机器人学基础数学基础蔡自兴PPT教案

1、会计学1机器人学基础数学基础蔡自兴机器人学基础数学基础蔡自兴2Review2 Course Schedule Top 10 Robotics News of 2008 Development of Robotics Structure, Feature, and Classification of Robots Robotics and AIFundamentals of Robotics第1页/共47页3Contents Representation of Position and Attitude Coordinate Transformation Homogeneous Transfor

2、mation Transformation of Object General Rotation Transformation 3Ch.2 Mathematical Foundations第2页/共47页44zyxApppp(2.1)2.1图 位置表示2.1 Representation of Position and Attitude第3页/共47页552.2图 方位表示333231232221131211rrrrrrrrrzyxRBABABAAB1;1AATABBBRRR2.1 Representation of Position and Attitude第4页/共47页66来描述，即(2

3、.9)BoAABRBpABopABR2.1 Representation of Position and Attitude 2.1 Representation of Position and Attitude第5页/共47页7Contents Representation of Position and Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation 7Ch.2 Mathematical Foundati

4、ons第6页/共47页882.2 Coordinate Transformation(2.10)BoABAppp2.3图 平移变换第7页/共47页99(2.11)2.4图 旋转变换ppBABAR2.2 Coordinate Transformation2.2 Coordinate Transformation第8页/共47页101010000),(cssczRz绕 轴旋转2.2 Coordinate TransformationcosABppxxsinABppyx00ABppzx2.2 Coordinate Transformation第9页/共47页1111(2.8)10000),(cssc

5、zRcsscxR00001),(csscyR00100),(2.7)(2.6)z绕 轴旋转2.2 Coordinate TransformationRotation about an axis2.2 Coordinate Transformation第10页/共47页1212(2.13)2.5图 复合变换BoABABARppp2.2 Coordinate Transformation2.2 Coordinate Transformation第11页/共47页1313ABR2.2 Coordinate Transformation解：303000.8660.5 0,30303000.50.866

6、 0001001ABcsRR zsc 2.2 Coordinate Transformation第12页/共47页1414ABR2.2 Coordinate Transformation解：01260ABp2.2 Coordinate Transformation第13页/共47页1515ABR2.2 Coordinate Transformation解：00.8660.5 031211.0980.50.866 07613.562001000AABABBpR pp 2.2 Coordinate Transformation第14页/共47页16Contents Representation o

7、f Position and Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation 16Ch.2 Mathematical Foundations第15页/共47页172.3 Homogeneous Transformation of the Coordinate Frames 齐次坐标变换172.3 Homogeneous Transformationxpyz 1xwxywypzwzw 第16页/共47页181

8、8(2.14)|0 0 0|111ABAABBoRpppppBABAT(2.15)BoABABARppp2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第17页/共47页1919解：2.3 Homogeneous Transformation of the Coordinate Frames0.8660.5 0120.50.866 060010010001AAABBoBRTp2.3 Homogeneous Transformation第18页/共47页2020解：2.3 H

9、omogeneous Transformation of the Coordinate Frames0.8660.5 012311.0980.50.866 06713.562001000000111AABBT pp2.3 Homogeneous Transformation第19页/共47页2121(2.20)1000100010001),(cbacbaTrans100/010/( , , )001/00011axxawx wabyybwy wbvTrans a b cuczzcwz wcww 2.3 Homogeneous Transformation of the Coordinate F

10、rames2.3 Homogeneous Transformation第20页/共47页2222100000000001),(csscxRot100000001000),(csscyRot100001000000),(cssczRot2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第21页/共47页2323c90s900 073s90c900 037,90001 022000 111vRot zu 解：c900 s90032010077,90s900 c9002300011

11、1wRot yv 2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第22页/共47页2424解： 如果把上述两变换组合在一起,90,900 0 1 0721 0 0 0370 1 0 0230 0 0 111wRot yRot zu 2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第23页/共47页25252.6图 旋转次序对变换结果的影响(a),90,

12、90Rot yRot z(b),90,90Rot zRot y2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第24页/共47页2626解： 把上述三变换组合在一起4, 3,7,90,900 0 141 0 030 1 070 0 01TransRot yRot z2.7图 平移变换和旋转变换的组合2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第25页

13、/共47页2727解：4, 3,7,90,900 0 14761 0 03340 1 072100 0 0111tTransRot yRot zu 则2.7图 平移变换和旋转变换的组合2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第26页/共47页2828例例2.5 （Example 2.5）2.3 Homogeneous TransformationAZAY,R Solution: Suppose the first rotation converts AP -AP,

14、and the second rotation converts AP-A P”. Then we have: AAzPRP AAyPRP AAyzPRRP第27页/共47页2929Example 2.52.3 Homogeneous TransformationAZAY,R Solution: AAyzPRRP ,yzRRR 00,010000001cscsc cc ssRscscscs cs sc 第28页/共47页30Contents Representation of Position and Attitude Coordinate Transformation Homogeneous

15、 Transformation Transformation of Object General Rotation Transformation 30Ch.2 Mathematical Foundations第29页/共47页31312.4 Transformation of Objects第30页/共47页3232001410004,0,0,90,9001000001TTransRot yRot z2.4 Transformation of Objects第31页/共47页333311111144000011111144664411111100220044000011111110000010

16、000141002.4 Transformation of Objects第32页/共47页3434(2.28)ppCBCBTpppCBCABBABATTT(2.27)2.4 Transformation and Inverse Transformation of ObjectABTBCT定义复合变换 ：101010BoACoBABBCABCoBBCBoAABBCABACRRRRRTTTpppp2.4 Transformation of Objects第33页/共47页35352.4 Transformation and Inverse Transformation of ObjectABTB

17、ATABTBATABRABopBARBAop2.4 Transformation of Objects|0 0 0|1AAABBoBRTp|?0 0 0|1BBBAAoARTp第34页/共47页36362.4 Transformation and Inverse Transformation of ObjectABTBATABRABopBARBAopBBBoABoAoRAABpppBATAoABoBBoRR BAAppp|0 |1ATBBBABoARRTAp2.4 Transformation of Objects第35页/共47页3737例2.6 （Example 2.6）2.3 Homog

18、eneous Transformation100102030001ABcsTscSolution:1001023|0230 |10001ATBBBABoAcscsRRTscscApBAT第36页/共47页38382.9图 变换方程及其有向变换图(a) 机械手与环境间的运动关系(b) 对应的有向变换图2.4 Transformation and Inverse Transformation of Object2.4 Transformation of ObjectsTTTTGTSGBSBT(2.37)第37页/共47页39Contents Representation of Position a

19、nd Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation 39Ch.2 Mathematical Foundations第38页/共47页4040(2.39)2.5 General Rotation Transformation1000000zzzyyyxxxaonaonaonCkjifzyxaaa(2.38)2.5 General Rotation Transformation 通用旋转变换通用旋转变换 绕矢

20、量 f 旋转等价于绕坐标系 C的 z 轴旋转(2.40),(),(zcRotRotf第39页/共47页4141(2.41)CST TCS1S), z(CRotT),(RotfTC), z(CRotT),(Rot1f(2.42)2.5 General Rotation Transformation 2.5 General Rotation Transformation第40页/共47页4242(2.43)1C), z(CRot),(Rotf00( , )00001xxyxzzxyxyzyyzyxxzyyzxzzf f verscf f versf sf f versf sf f versf sf f verscf f versf sRotf f versf sf f versf sf f verscf(2.45)2.5 General Rotation Transformation 2.5 General Rotation Transformation第41页/共47页43432.5 General Rotation Transformation 2.5 Genera