本論文在討論六軸關節式機器人，從正向運動學的方程式推得逆向運動學的微分方程。並且透過數值方法計算，在達到希望的位置與姿態，所對應六個軸的角度為何，簡單來說就是用數值法解逆向運動學。
為了建立六軸關節式機械手臂的數學模型，我們先找出在空間中移動的三軸手臂的正向運動學方程，並將此方程進行微分，得到逆向運動學的微分方程。給定一條運動路線，就有微分方程與其初始條件。用Euler method與Runge-Kutta數值法，解出每個時刻所對應六個軸的角度，並帶回正向運動學方程驗證。
再來為了推廣到六軸，後三軸控制了手臂的旋轉，所以我們訂了手臂的坐標系。並且使用無窮小旋轉的概念，推導出後三軸旋轉時的非線性微分方程組。再來分成兩個部份分析，其一是位置不動，只有姿態改變、其二是位置移動但姿態不變。綜合上述兩種模式，就可用數值法推算出機械手臂在空間中，任意位置與姿態所對應六個軸的角度。
最後將計算的結果輸入上銀機械手臂，實際測試手臂是否依照我們希望的軌跡與姿態運動。

In this paper, the six-axis articulated robot is discussed, and the differential equation of inverse kinematics is derived from the equation of forwarding kinematics. And through numerical calculation, when the desired position and attitude are achieved, what are the angles of the corresponding six axes, in simple terms, the numerical method is used to solve the inverse kinematics.

To establish the mathematical model of the six-axis articulated mechanical arm, we first find out the forward kinematics equation of the three-axis arm moving in space and differentiate this equation to obtain the differential equation of inverse kinematics. Given a motion route, there are differential equations and their initial conditions. Using the Euler and Runge-Kutta method, the angles of the six axes corresponding to each moment are solved and brought back to the forward kinematics equation for verification.

To generalize to six axes, the last three axes control the rotation of the arm, so we set the coordinate system of the arm. And using the concept of infinitesimal rotation, the nonlinear differential equations system of the latter three-axis rotation is deduced. Then it is divided into two parts for analysis, one is that the position does not move, only the pose changes, and the other is that the position moves but the pose does not change. Combining the above two modes, the numerical method can be used to calculate the angles of the six axes corresponding to any position and attitude of the robotic arm in space. Finally, input the calculated results into the HIWIN robotic arm, and test whether the arm moves according to the trajectory and pose we want.

[1] 上銀科技(2022)。關節式機器手臂RA605-710-GC本體使用手冊。台中市：上銀科技股份有限公司。
[2] 上銀科技(2022)。機器人軟體開發套件使用手冊。台中市：上銀科技股份有限公司。
[3] Stephen H.Friedberg,Arnold J.Insel,Lawrence E.Spence，Linear Algebra.Internation 4th Edition.Prentice Hall,New Jersey,2003.
[4] Kenneth Hoffman,Ray Kunze，Linear Algebra.2nd Edition. Edition.Prentice Hall,New Jersey ,1971.
[5] Richard L.Burden,J.Douglas Faires，Numerical Analysis 9th Editon.Cengage Learning,Boston,2011
[6] James Stewart.Calculus.8th Edition.Cengage Learning,Canada,2015.
[7] 佘步雲(編譯)(2016)。C++程式設計藝術(第九版)。新北市：台灣培生教育。(Paul Deitel,Harvey Deitel)
[8] Richard M.Murray,Zexiang Li,S.Shankar Sastry，A Mathematical Introduction to Robotic Manipulation.CRC press,Berkeley CA,1994.
[9] OOSGA，徹底理解工業4.0的定義、九大科技、以及八大應用領域
https://zh.oosga.com/pillars/industry40/
[10] OOSGA，工業自動化是甚麼？徹底解析自動化工廠的背後技術
[11] 大和有話說，工業機器人之五大機械結構及關鍵零組件分析
https://reurl.cc/1ZyX1m
[12] 林沛群，機器人學(Robotics(1))，Coursera課程
[13] 蔡自兴(2000)，機器人學，北京：清華大學出版社。