Properties of the Kinematics Model of 5-Axis CNC Machines (Milling Robots)

Nội dung text: Properties of the Kinematics Model of 5-Axis CNC Machines (Milling Robots)

1. Tuyển tập Hội nghị khoa học toàn quốc lần thứ nhất về Động lực học và Điều khiển Đà Nẵng, ngày 19-20/7/2019, tr. 201-205, DOI 10.15625/vap.2019000279 Properties of the Kinematics Model of 5-Axis CNC Machines (Milling Robots) Chu Anh MY* and Nguyen Van CONG *Department of Special Robotics and Mechatronics, Le Quy Don Technical University 236 Hoang Quoc Viet, Hanoi, Vietnam Advanced Technology Center, Le Quy Don Technical University 236 Hoang Quoc Viet, Hanoi, Vietnam E-mail: mychuanh@yahoo.com Abstract A 5-axis CNC machine is similar to two cooperating robots, one robot carrying the workpiece and one robot carrying the tool. The 5-axis CNC machines are also called 5-axis milling robots which have been designed in a large variety of kinematic configurations. This paper addresses a generalized kinematics model and it is properties. First, a generic kinematic chain of 5-axis machine is synthesized as a unified kinematic chain of two collaborative robots in order to formulate the generalized kinematic equation for the machines. Second, two important properties of the kinematics model are proved in a generalized case that are useful for several purposes, i. e machine structure comparison, postprocessor development, etc. Last, it was shown (a) that, by using the properties of the kinematics model, the forward and inverse kinematic equations for the two rotary axes of 5-axis CNC machines can be established in an effective and simplified manner. Keywords: 5-axis CNC machine, Milling robot, Kinematics Modelling 1. Introduction In recent decades, 5-axis CNC machining has been one of the most modern and effective material removal technologies used in manufacturing industries. Several types of the 5-axis CNC machines have been used for machining complex parts such as molds, turbine blades, automotive and aerospace parts whose geometries are (b) typically defined by complex surfaces. In essence, a 5-axis CNC machine is similar to two Fig. 1 a) The 5-axis CNC machine Maho 600e, b) The kinematic cooperating robots, one robot carrying the workpiece and chain of the machine Mahoo 600e one robot carrying the tool. Therefore, the 5-axis CNC Due to the 5-axis machines have been designed in a machines have been also called milling robots. Fig.1 large variety of kinematic structures, establishing a shows the structure and kinematic chain diagrams of a generalized kinematics equation for all the machines is 5-axis CNC machine Maho 600e. challenging. A typical 5-axis CNC machine has three translational Years ago, a number of efforts [1-12] have been joints (axes) X, Y and Z, and two rotary joints (axes) AB, made to study the kinematics modelling to construct AC or BC. The notations A, B and C imply the rotary postprocessors for individual machines. Some attempts axes around the linear axes X, Y and Z, respectively. The have been made to formulate the kinematics equation for rotary axes are usually implemented nearest either to the the 5-axis CNC machines with two orthogonal rotary workpiece or to the tool. In practice, there exists 108 axes [1-5] and the machines with two non-orthogonal feasible configurations of 5-axis CNC machine which are rotary axes [6-9]. The kinematics equation of the the feasible combinations of joint sequences. machine Maho 600e was studied by Munlin et al. [1],
2. Chu Anh My & Nguyen Van Cong when they worked on the toolpath optimization around general kinematic diagram of the two kinematic chains singularities. The kinematics modelling for the spindle - are presented in Fig. 2. tilting machines, the table- tilting machines and the T Let’s denote q = qqqqq table/spindle - tilting machines was investigated by Lee  12345  & She [2]. In addition, with the purpose of tool where qi X,,,,, Y Z A B C as a vector of the orientation control, the inverse kinematics equations for generalized coordinates of the system; O x y z is the machines with two rotary axes AC on the workpiece 0 0 0 0 defined as a reference coordinate system; O x y z is a carrying chain, and the machines with two rotary axes t t t t AB on the tool carrying chain was formulated by Farouki tool coordinate system, and Owwww x y z is a workpiece et al. [3]. To develop a kinematics model incorporated coordinate system. The system Ot x t t yt z is located at the with the tool inclinations, the kinematics equation for the tooltip and oriented parallel with O x y z . The system machine type XYZAC was established by Xu et al. [4]. 0 0 0 0 O x y z is usually placed on the workpiece and In literature, there has been a number of studies wwww related to the kinematic modelling of the 5-axis machines oriented parallel with O0 x 0 y 0 0z as well. consisting of non-orthogonal rotary axes. My [6] and My & Bohez [7] focused on the kinematics equation of the 5-axis machines DMU 50e and DMU 70e for the postprocessor implementation and the singular tool path error minimization. Also, Sứrby [8] and She & Huang [9] q1 studied the forward and inverse kinematic equations for the 5-axis machines with nutating table and nutating spindle. Regarding to the generalization of the kinematics model for the 5-axis CNC machines, there have existed some research works [10-12]. The kinematics module of a general 5-axis machine, to which two rotary movements on the workpiece table and two rotary movements on the spindle are added, was studied by She and Chang [10]. A general coordinate transformation matrix for all 5-axis machines with two rotary axes was derived by Tutunea-Fatan and Feng [11]. Another generic kinematics model was formulated by Yang & Altintas [12] who use the Screw theory. Fig. 2 A kinematic diagram of the general 5-axis milling It is important to note that the kinematic equation robot formulated in [10] is written with a general 7-axis configuration since two more revolute joints were added For describing the motion of each joint of a machine, onto the kinematic chain of a 5-axis CNC machine. In a homogeneous transformation matrix can be written as [12], the kinematics equation is expressed in a complex follows: form of exponential functions product. Different from Eti such kinematics modelling methods, in this paper, a new , for a prismatic joint i 01 method is investigated to formulate the forward and Dii(q ) = (1) inverse kinematic equation for the machines in a Siiτ , for a revolute joint i generalized and effective manner. The formulation of the 01 kinematics model is based on the kinematics modelling where E is an identity matrix, t i is a translational approach for two cooperative robot arms. Based on the vector, and τi characterizes the offset distances between obtained kinematics equation, two important properties the joint axis of qi and a corresponding axis of the are proved which are presented in this paper as well. machine coordinate system ( Ox00, Oy00 or Oz00). The These important results of the study could be useful for matrices Si are the basic rotation matrices when several purposes, especially when developing a qAi = , qBi = and qCi = . Note that if the join axis of generalized postprocessor for a wide spectrum of the a revolute joint q is inclined at an angle , and a i 5-axis CNC machines in manufacturing industries. matrix S ( ) represents the rotation of the inclination, R the matrix Sii(q ) must be additionally multiplied by ( ) ( ) 2. Kinematics modelling of a general 5-axis SR and SR − in the left and right sides of milling robot Sii(q ) , respectively. In the reference frame , the cumulative In this research, a 5-axis CNC machine is treated as transformation matrices for the two kinematic chains are two collaborative manipulators, one manipulator carries calculated as follows, respectively: the cutter and one manipulator caries the workpiece. The DDDD0t= n+ 1(q n + 1) n + 2( q n + 2) 5( q 5 ) (2)
3. Properties of the Kinematics Model of 5-Axis CNC Machines (Milling Robots) Table 1. Constant matrix DDDD01111wnnnn= (qqq) −−( ) ( ) (3) It is worth to note that if the two kinematic chains are A B C qu unified through the reference frame O0 x 0y 0 0z , the closed loop mechanism of the machines can be represented as a serial open mechanism of a single kinematic chain. In this 0 1 0 1 0 0 1 0 0 manner, the joint sequence of the unified chain is 0 0 1 0 0 1 0 1 0 q1 , q2 , q3 , q4 and q5 . The motion of the tool relative to the workpiece is thus described by the following kinematic relationship: Rewriting Eq(11) in a form of five independent −1 DDDwtwt= ( 00) equations yields (4) −−11 p g= q ( ) , (10) = DDDD111155(qqqq) nnnn( ) ++( ) ( ) where Since n 0 ,. . . ,5 , the matrix Dwt can be rewritten T as follows: p p= p TR , and (11) Dwt = ΘΘΘΘΘ1122334455(qqqqq) ( ) ( ) ( ) ( ) (5) T q q= q TR . (12) where Eu i Thus, the differential kinematic equation for the 5-axis , for a prismatic joint i 01 machines can be written as follows: Θii(q ) = (6) Riiτ , for a revolute joint i p J= q , (13) 01 With respect to all the joints of the tool carrying chain, where J55 is the Jacobian matrix. utii= ,for a prismatic joint i pp TT RSii= ,for a revolute joint i qq J = TR With respect to all the joints of the workpiece carrying ppRR (14) chain, qqTR ut=− ,for a prismatic joint i JJTTTR ii = T JJRTRR RSii= ,for a revolute joint i Finally, the kinematics model of the machine is Eq(13) is the differential kinematic equation that relates formulated as follows: the joint velocities, the tool velocity and the Jacobian RpwtT matrix which characterize the structure of the machines. Dwt = , (7) 01 T 3. Properties of the kinematics model where pT = x y z is the tooltip position. The three direction cosines of the tool axis vector i , j and k Prosperity 1. The forward kinematic equations for are the three entries of the last column of the matrix R . wt the two rotary axes of all 5-axis machines can be Let’s denote X = x y z i j kT as the tool formulated directly with only two rotation matrices Ru posture, the forward kinematic equation of the system can and R describing the motion of the primary revolute be written as follows: v joint ( qu ) and the secondary revolute joint ( qv ), X= f( q) (8) regardless of where the primary and the secondary rotary Let qu and qv be the joint variables of the primary axes are in the generalized kinematic chain of the 5-axis revolute joint and the secondary revolute joint, where CNC machines. In other words, the tool orientation vector T uvu vand,1,2,3,4,5( )  . Let qT = XYZ  p R can be calculated with Eq(15), and the direction T and qR= qq u v  be the vectors of the three prismatic cosines i , j and k are calculated with Eq(16). Both joint variables and the two revolute joint variables, the calculations are independent of the three prismatic T respectively. Let pR =   denote the orientation of joints variables X, Y and Z. the tool axis. The parameters  and are the two pR= ΦR u R v Γ (15) independent direction cosines selected from three T i j k = RRΓ (16) dependent direction cosines ( i , j and k ) of the tool axis   uv vector. The parameters  and must be selected so that T Note that Γ = 0 0 1 . both of them are expressed in terms of both qu and qv . T pR = Φi j k (9) where Φ is a constant matrix in Table 1.
4. Chu Anh My & Nguyen Van Cong Proof: Based on the rules of the block matrix which makes it possible to formulate the Jacobian multiplication, the following block matrix multiplications determinant for the machines. By using Property 2, can be calculated: D e t (J55 ) can be formulated as the determinant of a 22 EuEuEu iiT−1 matrix with a dimension of only. = (17) Proof: As discussed in the Proof of Property 1, it is 010101 clearly seen that all the transformation matrices R τEuRR uτ + iiiiiii ++11 Θ (q ) in Eq(8) are expressed as particular block = (18) ii 010101 matrices. Hence, multiplying two or three successive EuR τRuτ + translational transformation matrices, in different orders, iiiiii−−11= (19) yields a matrix in the same form of the multiplied 010101 matrices. However, multiplication of a translational It can be seen from Eq(17) that multiplying two transformation matrix with a rotational transformation translational transformation matrices results in a matrix is not commutative. Thus, the result of the matrix transformation matrix in the same form of the multiplied chain multiplication in Eq(5) depends on where the two matrices. Eqs(18,19) show that multiplying a rotational transformation matrices are in the matrix chain. translational transformation matrix with a rotational Theoretically, there exist seven different sequences of the transformation matrix, in a different order, the rotation joint variables (Fig. 3) that correspond to seven different block matrix R i in the rotational transformation matrix results of the matrix chain multiplication. is not transformed. Consequently, Eq(7) can be rewritten Workpiece → Tool as follows: S1 q q q q q RuRu u v 3 4 5 D = uRuvRv wt S2 q2 qv 0101 S3 q4 RuvuRvRu RR uu + S4 q = (20) 1 S5 q 01 3 S6 q R Rp 5 uvT S7 q = 5 01 In Eq(20) u and u are the vectors yielded by Fig. 3 Seven sequences of the joint variables Ru Rv the multiplications of a rotational transformation matrix The family of the 5-axis machines with both rotary with the translational transformation matrices, axes on the table is represented by the sequence of the respectively. joint variables S1. The sequence S3 represents the group Since the tool axis vector points in the axis z of the t of 5-axis machines with the primary rotary axis on the coordinate system Oxyz , its direction cosines i , j tttt table, and the secondary one on the spindle head. The and k are the three entries of the last row of the rotation sequence S5 denotes the 5-axis machines with both rotary matrix RRuv. Therefore, T axes on the tool chain. The sequence S2 represents for all ijk  = RRuvΓ , (21) the cases in which the rotational transformation matrix Substituting Eq(21) into Eq(9) yields Eq(13) and Θvv(q ) is in between two translational transformation completes the proof. matrices, meanwhile the matrix Θuu(q ) is the first matrix of the matrix chain. Similarly, as for the sequence S6, It is also important to note that, p R calculated with T both the matrices Θuu(q ) and Θvv(q ) are in between Eq(21) is a function of only qR= qq u v  , and it is independent of q = XYZT . As a consequence of a couple of translational transformation matrices; the T sequence S7 implies that and are Property 1, Θuu(q ) Θvv(q ) adjacent, but both the matrices are in the middle of the pR matrix chain. J0RT == (22) qT With the seven chains of the transformation matrices, the seven results of the matrix chain multiplication can be archived accordingly, where the tooltip position Property 2. The determinant of the Jacobian matrix T p = x y z is expressed as follows: J of the generalized 5-axis kinematics model can be T   S1. pR=++ R qR ττ directly calculated by using only the kinematics Tuv Tu vu S2. p=−+ RRq ++ RRuu ++ Ruu R ττ sub-model of the two rotary axes. In other words, Tu v Tu vuu v u ( 2 32) 3 ( ) S3. pRTu=+++ qR Tu vuu v t ττR R τ Det(JJ) = Det ( RR ) (23) S4. pTu=+ Tuu R qE v −+++ uu( R v ut+ )( uR12) τ τ R R τ (24) To evaluate the kinematic performances of a 5-axis S5. pT= q T + R uτ v + τ u + R u R v τ t machine, the Jacobian determinant is often needed. S6. pT= RRq u v T − RRu u v( 1 + u 3) + Ru u 3 + R uτ v + τ u + u 1 However, formulation of the Jacobian determinant in a S7. pT= R u R v q T +( E − R u R v)( u12 + u) + R uττ v + u generalized case is challenging, since the matrix J has a It is clear that, except from the Jacobian determinant dimension of 55 , and the determinant is a function of of the first term of all the expressions (S1-S7), the all five joint variables. Hence, this property is important Jacobian determinant of all the other terms equal to zero
5. Properties of the Kinematics Model of 5-Axis CNC Machines (Milling Robots) because the Jacobian matrix of a constant vector is a zero References matrix, and the determinant of a Jacobian matrix [1] M. Munlin, S.S. Makhanov, E.L. Bohez, Optimization containing one zero-column equals to zero. of rotations of a five-axis milling machine near Consequently, for the expressions S1, S2, S6 and S7, stationary points, Computer-Aided Design. 36 (12) the Jacobian JTT can be calculated with Eq(25). For S3 (2004) 1117-28. and S4, JTT is calculated with Eq(26), and for S5, JTT [2] R.S. Lee, C.H. She, Developing a postprocessor for is calculated with Eq(27). three types of five-axis machine tools, The International Journal of Advanced Manufacturing  RRq pT ( uvT ) Technology. 13 (9) (1997) 658-65. JTT == qqTT (25) [3] R.T. Farouki, C.Y. Han, S. Li, Inverse kinematics for = RR optimal tool orientation control in 5-axis CNC uv machining, Computer Aided Geometric Design. 31 (1) (2014) 13-26.  Rq pT ( uT) [4] H.Y. Xu, L.A. Hu, T. Hon-Yuen, K. Shi, L.A. Xu, A JTT == qqTT (26) novel kinematic model for five-axis machine tools and = R its CNC applications, The International Journal of u Advanced Manufacturing Technology. 67(5-8) (2013) 1297-307.  q pT ( T ) [5] J. Yun, Y. Jung, T. Kurfess, A geometric JTT == qqTT (27) postprocessing method for 5-axis machine tools using = E locations of joint points, International journal of precision engineering and manufacturing. 14 (11) Note that DetDetDet(RREuv) === ( ) ( ) 1 . (2013) 1969-77. Hence, for all the cases [6] C.A. My, Integration of CAM systems into multi-axes computerized numerical control machines, D e t (JTT ) =1 (28) Proceedings of the Second IEEE International Conference on Knowledge and Systems Engineering On the other hand, the determinant of the block (KSE), Hanoi, Vietnam, 2010, pp. 119-124. matrix J can be calculated as follows: [7] C.A. My, E.L.J. Bohez, New algorithm to minimise kinematic tool path errors around 5-axis machining DetDet(JJJJJ) =−( TTRRTRRT ) (29) singular points, International Journal of Production Research. 54 (20) (2016) 5965-75. Substituting Eq(22) and Eq(28) into Eq(29) yields [8] K. Sứrby, Inverse kinematics of five-axis machines Eq(23) and completes the proof. near singular configurations, International Journal of Machine Tools and Manufacture. 47 (2) (2007) 299-306. 4. Conclusions [9] C.H. She, Z.T. Huang, Postprocessor development of a five-axis machine tool with nutating head and table Owing to the particular architecture of the 5-axis configuration, The International Journal of Advanced CNC machines, the generic kinematics model of the Manufacturing Technology. 38(7-8) (2008) 728-40. machines has two special properties, as compared with [10] C.H. She, C.C. Chang, Design of a generic five-axis other CNC machine types and industrial robots. In this postprocessor based on generalized kinematics model paper, these properties have proved in a generalized case. of machine tool, International Journal of Machine The use of Properties 1 and 2 is very helpful for several Tools and Manufacture. 47 (3-4) (2007) 537-45. purposes, especially for modeling the kinematic tool path [11] [22] O.R. Tutunea-Fatan, H.Y. Feng, Configuration analysis of five-axis machine tools using a generic error around singularities of the 5-axis CNC machines kinematic model, International Journal of Machine that was presented in [7]. By applying the properties, the Tools and Manufacture. 44 (11) (2004) 1235-43. singular points of any 5-axis CNC machine can be [12] J. Yang, Y. Altintas, Generalized kinematics of effectively identified and analysed with Det (JRR ) = 0 , five-axis serial machines with non-singular tool path instead of Det (J) = 0 . Moreover, since Det (JRR ) is generation, International Journal of Machine Tools a function of merely one variable qv , the computational and Manufacture. 75 (2013) 119-32. complexity of the algorithm constructed for minimizing the kinematic tool path error in [7] is significantly reduced. Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 107.04-2017.09.