1、1,中南大学 蔡自兴，谢 斌 zxcai, 2010,机器人学基础 第二章 数学基础,1,Ch.2 Mathematical Basis,Fundamentals of Robotics,Fundamentals of Robotics,2,Review,2, Course Schedule Top 10 Robotics News of 2008 Development of Robotics Structure, Feature, and Classification of Robots Robotics and AI,Fundamentals of Robotics,3,Content
2、s, Representation of Position and Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation,3,Ch.2 Mathematical Foundations,4,4,Ch.2 Mathematic Basis,2.1 Representation of Position and Attitude 位置和姿态的表示Description of Position,2.1 Represent
3、ation of Position and Attitude,5,5,2.1 Representation of Position and Attitude,Description of Orientation,2.1 Representation of Position and Attitude,6,6,Description of Frames相对参考系A，坐标系B的原点位置和坐标轴的方位，分别由位置矢量(Position Vector) 和旋转矩阵(Rotation Matrix) 描述。这样，刚体的位姿（位置和姿态）可由坐标系B来描述，即,2.1 Representation of P
4、osition and Attitude,2.1 Representation of Position and Attitude,7,Contents, Representation of Position and Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation,7,Ch.2 Mathematical Foundations,8,8,2.2 Coordinate Transformation 坐标变换,平移
5、坐标变换 (Translation Transform),2.2 Coordinate Transformation,9,9,旋转坐标变换 (Rotation Transform),2.2 Coordinate Transformation,2.2 Coordinate Transformation,10,10,Rotation about an axis,2.2 Coordinate Transformation,2.2 Coordinate Transformation,11,11,2.2 Coordinate Transformation,Rotation about an axis,2
6、.2 Coordinate Transformation,12,12,复合变换 (Composite Transform),2.2 Coordinate Transformation,2.2 Coordinate Transformation,13,例2.1 已知坐标系B的初始位姿与A重合，首先B相对于坐标系A的zA轴转30，再沿A的xA轴移动12单位，并沿A的yA轴移动6单位。求位置矢量ApB0和旋转矩阵 。假设点p在坐标系B的描述为Bp=3,7,0T，求它在坐标系A中的描述Ap。,13,2.2 Coordinate Transformation,解：,2.2 Coordinate Tran
7、sformation,14,例2.1 已知坐标系B的初始位姿与A重合，首先B相对于坐标系A的zA轴转30，再沿A的xA轴移动12单位，并沿A的yA轴移动6单位。求位置矢量ApB0和旋转矩阵 。假设点p在坐标系B的描述为Bp=3,7,0T，求它在坐标系A中的描述Ap。,14,2.2 Coordinate Transformation,解：,2.2 Coordinate Transformation,15,例2.1 已知坐标系B的初始位姿与A重合，首先B相对于坐标系A的zA轴转30，再沿A的xA轴移动12单位，并沿A的yA轴移动6单位。求位置矢量ApB0和旋转矩阵 。假设点p在坐标系B的描述为Bp
8、=3,7,0T，求它在坐标系A中的描述Ap。,15,2.2 Coordinate Transformation,解：,2.2 Coordinate Transformation,16,Contents, Representation of Position and Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation,16,Ch.2 Mathematical Foundations,17,已知一直角坐标系中的某点
9、坐标，则该点在另一直角坐标系中的坐标可通过齐次坐标变换求得。 所谓齐次坐标就是将一个原本是 n 维的向量用一个 n+1 维向量来表示。一个向量的齐次表示是不唯一的，比如齐次坐标8,4,2、4,2,1表示的都是二维点2,1。 齐次坐标提供了用矩阵运算把二维、三维甚至高维空间中的一个点集从一个坐标系变换到另一个坐标系的有效方法。,2.3 Homogeneous Transformation of the Coordinate Frames 齐次坐标变换,17,2.3 Homogeneous Transformation,18,18,Homogeneous TransformationMatrix
10、Form：,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous Transformation,19,例2.2 已知坐标系B的初始位姿与A重合，首先B相对于坐标系A的zA轴转30，再沿A的xA轴移动12单位，并沿A的yA轴移动6单位。假设点p在坐标系B的描述为Bp=3,7,0T，用齐次变换方法求它在坐标系A中的描述Ap。,19,解：,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous Transformation,
11、20,例2.2 已知坐标系B的初始位姿与A重合，首先B相对于坐标系A的zA轴转30，再沿A的xA轴移动12单位，并沿A的yA轴移动6单位。假设点p在坐标系B的描述为Bp=3,7,0T，用齐次变换方法求它在坐标系A中的描述Ap。,20,解：,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous Transformation,21,21,Homogeneous Transformation of Translation 空间中的某点用矢量ai+bj+ck描述，该点也可表示为：对已知矢量 u=x,y,z,wT
12、进行平移变换所得的矢量 v 为：,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous Transformation,22,22,Homogeneous Transformation of Rotation,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous Transformation,23,23,例2.3 已知点 u=7i+3j+2k，将 u绕 z 轴旋转90得到点 v，再将点 v 绕 y轴旋转90得到点w，求
13、点v、w的坐标。,解：,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous Transformation,24,24,例2.3 已知点 u=7i+3j+2k，将 u绕 z 轴旋转90得到点 v，再将点 v 绕 y轴旋转90得到点w，求点v、w的坐标。,解：,如果把上述两变换组合在一起,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous Transformation,25,25,若改变旋转次序，首先使 u 绕 y
14、轴旋转90，再绕 z 轴旋转90，会使 u 变换至与 w 不同的位置w1。,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous Transformation,26,26,例2.4 已知点 u=7i+3j+2k，将 u绕 z 轴旋转90得到点 v，再将点 v 绕 y轴旋转90得到点w，最后进行平移变换4i-3j+7k，求最终的坐标。,解：,把上述三变换组合在一起,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous
15、Transformation,27,27,例2.4 已知点 u=7i+3j+2k，将 u绕 z 轴旋转90得到点 v，再将点 v 绕 y轴旋转90得到点w，最后进行平移变换4i-3j+7k，求最终的坐标。,解：,则,2.3 Homogeneous Transformation of the Coordinate Frames,2.3 Homogeneous Transformation,28,28,We are given a single frame A and a position vector AP described in this frame. We then transform A
16、P by first rotating it about by an angle , then rotating about by an angle . Determine the 3 3 rotation matrix operator, , which describes this transformation.,例2.5 （Example 2.5）,2.3 Homogeneous Transformation,Solution: Suppose the first rotation converts AP -AP, and the second rotation converts AP-
17、A P”. Then we have:,29,29,We are given a single frame A and a position vector AP described in this frame. We then transform AP by first rotating it about by an angle , then rotating about by an angle . Determine the 3 3 rotation matrix operator, , which describes this transformation.,Example 2.5,2.3
18、 Homogeneous Transformation,Solution:,30,Contents, Representation of Position and Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation,30,Ch.2 Mathematical Foundations,31,31,2.4 Transformation and Inverse One of Object 物体的变换及逆变换,Descr
19、iption of position of an object 我们可以用描述空间一点的变换方法来描述物体在空间的位置和方向。例如，下图所示物体可由坐标系内固定该物体的六个点来表示。,2.4 Transformation of Objects,32,32,2.4 Transformation and Inverse Transformation of Object 物体的变换及逆变换,Description of position of an object如果首先让物体绕z轴旋转90，接着绕y轴旋转90，再沿x轴方向平移4个单位，则该变换可描述为：,2.4 Transformation of
20、Objects,33,33,2.4 Transformation and Inverse Transformation of Object 物体的变换及逆变换,Description of position of an object上述楔形物体的六个点变换如下：,2.4 Transformation of Objects,34,34,Compound Transformation 给定坐标系A，B和C，若已知B 相对A 的描述为 ， C 相对B的描述为 ，则,2.4 Transformation and Inverse Transformation of Object,定义复合变换 ：,2.4
21、 Transformation of Objects,35,35,Inverse Transformation从坐标系 B 相对A的描述 ，求得坐标系 A 相对B的描述 ，是齐次变换求逆问题。对于给定的 ，求解 ，等价于给定 和计算 和 。,2.4 Transformation and Inverse Transformation of Object,2.4 Transformation of Objects,36,36,Inverse Transformation对于给定的 ，求解 ，等价于给定 和计算 和 。,2.4 Transformation and Inverse Transform
22、ation of Object,2.4 Transformation of Objects,37,37,Given a transformation matrix:Find,例2.6 （Example 2.6）,2.3 Homogeneous Transformation,Solution:,38,38,Transform Equations,2.4 Transformation and Inverse Transformation of Object,2.4 Transformation of Objects,39,Contents, Representation of Position a
23、nd Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation,39,Ch.2 Mathematical Foundations,40,40,设想 f 为坐标系C的 z 轴上的单位矢量,2.5 General Rotation Transformation,2.5 General Rotation Transformation 通用旋转变换,绕矢量 f 旋转等价于绕坐标系 C的 z 轴旋转,41,41,如果已知以参考
24、坐标描述的坐标系T，那么能够求得以坐标系C描述的另一坐标系S，因为 T 绕 f 旋转等价于绕坐标系 C 的 z 轴旋转：,2.5 General Rotation Transformation,2.5 General Rotation Transformation,42,42,2.5 General Rotation Transformation,2.5 General Rotation Transformation,43,43,Equivalent rotation angle and axis给出任一旋转变换，能够由式(2.45)求得进行等效旋转角的转轴,2.5 General Rotat
25、ion Transformation,2.5 General Rotation Transformation,44,44,Equivalent rotation angle and axis若已知旋转变换：,2.5 General Rotation Transformation,2.5 General Rotation Transformation,令,45,45,Equivalent rotation angle and axis,2.5 General Rotation Transformation,2.5 General Rotation Transformation,46,46,2.6 Summary 小结,Mathematical Foundations of Robotics Representation of Position and Attitude Coordinate and Homogeneous Transformation Transformation of Object General Rotation Transformation Mathematical Tools for Kinematics, Dynamics, and Control of Robotics.,2.6 Summary,
