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ple tuning process to determinate the PID controller
gains. However, these do not provide a good control
performance in all cases.
A MDOP can be solved by converting it into a non-
linear programming (NLP) problem (Kraft, 1985),
(Goh and Teo, 1988) and using the Goal Attainment
method (Liu et al., 2003) for the resulting problem.
Two transcription methods exist for the MDOP prob-
lem: the sequential and the simultaneous methods
(Betts, 2001). In the sequential method, only the
control variables are discretized; this method is also
known as the control vector parameterization. In the
simultaneous method the state and control variables
are discretized resulting in a large-scale NLP problem
which usually requires special solution strategies.
Current research efforts in the field of power trans-
mission of rotational propulsion systems, are dedi-
cated to obtain low energy consumption with high
mechanical efficiency. An alternative solution to this
problem is the so called continuously variable trans-
mission (CVT), whose transmission ratio can be con-
tinuously changed in an established range. There are
many CVT’s configurations built in industrial sys-
tems, especially in the automotive industry due to the
requirements to increase the fuel economy without
decreasing the system performance. The mechani-
cal development of CVT’s is well known and there
is little to modify regarding its basic operation prin-
ciples. However, research efforts go on with the con-
troller design and the CVT instrumentation side. Dif-
ferent CVT’s types have been used in different in-
dustrial applications; the Van Doorne belt or V-belt
CVT is the most studied mechanism (Shafai et al.,
1995), (Setlur et al., 2003). This CVT is built with
two variable radii pulleys and a chain or metal-rubber
belt. Due to its friction-drive operation principle, the
speed and torque losses of rubber V-belt are a dis-
advantage. The Toroidal Traction-drive CVT uses
the high shear strength of viscous fluids to transmit
torque between an input torus and an output torus.
However, the special fluid characteristic used in this
CVT becomes the manufacturing process expensive.
A pinion-rack CVT which is a traction-drive mecha-
nism is presented in (De-Silva et al., 1994), this CVT
is built-in with conventional mechanical elements as
a gear pinion, one cam and two pair of racks. The
conventional CVT manufacture is an advantage over
other existing CVT’s.
In this paper the parametric optimal design of a
pinion-rack CVT is stated as a MDOP to obtain a set
of optimal mechanical and controller parameters of
the CVT and, a higher mechanical efficiency and a
minimal energy controller. This paper is organized
as follows: The description and the dynamic CVT
model are presented in Section 2. The design vari-
ables, performance criteria and constraints to be used
in the parametric CVT design are established in Sec-
tion 3. Section 4 presents some optimization results
and discuss them. Section 5 presents some conclu-
sions and future work.
2 DESCRIPTION AND DYNAMIC
CVT MODEL
In order to apply the design methodology proposed
in this paper, the pinion-rack CVT presented in (De-
Silva et al., 1994) is used. The pinion-rack CVT,
changes its transmission ratio when the distance be-
tween the input and output rotation axes is changed.
This distance is called “offset” and will be denoted by
“e”. This CVT is built-in with conventional mechan-
ical elements as a gear pinion, one cam and two pair
of racks. Inside the CVT an offset mechanism is inte-
grated. This mechanism is built-in with a lead screw
attached by a nut to the vertical transport cam. Fig. 1
depicts the main mechanical CVT components.
Figure 1: Main CVT mechanical components
The dynamic model of a pinion-rack CVT is pre-
sented in (Alvarez-Gallegos et al., 2005). Ordinary
differential equations (1), (2) and (3) describe the
CVT dynamic behavior. In equation (1): T
m
is the
input torque , J
1
is the mass moment of inertia of the
gear pinion, b
1
is the input shaft coefficient viscous
damping, r is the gear pinion pitch circle radius, T
L
is the CVT load torque, J
2
is the mass moment of
inertia of the rotor, R is the planetary gear pitch cir-
cle radius, b
2
is the output shaft coefficient viscous
damping and θ is the angular displacement of the ro-
tor. In equations (2) and (3): L, R
m
, K
b
, K
f
and n
represent the armature circuit inductance, the circuit
MULTIOBJECTIVE OPTIMAL DESIGN OF STRUCTURE AND CONTROL OF A CONTINUOUSLY VARIABLE
TRANSMISSION
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