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[.]

Tạp chí Khoa học Kỹ thuật – ISSN 1859-0209 IDENTIFYING THE MOTION LAW OF CYLINDERS ACCORDING TO GIVEN TRAJECTORY OF THE REINFORCED CONCRETE CUTTING DISC MOUNTED ON HYDRAULIC EXCAVATOR Trong Cuong Le1, Quang Hung Tran1 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 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, bucketarm 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 this requires solving the problem of backward kinematic of a excavator with a replacement its bucket by a reinforced concrete cutting disc Kinematic of hydraulic excavator mounted on concrete cutting disc The kinematic model of hydraulic excavator attached concrete cutting equipment is shown in Fig 19 Journal of Science and Technique – ISSN 1859-0209 Fig Kinematic model of hydraulic excavator fitted with reinforced concrete cutter Considering the excavator is a system consisting of 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 Mechanical parameters of excavator, ai=Oi-1Oi Link i i di qi 0 0 a1 a2 a3 a4 0 0 1 2 3 4 The Denavit-Hartenberg matrix is relatively between link (i) and (i-1): cosi -cos isini sin isini a i cosi   sin cos cos -sin cos a sin  i i i i i i i Dii-1 =  với i    sin i cos i di    0   20 (1) Tạp chí Khoa học Kỹ thuật – ISSN 1859-0209 The transformation matrix of k link with fixed coordinates is calculated: c1 k s k k i D0 = Di-1   k  i=1   -s1 k c1 k 0 a1c1   a k c1 k  a1s1   a k s1 k  với    c1 k =cos(1 + +k )  s1 k =sin(1 + +k ) (2) From there, the coordinates of the cutting disc center O3 are determined:  xO3 yO3   a1c1 +a 2c12  a 3c123 h +a1s1 +a 2s12  a 3s123  (3) Notation of the generalized vector of the coordinate system   [ 1 2 3 ]T and the vector of coordinates of the center of the disc x   x O3 yO3  The results of forward T kinematic x  () are based on: ()= 1 () 2 ()   x O3 T yO3  T (4) 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: x    J()  (5) where J()- 2x3 matrix Jacobi, with: J()    J11 J12    J 21 J 22 J13  J 23  (6) in which Jij  i /  j , J11= a1s1  a 2s12  a 3s123 , J12= a 2s12  a 3s123 , J13= a 3s123 , J21= a1c1  a 2c12  a 3c123 , J22= a 2c12  a 3c123 , and J23= a 3c123 Suppose the inverse matrix of the rectangular matrix J() has the form: J  ()  J T () J()J T ()  1 (7) Multiply the two expressions of equation (5) with J  ((t)) , we get: J  ((t))x(t)  (t) (8) The accelerating vectors of the generalized coordinates are defined by the two-sided derivatives Eq (8): (t)  J  ((t))x(t)  J  ((t))x(t) (9) 21 Journal of Science and Technique – ISSN 1859-0209 Define the matrix J  ((t)) as follows: J  ()J()J T ()  J T () (10) Two-sided derivatives of expression (10), receive: J  ()J()J T ()  J  () J()J T ()  J()J T ()   J T () (11) Transformation (11) receives the matrix J  ((t)) :   J  ()  J T ()  J  () J()J T ()  J()J T ()  J()J T ()  1 (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 t k 1  t k  t with k = 1,2,…,N-1 Apply the Taylor expansion to  k 1 round  k , the received: k 1    t k  t   k  k t  k  t   (13) Substituting Eq (9) into Eq (13) and neglecting infinitely greater than 1, Eq (13) becomes: k 1  k  J  (k )xk t với k=1,2,…,N-1 (14) So we have found the law of angular displacement of the driven cylinder 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: xM k k yM 0  rOk 1  A rM T (15) where Ok-1 - joint of k link with k-1 link; A 0k - the rotation matrix of the k link is defined as the transfer matrix D 0k ; rMk - 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: 22 Tạp chí Khoa học Kỹ thuật – ISSN 1859-0209 rC0 =  x C yC 0  c1 -s1  lCcosα C  lCs1C  T 0 =rO1 +A10 rC1 = 0  +  s1 c1   lCsinα C  =  lCs1C         0   0      (16) Where C  CO0O1; lC  AC Displacement of the boom cylinder: x1  x  x B    y C  y B   B0 C0  C x1x  x1y  B0 C0 2 (17) where B0 C0 - 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  c1 -s1  l D cosα D  l Ds1D  rD =  x D y D 0 =rO1 +A rD = 0  +  s1 c1   l Dsinα D  =  l D s1D         0   0      a1c1  c12 -s12   l E cos E  a1c1  l E c12E  2 rE =rO2  A rE   a1s1    s12 c12  l E sin  E    a1s1  l Es12E                 T 1 (18) (19) where D  DO0O1; E  EO1O2 ; lD  AD; lE  FE Displacement of the bucket-arm cylinder: x2  x  x D    y E  y D   D0 E  E x x  x y  D0 E 2 (20) where D E - 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(x G,yG) and point I(xI,yI) coordinates are required Point G on link is determined by the formula: a1c1  c12 rG0 =rO2  A 02 rG2   a1s1    s12       -s12 c12 0   lG cos G  a1c1  lG c12G   lG sin  G    a1s1  l G s12G           0 (21) Point I of the four interlocking mechanism is determined by the coordinates of point H on link and point J on link 3, which is determined: 23 Journal of Science and Technique – ISSN 1859-0209 a1c1  l H c12H   l H sin  H    a1s1  l Hs12H           0 (22) a1c1 +a c12   lJ cos J  a1c1 +a 2c12  l J c123J  rJ0 =rO3  A 30 rJ3   a1s1 +a 2s12   A 30 l J sin  J    a1s1 +a 2s12  l J s123J              0 (23) a1c1  c12 rH0 =rO2  A 02 rH2   a1s1    s12       0   l H cos H  -s12 c12 where G  GO1O2 ; H  HO1O2 ; J  JO2O3 ; lG  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:  x  f f  y   I x I  H B A2 H fB   y  f  f x  I A B I 2 2 2  rJ  x J  y J  rH  x H  y H f  A 2(y J -y H )   x -x f B  J H 2(y J -y H )    x  f (f  y )  f   f (f  y )  x  r   I  H B A H   B   B A H   H  H  (24) Displacement of the cutting disc cylinder: x3  x  x G    yI  yG   I x 3x  x 3y  G I 2 (25) where G I - 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: xk  x k1x k1  yk1yk1   x k x k  yk yk  xk 2 2  x k x k1x k1  x k1  y k1y k1  y k1  (x k x k  x k  yk yk  yk )  xk   xk    x x x  y y  (x x  y y )   k k1 k1 k1 k1 k k k k  24 (26) (27) Tạp chí Khoa học Kỹ thuật – ISSN 1859-0209 where k = 1, 2, and k1 are the index of the endpoint and k2 is the index of the corresponding starting point of the cylinders, ie with k = (11 is point C, 12 is point B); k = (21 is point E, 22 is point D) and k = (31 is point I and 32 is point G) 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    Fig The trajectory of the center of the cutting disc x O3 = 5.6895-0.035t x = 5.6895-0.035t ; (II) O3 y O3 =1.2941+0.035t y O3 =1.2941 x =5.3395-0.35cos(t/T)  x = 5.6895; (III)  O3 (IV) O3 y O3 =1.2941+0.35sin(t/T)  y O3 = 1.2941+0.035t (I) (28) 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 Fig Law of angular displacement of cylinders 25 Journal of Science and Technique – ISSN 1859-0209 The disk center position error, figure 4, for all four cases, does not exceed 10-4m, T c c  is calculated according to the value  of the law of the (The value xc   x O3 yO3 T kinematics model (see Section 2), and x   x O3 yO3  is the law mentioned in Eq (28)) Fig Location error of center disc cutting over time 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 Nguyễn Văn Khang Động lực học hệ nhiều vật KH&KT, Hà Nội, 2007 Nguyễn Văn Khang, Vũ Liêm Chính, Phan Nguyên Di Động lực học máy Hà Nội, 2001 Nguyễn Doãn Phước Lý thuyết điều khiển nâng cao KH&KT, 2005 T R Kurfess Robotics and Automation Handbook CRC Press, 2005 J J Craig Introduction to Robotics: Mechanics and Control Pearson Prentice Hall, New Jersey, 2005 26 Tạp chí Khoa học Kỹ thuật – ISSN 1859-0209 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 xi lanh dẫn động cần, tay gầu, đĩa cắt trình cắt theo quỹ đạo cho trước sở xây dựng mơ hình động học máy đào thủy lực lắp thiết bị cắt bê tông cốt thép Kết tính tốn nhận có sai số nằm 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  27