This paper presents a point of view on determining reasonable dimensions when designing open kinematic chain robots. With two robots having the same configuration and reach, if the resolution of joint variables is the same but the length of links is different the response to endpoint accuracy will be different. So the rational strategy is that the strategy has dimensions increasing the desired accuracy without increasing the cost of manufacturing the robot hardware.
Nghiên cứu khoa học công nghệ DETERMINING REASONABLE DIMENSIONS IN ROBOT STRUCTURE DESIGN Le Thi Thu Thuy1*, Pham Thanh Long1 Abstract: This paper presents a point of view on determining reasonable dimensions when designing open kinematic chain robots With two robots having the same configuration and reach, if the resolution of joint variables is the same but the length of links is different the response to endpoint accuracy will be different So the rational strategy is that the strategy has dimensions increasing the desired accuracy without increasing the cost of manufacturing the robot hardware The justifiable link dimension can be determined by setting the different link length ratios and then examining them in the same boundary condition, the robot kinematic model will be used at this stage The application of the point of view introduced in this paper into the design will bring about a reasonable cost as it does not require increasing the resolution of the joint variables in the robot control, nor does require tightening of the tolerance in mechanical manufacturing Keywords: Reasonable link length; Kinematic problem,manufacturing tolerance; Open chain kinematic robot I INTRODUCTION In experimental studies, authors have shown some parameters which affected the orbit accuracy in robot control [1][2][3][4][6][7][8] In additions, studies on robot’s tolerance design [5] have shown a method of determining tolerance for DH parameters by quantitative values when nominal size of DH parameter is given However, the quantitative calculation of these DH parameters is rarely mentioned To be linked into kinematic sequence to the required range, robot must achieve both of the accuracy and the precision This is a complex problem and experimental planning techniques can’t solve this problem This problem is independent of both of the above problems and requires a different method to proceed There is no doubt that robots which have same structure and range but different DH sizes will give different endpoints accuracy when using the same resolution of joint variables This meant that, just by determining the reasonable length of each link in the structure, the precision of these robots has considerably improved when compared to others So this is a measure not only to increase the cost of manufacturing and assembly but also to increase the precision of the endpoint II BASIC OF DETERMINING THE TOLERANCE OF JOINT VARIABLES Assuming that the unit movement of endpoint was given, we have to determine the resolution of generalized coordinates of joints The kinematic equations system of robot can be written as follow: f i ( qi , d i , a i ) (1) i 1 n Where: di, are initial approximation values of the nominal dimension in the kinematic sequence qi is generalized coordinate of ith joint n is the number of generalized coordinates to determine the position and orientation of the last link The current position of the point in robot’s workspace is written as: Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san FEE, 08 - 2018 189 Cơ học – Cơ khí động lực pi ( xi , yi , zi ,i , i , i ) (2) (xi,yi,zi) is expressed the position of the last link (αi,βi,γi) is expressed the orientation of the last link The value of generalized coordinates for this state is found by solving the following equations system: f i ( qi , d i , a i ) p i (3) i 1 n In the joint space, Pi can be written as: Where: pi ( q1 , q , , q6 ) ( i ) (4) Figure The sphere describes the precision of the last link at work Pi+1 is any point in the sphere that describes the desired precision of the last link: pi 1 ( xi 1 , yi 1 , zi 1 , i 1, i 1, i 1 ) (5) pix dx Or: pi 1 piy dy (6) piz dz Replace to (3), the value of the generalized coordinate can be rewritten: pi 1 ( q1 , q2 , , q6 ) ( i 1) (7) And each joint variable made a move: q1 q1( i 1) q1( i ) ( i 1) (i ) q6 q6 q6 (8) nj is the resolution (n is the number of pulses / rotation) and is calculated as follow: 2 n j (i 1) qj q j (i ) (9) After calculating the value of nj and referring to the types of sensors, types of motors from manufacturers we can choose the reasonable resolution This value ensures that the endpoint and DH parameters (di, ai) are always in the given error sphere with the given radius 190 L T T Thuy, P T Long, “Determining reasonable dimensions in robot structure design.” Nghiên cứu khoa học công nghệ III BASIC OF DETERMINING THE REASONABLE DIMENSIONS OF THE LINK di and calculated in (1) will get for the following equation: f i ( qi qi , d i , a i ) pi pi i n (10) Assume that values of di and are changed and qi qi calculated in (8) n (a ' d ' i i ) const (11) i 1 Put (11) into (10), the position of endpoint is moved a distance: f i ( qi qi , d ' i , a ' i ) pi ' pi i n (12) d ' i , a ' i are calculated to improve d i , a i (more reasonable than old values) In case ' pi pi the alternative will be eliminated At first, ai, di are initial approximation and unreasonable values Assume that these dimensions follow a rule defined by the designer: d1 : d : d : x : x (13) Where x 0,1 is any initial approximation value Scanning this range, the designer can show xi to have min ' pi Except calculated as follow (8), qi can also be selected as a reference value, such as resolution of motors, sensors or expectation angle used on robots follow standard catalogues If there is min ' pi at the boundary of the reference area [0,1] the structure of robot is unrealistic because the length of a link in two ends of the kinematic sequence is very small We have to provide some structural constraints such as the minimum length of the first link or the last link IV ILLUSTRATED CALCULATION ON KUKA ROBOTS Figure DH table of a KUKA robot with rotary joints Case 1: According to the technique introduced in [9], the tolerance of this robot can be determined with specific dimensions given in followed table: d1 = 335 (mm) a1 = 75 (mm) a3 = 90 (mm) a2 = 270 (mm) d4 = 295 (mm) d6 = 80 (mm) Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san FEE, 08 - 2018 191 Cơ học – Cơ khí động lực The target is the accuracy when controlling an open circuit does not exceed a sphere with r = 0.5 (mm) Calculations on a grid of 26 nodes in the workspace of robot give the resolution of joint variables as shown in figure Re-standardize to the resolution parameters of standard angle sensors to have following specific parameters: - Using encoder at q1, q2, q3 we have the same resolution and n = 4000 - Using encoder at q4, q5, q6 we have the same resolution and n = 2000 After determining tolerances of links and joints by GRG, the results are as follows: - q1, q2, q3: q = 0,001571 q = -0.000785 ÷ 0.000785 - q4, q5, q6: q = 0,003141 q = -0.00157 ÷ 0.00157 Test the radius of error on the forward dynamic equations of the robot with above parameters to determine the maximum radius of error Use the statistical results to automatically determine rmax (dist) = 0.490855 mm < r (0.5mm) Figure The calculated result of the resolution of variables Figure The statistical result of radius of error when test meets the accuracy 192 L T T Thuy, P T Long, “Determining reasonable dimensions in robot structure design.” Nghiên cứu khoa học công nghệ Case 2: Maintain tolerances of variables as described above but change dimensions of links shown in table below and ensure that robot’s range remains constant d1 = 375 (mm) a1 = 115 (mm) a2 = 270 (mm) a3 = 90 (mm) d4 = 295 (mm) d6 = 40 (mm) Test the response accuracy at the same positions in case According to the statistic of error radius provided by software, the largest error radius decreases with respect to the result in case rmax(dist) = 0.383303 (mm) < 0.490855 (mm) This change is more positive than the original plan Figure Test result of response accuracy Case 3: Maintain tolerances of variables as described above but change dimensions of links shown in table below and ensure that robot’s range remains constant d1 = 290 (mm) a1 = 30 (mm) a2 = 270 (mm) a3 = 90 (mm) d4 = 295 (mm) d6 = 125 (mm) The endpoint did not reach the desired accuracy with max (dist) = 0.627612 (mm)> 0.5 mm The number of points that exceeds the allowable error tolerance is 8.23% This change make the response accuracy decrease V CONCLUSION Using predictive resolution motors and sensors is common in robot designs To response the accuracy of endpoints, besides length tolerances of links, the nominal dimension of links also determines the accuracy and precision of robot The calculation of the reasonable values of the length of links (nominal dimension) brings obviously economic benefits because it does not increase the cost of fabrication and assembly The illustration of the calculation process and the comparison with the standard KUKA robot show that our idea is completely exact and highly industrialized The problem can also be developed in terms of determining the different rules of length ratios among links of different robots so that they meet one or more specific criteria of designers In case of increasing the accuracy of open-loop control without constraint of the manufacturing tolerance on a known configuration, we only have to simply adjust the nominal dimensions of links Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san FEE, 08 - 2018 193 Cơ học – Cơ khí động lực REFERENCES [1] S D Dao and K Abhary, “Determination of the Significance of Tolerance Parameters on Robot Performance Using Taguchi’s Tolerance Design Experiment,” Appl Mech Mater., vol 229–231, pp 2100–2105, 2012 [2] B K Rout and R K Mittal, “Tolerance design of manipulator parameters using design of experiment approach,” Struct Multidiscip Optim., vol 34, no 5, pp 445– 462, 2007 [3] Y H Andrew Liou, P P Lin, R R Lindeke, and H D Chiang, “Tolerance specification of robot kinematic parameters using an experimental design technique-the Taguchi method,” Robot Comput Integr Manuf., vol 10, no 3, pp 199–207, 1993 [4] B K R R K Mittal, “Simultaneous selection of optimal parameters and tolerance of manipulator using evolutionary optimization technique,” pp 513–528, 2010 [5] T T Trung, L W Guang, and P T Long, “Tolerance Design of Robot Parameters Using Generalized Reduced Gradient Algorithm,” vol 5, no 2, 2017 [6] L.Y.H Andrew, P.P Lin, R.R Lindeke, H.D Chiang, “Tolerance specification of robot kinematic parameters using an experimental design technique the Taguchi method,” Robotics and Computer-Integrated Manufacturing, vol 10, no 3(1993), pp 199-207 [7] B.K Rout, R.K Mittal, “Tolerance design of robot parameters using Taguchi method,” Mechanical Systems and Signal Processing, vol 20, no (2006), pp 1832-52 [8] B.K Rout, R.K Mittal, “Tolerance design of manipulator parameters using design of experiment approach,” Structural and Multidisciplinary Optimization, vol 34, no (2007) pp 445-462 [9] Phạm Thành Long, Lê Thị Thu Thủy, “Xác định hệ số đồng dạng dung sai tính tốn độ xác robot theo lớp kết cấu,” Hội nghị khoa học tồn Quốc “ứng dụng cơng nghệ cao vào thực tiễn”, Thái Nguyên, 5/2017, pp.40-48 TÓM TẮT XÁC ĐỊNH KÍCH THƯỚC KHÂU HỢP LÝ TRONG THIẾT KẾ KẾT CẤU ROBOT Bài báo giới thiệu luận điểm xác định kích thước khâu hợp lý thiết kế robot chuỗi động học hở Khi hai robot có cấu hình tầm với nhau, độ phân giải biến suy rộng chiều dài khâu khác đáp ứng độ xác điểm cuối khác Như vậy, phương án hợp lý phương án có kích thước làm gia tăng độ xác mong muốn mà khơng tăng chi phí chế tạo phần cứng robot Kích thước khâu hợp lý nhận biết thơng qua ấn định tỉ lệ phân chia chiều dài khâu khác sau tiến hành khảo sát chúng điều kiện biên, mơ hình động học robot sử dụng giai đoạn Việc ứng dụng luận điểm giới thiệu báo vào thiết kế đem lại chi phí hợp lý khơng đòi hỏi tăng cường độ phân giải biến khớp điều khiển robot, khơng đòi hỏi xiết chặt dung sai chế tạo khí Từ khóa: Chiều dài khâu hợp lý; Bài toán động học; Dung sai chế tạo; Robot chuỗi động học hở Received date, 18th April, 2018 Revised manuscript, 10th September, 2018 Published, 20th September, 2018 Author affiliations: * 194 Thai Nguyen University of Technology Corresponding author: hanthuyngoc@gmail.com L T T Thuy, P T Long, “Determining reasonable dimensions in robot structure design.” ... Determining reasonable dimensions in robot structure design. ” Nghiên cứu khoa học công nghệ Case 2: Maintain tolerances of variables as described above but change dimensions of links shown in. .. always in the given error sphere with the given radius 190 L T T Thuy, P T Long, Determining reasonable dimensions in robot structure design. ” Nghiên cứu khoa học công nghệ III BASIC OF DETERMINING. .. Nguyen University of Technology Corresponding author: hanthuyngoc@gmail.com L T T Thuy, P T Long, Determining reasonable dimensions in robot structure design. ”