Fpga realization of forward kinematics and inverse kinematics for five axis articulated robot arm

Southern Taiwan University of Science and Technology Graduate School of Electrical Engineering Ph.D Dissertation FPGA Realization of Forward Kinematics and Inverse Kinematics for Five-Axis Articulated Robot Arm Graduate Student: Bui Thi Hai Linh Advisor：Ying-Shieh Kung July, 2015 i Acknowledgments I would like to express my deepest gratitude to all of my teachers in Department of Electrical Engineering-Southern Taiwan University, especially Prof Ying-Shieh Kung for his patience and guidance throughout my research since 2009 when I study the Master degree until now when I study the Doctoral degree His guidance and inspiration have provided an invaluable experience that will help me in my career I would also like to thank my lab-mates for their help and advice They are also the people who make me have unforgettable time and sweet memories in Taiwan Finally, I am grateful to my family for their constant love and support, and encouragement ii Abstract This dissertation presents a study of the forward and inverse kinematics for a fiveaxis articulated robot arm based on Field Programmable Gate Array (FPGA) technology Some trigonometric functions using Look-Up Table (LUT) and Taylor series method are used in hardware implementation to speed up of tracking the motion trajectories applied for forward kinematics and invers kinematics for five-axis articulated robot arm Firstly, the forward kinematics and inverse kinematics of five-axis articulated robot arm are derived Secondly, the computations algorithms and its hardware implementation are described Thirdly, Very high speed integrated circuits Hardware Description Language (VHDL) is applied to describe the overall hardware behavior of forward and inverse kinematics Additionally, Finite State Machine (FSM) is applied for reducing the hardware resource usage Further, to verify the correctness of the forward and inverse kinematics for five-axis articulated robot arm, a co-simulation work is constructed by Modelsim and Matlab Simulink The forward and inverse kinematics hardware is run by Modelsim and a test bench which generates stimulus to Modelsim and displays the output response that is taken in Simulink Under this design, the combination of the forward and inverse kinematics simulation for tracking the motion trajectories is adopted Fourthly, the design of forward and inverse kinematics IPs for five-axis robot arm is implemented by a single FPGA Additionally, a Nios II processor can be embedded into FPGA to construct a System on a Programmable Chip (SoPC) developing environment Programs in Nios II processor are coded in C language and IPs digital hardware is described by VHDL The Man-Machine Interface (MMI) developed by Visual Basic language which displays the results of computations kinematics in FPGA into decimal number for easy checking the correctness of results Therefore, the digital hardware/software co-design based on the SoPC is suitable for the development of the forward and inverse kinematics for five-axis articulated robot arm Finally, an experiment system has been built up as well as some experimental results have been demonstrated to verify the effectiveness and correctness of computations for forward and inverse kinematic which is applied to the real five-axis articulated robot arm iii 摘要 本文基於 FPGA（現場可程式邏輯閘陣列）技術提出了前向和逆向運動學的五 軸模組化機械臂之研究。首先，推導五軸關節型機械手臂的前向運動學和逆向運動 學。其次，針對演算法和硬體實現進行了描述。第三，超高速積體電路硬體描述語 言（VHDL）被用於描述前向和反向運動學的整體硬體行為。此外，運用有限狀態機 器（FSM）以減少硬體資源的使用。為了驗證五軸關節型機械手臂的前向和逆向運 動學的正確性，將結合 Modelsim 和 Matlab Simulink 進行模擬。前向和逆向運動學的 硬體是由 Modelsim 執行，而 Modelsim 測試平台產生的輸入訊號及輸出響應將顯示 在 Simulink 中。根據這樣的設計，前向及逆向運動學及運動軌跡追蹤可以在數微秒 內完成。第四，五軸機械臂的前向和逆向運動學 IP 設計將由單顆 FPGA 實現。此 外，Nios II 處理器可以嵌入到 FPGA 中以建構 SoPC（在系統可編程片）開發環境。 在 Nios II 處理器的應用程式以 C 語言撰寫，而 VHDL 將用於描述前向和逆向運動學 的數位硬體電路。另外，本論文也利用 visual basic 開發一套人機介面(MMI) ，此將 呈現前向和逆向運動學 IP 計算後的結果。因此，基於所述，SoPC 將適合發展五軸機 械手臂的前向和逆向運動學的硬體/軟體共同設計環境。最後，將建立一個實驗系統 並有實驗結果來證實應用於五軸模組化機械手臂的前向及逆向運動學計算的有效性 和正確性。 Table of Content Acknowledgments i Abstract iii 摘要 iv Table of Content iv List of Figures vii iv List of Tables x List of Symbols xi Chapter Introduction - 1.1 Research background & literature survey - 1.2 The motivation of the study - 1.3 The structure of thesis - Chapter Mathematical description of kinematics and motion trajectories planning - 2.1 Introduction of five-axis articulated robot arm - 2.2 Review of kinematics - 10 2.2.1 Rotating coordinate system - 12 2.2.2 Homogeneous coordinates - 13 2.2.3 Coordinates architecture - 15 2.2.4 Rotary joint coordinates architecture - 16 2.3 Robot kinematics of five-axis articulated robot arm - 18 2.3.1 Forward kinematics - 20 2.3.2 Inverse kinematics - 24 2.4 The computation of point-to-point motion control - 29 2.4.1 Five axes trajectory planning - 32 2.4.2 The formulas of motion trajectories - 33 2.4.2.1 Linear motion trajectory - 34 2.4.2.2 Circular motion trajectory - 34 2.4.2.3 Star motion trajectory - 35 2.4.2.4 Window motion trajectory - 36 Chapter Hardware implementation of forward kinematics and inverse kinematics - 38 3.1 Introduction and literature review - 38 3.2 Review of VHDL and Q-format design - 42 3.3 An example of Sum of Product - 44 3.4 Trigonometric functions - 48 3.4.1 Computation algorithm of Sine and Cosine functions - 49 3.4.2 Computation algorithm of Arctangent function using Taylor series expansion - 50 3.4.3 Computation of Arccosine function using Taylor series expansion method - 54 3.5 Design of hardware implementation for forward kinematics and inverse kinematics - 56 3.5.1 Forward kinematics and inverse kinematics design in VHDL using Q-format - 56 v 3.5.2 FSM for forward kinematics and inverse kinematics - 57 Chapter Modelsim/Simulink co-simulation of forward/inverse kinematics for five-axis articulated robot arm - 63 4.1 Introduction of Modelsim/Simulink co-simulation - 63 4.2 Co-Simulation cases using Modelsim/ Simulink - 67 4.2.1 Sum of Product simulation results - 68 4.2.2 Sine and cosine functions co-simulation results - 69 4.2.3 Arctangent and arccosine functions co-simulation results - 72 4.3 Modelsim/Simulink co-simulation of forward/inverse kinematics - 74 4.4 Simulation results in Modelsim/Simulink of tracking motion trajectories - 79 4.4.1 Linear motion trajectory - 81 4.4.2 Circular motion trajectory - 82 4.4.3 Star motion trajectory - 82 4.4.4 Window motion trajectory - 85 Chapter FPGA realization of forward/inverse kinematics for five-axis articulated robot arm - 88 5.1 Introduction - 88 5.2 Description of SoPC builder design - 89 5.2.1 DE2 115 board - 89 5.2.2 Nios II embedded processor - 92 5.3 FPGA implementation of forward kinematics and inverse kinematics - 95 5.4 Applying to real robot arm - 98 5.4.1 Hardware implementation system - 100 5.4.1.1 CAN bus interface - 100 5.4.1.2 Overall hardware system - 102 5.4.2 Experimental results - 104 5.4.2.1 Linear motion trajectory - 105 5.4.2.2 Circular motion trajectory - 106 5.4.2.3 Star motion trajectory - 106 5.4.2.4 Window motion trajectory - 107 Chapter Conclusion and future works - 112 6.1 Conclusion - 112 6.2 Future works - 113 References - 114 vi Biography - 122 Academic Publications - 123 - List of Figures Figure 2.1 Electrical · Rotary Actuators · Universal Rotary Actuators - Figure 2.2 The sectional diagram - Figure 2.3 The five-axis articulated robot arm - 10 Figure 2.4 The definition of standard Denavit-Hartenberg link parameters [37] - 11 Figure 2.5 The end-effector - 13 Figure 2.6 The coordinate architecture - 16 Figure 2.7 The location of three-vectors n, o, a - 16 Figure 2.8 Robot coordinates indicate - 17 Figure 2.9 The relationship coordinates between two joints - 17 Figure 2.10 The schematic representation of forward and inverse kinematics - 19 Figure 2.11 The link coordinate system of a five-axis articulated robot arm (general) - 19 Figure 2.12 The link coordinate system of a five-axis articulated robot arm (details) - 20 Figure 2.13 The base and first link coordinate schematic - 21 Figure 2.14 The first link and the second link coordinate schematic - 21 vii Figure 2.15 The sencond link and the third link coordinates schematic - 22 Figure 2.16 The third link and the fouth link coordinates schematic - 22 Figure 2.17 The fifth link and the fouth link coordinates schematic - 23 Figure 2.18 LFPB trajectory (a) velocity; (b) acceleration - 31 Figure 2.19 Minimum-time trajectory (a) velocity; (b) acceleration - 32 Figure 2.20 Circular motion trajectory tracking - 34 Figure 2.21 Star motion trajectory tracking - 36 Figure 2.22 Window motion trajectory tracking - 37 Figure 3.1 Parallel processing using three multipliers and two adders execute by one step - 46 Figure 3.2 Sequential processing using one multiplier and one adder execute by five step - 47 Figure 3.3 VHDL code for computing the sum of product - 47 Figure 3.4 Three cases for computing the sum of product - 48 Figure 3.5 FSM for computing the sine function - 49 Figure 3.6 FSM for computing the cosine function - 50 Figure 3.7 Compute   a tan 2( y / x ) of each region in X-Y coordinates - 52 Figure 3.8 The diagram to compute a tan 2( y / x ) function - 53 1 Figure 3.9 FSM to compute tan ( x) function - 53 Figure 3.10 Computing range of   cos 1 ( x) - 54 Figure 3.11 Hardware implementation diagram for arccosine function - 55 1 Figure 3.12 FSM for computing the cos ( x) function (range from 450 to 900) - 56 Figure 3.13 Block diagram for (a) forward kinematics and (b) inverse kinematics - 57 Figure 3.14 FSM for computing the Forward kinematics - 60 Figure 3.15 FSM for computing the Inverse kinematics - 61 Figure 3.16 Forward kinematics computations time in FPGA - 62 Figure 3.17 Inverse kinematics computations time in FPGA - 62 Figure 4.1 Project flow - 66 Figure 4.2 Simulation flow working library - 67 Figure 4.3 Modelsim co-simulation of Sum of Product with 16bit Q0 - 68 Figure 4.4 Modelsim co-simulation of Sum of Product with 16bit Q15 - 69 Figure 4.5 Modelsim co-simulation of Sum of Product with 16bit Q24 - 69 Figure 4.6 The co-simulation in Modelsim/Simulink of Sine function - 71 Figure 4.7 The co-simulation in Modelsim/Simulink of Cosine function - 71 Figure 4.8 The co-simulation in Modelsim/Simulink of Arctangent function - 73 Figure 4.9 The co-simulation in Modelsim/Simulink of Arccosine function - 73 Figure 4.10 Setting complier - 75 Figure 4.11 Setting the “220pack.vhd” and “220model.vhd” compile to library “lpm” - 75 Figure 4.12 Compiled all files successfully and run Modelsim - 76 Figure 4.13 Co-simulation architecture of forward kinematics using Modelsim/Simulink - 76 Figure 4.14 Co-simulation architecture of inverse kinematics using Modelsim/Simulink - 77 Figure 4.15 Forward kinematics and Inverse kinematics co-simulation in Modelsim/Simulink - 80 viii Figure 4.16 The results of linear motion trajectory co-simulation in Modelsim/Simulink - 81 Figure 4.17 Error of linear motion trajectory co-simulation in Modelsim/Simulink - 82 Figure 4.18 The results of circular motion trajectory co-simulation in Modelsim/Simulink - 83 Figure 4.19 Error of circular motion trajectory co-simulation in Modelsim/Simulink - 83 Figure 4.20 The results of star motion trajectory co-simulation in Modelsim/Simulink - 84 Figure 4.21 Error of star motion trajectory co-simulation in Modelsim/Simulink - 84 Figure 4.22 The results of window motion trajectory simulation in Modelsim/Simulink - 85 Figure 4.23 Error of window motion trajectory simulation in Modelsim/Simulink - 86 Figure 5.1 The architecture system of hardware implementation - 89 Figure 5.2 The DE2-115 board (top view) [76] - 91 Figure 5.3 The Nios embedded processor - 92 Figure 5.4 The Nios Development Tool Flow - 93 Figure 5.5 A Nios II system implemented on the DE2 board [76] - 94 Figure 5.6 Embedded components information flow - 95 Figure 5.7 Architecture of forward kinematics IP and inverse kinematics IP base on FPGA - 96 Figure 5.8 Design of forward kinematics IP, inverse kinematics IP and Nios II processor - 97 Figure 5.9 Nios II IDE environments - 97 Figure 5.10 The man-machine interface programmed in Visual Basic - 98 Figure 5.11 Five-axis articulated robot arm by PowerCube - 99 Figure 5.12 CAN-bus adaptor - 100 Figure 5.13 Block-circuit diagram of CAN-USB-Mini module - 101 Figure 5.14 Physical Connection for CAN bus [77] - 102 Figure 5.15 Block of system architecture - 103 Figure 5.16 Five-axis articulated robot arm and MMI - 104 Figure 5.17 The feedback of window motion trajectory on MMI - 104 Figure 5.18 The tracking response of linear motion trajectory by PowerCube robot arm - 105 Figure 5.19 Tracking error of linear motion trajectory - 106 Figure 5.20 The tracking response of circular motion trajectory by PowerCube robot arm - 107 Figure 5.21 Tracking error of circular motion trajectory - 108 Figure 5.22 The tracking response of star motion trajectory by PowerCube robot arm - 108 Figure 5.23 Tracking error of star motion trajectory - 109 Figure 5.24 The tracking response of window motion trajectory by PowerCube robot arm - 109 Figure 5.25 Tracking error of window motion trajectory - 110 - ix List of Tables Table 2.1 Denavit-Hartenberg parameters;their physical meaning, symbol and formaldefinition- 12 Table 2.2 Denavit-Hartenberg parameters - 20 Table 1The co-simulation Modelsim and Matlab Simulink of Sine function - 72 Table 4.2 The co-simulation Modelsim/Simulink of Cosine function - 72 Table 4.3 The evaluation results for arctangent function - 74 Table 4.4 The evaluation results for arccosine function - 74 Table 4.5 Two cases simulation results of inverse kinematics from Modelsim and Matlab - 78 Table 4.6 Two cases simulation results of forward kinematics from Modelsim and Matlab - 78 Table 4.7 Mean square error of simulation results - 87 Table 5.1 The datasheet of PowerCube modules type - 99 Table 5.2 Mean square error of experimental results - 110 - x x-axis error (mm) y-axis error (mm) z-axis error (mm) Star tracking error x-error -1 time (s) 6.2 y-error -2 time (s) 6.2 1.5 z-error -1.5 time (s) 6.2 Figure 5.23 Tracking error of star motion trajectory 365 command tracking z 360 355 350 345 -100 -150 350 300 -200 y 250 -250 200 -300 150 x Figure 5.24 The tracking response of window motion trajectory by PowerCube robot arm - 109 - x-axis error (mm) y-axis error (mm) Window tracking error x-error -1 time (s) 7.6 2.5 y-error z-axis error (mm) 2.5 time (s) 7.6 z-error -1 time (s) 7.6 Figure 5.25 Tracking error of window motion trajectory The experimental results of tracking motion trajectories are shown in Fig.5.18~Fig.5.25 which give the good performances of five-axis robot arm for tracking motion trajectories as well As we studied in Chapter 4, to evaluate the errors of tracking motion trajectories by real robot arm, MSE is considered here By collecting all of feedback data points of real robot arm and the Matlab commands, by comparing both of them, we obtain the different error and squared average the error data points of three axes x, y, z respectively The MSE of tracking responses by real robot arm are summarized in the Table 5.2 Table 5.2 Mean square error of experimental results (mm) Mean Square Error x-axis y-axis z-axis Linear 0.282 0.331 0.458 Circular 0.603 0.954 1.021 Star 0.623 1.125 0.936 Window 0.779 1.224 0.812 - 110 - According to the MSE error which measured in millimeter as shown in Table 5.2, the linear tracking response is the best performance and so on the circular tracking is better than star and window’s response; star tracking response gives smoother performance than window’s in x-axis and y-axis respectively The realization of forward kinematics and inverse kinematics for real robot arm demonstrated our correctness and effectiveness work as well as good performance results - 111 - Chapter Conclusion and future works 6.1 Conclusion The computations of forward kinematics and inverse kinematics for five-axis articulated robot arm based on FPGA have been successfully demonstrated in this dissertation Finite state machine is applied for reducing the hardware resource usage The design of forward/inverse kinematics IPs based on FPGA and the man-machine interface is developed The high speed computational power and reasonable accuracy apparently increases the motion performance of five-axis articulated robot arm for tracking motion trajectories Through the co-simulation of ModelSim and Matlab Simulink, the accuracy refer to some examples and some motion trajectories performances of forward kinematics and inverse kinematics with error of the end-effector endx, endy, endz are less than 0.06 mm and the error for θ1~θ5 are less than 0.010.The executing times for the computations of forward kinematics and inverse kinematics in FPGA are only 680ns and 940ns Further, the FPGA (Altera Cyclone IV) resource usage for the realization of the forward kinematics IP are 1,575 Les, 30,720 RAM bits while for the inverse kinematics IP are 9,400 Les, 84,224RAM bits The experimental results of error of the end-effector endx, endy, endz are less than mm when tracking the motion trajectories as well This dissertation summarized the completion of the research as following: The computations of forward kinematics and inverse kinematics for five-axis articulated robot arm are derived The digital hardware implementation of sine, cosine, artangent2 and arccosine are completed Co-simulation Modelsim/Simulink the combination of forward kinematics and inverse kinematics for tracking the motion trajectories is completed The implementation of forward kinematics and inverse kinematics IPs based on FPGA is completed Overall hardware system is built up with real five-axis articulated robot arm by Power Cube and man-machine interface is developed by Visual Basic language to able to perform and test the results of proposed forward and inverse kinematics - 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120 - [76] http://www.altera.com/ [77] Z L Song, “Research on Remote Control of Reconfigurable Modular Robotic System,” Master thesis, Mechanical Engineering Program University of Ontario Institute of Technology August, 2009 - 121 - Biography Name : Bui Thi Hai Linh Place of Birth : Thai Nguyen, Vietnam Date of Birth : January 31st, 1986 Education and Certificates ：Southern Taiwan University of Science and Technology [2011-2015] Doctoral Program in Electrical Engineering Southern Taiwan University of Science and Technology [2009-2011] Master Program in Electrical Engineering Thai Nguyen University of Technology [2004-2009] Undergraduate Program in Electrical Engineering - 122 - Academic Publications [1] Bui Thi Hai Linh and Ying-Shieh ung, “ igital Hardware Realization of Forward and Inverse Kinematics for a Five-A is Articulated Robot Arm,” Mathematical Problems in Engineering, Article ID 906505, in press, 2015 (SCI) [2] Ying-Shieh Kung , Ming-Kuang Wu , Hai Linh Bui Thi and , Tz-Han Jung , FengChi Lee and Wen-Chuan Chen , "FPGA-based hardware implementation of arctangent and arccosine functions for the inverse kinematics of robot manipulator,” Engineering Computations, Vol 31 Iss: 8, pp.1679-1690, 2014 (SCI) [3] Ying-Shieh Kung, Bui Thi Hai Linh, Ming-Kuang Wu, Feng-Chi Lee and WenChuan Chen, “FPGA-Realization of Inverse Kinematics Control IP for Articulated and SCARA Robot,” Design and Computation of Modern Engineering Materials, ISBN: 978-3-319-07383-5, DOI: 10.1007/978-3-319-07383-5_15, 2014 (EI) [4] Ying-Shieh Kung, Ming-Kuang Wu, Bui Thi Hai Linh, Tz-Han Jung, Shin-Hon Lee and Wen-Chuan Chen, “ esign of Inverse Manipulator,” CACS International inematics IP for a Si -Axis Articulated Automatic Control Conference, DOI: 10.1109/CACS.2013.6734150, pp.300-305, Taiwan, December 2012 - 123 - ... implementation of forward kinematics and inverse kinematics Chapter 4: Modelsim/Simulink co-simulation of forward/ inverse kinematics for five- axis articulated robot arm Chapter 5: FPGA- realization of forward/ inverse. .. up of tracking the motion trajectories applied for forward kinematics and invers kinematics for five- axis articulated robot arm Firstly, the forward kinematics and inverse kinematics of five- axis. .. description of the forward kinematics and inverse kinematics and its motion trajectory planning for five- axis articulated robot arm 2.1 Introduction of five- axis articulated robot arm A robot manipulator