Differential evolution techniques for the structure-control design of a five-bar parallel robot
ABSTRACT The present work deals with the use of a constraint-handling differential evolution algorithm to solve a nonlinear dynamic optimization problem (NLDOP) with 51 decision variables. A novel mechatronic design approach is proposed as an NLDOP, where both the structural parameters of a non-redundant parallel robot and the control parameters are simultaneously designed with respect to a performance criterion. Additionally, the dynamic model of the parallel robot is included in the NLDOP as an equality constraint. The obtained solution will be a set of optimal geometric parameters and optimal PID control gains. The optimal geometric parameters adjust the dynamic and the kinematic parameters, optimizing then, the link shapes of the robot. The proposed mechatronic design approach is applied to design simultaneously both the mechanical structure of a five-bar parallel robot and the PID controller.
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ABSTRACT: Several extensions to evolutionary algorithms (EAs) and particle swarm optimization (PSO) have been suggested during the last decades offering improved performance on selected benchmark problems. Recently, another search heuristic termed differential evolution (DE) has shown superior performance in several real-world applications. In this paper, we evaluate the performance of DE, PSO, and EAs regarding their general applicability as numerical optimization techniques. The comparison is performed on a suite of 34 widely used benchmark problems. The results from our study show that DE generally outperforms the other algorithms. However, on two noisy functions, both DE and PSO were outperformed by the EA.Evolutionary Computation, 2004. CEC2004. Congress on; 07/2004
Conference Proceeding: A comparative study of differential evolution variants for global optimization.[show abstract] [hide abstract]
ABSTRACT: In this paper, we present an empirical comparison of some Differential Evolution variants to solve global optimization problems. The aim is to identify which one of them is more suitable to solve an optimization problem, depending on the problem's features and also to identify the variant with the best performance, regardless of the features of the problem to be solved. Eight variants were implemented and tested on 13 benchmark problems taken from the specialized lit- erature. These variants vary in the type of recombination operator used and also in the way in which the mutation is computed. A set of statistical tests were performed in order to obtain more confidence on the validity of the re- sults and to reinforce our discussion. The main aim is that this study can help both researchers and practitioners in- terested in using differential evolution as a global optimizer, since we expect that our conclusions can provide some in- sights regarding the advantages or limitations of each of the variants studied.Genetic and Evolutionary Computation Conference, GECCO 2006, Proceedings, Seattle, Washington, USA, July 8-12, 2006; 01/2006
Article: Five-Bar Grashof CriteriaJournal of Mechanisms Transmissions and Automation in Design. 01/1986; 108(4).
Vol. 42, No. 6, June 2010, 535–565
Differential evolution techniques for the structure-control design
of a five-bar parallel robot
Miguel G. Villarreal-Cervantesa*, CarlosA. Cruz-Villara, JaimeAlvarez-Gallegosaand
aCinvestav-IPN, Departamento de Ingeniería Eléctrica, Av. IPN 2508, Apdo. Postal 14-740,
07300 México D.F., México;bCIDETEC-IPN, Departamento de Posgrado, Area de Mecatrónica,
Av. Juán de Dios Bátiz s/n Esq. Miguel Othon de Mendizabal, Unidad Profesional Adólfo López Mateos,
07700, México D.F., México
(Received 8 April 2009; final version received 1 September 2009)
The present work deals with the use of a constraint-handling differential evolution algorithm to solve a
approach is proposed as an NLDOP, where both the structural parameters of a non-redundant parallel
robot and the control parameters are simultaneously designed with respect to a performance criterion.
Additionally, the dynamic model of the parallel robot is included in the NLDOP as an equality constraint.
The obtained solution will be a set of optimal geometric parameters and optimal PID control gains. The
optimal geometric parameters adjust the dynamic and the kinematic parameters, optimizing then, the link
shapes of the robot. The proposed mechatronic design approach is applied to design simultaneously both
the mechanical structure of a five-bar parallel robot and the PID controller.
Keywords: differential evolution; parallel robot; integrated design; dynamic optimization problem
The integration of mechanical and electronic elements in mechatronic systems has originated
a multidisciplinary framework for their design (Youcef-Toumi 1996). In this multidisciplinary
design approach a whole description of the system behaviour is needed, where the mechanical
behaviour and the dynamic performance must be considered. Some works in mechatronic system
design propose an integrated design approach in order to fulfil the multidisciplinary framework
(Alvarez-Gallegos et al. 2005, Portilla-Flores et al. 2007, Cruz-Villar et al. 2009). Hence, the
parameters of the system. In most cases, the NLDOP implicates a set of criteria to satisfy in order
to measure the system performance, such that a multiobjective optimization problem (MOP) is
formulated.A typical method to handle MOPs consists of transforming the original problem into
*Corresponding author. Email: email@example.com
ISSN 0305-215X print/ISSN 1029-0273 online
© 2010 Taylor & Francis
M.G. Villarreal-Cervantes et al.
a weighted sum approach (Osyczka 1984). This transformation will be easier to solve than the
original multiobjective problem.
Several mathematical programming methods such as sequential quadratic programming (SQP)
are used to solve an NLDOP. However, a weakness of these methods is that they can be attracted
by local minima in the neighbourhood of the starting point. Moreover, these methods are able
to produce only one possible solution. In order to save this problem, various points are needed
to initialize the solution search, nevertheless a considerable sensitivity to its initial search point
is observed on the algorithm convergence (Cruz-Villar et al. 2009). Also, these methods involve
computing the gradient and the Hessian of the objective function and constraints, which implies
that continuity of the second order must be ensured (Portilla-Flores et al. 2007).
Hence, stochastic methods such as evolutionary algorithm (EAs), genetic algorithms (GAs)
or particle swarm optimization (PSO) can be used in order to overcome the above drawbacks.
That is, the development of efficient optimization algorithms remains as an active research area.
Some advantages are obtained when stochastic methods are used to solve this kind of opti-
mization problem: (1) these methods are population-based methods – therefore a global minima
solution can be reached, although not in all kinds of problem; (2) in order to start the search,
additional information is not necessary, i.e. gradients, Hessian matrices, initial search points, etc.;
(3) with these methods, complex problems can be solved, meaning that the optimization prob-
lem can include discontinuous physical models; (4) finally, these methods are independent of
the problem characteristics. That is, these methods can be used and/or adapted to a large set of
Several extensions of the EAs and PSO have been suggested during the last decades offering
differential evolution (DE) has shown performance superior to that of PSO and EAs in the widely
i.e. it is able to reproduce similar results consistently over many trials, whereas the performance
of PSO is more dependent on the randomized initialization of the individuals. As a result, PSO
Therefore, the DE algorithm is used in this article.
a special recombination operator that performs a linear combination of a number of individuals
is deterministic between the parent and the child (the best of them remain in the next population).
Moreover, the DE algorithm is easy to implement due to the design variables being encoded
as floating point variables and few control parameters being required. The DE algorithm can
efficiently find the neighbourhood of the global optimum. DE has been successfully applied to
optimize mechanical systems (Deb and Kain 2003,Alvarez-Gallegos et al. 2005, Shiakolas et al.
2005, Portilla-Flores et al. 2007, Cruz-Villar et al. 2009).
in Alvarez-Gallegos et al. (2005) and Portilla-Flores et al. (2007). In such a methodology, the
mechatronic design problem is stated as a dynamic optimization problem. The concurrent design
the parametrical optimal design and the proportional integral (PI) controller gains of a CVT are
of the controller, are considered together with two system performance criteria. The resulting
problem is solved using two methods: a mathematical programming method and an heuristic
approach. In Saravanan and Ramabalan (2008) a general method for computing minimum cost
cost function with constraints namely joint positions, velocities, jerks and torques by considering
dynamic equations of motion. A clamped cubic spline curve is used to represent the trajectory
and a nonlinear constrained optimization problem is presented. The cost function is a weighted
balance of transfer time, mean average of actuators efforts and power, singularity avoidance, joint
jerks and joint accelerations.
Traditionally, serial robotic systems (Lung-Wen 1999) have been used in factories for high-
precision routine operations. None the less, parallel robots, which consist of one or more closed
Those advantages involves high stiffness, speed and acceleration, good dynamic characteristics
and precise positioning capability (Hunt 1983). However, the main disadvantages are more sin-
gularity configurations and the lack of well developed tools for analysis, synthesis, control and
trajectory planning. Hence, the parallel robot design is a huge challenge because it is not an easy
task to find the mechanical and the control parameters that improve the system performance.
process is one of the most used methodologies to design parallel robots. That methodology is
usually composed of two design phases: the mechanical design, in which the structural design
both design (Youcef-Toumi 1996). On the other hand, a design approach for a five-bar parallel
robot with nonlinear PD controller gains is stated for the trajectory tracking problem in Ouyang
et al. (2003). The key idea of this approach is to find the dynamic (masses, mass centre lengths,
mass centre angles, inertias) and kinematic (link lengths) parameters of the links that produce the
best mass distribution in the parallel robot. The main idea of finding the best mass distribution is
to simplify the dynamic model. Once the dynamic and the kinematic parameters are found, the
control strategy is designed. But that results in a lack of integration in the design, which typically
requires several redesigns to achieve an aceptable system performance.
Another way to state the parallel robot design for the tracking of few points in the Cartesian
space is proposed in Affi et al. (2007). In that work, the authors take into account the Cartesian
positioning error, the maximum current variation and the current fluctuation to be optimized
in order to find the dynamic and kinematic parameters of a four-bar parallel robot. In order to
simplify the control strategy and minimize the power consumption of the motor, the velocity of
the crank is set as a constant. This means that the dynamic model (dynamic behaviour) is not
considered in the optimization process. So, the optimization problem is an approximation of the
real design problem and this problem results in a static one.A static optimization problem (Betts
in the optimization problem, the time variable is not included and/or the system dynamics is not
considered. The system dynamics is a function where the time variable is considered and this
dynamics is represented by nonlinear differential equations (NLDEs) and described by a set of
dynamic variables referred to as the state equations. The NLDEs describe the dynamic behaviour
of the system, such as mass and energy balances, and algebraic equations that ensure physical
and electrical relations.
On the other hand, singularity configurations are some of the most significant and critical
robots near a singular point can cause severe destruction of the robot. Several articles (Klein and
Blaho 1987,Yoshikawa 1990, Kim 2006) have established performance indexes derived from the
Jacobian matrix in order to avoid singularities. Optimal robot configurations (optimal position)
and optimal link lengths are found with those indexes. In Kim (2006), a kinematic design (static
optimization problem) based on the tracking task of a few Cartesian points for a parallelogram
five-bar parallel robot is proposed. In that work, the optimal link lengths are only found in one
single design step without considering the control parameters.
M.G. Villarreal-Cervantes et al.
In Ouyang et al. (2003),Affi et al. (2007) and Kim (2006), the optimal dynamic and kinematic
parameters of the links do not give to the designer an idea of the link shapes, therefore other
optimization process will be required for the optimal geometric design of the links. In addition,
in order to compute the inertias of the links, the link shapes must be known. Nevertheless, in
those works the following assumption is considered: the inertias were computed as if the links
were cylindrical or rectangular geometries due to the unknown shapes of links. Also, a static
optimization problem is stated since the system dynamics is not taken into consideration as an
equality dynamic constraint (Betts 2001).As a result, the optimal design found does not provide
the best performance in the real world due to the lack of system dynamics in the optimization
problem and due to the above assumption.
The main contributions of the present work is to state a mechatronic design as a nonlinear
dynamic optimization problem. The mechatronic design must consider the synergy between the
mechanical design of a non-redundant parallel robot and the control design. The target of the
mechatronic design is to find simultaneously the optimum structure-control parameters so that
the end-effector of the parallel robot can track a desired trajectory without singularity configu-
rations. In addition, this work deals with the use of a constraint-handling differential evolution
(CHDE) algorithm (Mezura-Montes et al. 2004) to solve such nonlinear dynamic optimization
algorithm is included for saving time in the evaluation of the performance function. The obtained
geometric parameters adjust the dynamic and kinematic parameters and the link shapes of the
robot such that other optimization processes are not required to adjust the geometry of the robot
links (link shapes). The mechatronic design is efficiently applied to the mechanical design of the
five-bar parallel robot and the PID control design.
This article is organized as follows: the five-bar parallel robot with its controller is presented
in Section 2. The formulation of the mechatronic design approach is stated in Section 3, where
results and discussions of the optimal structure-control design parameters for the five-bar parallel
robot are presented too. Finally, the conclusions of this approach are presented in Section 6.
2.Five-bar parallel robot
The five bar parallel robot is a non-redundant robot consisting of five revolute joints with two
of them active (Figure 1). The reduced model method of closed chain mechanisms presented in
Ghorbel et al. (2000) is employed to obtain the dynamic model of the five-bar parallel robot. The
dynamic parameters of the ith link are the mass mi, the distance of the mass centre lci, the inertia
Iiand the angle of the mass centre γi. The kinematic parameter is the link length asi. In Figure 1
these parameters are shown.
The dynamic model of the five-bar parallel robot is shown in Equation (1), where q =
[q1,q2]T∈ R2, ˙ q = [˙ q1, ˙ q2]T∈ R2areindependentgeneralizedcoordinatevectorsofpositionand
velocity of the actuated links, respectively. q?= [q1,q2,q3,q4]T∈ R4, ˙ q?= [˙ q1, ˙ q2, ˙ q3, ˙ q4]T∈
R4are dependent coordinate vectors of position and velocity. q3 and q4 are the angu-
lar positions of unactuated links and u = [u1,u2]T∈ R2is the applied generalized force
vector. D(q?) = ρ(q
C(q?, ˙ q?) = ρ(q
matrix, g(q?) = ρ(q
?) ∈ R2×2is a symmetric positive definite inertia matrix,
?, ˙ q
?) + ρ(q
?) ˙ ρ(q
?) ∈ R2×2is the Coriolis and centrifugal
?) ∈ R2is the gravity vector, ¯ c = [¯ xp, ¯ yp]Tand
¯ c = [
Figure 1.Schematic diagram of the five-bar parallel robot.
the position and velocity in the Cartesian space of the desired trajectory to be executed by the
end-effector of the robot. σ(q) involves a parametrization of q → q?,
when the parallel robot is virtually cut open at the end-effector (Ghorbel et al. 2000).
?) ∈ R4×4, C
?) ∈ R4×4, g
?) ∈ R4are the motion equations of two serial robots
¨ q = D−1(q?)(−C(q?, ˙ q?)˙ q − g(q?) + u)
˙ q?= ρ(q?)˙ q
The main characteristic of the parallel robot dynamic model that makes the difference to serial
robot models (Spong andVidyasagar 2004), is that there exist more singularity configurations in
the workspace and it is defined on a compact set where closed-chain constraints are satisfied and
singularity configurations are avoided. A manipulator is said to be at a singularity configuration
when the Jacobian matrix loses its full rank (Lung-Wen 1999). Therefore, at a singularity config-
uration, a manipulator may lose one or more degrees of freedom, and it will not be able to follow
some trajectories in the end-effector space. It follows that all control laws for open chain robots
could also be applied with the only restriction that the guaranteed (Lyapunov) stability conclu-
sions will be local (Ghorbel et al. 2000). In addition, some parallel robot dynamic models have
implicit equations requiring a numerical method to solve. For the particular case of the five-bar
parallel robot, the dynamic model is an explicit equation.
M.G. Villarreal-Cervantes et al.
In the last decades, researchers have been developing new and sophisticated control laws in order
to improve the system performance from the control system standpoint. Nevertheless, in most of
the cases the control system calibration is quite difficult, requiring previous experience to tune
the control system.
design focuses on developing neither new control strategies nor a new mechanical system. The
idea of a mechatronic design is to find a compromise between the control system design and the
mechanical system design in order to improve the mechatronic system performance, considering
control laws easy to implement and a predefined robot structure.Therefore, based on the idea of a
mechatronic design, two PID controllers are chosen, since the PID controller has an appropriate
performance in most of the applications and it is widely accepted by several industrial control
processes despite of the development of many others control strategies. Therefore with the PID
controller, the input vector u = [u1,u2]Tis defined in Equation (2):
Due to the integrator of the PID controller, the closed-loop system order is increased by one for
each control input. So, the dynamic model of the five-bar parallel robot (1) with its PID controller
(2) can be formulated in state variables as nonlinear differential Equations (3), which must be
solved by a numerical method such as the Runge–Kutta method:
ui= kpiei+ kii
eidt + kdi˙ ei,
for i = 1,2.
˙ x =dx
= f(x, ¯ x,p,u),
where x = [q1, q2, ˙ q1, ˙ q2,?tf
x3, ¯ x4− x4]Taretheangularpositionandvelocityerrorvectorsoftheactuatedangles,respectively,
t is the time, tf∈ R is the final time and p ∈ Rnpis a vector of all design variables where the
mechanical and controller parameters of the system are included.
0e2dt]T= [x1,...,x6]T∈ R6and ¯ x ∈ R6are the current
and desired state variable vectors, e = [e1,e2]T= [¯ x1− x1, ¯ x2− x2]Tand ˙ e = [˙ e1, ˙ e2]T= [¯ x3−
3.Structure–control integrated design approach
Dynamic optimization is the process of determining control and state histories for a dynamic
system over a finite time period to minimize a performance index (Bryson 1999). That is, an
optimal control problem (OCP) for a dynamic continuous system consist in finding the time
history of a control vector u(t) for t0< t < tfto minimize a performance index of the form:
J = ?[x(tf)] +
x(t0) = x0
with t0, tfand x0specified.
In the general optimal problem, an n-dimensional state vector stated by x(t) describes the
behaviour of the system at time t.Also, an m-dimensional control vector u(t) determines the time
rate of change of the state vector through the set of ordinary differential equations (ODEs) stated
by Equation (5). p is the time-independent parameter vector.
Dynamic optimization problems can be solved either by variational approaches or by applying
problem (Kameswaran and Biegler 2006). The variational approaches are focused on obtaining
a solution to the classical necessary conditions for optimality (Bryson 1999). These approaches
are also known as indirect methods. The second approaches are the methods that discretize the
original continuous-time formulation, and can be divided into two categories, according to the
level of discretization. The first category consists of methods that discretize only the control
profiles (partial dicretization methods) and the second one consists of the methods that discretize
solved either by dynamic programming or by applying a nonlinear programming (NLP) strategy
and they are known as control parametrization methods (Betts 2001). A basic characteristic of
values, is obtained by integration. The main advantage of these approaches is that they generate
smaller discrete problems than full discretization methods. The full discretization problem can
only be solved by NLP but the basic characteristic is that they solve the ODEs only once, at the
optimum. The full discretization method is larger and may require special solution techniques
(Kameswaran and Biegler 2006).
the infinite-dimensional OCP into a finite-dimensional optimization problem or NLP problem,
where the trajectory generation (state vector x(t)) and the feedback design of a control system
(control vector u(t)) can be considered as two separate problems. That is, the design of a control
system can be computed based on a control vector parametrization (CVP) while the trajectory
generation is subject to fulfilling the set of ODEs stated by Equation (5).The control parametriza-
tion method is used in several works, such as Sadegh and Driessen (1999) and Cruz-Villar et al.
trol vector parametrization is stated in Equation (2). Without loss of generality in the integrated
design framework, the problem stated in Equations (4) and (5) is rewritten as:
¯ J(x, ¯ x, ¯ c,
¯ J =
L(x, ¯ x, ¯ c,
= f (x, ¯ x,p,t),
¯ c,p,t) < 0,
hk(x, ¯ x, ¯ c,
x(t0) = x0
gj(x, ¯ x, ¯ c,
for j = 1,...,ng
for k = 1,...,nh,
¯ c,p,t) = 0,
where p ∈ Rnpis a vector of all design variables involving the geometric mechanical parameters
of the five-bar parallel robot and the PID controller gains, x ∈ Rnand ¯ x ∈ Rnare the current
and desired state variable vectors for the generalized coordinates and generalized velocities of
the system dynamics, respectively, x0∈ Rnis the initial state vector required for solving the
nonlinear differential equations (NLDEs) (7), t ∈ R is the time variable, ¯ c ∈ Rncand ¯ c ∈ Rncare
M.G. Villarreal-Cervantes et al.
the position and velocity vectors in the Cartesian space of the desired trajectory to be tracked by
the system, f(x, ¯ x,p,t) : Rn+np→ Rnis a nonlinear vector function, and L(x, ¯ x, ¯ c,
Rn+np→ R, gj(x, ¯ x, ¯ c,
functions which correspond to the performance criterion, inequality and equality constraints
referred to the system design, respectively. Then, the nonlinear dynamic optimization problem
(NLDOP) for the mechatronic design of the five-bar parallel robot is to find the design variables
p that minimize problem (6) subject to inherent design constraints (7)–(9).
In the next subsections, the performance criterion for the structure-control integrated design of
the five-bar parallel robot is presented. Moreover, the design variable vector and the constraints
of the NLDOP (6)–(9) are detailed.
¯ c,p,t) :
¯ c,p,t) : Rn+np→ R and hk(x, ¯ x, ¯ c,
¯ c,p,t) : Rn+np→ R are smooth
Singularity configuration is a typical problem in parallel robots. In a singular configuration, a
manipulator may lose one or more degrees of freedom, and it will not be able to follow some
cannot be controlled.
According to the singularity classification (Liu et al. 2006) of the five-bar parallel robot, there
are three kinds of singularity: the first kind, called the stationary singularity, occurs when the
end-effector reaches its workspace limit, that is, when the angles q1and q3or q2and q4are
collinear (see Figure 1). The second kind of singularity, called the uncertainty singularity, arises
when the robot gains an additional degree of freedom (d.o.f.). This kind of singularity is usually
inside the workspace and it occurs when the angles q3and q4are collinear. The last kind of
singularity is of a different nature from the first two, since it is not only configuration–but also
architecture-dependent, and it occurs when all links are collinear.
large manipulability measure means singularities remoteness and a small one means singularities
robots (Mayorga and Carrera 2006). As the five-bar parallel robot is a non-redundant robot, the
manipulability measure is a useful criterion to avoid singularity configurations. Nevertheless, the
manipulability measure cannot be used as the only performance criterion in a design approach
where both the mechanical and control parameters (mechatronic design) are simultaneously con-
carried out. So, the following performance criterion (10) is proposed in order to satisfy trajectory
tracking without singularity configurations:
¯ J =
sin(q1− q2+ q3− q4)
sin(q1− q2+ q3− q4)
q3= sin−1a4sinq4+ a2sinq2− a1sinq1
sin(q1− q2+ q3− q4)
sin(q1− q2+ q3− q4)
A = a2
− 2a5a1cos(−q1) − 2a4a2cosq2− 2a4a5+ 2a4a1cosq1
B = 4a4a2sinq2− 4a4a1sinq1
C = a2
− 2a5a1cos(−q1) + 2a4a2cosq2+ 2a4a5− 2a4a1cosq1.
3+ 2a2a5cos(q2) − 2a2a1cos(q2− q1)
3+ 2a2a5cos(q2) − 2a2a1cos(q2− q1)
In performance criterion (10), the first term involves the position errors where e = [ ¯ x1−
x1, ¯ x2− x2]Tis the vector of angular position errors. The second term is the manipulability
measure where det (•) and tr (•) are the determinant and the trace of the matrix (•), and J is the
error found in the trajectory tracking for the time interval [0,tf]. So, performance criterion (10)
Let us consider Figure 2. In this article an octagonal shape of the links is used because the
geometric parameters of the links can modify in a wide range the dynamic parameters of the
links (i.e. the mass mi, the mass centre length lci, the mass centre angle γi) and the kinematic
parameters (i.e. the link length asi). The geometric parameters of the ith link, asi, bsi, csi, dsi,
esi, fsi, gsi, hsi, isi, jsi, ksi, for the five-bar parallel robot are considered as the structural design
parameters to be optimized. Those parameters are selected because they can modify the dynamic
and kinematic parameters of the links. Once the optimization process is finished, the optimum
if one selects only the dynamic and kinematic parameters, another optimization process would
be required in order to find the structure shape.
for the five-bar parallel robot.
M.G. Villarreal-Cervantes et al.
as the control design parameters to be optimized. So, the design parameter vector p ∈ R51which
involves structural and control design parameters results as in (12):
p = [as1···as5,bs1···bs4,cs1···cs4,ds1···ds4,es1···es4,fs1···fs4,
NotethatinFigure2thegeometricparametersfsi,gsi,hsi,isi,jsi,ksi,i = 1,...,4,canchange
the mass centre angle γiof the ith link in a closed interval [0,±π]. If these parameters are not
taken into account, the mass centre angle of the links could be changed to the following discrete
γi, Ii) of the ith link in terms of geometric parameters, it is necessary to consider the following.
• The coordinate origin is placed at the intersection of the x–y-axis and in the C hole as shown
in Figure 2.
• The ith link can only move at the x–y-plane.
• There is a shaft in the B hole at links 1 and 3. A cylindrical support is placed on the C hole of
link 1 and two cylindrical supports are placed at the B and C holes of link 2, respectively.
• Links and shafts are made of aluminium and steel, respectively. The density of the aluminium
and the steel are ρal= 2710kgm−3and ρst= 7840kgm−3, respectively.
• The links are divided in simple geometries in order to simplify the computing of their dynamic
parameters. Therefore, the ith link is divided into nine parts, Ai, Bi, Ci, Di, Ei, Fi, Gi, Hi, Ii,
as shown in Figure 2.
• The ith mass is concentrated at the mass centre of the ith link.
in next subsections.
The masses of links are shown in Equations (13), where mAi,...,mIiare masses of the A, …, I
the masses of cylindrical supports which are placed on the B and C holes. rB= 3.175 × 10−3m
is the radius of the B hole, rcsB= 15.875 × 10−3m and rcsC= 19.05 × 10−3m are the radii of
cylindrical supports which are placed on the B and C holes, respectively, Lcs= 0.010m is the
length of the cylindrical support which is placed on the C hole, and ri= 0.006m for i = 1,2 and
ri= 0.0025m for i = 3,4 are the radii of cylindrical supports which are placed on the C hole.
m1= mT1+ mxB1+ mcsC1
m2= mT2+ mxB2+ mcsC2+ mcsB
m3= mT3+ mxB3
mTi= mAi− mBi− mCi+ mDi+ mEi− mFi− mGi− mHi− mIi
mxB1= ρstπrB2(es1+ es3)
mxB3= ρstπrB2(es3+ es4)
mcsCi= ρalπ[rcsC2− ri2]Lcs
mcsB= ρalπ[rcsB2− rB2]es3
mAi= ρal[(asi+ bsi+ dsi)(csi)(esi)]
mDi= ρal[(asi+ bsi+ dsi)(fsi)(esi)]
mEi= ρal[(asi+ bsi+ dsi)(gsi)(esi)]
mxB2= ρst[π(rB)2(es2+ es3+ es4)].
Length lciand angle γiof the mass centre
The mathematical expression for the length lciand the angle γiof the mass centre for the ith link
is shown in Equation (14). (xCMi,yCMi) are the x–y-coordinates of the total mass centre of the
link, (xAi,yAi),...,(xIi,yIi), (xxBi,yxBi), and (xcsB,ycsB) are the x–y mass centre coordinates
for theAi, …, Iipart, for the shaft placed on the B hole and for the cylindrical support placed on
the B hole, respectively.
(xCMi)2+ (yCMi)2,γi= Atan2
xTi= xAimAi− xBimBi− xCimCi+ xDimDi+ xEimEi− xFimFi− xGimGi
− xHimHi− xIimIi
yTi= yAimAi− yBimBi− yCimCi+ yDimDi+ yEimEi− yFimFi− yGimGi
− yHimHi− yIimIi
xCM2=xT2+ xxB2mxB2+ xcsBmcsB
yHi= yIi= −
M.G. Villarreal-Cervantes et al.
xxB2= xscB2= as2
xDi=(asi+ bsi+ dsi)
xEi=(asi+ bsi+ dsi)
yCM2=yT2+ (yxB2)(mxB2) + (ycsB)(mcsB)
yAi= yBi= yCi= yxBi= ycsCi= xCi= xcsCi= 0
xGi= (asi+ dsi) −1
xIi= (asi+ dsi) −1
yxB2= ycsC2= ycsB= yxB3= xcsB= xcsC2= 0
xAi=(asi+ bsi+ dsi)
where IAi,...,IIiare the inertias of theA, …, I geometric shapes, respectively, IxBiis the inertia
of the shaft placed at the Bihole, IcsBand IcsC2are the inertias of cylindrical supports which are
placed at the B and C holes, respectively, and P#is an equation where the sign # represents the
appropriate subscript, for example PAi, PBi, etc.
I1= IT1+ IxB1+ IcsC1
I2= IT2+ IxB2+ IcsC2+ IcsB
I3= IT3+ IxB3
ITi= IAi− IBi− ICi+ IDi+ IEi− IFi− IGi− IHi− IIi
?rcsB2+ rB2?+ PcsB
??asi+ bsi+ dsi
??asi+ bsi+ dsi
??asi+ bsi+ dsi
?rcsC2+ ri2?+ PcsC1
?rcsC2+ ri2?+ PcsC2.
M.G. Villarreal-Cervantes et al.
The constraints in the parallel robot design involve: the maximum torque of the actuators (16)
which establishes the real torque limits; the mobility of the robot (17) which is stated by the
five-bar Grashof criterion; the real limits in the geometric design variables (18); the trajectory
specification (19) to be followed by the end-effector of the five-bar parallel robot; and the parallel
robot dynamic model (3).
|ui(t)| ≤ ui max,
as5+ as1+ as2− as3− as4< 0
as1+ as2− as3< 0
as1+ as2− as4< 0
as5− asmax≤ 0
−as2+ asmin< 0
bsimin≤ bsi≤ bsimax
dsimin≤ dsi≤ dsimax
fsimin≤ fsi≤ fsimax
gsimin≤ gsi≤ gsimax
hsi+ isi≤ bsi+ asi+ dsi
jsi+ ksi≤ bsi+ asi+ dsi
for i = 1,...,4,
for i = 1,2, 0 ≤ t ≤ tf
as3− as5< 0
as4− as5< 0
− as1+ asmin< 0
csimin≤ csi≤ csimax
esimin≤ esi≤ esimax
dsimin≤ dsi≤ dsimax
whereuiistheappliedtorqueatthemotori,ui max= 5Nmisthemaximumtorquefortheithmotor
due to the maximum actuator torque andasiis theith link length.asmax,asminare the maximum and
minimum lengths. The maximum and minimum values of the geometric parameters are chosen
as follows: asmax= 0.5m, asmin= 0.035m; bsimin= 0.01905m, bsimax= 0.3m, csimin= 0.0381m,
csimax= 0.10m,fori = 1,2;bsimin= 0.015m,bsimax= 0.3m,csimin= 0.03175m,csimax= 0.10m,
for i = 3, 4; dsimin= 0.015m, dsimax= 0.3m, esimin= 0.00635m, esimax= 0.03m, fsimin= 0m,
fsimax= 0.1m, gsimin= 0m, gsimax= 0.1m, for i = 1,...,4.
linkage. Therefore, the second and third kind of singularity are avoided (Liu et al. 2006).
On the other hand, a circle in the Cartesian space is proposed to be tracked by the end-effector
of the parallel robot. Nevertheless, whatever other trajectory could be chosen, and in order to
verify that statement, a Lissajous curve is proposed too. Those trajectories are shown in Figure 3.
The mathematical expression is shown in Equations (19), where t is the time, frec= 0.2Hz
is the frequency at which the trajectory is completely tracked, r = 0.1m, ¯ c = [¯ xp, ¯ yp]Tand
¯ c =
the Cartesian space. cx= 0.15m, cy= 0.3m are the Cartesian coordinates of the intersection of
two lines or the circle centre for the Lissajous curve or for the circle, respectively (see Figure 3).
ˆ a = 2 or ˆ a = 0 for the Lissajous curve or for the circle, respectively.
¯ xp= cx+ r cos(2πfrect)
¯ xp= −2πfrecr sin(2πfrect)
¯ yp]Tare vectors which involve the position and the velocity of the desired trajectory in
¯ yp= cy+ r sin(ˆ a2πfrect)
¯ yp= ˆ a2πfrecr cos(ˆ a2πfrect)
Figure 3.Desired trajectories: (a) Lissajous curve; (b) circle.
Both the inverse kinematic IK and the Jacobian J of the parallel robot are required in order to
to joint coordinates.The direct kinematic DK and the inverse of the Jacobian J−1are required in
order to transform the position and the velocity from the current joint coordinates to the current
position c = [xp,yp]Tand velocity
robot the IK and DK are developed in Liu et al. (2006).
The last constraint includes the parallel robot dynamic model (3). This constraint and the
constraint stated in Equation (16) are dynamic constraints which are modified when the time is
changed (the time variable is included in those constraints), making the difference with respect
to a static optimization. The system dynamic (3) must be solved as NLDEs by a numeric method
in order to provide the temporary behaviour of the state vector x in the time interval [0,tf]. The
initial state vector for the NLDEs is proposed as x(t0) = [1.7453, 0, 0, 0, 0, 0]Twith final time
·c = [˙ xp, ˙ yp]TCartesian coordinates. For the five-bar parallel
Formulation of the five-bar parallel robot design as an NLDOP
The NLDOP consist of finding the optimal design variable vector p∗(12) which involves
the optimal geometries of the parallel robot’s links and the optimal PID controller gains, that
simultaneously minimize (20) (minimize the position error and move away from singularity con-
figurations), subject to constraints at the parallel robot dynamic model (3), the motor torque (16),
the Grashof criterion (17), the geometric limits of the links (18), and the desired trajectory to be
executed in the Cartesian space (19). Therefore, the NLDOP can be formulated as follows:
¯ J =
(i) the system dynamics (3);
(ii) the maximum applied power for the motor (16);
(iii) the Grashof criterion (17);
(iv) the geometric limits of the links (18);
(v) the desired trajectory (19).