In this paper, we propose an efficient method for solving the kinematic problem for asymmetrical parallel manipulators. By solving, we converted the kinematic problem to the optimal form. Mathematical models are obtained by using loop of vector equation as other parallel manipulators.
Nghiên cứu khoa học công nghệ SOLVING THE KINEMATICS PROBLEM FOR ASYMMETRICAL PARALLEL MANIPULATOR BASED ON GRG METHOD Pham Thanh Long*, Le Thi Thu Thuy, Duong Quoc Khanh Abstract: In this paper, we propose an efficient method for solving the kinematic problem for asymmetrical parallel manipulators By solving, we converted the kinematic problem to the optimal form Mathematical models are obtained by using loop of vector equation as other parallel manipulators The example shown in this paper shows the applicable possibility of asymmetrical parallel manipulator The joint variables as well as the sub parameters of each leg are accurately and fully defined This method also does not require initial approximation values as Newton-Raphson method, which is a great advantage of the Banana method of kinematic problems Keywords: Asymmetrical parallel mainipulator; Kinematic problem sub parameter; Joint variable; Optimal I INTRODUCTION Parallel manipulators with advantages of stiffness, accuracy and excelled dynamic power due to having spare parallel transmission structure increasingly present in engineering Small workspace, difficult design and control make parallel manipulators hard to widespread practical application One of difficulties is to solve the kinematic problem of the parallel manipulators some authors have delved into symmetrical structures [1][2][3][4][5][6][7] In asymmetric categories, different structures in legs’ configuration make the complexity of problem increase dramatically Obviously, the above-mentioned achievements are limited, so the new research is needed to solve this issue In the field of kinematic of parallel manipulators, the Newton – Raphson method has been used most widely [8][9][10] but biggest drawback is suitable initial approximation values In additions, this method is just applied favorably to standard driving robots while with deficiency or residual driving robots, the inversion of a non-square matrix will need more time With transcendental mathematical structures as the kinematic equation system of parallel manipulators, GRG method is particularly suitable when the problem is resolved by optimal solution [11][12][13] This paper presents a highly generalized method for solving the kinematic problem of asymmetrical parallel manipulators This method is based on the GRG algorithm [14][15] efficiently applied in serial manipulators This theory has been also shown to work on symmetrical parallel manipulators [16][17] II MATHEMATICAL MODEL 2.1 Types of joints are used in parallel structures Unlike serial structures, there are two types of joints in parallel structures: active and passive joints Active joints are 5-type joints connected to a motor: P (Prismatic) or R joint (Revolute) Passive joints are not connected to a motion source, they transfer power and are usually 4-type joints (H - helix joint, C - Cylinder joint, U - Universal joint) or 3-type joints (S - Sphere joint) Structures of each joints as well as characteristic parameters are described in table Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san FEE, 08 - 2018 195 Cơ học – Cơ khí động lực Table Structure and parameters of types of joints used on parallel manipulators Types of joints P joint R joint H joint C joint U joint S joint Symbol Parameter l (mm) q(rad) l(mm), q(rad) l(mm), q(rad) q1, q2(rad) q1, q2, q3(rad) These parameters will be entered into the mathematical model of manipulator’s kinematic sequence 2.2 Principles of modeling general kinematic chain by a loop of vectors Figure General schema for a certain leg A closed loop of vectors is set onto any leg of robot which obtains both fixed and mobile coordinate system, as shown in figure Starting point is selected as the origin of the fixed coordinate system; Destination is the origin of mobile one The following loop of vectors is based on orientation of component vectors gained from radial parameters and the direction cosine of each vector is set as follows: A1 A2 An I (1) The equation (1) is a mathematical model of the current leg In asymmetrical structure, (1) cannot be traced to the other legs Equation of different legs needs to follow this principle III SOLUSION 3.1 Conversion of equivalent problems A traditional kinematic problem is solving the nonlinear, transcendental equations Analytical techniques considered as less efficient when applied in the reality because of the subjectivity of solvers, structural characteristics are not discussed in this paper Numerical methods are highly applicable but some restrictions still exist For instance, the Newton-Raphson method has problem with initial approximation values [18], GA method is limited by solving time and accuracy of results when applied in form of the Rosenbrocbanana function [19] 196 P T Long, L T T Thuy, D Q Khanh, “Solving the kinematics … based on GRG method.” Nghiên cứu khoa học công nghệ In this section, the GRG method [20] is presented to investigate the kinematic problems of asymmetric parallel manipulators The closed loop equation (1) for a leg can be written as follow: f j ( qi ,i , li ) pi i n, j m (2) Where: i is the index of the i th link on the j th leg; qi , i is direction cosine parameter of the i th link on the j th leg; li is the length of the i th link on the j th leg; A nonlinear transcendental equation system exists when (2) is written for m full legs This system can be square or non - square depending on the manipulator’s configurations (standard, miss or spare driving) With standard driving systems, the number of DOFs equals the number of active links, the number of variables of equation system is the same number of equations It is favorable to use Newton-Raphson method to solve the problem when the suitable initial approximation is found In case the driving structure is not considered, the equation system (2) is transformed as: m Min ( f j ( qi , i , li ) pi ) j 1 i n, j m (3) This problem is presented in the optimal form so it allows adding more constrains such as selecting the control roots instead of pure mathematical roots The main problem here is determining a suitable method to solve (3) when the manipulator owns a large number of legs 3.2 GRG method According to [12], the GRG method has some characteristics as follow: - Use the derivative algorithm resulted in high converging speed; - Do not need the initial approximation value; - Converging speed and the accuracy depend on how to calculate the difference; Comparing three methods included GA, SQP and GRG [19] shown that GRG method is absolutely suitable for the target functions in forms of Rosenbroc-banana This method was used to solve kinematic problem of symmetrical parallel manipulators by modifying variable to downgrade target functions [18] The GRG method has been shown to work on solving problems of form (3) IV ILLUSTRATION ON AN ASYMMETRICAL PARALLEL MANIPULATOR 4.1 Modeling by vector loop method Figure An asymmetrical parallel manipulator (a) and its graph (b) Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san FEE, 08 - 2018 197 Cơ học – Cơ khí động lực An asymmetrical parallel manipulator is a manipulator with different leg configurations This is resulted from technology and asymmetrical workspace The studied system is shown in fig.2 Leg A has URU configuration with 5-DOF, driving R joint; Leg B has RSS configuration with 7-DOF, driving revolute joint (R-joint) However two spherical joints (S-joint) stand side by side and limit themselves to lose 1-DOF, leading to this joint is equivalent to universal joint (U-joint) Leg C has UPU configuration with 5-DOF, driving P joint; As each leg has different configuration, this manipulator has the asymmetrical structure and the modelization is proceeded with each specific leg The base frame and the mobile platform frame are denoted by O0 and O1 coordinate frames, respectively The triangles A1A2A3 and B1B2B3 are equilateral, separating the closed loop of leg A is shown in Figure Figure Diagram of a closed loop of leg A p( px , p y , pz ) and RRPY f ( , , ) are the position and the orientation of O1x1y1z1 coordinate frame with respect to the frame O0x0y0z0 respectively As shown in figure 3, the closed loop equation can be written as: a b RRPY n m p (4) Rewrite (4) as an expansion using direction cosine matrixes: a.c( qA1 ) c( qA2 ) b.c( q A1 qA3 ) c( q A2 ) c c a.c( q )s( q ) b.c( q q )s( q ) c s A1 A2 A1 A3 A2 a.s( qA1 ) b.s( q A1 qA3 ) s s s c c s s s s c c s c c s c s s xB1 x A1 px c s s s c y B1 y A1 p y c c zB1 z A1 pz (5) Closed loop of leg B is shown in Figure Figure Diagram of a closed loop of leg B 198 P T Long, L T T Thuy, D Q Khanh, “Solving the kinematics … based on GRG method.” Nghiên cứu khoa học công nghệ The closed loop equation can be written as: c d RRPY n p m (6) Rewrite (6) as an expansion using direction cosine matrixes: c.c( qB ) c( qB1 ) d c( qB1 qB ) c c c.s( q ) c s B2 c.c( qB ) s ( qB1 ) d s( qB1 qB ) s s s c c s s s s c c s c c s c s s xB p x x A2 c s s s c y B p y y A2 c c z B pz z A2 (7) Closed loop of leg C is shown in Figure Figure Diagram of a closed loop of leg C The closed loop equation can be written as: m lc RRPY n p (8) Rewrite (8) as an expansion using direction cosine matrixes: x A3 lc c( qC1 )c( qC ) c c y l c( q ) s( q ) c s C1 C2 A3 c z A3 lc s( qC1 ) s s s c c s s s s c c s c c s c s s xB px c s s s c y B p y (9) c c zB pz Equations (5) (7) and (9) can be gathered into an equation system included equations with parameters analyzed as in table 2: Parameter Table Parameters and their meaning in the mathematical model Definition Forward problem Inverse problem p( p x , p y , pz ) Mobile platform position calculate given RRPY ( , , ) qA1 , qA2 , qA3 Mobile platform direction Robot’s texture parameters Direction cosines of leg A (calculate)* given calculate qA1, qA2 (given)* given calculate qB1 , qB , qB Direction cosines of leg B calculate qB1, qB2 calculate calculate given given calculate given given a, b, c, d, m, n qC1 , qC Direction cosines of leg C lC Length of leg C ( x A1 , y A1 , z A1 ) Coordinates of A1 in the frame O0 Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san FEE, 08 - 2018 199 Cơ học – Cơ khí động lực Coordinates of A2 in the frame O0 ( x A3 , y A3 , z A3 ) Coordinates of A3 in the frame O0 ( xB1 , yB1 , zB1 ) Coordinates of B1 in the frame O1 given given given given given given ( xB , yB , zB ) Coordinates of B2 in the frame O1 ( xB , yB , zB ) Coordinates of B3 in the frame O1 given given given given ( x A2 , y A2 , z A2 ) Note: - ()* this parameter can’t be controlled because of lack of DOF - Each manipulator’s leg has only one active joint (5-type joint), others are passive joints Thus, qA3 is the joint variable of leg A, two other parameters are subparameters - qB3 is the joint variable of leg B, two other parameters are sub-parameters; - lC is the joint variable of leg C, two other parameters are sub-parameters; We can control variables, sub-parameters are to calculate, not to be controlled Similarly, because of having three DOF, if the manipulator position, presented by p ( p x , p y , pz ) , is chosen to control, the direction control, presented by RRPY ( , , ) , has to be ignored and vice versa The next section will show the calculation method to control the position of manipulator mentioned above 4.2 Solving the kinematic problem by the GRG method 4.2.1 The inverse kinematic problem P (Px, Py, Pz) expressed the position of the origin O1 is given, parameters unknowns are: - Controlled variables of each leg: qA3, qB3, lC; - Direction cosines of each leg: qA1, qA2, qB1, qB2, qC1, qC2; Figure Illustrate the result of an inverse dynamic problem at a studied point According to results of program, the value of target in problem (2) at the B20 (column B with respect to row 20) is small enough (1.06E-17) The convergence of the problem is achieved and corresponding solutions are shown on line in figure 200 P T Long, L T T Thuy, D Q Khanh, “Solving the kinematics … based on GRG method.” Nghiên cứu khoa học công nghệ 4.2.2 The forward kinematic problem qA3, qB3, lC are joint variables given, parameters unknown are: - The position of reference system O1 in reference system O0 or determine P (Px, Py, Pz); - Direction cosines of legs: qA1, qA2, qB1, qB2, qC1, qC2; Figure Illustrate the result of a forward kinematic problem at a studied point The convergence of the forward kinematic problem is achieved at the studied point when joints variables are given and the position and the orientation of each leg are absolutely determined As a result of the problem, the accuracy is highly achieved because the value of the target function in B20 is approximately TT x y z p1 150.3508 245.2768 p2 68.8344 -111.4814 259.4241 p3 99.0695 -39.0371 285.7916 p4 99.0695 39.0371 314.2084 p5 68.8344 111.4814 340.5759 p6 150.3508 354.7232 p7 -68.8344 111.4814 340.5759 p8 -99.0695 39.0371 314.2084 p9 -99.0695 -39.0371 285.7916 p10 -68.8344 -111.4814 259.4241 Figure The trajectory of 10 points in the workspace Solving the problem by this proposed algorithm, the variation law of joints variables and sub-parameters of the manipulator is corresponding shown in fig.9,10: tt qa3 qb3 lc qa1 qa2 qb1 qb2 qc1 qc2 P1 1.369845 -2.24626 -278.834 1.900616 1.570796 2.618677 0.431062 -2.06646 -4.37978 P2 0.683601 -2.10415 -294.245 2.323012 -0.72903 2.315972 -0.4723 -1.07937 -4.52745 P3 0.671149 -2.00484 -298.14 2.277265 0.11021 2.129291 -0.21511 -1.28198 -3.9959 P4 0.801918 -1.91578 -319.428 2.135596 0.732115 2.104031 0.046807 -1.38978 -2.89501 P5 0.951818 -1.82866 -352.311 1.973021 1.167852 2.195193 0.292421 -1.31197 -1.85788 Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san FEE, 08 - 2018 201 Cơ học – Cơ khí động lực P6 1.028283 P7 0.951818 P8 0.801918 P9 0.671149 P10 0.683601 P1 -1.36985 -1.79641 -378.704 1.881067 1.570796 2.407639 0.431062 -1.21302 -1.23819 -1.90095 -368.843 1.973021 1.973741 2.626449 0.292421 -1.17675 -0.65694 -2.0193 -345.244 2.135596 2.409478 2.756978 0.046807 -1.14354 -0.09828 -2.11545 -325.649 2.277265 3.031382 2.841925 -0.21511 -1.07085 0.42268 -2.19297 -313.851 2.323012 3.870626 2.877707 -0.4723 -0.97301 0.883012 -2.24626 -278.834 3.270461 4.712389 2.618677 0.431062 -2.06646 1.903404 pa3 pb3 radian -1 -2 -3 point 10 11 -250 lc mm -300 -350 -400 point 10 11 Figure Graph showing the variation law of qA3, qB3 and lC qa1 qa2 qb1 qb2 qc1 qc2 radian -1 -2 -3 -4 -5 point 10 11 Figure 10 Graph showing the variation law of sub-parameters V CONCLUSION In this paper, we have shown that with asymmetrical structures, resolving kinematic problem by optimal form to use the GRG method is shown to work for both forward and inverse problem The accuracy is highly achieved It is important that this method allows 202 P T Long, L T T Thuy, D Q Khanh, “Solving the kinematics … based on GRG method.” Nghiên cứu khoa học công nghệ proceeding on deficiency or residual motion systems In contrast to Newton-Raphson method, this method does not have difficulties to inverse non-square matrixes as a result of deficiency and residual structures and require initial approximation values Especially, when considering the problem in optimal form, the condition to select the control solution can be used as boundary conditions, saving time for the kinematic data preparation of users REFERENCES [1] J Wang, X Liu, and C Wu, “Optimal design of a new spatial 3-DOF parallel robot with respect to a frame-free index,” Sci China, Ser E Technol Sci., vol 52, no 4, pp 986–999, 2009 [2] L Rey and R Clavel, “The Delta Parallel Robot,” pp 401–417, 1999 [3] G Cheng, S R Ge, and J L Yu, “Sensitivity analysis and kinematic calibration of 3-UCR symmetrical parallel robot leg,” J Mech Sci Technol., vol 25, no 7, pp 1647–1655, 2011 [4] H B Choi, A Konno, and M Uchiyama, “Analytic singularity analysis of a 4-DOF parallel robot based on jacobian deficiencies,” Int J Control Autom Syst., vol 8, no 2, pp 378–384, 2010 [5] H B Choi, A Konno, and M Uchiyama, “Closed-form forward kinematics solutions of a 4-DOF parallel robot,” Int J Control Autom Syst., vol 7, no 5, pp 858–864, 2009 [6] B Achili, B Daachi, A Ali-Cherif, and Y Amirat, “A c5 parallel robot identification and control,” Int J Control Autom Syst., vol 8, no 2, pp 369–377, 2010 [7] Y Yun and Y Li, “Design and analysis of a novel 6-DOF redundant actuated parallel robot with compliant hinges for high precision positioning,” Nonlinear Dyn., vol 61, no 4, pp 829–845, 2010 [8] C Yang, Q Huang, P O Ogbobe, and J Han, “Forward Kinematics Analysis of Parallel Robots Using Global Newton-Raphson Method,” 2009 Second Int Conf Intell Comput Technol Autom., pp 407–410, 2009 [9] L Sun et al., “Forward kinematics analysis of parallel manipulator using modified global Newton-Raphson method,” Huagong Xuebao/CIESC J., vol 60, no 2, pp 444–449, 2009 [10] H L J.M Selig, “A geometric Newton-Raphson method for Gough-Stewart Platforms.” [11] “Gradient Projection and Reduced Gradient Methods,” Report, pp 176–182 [12] G A Gabriele and K M Ragsdell, “The Generalized Reduced Gradient Method : A Reliable Tool for Optimal Design,” J Eng Ind., vol 99, no 2, pp 394–400, 1977 [13] Y Smeers, “Generalized reduced gradient method as an extension of feasible direction methods,” J Optim Theory Appl., vol 22, no 2, pp 209–226, 1977 [14] H L S C H Kang, “A Study of Generalized Reduced Gradient Method with Different Search Directions,” Idea, vol 1, no 1, pp 25–38, 2004 [15] R Sancibrian, “Improved GRG method for the optimal synthesis of linkages in function generation problems,” Mech Mach Theory, vol 46, no 10, pp 1350– 1375, 2011 [16] L W G and P T L Trang Thanh Trung, “A New Method to Solve the Kinematic Problem of Parallel Robots Using an Equivalent Structure,” Int Conf Mechatronics Autom Sci 2015)Paris, Fr., pp 641–649, 2015 [17] L W G and P T L Trang Thanh Trung, “A New Method to Solve the Kinematic Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san FEE, 08 - 2018 203 Cơ học – Cơ khí động lực Problem of Parallel Robots Using General reduce Gradient algorithm,” J Robot Mechatronics, vol 28-N03, 2016 [18] T T Trung, “Optimization Analysis Method of Parallel Manipulator Kinematic Model,” a dissertation for the degree of doctor, Đại học Hoa Nam, Quảng Châu, Trung Quốc, 2018 [19] T T L Phạm Thành Long, “A new method based on kinematics of robots to analyze the kinematics of persian joint,” J Environ Sci Eng., pp 55–59 [20] L T T T Phạm Thành Long, Nguyễn Hữu Công, “Ứng dụng phương pháp giảm gradient tổng quát kỹ thuật robot,” 2017 TĨM TẮT BÀI TỐN ĐỘNG HỌC CỦA ROBOT SONG SONG CẤU TRÚC BẤT ĐỐI XỨNG TRÊN CƠ SỞ PHƯƠNG PHÁP GRG Bài báo giới thiệu phương pháp hiệu cho giải toán động học robot song song bất đối xứng Để giải tốn này, chúng tơi chuyển tốn động học sang dạng tối ưu Mơ hình tốn đạt cách sử dụng phương trình vòng véc tơ giống robot song song khác Qua ví dụ ứng dụng trình bày cho thấy khả ứng dụng robot song song cấu trúc bất đối xứng hoàn toàn khả quan Toàn biến khớp tham số phụ chân xác định đầy đủ xác Phương pháp khơng đòi hỏi cung cấp giá trị xấp xỉ đầu phương pháp Newton-Raphson yêu cầu, lợi lớn phương pháp dạng hàm Banana tốn động học robot Từ khóa: Robot song song bất đối xứng; Bài toán động học; Tham số phụ; Biến khớp; Tối ưu Received date, 18th April, 2018 Revised manuscript, 10th August, 2018 Published, 09th September, 2018 Author affiliations: * Thai Nguyen University of Technology Corresponding author: kalongkc@gmail.com 204 P T Long, L T T Thuy, D Q Khanh, “Solving the kinematics … based on GRG method.” ... versa The next section will show the calculation method to control the position of manipulator mentioned above 4.2 Solving the kinematic problem by the GRG method 4.2.1 The inverse kinematic problem. .. Solving the kinematics … based on GRG method. ” Nghiên cứu khoa học công nghệ In this section, the GRG method [20] is presented to investigate the kinematic problems of asymmetric parallel manipulators... (1.06E-17) The convergence of the problem is achieved and corresponding solutions are shown on line in figure 200 P T Long, L T T Thuy, D Q Khanh, Solving the kinematics … based on GRG method. ”