474-Article Text-1982-1-10-20221027.Pdf

ĐỊNH HƯỚNG VÀ ĐIỀU KHIỂN ANTEN MẠNG PHA Tạp chí Khoa học và Kỹ thuật – ISSN 1859 0209 19 IDENTIFYING THE MOTION LAW OF CYLINDERS ACCORDING TO GIVEN TRAJECTORY OF THE REINFORCED CONCRETE CUTTING DISC M[.]

IDENTIFYING THE MOTION LAW OF CYLINDERS ACCORDING

TO GIVEN TRAJECTORY OF THE REINFORCED CONCRETE CUTTING DISC MOUNTED ON HYDRAULIC EXCAVATOR

Trong Cuong Le 1 , Quang Hung Tran 1

1Le Quy Don Technical University

Abstract

This paper presents a method of determining the displacement law of cylinders driving boom, bucket-arms, cutting disc in the cutting process in a given trajectory based on the formulated kinetic model of hydraulic excavator with reinforced concrete cutting equipment Calculation results received with errors within the allowable range

Keywords: Hydraulic excavator kinematics; reinforced concrete cutting disc

1 Introduction

Hydraulic excavator equipped with reinforced concrete cutting disc is used for cutting girder and reinforced concrete slabs in the case of construction, renovation and demolition of reinforced concrete structures The cutting disc is inserted into the bucket position by means of a cutting disc set The cutting disc is driven by a hydraulic motor Moving the cutting disc during the work by the boom, bucket-arm and cutting disc cylinders (Fig 1) should be in accordance with the actual cut requirements, cut thickness, conductivity The problem is the simultaneous control of the boom, bucket-arm and cutting disc cylinders to meet the cutting trajectory and the required cutting speed with the allowable tolerance Controlling the reinforced concrete cutting process

in the cab or remote can be difficult to achieve because the error is too large It is necessary to programmatically control the cylinders in a given cutting path To solve this problem, it is necessary to determine the motion of the cylinders in the given trajectory of the cutting disc To do this requires solving the problem of backward kinematic of a excavator with a replacement its bucket by a reinforced concrete cutting disc

2 Kinematic of hydraulic excavator mounted on concrete cutting disc

The kinematic model of hydraulic excavator attached concrete cutting equipment

is shown in Fig 1

Fig 1 Kinematic model of hydraulic excavator fitted with reinforced concrete cutter

Considering the excavator is a system consisting of 5 links absolutely hard: chassis-cabin (0); boom (1); arm (2); cutting disc rack (3); cutting disc (4) These units are linked together by hinge joints Oi-1 The fixed coordinate system is O0x0y0z0 is located at the rotating floor axis Coordinate systems O1x1y1z1, O2x2y2z2, O3x3y3z3 and

O4x4y4z4 originate at the point of attachment between the machine and the cutting device, Oizi axes are perpendicular to the plane of expression

Tab 1 Mechanical parameters of excavator, a i =O i-1 O i

The Denavit-Hartenberg matrix is relatively between link (i) and (i-1):

i

i-1

cos -cos sin sin sin a cos sin cos cos -sin cos a sin

D =

The transformation matrix of k link with fixed coordinates is calculated:

1 k 1 k 1 1 k 1 k k

1 k 1 k 1 1 k 1 k

0 i-1

i=1

 



c =cos( + + )

s =sin( + + )





From there, the coordinates of the cutting disc center O3 are determined:

xO3 yO3  a c +a c1 1 2 12a c3 123 h +a s +a s0 1 1 2 12a s3 123 (3)

Notation of the generalized vector of the coordinate system T

1 2 3

   

vector of coordinates of the center of the disc  T

O3 O3

x y



x The results of forward kinematic x ( ) are based on:

The goal of the backward kinematics problem is to determine the values of  to obtain x on demand, ie to establish a relation 1( )x From Eq (4), take the derivative of two- sides of the equation x ( ) over time:

J( )





where J()- 2x3 matrix Jacobi, with:

11 12 13

21 22 23

J( )





in which Jij   i/ j , J11= a s1 1a s2 12a s3 123 , J12= a s2 12a s3 123 , J13= a s3 123 ,

J21=a c1 1a c2 12a c3 123, J22=a c2 12a c3 123, and J23=a c3 123

Suppose the inverse matrix of the rectangular matrix J() has the form:

1

J ( )  J ( ) J( )J ( )     (7) Multiply the two expressions of equation (5) with J ( (t))  , we get:

J ( (t)) (t)  x (t) (8) The accelerating vectors of the generalized coordinates are defined by the two-sided derivatives Eq (8):

(t)J ( (t)) (t) J ( (t)) (t) x   x

Define the matrix J ( (t))  as follows:

J ( )J( )J ( )    J ( ) (10) Two-sided derivatives of expression (10), receive:

J ( )J( )J ( ) J ( ) J( )J ( ) J( )J ( )             J ( ) (11) Transformation (11) receives the matrix J ( (t))  :

J ( )   J ( ) J ( ) J( )J ( ) J( )J ( )            J( )J ( )   (12)

To determine (t) in Eq (8) and Eq (9), divide the operating time of the cutting device [0 T] to N by approximately:

t T / N

  , we have tk 1   tk t with k = 1,2,…,N-1

Apply the Taylor expansion to k 1 roundk, the received:

k 1  tk   t k   k t k t 

Substituting Eq (9) into Eq (13) and neglecting infinitely greater than 1, Eq (13) becomes:

k 1  k J ( k)xkt

So we have found the law of angular displacement of the driven cylinder

3 Driving law of the cylinder

The relationship between angular displacement and displacement of cylinders is the law to look for The coordinates of any point M on the unit k are defined as follows:

M M Ok 1 0 M

x y 0 r  A r (15)

where Ok-1 - joint of k link with k-1 link; k

0

A - the rotation matrix of the k link is defined

as the transfer matrix k

0

D ; k M

r - the coordinates of the point M on the mobile coordinate system attached to the k link

3.1 Displacement of the cylinders

a) Displacement of the boom cylinder

Point B has a constant coordinate [xB yB], the coordinates of point C(xC yC) are determined:

 T 1 1 C C C 1C

(16)

Where   C CO O ; l0 1 CAC

Displacement of the boom cylinder:

x  x  x  y  y  B C  x  x  B C (17)

where B C0 0 - the distance between points B and C at the beginning

b) Displacement of the bucket-arm cylinder

To determine the displacement of the bucket-arm cylinder, the coordinates D(xD,yD) and point E(xE,yE) are required Because point D on link 1, and point E on link 2, should:

0 c -s 0 l cosα l s

r = x y 0 =r +A r = 0 + s c 0 l sinα = l s

(18)

a c c -s 0 l cos a c l c

r =r A r a s s c 0 l sin a s l s

(19)

where   D DO O ;0 1   E EO O ; l1 2 DAD; lEFE

Displacement of the bucket-arm cylinder:

where D E0 0- distance between D and E points at start time

c) Displacement of the cutting disc cylinder

To determine the displacement of the cutting disc cylinder, a point G(xG,yG) and point I(xI,yI) coordinates are required Point G on link 2 is determined by the formula:

(21)

Point I of the four interlocking mechanism is determined by the coordinates of point H on link 2 and point J on link 3, which is determined:

1 1 12 12 H H 1 1 H 12H

a c c -s 0 l cos a c l c

r =r A r a s s c 0 l sin a s l s

(22)

a c +a c l cos a c +a c l c

r =r A r a s +a s A l sin a s +a s l s

(23)

where   G GO O ;1 2   H HO O ;1 2   J JO O ; l2 3 GFG; lH FH; lJ KJ

By the analytical method we find the coordinates of point I which is the intersection of the center circle J(xJ,yJ) of the radius rJ = IJ and the center circle H(xH,yH)

of radius rH = HI:

B

A

J H

J H B

J H

x f f y x

f 1

y f f x

r x y r x y f

2(y -y )

x -x f

2(y -y )

x f (f y ) f 1 f (f y ) x r





 





























Displacement of the cutting disc cylinder:

x  x  x  y  y  x  x  G I (25)

where G I0 0- the distance between points I and G at the initial time

3.2 Velocity and acceleration of motion of cylinders

Based on the law of motion of the cylinders shown in Eq (17), Eq (20) and

Eq (25), the velocity and acceleration of the cylinders are as follows:

k

x 1

x x y y x x y y x

k

k k1 k1 k1 k1 k1 k1 k 2 k 2 k 2 k 2 k 2 k 2 2

k k k1 k1 k1 k1 k 2 k 2 k 2 k 2

x

1





(27)

where k = 1, 2, 3 and k1 are the index of the endpoint and k2 is the index of the corresponding starting point of the cylinders, ie with k = 1 (11 is point C, 12 is point B);

k = 2 (21 is point E, 22 is point D) and k = 3 (31 is point I and 32 is point G)

4 Results and discussion

Structural parameters: a1 = 4.015 m;

a2 = 1.887 m; a3 = 0.450m; 1 = π/6; 2 = 7π/4;

3 = 23π/12; xB = 0.535 m; yB = 0.235 m;

lC = 2.041 m; lD = 2.579 m; lE = 0.458 m;

lG = 0.48 m; lJ = 0.393 m; C = 0.192;

D = 0.297; E = 2.739; G = 1.396; J = 1.064

Consider four cases: Verticality cut (I),

horizontal cut (II), tilted cut at a 45 degree

angle (IV) and curved cut (IV) with

mathematical expressions describing the law

of change of point O3 as Eq (28) The given

trajectory is shown in Fig 2

Fig 2 The trajectory of the center

of the cutting disc



O3 O3

O3 O3 =

y

x = 5.6895-0.035t x = 5.6895-0.035t

y =1.2941+0.035t y =1.2941

5.6895 x =5.3395-0.35cos(t /T)

1.2941+0.035t y =1.2941+0.35sin(t /T)











Using the calculation steps described above, the law of angular displacement of

cylinders so that the center of the disc is moving according to the given trajectories during T = 30s is shown in Fig 3

Fig 3 Law of angular displacement of cylinders

The disk center position error, figure 4, for all four cases, does not exceed 10-4m, (The value xcxcO3 ycO3Tis calculated according to the value  of the law of the

kinematics model (see Section 2), and xxO3 yO3Tis the law mentioned in Eq (28))

Fig 4 Location error of center disc cutting over time

5 Conclusion

- By building algorithms, we have found intermediate transfer matrices and motion laws of the boom, the bucket-arm and the cutting disc cylinders to meet the coordinates of the disc in the given trajectory;

- Depending on the method of cutting, the value of the error varies, this value is less than 10-4m, within the tolerance limit for the requirement;

- Research results may serve as a basis for the design of automatic control of hydraulic cylinders of excavator mounted on reinforced concrete cutters

References

1 Nguyễn Văn Khang Động lực học hệ nhiều vật KH&KT, Hà Nội, 2007

2 Nguyễn Văn Khang, Vũ Liêm Chính, Phan Nguyên Di Động lực học máy Hà Nội, 2001

3 Nguyễn Doãn Phước Lý thuyết điều khiển nâng cao KH&KT, 2005

4 T R Kurfess Robotics and Automation Handbook CRC Press, 2005

5 J J Craig Introduction to Robotics: Mechanics and Control Pearson Prentice Hall, New

Jersey, 2005

XÁC ĐỊNH QUY LUẬT CHUYỂN ĐỘNG CỦA CÁC XI LANH THEO QUỸ ĐẠO CHO TRƯỚC CỦA ĐĨA CẮT BÊ TÔNG CỐT THÉP

LẮP TRÊN MÁY XÚC THỦY LỰC

Tóm tắt: Bài báo trình bày phương pháp xác định quy luật chuyển vị các xi lanh dẫn

động cần, tay gầu, đĩa cắt trong quá trình cắt theo quỹ đạo cho trước trên cơ sở xây dựng mô hình động học của máy đào thủy lực lắp thiết bị cắt bê tông cốt thép Kết quả tính toán nhận được có sai số nằm trong phạm vi cho phép

Từ khóa: Động học máy xúc thủy lực; đĩa cắt bê tông cốt thép

Received: 08/9/2017; Revised: 26/01/2018; Accepted for publication: 12/6/2018

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