Five-Axis Interpolation of Continuous Short Linear Trajectories for 3[PP]S-XY Hybrid Mechanism by Dual Bezier Blending

doi:10.1007/s12204-015-1688-6

Abstract

Abstract: A novel five-axis real-time interpolation algorithm for 3[PP]S-XY hybrid mechanism is proposed in this paper. In the algorithm, the five-axis tool path for controlling this hybrid mechanism is separated into two sub-paths. One sub-path describes the movement of 3[PP]S parallel kinematic mechanism module, and the other one describes the movement of XY platform. A pair of cubic Bezier curves is employed to smooth the corners in those two sub-paths. Based on the homogenous Jacobian matrix of 3[PP]S mechanism, a relationship between the position errors of every driving joint in hybrid mechanism and the position deviation of the tool tip center point at the moving platform is established. This relationship is used to estimate the approximation error for the corners smoothing according to the accuracy requirement of tool tip center in interpolation. Due to the high computational efficiency of this corner smoothing method, it is integrated into the look-ahead module of computer numerical control (CNC) system to perform online tool path smoothing. By performing the speed planning based on a floating window scheme, a jerk limited S-shape speed profile can be generated efficiently. On this basis, a realtime look-ahead scheme, which is comprised of path-smoothing and feedrate scheduling, is developed to acquire a speed profile with smooth acceleration. A monotonic cubic spline is employed for synchronization between those two smoothed sub-paths in tool path interpolation. This interpolation algorithm has been integrated into our own developed CNC system to control a 3PRS-XY experimental instrument (P, R and S standing for prismatic, revolute and spherical, respectively). A club shaped trajectory is adopted to verify the smoothness and efficiency of the five-axis interpolator for hybrid mechanism control.

摘要： A novel five-axis real-time interpolation algorithm for 3[PP]S-XY hybrid mechanism is proposed in this paper. In the algorithm, the five-axis tool path for controlling this hybrid mechanism is separated into two sub-paths. One sub-path describes the movement of 3[PP]S parallel kinematic mechanism module, and the other one describes the movement of XY platform. A pair of cubic Bezier curves is employed to smooth the corners in those two sub-paths. Based on the homogenous Jacobian matrix of 3[PP]S mechanism, a relationship between the position errors of every driving joint in hybrid mechanism and the position deviation of the tool tip center point at the moving platform is established. This relationship is used to estimate the approximation error for the corners smoothing according to the accuracy requirement of tool tip center in interpolation. Due to the high computational efficiency of this corner smoothing method, it is integrated into the look-ahead module of computer numerical control (CNC) system to perform online tool path smoothing. By performing the speed planning based on a floating window scheme, a jerk limited S-shape speed profile can be generated efficiently. On this basis, a realtime look-ahead scheme, which is comprised of path-smoothing and feedrate scheduling, is developed to acquire a speed profile with smooth acceleration. A monotonic cubic spline is employed for synchronization between those two smoothed sub-paths in tool path interpolation. This interpolation algorithm has been integrated into our own developed CNC system to control a 3PRS-XY experimental instrument (P, R and S standing for prismatic, revolute and spherical, respectively). A club shaped trajectory is adopted to verify the smoothness and efficiency of the five-axis interpolator for hybrid mechanism control.

CLC Number:

SHI Jing (石璟), BI Qingzhen (毕庆贞), WANG Yuhan* (王宇晗). Five-Axis Interpolation of Continuous Short Linear Trajectories for 3[PP]S-XY Hybrid Mechanism by Dual Bezier Blending[J]. Journal of shanghai Jiaotong University (Science), 2016, 21(1): 90-102.

References 33

[1]	TUTUNEA-FATAN O R, BHUIYA M S H. Comparingthe kinematic efficiency of five-axis machinetool configurations through nonlinearity errors [J].Computer-Aided Design, 2011, 43(9): 1163-1172.
[2]	XIE F G, LIUX J, ZHANG H, et al. Design andexperimental study of the SPKM165, a five-axis serialparallelkinematic milling machine [J]. Science ChinaTechnological Sciences, 2011, 54(5): 1193-1205.
[3]	GELDART M, WEBB P, LARSSON H, et al. Adirect comparison of the machining performance of avariax 5 axis parallel kinetic machining centre withconventional 3 and 5 axis machine tools [J]. InternationalJournal of Machine Tools & Manufacture, 2003,43(11): 1107-1116.
[4]	HARIB K H, SHARIF ULLAH A M M, HAMMAMIA. A hexapod-based machine tool with hybridstructure:Kinematic analysis and trajectory planning[J]. International Journal of Machine Tools & Manufacture,2007, 47(9): 1426-1432.
[5]	ALTUZARRA O, MART′IN Y S, AMEZUA E, etal. Motion pattern analysis of parallel kinematic machines:A case study [J]. Robotics and Computer-Integrated Manufacturing, 2009, 25(2): 432-440.
[6]	KANAAN D, WENGER P, CHABLAT D. Kinematicanalysis of a serial-parallel machine tool: Theverne machine [J]. Mechanism and Machine Theory,2009, 44(2): 487-498.
[7]	LIU H, HUANG T, MEI J, et al. Kinematic designof a 5-DOF hybrid robot with large workspace/limbstrokeratio [J]. Journal of Mechanical Design, 2007,129(5): 530-536.
[8]	HUNAG T, LI M, ZHAO X, et al. Conceptual designand dimensional synthesis for a 3-DOF module ofthe TriVariant: A novel 5-DOF reconfigurable hybridrobot [J]. IEEE Transactions on Robotics, 2005, 21(3):449-456.
[9]	HUANG P, WANG J, WANG L, et al. Identificationof structure errors of 3-prs-xy mechanism with regularizationmethod [J]. Mechanism and Machine Theory,2011, 46(7): 927-944.
[10]	WAHL J. Articulated tool head [P]. United StatesPatent: US6431802B1. 2001-08-13.
[11]	LI Y G, HUANG T, LIU H T, et al. Design of a3-DOF PKM module for large aircraft component machining[C]//Proceedings of the 7th World Congresson Intelligent Control and Automation. Chongqing,China: IEEE, 2008: 370-375.
[12]	ERKORKMAZ K, YEUNG C H, ALTINTAS Y.Virtual CNC system. Part II. High speed contouringapplication [J]. International Journal of Machine Toolsand Manufacture, 2006, 46(10): 1124-1138.
[13]	YAU H T, WANG J B. Fast Bezier interpolatorwith real-time lookahead function for high-accuracymachining [J]. International Journal of Machine Toolsand Manufacture, 2007, 47(10): 1518-1529.
[14]	ZHANG L B, YOU Y P, HE J, et al. The transitionalgorithm based on parametric spline curve for highspeedmachining of continuous short line segments [J].The International Journal of Advanced ManufacturingTechnology, 2010, 52(1-4): 245-254.
[15]	JAHANPOUR J. High speed contouring enhancedwith C2 PH quintic spline curves[J]. Scientia Iranica,2012, 19(2): 311-319.
[16]	BI Q Z, WANGY H, ZHU L M, et al. Apracticalcontinuous-curvature B′ezier transition algorithm forhigh-speed machining of linear tool path [J]. LectureNotes in Artificial Intelligence, 2011, 7102: 465-476.
[17]	ZHAO H, ZHU L M, DING H. A real-timelook-ahead interpolation methodology with curvaturecontinuousB-spline transition scheme for CNC machiningof short line segments [J]. International Journalof Machine Tools and Manufacture, 2013, 65: 88-98.
[18]	PECHARD P Y, TOURNIER C, LARTIGUE C,et al. Geometrical deviations versus smoothness in 5-axis high-speed flank milling [J]. International Journalof Machine Tools and Manufacture, 2009, 49(6): 454-461.
[19]	ZHENG G, BI Q Z, ZHU L M. Smooth tool path generationfor five-axis flank milling using multi-objectiveprogramming [J]. Journal of Engineering Manufacture,2011, 226(2): 247-254.
[20]	BEUDAERT X, PECHARD P Y, TOURNIER C.5-Axis tool path smoothing based on drive constraints[J]. International Journal of Machine Tools and Manufacture,2011, 51(12): 958-965.
[21]	SIEMENS 6FC5095-0AB10-0BP1, Milling with SINUMERIK5-axis machining [S].
[22]	WANG L, CAO J. A look-ahead and adaptive speedcontrol algorithm for high-speed CNC equipment [J].TheInternational Journal of Advanced ManufacturingTechnology, 2012, 63(5-8): 705-717.
[23]	LIU X J, BONEV I A. Orientation capability, erroranalysis, and dimensional optimization of two articulatedtool heads with parallel kinematics [J]. Journal ofManufacturing Science and Engineering, 2009, 130(1):011015-1-9.
[24]	BRIOT S, BONEV I A. Singularity analysis of zerotorsionparallel mechanisms [C]//IEEE/RSJ InternationalConference on Intelligent Robots and Systems.Nice, France: IEEE, 2008: 1952-1957.
[25]	WANG Y, HUANG T, GOSSELIN C M. Interpolationerror prediction of a three-degree parallel kinematicmachine [J]. Journal of Mechanical Design, 2004,126(5): 932-937.
[26]	HUANG P, WANG J S, WANG L P, et al. Identificationof structure errors of 3-PRS-XY mechanismwith Regularization method [J]. Mechanism and MachineTheory, 2011, 46(7): 927-944.
[27]	FAN L Z, ELATTA A Y, LI X P. Kinematic calibrationfor a hybrid 5 DOF manipulator based on 3-rpsin-actuated parallel manipulator [J]. The InternationalJournal of Advanced Manufacturing Technology, 2004,25(7-8): 730-734.
[28]	SHI J, WANGY H, ZHANG G, et al. Optimal designof 3-DOF PKM module for friction stir welding [J].The International Journal of Advanced ManufacturingTechnology, 2013, 66(9-12): 1879-1889.
[29]	SHI J, WANG Y H, HAN X H. Design a hybridmachine tool for plane curve friction stir welding[C]//Proceedings of 2011 International Conference onMechanical Engineering and Technology. London, UK:[s. n], 2011: 113-118.
[30]	FRITSCH F N, CARLSON R E. Monotone piecewisecubic interpolation [J]. SIAM Journal on NumericalAnalysis, 1980, 17 (2): 238-246.
[31]	LENG H B, WU Y J, PAN X H. Research on flexibleacceleration and deceleration method of NC system[C]//International Technology and Innovation Conference.[s. l.]: IEEE, 2007: 1953-1956.
[32]	CHEN J H, YEH S S, SUN J T. An S-curve acceleration/deceleration design for CNC machine tools usingquintic feed rate function [J]. Computer-Aided Design& Applications, 2011, 8(4): 583-592.
[33]	LEE A, LIN M, PAN Y, et al. The feedrate schedulingof nurbs interpolator for cnc machine tools [J].Computer-Aided Design, 2011, 43(6): 612-628.