVT, intended to overcome some of the aforementioned limi-
tations of existing CVT designs, is marked by its simple kinematic design and IVT
characteristics, i.e., the ability to transition smoothly between the forward, neutral,
and reverse states without the need for any brakes or clutches.
Because the S-CVT transmits power via rolling resistance between metal on
metal, it has limitations on the overall transmitted torque, which is effectively de-
termined by the static coefficient of friction and the magnitude of the normal forces
applied to the sphere. Due to this torque limitation, the S-CVT is not intended
for automobiles and other large capacity power transmission applications. Target
applications for the S-CVT include mobile robots, household electric appliances,
23
small-scale machine tools, and other applications with moderate power transmis-
sion requirements. Although the current design of the S-CVT is based on friction
drive designs, it is our expectation that the power capacity of the S-CVT can be
increased by the use of traction oil, an issue which we do not pursue further in this
dissertation.
Other spherical CVT structures have been proposed for use in passive mobile
robots and for use as nonholonomic joints in robot manipulators. Carl A. Moore et
al. (1999) have reported a 3R passive robot, called the Cobot. The Cobot adopts a
rotational CVT to provide smooth, hard virtual surfaces for passive haptic devices in
place of conventional motors. Its rotational CVT consists of a sphere caged by four
rollers, and adopts the joint speeds and task space speeds along with the steering
angles as control inputs [54]. Another application can be found in underactuated
manipulators, designed by Søerdalen et al. (1994). This work proposes a new type
of manipulator architecture using a CVT-type robot joint that takes advantage of
the inherent nonholonomy of the CVT [55]. Although these systems are designed
to manipulate the speed ratio using a CVT mechanism, their main purpose is not
for power transmission to improve the energy efficiency. Furthermore, the shifting
mechanism of the S-CVT is quite different from these previous designs, as will be
described below.
In this chapter, the conceptual design and operating principles of the S-CVT
are described together with a detailed kinematic and dynamic analysis of its perfor-
mance. Section 2 describes the basic kinematic structure and operating principles
of the S-CVT. In Section 3, we examine the dynamics of the S-CVT by deriving
the equations of motion and its shifting mechanism. Finally in Section 4, we exam-
ine the reaction forces on the S-CVT, in particular those exerted normally on the
variator and discs.
24
2.2 Kinematics of S-CVT
2.2.1 Structure
The S-CVT is composed of three pairs of input and output discs, variators, and a
sphere (see Figure 2.1). The input discs are connected to the power source, e.g., an
engine or an electric motor, while the output discs are connected to the output
shafts. The sphere, which is the main component of the S-CVT, transmits power
from the input discs to the output discs via rolling resistance between the discs and
the sphere. The variators, which are connected to the shifting controller, are in
contact with the sphere like the discs, and constrain the direction of rotation of the
sphere to be tangent to the rotational axis of the variator.
Figure 2.1: Standard structure of S-CVT.
25
The speed and torque transmission ratios of the S-CVT vary with the angular
displacements of the variators; this will be described in further detail in the following
subsection on the operating principles of the S-CVT. To transmit power from the
discs to the sphere or from the sphere to the discs, a device that supplies a normal
force to the sphere, such as a spring or hydraulic actuator, must be installed on each
shaft. As can be seen in Figure 2.1, the structure and components of the S-CVT
are simple enough to allow for a considerable reduction in size and weight compared
to conventional transmissions. The orientations of the input and output shafts can
also be located freely using rollers at arbitrary positions rather than discs.
2.2.2 Operating Principles
When the input device is actuated by a power source, the input disc rotates about the
input shaft. This rotation in turn causes a rotation of the sphere, due to the condition
of rolling contact without slip between the input discs and the sphere. Rotation of
the sphere in turn causes a rotation of the output discs, and subsequently of the
output shaft. In the absence of any contact between the sphere and the variator, the
axis of rotation of the sphere will largely be determined by an equilibrium condition
among the various contact and load forces being applied to the sphere.
The role of the variator is to control the axis of rotation of the sphere. Specifically,
referring to Figure 2.2, the variator contacts the sphere at a point (marked by P )
located directly above the sphere center. Since the variator rotates about an axis
normal to the variator disc and passing through the variator center (marked by C1,
C2), it follows that the contact point between the variator and the sphere undergoes
a linear velocity in a direction tangential to the variator disc (marked by V1, V2).
By adjusting the location of the variator (from C1 to C2) it is therefore possible to
control the axis of rotation of the sphere (from ω1 to ω2); the axis will be parallel
26
V1
V2
∆ V
∆θ
ω 1
ω 2
P
C2
C1
Figure 2.2: Velocity constraint diagram.
to the line between the variator center and the sphere-variator contact point, and
passing through the sphere center.
By varying the axis of rotation of the sphere, it is in turn possible to vary the
radius of rotation of the contact point between the input disc and the sphere, Ri,
as well as the radius of rotation of the contact point between the output disc and
the sphere, Ro (see Figure 2.3). In this way the speed-torque ratio of the S-CVT
can be adjusted. Figure 2.3 shows the various alignments of the variator for the
forward, neutral, and reverse states of the output shaft of the S-CVT. The neutral
state, which corresponds to zero rotation of the output disc, is achieved when Ro
becomes zero. As apparent from the figure, the forward, neutral, and reverse states
can all be achieved by smoothly manipulating the variator alignment , without the
need for any additional clutches or brakes.
Assuming roll contact without slip, the speed and torque ratio between the input
27
and output discs is related to the variator angle by the following relations:
ωout
ωin=
riro
tan θ (2.1)
Tout
Tin=rori
cot θ (2.2)
where θ is the angular displacement of the variator, ωin and ωout are the respective
angular velocities of the input and output shafts, Tin and Tout are the respective
input and output torques, and ri and ro are the respective radii of the contact
points of the input and output discs (see Figure 2.3). There are two design variables
that prescribe the transmission ratio: the ratio of the input and output contact
ri
ro
RRi
Ro
ri
ro
RRi
ri
ro
R
Ri
Ro
variator angle
Figure 2.3: Operating principles of S-CVT.
28
-50 -30 -10 10 30 50
0.10.30.50.70.91.1
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Figure 2.4: Ideal speed ratio of S-CVT.
radiiriro, and the variator angle θ. From Equation (2.1) it is apparent that a large
range of available transmission ratios is possible even with a sphere of small radius.
Assuming that there is no slip or other physical effects, the ideal speed ratio of the
input speed to output speed is shown in Figure 2.4.
Although ideally an infinite torque ratio is possible with the S-CVT as seen in
Equation (2.2), in practice there is a limit to the torque that can be transmitted
because power transmission occurs from rolling resistance of metal on metal. Figure
2.5 shows a plot of the torque ratio as a function of the variator angle, for a given
fixed input torque. The actual torque ratio of the S-CVT will lie somewhere in the
operating region as indicated in the figure because of power loss due to friction, slip,
heat generation. The limiting torque Tmax is determined by the static coefficient of
friction µs and the normal force N exerted by the output disc spring mechanism on
29
the sphere according to the relation Tmax = rµsN , where r is the contact radius of
the disc. When either the input or output torque applied at the disc-sphere contact
exceeds this limit, slippage can occur. Taking into account this limiting torque, the
output torque for a given input torque Tin is given as follows:
Tout = Tmax · sat(Tin
Tmax
rori
cot θ)− TLoss (2.3)
where the saturation function sat(·) is defined by
sat(x) =
sgn(x) if |x| > 1
x if |x| ≤ 1
,
and TLoss is the torque loss in S-CVT. Though assuming roll contact without slip
(i.e., the speed ratio can be realized as the ideal case), the torque loss cannot be
zero, because there exist some torque losses resulted from the spin moments and
internal loads, etc., which will be discussed in Chapter 3 and 4.
Figure 2.5: Transmittable torque.
30
2.3 Dynamics of S-CVT
2.3.1 Motion of Sphere
To investigate the shifting mechanism of the S-CVT, we designate a reference frame
XYZ situated at the center of the sphere, and a moving reference frame xyz, with
z coinciding with the spin axis of the sphere (see Figure 2.6). The various external
forces acting on the sphere are also shown in this figure, neglecting the normal
forces exerted on the sphere-discs contact points to hold the sphere and the weights
of sphere and discs.
We define the driving forces which are delivered from the input discs as FZi1 and
FZi2, and the reaction forces exerted by the load torque from the output discs as
Figure 2.6: Coordinate system and forces on S-CVT.
31
FZo1 and FZo2. Ftv1 and Ftv2 denote the forces generated by the shifting actuator
acting at the sphere-variator contact points. The remaining reaction forces at the
input and output discs and variators are respectively denoted by FXi1, FXi2, FY o1,
FY o2, Fnv1, Fnv2.
Assuming that the sphere center does not move and the rotational axis of the
sphere lies on the xy plane, the force equilibrium conditions for each coordinate are
as follows:
FXo1 − FXo2 + FXi1 − FXi2 + (Fnv1 − Fnv2) cos θ − (Ftv1 − Ftv2) sin θ
FY i1 − FY i2 + FY o2 − FY o1 + (Fnv1 − Fnv2) sin θ + (Ftv1 − Ftv2) cos θ
FZo1 − FZo2 + FZi1 − FZi2 + FZv1 − FZv2
= 0. (2.4)
With respect to the specified coordinate frames, we can derive the dynamic equations
relating the angular momentum change with the resultant moment acting on the
sphere, i.e.,
d
dtHo =
∑
Mo
where Ho is the angular momentum and∑
Mo is the resultant moment. The
angular momentum of the sphere is given by:
Ho = Is × ω =2
5msR
21× ω
where ω is the angular velocity of the sphere, Is is its mass moment of inertia, ms
the mass, and R the radius. Expressing the angular momentum of the sphere in
terms of the moving coordinate frames, we obtain the derivatives of this momentum
and the resultant moments, leading to the following set of second-order differential
equations:
Is
θω
θ
ω
=
(Fnv1 + Fnv2)R − (FZi1 + FZi2)R sin θ − (FZo1 + FZo2)R cos θ
(FY o1 + FY o2 + FXi1 + FXi2)R
−(Ftv1 + Ftv2)R − (FZo1 + FZo2)R sin θ + (FZi1 + FZi2)R cos θ
(2.5)
32
where ω is the spinning rate of the sphere.
Considering the attributes of the external forces, as stated earlier, we can restate
those forces in Equation (2.5) as follows:
FZi1 + FZi2 = Fi,
FZo1 + FZo2 = Fo,
Ftv1 + Ftv2 = Ft,
Fnv1 + Fnv2 = Fn.
In the above equations, Fn should not be regarded as an active force for shifting,
but rather as a loss-like force acting to resist any variator displacements. Examining
the reaction forces at the input and output discs caused by changes in the sphere
axis of rotation, we can also conclude that the magnitudes of these forces must be
equal, otherwise the sphere will be distorted:
FXi1 = FXi2 = FY o1 = FY o2 = D. (2.6)
The relevant forces can therefore be summarized as follows:
Fi = Driving force delivered from the input discs;
Fo = Reaction force caused by the output discs connected to the load torque;
Ft = Shifting force on sphere delivered from the variator in the tangential direction;
Fn = Loss-like reaction force exerted normally on variator;
D = Reaction force on sphere generated by the shifting.
2.3.2 Shifting Dynamics
To establish the dynamic relations between the sphere and variator, we first define
the forces on the upper sphere-variator contact point and the connected shifting
33
Figure 2.7: Forces on variator.
actuator (see Figure 2.7). To permit spinning motion of the variator, bearings are
located on the connecting rod, which connects the shifting actuator and variator.
In this figure, θ denotes the angular displacement of the shifting actuator, which
consists of the same number of variators, m is the mass of the variator, and ε the
eccentric distance between the centers of the shifting actuator shaft and variator. In
addition, Fsv1 is the shifting force delivered by the shifting actuator, a is the linear
acceleration of the variator center, and ωv1 the rotational speed of variator.
Using the velocity constraint on the sphere-variator contact point, one can obtain
the rotational speed of variator ωv1
ωv1 = θ +R
εω (2.7)
34
where ω is the spinning rate of the sphere, and R the sphere radius as defined earlier.
Let the moment of inertia of the shifting actuator shaft and connecting rod be Ia,
and that of the variator be Iv. The shifting force delivered from the variator onto
the sphere in the tangential direction (Ftv1) can be written as
Ivωv1 = εFtv1. (2.8)
By the force relation Fsv1 = ma+Ftv1, and using the fact that the linear acceleration
of the variator a = ε θ, as well as Equations (2.7), (2.8), we can express the shifting
torque εFsv1,
εFsv1 = (Iv +mε2 + Ia) θ + IvR
εω. (2.9)
We assume that the lower variator always runs synchronously with the upper
one; then the total shifting force Fs = 2Fsv1 and ωv1 = ωv2. Rearranging the
equations of the sphere and variator (2.5) and (2.9), we obtain the following set of
second-order differential equations for the S-CVT:
2 (Ia+Iv+mε2)ε 2RIv
ε2
2 Ivε
IsR + 2RIv
ε2
θ
ω
=
Fs
Fi cos θ − Fo sin θ
. (2.10)
The reaction forces are given by
Fn =IsR
θω + Fi sin θ + Fo cos θ, (2.11)
D =1
4
IsR
θ, (2.12)
Ft = 2Ivε(θ +
R
εω). (2.13)
2.4 Reaction Forces of the S-CVT
The two main reaction forces of the S-CVT are Fn and D, which are exerted respec-
tively at the contact points between the sphere and discs, and sphere and variators.
35
In order to prevent slippage, they must be smaller than the maximal friction force.
In this section we derive analytic expressions for Fn and D, and examine their effect
on the performance of the S-CVT.
2.4.1 Normal Reaction Force Exerted on the Variator: Fn
Fn, the reaction force which is exerted normally on the variators, can be considered
as a loss force, and restricts the available gear ratios. Since Fn acts ultimately on
the bearings located within the connecting rod, which connects the shifting actuator
and variator (see Figure 2.7), it can therefore cause excessive bearing normal forces
and bending moments on the variator and connected shafts.
From Equation (2.11), Fn at steady state becomes
Fn = Fi sin θ + Fo cos θ. (2.14)
From the fact that the shifting effort Fs is zero at steady state, the relation between
Fi and Fo of Equation (2.10) becomes
Fi cos θ = Fo sin θ.
Substituting Fi into Equation (2.14), Fn becomes
Fn =Fo
cos θ. (2.15)
Beyond a certain variator angle, the magnitude of Fn becomes larger than the max-
imal friction force which is determined by the static coefficient of friction µs and
the normal force N ; slippage therefore occurs at the sphere-variator contact point
(similar to the limiting torque Tmax).
During transient states, the dynamics of sphere and variator θω influences the
magnitude of Fn additionally. More than any other reaction forces on the S-CVT,
36
Fn varies considerably together with input/output force and the shifting dynamics;
thus it contributes the limit of available gear ratios of S-CVT. The allowable range
of Fn during transient states is
Fn =Fo
cos θ+IsRθω ≤ µsN .
Rearranging this, we obtain a range for the gear ratio θ:
| θ | ≤ cos−1( Fo
µsN − IsR θω
)
. (2.16)
In order to increase the range of available gear ratios, one can reduce the internal
load and hence increase the output force Fo, or decrease the shifting response θ, as
well as improve material properties with respect to µs, N .
2.4.2 Shifting Reaction Force on the Sphere: D
There are four contact points between the sphere and input/output discs in the S-
CVT (see Figure 2.1). When shifting (i.e., changes in gear ratio) occurs, the reaction
force D, which resists the angular momentum change of the sphere, appears at each
contact point. The reaction force D is normally exerted on the discs, and it acts
directly on the bearings located within the input/output shafts; thus it can be
considered as loss force like Fn.
Moreover from Equation (2.12),D is related with the shifting response θ, and acts
to restrict the available shifting response. As is the case for Fn, slippage resulting
from a reaction force larger than the maximal friction force makes the S-CVT unable
to transmit power; therefore the following inequalities must be hold:
D ≤ µsN , θ ≤ 4R
IsµsN . (2.17)
From this relation, we can conclude that the shifting response is constrained by the
material properties µs, the sphere geometries R, Is, and the normal force N .
37
2.5 Summary
In this chapter, we have presented the design and analysis of a newly developed
spherical continuously variable transmission (S-CVT) focusing on its basic structure
and operating principles, shifting mechanism, and its reaction forces. The S-CVT is
intended to overcome some of the limitations of existing CVTs, e.g., difficult shifting
controller design, and the necessity of a large-capacity and typically inefficient shift-
ing actuator. It is marked by its simple configuration, infinite variable transmission
(IVT) characteristics and realization of forward, neutral, and reverse states without
any brakes or clutches.
Because the S-CVT transmits power through rolling resistance between metal on
metal, torque limitations prevent current versions of the S-CVT from being applied
to large capacity power transmission systems like passenger cars. However, our
study suggests that it can be well-suited for applications involving small mechanical
systems such as mobile robots, household electric appliances, small-scale machining
centers, etc.
Finally, we have investigated the reaction forces which are exerted normally on
the variator and discs. Both Fn and D constitute sources of power loss for the
S-CVT; in particular, the magnitude variation of Fn along the variator angle is
steeper than any other forces on S-CVT. Moreover Fn can be a dominant factor in
determining the available range of gear ratios of the S-CVT. The shifting reaction
force D is related with the shifting response θ and acts to restrict the available
shifting response.
38
Chapter 3
Prototype Design and
Experimental Results
3.1 Introduction
In designing a transmission, one must consider both the power capacity of the trans-
mission and power source as well as the load conditions. In this chapter, we first
define the design objectives of the S-CVT, taking into account its inherent charac-
teristics such as the power transmission mechanism based on friction force, shifting
mechanism, and operating principles.
The proposed S-CVT is intended for use in small capacity mechanical systems,
e.g., mobile robots, household electric appliances, small-scale machine tools, and
other applications with moderate power transmission requirements. In determining
the hardware specifications of the S-CVT, practical issues such as the amount of
normal force required to assure rolling resistance at the contact points of the S-
CVT, and the capacity of the shifting actuator that can realize the desired shifting
39
response, and the range of available gear ratios must all be considered. Based on the
kinematic and dynamic analysis results of the previous chapter, we have designed
and built the following S-CVT prototype.
In this chapter, we present the prototype hardware specifications for the S-CVT,
and an analysis of its performance. Using a prototype testbench, we obtain ex-
perimental results that serve to validate the operating principles and performance
of the S-CVT. In Section 2, we discuss various issues in the mechanical design of
the S-CVT, focusing on the normal force loading device and the shifting actuator
capacity. To assure rolling resistance force at the contact points of the S-CVT, we
adopt compressible springs because of their simple structure and the ease in ad-
justing the preset load. To determine the shifting actuator capacity, we derive the
Figure 3.1: 3-dimensional concept view.
40
numerical relationship between the necessary power and the shifting demand using
the previous dynamic analysis results. Section 3 shows the hardware specifications
and schematic drawings of the prototype S-CVT. In Section 4, we present experi-
mental results on the actual transmission ratios and power efficiency obtained from
the prototype testbench. Finally, we briefly address the strength and life estimation
of the S-CVT, based on the well-known ball-bearing life theory.
3.2 Issues in Mechanical Design
Among the relevant issues in designing the S-CVT, we will focusing in particular
on the normal force loading device and the capacity of the shifting actuator. The
considered issues are mainly related to power capacity, namely the maximal trans-
mittable force and the shifting actuator design.
3.2.1 Normal Force Loading Device
In order to assure rolling resistant force at the contact points of the S-CVT, an appro-
priate normal force should be applied on the sphere and discs. Compressible springs
are employed at each shaft, which are connected to the variators and input/output
discs, to make the mechanical structure simple and to adjust the amount of normal
force easily (see Figure 3.2). Since the spring force is closely related with the limit
of transmittable force, we need to measure and adjust it. Using a set-screw, the
amount of normal force can be adjusted by fixing the preset displacement of the
spring. To set an accurate spring force, strain gauges are attached to each relevant
shaft.
Regarding the amount of normal force, the larger spring force increases the trans-
mittable force. However, applying too large normal force causes yielding and plastic
41
Figure 3.2: Normal force loading device using a spring.
deformation of the sphere and discs; careful consideration for the normal stress on
the contact regions must be carried out. In this study, we have designated the nor-
mal force amount as 100 kgf using the corresponding finite element analysis results
obtained by ANSYS.
3.2.2 Capacity of Shifting Actuator
In order to determine the capacity of the shifting actuator, it is necessary to inves-
tigate the variation of shifting force Fs along with the desired performance. From
Equation (2.10), in steady state Fs is zero and the input-output force relation be-
comes Fi cos θ = Fo sin θ. To achieve shifting (i.e., gear ratio change), a non-zero Fs
must be induced by the shifting actuator in an appropriate manner.
For example, we consider the case when shifting occurs by the amount θd at a
42
certain steady state instant. At the beginning of shifting, we can assume that the
input-output force relation still holds. Rearranging Equation (2.10), the shifting
force Fs becomes
Fs = 2(Ia + Iv +mε2)
ε− 2 I2vR
2
ε(Isε2 + 2 IvR2)
θd . (3.1)
As seen in Equation (3.1), Fs necessary for shifting is determined by R, θd, and
the mass moments of inertia of the sphere Is, variator Iv, and connected elements
Ia +mε2. Considering that shifting forces of other traction or belt drives must be
large enough to resist the traction or friction force, which is generated directly by
the transmitted torque, the overall magnitude of shifting force of the S-CVT will
likely be much smaller than that of other existing CVTs.
The necessary power Ps of the shifting actuator is calculated using Equation
(3.1):
Ps = εFsθd = 2
(Ia + Iv +mε2)− 2I2vR2
(Isε2 + 2 IvR2)
θd θd (3.2)
where θd is the corresponding angular velocity to the required shifting demand θd.
3.3 Prototype Specifications
Based on the numerical investigation results from the previous studies, we have des-
ignated the hardware specifications of the S-CVT prototype. The overall layout of
the power transmission is shown in Figure 3.3. Because of the maximum limiting
torque, a reduction gear with a ratio of three is added to the prototype; this ratio
also includes a safety factor. This additional reduction gear can be eliminated by
improving the material properties such as the static coefficient of friction and in-
creasing the normal force at the contact point. The final assembly drawing is shown
43
Driving Motor
PivotMotor
z=36z=18
z=18z=18
z=36
z=72
z=36
z=26
z=52 z=
26z=
52
z=26
z=39
z=26
z=39
Variator
Sphere
Wheel
Wheel
10
VariatorMotor
z=18 z=66
z=66z=18
1313
Input disc
Figure 3.3: Schematic diagram of S-CVT.
in Figure 3.4. In Figure 3.3 and 3.4, a dc motor referred to as the pivot motor, and
internal gears are included in the power-flow line of the S-CVT. These elements are
added in order to make each output shaft rotate in opposite directions. This novel
pivot mechanism is proposed for the application to the CVT-based mobile robot,
which is the main subject of Chapter 7.
44
load springhousing
Figure 3.4: Assembly drawing of S-CVT.
45
Element Nomenclature and Specification Material
Spheremass (ms) = 0.882 kg
radius (R) = 30 mm
Steel ball
of ball bearing
Input/output disc
Variator
mass (m) = 0.095 kg
radius (r) = 16 mm
contact radius (ri/o, ε) = 10 mm
SNCM 8 class
Input/output shafts SCM 4 class
Gears Refer to Figure 3.3 SCM 21 class
Mass moments
of inertia
Sphere (Is) = 3.1758× 10−4 kg ·m2
Input parts (Iin) = 2.3581× 10−5 kg ·m2
Output parts (Iout) = 3.8609× 10−4 kg ·m2
Variator (Iv) = 1.0514× 10−5 kg ·m2
Variator connected parts (Ia) = 1.0585× 10−4 kg ·m2
Table 3.1: Specifications of prototype.
The detailed specifications for numerical studies and experiments are shown in
Table 3.1. The prototype S-CVT has been built and is shown in Figure 3.5.
Figure 3.5: S-CVT prototype.
46
3.4 Experimental Results
In order to validate the operating principles and performance of the S-CVT, we have
built a testbench for it. Two eddy-current type AC servo motors (input: 3-phase
AC, 122 V , 9 A; output: 1500 Watts; rated speed: 2000 rpm) are used for a driving
power source and a driven load generator. In the testbench (see Figure 3.6), the
variator angle is controlled by a dc stepped motor with an angular resolution of
0.024/pulse. The rotational speeds of the input and output shafts are measured
through incremental optical encoders attached to the shafts.
3.4.1 Performance of S-CVT
Setting the external load torque to zero, we observe the output speed together with
the variator angle displacement while the input speed is set respectively to 748,
Figure 3.6: Testbench of S-CVT.
47
1502, and 2001 rpm. A steady-state speed ratio curve of the S-CVT is extracted
for the no-load condition (see Figure 3.7 (a)). Note that the overdrive of the output
speed, which implies that the output speed is faster than the input speed, occurs
when the variator angle exceeds 50. In addition, there is a large deviation between
the ideal value and the test result beyond a variator angle of 65, which indicates
the onset of slippage. These less than ideal output speeds arise from the increase of
reaction force normally exerted on variator Fn, which is described in Section 2.4. In
the experimental result, moreover, there must be a certain amount of internal load
induced by manufacturing and other errors, which makes Fo large (see more details
in Section 2.4).
Using slip-ring type torque sensors, we have also observed the output torque
together with the variator angle displacement by adjusting input/output torque to
realize the pre-obtained steady state speed ratio (see Figure 3.7 (b)). The actual
torque ratio is limited to under 20, which is determined mainly by the static coeffi-
0 10 20 30 40 50 60 70
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
(a) Speed ratio of S-CVT.
0 10 20 30 40 50 60 70
-5
0
5
10
15
20
25
30
35
40
45
(b) Torque ratio of S-CVT.
Figure 3.7: Experimental results.
48
0 10 20 30 40 50 600
10
20
30
40
50
60
70
80
90
100
Figure 3.8: Power efficiency of S-CVT.
cient of friction and the exerted normal force.
Finally, we calculate the power efficiency of the S-CVT using the obtained speed
and torque ratios (see Figure 3.8). The efficiency is almost 85% for variator an-
gles under 15, while the average efficiency beyond this angle is about 65%. The
power efficiency of the prototype S-CVT is somewhat low; this is mainly due to the
manufacturing errors including bearing friction loss, gear backlash, etc. From exper-
iments with the prototype S-CVT, we have also found that slight misalignments of
the shafts may cause bending moments in the shafts and discs, resulting in increased
bearing friction loss and slippage, although for applications accurate shaft alignment
will have to be separately addressed.
3.4.2 Strength and Life Prediction of S-CVT
Perhaps the most common form of mechanical failure in friction and traction drives
is by wear. The laws governing the overall friction and wear between two surfaces
49
seem to depend primarily on the total force transmitted across the two surfaces
rather than on the local distributions of stress and strain, and one may assume that
the two bodies in contact are perfectly rigid. In studying the details of the actual
mechanism of wear and friction, however, one must take into account the extremely
small areas of actual load contact between two bodies and the elastic and plastic
deformations in these regions. Another factor which must be considered in studying
the detailed mechanism is the surface condition of the metal, since this condition
may be such that these local points of contact behave in a manner quite different
from that of the same material in bulk form. It has been generally accepted that
the addition of a reasonable tangential force to a rolling contact has no appreciable
effect on drive life. This is so only when spin is almost entirely absent.
Dawe and Lohr (1993) reported that application of a realistic tangential trac-
tion force at the contacts does not seem to cause dramatic reduction in life, and
circular contacts appear to offer the best efficiency concerning durability [56]. The
well known basic formula for the fatigue life of a rolling bearing was presented by
Lundberg and Palmgren (1947) [57]. According to the theory, the fatigue life of
rolling elements was caused by the maximum shear stress in the sub-surface of the
rolling contacts.
Machida et al. (1991) reported that rolling fatigue life of a traction drive is
inverse proportional to the cube of a load using the ball-bearing theory [30]. This
implies that a higher value of the contact force cannot be used simply to transmit
higher torque. Machida and Tanaka (1991) also presented an in-depth study with
experiments of oil film and surface damage in traction drives [58]. With another
traction drive mechanism, Coy et al. (1981) presented a contact fatigue life analysis
method for multiroller traction drives [59]. The method was based on the Lundberg-
Palmgren analysis, and also used life adjustment factors for materials, processing,
50
Input torque range 0.5 - 4 kgf · cm
Variator angle change 0 − 60
Normal force range 100 - 200 kgf
Table 3.2: Endurance test condition.
lubrication, and effect of traction.
Adopting these previous results, the strength and life prediction of the S-CVT
can be performed based on ball-bearing theory. However, there must be a prior
investigation of the normal and shear stresses at the contact points; these will be
discussed in the following chapter, with a detailed analysis of spin moments as well
as several numerical results by ANSYS.
Figure 3.9 shows the input disc surface after 107 revolutions under the following
test conditions (see Table 3.2). In this figure, there appears a circular deflection
track along the contact points with the sphere. The deflection width reaches almost
2 mm, and the depth reaches about 0.005 mm. Considering that the ideal elastic
Figure 3.9: Endurance test result of input disc.
51
deformation is 2.4× 10−3 mm, further described in Section 4.3, the resulting stress
distribution (shear and normal stresses) must be beyond the elastic range. Moreover
from the wide deflection track, each shaft has not been aligned exactly and bending
moment and slip loss must be resulted in the shafts and discs. However there is no
evidence of surface flaking on the tested input disc.
3.5 Summary
In this chapter, we have discussed typical issues on the S-CVT hardware design,
focusing on the normal force loading device and the shifting actuator capacity. To
assure rolling resistant force at the contact points of the S-CVT, we adopted com-
pressible springs in the normal force loading device, because of their simple structure
and the ease in adjusting the preset load. To determine the shifting actuator ca-
pacity, we presented a numerical relationship between the necessary power and the
shifting response demand in explicit form, using the previous dynamic analysis re-
sults.
We presented hardware specifications and some drawings for the S-CVT pro-
totype. In the prototype, an additional reduction gear was added for increasing
the transmittable limiting torque and internal gears for the mobile robot applica-
tion. Experimental results on the actual transmission ratios and power efficiency
obtained with the testbench of S-CVT were represented in order to validate the op-
erating principles and performance. Finally, we briefly addressed the strength and
life estimation for the S-CVT based on methods for predicting ball-bearing life.
52
Chapter 4
Slip Analysis of the Spherical
CVT
Like other friction and traction drives, slippage takes place whenever the transmitted
force (or torque) exceeds the limiting value at the contact points of the S-CVT.
Slippage causes wear, heat, power loss, and even failure of the power transmission;
therefore it is a critical problem in the design and control of the S-CVT. Although
many efforts have been dedicated to compensate for slip in a variety of mechanical
systems or ages, an accurate analysis of the mechanism of slippage is still an open
topic of research.
There are two main sources of power loss in the S-CVT, excluding the losses due
to its mechanical structure, bearings and shifting actuator (particularly in traction
drives, shifting actuator is composed of hydraulics). One is “slip loss” on the contact
points between the sphere and discs resulting from slippage in the rolling directions.
Once the rolling directional slippage occurs at these contact points, the transmitted
power becomes different from the desired value; the power transmission can even
53
fail.
The other one is called “spin loss,” which is also one of the main design issues
in traction drives also (see Figure 4.1) [60]-[62]. Spin loss results from the elastic
contact deformation of rotating bodies that have different rotational velocities. To
reduce the spin loss in traction and friction drives, many designers have investigated
different approaches to optimal contact geometry design, normal load application,
and controller design.
In this chapter a modified classical friction model, which describes pre-sliding
displacement friction, is presented to model the spin loss in the S-CVT. We also
perform a quantitative analysis of spin loss in the S-CVT along with an in-depth
study of velocity fields and the normal pressure distribution generated on the contact
regions. Finally we study the contact analysis of slip, when a shear force resulting
from friction occurs on the contact surface.
Figure 4.1: Spin loss in traction drives.
54
4.1 Friction Model Review
The study of friction has a long history in the fields of mechanics, metallurgy, tribol-
ogy, and control. Using both theory and experimentation, researchers have devel-
oped several different models of the structure and dynamics of friction. Pure rolling
friction conditions occur when the contact between two surfaces is a point. However,
according to Rabinowicz (1965), the contact region between two surfaces is typically
of larger area than a point due to an elastic (and possibly plastic) deformation [63].
Classical friction, also referred to as the stick-slip model , is the earliest and most
widely used model of friction. The three components of classical friction ≡ kinetic
friction, viscous friction, and static friction ≡ are illustrated on the friction versus
velocity graph in Figure 4.2. Although kinetic friction simply provides a constant
retarding force on rubbing surfaces, it also introduces a discontinuity at zero velocity.
As a result, servomechanisms performing bi-directional tasks will be subject to the
discontinuity during every velocity reversal. The discontinuous behavior of kinetic
Figure 4.2: Classical model of static, kinetic, and viscous friction.
55
friction can be classified as a “hard nonlinearity”; it is well-known that a closed
loop system with a hard nonlinearity can produce a limit cycle, i.e., self-sustained
oscillation, that would lead to poor control accuracy.
Viscous friction results from the viscous behavior of a fluid lubricant layer be-
tween two rubbing surfaces. As shown in Figure 4.2, viscous friction is represented
as a linear function of slip velocity. Static friction is the force required to initiate
motion from rest. Typically, the magnitude of static friction is greater than that of
kinetic friction, which can lead to intermittent motion known as “stick-slip”. Stick-
slip manifests itself as repeated sequences of sticking between two surfaces with static
friction, followed by sliding or slipping with kinetic friction. In the servomechanism
control problem, stick-slip can diminish control accuracy; stick-slip limit cycling can
be avoided if damping and stiffness are sufficiently high, however. The equations for
the classical lumped friction force model Ff are as follows:
Ff =
Fk sgn(V ) + µV if V 6= 0
Fs sgn(F ) if V = 0
. (4.1)
Here Fs and Fk denote static and kinetic friction, respectively. Equation (4.1) shows
that Ff depends on the slip velocity V and coefficient of viscous friction µ.
Contrary to the predictions derived from the classical friction model, researchers
including Courtney-Pratt and Eisner (1957) and others have found experimentally
that small relative displacements between two bodies in contact occur when the
applied tangential force is less than the static friction [64]. Dahl (1977) provided a
model of this pre-sliding displacement phenomenon [65], known as the “Dahl model,”
that assumes friction force is a function of displacement x and time t such that
dFf (x, t)
dt=∂Ff (x, t)
∂xx+
∂Ff (x, t)
∂t, (4.2)
56
with ∂Ff (x, t)/∂t = 0, and
∂Ff (x, t)
∂x= σ|1− Ff
Ffc· sgn(x)|i . (4.3)
σ and Ffc are as shown in Figure 4.3 (a), and i is an exponent that Dahl empirically
derived to be approximately 1.5; however, it cannot be applied when the velocity
xÀ 0.
While the simple static plus kinetic friction model offers an intuitive explanation
for the possibility of stick-slip oscillations, it does not offer adequate justification
for the existence of these limit cycles in the wide range of conditions under which
they have been observed. However, several researchers have found a source for this
discrepancy in the Stribeck effect , and experimentally derived a model of friction
variation with velocity as depicted in Figure 4.3 (b). The implication of the Stribeck
effect for servomechanism dynamics includes an increased likelihood of stick-slip limit
cycling at low velocities. Among many empirical models derived for the friction
(a) Dahl effect. (b) Stribeck friction.
Figure 4.3: Pre-sliding displacement phenomenon.
57
incorporating the Stribeck effect, the following is the most popular:
Ff (V ) = Fk · sgn(V ) + µV + (Fs − Fk)e−(V/Vstr)2 · sgn(V ) (4.4)
where Vstr is the critical Stribeck velocity.
Leonard and Krishnaprasad (1992) presented a comparative investigation of fric-
tion compensating control strategies designed to improve low-velocity position track-
ing performance in the presence of velocity reversals for servomechanisms [66]. In
their work, the various controller designs incorporate different friction models rang-
ing from classical friction and Stribeck friction to the Dahl friction model. They
have claimed the Dahl model proved to be significant for the friction compensating
control problem with repeated zero-velocity crossings.
Hu (1994) used Karnopp’s friction model, where the stiction zone is broadened to
an interval around zero velocity, for the position control of a servo-system containing
friction [67]. Lee et al. (1999) presented a numerical study with an extended Dahl’s
friction model [68]. In this study, they performed a comparative numerical analysis
of the computational cost and modelling efforts between the classical stick-slip model
and an extended Dahl’s friction model for an automatic transmission system analysis.
Besides the above analyses of friction mechanisms, control engineers have used
open-loop smoothing techniques, such as dither and pulse-width modulation, to
compensate for friction in mechanical systems. However, these techniques have
disadvantages, e.g., dither can cause mechanical problems such as fatigue by exciting
vibrations in manipulators. As an alternative to these techniques, recent works have
brought to the forefront adaptive and estimation-based control techniques for the
compensation of friction in mechanical systems [69]-[72].
Lee and Tomizuka (1996) presented a controller structure for robust high-speed/
high-accuracy motion control systems [13]. In their control system, the friction com-
58
pensator is based on the experimental friction model and compensates for nonlinear
friction that is not modeled. Vedagarbha, Dawson, and Feemster (1999) have de-
signed an observer-based exact model knowledge position tracking controller for a
second-order mechanical system with nonlinear load dynamics and the nonlinear
dynamic friction model proposed by C. Canudas et al. [73] in their recent work [74].
In traction drive fields, the slip of traction oil within the contact region is an
important issue in determining torque capacity, power loss, the shifting mechanism,
and mechanical life. Although the fundamental mechanisms of traction differs from
that of friction, the kinetic characteristics are similar; therefore many engineers use
the terminology of friction to construct numerical models for traction drives [18],
[19], [21], [23], [24], [27], [29], [30], [75].
4.2 Modified Friction Model for S-CVT
There is spin loss in S-CVT like traction drive CVTs; this is described in the fol-
lowing section in detail, where the pre-sliding effect in the vicinity of zero relative
velocity is considered. Friction models based on Dahl’s show difficulties in numerical
integration due to the high stiffness and damping coefficients; moreover they must be
obtained through a careful experimental analysis [68]. Thus, the classical stick-slip
friction model is adopted to the S-CVT system, because velocity reversal seldom oc-
curs in the S-CVT. We propose a modified classical friction model including Stribeck
effect like
Ff =
[
(µs − µk) exp−(∆V
Vstr)2
+ µk
]
P · sgn(∆V ) (4.5)
where µs, µk are static and kinetic coefficients of friction, respectively. Here we
neglect viscous friction, as there is no lubricant layer in the S-CVT. For typical
59
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-10
-5
0
5
10
Figure 4.4: Proposed friction model.
values of friction model parameters, the friction force versus slip velocity of the
proposed model is depicted in Figure 4.4.
4.3 Spin Loss of the S-CVT
4.3.1 Velocity Fields on the Contact Surface
The friction force on the contact surface is determined by the normal force and
friction coefficients. Considering that the kinetic coefficient of friction is related to
the relative velocity ∆V between two rotating bodies, we first investigate the relative
velocity field on the contact surface. In this subsection, Hertzian results for elastic
deflection are employed to construct the geometric parameters of the contact surface.
Hertzian theory deals with the general solution of the elastic contact problem of rigid
bodies [76], [77].
60
Contact of Rotating Bodies with Different Radii of Curvature
Consider two solid bodies in contact under a normal force P (Figure 4.5). In this fig-
ure, Rx1, Ry1 and Rx2, Ry2 denote their radii of curvature respectively. The material
properties (e.g., Young’s modulus and Poisson’s ratio) of each body are E1, ν1 and
E2, ν2 respectively. We set the reference frame XYZ to be on the contact point,
and the local coordinate frame ξηz on the deformed contact surface such that its
origin coincides with the contact center. According to Hertzian analysis [77], the
contact surface in this case produces an elliptic shape with principal axes a, b and
corresponding normal deflections δx, δy; these can be approximated as
a = 1.109× 3
√
P
E· Ry1Ry2
Ry1 +Ry2, b = 1.109× 3
√
P
E· Rx1Rx2
Rx1 +Rx2, (4.6)
δx = 2.64× 3
√
P 2
E2· Rx1 +Rx2
Rx1Rx2, δy = 2.64× 3
√
P 2
E2· Ry1 +Ry2
Ry1Ry2, (4.7)
Figure 4.5: Contact of two bodies with different curvature.
61
2
E=
(1− ν12)
E1+
(1− ν22)
E2, (4.8)
where E is the equivalent Young’s modulus.
Suppose that body 1 and body 2 have angular velocities ω1 and ω2, respectively.
Here we assume that the contact center does not dislocate from the original contact
center (i.e., roll without slip in the rolling direction). Then we can derive the
relative velocity field ∆V(ξ, η) = V1(ξ, η) − V2(ξ, η) in the contact surface with
pure rotational motion in R3, using the relevant curvature radii as follows:
∆V(ξ, η) =
(Ry1 − δy2 )ω1Y − (Ry2 − δy
2 )ω2Y + η(ω1Z − ω2Z)
(Rx1 − δx2 )ω1X − (Rx2 − δx
2 )ω2X + ξ(ω1Z − ω2Z)
T
(4.9)
where ωiX,Y,Z is the rotational velocity component in the reference frame of each
body.
Contact of Disc and Sphere: S-CVT Case
Figure 4.6 shows the contact surface between the sphere and upper variator. In this
figure, we set a local coordinate frame ξyη, in the directions of xyz as shown in
Figure 2.6, at the center of the contact surface S; the rolling direction is in the ξ
direction. Taking into account that the two contact bodies in the S-CVT are disc
and sphere, one can let Rx1 = Ry1 = ∞ and Rx2 = Ry2 = R in Equations (4.6)-
(4.8). Furthermore, supposing there is no bending deformation of the variator along
the x, z axes, the normal deflection δ and contact surface radius c can be calculated
as
c = 1.109× 3
√
P
E·R , δ = 2.64× 3
√
P 2
E2· 1R
. (4.10)
To obtain each velocity field, we first recall that the sphere has a pure rotational
speed of ω in the z direction and the variator a rotational speed of ωv in the y
62
Figure 4.6: Velocity vector field on contact point.
direction. The velocity field of the sphere V1(ξ, η) and that of variator V2(ξ, η) on
the contact surface can be obtained as follows:
V1(ξ, η) = [ (R− δ)ω, 0 ] ,
V2(ξ, η) = [ (ε+ η)ωv, − ξωv ] ,(4.11)
where ε is the distance between the contact surface center and variator center. Con-
sequently, the relative velocity field ∆Vv(ξ, η) can be derived as
∆Vv(ξ, η) = [ (R− δ)ω − (ε+ η)ωv, ξωv ] . (4.12)
Similarly, the relative velocity fields at the other contact points (input and output
discs) can be obtained.
63
As shown in Equation (4.12), there is a relative velocity component of ∆Vv in the
η direction, ∆Vη, on the contact surface. However ∆Vη(ξ, η) does not accumulate
total relative velocity in the η direction, because it is symmetric along the η axis.
Therefore, ∆Vη(ξ, η) contributes to spin along the direction normal to the ξη plane.
In the case of ∆Vξ(ξ, η), the relative speed of −(δω + ηωv) occurs in the rolling
direction (recall the rotational speed relation of Rω = εωv). Note that δ becomes
small enough to be neglected compared R (for example, δ = 2.4 × 10−3 mm, c =
0.59 mm, and R = 30 mm for the case of the S-CVT prototype); therefore ∆Vξ(ξ, η)
can be approximated to be−ηωv. The contribution of ∆Vξ(ξ, η) is also a spin, similar
to ∆Vη(ξ, η).
The vector diagram of relative velocity is obtained using typical values of ω, ωv,
ε, R, P , and E that correspond to the S-CVT prototype specification (see Figure
-0.0006 -0.0004 -0.0002 0.0000 0.0002 0.0004 0.0006
-0.0006
-0.0004
-0.0002
0.0000
0.0002
0.0004
0.0006
Figure 4.7: Typical relative velocity vector diagram.
64
4.7). The spin velocity field can be found straightforwardly, although there are no
excessive forces that cause slippage. From this result, we can be assured that there
must be spin in the contact surface around the origin of the local coordinate frame
of the contact point in the S-CVT regardless of the existence of shear force resulting
slippage.
4.3.2 Normal Pressure Distribution
Now we consider the normal pressure distribution on the contact patch to obtain the
friction force as well as to analyze the strength of the S-CVT. A Hertzian pressure
distribution develops in the circular shaped contact patch (with radius c) between
the sphere and disc. The pressure at each point in the contact surface is known to
be
p (ξ, η) =3
2
P
πc3
√
c2 − ξ2 − η2 . (4.13)
The maximal normal pressure pmax is located at the center of the contact surface;
at the boundaries, the normal pressure p becomes zero. The mean value of normal
pressure, pmean, equals the normal contact force P divided by the area of the contact
surface:
pmean =P
πc2=
1
1.1092π
3
√
PE2
R2.
Using the relation pmax = 1.5 pmean, the maximal normal pressure can be calculated
as
pmax = 0.3883
√
PE2
R2. (4.14)
The maximal pressure pmax should not exceed the yield strength of the sphere
and disc. For design purposes, the maximal normal pressure values for several com-
binations of contact bodies are illustrated in Table 4.1. On the same load condition,
65
Table 4.1: Maximal normal pressure comparison.
the maximal pressure values for the S-CVT (i.e., sphere and disc contact) reaches
almost 1.588 times that of the case that is inscribed in a circular body more than
twice the radius.
4.3.3 Quantitative Analysis of Spin Loss
Relative velocities resulting from the elastic contact of rotating bodies usually give
rise to friction mechanisms; in which case friction moments (spin loss) occur in the
66
contact region. Once there is an elastic or plastic deformation at the contact points
of bodies with different rotational velocities, there must occur a spin loss. Spin loss
is not caused by changes in the applied or exerted forces, but by the difference in
rotating velocities and geometric properties of the contact bodies. Therefore spin
loss always exists in traction and friction drives; only the amount is different for
each drive [5], [19], [20], [23], [27], [32], [33], [35], [36], [61], [62], [75].
Consider the infinitesimal area at the contact surface S, with the friction force
of the ith area in the rolling direction (ξ direction) denoted Fξi, and Fηi the force
in the η direction as shown in Figure 4.8. The total friction forces Fξ and Fη can be
obtained using the following equations;
Fξ =
∫ c
−c
∫ c
−cFξi(ξ, η) dξdη , Fη =
∫ c
−c
∫ c
−cFηi(ξ, η) dξdη .
Recall that the normal pressure distribution has symmetries along the ξ and η axes,
and that ∆Vξ in Equation (4.12) varies along the η direction (neglecting δ), and ∆Vη
along the ξ direction; there are no total relative velocities in the ξ and η directions.
Figure 4.8: Friction forces at the infinitesimal area of the contact surface.
67
Therefore one can conclude that Fξ and Fη become zero.
Using the proposed friction model in Equation (4.5), the spin moment Tspin for
the variator can be calculated as
Tspin =
∫ c
−c
∫ c
−c(ηFξi + ξFηi) dη dξ (4.15)
where
Fξi = [(µs − µk) exp−(∆VξVstr
)2+ µk] p(ξ, η) · sgn(∆Vξ) ,
Fηi = [(µs − µk) exp−(∆VηVstr
)2+ µk] p(ξ, η) · sgn(∆Vη) ,
∆Vξ = −ηωv ,
∆Vη = ξωv .
Rearranging and integrating by parts, Equation (4.15) becomes
Tspin =3Pc
4π
[
µk + (µs − µk)2
c4Vstr
2
ωv2
[
c2 − Vstr2
ωv2(1− exp−( cωv
Vstr)2)
]]
(4.16)
where c is the radius of the contact surface, which can be calculated using Equation
(4.10).
To investigate the amount of spin loss at the contact points of the S-CVT, we cal-
culate the respective spin losses using Equation (4.16) for the input and output discs
and variators with typical values of µk, µs, Vstr, P . Figure 4.9 shows the numerical
results for spin loss at an input speed of 3000 rpm along the variator angle change,
and the speed changes in the discs and variator. Spin moments occur at six contact
regions in the S-CVT, and the total amount of spin loss reaches almost 0.076 N ·m.
Spin moments decrease as the variator angle increases, except for the input discs
whose spin moments remain constant (because there are no speed changes in the
input discs). The gross spin loss decreases as the input speed rises, that is due to
the characteristics of our friction model: the friction force at zero relative speed has
the maximal value.
68
0 10 20 30 40 50 60 70
0
2000
4000
6000
8000
decreasing as theinput speed increases
0.020
0.025
0.07
0.08
Figure 4.9: Spin losses on S-CVT at input speed of 3000 rpm.
The average value of spin loss of our numerical results is almost 0.072 N · m.
Considering the input torque is limited under the static friction torque of 1.962 N ·m,
the ratio of spin loss to static friction torque is almost 3.67%. Considering normal
operating conditions, at which the input torque is smaller than the limiting torque,
one can note that the ratio of spin loss becomes much greater. To reduce this loss,
it is helpful to operate the S-CVT with high input speeds; the increased relative
velocity reduces the relevant friction force.
69
4.4 Slip Motion of the S-CVT
4.4.1 Stick and Slip States
When there are stick states at all contact points of the S-CVT, the exerted tangential
force (e.g., driving, load, or shifting force) must be smaller than the static friction
force. When leaving this stick condition, for example when a certain tangential force
becomes large enough to cause slip in the S-CVT, the relative velocity grows, and
the transmitted force is limited by the kinetic friction force as Equation (4.5).
The dynamics of the S-CVT forms a set of second-order differential equations
as shown in Equation (2.10). However, when slippage occurs at any contact point
of the S-CVT, the dynamic motion of the S-CVT is determined mainly by friction
forces; therefore the whole equations of motion change.
4.4.2 Slip Loss of the S-CVT
Slip loss is caused by rotational slippage at the contact points, mainly by changes
in the transmitted forces. Thus slip loss Tslip can be defined as the torque difference
between the driver and the driven (in the case of the S-CVT, the sphere and discs):
Tslip = Rdriver · Fdriver − Rdriven · Ff (4.17)
where Rdriver, Rdriven are the effective contact radii of the driver and the driven
respectively, Fdriver is the driving force, and Ff is a kinetic friction force.
4.4.3 Slip Involved Contact Analysis
The contact analysis of rotating bodies in rolling conditions were discussed in Section
4.3. The results of that section are based on the assumption that the contact center
does not dislocate from the original contact center. However, when slippage in the
70
rolling direction is induced, the contact center must be dislocated, together with
changes in the normal pressure distribution according to the moment equilibrium
condition at the contact point.
When the tangential force becomes large enough to cause slippage (i.e., larger
than the static friction force), there must be a shear force on the contact region
and thus a reactive moment about the center of the sphere (see Figure 4.10). To
satisfy the moment equilibrium about the sphere center by the reactive moment and
tangential shear force, the contact center must be moved by ε in the rear direction.
The amount of dislocation ε can be calculated as
ε =R∆F
P. (4.18)
As a result of this contact center dislocation, the overall normal pressure distri-
bution must be shifted accordingly, which also brings about changes in the friction
force. The change in normal pressure distribution is depicted in Figure 4.11; in the
Figure 4.10: Dislocation of contact center.
71
Figure 4.11: Change of normal pressure distribution in XZ plane.
XZ plane. The normal pressure distribution will be distorted as shown in this fig-
ure; pmax will be larger considering there is no change in the amount of total normal
force. The applied shear force ∆F exposes a boundary between the stick and slip
region of the contact surface. The friction force Ff can be roughly determined by
the relation Ff = µN ; the increased normal pressure causes a larger static friction
force which can resist the applied shear force. Therefore above a certain value of
normal pressure, slippage does not occur.
4.5 Summary
There are two main sources of power loss resulting from slippage in the S-CVT, spin
and slip loss. Spin loss, which is also a main design issues in traction drives, results
from the elastic contact deformation of rotating bodies having different rotational
velocities. Slip loss is generated at the contact points between the sphere and discs in
their rolling direction. Once slippage occurs at those contact points, the transmitted
power becomes different from the desired value, and the power transmission can even
fail.
72
To analyze the losses resulting from slippage, we first reviewed previous analyses
of the friction mechanism. We proposed a modified classical friction model that
describes the friction behavior of the S-CVT including Stribeck (i.e., pre-sliding)
effect. We also performed an in-depth study for the velocity fields generated at the
contact regions along with a Hertzian analysis of deflection. Hertzian results were
employed to construct the geometric parameters and normal pressure distributions
of the contact surface with respect to elastic and plastic deformations.
With analytic formulations of the relative velocity field, deflection, and friction
mechanism of the S-CVT, we carried out a quantitative analysis of spin loss. As a
result, an explicit model of spin loss was developed. Spin loss is one of the main
design issues in traction and friction drive designs, and our results can provide an
effective means of measuring and predicting spin loss.
We also described some issues related to the slip loss of the S-CVT resulting
from slippage in the rolling direction. When slippage occurs at any contact point
of the S-CVT, the dynamic motion of the S-CVT is determined mainly by friction
forces; therefore the whole equations of motion change. To predict the behavior of
the S-CVT in stick-slip states, it is important of an instantaneous investigation of
those states using information on velocities, accelerations, and forces at the contact
points. Finally we briefly described the contact analysis related to slip, when a shear
force resulting from friction occurs at the contact surface.
73
Chapter 5
Shifting Controller Design via
Exact Feedback Linearization
5.1 Introduction
The most important role of the shifting controller for CVTs is the realization of the
target gear ratio, which is directly related to the input/output ratio of power. When
the shifting command for a certain gear ratio is given, the shifting system must be
stabilized so as to realize the demanded gear ratio with the desired performance
(e.g., little shifting effort, short settling time, etc.). The shifting command of a
CVT can be either a final value or a trajectory of the target gear ratio. According
to the shifting command, the shifting controller design task is denoted as stabilizer
(or regulator) design for the former and tracker (or servo) design for the latter.
In control theory, a basic problem is how to use feedback in order to modify
the original internal dynamics of a controlled plant so as to achieve some prescribed
behavior. In particular, feedback may be used for the purpose of imposing, on the as-
74
sociated closed-loop system, the (unforced) behavior of some prescribed autonomous
linear system. When the plant is modeled as a linear time-invariant system, this is
known as the problem of pole placement, while in the more general case of a non-
linear model, this is known as the problem of feedback linearization (see [78]-[81]).
Feedback linearization is an approach to nonlinear control design which has at-
tracted a great deal of research interest in recent years. The central idea is to
algebraically transform a nonlinear system dynamics into a (fully or partly) linear
one, so that linear control techniques can be applied. This differs entirely from
conventional linearization in that feedback linearization is achieved by exact state
transformations and feedback, rather than by linear approximations of the dynamics
(i.e., Jacobian linearization).
The shifting system of the S-CVT has second-order nonlinear dynamics, and
the original open-loop system reveals unstable characteristics. In order to cancel
nonlinearities of the S-CVT shifting system, and to make it stable and have good
tracking characteristics, we develop a feedback controller based on the exact feed-
back linearization method in this chapter. We first investigate the instability of the
original shifting system using Lyapunov’s indirect method in Section 2. Section 3
briefly reviews the differential geometric preliminaries for the formal description of
the feedback linearization method. In Section 4, we address the input-state feedback
controller design of the S-CVT shifting system. Finally, we investigate the stabi-
lizing and tracking performance of the dedicated shifting controller by numerical
simulation.
75
5.2 Stability Analysis of S-CVT Shifting System
Lyapunov’s (indirect) linearization method is involved with the local stability of a
nonlinear system. It is a formalization of the intuition that a nonlinear system should
behave similarly to its linearized approximation for small range motions. Because all
physical systems are inherently nonlinear, Lyapunov’s linearization method serves as
the fundamental justification of using linear control techniques in practice, i.e., that
stable design by linear control guarantees the stability of the original physical system
locally .
Theorem 5.1 (Lyapunov’s (indirect) linearization method) Let x = 0 be an
equilibrium point for the nonlinear system
x = f(x) (5.1)
where f : D → Rn is continuously differentiable and D is a neighborhood of theorigin. Let
J =∂f
∂x(x) |x=0
Then,
1. The origin is asymptotically stable if Re(λi) < 0 for all eigenvalues of J.
2. The origin is unstable if Re(λi) > 0 for one or more of the eigenvalues of J. ♦
For the detailed proof of this theorem, see pp. 127-130 of [80].
To determine the stability of the S-CVT shifting system, we first restate the
shifting dynamics in Equation (2.10) into state-space form (5.1). Here we replace
the state x3 (the rotational speed of sphere) by a matrix transformation, because
it does not affect the shifting dynamics. Letting x1 = θ, x2 = θ be the states, the
76
corresponding state-space equation is the following second-order state equation:
x1 = x2
x2 =1
Da22Fs − a12(Fi cosx1 − Fo sinx1)
(5.2)
where
a11 a12
a21 a22
=
2 (Ia+Iv+mε2)ε 2RIv
ε2
2 Ivε
IsR + 2RIv
ε2
, D = a11a22 − a12a21 .
Using the trigonometric transformation, i.e.,
a sinx+ b cosx =√
a2 + b2 sin(x+ φ), φ = tan−1(b
a)
Equation (5.2) can be written
x1 = x2
x2 =1
Da12
√
F 2i + F 2
o sin(x1 − φ) + a22Fs(5.3)
where
φ = tan−1(Fi
Fo) .
Considering D =2Is(Ia + Iv +mε2)
εR+
4RIv(Ia +mε2)
ε3is always larger than
zero, the equilibrium point is given by
x∗1 = φ = tan−1Fi
Fo, x∗2 = 0, F ∗
s = 0 . (5.4)
We can say that the equilibrium point of interest is x∗ = (φ, 0). Physically, this
point corresponds to the steady state of the shifting system in which shifting does
not occur.
The Jacobian matrix J of the shifting system (5.3) linearized about the equilib-
rium point becomes
J =
0 1a12D
√
F 2i + F 2
o 0
(5.5)
77
the eigenvalues of J are λi = ±√
a12D
√
F 2i + F 2
o . Hence the linearized system is
unstable, and therefore so is the shifting system of the S-CVT at this equilibrium
point.
Physically this means that when the input and output force relation (Fi cos θ =
Fo sin θ) is broken (i.e., steady state is destroyed) by some disturbances from the
input or output force, it must be followed by a change in variator angle (gear ratio)
from the shifting actuator, or by a change in input force from the power source
controller. In order to make the shifting system stable, one can conclude that an
appropriate feedback controller is necessary. In the following sections, we will discuss
the design of a feedback controller based on the exact feedback linearization method.
5.3 Differential Geometric Preliminaries
We start by recalling some differential geometric preliminaries (see [78]-[81]); we
then apply these tools to the input-state linearization of our shifting system.
Lie Derivative: Let h : D → R be a smooth scalar function, and f : D → Rn be
a smooth vector field on Rn. The Lie derivative of h with respect to f or along f ,
written as Lfh, is defined by
Lfh = ∇h · f .
Repeated Lie derivatives can be defined recursively by
L0fh = h,
Lifh = Lf (L
i−1f h) = ∇(Li−1
f h) · f .
Similarly, if g is another vector field, then the scalar function LgLfh is
LgLfh = ∇(Lfh) · g .
78
Lie Bracket: Let f and g be two vector fields on Rn. The Lie bracket of f and g
is a third vector field defined by
[f ,g] = ∇g · f −∇f · g .
The Lie bracket [f ,g] is commonly written as adfg, where ad stands for “adjoint.”
Repeated Lie brackets can then be defined recursively by
ad0fg = g,
adifg =[
f , adi−1f g]
.
Diffeomorphism: A mapping T : Rn → Rn, defined in a region Ω, is called a
diffeomorphism if it is smooth, and if its inverse T−1 exists and is smooth. If the
region Ω is the whole space Rn, then T (x) is called a global diffeomorphism. Global
diffeomorphisms are rare, and therefore one often looks for local diffeomorphism,
i.e., for transformations defined only in a finite neighborhood of a given point. A
diffeomorphism can be used to transform a nonlinear system into another nonlinear
system in terms of a new set of states.
Distribution: Let f1, f2, . . . , fk be vector fields on D ⊂ Rn. At any fixed point
x ∈ D, f1(x), f2(x), . . . , fk(x) are vectors in Rn and
4(x) = spanf1(x), f2(x), . . . , fk(x)
is a subspace of Rn. To each point x ∈ Rn, we assign a subspace 4(x). We will
refer to this assignment by
4(x) = spanf1, f2, . . . , fk
which we call a distribution.
79
Involutive Distribution: A distribution 4 is involutive if
g1 ∈ 4 and g2 ∈ 4 ⇒ [g1, g2] ∈ 4 .
If 4 is a nonsingular distribution on D, generated by f1, f2, . . . , fr, then it can be
verified that 4 is involutive if and only if
[fi, fj ] ∈ 4 , ∀ 1 ≤ i, j ≤ r .
Theorem 5.2 (Frobenius Theorem) Let f1, f2, . . . , fr be a set of linearly inde-
pendent vector fields. The set (equivalently, a nonsingular distribution) is completely
integrable if and only if it is involutive. ♦
Consider an affine nonlinear single input system
x = f(x) + g(x) · u . (5.6)
With this system, the input-state linearization problem can be stated as follows:
Find u = α(x) + β(x) · ν and z = T (x)
such that
z1 = z2,
z2 = z3,...
zn = ν.
(5.7)
The following theorem provides a definite criteria for the existence of the input-
state linearization solution and constitutes one of the most fundamental results of
feedback linearization theory.
Theorem 5.3 (Input-State Linearizable Condition of Single Input System)
The nonlinear system (5.6), with f(x) and g(x) being smooth vector fields in Rn, is
80
input-state linearizable if and only if there exists a region Ω such that the following
conditions hold:
1. the vector fields g, adfg, . . . , adn−1f g are linearly independent in Ω ,
2. the set g, adfg, . . . , adn−2f g is involutive in Ω . ♦
For the detailed proof of this theorem, see pp. 568 of [80] and pp. 239-241 of [81].
The proof of this theorem leads to important relations that can be deduced from the
independent conditions of Lie brackets, which suggests an implicit way of obtaining
an appropriate diffeomorphism z = T(x) as follows:
∇z1 · adkf g = 0 k = 0, 1, . . . , n− 2 ,
∇z1 · adn−1f g 6= 0 .(5.8)
Moreover, recursive application of the Lie bracket to the zn equation yields
α(x) = − Lnf z1
LgLn−1f z1
, β(x) =1
LgLn−1f z1
. (5.9)
5.4 Shifting Controller Design via Input-State Lineariza-
tion
Based on the above differential geometric definitions and theorems, input-state lin-
earization of the shifting system of the S-CVT has been performed via the following
steps:
1. Construct the vector fields g, adfg, . . . , adn−1f g for our system.
2. Check the controllable and involutive conditions.
3. Find the first new state z1 from Equation (5.8).
4. Compute the diffeomorphism that transforms the state x into the new state z,
T (x) =[
z1 Lfz1 . . . Ln−1f z1
]T, and the input transformation using Equation (5.9).
81
First, we put the shifting system dynamics into the affine nonlinear control sys-
tem form (5.5) in order to obtain the corresponding vector fields f and g. Here we
consider the shifting force Fs to be the control input u. Then f and g of the shifting
dynamics can be written
f =
[
x2a12D
√
F 2i + F 2
o sin(x1 − φ)
]T
, g =[
0a22D
]T. (5.10)
Knowing that the system order n = 2 and ∇g = 0, the corresponding Lie bracket
then becomes
adfg = ∇g · f −∇f · g
= 0−
0 1a12D
√
F 2i + F 2
o cos(x1 − φ) 0
0a22D
=[
−a22D
0]T
.(5.11)
5.4.1 Controllability and Linearizability
We say the system is controllable if, using appropriate control inputs, the states can
be moved in any direction in the state space. For a linear system such as
x = Ax+Bu
controllability is a property of the pair (A,B) and can be checked as follows.
The pair (A,B) is controllable if and only if the rank of controllability matrix ,
C, is n (n is the system order, i.e., dimension of A), where C is given by
C =[
B AB A2B . . . An−1B]
To determine the controllability of nonlinear systems of the form (5.5), the control-
lability matrix C in the linear system is replaced by
[
g adfg ad2fg . . . adn−1f g]
(5.12)
82
In order to determine the controllability of the shifting system of the S-CVT,
we investigate the rank of the controllability matrix using the results of Equations
(5.10) and (5.11):
rank
0 −a22D
a22D
0
= 2.
Hence, we can say that the shifting system of the S-CVT is controllable. Further-
more, since the vector fields g, adfg are constant (i.e., its Lie derivatives are zero),they form an involutive set. Therefore the shifting system is input-state linearizable.
5.4.2 Input-State Linearization
Now we are ready to perform input-state linearization with the new states. First we
find a diffeomorphism T(x)that can transform the original shifting dynamics into
the linearized system. Using the results of Equation (5.8), the necessary conditions
for the first state z1 are
∂z1∂x16= 0,
∂z1∂x2
= 0 .
Thus z1 must be a function of x1 only. Among the various candidates for z1, the
simplest solution is z1 = x1 − φ. The other state can be obtained from z1
z2 = ∇z1f = x2 .
The corresponding diffeomorphism T(x) can be obtained as
z = T(x) =
x1 − φ
x2
. (5.13)
Accordingly the input transformation in Equation (5.9) is
u =ν −∇z2f∇z2g
83
which can be written explicitly as
u =D
a22ν − a12
D
√
F 2i + F 2
o sin(x1 − φ) . (5.14)
As a result of the above state and input transformations, we end up with the fol-
lowing set of linear equations
z1 = z2 , z2 = ν (5.15)
ν =a12D
√
F 2i + F 2
o sin(x1 − φ) +a22D
u . (5.16)
thus completing the input-state linearization.
5.5 Shifting Controller Design
By the above input-state linearization results, we now perform the shifting controller
design which can stabilize the shifting system according to the shift command and
track the demanded variator angle trajectory.
Stabilizing Controller Design
Since the new dynamics (5.15) is linear and controllable, it is well known that the
linear state feedback control law
ν = −k1z1 − k2z2
can guarantee asymptotic stability by selecting feedback gains k1 and k2 so as to
satisfy the Hurwitz condition. The linearized system can be written
z1 + k2z1 + k1z1 = 0 . (5.17)
84
Tracking Controller Design
For the case of the tracking problem, it is desired to have the variator angle θ track
a prescribed trajectory θd. Then the input ν is designed as
ν = ¨z1d − k1e− k2e (5.18)
where e = z1 − z1d and z1d = θd − φ. Therefore, the tracking problem of linearized
shifting dynamics transforms into the following error dynamics:
e+ k2e+ k1e = 0 . (5.19)
In order to guarantee asymptotic tracking performance of the shifting system, one
may check whether the gain selection k1, k2 can satisfy the Hurwitz condition, sim-
ilarly to the case of stabilizer design. The Hurwitz condition, however, offers only a
set of inequalities for the feedback gains, and the gain selection within these bound-
aries must be achieved using other criteria.
Gain Selection
The resulting closed-loop dynamics of the shifting system (5.17), (5.19) can be
viewed as the canonical form of a general second-order oscillation problem:
s2 + 2 ζωns+ ωn2 = 0 .
Hence one can give physical meaning to the feedback gains as the respective damping
ratio ζ and the natural frequency ωn. The relation between the feedback gains and
ζ, ωn are simply
k1 = ωn2, k2 = 2 ζωn . (5.20)
85
Case A k1 = 100, k2 = 20
Case B k1 = 50, k2 = 10√
2
Table 5.1: Candidates for k1, k2.
Therefore, we can deduce the relation of the gains k1, k2 as follows:
k2 = 2ζ√
k1 . (5.21)
Based on previous well-known research results [82], [83] on the vibration of
second-order systems, we consider two cases of k1, k2 (see Table 5.1). In this study,
we desire our shifting controller to provide the most rapid response according to the
shifting command without overshoot; we designate the settling time of the shifting
system (the time in reaching the new equilibrium state) to be less than 1 second.
Hence, we select the system damping ratio ζ to 1, which corresponds to the case
of critical damping . For a given initial excitation, a critically damped system tends
to approach the equilibrium position the fastest without any overshoot. Moreover,
these feedback gains guarantee the asymptotic stability and tracking performance
of the S-CVT shifting system.
Numerical Results
We first investigate the stability of the shifting system with the proposed feedback
gains. To do this, we simulate the behaviors of the shifting system numerically.
For the simulation conditions, we set the initial states of the system to
θ = 30, θ = 0, Fi = 1, Fo =√3 .
This initial condition is one of the equilibrium points of the shifting system. At
this instant, however, the output force suddenly changes from√3 to 1. Thus the
86
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
30
33
36
39
42
45
(a) Variator angle.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
Fs
(b) Control.
Figure 5.1: Stability of the S-CVT shifting system.
input-output force equilibrium no longer holds, and the gear ratio (i.e., the variator
angle θ) of the S-CVT must be changed into a new equilibrium state which makes
the system stable. Figure 5.1 shows the numerical results of the system behavior
and corresponding control from Equations (5.14), (5.17). As expected, both cases
of feedback gains show the asymptotic stability of the system. The variator angles
for each case change from the initial state into the new equilibrium point θ = 45.
Next we investigate the tracking performance of the shifting system as follows.
For the reference trajectory of the variator angle, we consider a sinusoidal functionπ
3sin(
π
2t− φ) (see Figure 5.2 (a)), with the initial states of the system chosen as
θ = 45, θ = 0, Fi = 1, Fo = 1 .
Maintaining the input-output force as the initial values, the calculated variator angle
changes are depicted in Figure 5.2 (b) using Equation (5.18). As expected, both
cases of feedback gains show asymptotic convergence of tracking error. The relevant
87
0 2 4 6 8 10
-60
-40
-20
0
20
40
60
(a) Reference shifting command θd.
0.0 0.3 0.6 0.9 1.2 1.5 1.8
-60
-40
-20
0
20
40
60
0.0 0.2 0.4 0.6 0.840
45
50
55
60
(b) Variator angle changes.
Figure 5.2: Tracking performance of the S-CVT shifting system.
tracking error and corresponding shifting effort are shown in Figure 5.3.
For both cases, the system responses match our predefined performance measure.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
(a) Tracking error.
0 2 4 6 8 10
-0.2
0.0
0.2
0.4
0.6
Fs
(b) Control.
Figure 5.3: Tracking error and corresponding control.
88
From the numerical results, we select the feedback gains for case B, although the
shifting response for case A is faster than that for case B (the time to reach the new
equilibrium variator angle 45 in case A is almost 0.7 second, while for case B it is
almost 1.0 second). The shifting effort (i.e., control effort) for case B maintains a
small value and varies monotonically compared to case A.
Using the selected feedback gains, we reconsider the stability of the system.
The overall system behavior during a gear ratio change is determined from the
S-CVT dynamics (2.10). The rotational speeds of the input, output, and sphere
are depicted in Figure 5.4, using the initial condition and corresponding control in
stability analysis.
0.0 0.2 0.4 0.6 0.8 1.0 1.2900
1200
1500
1800
2100
2400
2700
Figure 5.4: System behaviors of S-CVT during the gear ratio change.
89
5.6 Summary
Due to the nonlinearity of the S-CVT shifting dynamics, the original open-loop
system is inherently unstable. Hence a feedback controller is necessary to make the
system stable and to achieve effective tracking performance. To do this, we designed
a feedback controller that cancels nonlinearities and transforms the original nonlinear
system dynamics into a stable and controllable linear one, based on the input-state
linearization method.
In this chapter, we showed the instability of the original S-CVT shifting system
using Lyapunov’s linearization method. We also briefly reviewed the mathemati-
cal background for the formal description of the input-state linearization method.
With this background, we performed the input-state linearization of our system,
and designed a feedback controller which achieves asymptotic stability and effec-
tive tracking performance of the S-CVT shifting system. In selecting the feedback
gains of the proposed controller, we considered our linearized shifting dynamics as a
canonical second-order oscillation problem. In order to achieve a predefined shifting
performance, we then set the feedback gains; comparing the numerical results of
the shifting effort (i.e., control effort) and the settling time. Finally we presented
numerical results which demonstrate shifting controller performance with respect to
stability and tracking.
90
Chapter 6
Optimal Control of an S-CVT
equipped Power Transmission
6.1 Introduction
Among the various advantages of a CVT, the most prominent is its ability to run the
power source at the power efficient regime. Furthermore, in most power sources such
as internal combustion engines and electric motors, optimal efficiency lies at a certain
operating point. As reviewed in Chapter 1, many control engineers endeavor to find
an effective way of controlling a CVT’s gear ratio to maintain the power source at
the most efficient point and realize shifting commands in the desired manner.
Optimal control of a CVT equipped power transmission is then defined as the
problem of finding a gear ratio which can minimize the energy consumption of the
power source without any losses in output performance. Hence, in order to design
a minimum energy control law for a CVT, one must first investigate the efficiency
characteristics of the power source as well as define the target performance. Driven
91
mainly by automotive engineers, various control approaches have been tried and
realized. There are two major issues in controlling CVTs to achieve efficiency and
performance objectives: power source consolidated control, and establishing the
shifting map (the variogram), which is a look-up table of the speed relations between
the power source and output.
The S-CVT is intended for use in small power capacity power transmissions; thus
a dc motor is considered as the power source in this study. DC motors are designed
to be very efficient at their rated speeds, and it is now generally believed that there
is very little room for improvement in terms of hardware performance. With recent
advances in power electronics, the motor drivers that supply the input voltage or
current are also now extremely efficient, compared to previously used analog drivers,
enough to be used as variable speed drives [84], [85]. In addition, dc motor optimal
control algorithms that take into account the load and other operating characteristics
have been developed [86], [87], further reducing overall power consumption.
In this chapter, we present a minimum energy control law for the S-CVT con-
nected to a dc motor. To do this, in Section 2 we first describe the general power
efficiency characteristics of a dc motor using the well-known dc motor dynamic equa-
tions. In Section 3 we present the results of a numerical investigation of the possi-
bility of energy saving using the S-CVT benchmarked against a standard reduction
gear, taking into account the equations of motion of the S-CVT equipped power
transmission system and an ideal motor model. In addition, a computed torque
control algorithm for the S-CVT is proposed. Section 4 deals with the minimum
energy control design via a B-spline parameterization of the trajectories. Finally we
show some numerical results of energy savings using the proposed minimum energy
control law.
92
6.2 Power Efficiency of a DC Motor
6.2.1 DC Motor Dynamics
We now consider a general armature-controlled dc motor as shown in Figure 6.1, in
which the field current is held constant. We adopt the following nomenclature:
Ra = armature resistance [ohm],
La = armature inductance [henry],
ia = armature current [ampere],
if = field current [ampere],
ea = applied armature voltage [volt],
eb = back-emf (electromotive force) [volt],
ωM = angular velocity of the motor [rad/sec],
ωo = angular velocity of the CVT output shaft [rad/sec],
Ieq = equivalent moment of inertia of the motor and load referred to
the motor shaft [kg ·m2],
TM = motor torque [N ·m],
Tload = load torque [N ·m].
Figure 6.1: Diagram of an armature-controlled dc motor.
93
Then the circuit equation is
Ladiadt
+Raia + eb = ea . (6.1)
The induced voltage eb is directly proportional to the speed of the motor ωM , or
eb = keωM (6.2)
where ke is a back emf constant. The motor torque TM is directly proportional to
the armature current:
TM = kia (6.3)
where k is a motor-torque constant.
Referring to Figure 6.1, we consider an S-CVT equipped power transmission.
Using a gear train including CVTs at the motor shaft has the effect of reducing not
only the load torque by the gear ratio, but also the equivalent inertia by a square of
the gear ratio; the motor dynamic equation becomes
Ieqα2
dωM
dt= TM −
TLoad
α, ωo = ωM
1
α(6.4)
where α represents the reduction gear ratio. In the case of the S-CVT, however,
α is replaced by the torque ratio of the S-CVT, i.e., α = cot θ, so that the motor
dynamic equation becomes
Ieq tan2θdωM
dt= TM − TLoad tan θ, ωo = ωM tan θ . (6.5)
Assuming that the armature inductance La in the circuit is small enough to
neglect, we obtain the following equation from (6.1):
Raia + eb = ea
94
thus the motor torque can be written as
TM =k(ea − keωM )
Ra. (6.6)
Finally, the differential equation for the speed of the output shaft ωo (6.5) becomes
Ieqdωo
dt+kkeRa
ωo cot2 θ =
keaRa
cot θ − TLoad . (6.7)
6.2.2 Power Efficiency of a DC Motor
In this section, we consider the efficiency characteristics of a general armature-
controlled dc motor in Figure 6.1. The torque produced by a dc motor is directly
proportional to the armature current (6.3); when the equivalent inertia and/or the
load torque applied at the motor shaft is increased, the armature current must also
be increased. Rearranging the above equations and using the fact that the value
of ke is equal to k, the relationship between the mechanical power and the electric
power is
TMωM = eaia − ia2Ra . (6.8)
In the above equation, the ia2Ra term represents the electric power-loss, called the
armature-winding loss, generally dissipated through heat generation.
From Equation (6.1), at certain values of the armature voltage, decreasing the
armature current will increase the value of the back-emf and the motor speed. Based
on these observations, one can notice that motors have their highest efficiency in the
low-torque, high-speed region. Figure 6.2 depicts the power efficiency of a general
dc motor with respect to the motor speed and the load torque. As can be seen from
the graph, motor efficiency is highest in the low-torque, high-speed region (indicated
in dark blue), with the efficiency dropping off steeply in other regimes. To enhance
95
Figure 6.2: Efficiency of an armature-controlled dc motor.
the power efficiency of a dc motor, it is clearly advantageous to operate it in this
region of maximum efficiency.
6.3 Investigation of S-CVT Energy Savings
In this section, we perform a numerical investigation of the energy savings of the
S-CVT. As a comparative benchmark, we consider two power transmission systems
driven by the same dc motor: one driven by a reduction gear, the other driven by
the S-CVT. A typical output speed profile is given to each system, and then the
variations of electricity (e.g., ampere and voltage) are calculated. In the case of a
reduction gear, the motor speed is controlled in order to follow the given output
speed profile. However, in the case of the S-CVT, one can control either the gear
ratio (i.e., the variator angle) or the motor speed. In this study we manipulate the
96
Rated voltage 12 V olts
Rated power 60 Watts
Motor-torque constant 0.0272 N · m/A
Back emf constant 0.0272 volt · sec/radRotor winding resistance 0.48 Ohm
Stall torque 0.68 N · m
Table 6.1: Characteristic coefficients of dc motor.
output speed by the variator angle only, choosing to operate the motor at its most
efficient regime by effectively treating the armature voltage ea as its rated value.
For the numerical investigation, we assign a load torque TLoad of 0.07 Nm, and
an equivalent inertia with respect to the input shaft Ieq of 0.01 kgm2; for these
values, the stall torque is calculated to be 1.6 Nm. We therefore choose a dc motor
with a power rating of 60 Watts (see Table 6.1 for the detailed specifications of the
dc motor) and a gear ratio of four for the reduction gear case. The desired output
speed profile is chosen to be a sinusoid with a magnitude of 500 rpm and a period
of 40 seconds (See Figure 6.3).
0 10 20 30 40-600
-400
-200
0
200
400
600
Figure 6.3: Target profile of output speed.
97
6.3.1 Control Design based on the Computed Torque Method
In this section, we design a control based on the computed torque control method
which is used widely in robotics and other engineering fields. The computed torque
control compensates for tracking errors by using feedback information about the dif-
ferences between the predefined objective trajectories of position, speed, acceleration
and the estimated actual trajectories.
From the given output speed profile ωo(t), the computed motor torque TM is
calculated using Equation (6.6) and the relation of ωo between ωM :
TM =k
Ra(ea − keωo cot θ) .
This motor torque must be balanced by the torque due to the equivalent inertia and
the load (6.5), i.e.,
k
Ra(ea − keωo cot θ) = (Ieq
dωo
dt+ TLoad) tan θ .
Rearranging the above equation yields
kkeRa
ωo cot2 θ − kea
Racot θ + Ieq
dωo
dt+ TLoad = 0 . (6.9)
Hence the variator angle profile can be obtained by solving the second-order poly-
nomial Equation (6.9). Given the desired output speed profile ωo(t) as shown in
Figure 6.3, and assuming a fixed armature voltage of 12 V olts, we determine the
trajectory of the variator angle θ(t) (see Figure 6.4).
Comparing the results with the sinusoidal shape of the target output speed profile
in Figure 6.3, the variator angle time profile is seen to have a sharper gradient during
the rise stage. This difference can be accounted for by the acceleration rate of the
output speed. In general reduction gear equipped transmissions, the input speed (the
motor speed here) varies directly with the output speed; the required acceleration
98
0 10 20 30 40-8
-6
-4
-2
0
2
4
6
8
Figure 6.4: Computed variator angle time profile.
rate of the output speed is generated entirely by the input torque (in this case the
motor torque). However, CVTs can manipulate the gear ratio according to the
output speed. Therefore the variator angle profile of the S-CVT is affected by the
shape of the desired output speed profile and the necessary acceleration rate. The
magnitude of the angle variation, which for our case can be regarded as the control
effort, is small enough such that it can be implemented even with a relatively efficient
small-capacity shifting actuator.
6.3.2 Numerical Results
Using the equations derived previously, we now perform a numerical study of the
energy consumption rates for each system. In Figure 6.5 (a), the initial motor speed
of the S-CVT system is set to about 4200 rpm, which is obtained from a zero-load
condition for the dc motor considered here. While the motor speed for the reduction
gear case varies depending on the output speed profile, for the S-CVT it remains
close to 4000 rpm which is nearly the nominal speed for the zero-load condition. The
99
0 10 20 30 40
-2000
-1000
0
1000
2000
3000
4000
5000
(a) Motor speed.
0 10 20 30 40-0.30
-0.15
0.00
0.15
0.30
0.45
(b) Motor torque.
Figure 6.5: Motor behaviors; reduction gear vs. S-CVT.
torque exerted on the motor for each case is calculated in Figure 6.5 (b). Generally,
maximal torque is necessary when the motor first begins to rotate. However, for the
S-CVT, almost zero torque is exerted when the motor starts rotating. This reduction
in load torque is a consequence of the infinite torque multiplication characteristics
of the S-CVT, which can be realized in the vicinity of zero variator angle.
0 10 20 30 40
0
20
40
60
80
100
120
140
Figure 6.6: Power consumption; reduction gear vs. S-CVT.
100
Reduction gear S-CVT
1783.2 Joules 957.4 Joules
Table 6.2: Energy consumption; reduction gear vs. S-CVT.
The consumed energy for each case is calculated using the above results and the
relation
Energy =
∫
| ea(t)× ia(t) | dt.
We regard negative values of current and voltage as part of the overall consumed
energy. From Figure 6.6, we have calculated the energy consumption in Table 6.2
consequently. Our results suggest that in principle, the consumed energy for the
case with the S-CVT is less than the other case by almost 46.3%, although in actual
implementations the effects of friction, backlash, and other sources of loss will have
to be considered in more detail.
6.4 Minimum Energy Control via a B-Spline Parame-
terization
In this section, we describe the minimum energy control design via a B-Spline pa-
rameterization. The minimum energy control problem of an S-CVT equipped power
transmission is defined as follows:
Find the optimal control u that minimizes J =
∫ tf
t0
iaeadt
subject to Ieqωo = −kkeRa
ωou2 +
keaRa
u− TLoad
(6.10)
where u is the gear ratio, i.e., cot θ. The boundary conditions can be expressed
in various forms according to the target performance. In this section, we consider
101
the more complicated case of the S-CVT application for some position changers,
e.g., a mobile robot, a vehicle, a positioning table, etc. For this case, the boundary
conditions are given as follows:
ωo(t0) = ωo(tf ) = 0, s(t0) = 0, s(tf ) = d
where d is the desired displacement, s(t) represents the displacement profile, and to,
tf represent the initial and final times respectively.
6.4.1 B-Spline Parameterization
A solution to the above optimal control can be found by assuming that the displace-
ment profile s(t) is parameterized by a B-spline. The B-spline curve depends on
the basis functions Bi(t) and the control points p = [p1 . . . pn] with pi ∈ R . The
displacement profile then has the form s = s(t,p) with
s(t,p) =
n∑
i=1
Bi(t)pi (6.11)
Using this formulation (6.11), ωo, ωo and u, which are functions of t and p, can
be written as
ωo(t,p) =1
r
∂
∂ts(t,p), ωo(t,p) =
1
r
∂2
∂t2s(t,p),
and
u(t,p) =kea +
√
D(t,p)
2kkeωo(6.12)
where D(t,p) = k2e2a − 4Rakkeωo(TLoad − Ieqωo), r is the conversion factor from a
rotational speed into a linear speed (for the cases of mobile robots and vehicles this
means the wheel radius). The control u is determined from Equation (6.12), but an
102
additional inequality constraint D(t, P ) ≥ 0, ∀t ∈ [t0, tf ] must also be satisfied. In
order to satisfy the boundary conditions, we set p1, p2 to zero and pn−1, pn to d.
Setting the input voltage ea to a constant value by the same reason as in the
previous section, the armature current ia can be calculated from Equations (6.5)
and (6.6):
ia(t,p) =ea − keωou
Ra
Hence the original optimal control problem is converted into a parameter optimiza-
tion problem as follows:
minimize J(p) = ea
∫ tf
t0
ia(t,p)dt
subject to D(t,p) ≥ 0 , ∀t ∈ [t0, tf ] .
(6.13)
6.4.2 Gradients of the Objective Function and Constraint
To apply various parameter optimization algorithms (i.e., steepest descent, modified
Newton method, quasi-Newton method, penalty method, etc.) to this problem
(6.13), we must formulate the gradients of the objective function and constraint
because almost all optimization algorithms require gradients of the objective function
and constraint.
The gradient of the objective function is
∂J
∂pi=
∫ tf
t0
∂ia∂pi
eadt
where the partial derivatives of ia are as follows:
∂ia∂pi
= − keRa
(∂ωo
∂piu+ ωo
∂u
∂pi
)
The derivatives of ωo and u are obtained from the fact that
∂s
∂pi= Bi(t)
103
Since the constraint in Equation (6.13) is represented in the form of inequality,
we can just know whether the constraint is effective or ineffective. In order to find
the gradient of the constraint, we now propose the new constraint by defining a new
function, g(t,p), g(p), as follows:
g(t,p) =
−D(t,p) if D(t,p) < 0
0 if D(t,p) ≥ 0
g(p) =
∫ tf
t0
g(t,p)dt .
Figure 6.7 illustrates the interpretation of g(p). With these definitions, it is apparent
that the constraint in Equation (6.13) is equivalent to the following constraint:
g(p) = 0
It is difficult to solve this constraint analytically; however the gradient is well defined.
αβ
Figure 6.7: Interpretation of g(p).
104
g(p) can be redefined as
g(p) =∑
j
∫ βj
αj
−D(t,p)dt
where D(t,p) < 0 for ∀t ∈ [αj , βj ] and D(αj ,p) = D(βj ,p) = 0. The gradient of
g(p) can be defined as follows:
∂g
∂pi=
∑
j
∂
∂pi
∫ βj
αj
−D(t,p)dt
=∑
j
(
∫ βj
αj
−∂D(t,p)
∂pidt−D(βj ,p)
∂βj∂pi
+D(αj ,p)∂αj
∂pi
)
=∑
j
∫ βj
αj
−∂D(t,p)
∂pidt .
The gradient is now rewritten as follows:
∂g
∂pi=
−∂D(t,p)
∂piif D(t,p) < 0
0 if D(t,p) ≥ 0
, (6.14)
∂g(p)
∂pi=
∫ tf
t0
∂g(t,p)
∂pi. (6.15)
Because it is difficult to find αj , βj for a given p, we can alternatively use Equation
(6.15) to numerically calculate the gradient of the constraint.
6.4.3 Numerical Results
We determine by simulation the power consumption for a minimum energy control
and a comparative computed torque control (see more detail in Section 6.3). The
comparative control is designed to manipulate the output speed in a sinusoidal
105
fashion, satisfying the boundary conditions. To satisfy the boundary conditions the
displacement profile is described as follows:
s(t) =d
2
(
1− cosπt
tf
)
From this relation, ωo, ωo are derived by differentiation, and the control u is obtained
from Equation (6.9).
Based on the mathematical models presented in the previous sections, we have
developed a simulation program with MATLAB. This program uses Simpson’s rule
for integration and the BFGS quasi-Newton method for optimization. We assign
the final time tf to be 5 seconds, and the desired displacement d to be 8 meters.
Figure 6.8 depicts the corresponding variator angle time trajectories which are
directly related with the controls for each case. In this figure, the optimized variator
angle is much flatter than in the case of the computed torque control. The resulting
motor speed and torque are calculated in Figure 6.9. As can be seen in Figure 6.9,
in the minimum energy control case the variation of the motor speed is smaller and
0 1 2 3 4 5
0
5
10
15
20
Figure 6.8: Optimal variator angle time profile.
106
0 1 2 3 4 53000
3200
3400
3600
3800
4000
4200
4400
4600
4800
5000
(a) Motor speed.
0 1 2 3 4 5-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
(b) Motor torque.
Figure 6.9: Motor behaviors with the minimum energy control.
the motor torque is closer to zero compared to the other controller.
Figure 6.10 shows the output behavior of the S-CVT equipped power transmis-
sion. From this figure, one can see that the minimum energy controller accelerates
the output faster than the computed torque control. Consequently, we have calcu-
0 1 2 3 4 5
0
2
4
6
8
(a) Displacement.
0 1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
2.5
(b) Output speed.
Figure 6.10: Output behaviors with the minimum energy control.
107
Minimum energy control Computed torque control
43.37 Joules 57.16 Joules
Table 6.3: Energy consumption with the minimum energy control.
lated the energy consumption in Table 6.3. The optimized energy consumption is
less than that of the other case by almost 24.1%
6.5 Summary
Using an ideal motor model, we carried out a numerical study on the energy efficiency
of an S-CVT equipped power transmission system, and compared the results with
that of a standard reduction gear. The S-CVT was intended to primarily for use
in small power capacity transmissions, thus a dc motor was considered here as the
power source. In this chapter, we presented a minimum energy control law for the
S-CVT in a typical power transmission with a dc motor.
To do this, we first described the general power efficiency characteristics of a dc
motor using well-known dc motor dynamics relations. We then presented the nu-
merical results for the investigation of the S-CVT energy saving possibility, bench-
marked against a standard reduction gear. In addition, we proposed a computed
torque control algorithm for the S-CVT. Section 4 deals with the minimum energy
control design via a B-spline parameterization. By parameterizing the displacement
profile in terms of a B-spline, the optimal control problem is converted into a pa-
rameter optimization problem involving the B-spline control points. Finally, to show
the effectiveness of the developed minimum energy control law, computer simulation
results using a computed torque control and an optimal control law for the same
system are addressed.
108
Chapter 7
Case Study: An S-CVT based
Mobile Robot
We propose an S-CVT based mobile robot (named as MOSTS: Mobile rObot with a
Spherical Transmission System) to put the various advantages of S-CVT including
the originally intended CVT characteristic of energy efficiency into practical use.
In this chapter, we first address the motivation for applying the S-CVT to a
wheeled mobile robot, by first reviewing the current hardware designs of mobile
robots and their power efficiency in Section 1. Section 2 shows the hardware design
of our S-CVT based mobile robot. In addition, we propose a novel pivot mechanism
that uses an internal gear and an uncontrolled dc motor. In Section 3, we perform
numerical simulations and experiments to validate the robot’s operation, the CVT
characteristics, and its energy saving possibility.
109
7.1 Motivation for Mobile Robot Applications
In recent years, there has been an explosion of research activity in mobile robots,
driven in part by the focus on service robots and their applications, e.g., patient
transportation, autonomous security services, mobile platforms for manipulators,
etc.A large part of the mobile robot literature addresses issues in their planning and
control, taking into account the non-holonomy generated by the wheels. In contrast
to the literature on mobile robot motion planning, relatively little attention has been
given to hardware platforms and other physical aspects of mobile robots [88], [89].
While the mechanical hardware specifications for mobile robots vary widely, gen-
erally a wheeled mobile robot requires at least two actuators for moving about in
the plane, each with a dedicated controller (see Table 7.1). Wheel drives, generally
known as differential drives, are said to be omnidirectional mobile robots as they
can move about arbitrarily in the planar workspace. Track drives, using tracks (or
caterpillars), use at least four track-drive motors with idle track-wheels. Although
track drives can be used in the desert, muddy or undeveloped grounds as they can
move on rough surfaces, owing to the track characteristics they show low power
efficiency.
Number of
Drive Motors
Turning
DeviceWorkspace Features
Wheel
Using
Differential Gear1
Steering motor
is necessaryPlane
Controllers
for each motor
Drives Differential
Drives
Equal to
number of wheels
Unnecessary,
Pivot/SteeringPlane
Controllers
for each motor
Track Drives at least 4Unnecessary,
Pivot/Steering
Rough
surfaces
Controllers
for each motor,
low efficiency
Legged RobotsEqual to
number of jointsUnnecessary
Rough
terrain
Controllers
for each joint
Table 7.1: Hardware specifications of general mobile robots.
110
Besides the hardware specification aspects, mobile robots typically use electric
motors as actuators, in particular dc motors, because of their relatively simple con-
trol features, and the fact that power can be supplied from battery sources. Despite
advances in motor efficiency, the runtime of mobile robots is still limited by their
batteries and reliance on load conditions. Furthermore, general dc motors have their
best power efficiency in the regime of low-torques and high speeds. Thus, it is clear
that the overall power efficiency of a mobile robot depends on that of the dc motor
which is adopted in the robot. Consequently to prolong the robot’s run time, it is
advantageous to operate the motor in this region of maximum efficiency.
Hence, to improve the run time and to avoid having to use an oversized motor,
general wheeled mobile robots and vehicles typically use gear reduction [89]. A
reduction gear reduces the load torque and increases the motor speed by the selected
gear ratio. Although automobiles have a finite range of available gear ratios, it is
generally impractical to equip mobile robots with standard transmission devices due
to manufacturing costs, space, and other limitations.
In the case of reduction gears, the motor is operated in the low-efficiency region
when accelerating or decelerating the mobile robot due to its fixed gear ratio. The
CVT allows for infinite ranges of gear ratio, and offers the possibility of much im-
proved energy efficiency and performance. Moreover, it allows the motor to deliver
a range of torques at its most efficient speed (the so-called rated speed) while the
mobile robot moves, by changing its gear ratio continuously.
111
7.2 MOSTS: An S-CVT Mobile Robot
7.2.1 Pivot Device for Planar Accessibility
As previously seen in typical mobile robot designs, an additional controlled actua-
tor, such as a steering wheel or a motor for differentiating each wheel velocity, is
necessary in order to move a mobile robot in the plane. Employing a novel pivot
device, however, we can eliminate the need for an additional steering actuator and
controller. To change its heading direction, MOSTS turns about its center (or piv-
ots) by rotating one of the wheels in the reverse direction. For this to occur, we have
been inspired by the fact that the S-CVT can locate arbitrarily the orientation of
the output shaft. To achieve this, it is necessary to locate one of the output shafts
on the opposite side of the sphere (see Figure 7.1 (a)).
For this operation we have adopted an internal gear driven by a simple actuator
(see Figure 7.1 (b)), e.g., a limit switch used in automated windows, and an uncon-
(a) Pivot inspiration. (b) Realization by use of an internal gear.
Figure 7.1: Pivot device for planar accessibility of MOSTS.
112
trolled motor; this is publicized in the form of patent, pending by the Office of the
Patent Administration of Korea [90]. Using simple analog devices, we build a pivot
switch that can be turned off according to a pre-set current limit.
The electric circuit diagram of the pivot switch including the driving motor cir-
cuit is shown in Figure 7.2. In the electric circuit of the pivot switch and driving
motor, one can find that there is no speed controller for the driving motor and pivot
Signal +5V, 0V
Relay 2 levelSetting
MDriving Motor
-
+
741-
+
C3117
10K Fm330D560
W47R
W5
2.0 W
1K
Amp.out1A=0.1V
Relay2 (12V)
W5
70W
Relay1(10V)
+12V
NORelay 1
10K
NORelay 2
0.5K
Offset Control
C3989
+12V
+12V
5.01
5.0==
K
KGain
Sig. +5V,0V
4.7K
M
741-
+
741-
+
+12V
D560
D560
D560
W5
2.0 W
NO NO
NC NC
C C
C C
NONO
+12
Relay2
Relay3
A B
5K
5K
+12
+12
+12
Offset1K
18K
Stop levelControl
103¡þ
1.4K
Relay1
Relay1 Relay3
C
NO NC Relay2
Pivot Switch
D 560
+12
10K
Fm220
A
B
Figure 7.2: Electric circuit diagram of pivot switch and driving motor.
113
motor. During pivot motion, each wheel rotates in opposite directions with the
same magnitude, while the driving motor rotates continuously without any changes
of state. The amount of pivot angle is determined by the amount of angular dis-
placements of each wheel, which is controlled by the shifting actuator, or variator.
Moreover, if a controlled actuator is used to rotate the movable output shaft, steer-
ing motion can be obtained. Designed in this fashion, MOSTS has the capability
to move in the plane with one drive motor, one controller for the S-CVT, and one
switching actuator.
7.2.2 Prototype Design
For the construction of the mobile robot platform, we have set the following perfor-
mance targets:
1. A top speed of 5 m/sec;
2. A maximum ascending angle of 10;
3. A combined vehicle-payload mass of 50 kgs.
To satisfy these goals, we begin by specifying the static-load conditions. The total
resistive force on the wheel is given by
Fresistant = W × sin θ + C1 × cosβ
where we assume the drag force coefficient C1 to be 0.75 kgf based on typical
values for the friction coefficient between the wheel and the ground, β represents
the ascending angle, and W is the weight of the mobile robot platform. The static
friction coefficient between the sphere and the output disc is assumed to be 0.12,
while the distance between the disc center and the sphere-disc contact point is set
to be 10 mm. The normal force exerted on the contact point is set to be 80 kgf , or
114
Rated voltage 12 V olts
Rated power 150 Watts
Motor-torque constant 0.0164 N ·m/A
Back emf constant 0.0164 volt · sec/rad
Rotor winding resistance 0.117 Ohm
Stall torque 2.03 N ·m
Table 7.2: DC motor charateristic coefficients of MOSTS.
equivalently 784.8 N . The maximum torque that can be transmitted by the S-CVT
in this case becomes 0.942 Nm.
From the above hardware specifications and material properties of the S-CVT,
we choose a specific dc motor that produces a power of 150 Watts with 12 V olts
under nominal operating conditions as the driving motor (see the details provided
in Table 7.2).
The body of the mobile robot is designed to have a cylindrical shape, and a caster
wheel is added to provide stable support. The internal body consists of three layers:
a mechanical base for the transmission system, an intermediate layer for the battery
pack and controller, and a top layer for peripherals and accessories, e.g., navigation
sensors, manipulators. Rotary encoders sensing the speeds of the input and output
shafts are also included. The overall size of the platform is 260 mm in radius, and
500 mm in height (see Figure 7.3).
7.3 Numerical and Experimental Results
In this section, we present numerical and experimental results that demonstrate the
operation of MOSTS, and the energy savings possible from the use of the S-CVT
115
Figure 7.3: Hardware prototype of MOSTS.
N
S
EW
Figure 7.4: The desired trajectory.
116
mechanism over standard reduction gears.
The reference path is shown in Figure 7.4; there are three linear movements
and two pivot motions during 22 seconds. The distance traversed by the robot is 20
meters. During the pivot motion, there is an auxiliary 2 second period for actuating
the pivot switch, which is necessary to move one of the output shafts of the S-CVT
to the opposite direction.
With this reference trajectory, we calculate the necessary wheel velocity profile
satisfying the time constraints by using a sine function (see Figure 7.5). The pivot
motions in the path are specified as a 90 counter-clockwise rotation, followed by a
90 clockwise rotation.
First, we calculate the value of the output speed acceleration from the driving
pattern under the assumption that the input voltage is held constant at 12 V olts.
The exerted load torque is set to 2.5215 Nm, and the equivalent inertia with respect
to the motor shaft is set to 0.01 kgm2. With these values and the output speed, we
0 5 10 15 20 25-100
-50
0
50
100
150
200
250
300
Figure 7.5: Calculated wheel velocity profile.
117
0 5 10 15 20 25
-2
0
2
4
6
8
Figure 7.6: Trajectory of variator angle.
extract the necessary variator angle θ by a computed torque control algorithm in
Section 6.3. Finally, the trajectory of the variator angle is presented in Figure 7.6.
7.3.1 Numerical Results
Using the equations derived in the previous chapter, we have developed a simulation
program that computes the motor speed, produced torque, and the power consump-
tion. We use the Runge-Kutta fourth-order algorithm for numerical integration in
the simulation program.
In Figure 7.7 (a), the initial motor speed is about 7000 rpm, which is obtained
from the no-load condition of the dc motor considered here. During the whole
operation period, the motor speed varies freely between 6500 rpm and 7000 rpm
regardless of the behavior of the robot (stop, start, and pivot motions), which are
almost the nominal speeds under a no-load condition. The necessary motor torque
is calculated in Figure 7.7 (b). Generally, maximal torques are necessary when the
118
0 5 10 15 20 25
6600
6700
6800
6900
7000
(a) Motor speed.
0 5 10 15 20 25-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
(b) Motor torque.
Figure 7.7: Motor behaviors of MOSTS.
motor of general mobile robots starts rotating. However, in the case of our robot,
almost zero torque is exerted at the start, and the torque variations are quite small
during the whole period.
To investigate the increase in energy savings, we calculate the energy consump-
tion rate of our mobile robot for the reference trajectory. As a benchmark, we
consider a differential drive type mobile robot having a reduction gear unit with a
gear ratio of six under the same load condition. The differential drive type robot
considered has two driving motors of 150 Watts at each wheel shaft and follows the
same reference trajectory. Consequently, we calculate the energy consumption rates
for each case using the following equation:
Energy =
∫
| ea(t)× ia(t) | dt.
From Figure 7.8, we calculate the total energy consumption to be 1389.61 Joules
for the differential drive with reduction gear unit and 727.86 Joules for our CVT-
119
0 5 10 15 20 25-10
0
10
20
30
40
50
60
70
80
90
100
110
Figure 7.8: Power consumption; MOSTS vs. differential drive.
based mobile robot. Our mobile robot equipped with the S-CVT consumes less than
47.6% of the energy consumed by the differential drive, a significant improvement
in energy efficiency.
7.3.2 Experimental Results
Using the sequential manipulation of the variator angle according to calculated val-
ues of Figure 7.6, we experimentally determined the actual energy consumption of
MOSTS under the same reference trajectory mentioned above. The actual energy
consumption is 1294.92 Joules, which is larger than the ideal case by 567.06 Joules.
However, this is still smaller than the calculated energy consumption of 1389.61
Joules for the differential drive case (the actual energy consumption for this case
will most likely be significantly higher than the calculated ideal rate).
To investigate the reason behind this difference in total energy consumption,
120
0.0 0.5 1.0 1.5 2.06200
6300
6400
6500
6600
6700
6800
6900
7000
(a) Motor speed.
0.0 0.5 1.0 1.5 2.0-1
0
1
2
3
4
5
6
7
8
(b) Motor induced current.
Figure 7.9: Experimental results.
the induced motor current and the actual motor speed for the first two seconds are
depicted in Figure 7.9. As the reference motion trajectory considered here has five
repetitive sequences (see Figure 7.5, 7.6, and 7.7), it is sufficient to investigate the
first two second period experimental results. Observe that the initial motor current
is almost 4 Amperes, whereas the ideal value is almost zero. This initial induced
motor current is mainly due to the power loss resulting from manufacturing errors
including bearing friction, gear backlash, etc.Consequently, this power loss makes
the driving motor run at lower speeds, causes the overall power efficiency to decrease.
MOSTSsimulation result
experimental result
727.86 Joules
1294.92 Joules
Differential drive
with reduction gear
simulation result
experimental result
1389.61 Joules
??
Table 7.3: Energy consumption; MOSTS vs. differential drive.
121
7.4 Summary
In this chapter, we have presented the design of a CVT-based mobile robot using
a minimal number of actuator and control components, by taking advantage of
the typical characteristics of the S-CVT. Such a CVT-based mobile robot has the
advantage of being able to operate the motors in their regions of maximum efficiency,
thereby prolonging the total run time of the robot. The addition of a novel pivot
device also enables the mobile robot to achieve steering (more precisely, changing
its heading direction) by using only a single drive motor and controller, unlike most
existing mobile robot platforms.
We also perform an in-depth analysis of the energy efficiency of our mobile
robot taking into account features of the dc motors, the S-CVT, and the mobile
robot dynamics. The results are benchmarked numerically with a differential drive
type mobile robot equipped with a reduction gear. Furthermore, we perform an
experiment using the prototype robot to verify the robot’s operation and the CVT
characteristics. The numerical and experimental results show that our mobile robot
with S-CVT consumes power less than differential drive type robots.
122
Chapter 8
Conclusion
In this thesis we have performed a comprehensive study on the design, analysis, and
control of the Spherical CVT. Based on these results, we can conclude that proposed
Spherical CVT shows attractive advantages, such as compact and simple design and
relatively simple control features, effective in particular for mechanical systems in
which excessively large torques are not required and we have performed theoretic
and practical works which could confirm these advantages with a case study of a
Spherical CVT-based mobile robot.
The important conclusions from this work are summarized as follows.
• The S-CVT is marked by its simple configuration, infinitely variable transmis-
sion (IVT) characteristics and realization of the smooth transitions between
forward, neutral, and reverse states without any brakes or clutches. The power
transmission mechanism is based on dry rolling friction between the contact
bodies of the sphere and discs. Its practical applications are currently lim-
ited to small power capacity mechanical systems, though adopting traction
fluid can increase the maximum torque of the S-CVT so as to make its use in
123
traction drive possible.
• The S-CVT is intended to overcome some of the limitations of existing CVTs,
e.g., difficult shifting controller design, and the necessity of a large-capacity
and typically inefficient shifting actuator. The analysis results on operating
principles, transmission ratios and power efficiency of S-CVT have been verified
by experimental results obtained with the testbench.
• Spin loss, which is one of the main design issues on traction drives, is analyzed
from its physical mechanism to a quantitative explicit formulation. To analyze
this, we have proposed a modified classical friction model, which can describe
the friction behavior of the S-CVT including pre-sliding effects (i.e., Stribeck
effects). Additionally, we have performed an in-depth study of velocity fields
generated at the contact regions along with a Hertzian analysis of deflection.
• To stabilize and achieve effective tracking performance we have designed a
feedback controller, which can cancel typical nonlinearities and transform the
original nonlinear system dynamics into a stable and controllable linear one,
based on the input-state linearization method. The designed feedback shifting
controller shows asymptotic stability and tracking performances; the settling
time is smaller than 1 second, the shifting effort varies monotonically and
keeps small value.
• Using an ideal motor model, we have presented the numerical results for the
investigation of the S-CVT energy saving possibility, benchmarked against a
standard reduction gear. The minimum energy control design via a B-spline
parameterization is carried out by parameterizing the displacement profile in
terms of B-splines; the original optimal control problem is converted into a
124
parameter optimization problem involving the B-spline control points. To show
the effectiveness of the developed minimum energy control law, simulation
results using a computed torque control and an optimal control law for the
same system are addressed.
• We have presented the design of a CVT-based mobile robot using a minimal
number of actuator and control components, by taking advantage of the typi-
cal characteristics of S-CVT. The addition of a novel pivot device also enables
the mobile robot to achieve steering (more precisely, changing its heading di-
rection) by using only a single drive motor and controller, unlike most existing
mobile robot platforms.
• The energy efficiency of our mobile robot is benchmarked numerically with
a differential drive type mobile robot equipped with a reduction gear. Fur-
thermore, we perform an experiment using the prototype robot to verify the
realization of robot’s operation and the CVT characteristics. The numerical
and experimental results show that our mobile robot with S-CVT consumes
the electric power less than that of a differential drive type robot, significantly.
125
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