Coordinate Systems. Ioannis Rekleitis

Transcription

1 Coodinate Systems Ioannis ekleitis

2 Position epesentation Position epesentation is: P p p p x y z P CS-417 Intoduction to obotics and Intelligent Systems

3 Oientation epesentations Descibes the otation of one coodinate system with espect to anothe CS-417 Intoduction to obotics and Intelligent Systems 3

4 otation Matix Wite the unit vectos of in the coodinate system of. otation Matix: CS-417 Intoduction to obotics and Intelligent Systems

5 Popeties of otation Matix T T 1 I 3 T CS-417 Intoduction to obotics and Intelligent Systems 5

6 Coodinate System Tansfomation M p p p 1 x y z 03 T 1 1 whee is the otation matix and T is the tanslation vecto CS-417 Intoduction to obotics and Intelligent Systems 6

7 otation Matix The otation matix consists of 9 vaiables but thee ae many constaints. The minimum numbe of vaiables needed to descibe a otation is thee. CS-417 Intoduction to obotics and Intelligent Systems 7

8 otation Matix-Single xis x y x 1 0 cos sin 0 sin cos cos 0 sin sin 0 cos cos sin 0 sin cos CS-417 Intoduction to obotics and Intelligent Systems 8

9 Fixed ngles One simple method is to pefom thee otations about the axis of the oiginal coodinate fame: -- fixed angles z y x cos cos cos sin sin sin cos cos sin cos sin sin sin cos sin sin sin cos cos sin sin cos cos sin sin cos sin cos cos Thee ae 1 diffeent combinations CS-417 Intoduction to obotics and Intelligent Systems 9

10 Invese Poblem Fom a otation matix find the fixed angle otations: cos sin thus: cos cos sin sin sin cos cos sin cos sin sin cos sin sin sin cos cos sin sin cos cos sin sin cos sin cos cos tan tan tan 1 3 cos cos cos cos CS-417 Intoduction to obotics and Intelligent Systems 10

11 Eule ngles : Stating with the two fames aligned fist otate about the axis then by the axis and then by the axis. The esults ae the same as with using fixed angle otation. Thee ae 1 diffeent combination of Eule ngle epesentations CS-417 Intoduction to obotics and Intelligent Systems 11

12 Eule ngles Taditionally the thee angles along the axis ae called oll Pitch and aw CS-417 Intoduction to obotics and Intelligent Systems 1

13 Eule ngles Taditionally the thee angles along the axis ae called oll Pitch and aw oll CS-417 Intoduction to obotics and Intelligent Systems 13

14 Eule ngles Taditionally the thee angles along the axis ae called oll Pitch and aw Pitch CS-417 Intoduction to obotics and Intelligent Systems 14

15 Eule ngles Taditionally the thee angles along the axis ae called oll Pitch and aw aw CS-417 Intoduction to obotics and Intelligent Systems 15

16 Eule ngle concens: Gimbal Lock Using the convention ( ) ( ) multiples of 360 (7 0 0 ) ( ) singula alignment (Gimbal lock) ( ) ( ) bistable flip CS-417 Intoduction to obotics and Intelligent Systems 16

17 xis-ngle epesentation epesent an abitay otation as a combination of a vecto and an angle V CS-417 Intoduction to obotics and Intelligent Systems 17

18 Quatenions e simila to axis-angle epesentation Two fomulations Classical ased on JPL s standads W. G. eckenidge Quatenions - Poposed Standad Conventions JPL Tech. ep. INTEOFFICE MEMONDUM IOM voids Gimbal lock See also: M. D. Shuste suvey of attitude epesentations Jounal of the stonautical Sciences vol. 41 no. 4 pp Oct. Dec CS-417 Intoduction to obotics and Intelligent Systems 18

19 Quatenions Classic notation JPL-based CS-417 Intoduction to obotics and Intelligent Systems 19 k j i j ik ki i kj jk k ji ij k j i 1 j ik ki i kj jk k ji ij ijk k j i 1 k j i cos sin sin sin 4 4 k k k z y x p 1 I p z y x cos sin cos sin cos sin cos Vecto Notation See also: N. Tawny and S. I. oumeliotis Indiect Kalman Filte fo 3D ttitude Estimation Univesity of Minnesota Dept. of Comp. Sci. & Eng. Tech. ep Mach 005.

20 Coodinate fames on P CS-417 Intoduction to obotics and Intelligent Systems 0