Research Article
SDRE Control Applied to the Wheel Speed of a Compressed
Air Engine with Crank-Connecting-Rod Mechanism

Alexandre de Castro Alves,1,2 Angelo Marcelo Tusset,2

Jose Manoel Balthazar,1,3 Jeferson Jose de Lima,1 Frederic Conrad Janzen,2

Rodrigo Tumolin Rocha,2 and Airton Nabarrete3

1Sao Paulo State University, Bauru, SP, Brazil
2Federal University of Technology-Parana, Ponta Grossa, PR, Brazil
3Aeronautics Technological Institute, São José dos Campos, SP, Brazil

Correspondence should be addressed to Jose Manoel Balthazar; jmbaltha@ita.br

Received 6 March 2017; Revised 11 May 2017; Accepted 23 May 2017; Published 15 August 2017

Academic Editor: Mario Terzo

Copyright © 2017 Alexandre de Castro Alves et al. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.

Renewable energy sources for vehicles have been the motivation of many researches around the world.The reduction of fossil fuels
deposits and increase of the pollution in cities bring the need of more efficient and cleaner energy sources. In this way, this work
will present the application of a compressed air engine applied to a bicycle. The engine is composed of two pneumatic cylinders
connected to the bicycle wheel through a crank-connecting-rodmechanism. In order to control the velocity of the bicycle, a strategy
of control composed of two controls was implemented: a feedback and a feedforward control. For feedback control, the State-
Dependent Riccati Equation (SDRE) control and also a proportional-derivative (PD) control are considered, considering three
cases for velocity bicycle variation: 10 km/h, 20 km/h, and 30 km/h. The equations of motion of the system were obtained through
the Lagrangian energy method. Numerical simulations were performed in order to analyze the dynamics of the system and the
efficiency of the controllers.

1. Introduction

The use of energy has become every time more intense
through the society in the last decades; however, most of this
energy comes from nonrenewable resources like oil, natural
gas, and coils, that is, fossil fuels in general.

In order to convert these kinds of energy sources into
energy of movement, especially in vehicles, the main mecha-
nism of engines, which is responsible for this transformation
is the crank-connecting-rod.

The crank-connecting-rod mechanism has been windily
studied with the objective of increasing the engine perfor-
mance. Some of these studies consider the movement in
function of the system’s geometry, noises, and vibrations
induced by this mechanism [1–3].

Due to the limited current sources of fossil fuels, because
of the world demand, and the need of improving vehicular

performance, the study of new energy matrices like Hybrid
Renewable Energy Systems (HRES) has been of great interest
to numerous researchers [4, 5].

In this context, the compressed air systems become a very
interesting alternative. The compressed air systems allow the
energy recovery in the form of pressure, which can be applied
as an extra energy source to the combustion engines, which
characterizes a hybrid engine [6].

Pneumatic motors are very interesting in these applica-
tions because of their high force in relation to theirmasses [7].
In this way,many authors have been interested in the research
of the application of compressed air engines in small vehicles,
for example, motorcycles [8–10].

Therefore, this work proposes the application of a pneu-
matic motor to a bicycle as a main force generator. The
pneumatic engine is composed of two pneumatic cylinders
connected through a crank-connecting-rod mechanism to

Hindawi
Shock and Vibration
Volume 2017, Article ID 8340510, 14 pages
https://doi.org/10.1155/2017/8340510

https://doi.org/10.1155/2017/8340510


2 Shock and Vibration

the bicycle wheel. Hence, the force generated by the com-
pressed air is converted into angular movement of the wheel
and into linear movement of the bicycle.

The crank-connecting-rod mechanism converts the lin-
ear force of the pneumatic cylinders into torque applied to
the wheel considered as a single-degree-of-freedom system.
In addition, the dynamic model is nonlinear because of the
complexity of variables. Taking into account the fact that the
velocity of the bicycle will be controlled, this work presents
the application of the SDRE control and a PD control.

The SDRE control technique is a suboptimal control,
which searches for local stabilities of a system [11]. The
advantage of this control technique is that it does not cancel
possible benefits provided by nonlinearities of the system,
due to the fact that it is not necessary to linearize the system
when applying this technique [12–16]. Among successful
techniques implemented in real applications, there is the
classical proportional-derivative (PD) controllers [17–19].

The next sections will show the mathematical modelling
of the system composed of the wheel and the pneumatic
engine composed of the crank-connecting-rod and the
pneumatic cylinders. The SDRE and PD controllers will be
presented and numerical simulations will be performed in
order to analyze the system dynamics.

The remainder of this paper is organized as follows.
Section 2 presents the literature related to the engineering
problemand itsmathematicalmodelling through subsections
with the equations of motion and the control designs for
SDRE and PD. In Section 3 the parameters and numerical
results of the simulations with the results of applied controls
SDRE and PD are presented with the discussions of the
errors. Section 4 presents the final conclusions for the study
presented in this article. In addition, posttextual elements that
make up the structure of the work are described. Acknowl-
edgments and references for literature are presented in
sequence. Finally, the technical terms are presented through
a glossary section with a large volume of presented terms.

2. The Engineering Problem Design and
Mathematical Modelling

By applying compressed air force (𝐹a) to the crank-
connecting-rod mechanism, it changes the position of the
connecting-rod and generates the angular displacement of
the crank. It is considered that the links of the system have
their mass distribution proportional to a mass concentered
in the Center of Gravity (CG) of each link.

Figure 1 presents a schematic draw of the crank-
connecting-rodmechanism.Thismechanism has restrictions
to move in the vertical direction because of the cylinder but
can translate free in 𝑥 direction. This vertical displacement
restriction enables the system tomove in relation to its length
(𝑠CGcy), which generates the angular movement because of its
connection to the connecting-rod.

These mechanisms presented in Figure 1 are a single-
degree-of-freedom system with the displacement of the
pneumatic cylinder varying the Top Dead Center (TDC) up
to the Bottom Dead Center (BDC), thus rotating the output
crank in 2𝜋 rad.The linear position of the cylinder in relation

R

y

r

x

q

s

BDCTDC
x

CGcr
CGwh CGcy

CGco

xCGcy sCGcy

Q





Fcr
Fa

Fcocr

Fcyco

Fg

Fcy

Ff



CGcol

l

Figure 1: Schematic draw of a compressed air engine composed of
pneumatic cylinder and a crank-connecting-rod.

to its CG in the plane 𝑥𝑦 (𝑥CGcy) through the length of CG
(𝑠CGcy) can be described as presented in

𝑥CGcy = 𝑟 cos 𝜃 + 𝑙 cos𝜙 − 𝑠CGcy. (1)

In the way to determine the linear position of (𝑥CGcy) in
relation to the angular position of the crank 𝜃 and the linear
speed of the cylinder, the following geometrical relations are
made out as presented in (2) to (4) [20].

𝑞 = 𝑟 sin 𝜃 = 𝑙 sin𝜙 (2)

sin𝜙 = 𝑟𝑙 sin 𝜃 (3)

cos𝜙 = (1 − sin2𝜙)1/2 = (1 − (𝑟𝑙 sin 𝜃)
2)1/2 (4)

𝑥CGcy = 𝑟 cos 𝜃 + 𝑙 (1 − 𝑟2𝑙2 sin2𝜃)
1/2 − 𝑠CGcy

= 𝑟 cos 𝜃 + (𝑙2 − 𝑟2sin2𝜃)1/2 − 𝑠CGcy
(5)

VCGcy = ̇𝜃(−𝑟 sin 𝜃 − 𝑟2 sin 𝜃 cos 𝜃
(𝑙2 − 𝑟2sin2𝜃)1/2) . (6)

In kinematics, (5) presents the (𝑥CGcy) position in relation
to 𝜃 and (6) shows the (VCGcy) linear velocity, which is further
used to calculate the kinetic energy.

2.1. The Modelling of the Governing Equations of Motion. In
this subsection, the mathematical equations that emulate the
motion of the nonlinear connecting-rod-crank system with
horizontal geometry and monocylinder motor are developed
using the energy method of Lagrange, where its function is
represented by

L (𝜃, ̇𝜃, 𝑡) = 𝑇 − 𝑉. (7)



Shock and Vibration 3

The Lagrangian is expressed in terms of the generalized
coordinate 𝜃, the masses concentrated in the CG of each link,
the geometry, and the rigidmaterials.Therefore, the equation
of Euler-Lagrange is described as (8), where𝑄𝜃 represents the
nonconservative forces.

𝑑𝑑𝑡 (𝜕L𝜕 ̇𝜃 ) − 𝜕L𝜕𝜃 = 𝑄𝜃. (8)

In this way, the resulting force 𝐹R composed of the non-
conservative forces in function of the virtual displacement
can be obtained by the virtual work of

𝑄𝜃 = 𝐹R (𝜕𝑥CGcy𝜕𝜃 )
= 𝐹R (−𝑟 sin 𝜃 − 𝑟2 sin 𝜃 cos 𝜃

(𝑙2 − 𝑟2sin2𝜃)1/2) ,
(9)

where

𝐹R = 𝐹a − 𝐹f − 𝐹cae. (10)

Thenonconservative force𝐹cae is considered as a damping
force, considering the whole damping forces of the system,
which are 𝐹cy; 𝐹cyco; 𝐹cocr, and 𝐹cr (Figure 1). Moreover, the
compressed air engine is considered as a viscous-damping,
that is, a relation proportional to the angular velocity of the
wheel of the system multiplied by a constant, as given by

𝐹cae = 𝑐cae ̇𝜃. (11)

The 𝐹a force is another dissipative force of the system. It
is provided due to two dissipative forces of the system, one is
from the ownmotion of themechanism, and the other is from
the transmission of the wheel motion to the bicycle, which
dissipate energy of the system. The connecting-rod crank
system is excited by the air force of the pneumatic pressure
via a control signal to a pressure regulator valve. Therefore,
the air force 𝐹a is proportional to the area of the cylinder and
air pressure 𝑃a determined by the control SDRE, given by

𝑃a (𝑥1) = 𝐹a (𝑥1)(𝜋 (𝑑2/4)) , (12)

where 𝑃a is the nonconservative pressure which generates the
force 𝐹a of the air that excites the movement of the system
with 𝑑 being the cylinder diameter of the compressed air
engine.

The friction force𝐹f depends on the pressure, tire section,
and wheel length, as much as the type of the contact surface.
The considered value to this resistance force of the bearing is
approximately 9N to big section tires and 2.3N to racing tires
of thin section, low friction, and high performance [21]. 𝐹f is
proportional to the normal force (𝐹N), whose component is
generated by the angle of inclination of the ground (𝛽) applied
to themass of the bike (𝑚b) and themass of the cyclist (𝑚c) as
function of gravity (𝑔).Therefore, (13) depends on the rolling
coefficient (𝑐r) in function of the masses which varies with

the following types of parameters: bearings, ground, tire, and
hoop, as well as tire pressure [22].

𝐹f = 𝑐r cos (𝜓)
𝐹N = 𝑐r cos (𝜓) (𝑚c + 𝑚b) 𝑔. (13)

In this way, all forces relation of (8) of Euler-Lagrange
to the calculus of resulting work of nonconservative forces
was defined. However, to determine the full equations of
motion, it is needed to estimate the conservative forces and
energies that balance the mechanism movement. Thus, for
the analysis of the equilibrium of the conservative forces, the
kinetic energy 𝑇 for translational movements and rotation of
the inertial restrictions of the mechanismmust be calculated.
Therefore, the CG of each link is used as a way to interpret
the individual motions of each link, in function of its links
and combined restrictions. Each link moves by rotation
and translation motions or still with their combination
around their CG. The potential energy 𝑉 is evaluated by its
contribution from the gravitational potential energy, favoring
or not the movement of each position of the links. Therefore,
the Lagrangian of the system is calculated for the energies of
the three links that have movements in relation to the fixed
link of the mechanism, as follows:

L = (𝑇cy + 𝑇cr + 𝑇co) − (𝑉co + 𝑉cr + 𝑉cy) . (14)

The kinetic energy 𝑇cy of the monocylinder possesses
an inertia for the translation movement equivalent to that
of a cylindrical bar. The kinetic energy 𝑇cr of the crank
possesses an inertia of the rotation movement equivalent to
that of a solid cylinder/disk (shaft and crank connection)
and thin hoop (wheel and radius). The kinetic energy 𝑇co
of the connecting-rod has inertia of the rotation movement
equivalent to that of a solid cylinder/disk (shaft and crank
connection) and thin hoop (wheel and radius). The gravita-
tional potential energy𝑉co of the connecting-rod is generated
by the vertical movement of its mass 𝑚co on the two-
dimensional plane 𝑥𝑦. However, the gravitational potential
energies 𝑉cy and 𝑉cr do not consider vertical movement of
their masses on the two-dimensional plane 𝑥𝑦, then not
generating potential energy.

Therefore, one should calculate the kinetic energy of all
links of the mechanism in function of their masses and
only the potential gravitational energy of the connecting-
rod, which is the unique link that generates vertical motion.
The kinetic energy of the crank just considered the rotational
motion, because the crank possesses a rigid coupling with the
wheel by means of hoops free to rotate through the bearing
coupled to the fixed link of themechanism.Thus, the calculus
of the kinetic energy of the crank is defined as an inertial
system equivalent to a solid cylinder and disk for the shaft
and the crank combined with the inertia of thin hoops for
the wheel and your radius, as follows:

𝑇cr = 12 ((12 ((𝑚cr) (𝑟)2) ̇𝜃2) + (((𝑚ra) (𝑅2 )
2) ̇𝜃2)

+ (((𝑚wh) (𝑅)2) ̇𝜃2)) .
(15)



4 Shock and Vibration

For the calculus of energies it is necessary to define the
geometry for the components because the masses generate
the moment of inertia equivalent to the components of the
compressed air engine. Thus, the mass 𝑚cr related to the
moment of inertia is equivalent to a solid cylinder (shaft) and
a disk (crank) related to the geometry and radius (𝑟). The
mass 𝑚ra related to the moment of inertia is equivalent to
a thin hoop with the distribution in function of the wheel
middle radius (R/2). The mass𝑚wh related to the moment of
inertia is equivalent to a thin hoop for the hoop of the wheel
with tire which is related to the wheel radius (𝑅). The mass𝑚co, which is related to the moment of inertia, is equivalent
to a slender rod for the connecting-rod and is related to
the crank and the cylinder through their angular and linear
velocities. Moreover, the mass 𝑚wh related to the moment of
inertia is equivalent to a thin hoop for the hoop of the wheel
with tire and is related to the wheel radius (𝑅).

However, the kinetic energy of the connecting-rod is
calculated considering the momentum of inertia of the
masses as to rotational motion and to translational motion,
and the connecting-rod possesses two degrees-of-freedom of
motion in𝑥𝑦plane.Thus, the calculation of the kinetic energy
of the connecting-rod was defined as an equivalent inertia
system to a slender rod with axis through end, rotating and
translating during the motion in the plane 𝑥𝑦, as given by

𝑇co
= 12 (((𝑚co) VCGco2) + ((13 (𝑚co) ( 𝑙2)

2) ̇𝛽2)) . (16)

Transforming the energy equations in terms of gener-
alized coordinates, trigonometric relations are used to the
angles of 𝜃 and 𝛽 (congruent and obtuse) to use in relative
velocity of the connecting-rod in function of the generalized
coordinates. As the connecting-rod link is a two-degrees-
of-freedom system, there is two relative velocities related
to the crank and its coordinate 𝜃. Therefore, the relative
velocity of the connecting-rod considers the translational and
rotational motions which is developed by using the cosines’
law, denoting [23, 24]

VCGco
2 = 𝑟2 ̇𝜃2 + 𝑙CGco2 ̇𝛽2 + 2𝑟𝑙CGco cos (𝜃 − 𝛽) ̇𝜃 ̇𝛽, (17)

where

cos (𝜃 − 𝛽) = cos 𝜃 cos𝛽 + sin 𝜃 sin𝛽
cos𝛽 = cos𝜙 = (1 − (𝑟𝑙 sin 𝜃)

2)1/2

= (𝑙2 − 𝑟2sin2𝜃)1/2
𝑙

sin𝛽 = (−𝑟𝑙 sin 𝜃)
̇𝛽 = − ̇𝜃( 𝑟 cos 𝜃

(𝑙2 − 𝑟2sin2𝜃)1/2) .

(18)

mcocy

mcocr mco

CGcocr

CGcocr

CGcocy

CGco

l

CGcocyl

Figure 2: Equivalent dynamic model of concentrated masses of the
connecting-rod.

However, because the relative velocity is a very complex
term and may be spread in many terms to represent the
kinetic energy of the connecting-rod, a simplification of
the model is necessary and represents the movement of
the system as much as using the complex term. Thus, the
technique of concentrated masses of an equivalent dynamic
model [20] can be applied when three requirements are
evaluated to the dynamical equivalence, which are as follows:
the final mass of the model must be equal to the total mass
of the original link; the CG of connecting-rod must stay in
the same original position of the link; the final momentum
of inertia of the masses must be equal to the initial link, as
shown in Figure 2.

Through this equivalent dynamic model of concentrated
masses, the complex rotational and translation motions of
the connecting-rod are converted into a rotational motion
of the crank and a translational motion of the cylinder.
This conversion generates equivalence when the place of the
percussion centers of the equivalent masses related to the
original link mass is determined. In the typical geometry of
a typical connecting-rod, because its masses are higher to
the forces of connection with the crank (width and height)
its percussion center is closer to the connection extremity
of the crank than the CG. Therefore, this geometry allows
simplifying and concentrating the masses in the connections
with the links and present an error relatively small in the
precision of this dynamicmodel [20].These connectionswith
links are denoted by

𝑚cocr = 𝑚co
𝑙CGcocy(𝑙CGcocr + 𝑙CGcocy)

𝑚cocy = 𝑚co
𝑙CGcocr(𝑙CGcocr + 𝑙CGcocy) .

(19)

The connecting-rod of this work has typical geometry
with CGco placed with 𝑙CGcocr to 1/3 of the length 𝑙 of the
connection with the crank and 𝑙CGcocy to 2/3 of length 𝑙 of the
connection to the cylinder. Introducing these lengths in (19),
it is possible to obtain the partial value of the total ofmasses of
the connecting-rod coupled to the links by the connections,
as given by

𝑚cocr = 𝑚co
(2/3) 𝑙((1/3) 𝑙 + (2/3) 𝑙) = 𝑚co

23 𝑙 (20)

𝑚cocy = 𝑚co
(1/3) 𝑙((1/3) 𝑙 + (2/3) 𝑙) = 𝑚co

13 𝑙. (21)



Shock and Vibration 5

This geometric distribution for the connecting-rod results
in the equivalent masses for application on the crank with the
equivalent mass 𝑚cocr equal to 1/3 of 𝑚co and the equivalent
mass to application in the cylinder𝑚cocy equal to 2/3 of𝑚co.
In this way of distribution of applied masses to the links
of crank (20) and cylinder (21), with their pure motions of
rotation and translation, possibilities of the obtainment of the
kinetic energy equivalent of the connecting-rod link are given
by

𝑇co = 12 (((𝑚cocy) VCGcy2) + ((13 (𝑚cocr) (𝑟)2) ̇𝜃2))
= 12 (((𝑚co3 ) VCGcy2) + ((13 (2𝑚co3 ) (𝑟)2) ̇𝜃2)) . (22)

The kinetic energy of the cylinder is also considered the
translational movement. The angular position 𝜃 used (6),
which determines the linear position of the cylinder CG in
function of the crank’s rotational angle. Thus, the kinetic
energy of the monocylinder is given by

𝑇cy = 12 ((𝑚cy) VCGcy2)
= 12 ((𝑚cy) ̇𝜃2(−𝑟 sin 𝜃 − 𝑟2 sin 𝜃 cos 𝜃

(𝑙2 − 𝑟2sin2𝜃)1/2)
2) . (23)

The potential gravitational energy is given by the
connecting-rod link, which possesses the CG motion related
to the vertical displacement that generates a variation of
energy in relation to the position variation. By geometry,
the potential gravitation energy is considered as a vertical
projection of 𝑙CGco to the variation of the CG position of the
connection-rod link in function of the gravitational force,
given by

𝑉co = (𝑚co) 𝑔 (2𝑙3 sin𝜙) = (𝑚co) 𝑔 (2𝑙3 𝑟𝑙 sin 𝜃)
= (𝑚co) 𝑔 (2𝑟3 sin 𝜃) .

(24)

Therefore, (24) can be rewritten in function of the
gravitational constant force, given by

𝑉co = 𝐹g (2𝑟3 sin 𝜃) . (25)

Substituting the energy terms into (14) in terms of their
derivatives and applying Euler-Lagrange equation (8), the
governing equation of motion of the crank-connecting-rod
system in function of the conservatives and nonconservative
forces are obtained and arewritten in state-space form as (26),
where 𝜃 = 𝑥1, ̇𝜃 = �̇�1 = 𝑥2, and ̈𝜃 = �̈�1 = �̇�2.

�̇�1 = 𝑥2
�̇�2 = 𝑥22 (𝛿 (𝑥1) 𝜀 (𝑥1)) + 𝑥2 (𝛿 (𝑥1) 𝜍 (𝑥1))

+ (𝛿 (𝑥1) (𝐹a (𝑥1) 𝜁 (𝑥1) + 𝜉 (𝑥1))) ,
(26)

where

𝛿 (𝑥1) = 1 × ((𝑚wh) (36 sin𝑥14𝑅2𝑟4
− 72 sin 𝑥12𝑅2𝑙2𝑟2 + 36𝑅2𝑙4) + (𝑚ra)
⋅ (9 sin𝑥14𝑅2𝑟4 − 18 sin𝑥12𝑅2𝑙2𝑟2 + 9𝑅2𝑙4)
+ (𝑚cr) (18 sin𝑥14𝑟6 − 36 sin 𝑥12𝑙2𝑟4 + 18𝑙4𝑟2)
+ (𝑚co) (−12 cos𝑥12 sin𝑥14𝑟8 + 12 sin𝑥16𝑟8
+ 12 cos𝑥12 sin𝑥12𝑙2𝑟6 − 24 sin 𝑥14𝑙2𝑟6
+ 24𝜆 (𝑥1)3/2 cos𝑥1 sin𝑥12𝑟5 + 8 sin𝑥14𝑟6
+ 12 sin 𝑥12𝑙4𝑟4 − 16 sin𝑥12𝑙2𝑟4 + 8𝑙4𝑟2) + (𝑚cy)
⋅ (−36 cos𝑥12 sin𝑥14𝑟6 + 36 sin𝑥16𝑟6
+ 36 cos𝑥1 sin𝑥12𝑙2𝑟4 − 72 sin𝑥14𝑙2𝑟4
+ 72 cos𝑥1 sin𝑥12𝜆 (𝑥1)3/2 𝑟3
+ 36 sin 𝑥12𝑙4𝑟2))−1

𝜀 (𝑥1) = ((𝑚cy) (−72 cos𝑥1 sin𝑥15𝑟6
− 36𝜆 (𝑥1)1/2 cos𝑥12 sin 𝑥13𝑟5
− 36 cos𝑥13 sin𝑥1𝑙2𝑟4 + 108 cos𝑥1 sin𝑥13𝑙2𝑟4
− 72𝜆 (𝑥1)3/2 cos𝑥12 sin 𝑥1𝑟3
+ 36𝜆 (𝑥1)3/2 sin𝑥13𝑟3 − 36 cos𝑥1 sin𝑥1𝑙4𝑟2)
+ (𝑚co) (−24 cos𝑥1 sin𝑥15𝑟8
− 12𝜆 (𝑥1)1/2 cos𝑥12 sin 𝑥13𝑟7
− 12 cos𝑥13 sin𝑥1𝑙2𝑟6 + 36 cos𝑥1 sin 𝑥13𝑙2𝑟6
− 24 cos𝑥12 sin𝑥1𝜆 (𝑥1)3/2 𝑟5
+ 12 sin 𝑥13𝜆 (𝑥1)3/2 𝑟5 − 12 cos𝑥1 sin𝑥1𝑙4𝑟4))

𝜍 (𝑥1) = (−36𝑐cae𝜂 (𝑥1) 𝑟4 sin𝑥14 + 72𝑐cae𝜂 (𝑥1) 𝑟2
⋅ sin 𝑥12𝑙2 − 36𝑐cae𝜂 (𝑥1) 𝑙4)

𝜁 (𝑥1) = (36 sin𝑥14𝑟4𝜂 (𝑥1) − 72 sin𝑥12𝑙2𝑟2𝜂 (𝑥1)
+ 36𝑙4𝜂 (𝑥1))

𝜉 (𝑥1) = (𝐹g (−24𝑟5 cos𝑥1 sin𝑥14
+ 48𝑟3 cos𝑥1 sin𝑥12𝑙2 − 24𝑟 cos𝑥1𝑙4)



6 Shock and Vibration

+ 𝐹f (−36 sin𝑥14𝑟4𝜂 (𝑥1) + 72 sin𝑥12𝑙2𝑟2𝜂 (𝑥1)
− 36𝑙4𝜂 (𝑥1))) ,

(27)

where 𝜆(𝑥1) = (𝑙2 − 𝑟2sin 𝑥12) and 𝜂(𝑥1) =(−𝑟 sin𝑥1((−𝑟2 sin𝑥1 cos𝑥1)/𝜆(𝑥1)1/2)).
The application of the controller consists of substituting

the excitation torque generated by the air force 𝐹a induced by
the pressure 𝑃a (12), for the control signalU, given by

U = 𝐹a𝜁 (𝑥1) . (28)

Introducing the control signal in (26), it has the equation
of motion with the control system which can be denoted by

�̇�1 = 𝑥2
�̇�2 = 𝑥22 (𝛿 (𝑥1) 𝜀 (𝑥1)) + 𝑥2 (𝛿 (𝑥1) 𝜍 (𝑥1))

+ (𝛿 (𝑥1) (U + 𝜉 (𝑥1))) .
(29)

The control vectorU consists of two parts; Ufe and Ufo =−𝜑(𝑥1), where Ufo is the feedforward control and Ufe is the
feedback control obtained through the SDRE control signal,
as show in

U = Ufe + Ufo = Ufe − 𝜉 (𝑥1) . (30)

Therefore, changing the notation of the control signalU =
Ufe+Ufo in (29), the final equation ofmotion of the controlled
system can be rewritten by the form of

�̇�1 = 𝑥2
�̇�2 = 𝑥22 (𝛿 (𝑥1) 𝜀 (𝑥1)) + 𝑥2 (𝛿 (𝑥1) 𝜍 (𝑥1))

+ (𝛿 (𝑥1) (Ufe)) .
(31)

From (31), the movement of the system can be controlled
by the angular velocity ̇𝜃 = �̇�1 = 𝑥2, being the main
control variable, similar to an automatic vehicular driver.The
application of the control variables and description of the
process by the method of SDRE and PD controls will be
detailed in the next section.

2.2. SDRE Control Design. The technique of SDRE control
by nonlinear feedback has been applied in many nonlinear
problems, estimating the states satisfying the aiming func-
tions [25–28]. The SDRE controller uses the LQR (Linear
Quadratic Regulator) method to find the suboptimal gain in
function of State-Dependent Riccati Equations, recalculating
them along with the application on nonlinear systems of (32)
[11].

Therefore, the equation of motion of the crank-connect-
ing-rod of (31) can be rewritten in matrix form as given by

Ẋ = A (X)X + B (X)Ufe, (32)

where

A = [0 1
0 (𝑥2 (𝛿 (𝑥1) (𝜀 (𝑥1))) + (𝛿 (𝑥1) 𝜍 (𝑥1)))]

X = [𝑥1𝑥2]
B = [ 0

𝛿 (𝑥1)] .
(33)

The quadratic performance measure for the feedback
control (Ufe) problem is given by

J = ∫∞
0
(e𝑆𝑇Qe𝑆 + U𝑇feRUfe) 𝑑𝑡, (34)

where e𝑆 = [ 𝑥1−𝑥∗1𝑥
2
−𝑥∗
2

], 𝑥∗1 is the desired orbit, 𝑥∗2 is the desired
velocity, andQ(x) andR(x) are positive definitematrices.The
minimization of functional (34) implies the minimization of
the system deviation (32) of the desired state (e𝑆 = [ 𝑥1−𝑥∗1

𝑥
2
−𝑥∗
2

])
and of the applied feedback control (Ufe). Assuming full-state
feedback (Ufe), the control law is given by [29]

Ufe = R−1B (X)T P (X) e𝑆, (35)

where P(X) is the solution of the Riccati equation:

A (X)T P (X) + P (X)A (X)
− P (X)B (X)R−1B (X)T P (X) +Q = 0. (36)

Another important factor to consider is that the matrix
A(X) cannot violate the controllability of the system. Crank-
connecting-rod system (32) is controllable if the rank of the
matrixM is 2:

M = [B2×1 (X) A2×2 (X)B2×1 (X)] . (37)

The SDRE technique to obtain a suboptimal solution for
dynamic control problem has the following procedure [29]:

(1) Define the state-space model with the state-depend-
ent coefficients A(X) and B(X).

(2) Define the initial condition x(0) = x0 so that the
rank of M is 𝑛 and choose the coefficients of weight
matrices Q and R, where the matrices Q and R
determine the relative importance of parameter error
and energy expenditure.

(3) Solve P(X) for the Riccati equation which will be
described in the sequence of thismethod for the states
as a function of time.

(4) Define e𝑆 the objective functions for the calculation of
the parameter errors.

(5) Calculate the input signal Ufe.
(6) Integrate the equation obtained in step (1) and update

the state of the system as a function of time with the
results.



Shock and Vibration 7

(7) Calculate the rank of step (2) and if rank = 2 go to
step (3). However, if rank < 2, the matrix A(X) is not
controllable; therefore, you should use the last matrix
controllableA(X) that has been obtained, and thus go
to step (3).

2.3. PD Control Design. The PD control comprises one
control loop which regulates the suspension travel. The PD
controller operates according to the following equation [17–
19]:

Ufe = 𝑘𝑝𝑒𝑥
1
(𝑡) + 𝑘𝑑𝑒𝑥

2
(𝑡) , (38)

where 𝑒𝑥
2

= 𝑑𝑒𝑥
1

(𝑡)/𝑑𝑡, 𝑒𝑥
1

= 𝑥1 − 𝑥1∗, and 𝑒𝑥
2

= 𝑥2 − 𝑥2∗
and 𝑘𝑝 is the proportional gain and 𝑘𝑑 is the derivative gain
of the loop control, respectively.

For the determination of 𝑘𝑝 and 𝑘𝑑 gains, the autotuning
of the PD gains of the MathWorks� algorithm is considered.
To PD controllers, theMathWorks algorithm adjusts gains for
good balance between performance and robustness [30].

3. Numerical Results and Discussions

With the objective of keeping the angular velocity of the
crank constant, similar to an automatic vehicular driver, the
application of the control signal will be considered in a way
to keep the angular velocity 𝑥2∗ (8.415 rad/s, 16.83 rad/s, and
25.245 rad/s), and such velocity in the crank axis provides a
velocity of 10 km/h, 20 km/h, and 30 km/h. Considering the
initial conditions 𝑥1(0) = 𝜋 rad and 𝑥2(0) = 0 rad/s, it is
possible to estimate 𝑥∗1 as 𝑥∗1 ≈ 𝑥∗2 𝑡 + 4.67.

The application of the control signalU = Ufe+Ufo will be
considered two cylinders connected to the same connection
with the connecting-rod; that is, two cylinders will be applied
in symmetric form related to the 𝑦-axis.𝐹f obtained from (13) to asphalt floor rolling with tubular
tire 22 g 700 × 23mm with 275 kPa of pressure will be
considered, with the following parameters [31–33]: cyclist
mass 72 kg, bicycle mass 18 kg, and rolling coefficient 𝑐r =
0.006.

After these initial considerations, the parameters that will
be used for numerical simulations are listed in Table 1, whose
values are physical parameters of the connecting-rod crank
system.

Hence, considering the equation of motion obtained in
(32) and using the parameters of Table 1 and implementing
the control signal, numerical simulations are shown in the
following. The numerical simulations are carried out using
the method of Runge-Kutta of 4th order with a fixed step ofℎ = 0.001.
3.1. Numerical Results for SDRE Control. To determine the
feedback control (Ufe) used in (32) of themotion we consider
the following initial conditions and matrices: 𝑥1(0) = 𝜋 rad,𝑥2(0) = 0 rad/s, Q = [ 1 00 1000 ], and R = [1].

Figure 3 shows the results of the system of (32) applying
the SDRE control signal U with its results for states (𝑥1 and𝑥2) and velocity error (𝑒𝑥

2

) obtained for the casewith constant
velocity of 25.245 rad/s (30 km/h).

Table 1: Parameters for simulation.

Parameter Value Unit (SI)𝑟 0.10 m𝑙 0.30 m𝑅 0.33 m𝑑 0.025 m𝑆CGcy 0.12 m𝑚c 80 kg𝑚b 18 kg𝑚cy 0.34 kg𝑚co 0.55 kg𝑚cr 1.00 kg𝑚ra 0.50 kg𝑚wh 2.50 kg𝐹f 5.77 N𝑐r 0.006 —𝑐cae 0.02 N⋅s/rad𝑔 9.81 m/s𝜓 0 Degrees (∘)𝑥1(0) 𝜋 rad𝑥2(0) 0 rad/s

0 1 2 3 4 5 6 7 8 9 10
−5

0

5

10

15

20

25

t (s)

x1

x2

ex2

Figure 3: Double-cylinder with SDRE control by position and
constant velocity (𝑥2∗ = 25.245 rad/s = 30 km/h).

Figure 4 shows the results normalized for the case with
constant velocity of 25.245 rad/s (30 km/h), and to obtain the
real values it is needed to multiply U for 638N⋅m, 𝑃a for
80200 kPa, and 𝐹a for 39368.17N.

Themaximumvalue of control signal was 0.58Nm (0.92×10−3×638) to steady state and themaximum applied pressure
was 240.60 kPa (3×10−3×80200) which generates amaximum
force of 118.10N (3 × 10−3 × 39368.17) in pressure peaks
(Figure 4). However, there is another pressure peak during
the cycle of 2𝜋 of rotation which is of 80.20 kPa (1 × 10−3 ×80200) and it generates amaximum force of 39.37N (1×10−3×39368.17).

The pressures and forces values were obtained through
applying (28), (29), (30), and (31) which generates the



8 Shock and Vibration

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

U

t (s)

P； ；Ｈ＞ F；

−6
−5
−4
−3
−2
−1

0
1
2
3
4
5
×10−3

Figure 4: SDRE control signal for constant velocity (𝑥2∗ =25.245 rad/s = 30 km/h) with double-cylinder excited by pressure
(normalized).

−4
−2
0
2
4
6
8

10
12
14
16

0 1 2 3 4 5 6 7 8 9 10
t (s)

x1

x2

ex2

Figure 5: Double-cylinder with SDRE control by position and
constant velocity (𝑥2∗ = 16.83 rad/s = 20 km/h).

excitation force 𝐹a converted into pressure 𝑃a through (12).
This variation of force does not impact on the motor and
its machinery elements for there are the characteristics of
compressibility and damping of energy by compressed air
with the overpressure being equalized during the movement.

Figure 5 shows the results of the system of (32) applying
the control signal U of SDRE with their results for the states
(𝑥1 and 𝑥2) and the velocity error (𝑒𝑥

2

) obtained for the case
with constant velocity of 16.83 rad/s (20 km/h).

Figure 6 shows the results normalized for the case with
constant velocity of 16.83 rad/s (20 km/h), and to obtain the
real values it is needed to multiply U for 426N⋅m, 𝑃a for
56400 kPa, and 𝐹a for 27685.35N.

The maximum value of the control signal was 0.36Nm
(0.85×10−3×426) to steady state which generates amaximum
applied pressure of 225.6 kPa (4 × 10−3 × 56400) and a
maximum force of 110.7N (4 × 10−3 × 27685.35) in pressure
peaks (Figure 6). However, there is another pressure peak
during the cycle of 2𝜋 of rotation which is 25.9 kPa (0.46 ×

−6
−5
−4
−3
−2
−1

0
1
2
3
4
5

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

U

t (s)

P； ；Ｈ＞ F；

×10−3

Figure 6: SDRE signal control to constant velocity (𝑥2∗ =16.83 rad/s = 20 km/h) with double-cylinder excited by pressure
(normalized).

−2
−1
0
1
2
3
4
5
6
7
8
9

0 1 2 3 4 5 6 7 8 9 10
t (s)

x1

x2

ex2

Figure 7: Double-cylinder with SDRE control by position and
constant velocity (𝑥2∗ = 8.415 rad/s = 10 km/h).

10−3 × 56400) and generates a force of 12.73N (0.46 × 10−3 ×27685.35).
Figure 7 shows the results of the system of (32) applying

the SDRE control signalU with their results for the states (𝑥1
and 𝑥2) and the velocity error (𝑒𝑥

2

) obtained for the case with
velocity constant of 8.415 rad/s (10 km/h).

Figure 8 shows the results normalized for the case with
constant velocity of 8.415 rad/s (10 km/h), and to obtain the
real values it is needed to multiply U for 213N⋅m, 𝑃a for
37400 kPa, and 𝐹a for 18358.72N.

The maximum value of the control signal was 0.213Nm
(1 × 10−3 × 213) to steady state which generates a maximum
applied pressure of 205.7 kPa (5.5 × 10−3 × 37400) and a
maximum force of 100.97N (5.5×10−3×13358.72) in pressure
peaks (Figure 8). However, there is another pressure peak
during the cycle of 2𝜋 of rotation which is 10.84 kPa (0.29 ×10−3 × 37400) and it generates a force of 5.32N (0.29 × 10−3 ×18358.72).



Shock and Vibration 9

−8

−6

−4

−2

0

2

4

6

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

U

t (s)

P； ；Ｈ＞ F；

×10−3

Figure 8: SDRE control signal to constant velocity (𝑥2∗ =8.415 rad/s = 10 km/h) with double-cylinder excited by pressure
(normalized).

−5

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9 10
t (s)

x1

x2

ex2

Figure 9:Double-cylinderwith PDcontrol by position and constant
velocity (𝑥2∗ = 25.245 rad/s = 30 km/h).

3.2. Numerical Results for PD Control. Considering (33)
and parameters of Table 1, autotuning of PD gains of the
MathWorks algorithm obtains 𝑘𝑝 = −1.4 and 𝑘𝑑 =−4.3. Substituting the gains in (38), the feedback control is
obtained:

Ufe = −1.4 (𝑥1 − 𝑥∗1 ) − 4.3 (𝑥2 − 𝑥∗2 ) . (39)

Figure 9 shows the results of the system of (32) applying
the PD control signal U with their results for the states (𝑥1
and 𝑥2) and the velocity error (𝑒𝑥

2

) obtained for the case with
constant velocity of 25.245 rad/s (30 km/h).

Figure 10 shows the results normalized at steady state for
the case with constant velocity of 25.245 rad/s (30 km/h), and
to obtain the real values it is needed tomultiplyU for 105N⋅m,𝑃a for 54000 kPa, and 𝐹a for 26507.25N.

Themaximum value of control signal was 0.44Nm (4.2 ×10−3 × 105) at steady state which generates a maximum
applied pressure of 378 kPa (7×10−3×54000) and amaximum
force of 185.55N (7 × 10−3 × 23507.25) in pressure peaks

−10
−8
−6
−4
−2

0
2
4
6
8

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

U

t (s)

P； ；Ｈ＞ F；

×10−3

Figure 10: PD control signal for constant velocity (𝑥2∗ =25.245 rad/s = 30 km/h) with double-cylinder excited by pressure
(normalized).

−4
−2

0
2
4
6
8

10
12
14
16

0 1 2 3 4 5 6 7 8 9 10
t (s)

x1

x2

ex2

Figure 11: Double-cylinder with PD control by position and
constant velocity (𝑥2∗ = 16.83 rad/s = 20 km/h).

(Figure 10). However, there is another pressure peak during
the cycle of 2𝜋 of rotationwhich is 37.8 kPa (0.7×10−3×54000)
and it generates a maximum force of 18.55N (0.7 × 10−3 ×26507.25).

Figure 11 shows the results of the system of (32) applying
the PD control signal U with their results for the states (𝑥1
and 𝑥2) and the velocity error (𝑒𝑥

2

) obtained for the case with
velocity constant of 16.83 rad/s (20 km/h).

Figure 12 shows the results normalized for the case with
constant velocity of 16.83 rad/s (20 km/h), and to obtain
the real values it is needed to multiply U = 70Nm, 𝑃a for
39000 kPa, and 𝐹a for 19144.12N.

The maximum value of control signal was 0.33Nm (4.7 ×10−3×70) at steady state which generates a maximum applied
pressure of 312 kPa (8×10−3×39000) and amaximum force of
153.15N (8 × 10−3 × 19144.12) in pressure peaks (Figure 12).
However, there is another pressure peak during the cycle of



10 Shock and Vibration

−10
−8
−6
−4
−2

0
2
4
6
8

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

U

t (s)

P； ；Ｈ＞ F；

×10−3

Figure 12: PD control signal for constant velocity (𝑥2∗ = 16.83 rad/s
= 20 km/h) with double-cylinder excited by pressure (normalized).

−2
−1

0
1
2
3
4
5
6
7
8
9

0 1 2 3 4 5 6 7 8 9 10
t (s)

x1

x2

ex2

Figure 13: Double-cylinder with PD control by position and
constant velocity (𝑥2∗ = 8.415 rad/s = 10 km/h).

2𝜋 of rotation which is 37.8 kPa (0.7 × 10−3 × 54000) and it
generates amaximum force of 18.55N (0.7×10−3×26507.25).

Figure 13 shows the results of the system of (32) applying
PD control signalUwith their results for the states (𝑥1 and𝑥2)
and the velocity error (𝑒𝑥

2

) obtained for the casewith constant
velocity of 8.415 rad/s (10 km/h).

Figure 14 shows the results normalized for the case with
constant velocity of 8.415 rad/s (10 km/h), and to obtain the
real values it is needed to multiply U for 34.9N⋅m, 𝑃a for
21400 kPa, and 𝐹a for 10504.72N.

The maximum value of control signal was 0.22Nm
(0.0065 × 34.9) at steady state which generates a maximum
applied pressure of 235.4 kPa (0.011 × 21400) and a maximum
force of 115.55N (0.011 × 10504.72) in pressure peaks (Fig-
ure 14). However, there is another pressure peak during the
cycle of 2𝜋 of rotation which is 10.7 kPa (0.0005 × 21400) and
it generates a maximum force of 5.25N (0.0005 × 10504.72).

3.3. Discussions for SDRE and PD Control. The results of the
previous sections showed that the control strategy composed

−0.015

−0.01

−0.005

0

0.005

0.01

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

U

t (s)

P； ；Ｈ＞ F；

Figure 14: PD control signal for constant velocity (𝑥2∗ = 8.415 rad/s
= 10 km/h) with double-cylinder excited by pressure (normalized).

of feedback and feedforward controls is efficient in control-
ling the systems of both orbits (𝑥∗1 and 𝑥∗2 ) and that both
SDRE and PD controls were efficient in leading the system
to a desired orbit.

In this subsection the performance of the controllers will
be considered considering RMS (Root Mean Square) of the
absolute error and absolute errors in steady state.

Figure 15 shows the absolute error variation for the three
studied cases 𝑥2∗ = 8.415 rad/s, 𝑥2∗ = 16.83 rad/s, and 𝑥2∗ =
25.245 rad/s.

Considering the case that the desired velocity is𝑥2∗ = 8.415 rad/s, it has SDRE control |𝑒𝑥
2

|RMS_SDRE =0.000813 rad/s and error at steady state 𝑒R𝑥
2
_SDRE =0.017211%. Considering PD control, it has |𝑒𝑥
2

|RMS_PD =0.005659 rad/s and error at steady state 𝑒R𝑥
2
_PD = 0.125564%.

As can be observed, the SDRE control has as well the absolute
error as error at steady state smaller than that obtained with
PD control, with the error at steady state to SDRE control
being 86.1711% smaller than the error at steady state observed
to PD control.

When the desired velocity is𝑥2∗ = 16.83 rad/s, it has SDRE
control |𝑒𝑥

2

|RMS_SDRE = 0.003222 rad/s and error at steady
state 𝑒R𝑥

2
_SDRE = 0.035152%. Considering PD control, it

has |𝑒𝑥
2

|RMS_PD = 0.019867 rad/s and error at steady state𝑒R𝑥
2
_PD = 0.209257%. As can be observed, the SDRE control

has as well the absolute error as error at steady state smaller
than that obtained with PD control, and the error at steady
state to SDRE control is 83.20% smaller than the error at
steady state observed to PD control.

Moreover, considering the case that the desired velocity
is 𝑥2∗ = 25.245 rad/s, it has SDRE control |𝑒𝑥

2

|RMS_SDRE =0.007183 rad/s and error at steady state 𝑒R𝑥
2
_SDRE =0.051457%. Considering PD control, it has |𝑒𝑥
2

|RMS_PD =0.03820 rad/s and error at steady state 𝑒R𝑥
2
_PD = 0.260022%.

As can be observed, the SDRE control has as well the absolute
error as error at steady state smaller than that obtained with
PD control, and the error at steady state to SDRE control is
80.21% smaller than the error at steady state observed to PD
control.



Shock and Vibration 11

0

0.002

0.004

0.006

0.008

0.01

0.012

PD
SDRE

90 91 92 93 94 95 96 97 98 99 100
t (s)

|e
x
2
|

(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

PD
SDRE

90 91 92 93 94 95 96 97 98 99 100
t (s)

|e
x
2
|

(b)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

PD
SDRE

90 91 92 93 94 95 96 97 98 99 100
t (s)

|e
x
2
|

(c)

Figure 15: Absolute error variation |𝑒𝑥2 | = |𝑥2 − 𝑥∗2 |. (a) |𝑒𝑥2 | = |𝑥2 − 8.415|. (b) |𝑒𝑥2 | = |𝑥2 − 16.83|. (c) |𝑒𝑥2 | = |𝑥2 − 25.245|.

Therefore, the next section shows the conclusions about
this intense study.

4. Conclusions

In this work, a nonlinear dynamic model of a crank
connecting-rod systemmodelled using the method of energy
of Lagrange in function of its geometry was presented.
Numerical simulations were carried out to the study of the
dynamics of the system for SDRE and PD controls with the
system showing to be controllable for the two controllers.The
dynamic system was analyzed as a motor of double-cylinder,
with excitation by pressure along all the period of rotation by
means of pneumatic force.

To SDRE control, the peaks of force variated between 100
and 118N and, to PD control, these peaks were higher with
greater variation among the three velocities, varying from 115
to 185N. However, this higher difference of the peaks of force
to PD controller is function of the pressure profile distributed
along the course of the cylinder which uses fixed gains rather
than variable ones being dependent on the applied states to
SDRE control.

The geometry considered in this study has the potential
of application with hybrid motor with the human biome-
chanical motive energy. However, the viability of production
with pneumatic cylinders has to be considered, because they
can be coupled to a bicycle wheel. In this way, due to being
clean and possessing lowweight as fuel, the pneumatic energy
as a motor of rotational energies in wheels with traditional
geometry remains as an alternative of application.

The efficiency of the PD controller to the nonlinear
system showed an error of 0.12% (10 km/h) and 0.26%
(30 km/h) when applied as active control. In the case of the
SDRE controller to the nonlinear system showed an error of
0.01% (10 km/h) and 0.05% (30 km/h). The error at steady
state of SDRE control was 86.1711% (10 km/h) and 80.21%
(30 km/h) smaller in comparison to PD control.

Thus, the values of the errors showed that the velocity
controlled presented low values with less than 0.038 rad/s
for PD and 0.007 rad/s for SDRE control. In this way, both
controllers presented viable final values of implementation in
control of the automatic pilot to the pneumaticmotor applied
to a bicycle. However, to hybrid application of two motors
with nonideal behavior, the SDRE control should provide



12 Shock and Vibration

a better performance in function of their characteristics of
adaptation related to each state.

Glossary

: Symbol which represents the Center of
Percussion (CP)𝑚wh: Mass concentrated on the CG of the wheel
with tire relative to the link grounded

: Symbol which represents the Center of
Gravity (CG)

P(X): Matrix solution of the Riccati Equation for
SDRE

A(X): Matrix of the terms as a function of system
properties without influence on the
control𝑃a: Pressure of the air that generates the
movement on the cylinder

B(X): Matrix of the terms as a function of system
properties with influence on the control𝑞: Length of the height formed by the
geometry of the angles of the mechanism𝑐cae: Viscous-damping coefficient of all
damping forces in the compressed air
engine

Q: Matrix which determines the relative
importance of states and your errors

CGco: CG of the connecting-rod with freedom of
rotation and translation movement𝑄𝜃: Resultant momentum of forces associated
with the generalized coordinated applied
the wheel 𝑖

CGcocr: Center of gravity and percussion of the
mass of the cylinder relative to the
connecting-rod

R: Matrix for relative importance of energy
expenditure

CGcocy: Center of gravity and percussion of the
mass of the crank relative to the
connecting-rod𝑅: Radius of the wheel with rigid connection
to the crank link

CGwh = CGcr: Coincident CG of the wheel and crank
with connect by bearings for the rotation𝑟: Total length of the link crank

CGcy: CG of the cylinder with freedom of
translation movement𝑠: Total linear length of the cylinder link𝑐𝑟: Rolling coefficient of the bicycle in
circumstances determined𝑠CGcy: Linear length of the cylinder from the CG
that connects to link connecting-rod𝑑: Cylinder diameter of the compressed air
engine𝑇: Kinetic energy for translational
movements and rotation in function of the
inertia

e𝑆: Matrix of angular position errors and
velocity for SDRE control

𝑇co: Kinetic energy of the connecting-rod with
inertia of the rotation movement𝑒𝑥

1

: Value of the angular position error𝑇cr: Kinetic energy of the crank with inertia of
the rotation movement𝑒𝑥

2

: Value of the angular velocity error𝑇cy: Kinetic energy of the monocylinder with
inertia of the translation movement|𝑒𝑥

2

|RMS: Absolute error of the angular velocity
using the Root Mean Square (RMS)

U: The total control signal for each case𝑒R𝑥
2

: Maximum percentage error of the angular
velocity using the Root Mean Square
(RMS)

Ufo: The feedforward control for each case𝐹a: Force of the air that excites the motion of
the cylinder

Ufe: Feedback control obtained through the
PD or SDRE control signal𝐹cae: Resultant damping force equivalent of
compressed air engine𝑉: Potential energy gravitational𝐹cr: Force that dissipates energy by rotating
friction at the crank and the grounded link𝑉co: Gravitational potential energy of the
connecting-rod for the vertical motion𝐹cocr: Force that dissipates energy by rotating
friction at the connecting-rod and the
crank𝑉cr: Gravitational potential energy of the crank
generated for the vertical motion𝐹cy: Force that dissipates energy by translation
friction in the cylinder and the grounded
link𝑉cy: Gravitational potential energy of the
cylinder generated for the vertical motion𝐹cyco: Force that dissipates energy by rotating
friction at the cylinder and the
connecting-rod

VCGco: Linear velocity of the CG of the
connecting-rod relative to the crank and
cylinder velocities𝐹f : Force of friction transmitted by the wheel
to the ground for the pure rolling on a
bicycle

VCGcy: Linear velocity of CG cylinder to the
origin 𝑥𝑦𝐹g: Gravitational force that generates potential
energy in the mechanism

Ẍ: Vector of angular acceleration𝐹N: Normal force component for the angle 𝛽
Ẋ: Vector of angular velocities𝐹R: Nonconservative resultant force
X: Vector of angular positions𝑔: Acceleration of gravity𝑥CGcy: Linear position of CG cylinder to the

origin 𝑥𝑦
J: The cost functional of SDRE control



Shock and Vibration 13

𝑥𝑦: Plane system bidimensional of Cartesian
coordinates

KPD: Matrix of gains for PD control𝜃 = 𝑥1: Angular position of the link crank to the
origin 𝑥𝑦

KS: Matrix of gains for SDRE control̇𝜃 = �̇�1 = 𝑥2: Angular velocity of the link crank to the
origin 𝑥𝑦𝑙: Total length of the link connecting-rod̈𝜃 = �̈�1 = �̇�2: Angular acceleration of the link crank to
the origin 𝑥𝑦𝑙CGco: Length of the link connecting-rod to CG𝜃∗ = 𝑥1∗: Desired angular position 𝑥1𝑙CGcocr: Length of connecting-rod starting at the
CG/CP the crank̇𝜃∗ = 𝑥2∗: Desired angular velocity 𝑥2𝑙CGcocy: Length of connecting-rod starting at the
CG/CP the cylinder𝛽: Angular position of the connecting-rod
with horizontal reference in relation to the
crank

L: Function generated by Lagrangian
mechanicṡ𝛽: Angular velocity of the connecting-rod
with horizontal reference in relation to the
crank

M: Controllability matrix𝜙: Angular position of the connecting-rod
with horizontal reference in relation to the
cylinder𝑚b: Bicycle’s mass𝜓: Angle of inclination of the ground𝑚c: Cyclist’s mass𝛿(𝑥1): Term of matrices 𝐴(𝑋) and 𝐵(𝑋) with
parameters (𝑚wh,𝑚ra,𝑚cr,𝑚co,𝑚cy, and
geometric) and variable 𝑥1𝑚co: Mass concentrated on the CG of the
connecting-rod relative to the link
grounded𝜀(𝑥1): Term of matrix 𝐴(𝑋) with parameters
(𝑚co,𝑚cy) and variable 𝑥1𝑚cocr: Mass concentrated on the CG of the crank
relative to the connecting-rod𝜍(𝑥1): Term of matrix 𝐴(𝑋) with parameters
(𝑐cae) and variable 𝑥1𝑚cocy: Mass concentrated on the CG of the
cylinder relative to the connecting-rod𝜁(𝑥1): Term of matrix 𝐴(𝑋) with only the
geometric parameters and variable 𝑥1
which multiply 𝐹a𝑚cr: Mass concentrated on the CG of the crank
and shaft relative to the link grounded𝜉(𝑥1): Term of matrix 𝐴(𝑋) with parameters (𝐹g,𝐹f ) and variable 𝑥1𝑚cy: Mass concentrated on the CG of the
cylinder relative to the link grounded𝜆(𝑥1): Term which simplifies geometric
parameters and variable to the terms𝛿(𝑥1), 𝜀(𝑥1), 𝜍(𝑥1), 𝜁(𝑥1) and 𝜉(𝑥1)

𝑚ra: Mass concentrated on the CG of the wheel
middle radius relative to the link grounded𝜂(𝑥1): Term which simplifies geometric
parameters and variable to the terms𝜍(𝑥1), 𝜁(𝑥1) and 𝜉(𝑥1).

Conflicts of Interest

The authors declare that there are no conflicts of interest
regarding the publication of this paper.

Acknowledgments

The authors acknowledge support by CNPq (Grant 447539/
2014-0), CAPES, and ARAUCÁRIA Foundation, all Brazilian
research funding agencies.

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