機(jī)器人學(xué)基礎(chǔ)數(shù)學(xué)基礎(chǔ)蔡自興PPT教案

1、會計學(xué)1機(jī)器人學(xué)基礎(chǔ)數(shù)學(xué)基礎(chǔ)蔡自興機(jī)器人學(xué)基礎(chǔ)數(shù)學(xué)基礎(chǔ)蔡自興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頁/共47頁3Contents Representation of Position and Attitude Coordinate Transformation Homogeneous Transfor

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

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

4、ons第6頁/共47頁882.2 Coordinate Transformation(2.10)BoABAppp2.3圖 平移變換第7頁/共47頁99(2.11)2.4圖 旋轉(zhuǎn)變換ppBABAR2.2 Coordinate Transformation2.2 Coordinate Transformation第8頁/共47頁101010000),(cssczRz繞 軸旋轉(zhuǎn)2.2 Coordinate TransformationcosABppxxsinABppyx00ABppzx2.2 Coordinate Transformation第9頁/共47頁1111(2.8)10000),(cssc

5、zRcsscxR00001),(csscyR00100),(2.7)(2.6)z繞 軸旋轉(zhuǎn)2.2 Coordinate TransformationRotation about an axis2.2 Coordinate Transformation第10頁/共47頁1212(2.13)2.5圖 復(fù)合變換BoABABARppp2.2 Coordinate Transformation2.2 Coordinate Transformation第11頁/共47頁1313ABR2.2 Coordinate Transformation解：303000.8660.5 0,30303000.50.866

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

7、f Position and Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation 16Ch.2 Mathematical Foundations第15頁/共47頁172.3 Homogeneous Transformation of the Coordinate Frames 齊次坐標(biāo)變換172.3 Homogeneous Transformationxpyz 1xwxywypzwzw 第16頁/共47頁181

8、8(2.14)|0 0 0|111ABAABBoRpppppBABAT(2.15)BoABABARppp2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第17頁/共47頁1919解：2.3 Homogeneous Transformation of the Coordinate Frames0.8660.5 0120.50.866 060010010001AAABBoBRTp2.3 Homogeneous Transformation第18頁/共47頁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頁/共47頁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頁/共47頁2222100000000001),(csscxRot100000001000),(csscyRot100001000000),(cssczRot2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第21頁/共47頁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頁/共47頁2424解： 如果把上述兩變換組合在一起,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頁/共47頁25252.6圖 旋轉(zhuǎn)次序?qū)ψ儞Q結(jié)果的影響(a),90,

12、90Rot yRot z(b),90,90Rot zRot y2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第24頁/共47頁2626解： 把上述三變換組合在一起4, 3,7,90,900 0 141 0 030 1 070 0 01TransRot yRot z2.7圖 平移變換和旋轉(zhuǎn)變換的組合2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第25頁

13、/共47頁2727解：4, 3,7,90,900 0 14761 0 03340 1 072100 0 0111tTransRot yRot zu 則2.7圖 平移變換和旋轉(zhuǎn)變換的組合2.3 Homogeneous Transformation of the Coordinate Frames2.3 Homogeneous Transformation第26頁/共47頁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頁/共47頁2929Example 2.52.3 Homogeneous TransformationAZAY,R Solution: AAyzPRRP ,yzRRR 00,010000001cscsc cc ssRscscscs cs sc 第28頁/共47頁30Contents Representation of Position and Attitude Coordinate Transformation Homogeneous

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

16、000141002.4 Transformation of Objects第32頁/共47頁3434(2.28)ppCBCBTpppCBCABBABATTT(2.27)2.4 Transformation and Inverse Transformation of ObjectABTBCT定義復(fù)合變換 ：101010BoACoBABBCABCoBBCBoAABBCABACRRRRRTTTpppp2.4 Transformation of Objects第33頁/共47頁35352.4 Transformation and Inverse Transformation of ObjectABTB

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

18、eneous Transformation100102030001ABcsTscSolution:1001023|0230 |10001ATBBBABoAcscsRRTscscApBAT第36頁/共47頁38382.9圖 變換方程及其有向變換圖(a) 機(jī)械手與環(huán)境間的運(yùn)動關(guān)系(b) 對應(yīng)的有向變換圖2.4 Transformation and Inverse Transformation of Object2.4 Transformation of ObjectsTTTTGTSGBSBT(2.37)第37頁/共47頁39Contents Representation of Position a

19、nd Attitude Coordinate Transformation Homogeneous Transformation Transformation of Object General Rotation Transformation 39Ch.2 Mathematical Foundations第38頁/共47頁4040(2.39)2.5 General Rotation Transformation1000000zzzyyyxxxaonaonaonCkjifzyxaaa(2.38)2.5 General Rotation Transformation 通用旋轉(zhuǎn)變換通用旋轉(zhuǎn)變換 繞矢

20、量 f 旋轉(zhuǎn)等價于繞坐標(biāo)系 C的 z 軸旋轉(zhuǎn)(2.40),(),(zcRotRotf第39頁/共47頁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頁/共47頁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頁/共47頁43432.5 General Rotation Transformation 2.5 Genera