Recursive nonlinear identification an electromechanical manipulator using the MIMO NARX model

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Volume 2 Issue 1 (January 2015

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Recursive nonlinear identification an electromechanical manipulator using the MIMO NARX model

Anrafel Silva Meira^* Paulo Henrique de Miranda Montenegro

José Antônio Riul PPGEM, UFPB PPGEM, UFPB. PPGEM, UFPB Abstract— The objective this article is selects nonlinear models of multiple inputs and multiple outputs to an electromechanical manipulator using the MIMO NARX model. For the selection of models we used the error reduction rate (ERR) algorithm to find the most important terms of the model, and the sum of the square error for check the accuracy of the estimation. At the end are found multivariable and nonlinear models with parameters estimable at each sampling period for the robotic manipulator.

Keywords— Recursive identification, MIMO NARX model, error reduction rate (ERR), Robotic manipulator, Nonlinear model

I. INTRODUCTION

The manipulators are defined as combinations of rigid structural elements (links) connected to each other through a joint or joints, which at its end is placed a tool or device for performing the task (e.g. [1]).

For robotic manipulators project is usually necessary to have a mathematical model for the project then its controlling.

These models can be obtained using the physical laws (white box model) or using identification techniques using system input and output (black box models). White box models of these systems are nonlinear (e.g. [2]) and difficult to obtain when all the phenomena that occur are considered, while the black box modeling (eg. [3],[4]) can generate linear models and nonlinear.

Although the linear models are used to represent the mechanical manipulators, these are not the most appropriate, since such systems are multivariable, nonlinear, dynamic engagements and therefore these mathematical models and systems have uncertain parameters that vary over time (e.g. [5]). The Non-linear phenomena that occur are diverse and they may increase over time, such as friction and backlash between the links.

From the point of view of control, the traditional control techniques based on fixed gain control laws, on/off control, proportional, proportional derivative, proportional integral, proportional integral derivative, phase advance, phase delay and advance and delay phase (e.g. [6], [7]), do not have acceptable performance when disturbances and process variations are significant, forcing control strategies more elaborate (e.g. [8]).

Based on the problems described, non-linearity and parametric variation in robotic manipulators, then comes the need for mathematical models to the robotic manipulators that are nonlinear multivariable and can be used for online identification, models.

II. DESCRIPTION OF THE ROBOTIC MANIPULATOR

The robotic manipulator comprises three rotary joints and three links appointed 1, 2 and 3, as show in Fig. 1. The total displacement of the link 1 is 180º and of the link 2 is 110º, with each of these links being driven by a direct current motor, while the link 3 has its motion controlled by a mechanical system that always keeps the horizontal position

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Fig. 1 Robotic manipulator comprises three rotary joints and three links

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III. RECURSIVE IDENTIFICATION USING MIMONARX MODEL

The equation of nonlinear autoregressive model with exogenous variables NARX for m inputs and r outputs is given by equation (1) and (2) (e.g. [4], [9]):

( )k F (k1),, (kn_y), (k1),, (kn_u) ( )k

y y y u u e (1)

1( ) ( )

( )

 

 

 

 

 



m

y k k

y k

y

1( ) ( )

( )

 

 

 

 

 



r

u k k

u k

u

1( ) ( )

( )

 

 

 

 

 



m

e k k

e k

e (2)

where: F[] is a nonlinear function any, y(k) the vector of system output, u(k) the vector of system inputs, e(k) the vector of noises, ny the maximum lags in the outputs and nu the maximum lags output

The equation (1) can be written as m scalar equations, allowing the estimation is similarly to models with simple input and simple output (e.g. [10]):

1

1 1

( ) ( 1), , ( ), , ( 1), , ( ),

( 1), , ( ), , ( 1), , ( ) ( )

1, ,

     

    



  



m

r

i i y m m y

u r r u

y k F y k y k n y k y k n

u k u k n u k u k n e k

i m

(3)

Expanding F[] as a polynomial function of degree l gives the equation (4) (e.g. [4], [10]):

1 1 1 2 1 2 1 1

1 1 2 1 1 1

0

1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( )

; 1, ,

        

    

    

  

    

  



r r r r r

l l

M M M M M

i i i i

i i i i i i i i i i i i

i i i i i i i

r y u

y k k k k k k e k

M mn rn i m

(4)

where θ are system parameters and ψ are delayed outputs and inputs gives the equation (5)

1 1

1

1 1 1

1 1

1

1 2

...

... 1 1

... 1

... 1 2

... ...

( ) ( 1)

( ) ( )

( ) ( 1)

( ) ( )

( ) ( 1)

( ) ( )

( ) ( 1)

( ) ( )





 

  

   

    

 



y

y ym m

y ym

y ym u

y ym y ur r

n y

n

n n m y

n n

n n n u

n n n

n n n n r u

k y k

k y k n

k y k

k y k n

k u k

k u k n k u k

k u k n

(5)

The equation (4) has a number of terms which increase with the number of inputs and outputs and total delay of the system inputs and outputs, so the terms reduction has to be performed to simplify the model. The error reduction rate ERR algorithm is a good option to reduce terms of the NARX model.

The error reduction rate ERR (error reduction ratio) is an index related to each candidate term of the model, which indicates the improvement in the system identification by your inclusion, how much the greater this ratio greater your contribution to identification. The ERR algorithm is described as (e.g. [11]):

1. Set up the matrix Ψ of the equation (6) with the Mr candidate terms and N samples:

1 1

1 (1) (1) (1) (1)

1 (2) (2) (2) (2)

1 ( ) ( ) ( ) ( )

   

 

 

 

   

 

 



     



y y y u

n n n n

N N N N

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2. For η=1 and i=1...M do:

1

ⁱ  ⁱ _i

w w (7)

 

1 1

  

i T

i i

i T i

g g w

w w

Y (8)

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     

¹ ² ¹ ¹



i i T i

i T

g w w

ERR Y Y (9)

3. Choose to be the first term of the model, one with the highest ERR, which index will be called by h1.

1 1 1^h _h1

w w (10)

4. Now, for η=2,...,Mr, i =1,...,Mr, i≠h1,...,i≠hη-1 and 1≤j≤η-1 do:

 

,

 

  

i T

j i

i

j i T i

j j

w

w w

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1

, 1



   





 



i i

i j j

j

w w (12)

 

   



 



i T i

g w

w w

Y (13)

     

 ²  



i i T i

i T

g w w

ERR Y Y (14)

and choice to be the eighth term of the model, one with the highest ERR among others, which index will be named by hη 1

, 1

 



    





 ^h  ^h 



^h^j ^j

j

w w w (15)

After classification the terms of the model by ERR can select models using the best terms and estimate its parameters in real time at each sampling period by algorithm of recursive least squares (RLS). The estimator of recursive least squares with a forgetting factor λ is given by the set of equations (e.g. [3]):

( ) ( 1)

( ) ( 1) ( ) ( 1)



  

 

 ^T  

P k k

K k k P k k (16)

 

( ) ( 1) ( ) ( ) ( 1) ( 1)

 k  k K k y k ^T k  k (17)

 

( ) 1 ( 1) ( ) ( 1) ( 1)

   ^T  

P k P k K k k P k (18)

where: K(k) is the gain of the estimator; θ(k) is the vector of estimated parameters; P(k) is the covariance matrix and ψ(k) is the vector with delayed outputs and inputs of the model.

And to choose the amount necessary terms to the model, we use the sum of the squared error (SSE) given by equation (19):

[ ]²

1

( ) ˆ( )

N

k

SSE y k y k

=

å

- ⁽¹⁹⁾

where ŷ(k) is the estimated output.

IV. IDENTIFICATION OF THE ROBOTIC MANIPULATOR

Initially not have any information of the mathematical model of the electromechanical manipulator, and must choose an initial model for identification. First, we used the MIMO NARX model considering it as a polynomial expansion of equation (4) with degree l = 2, so for each link of the robot have been equation (20):

0 1 1 2 1 3 2 4 2

5 1 6 1 7 2 8 2

2

9 1 10 1 1 11 1 2

12 1 2 13 1 1 14 1

( ) ( 1) ( 2) ( 1) ( 2)

( 1) ( 2) ( 1) ( 2)

( 1) ( 1) ( 2) ( 1) ( 1)

( 1) ( 2) ( 1) ( 1) ( 1)

    

   

  

         

        

        

       

i i i i i

i

i i i i

i i i

y k y k y k y k y k

u k u k u k u k

y k y k y k y k y k

y k y k y k u k y k ₁

2

15 1 2 16 1 2 17 1

18 1 2 19 1 2 20 1 1

21 1 1 22 1 2 23 1 2

2

24 2 25 2

( 2)

( 1) ( 1) ( 1) ( 2) ( 2)

( 2) ( 1) ( 2) ( 2) ( 2) ( 1)

( 2) ( 2) ( 2) ( 1) ( 2) ( 2)

( 1) ( 1

  

 

 

        

         

   

i i i

i i

u k

y k u k y k u k y k

y k y k y k y k y k u k

y k u k y k u k y k u k

y k y k ₂ ₂₆ ₂ ₁

27 2 1 28 2 2 29 2 2

2

30 2 31 2 1 32 2 1

2

33 2 2 34 2 2 35 1

36 1

) ( 2) ( 1) ( 1)

( 1) ( 2) ( 1) ( 1) ( 1) ( 2)

( 2) ( 2) ( 1) ( 2) ( 2)

( 2) ( 1) ( 2) ( 2) ( 1)

(



  



    

         

        

        

 

i

i i i

i

y k y k u k

y k u k y k u k y k u k

y k y k u k y k u k

y k u k y k u k u k

u k ₁ ₃₇ ₁ ₂ ₃₈ ₁ ₂

2

38 1 39 1 2 40 1 2

2 2

41 2 42 2 2 43 2

1) ( 2) ( 1) ( 1) ( 1) ( 2)

( 2) ( 2) ( 1) ( 2) ( 2)

( 1) ( 1) ( 2) ( 2)

 

  

       

        

      

i i

i i i

u k u k u k u k u k

u k u k u k u k u k

u k u k u k u k

(20)

where: i=1 for link 1 and i=2 for link 2.

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www.ijirae.com For identification of the electromechanical manipulator used it the input signal of Fig. 2 to the link 1 and Fig. 3 to link 2, using the sampling time of 150 milliseconds, was obtained the output signals shown in Fig. 4 for the link 1 and Fig. 5 for the link 2.

Fig. 2 Input signal in the link 1 Fig. 3 Input signal in the link 2

Fig. 4 Output signal in the link 1 Fig. 5 Output signal in the link 2

The set of inputs and outputs, and then the algorithm of the error reduction rate is used to select the best terms to compose the model of robotic manipulator. The Table 1 show the result of using the error rate of the algorithm for the model equation for the link 1 and link 2, in them are placed the first seven terms in order of importance (those who contribute most to the estimation)

TABLEI

TERMS RANKED IN ORDER OR IMPORTANCE FOR THE LINK 1 AND 2 Order of importance Terms of the link

1

Terms of the link 2

1º y1(k-1) y2(k-1)

2º u1(k-2) u2(k-2)

3º u2(k-1) u2(k-1)

4º y1(k-2) y1(k-1)u2(k-2)

5º y2(k-2)u2(k-2) y2(k-2)u2(k-1)

6º y2(k-1)u1(k-1) y22

(k-2) 7º y1(k-1)u2(k-2) u1(k-1)u2(k-2)

Setting models for the link 1 and 2 selecting the first terms of Table 1 for each link, the first model being composed by the first term, the second by the first two and so on, calculated the sum of the square error of each model, using the estimated output by the estimator of the recursive least squares. The Table 2 show the sum of the square error for the models of the link 1 and 2.

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www.ijirae.com TABLEII

SUM OF THE SQUARE ERROR (SSE) FOR THE MODELS OF THE LINK 1 AND 2 Numbers of terms of the

model

SSE for link 1 SSE for link 2

only one 3326,2368733 1771,9041455

two 192,0237168 182,4379186

three 64,9959483 76,6566290

four 49,1707292 54,5581132

five 39,2020515 46,3242438

six 38,3495284 34,4611558

seven 34,6085118 29,6500457

It is observed that for both models of the link 1 and 2 have been a large reduction in the first embodiment of the SSE for the second and from second to third, however, for the other reduction sequences is somewhat lower, meaning that the improvement in estimation is not significant, so we selected models with only five terms, ensuring a small estimated error and a simplicity in the model to estimate at each sampling period.

The equation (21) and (22) show the models selected for the link 1 and 2 respectively:

1 1 1 1 1

1( )1 1( 1)2 1( 2)6 1( 2)7 2( 1)34 2( 2) 2( 2) 1( )

y k y k y k u k u k y k u k e k (21)

2 2 2 2 2

2( )3 2( 1)7 2( 1)8 2( 2)16 1( 1) 2( 2)33 2( 2) 2( 1) 2( )

y k y k u k u k y k u k y k u k e k (22)

The Fig. 1 and Fig. 2 show the comparison between real and estimated output of the link 1 and 2 respectively, using the equation 1 and 2.

Fig. 6 Comparison between real and estimated output of link 1

Fig. 7 Comparison between real and estimated output of link 2

V. CONCLUSIONS

The models obtained in the equation (21) and (22) are multivariable, nonlinear, and the estimates at each sampling period of the output signal of the link 1 and 2 seen in Fig. 6 and Fig. 7 are very close to the actual output, therefore, it is concluded that these models are suitable for the design of controllers, and which are better suited than linear models, since the ERR algorithm made a selection of the best in terms linear and nonlinear of the equation (20).

ACKNOWLEDGMENT

I thank will CNPQ for research funding

REFERENCES

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