• No results found

Vol 60, No 3 (2017)

N/A
N/A
Protected

Academic year: 2020

Share "Vol 60, No 3 (2017)"

Copied!
6
0
0

Loading.... (view fulltext now)

Full text

(1)

TECHNICAL UNIVERSITY OF CLUJ-NAPOCA

ACTA TECHNICA NAPOCENSIS

Series: Applied Mathematics, Mechanics, and Engineering Vol. 60, Issue III, September, 2017

GEOMETRICAL MODELLING USING MATRIX EXPONENTIAL

FUNCTIONS FOR A SERIAL ROBOT STRUCTURE

Claudiu SCHONSTEIN, Iuliu NEGREAN, Nicolae PANC

Abstract: In the paper, on the basis of formulations based on matrix exponential will be established the

direct and inverse geometry equations. The matrix exponentials functions in robotics will be applied for a serial robot structure. Unlike classical algorithms, the application of the exponential matrix functions presents some advantages in determining the direct geometry equations. The results obtained in geometrical modeling, are representing input data for kinematical modeling, all those being important data in the study of dynamic behavior of any mechanical robot system.

Key words: Matrix exponentials functions, mathematical modeling, geometry, control.

1.INTRODUCTION

Nowadays, significant resources are being spent in the development of mechanical robotic systems. A part of this development is based on mathematical studies, having the role of using new approaches for establishing and studying the parameters which are defining the dynamic behavior of any mechanical system.

Transfer matrix equations, specific to a serial kinematic chain, having rotation and translation kinematic joint, from the robot's mechanical structure, can be expressed by using Exponential Matrix Functions (ME), [1]-[4].

2. THE GEOMETRY EQUATIONS

In this section, will be presented the geometry equations for a serial structure, by using the matrix exponentials used for determine the locating matrices, according to [1]-[4].

2.1 Establishing of Locating Matrices by using of Matrix Exponentials

The matrix exponentials and their associated transformations are included in the algorithm of matrix exponentials (MEG) devoted to direct geometry equations, according to [2] and [4]. The main steps of the algorithm are presented in the following.

The matrix of the nominal geometry Mvn( )0 , corresponding to configuration θ( )0 is known:

( )

( )

[ ]

( ) ( )

[

]

{

0T

}

T

i T 0 i 6 1 n 0

vn Matrix p k i 1 n 1

M = = → +

×

+ .(1)

On the basis of this matrix is represented the kinematical structure of the robot, taken in study. The matrix of the nominal geometry is completed with the screw parameters

{

( ) ( )0

}

i 0 i ;v

k

also named the homogeneous coordinates. The new matrix is symbolized by ( )0**

vn

M , according to:

( ) ( )

[ ]

( ) ( )

[

]

{

0T

}

T

i T 0 i 6 1 n 0

vn Matrix p k i 1 n 1

M = = → +

×

+ . (2)

In keeping with MEG Algorithm, an outward loop is opened

(

i=1n

)

. The matrix exponentials devoted to direct geometry equations (DGM equations) are bellow obtained. The differential matrix Ai has the same expression for the both configurations θ( )0 and θ . Considering [2] and [4], this matrix shows as:

{

}

{

}

{

}

(

)

 

  

 

 

  

 

   

  

× × +

=

   

  

×

=

0 0

0 0

k 1 k

p k

0 0

0 0

v k

A

) 0 ( i i i

) 0 ( i ) 0 ( i i ) 0 ( i

) 0 ( i i ) 0 ( i i

∆ ∆

(2)

where

{

ki = zi;vi =

{

pi×

}

zi⋅ ∆ +i

(

1− ∆i

)

zi

}

are the screw parameters or homogeneous coordinates of the driving axis (i), according to [5] and [6], which by generalization becomes

{

ki =

{

x y zi; ;i i

}

;vi=

{

pi×

}

ki⋅ ∆ +i

(

1− ∆ ⋅i

)

ki

}

(4) The exponential of rotation matrix is:

( )

{

}

{

}

(

)

(

)

{

( )

}

(

)

{

( ) ( )

[

(

)

]

}

[ ]

             ≡ ⋅ − ⋅ + + ⋅ × + ⋅ ⋅ ≡ ≡ ⋅ ≡ ⋅ ⋅ × − R q c 1 k k q s k q c I q ; k R q k exp 1 i i i i T 0 i 0 i i i 0 i i i 3 i i i i i 0 i ∆ ∆ ∆ ∆ ∆

. (5)

The inverse of the exponential of rotation matrix is also expressed, in keeping with the next:

( )

{

}

{

}

(

)

(

)

{

( )

}

(

)

{

( ) ( )

[

(

)

]

}

[ ]

             ≡ ⋅ − ⋅ + + ⋅ × + ⋅ ⋅ ≡ ≡ ⋅ ≡ ⋅ × −

1 T i i T i i T 0 i 0 i i i 0 i i i 3 i i i T i i 0 i R q c 1 k k q s k q c I q ; k R q k exp ∆ ∆ ∆ ∆ ∆ . (6)

The defining expression for the column vector bi, is established with the following:

{

}

[

(

)

]

{

(

)

[

]

}

        ⋅ ⋅ − ⋅ ⋅ + + ⋅ − × + ⋅ = ) 0 ( i i i i T ) 0 ( i ) 0 ( i i i ) 0 ( i i 3 i v q s q k k q c 1 k q I b ∆ ∆ . (7)

Another matrix exponential, having a great significance for locating transformation, shows as:

( )

{

}

( )

{

}

{

}

                        × = =                 × = 1 0 0 0 b q k exp q 0 0 0 0 v k exp e i i i ) 0 ( i i 0 i 0 i i q i A ∆ ; (8)

{ }

{ }

{ }

{

{

( )

}

}

{ }

p exp

{

{

k( )

}

q

}

b .

exp ; q k exp R exp where ; 1 0 0 0 p exp R exp q A exp i 0 j 1 j i 0 k k k 0 k i 0 j j j 0 j i 0

j j j

= + = = = ⋅       ⋅ × = ⋅ × =       =       ∑ ⋅ ∆

∆ (9)

The inverse of the matrix exponential (8) is:

{ }

{ }

{ }

{

{

( )

}

}

{ }

{

{

( )

}

}

                             × − − = × − =       =               ∑ ⋅ −

= = = = 0 i j j j i k k k 0 k 0 i j j j 0 j 0 i j j j

b q k exp p exp q k exp R exp where 1 0 0 0 p exp R exp q A exp

. (10)

The exponentials expressions for the locating matrices, which define the position and orientation of the

{ }

n and

{

n+1

}

with respect to fixed frame

{ }

0 , are obtained as follows:

{

}

         + =       = =

= − 1 n ; n x where 1 0 0 0 p R T

T x0

x 1 i 1 ii 0

x ; (11)

( ) ( ) ( )

{

}

( ) ( )

{

}

( )

{

}

( )

{

} {

}

{

0; x n ;1; x n 1

}

. and ; p q k exp b q k exp p ; R q k exp R where ; T q A exp T e T x x 0 n 1

i i i

0 i n 1 i i 1 i 0

j j j

0 i 0 0 x n 1

i i i

0 i 0 x n 1 i 0 0 x n 1 i i i 0 0 x i q i A 0 x + = = =          ∑⋅ × + +                   ∑ × = ⋅       ∑ × ⋅ = ⋅       ⋅ = ⋅       = = = − = = = = ⋅

δ δ ∆ ∆ ∆ (12)

The inverse of the locating matrix (11) is:

                = =

= − − − 1 0 0 0 p R R T T T 0 x T 0 x 1 x i 1 1 ii 1 0

x ; (13)

( )

{

}

(

)

( )

{

}

                   ⋅ − ⋅ = = ⋅ − ⋅ =

= − = − − 1 n i i i 1 0 0 x 1 n i i i 1 0 0 x 1 0 x q A exp T q A exp T T

; (14)

and: ( )

{ }

{

( )

}

            ∑ × − ⋅ = = 1 n

i i i

0 i T 0 0 x T 0

x R exp k q

R;(15)

( )

{

}

( )

{

}

( )               ⋅                ∑ × − − −               ∑ × − = ⋅ − = = =−

x 0 1 n

i i i

0 i 1 n i i 0 1 i

j j j

0 i T 0 x p q k exp b q k exp p R δ ∆ ∆ .(16)

Remark: The MEG Algorithm, due to

computational advantages and independent of the reference can be applied for any robot structure.

3. APPLICANTION

(3)

3.1 Direct Geometry Modeling of 2TR robot For the proposed structure, according to, [1]-[4], algorithms based on the matrix calculation with exponential functions will form the basis of the homogeneous transformations in the Direct Geometry Modeling (DGM), and the exponential of the location matrix between the systems { }0 ® { }4

is determined first. These are characterizing the position and orientation (or location) of the final effector relative to the system attached to the fixed base of the robot. Thus, according to the MEG algorithm, the following steps are performed to determine the position and orientation of the end-effector.

• There is determined the nominal geometry matrix, ( )0

vn

M , which is describing the

( )0

q configuration of the robot, which for the

mechanical system taken into study is presented in Table 1.

( )0

2 vn

M **Î TR Table 1

As it can be observed from the Table 1, the screw parameters

{

( )0 ( )0

}

;

i i

k v are contained in

the nominal geometry matrix . [7]

• Is represented the kinematic schema of the 2TR type robot, in Figure 2, according to Table 1.

• According to the algorithm, is opened an external cycle for

(

i= ®1 3

)

. Within the cycle

is determined the matrices and exponential functions, specific to the direct geometry modeling, in accordance to (3)-(8).

For the first element, representing a prismatic joint, there are determined the following expressions for exponential matrix functions:

( )

{

0

}

( )0

1 1

1

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0

k v

A

 

 

×

 

 

= ≡

 

 

 

 

; (17)

( )

( )

{

}

{

}

0 1 1 1

3 0

1 1 1

1 0 0 0 1 0 exp 0 0 1

k q

e

l

k q

 

× ⋅ ⋅∆

   

 

 

   

≡ =

 

× ⋅ ⋅ ∆

 

 

 

(18)

( ) 1

1 0

3 1 1

1

1 0 0 0 0

0 1 0 0 0

0 0 1 1

b

q l q v

q

     

 

       

= ⋅ ⋅ =

      

⋅ ⋅

 

         

(19)

( )

{

}

( )

1 1

0 0

1 1

1 1

1 0 0 0 0 1 0 0 0 0 1 exp

0 0 0 0 0 0 0 1

A

e

k v

q q

⋅∆  

 

 

 

= ×

 

 

 

 

   

(20) 1

l

1

a

0

O

( )0 4

O

2

q

1

q

3

q

a

s n

0

z

0

x

0

y

2

l

2

a

( )0 2

O

( )0 1

O

( )0 3

O

( )0 1

k

( )0 2

k

( )0 3

k

(4)

According to Figure 2, the second joint of the 2TR structure, is also a prismatic joint, so for the kinetic element, the matrix exponentials are:

( )

{

0

}

( )0

2 2

2

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

k v

A

 

 

×

 

 

= ≡

 

 

 

 

(21)

( )

( )

{

}

{

}

0 2 2 2

3 0

2 2 2

1 0 0 0 1 0

exp 0 0 1

k q

e

l

k q

 

× ⋅ ⋅∆

   

 

 

   

= ≡

 

× ⋅ ⋅ ∆

 

 

 

; (22)

( )

2 2

2 0

3 2 2

1 0 0 1

0 1 0 0 0

0 0 1 0 0

q b

q l q v

     

 

       

= ⋅ ⋅ =

      

⋅ ⋅

 

         

(23)

{

}

( )

{

}

( )

( )

(

)

=

 

= ×

 

 

 

 

 

 

 

 

= =

 

 

 

2 2

2 2

0 0

2 2

2

2 0

2 2

2

exp

exp

0 0 0 0

1 0 0 ; 0 1 0 0

0 0 1 0 0 0 0 1

0 0 0 1

A q

e A q

k v

q

q

R k q b

. (24)

For the third kinetic element, consisting of a rotation joint, are determined the following expressions:

( )

{

0

}

( )0

1

3 3

3

1

0 0 0 0

0 0 1

0 0 0

0 0 0 0

0 0 0 0

k v l

A

a

 

 

×

 

 

= ≡

 

 

 

 

; (25)

( )

( )

{

}

{

}

( )

{

}

{

( )

}

(

)

0 3 3

0 3 3

2

0 0

3 3 3 3 3

exp

1

k q

e

k q

l k sq k cq

 

× ⋅

   

 

 

 

=

 

× ⋅

 

 

= + × ⋅ + × ⋅ −

(26)

( )0 3 3

3 3

3 3

1 0 0

0 0

k q

e cq sq

sq cq

 

× ⋅

   

 

 

=

 

 

; (27)

( )

{

}

(

)

( )

{

}

(

)

( )

0

3 3 3 3 0

2

3 0 3

3 3

3

1 3 1 1 3

1 1 3 1 3

1

0

I q k cq

b v

k q sq

a cq a l sq

l a sq l cq

+ × ⋅ +

 

= ⋅ =

+ × ⋅ − 

 

 

 

= ⋅ − + ⋅ 

 

+ ⋅ − ⋅

 

;(28)

{

}

( )

{ }

( )

(

( )

)

3 3

3 3

0

3 3 3

0 0

3 3

3

3 3 1 3 1 1 3

3 3 1 1 3 1 3

exp

; exp 0 0 0 1

0 0 0 0

1 0 0 0 0

0

0 0 0 1

A q

e A q

R k q b

k v

q

cq sq a cq a l sq

sq cq l a sq l cq

=

   

= =

×

 ⋅  

 

 

 

 

 

− ⋅ − + ⋅

 

=

+ ⋅ − ⋅

 

 

(29)

For the initial configuration θ(0) , the homogeneous transformation matrix between

{ } { }

0 → 4 reference frames, is determined as:

( ) ( ) ( )

+

 

 

  

≡ = 

 

 

 

 

 

2 2

0 0

1 40

0 40

1

0 0 1

1 0 0

0 1 0 0 0 0 1

0 0 0 1

l a

a

R p

T

l (30)

In keeping with (9)-(10), the situation matrix between

{ } { }

0 → 4 reference frames, which

expresses the position and orientation of the end effector, is expressed as follows:

[ ]

( )

[ ]

( )

( )

= ⋅

=

 

=

 

 

 

= ⋅

   

 

4

0

1 3

4

1

0 1

0 0 40

4

0 0 0 1

ii i

i A qi i

i

T T q

e T

R p (31)

where:

[ ]

{

( )

}

( )

=

  

=   × ⋅ ⋅ ∆ ⋅

 

3

0 0 0

40 4

1

exp i i i

i

R k q R (32)

( ) ( )

( )

  

 

× ⋅ ⋅∆

    × ⋅ ⋅∆

 

=   

  =

=

 

 

 

=  ⋅ ⋅ ⋅

 

 

 

1 0 3

0 3

0

0 1

1 i

kj q j j k qi i

i

j i

i i

(5)

Therefore, on the basis of (31)-(33) by performing matrix calculations, the following expressions are obtained for the 2TR structure:

[ ]

[ ]

[

]

( ) ( )

 

× ⋅

   

 

 

 

= = ⋅ = − 

   

 

 

0 0

3 3

0 4 0

3 3

40 4

3 3

0 0 1

0

0

k q

R

T e R cq sq

n s a

sq cq

(34)

( )

( )

(

)

( ) ( )

( )

 

×

   

   

× ×

       

   

= = + ⋅ + +

   

+ +

 

 

+ ⋅ ⋅ =

+

 

0 1 1 0

1 2 3

4

0 0 2 2 2

1 3

1 3 0

1

1 1

x k q

y

z

k q k q

p

p p b e b b

p

a d q

e e p a

l q

(35)

which represents the resultant matrix and the position vector, these being included in the following expression of the column vector of the operational variables [4]:

(

)

(

)

θ α β γ

 

 

= =

 

 

0

T

x y z

T

z y z

p p p

p

X (36)

known as direct geometry equations (DGM equations).

According to [4], in order to establish the orientation angles, αz, βy, γz for exact determination of the values, there is used the trigonometric functionAtan 2, defined by:

(

)

[

]

{

}

[

]

{

}

[

]

{

}

[

]

{

}

; 0; 0 ;

/ 2 ; 0; 0

tan2 ;

; 0; 0 ;

/ 2 ; 0; 0

s c

s c

x A s c

s c

s c

α α α

π α α α

α α

π α α α

π α α α

>

 

+ > <

 

 

= = 

+ < <

 

 

− + < ≥

 

 

(37)

Hence, in keeping with (37), results:

(

)

( )

(

)

(

3 3

)

3

tan2 ; tan2 1;0 2; 2;

tan2 ; tan2 ;

z z z

y

z z z

A s c A

A s c A sq cq q

α

α α

π

β

π

γ

γ γ

π

= = =

=

= = − − = +

(38)

The column vector of the operational coordinates, defined by (36), becomes [7]:

(

)

(

)

[

]

0

2 2 2

1 3

1 1

... ...

2 2

T

x y z

T

z y z

T

T

p p p p

X

a d q

a q

l q

ψ α β γ

π π π

 

 

 

= = =

 

 

 

 + +  

  

= + 

+

 

 

(39)

and characterizes the direct geometric modeling of the 2TR type robot taking into account.

3.2 Inverse Geometry Equations of 2TR serial structure

The inverse geometric modeling (MGI), or modeling of geometric control functions, consists of determining the column vector of generalized coordinates

(

q q q1, 2, 3

)

, which are

characterizing the movement in each kinetic link, expressed as:

[

]

{

}

( )

(

)

1 2 3

1 0 1

T

T

x y z x y z

q q q

f X f p p p

θ

α β γ

− −

≡ =

 

= ≡

(40)

In inverse geometric modeling, the position of the mobile system attached to the final effector in the characteristic point is known by numerical values. As an observation, the number of operational parameters varies, but cannot exceed the number of degrees of freedom. In keeping with (39), results:

2 2 2

1 1

x

z

q p a d

q p l

= − −

 

 

= −

  (41)

According to previous expression, the displacement on y0 axis is absent, namely:

1 .

y

p = −a =cst (42)

From the same relations belonging to the direct geometry, the third generalized coordinate is determined as being:

3 z

q =

γ

π

(43) The expressions (41) and (43) are the geometric control functions corresponding to the input data relating to the position of the characteristic point in the Cartesian space.

4. CONCLUSION

(6)

mathematical methods that are helping to establish the connection between the elements that determine the location of the end effector in the Cartesian space. Thus, by applying matrix methods based on matrix exponentials, were presented the direct geometry equations, which express the position and orientation of the characteristic point of the end effector with respect to the fixed reference system attached to the robot base.

As an important remark, is that the approach of geometry with exponential functions leads to essential advantages on the one hand, due to the writing in a compact form, easy to visualize in geometrical form, on the other hand it is noted as an essential advantage the lack of reference systems which introduces geometric restrictions, and in the precision study could lead to additional geometric errors.

5. REFERENCES

[1] Negrean, I., Negrean, D. C., Matrix Exponentials to Robot Kinematics, 17th International

Conference on CAD/CAM, Robotics and Factories of the Future, CARS&FOF 2001,Vol.2, pp. 1250-1257, Durban, South Africa, July 2001.

[2] Negrean,I.,Negrean,D.C.,The Locating Matrix Algorithm in Robot Kinematics, Cluj-Napoca,

October 2001, Acta Technica Napocensis, Series: Applied Mathematics and Mechanics. [3] Negrean, I., Schonstein, C., „Advanced studies

on matrix exponentials in robotics,” Acta

Technica Napocensis, Series: Applied Mathematics and Mechanics, Nr. 53, Vol. I, 2010, ISSN 1221-5872, pp. 13-18, Cluj-Napoca, Romania.

[4] I., Negrean, Mecanică Avansată în Robotică,

Editura UT PRESS, Cluj-Napoca, 2008, ISBN 978-973-662-420-9.

[5] Bernstein, D.S., So, W.G., Some explicit Formulas for the Matrix Exponential, IEEE

Transaction on Automatic Control, Vol. 38, 1993. [6] Park, F.C., Computational Aspects of the

Product-of-Exponentials Formula for Robot Kinematics,

IEEE Transaction on Automatic Control, Vol. 39, No. 3, 1994.

[7] Schonstein, C., Contribuții în dezvoltarea unei structuri robotizate hibride, PhD Thesis,

Cluj-Napoca, 2011.

Modelare geometrică cu funcţii exponenţiale de matrice pentru o structură serială de robot

Rezumat: În lucrare, pe baza formulărilor bazate pe exponenţiale de matrice se vor stabili ecuaţiile

geometriei directe şi inverse. Funcţiile exponenţiale de matrice în robotică vor fi aplicate în modelarea

matematică a unei structuri de robot serial. Spre deosebire de algoritmii clasici, aplicarea funcţiilor

exponenţiale de matrice prezintă câteva avantaje în determinarea ecuaţiilor geometriei directe. Rezultatele

obţinute în modelarea geometrică, reprezintă date de intrare pentru modelarea cinematică, toate acestea fiind

date importante în studiul comportării dinamice a oricărui sistem mecanic robotizat.

Claudiu SCHONSTEIN, Lecturer, Ph.D, Eng, Technical University of Cluj-Napoca, Department of

Mechanical Systems Engineering, E-mail: schonstein_claudiu@yahoo.com, No.103-105, Muncii Blvd., Office C103A, 400641, Cluj-Napoca.

Iuliu NEGREAN, Professor, Ph.D., Head of Department of Mechanical Systems Engineering, Technical

University of Cluj-Napoca, iuliu.negrean@mep.utcluj.ro, No.103-105, Muncii Blvd., +40-264-401616.

Nicolae PANC, Lecturer, Ph.D, Eng, Technical University of Cluj-Napoca, Department of Manufacturing

Figure

Figure 2 Kinematic schema of the 2TR serial robot Figure 1  The serial robot structure 2TR

References

Related documents

Political Parties approved by CNE to stand in at least some constituencies PLD – Partido de Liberdade e Desenvolvimento – Party of Freedom and Development ECOLOGISTA – MT –

For applicants who take over an existing centre, the minimum staffing to meet licensing criteria will apply and the applicant must ensure that there are sufficient trained teachers

The distributor may seek to limit the producer’s right to terminate until distributor has recouped its advance (assuming it has given the producer an advance.) Another

It is possible that a segment is stored at multiple peers in the set cover, in this case the requesting client can pick a peer based on some other criteria such as delay or number

• Storage node - node that runs Account, Container, and Object services • ring - a set of mappings of OpenStack Object Storage data to physical devices To increase reliability, you

Other ways that equality and diversity has been promoted in the college to raise awareness included College events to celebrate success and achievement such as Supported

In the previous sections, we dis- cuss the expectation that a neural network exploiting the fractional convolution should perform slightly worse than a pure binary (1-bit weights

Recording Data for New or Revised Course (Record only new or changed course information.) Course prefix (3 letters) Course Number (3 Digits) Effective Term (Example: