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 the School of Mechanical and Aerospace Engineering at Seoul National University February 2001
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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
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
References
[1] Youngdug Choi and Innwoog Yeo, 1999, Technical Report: Trends of Continu-
ously Variable Transmission, Technical Center, Daewoo Motor Company.
[2] Proc. of Inter. Conf. on Continuously Variable Power Transmissions CVT ’96 ,
Society of Automotive Engineers of Japan, Yokohama, Sept. 11-12, 1996.
[3] Proc. of the 7th Inter. Power Transmission and Gearing Conference, Design
Engineering Division, ASME, San Diego, California, Oct. 6-9, 1996.
[4] Proc. of Inter. Congress on Continuously Variable Power Transmissions CVT
’99 , Sponsored by I Mech E, JSAE, KIVI, SAE, VDI-EKV, Eindhoven, Sept.
16-17, 1999.
[5] Report of U. S. Department of Energy, 1982, Advanced Automotive Transmis-
sion Development Status and Research Needs, DOE/CS/50286-1.
[6] Philip G. Gott, 1991, Changing Gears: The development of the Automatic
Transmission, Society of Automotive Engineers.
[7] S. H. Loewenthal, 1982, A Historical Perspective on “Traction Drives and Re-
lated Technology” in Advanced Power Transmission Technology , NASA CP-
2210, pp. 79-108, Fisher, G. K., ed.
126
[8] S. H. Loewenthal, N. E. Anderson, D. A. Rohn, 1983, Advances in Traction
Drive Technology , NASA TM-83397, pp. 1-2.
[9] Tokyo Motor Show, 1999, Tokyo, Japan.
[10] Appleton’s Cyclopedia of Applied Mechanics, 1880.
[11] N. Moronuki and Y. Furukawa, 1988, “On the Design of Precise Feed Mechanism
by Friction Drive”, in Journal of the Japan Society for Precision Engineering ,
pp. 2113-2118, Vol. 11.
[12] K. Kato, 1990, “Fundamentals and Applications of Friction Drive”, in Journal
of the Japan Society for Precision Engineering , pp. 1602-1606, Vol. 9.
[13] H. S. Lee and M. Tomizuka, 1996, “Robust Motion Controller Design for High-
Accuracy Positioning Systems”, in Transactions on Industrial Electronics , pp.
48-55, Vol. 43, No. 1, Feb.
[14] W. S. Chang and K. Y. Toumi, 1998, “Modeling of an Omni-Directional High
Precision Friction Drive Positioning Stage”, in Proc. of Inter. Conf. on Robotics
and Automation , pp. 175-180, Leuven, Belgium.
[15] M. K. Kurosawa, M. Takahashi, and T. Higuchi, 1998, “Elastic Contact Con-
ditions to Optimize Friction Drive of Surface Acoustic Wave Motor”, in Trans-
actions on Ultrasonics, Ferroelectrics, and Frequency Control , pp. 1229-1237,
Vol. 45, No. 5, Sep.
[16] R. W. Carson, 1977, “100 years in review: Industrial Traction Drives”, in Power
Transmission Design, Oct.
[17] D. Dowson and G. R. Higginson, Elasto-Hydrodynamic Lubrication, Pergamon.
127
[18] J. L. Tevaarwerk and K. L. Johnson, 1979, “The Influence of Fluid Rheology
on the Performance of Traction Drives”, in Transactions of ASME , Vol. 101,
July, pp. 266-274.
[19] L. O. Hewko, 1969, “EHL Contact Traction and Creep of Lubricated Cylindrical
Rolling Elements at Very High Surface Speeds”, in Transactions of ASLE , 12.
[20] B. J. Hamrock and D. Dowson, 1976, “Isothermal Elastohydrodynamic Lubri-
cation of Point Contacts Part1-Theoretical Formulation”, in Transactions of
the ASME , April.
[21] L. Houpert, 1985, “New Results of Traction Force Calculations in Elastohydro-
dynamic Contacts”, in Transactions of the ASME , Vol. 107, April.
[22] C. R. Evans and K. L. Johnson, 1986, “The rheological properties of elasto-
hydrodynamic lubricants”, in Proc. of Mechanical Engineers, 200, c5, pp. 303-
324.
[23] M. O. A. Mokhtar et al., 1987, “Elastohydrodynamic Behavior of Elliptical
Contacts Under Pure Rolling Situations”, in Transactions of the ASME , Vol.
109, Oct.
[24] W. Hirst and J. W. Richmond, 1988, “Traction in Elastohydrodynamic Con-
tacts”, in IMechE , Vol. 202, No. C2.
[25] K. T. Ramesh, 1989, “On the Rheology of a Traction Fluid”, in Transactions
of the ASME , Vol. 111, Oct.
[26] L. Chang et al., 1989, “An Efficient, Robust, Multi-Level Computational Al-
gorithm for Elasto-hydrodynamic Lubrication”, in Transactions of the ASME ,
Vol. 111, April.
128
[27] F. Sadeghi and P. C. Sui, 1990, “Thermal Elastohydrodynamic Lubrication of
Rolling/Sliding Contacts”, in Transactions of the ASME , Vol. 112, April.
[28] S. Natsumeda, 1992, “On the Analysis for the EHL Concentrated Contact”, in
Journal of JSLE , Vol. 37, No.12.
[29] M. Muraki and S. Konishi, 1993, “Shear Behavior of Low-Viscosity Fluids in
EHL Contacts”, in Journal of JSLE , Vol. 38, No. 8.
[30] H. Machida and S. Aihara, 1991, “State of the Art of the Traction Drive CVT
Applied to Automobiles”, in Vehicle Tribology , Elsevier Science Publishers B.V.
[31] Y. Arakawa et al., 1999, “Development and Testing of CVT Fluid for Nissan
Troidal CVT”, in Proc. of Inter. Congress on Continuously Variable Power
Transmissions CVT ’99 , pp. 213-217, Eindhoven, Sept. 16-17, 1999.
[32] Thomas G, Fellows and Christopher J. Greenwood, 1991, “The Design and
Development of an Experimental Traction Drive CVT for a 2.0 Liter FWD
Passenger Car”, in Transactions of SAE , No. 910408.
[33] P. W. R. Stubbs, 1980, “The Development of a Perbury Traction Transmission
for Motor Car Applications”, in Journal of Mechanical Design, ASME.
[34] Lubomyr O. Hewko, 1986, “Automotive Traction Drive CVTs-An Overview”,
in Proc. of Passenger Car Meeting and Exposition, Sep. 22-26, Dearborn, Michi-
gan.
[35] Masaki Nakano and Toshifumi Hibi, 1991, “Large Capacity and High Efficiency
Toroidal Traction CVT”, in Proc. of JSME International Conference on Motion
and Powertransmissions, pp. 965-970, Nov. 23-26, Hiroshima, Japan.
[36] Hirohisa Tanaka and Tatsuhiko Goi, 1999, “Traction Drive of a High Speed
Double Cavity Half-Toroidal CVT”, in Proc. of Inter. Congress on Continuously
129
Variable Power Transmissions CVT ’99 , pp. 85-90, Eindhoven, Sept. 16-17,
1999.
[37] Haruyoshi Kumura et al., 1999, “Development of a Dual-Cavity Half-Toroidal
CVT”, in Proc. of Inter. Congress on Continuously Variable Power Transmis-
sions CVT ’99 , pp. 65-70, Eindhoven, Sept. 16-17, 1999.
[38] Hisashi Machida, 1999, “Traction Drive CVT up to date”, in Proc. of Inter.
Congress on Continuously Variable Power Transmissions CVT ’99 , pp. 71-76,
Eindhoven, Sept. 16-17, 1999.
[39] Hirohisa Tanaka and Masatoshi Eguchi, 1991, “Speed Ratio Control of a Half-
Toroidal Traction Drive CVT”, in Journal of JSME , pp. 276-279, Vol.57, No.
91-0375B.
[40] Thomas G, Fellows and Christopher J. Greenwood, 1991, ”The Design and
Development of an Experimental Traction Drive CVT for a 2.0 Liter FWD
Passenger Car”, in Transactions of SAE , No. 910408.
[41] H. Tanaka and T. Ishihara, 1984, “Electro-Hydraulic Digital Control of Cone-
Roller Toroidal Traction Drive Automatic Power Transmission”, in Journal of
Dynamic Systems, Measurement, and Control , pp. 305-310, Vol. 106.
[42] Hirohisa Tanaka and Hisashi Machida, 1991, “Stability of a Speed Ratio Control
Servomechanism for a Half-Toroidal Traction Drive CVT” in Proc. of JSME In-
ternational Conference on Motion and Powertransmissions, pp. 971-976, Nov.
23-26, Hiroshima, Japan.
[43] Hisashi Machida and Nobuhide Kurachi, 1990, “Study of Half Toroidal Contin-
uously Variable Transmission (1st Report, Analysis of Contact Force by means
of Loading Cam)”, in Journal of JSME , (C), Vol. 56, No. 525, May.
130
[44] Hisashi Machida and Nobuhide Kurachi, 1990, “Development of Half Toroidal
CVT -Part 1. Some Examples of Vehicle Test-”, in Transactions of JSAE , No.
45.
[45] Takahashi, S., 1998, “Fundamental study of low fuel consumption control
scheme based on combination of direct fuel injection engine and continuously
variable transmission”, in Proc. of the 37th IEEE Conf. on Decision and Con-
trol , pp. 1522-1529, Tampa, Florida.
[46] K.Sawamura et al., 1996, “Development of an Integrated Power Train Con-
trol System with an Electronically Controlled Throttle”, in Proc. of Society of
Automotive Engineering of Japan.
[47] L. R. Oliver and D. D. Henderson, 1972, “Torque Sensing Variable Speed V-belt
Drive”, in Transactions of SAE , No. 720708.
[48] F. S. Jamzadeh and A. A. Frank, 1982, “Optimal Control for Maximum Mileage
of a Flywheel Energy-Storage Vehicle”, in Transactions of SAE , No. 820747.
[49] C. Chan et al., 1984, “System Design and Control Considerations of Automotive
Continuously Variable Transmissions”, in Transactions of SAE , No. 8450048.
[50] Ilya Kolmanovsky, Jing Sun, and Leyi Wang, 1999, “Coordinated Control of
Lean Burn Gasoline Engines with Continuously Variable Transmissions”, in
Proc. of the American Control Conf., pp. 2673-2677, San Diego, California,
June.
[51] S. Liu and B. Paden, 1997, “A survey of today’s CVT control”, in Proc. of the
36th CDC Conference, pp. 4738-4743, San Diego.
[52] Lino Guzzella and Andreas Michael Schmid, 1995, “Feedback Linearization of
Spark-Ignition Engines with Continuously Variable Transmissions”, in IEEE
131
Transactions on Control Systems Technology , pp. 54-60, Vol. 3, No. 1, March.
[53] Jungyun Kim, Yeongil Park, and Joukou Mitsusida, 1998, “S-CVT: the New
Type CVT with Sphere”, in Proc. Autumnal Conf. the Korean Society of Pre-
cision Engneers, pp. 815-818, Nov, Seoul, Korea.
[54] Moore, C. A., Peshikin, M. A., and Colgate, J. E., 1999, “Design of a 3R Cobot
Using Continuously Variable Transmission”, in Proc. of IEEE International
Conference on Robotics and Automation, Detroit, Michigan, May.
[55] Søerdalen O. J., Nakamura Y., and Chung W. J., 1994, “Design of a nonholo-
nomic Manipulator”, in Proc. of IEEE International Conference on Robotics
and Automation, San Diego, CA.
[56] Daniel J. Dawe and Charles B. Lohr, 1993, “A High Ratio Multi-Moded Vehicle
Transmission Utilizing a Traction Toroidal Continuously Variable Drive and
Traction Planetary”, in Transactions of SAE , No. 932997.
[57] G. Lundberg and A. Palmgren, 1947, “Dynamic Capacity of Rolling Bearings”,
in Acta Polytechnica, Mechanical Engineering Series, 1, R. S. A. E. E., No. 3.
[58] H. Machida and H. Tanaka, 1991, “Oil Film and Surface Damage in Traction
Drive for Automobiles”, in Proc. of JSME International Conf. on Motion and
Powertransmissions, pp. 959-964, Nov. 23-26, Hiroshima, Japan.
[59] J. J. Coy, D. A. Rohn, and S. H. Loewenthal, 1981, “Constrained Fatigue Life
Optimization of a Nasvytis Multiroller Traction Drive”, in Jornal of Mechanical
Design, pp.423-429, Vol. 103, April.
[60] S. H. Loewenthal et al., 1981, “Evaluation of a High Performance Fixed-Ratio
Traction Drive”, in Transactions of the ASME , Vol. 103.
132
[61] Bor-Tsuen Wang and Robert H.Fries, 1989, “Determination of Creep Force,
Moment, and Work Distribution in Rolling Contact With Slip”, in Transactions
of the ASME , Vol. 111, Oct.
[62] H. Tanaka et al., 1989, “Spin Moment of a Thrust Ball Bearing in Traction
Fluid”, in Journal of JSME , 89-0148 B.
[63] E. Rabinowicz, 1965, Friction and Wear of Materials, John Wiley and Sons.
[64] J. Courtney and E. Eisner, 1957, “The Effect of a Tangential Force on the
Contact of Metallic Bodies”, Proc. of the Royal Society , Vol. A238, pp. 529-
550.
[65] P. R. Dahl, 1977, “Measurement of Solid Friction Parameters of Ball Bearing”,
Proc. of the 6th Annual Symposium on Incremental Motion, Control Systems
and Devices, pp.49-60, University of Illinois.
[66] N. E. Leonard and P. S. Krishnaprasad, 1992, “Adaptive Friction Compensation
for Bi-directional Low-velocity Position Tracking”, Proc. of the 31th Conf. on
Decision and Control , pp. 267-273, Tucson, Arisona, Dec. 1992.
[67] Xiaoming Hu, 1994, “On Control of Servo Systems Affected by Friction Forces”,
Proc. of the 33rd Conf. on Decision and Control , pp. 47-473, Lake Buena Vista,
FL, Dec. 1994.
[68] Chinwon Lee et al., 1999, “Comparison of Friction model on the variable DOF
system”, Proc. of Spring Conf. the Korean Society of Mechanical Engineers,
Vol. 3, pp. 134-140, Taegu, Korea.
[69] J. W. Gilbert and G. C. Winston, 1974, “Adaptive Compensation for an Optical
Tracking Telescope”, Automatica, Vol. 10, pp. 125-131.
133
[70] C. Walrath, 1984, “Adaptive Bearing Friction Compensation based on Recent
Knowledge of Dynamic Friction”, Automatica, Vol. 20, No. 6, pp. 717-727.
[71] J. Craig, 1988, Adaptive Control of Mechanical Manipulators , Addison-Wesley.
[72] C. Canudas, K. Astrom, and K. Braun, 1986, “Adaptive Friction Compensation
in DC Motor Drives”, in Proc. of the Inter. Conf. on Robotics and Automation,
pp. 1556-1561, April, San Francisco, CA.
[73] C. Canudas de Wit et al., 1995, “A New Model for Control of Systems with
Friction”, in Transactions of Automatic Control , Vol. 40, pp. 419-425, March.
[74] P. Vedagarbha, D. M. Dawson, and M. Feemster, 1999, “Tracking Control of
Mechanical Systems in the Presence of Nonlinear Dynamic Friction Effects”, in
Transactions on Control Systems Technology, pp. 446-456, Vol. 7, No. 4, July.
[75] Hirohisa Tanaka, 1986, “Power Transmission of a Cone Roller Toroidal Traction
Drive (1st report, Speed and Torque Transmission Efficiencies)”, in Journal of
Japan Society of Mechanical Engineers , No. 86-1182A.
[76] S. Timoshenko and J. N. Goodier, 1951, Theory of Elasticity , 2nd Ed., McGraw-
Hill Book Co.
[77] Sunmo Chung and D. C. Han, 1996, Standards of Mechanical Design,
Dongmyung-Sa, Korea.
[78] A. Isidori, 1995, Nonlinear Control Systems, 3rd Ed., Springer-Verlag.
[79] A. Isidori, Maria D. Di Benedetto, 1996, “Feedback Linearization of Nonlinear
Systems”, in The Control Handbook, Vol. 2, pp. 909-917, CRC and IEEE Press.