I. Ramos Arreguin (ed.) Automation and Robotics

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Towards an Automated and Optimal Design of Parallel Manipulators

153

orientation workspace of the selected solution. As shown by figure 7 and 8 an orientation of

°= 20θ is also feasible without violating the workspace requirement.

Figure 9 and 10 depict the displacement of the first actuator for a position error of 0.1 mm in

each direction and an orientation error of 0.05° in the two first Euler angles, as defined in

(Mbarek et.al, 2005). Moreover, figure 11 and 12 demonstrate that these errors induce a

displacement of 1e-4 m

Fig. 4: Flow-chart of the overall numerical procedure

throughout the workspace for

20θ . The accuracy requirement is therefore satisfied for

this actuator, since the resolution of the encoders is 1e-5 m. For simplicity of exposition, the

displacements of the other actuators are not represented in this work.

The velocity requirement is also satisfied. Indeed, the actuators velocities of each actuator is

less than the prescribed limit 1 m/s. Figure 13-16, depict the required velocities of the fifth

actuator. Finally, figure 17-20 depict the distribution of the condition number throughout

the workspace.

Fig 5: The prescribed workspace and the

workspace of the selected design solution for

°== 0,0

Fig 6: The prescribed workspace and the

workspace of the selected design solution

for

0,0,78.0

Fig 7: The prescribed workspace and the

workspace of the selected design solution for

°== 20,0

Fig 8: The prescribed workspace and the

workspace of the selected design solution

for °

20,0,78.0

Automation and Robotics

154

Fig 9:

δρ

for °

= 0,0

Fig 10:

δρ

for °=

0,0,78.0

Fig 11:

δρ

for °

20,0

Fig 12:

δρ

for °=

20,0,78.0

Fig 13: v

for °

= 0,0

Fig 14: v

for °=

0,0,78.0

Towards an Automated and Optimal Design of Parallel Manipulators

155

Fig 15: v

for

= 20,0

Fig 16: v

for

°=

20,0,78.0

Fig 17: The condition number distribution for

°== 0,0

Fig 18: The condition number distribution

for

0,0,78.0

Fig 19: The condition number distribution for

°== 20,0

Fig 20: The condition number distribution

for °

20,0,78.0

5. Conclusion

In this chapter, we investigated the optimal design problem of a parallel manipulator with

five degrees of freedom that will be used in a high-speed stitching unit as a guidance

Automation and Robotics

156

mechanism for a novel sewing head. First, we reviewed the kinematics of the manipulator.

Starting from the requirements list, we derived performance indices that allowed us to

evaluate the adequacy of the manipulator for this application. Finally, we developed a

numerical procedure that provides many design solutions, which satisfy all requirements.

Owing to the iterations that may occur during the design process, the designer may consider

different solutions. This allows one to take into account other requirements, such as

manufacturing capabilities, available actuators on the market, costs etc … The presented

robot stitching unit can be used to assemble reinforcement textiles for the aerospace,

automotive, rail vehicles and shipbuilding industry.

6. References

Kordi, M.T., T. Mbarek and B. Corves (2006). Synthesis of a new mechanism for needle

guidance in a sewing machine with single-sided sewing technique. In: Proceedings of

EuCoMes, the first European Conference on Mechanism Science. Obergurgl. Austria.

Mbarek, T., M. Nefzi and B. Corves (2005). Prototypische Entwicklung eines

Parallelmanipulators mit dem Freiheitsgrad fünf. In: Proceedings of Mechatronik

2005: Innovative Produktentwicklung. Wiesloch (Germany).

Gosselin, C.M. (1988). Kinematic Analysis and Programming of Parallel Robotic

Manipulators. Ph.D. Thesis, McGill University, Montréal, Canada.

Merlet, J.P. (2006). Parallel Robots (Solid Mechanics and Its Applications). Second Edition.

Springer, 2006.

Merlet, J.P. (1997). DEMOCRAT: A DEsign MethOdology for the Conception of Robot with

parallel ArchiTecture. In: Proceedings of the International Conference on Intelligent

Robots and Systems, IROS. Grenoble. France. Pp. 1630-1636

Merlet, J.P. (2005a). The necessity of optimal design for parallel machines and a possible

certified methodology. In: Proceedings of the 2

International Colloqium of the

Collaborative Research Center 562, Robotic Systems for Handling and Assembly.

Braunschweig. Germany. Pp. 7-20

Hao, F. and J.P. Merlet (2005). Multi-Criteria optimal design of parallel manipulators based

on interval analysis. Mechanism and Machine Theory 40.

Merlet, J.P. (2005b) Optimal design of robots. In: Online Proceedings of Robotics: Science and

Systems. Boston. USA.

Boudreau, R. and C.M. Gosselin (2001). La synthèse d'une plate-forme de Gough-Stewart

pour un espace atteignable prescrit. Mechanism and Machine Theory 36. Pp. 327-342

Ottaviano, E. and M. Ceccarelli (2001). Optimal design of CAPAMAN (Cassino parallel

manipulator) with prescribed workspace. In: Proceedings of Computational

Kinematics. Seoul. South Corea. Pp. 35-44.

Cecarelli, M. (2002). An Optimum Design of Parallel Manipulators: Formulation and

Experimental Validation. In: Proceedings of Collaborative Research Centre 562: Robotic

Systems for Handling and Assembly. Braunschweig. Germany. Pp. 47-64.

Lou, Y., Z. Dongjun and L. Zexiang (2005). Optimal Design of a Parallel Machine Based on

Multiple Criteria. In: Proceedings of the 2005 IEEE International Conference on Robotics

and Automation. Barcelona. Spain. Pp. 3219-2224

The Math Works, Inc. (2006). Optimization Toolbox. User’s Guide, Version 3. Pp. 4-12.

Identification of Continuous-Time Systems with

Time Delays by Global Optimization Algorithms

and Ant Colony Optimization

Janusz P. Paplinski

Technical University of Szczecin

Poland

1. Introduction

There are systems which have inherent time delay. If the time delay used for controller

design does not coincide with the real process time delay, than a close-loop system can be

unstable, or may cause efficiency lost (Bjorklund & Ljung, 2003; Boukas, 2003; Li & Wan,

2002). The identification of linear systems with unknown time delay is important and

should be treated as first task during system analysis and control design. This problem can

become more complicated for the multi-input single-output (MISO) system, where the

solution space is multi-modal.

The most of conventional system identification techniques, such as those based on the non-

linear estimations method, for example separable nonlinear least squares (SEPNLS) method,

are in essence gradient-guided local search methods. They require a smooth search space or

a differentiable performance index. The conventional approaches in the multi-modal

optimisation can easily fail in obtaining the global optimum and may be stopped at a local

optimum (Chen & Hung, 2001; Harada et al., 2003). One of the possible solution of this

problem is use of a SEPNLS methods with global optimisation elements (Chen & Wang,

2004), for example Global SEPNLS (GSEPNLS), known from the literature (Previdi &

Lovera, 2004).

New possibility in identification of systems with multi modal solution space is opened by

application of the computational intelligence methods (Papliński, 2004; Path & Savkin, 2002;

Shaltaf, 2004; Yang et al., 1997). Ant Colony Optimization (ACO) is one among them. Ants

are known as a social insects. They exhibit adaptive and flexible collective behavior to

achieve various tasks. The macro-scale complex behavior emerges as a result of cooperation

in micro-scale.

This paper considers the problem of parameter estimation for continuous-time systems with

unknown time delays from sampled input-output data. The iterative separable nonlinear

least-squares (SEPNLS) method and global separable nonlinear least-squares (GSNLS)

method (Westwick & Kearney, 2001) are presented. We have extended this method by using

the ACO. The ACO proposed in the paper is looking for the time delays. Another

parameters of linear system are obtained during evaluation of leaving pheromone, by using

the SEPNLS method.

Automation and Robotics

158

2. Problem description

The dynamic of continuous-time MISO system with unknown time delays can be described

as:

−

===

−τ

∑∑∑

i0 j1k1

a p x(t) b p u (t ) (1)

where

; ≠

b0;

p - differential operator;

(t) - j-th input;

- time delay of j-th input;

x - non-disturbed output of system;

We assume the parameters

n and m

are known.

The measurement output is disturbed by stochastic noise:

(t) x(t) v(t) (2)

The zero order hold is used



u(t) u(k) for

−

≤<(k 1)T t kT , (3)

where T – sampling period.

The problem studied here is as follows: how to estimate the time delays and the system

parameters from sampled data representation of the inputs and the noisy output.

3. SEPNLS estimation method

The low-pass pre filter Q(p) is used in order to obtain direct signal derivatives (Iemura et al.,

2004):

()

α+

Q(p)

(4)

where

is the parameter of the Q(p).

Using the pre-filter Q(p) and bilinear transformation in the system (1) we can obtain an

approximated discrete-time estimation model of the original system:

()

0y i iy jl j

nm 1u

i1 j1l1

ka(k) b k rk

−+

===

ξ=ξ= ξ −τ+

∑∑∑

(5)

where

() ()

=ξ

∑

iiv

r k a k , (6)

Identification of Continuous-Time Systems with Time Delays by Global Optimization

Algorithms and Ant Colony Optimization

159

==ρ+



≤

Δ<0 T (7)

and

()

()()

()

ini

−

−−

−

−−

⎟

⎠

⎞

⎜

⎝

⎛

++−

−+

⎟

⎠

⎞

⎜

⎝

⎛

[

]

y, u, vγ=

(8)

The model (5) can be written in the vector form:

() ( ) ()

ξ=ϕτθ+

k k, r k (9)

where

(

)

(

)

(

)

()

(

)

()

() ()

]

,...,

,,...,[,

runrumnun

umnyny

kkk

kkkk

rrj

τξτξτξ

τξξξτϕ

−−−

−−−=

+−

(10)

⎡

⎤

θ=

⎢

⎥

⎣

⎦

1n111m r1rm

a ,...,a ,b ,...,b ,...,b ,...,b (11)

The parameters of the model can be estimated as the minimizing arguments of the LS

criterion

()

kk 1

V, k,,

Nk 2

τ= ε θτ

−

∑

(12)

() ()

()

θτ =ξ −ϕ τθ

k, , k K,

(13)

The vectors of the time delays τ and linear parameters θ are estimated in a separable

manner. The linear parameters, when the time delays are known, can be obtained from

linear LS method:

() ()()

−

τ= τ τ

RN,fN,

(14)

where

()

()()

=ϕτϕτ

−

∑

kk 1

RN, k, k,

(15)

()

()()

τ= ϕ τξ

−

∑

kk 1

fN, k, k

(16)

The time delays

can be estimated as the minimizing arguments of the criterion

Automation and Robotics

160

()

=τ



arg min V (17)

where

()

kk 1

Vk,

Nk 2

=ετ

−

∑



(18)

and

() () ( )

()()

−

τ=ξ −ϕ τ τ τ



k, k k, R N, f N, (19)

The all time delays occurring in the system can be computed by the iterative algorithm

=τ +Δτ

l1 l l1

jjj

ˆˆˆ

(20)

where

() ()

−

⎡⎤

τ=−μ τ τ

⎢⎥

⎣⎦



l1 l ' l

jjjjj

ˆˆˆ

(21)

The parameter

determines the step size of the algorithm. The Hessian of the LS criterion

()



R , which permits to use information about the local curvature of the error-surface

(Westwick & Kearney, 2001), may be calculated as

()

=ψτ

−

∑



kk 1

Rk,

, (22)

and the gradient of LS criterion

()



can be obtained from:

()

()()

=− ψ τ ε τ

−

∑



kk 1

Vk,k,

(23)

The derivatives ψ

(k,τ) are defined as

()

()()

()

k, k,

k, R N, f N,

RN,

k, f N,

fN,

k, R N,

−

∂ε τ ∂ϕ τ

τ= = τ τ

∂τ ∂τ

∂τ

+ϕ τ τ

∂τ

+ϕ τ τ

∂τ



(24)

After suitable differentiations:

Identification of Continuous-Time Systems with Time Delays by Global Optimization

Algorithms and Ant Colony Optimization

161

()

(

)

[

()

−−+

−

∂ϕ τ

ϕτ= =

∂τ

−ξ − τ −ξ − τ

⎤

−ξ − τ

⎦





1xm 1xm 1xm

nmu nm1u

1xm 1xm

n1u

k, 0,0,...,0,

k, k,...,

k ,0 ,...,0

(25)

()

()()

−

ττ

∂τ

τ=

∂τ

=ϕτϕτ

−

∑

kk 1

RN,

RN, k, k,

(26)

()

(

)

()

()()

ττ

∂τ

τ=

∂τ

τ= ϕ τξ

−

∑

kk 1

fN,

fN, k, k

(27)

finally we can get

() ()

()()

()

() () () ()()

()

()()

T1 T 1

k, k, R N, f N,

k, R N, R N, R N, R N, f N,

k, R N, f N,

−

−−

−

ψτ=ϕ τ τ τ

⎡⎤

−

ϕτ τ τ+ τ τ τ

⎢⎥

⎣⎦

+ϕ τ τ τ

(28)

I. The SEPNLS algorithm

Let l = 0. Set the initial estimate of the time delays

(

)

(

)

=τ

ˆˆ

Compute Hessian

(

)

RN, and gradient

(

)

f N, of the linear LS criterion, for the

assumed time delays

(

)

- expressions (15) – (16)

Compute Hessian

()



R and gradient

()



V of LS criterion by harness equations (22) –

(28)

Compute new value of the time delays

by using the equation (20) and (21)

Check if the new time delays belong to permissible area. If not, let decrease the

increment of time delays

τ=υΔτ

l1 l1

ˆˆ

, where υ is a random variable, and go back to

step 4.

Check if the new value of LS criterion

()

(

)



V is smaller than the anterior

()

(

)

()

(

)

τ≤τ



l1 l

ˆˆ

V V . If not, decrease the increment of the time delays

τ=υΔτ

l1 l1

ˆˆ

, (29)

Automation and Robotics

162

where υ is a random variable, and go back to step 4.

If the stopping condition is satisfied, go to the next step 7, else let l = l + 1 and go back

to step 2.

Compute the linear parameter vector from linear LS criterion (14), and terminate

algorithm.

4. The global SEPNLS estimation method (GSNLS)

The SEPNLS method can converges to the local optimum. It is possible to apply stochastic

approximation (Bhart & Borkar, 1999) with convolution smoothing to the SEPNLS method

in order to reach the global optimum (Iemura et al., 2004). The estimate of the time delay in

GSNLS can be obtain as

() ()

()

l1 l l l

jj j j

ˆˆ ˆ ˆ

−

⎛⎞

=τ −μ τ τ +β η

⎜⎟

⎝⎠



(30)

where

Rη∈ is a random vector used to perturb τ. The values of η are generated by

Gaussian, uniform, or Cauchy distribution probability density function. The values of μ and

β control its variance.

The value of β has to be chosen large at the start of the iterations and decreased to zero

when the global minimum is reached (Yang et al., 2005). It can be obtain for

()

(

)

=β τ



(31)

where the LS criterion

()

(

)

V τ



is given by equation (18). The value of

()

(

)

V τ



decreases

during the sequence of iterations, which cases that disturbances of

decrease too.

5. Ant colony optimization (ACO)

Recently, researchers in various fields have showed interest in the behaviour of social

creatures to solve various tasks. The ants exhibit collective behaviour to perform task as

foraging, building a nests. These tasks can not be carried out by one individual. The macro-

scale complex behaviour emerges as a result of cooperation in micro-scale. This appears

with out any central or hierarchical control. A way of communicating between individuals

in colony is chemical substances called pheromones (Agosta, 1992; Bonabeau et al., 1999).

Ants looking for food lay the way back to their nest with a specific type of pheromone.

Other ants can follow the pheromone trail and find the way to the aliments. Pheromones

remain in the some way superimpose and intensify, concurrently the pheromones evaporate

in time and their intensity decreases. A specific map of pheromones is created on the search

space. This map is not unalterable and can bring into line with ambient. Each ant looks for

food independently of the others and moves from nest to source of food. There are a lot of

ways in which ants can go. Ants choose a way for theirs leverage three sources of

information

- the own experience

- the local information

- the pheromone trail.