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Wireless Ad Hoc Networks

by: Belle Wintheiser

Wireless Ad Hoc Networks ELEE 6399

Belle Wintheiser
University of Texas-Pan American (UTPA)
GPA 3.89

Wenjie Dong

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Wenjie Dong
Class Notes
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This 4 page Class Notes was uploaded by Belle Wintheiser on Thursday October 29, 2015. The Class Notes belongs to ELEE 6399 at University of Texas-Pan American (UTPA) taught by Wenjie Dong in Fall. Since its upload, it has received 20 views. For similar materials see /class/231312/elee-6399-university-of-texas-pan-american--utpa- in Electrical Engineering at University of Texas-Pan American (UTPA).

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Date Created: 10/29/15
Lecture 3 Basic Rotation Matrices and Applications of Rotation Matrices 091409 I THE BASIC ROTATION MATRICES Given a coordinate frame F1 01 7 zlylzl and its unit vectors on axes are 2391 17 07 0F jl 07170T k1 07071 1 Coordinate frame F2 01 7 z2y222 is obtained by rotating F1 an angle 6 around the zl axis Fig 1 What is the rotation matrix RN Bi Noting 2392 cos 67sin 670T jg 7 sin 6700s 67 0F k2 07071T then 1 0 0 cos6 7sin6 0 El 0 1 0 7 B2 sin6 cos6 0 0 0 1 0 0 1 cos6 7 sin6 0 R13 B132 sin6 cos 6 0 1 0 0 1 2 Coordinate frame F2 01 7z2y222 is obtained by rotating F1 an angle 6 around the zl axis What is the rotation matrix RM Bi n 11 Fig 1 Rotation about zo by an angle 6 l 0 0 RM 0 cos 6 7 sin6 2 0 sin 6 cos 6 3 Coordinate frame F2 01 7z2y222 is obtained by rotating F1 an angle 6 around the yl axis What is the rotation matrix RM Bi cos 6 0 sin 6 Ry 0 l 0 3 7 sin 6 0 cos 6 The rotation matrices 1 3 are called the basic rotation matrices It can be shown that 349 4 Rz6Rz Rz6 5 Rg 7Rze 6 Rm I 7 Rm6Rm Rm6 8 RI 7 RM 9 Biz0 I 10 ByeRy Ry 11 11 Byte 12 II APPLICATIONS OF ROTATION TRANSFORMATION A Coordinate Transformation Question Given two coordinate frames 01 7 zlylzl and 01 7 z2y222 the rotation matrix R is known For any point p its coordinates in coordinate frames 01 7 zlylzl and 01 7 z2y222 are p1 and p2 respectively What is the relationship between p1 and p2 ugh wll Solution Assume 2391 jl and k1 are three unit vectors in the axes of coordinate frame 017z1y121 and 2392 jg and k2 are three unit vectors in the axes of coordinate frame 01 7 z2y222 Then the vector p 10 W2 72 101 g 2 Owldnmte mule attuned to 1 ngd iniy Notlng lt m gt 2 1122 wwa p ltpt7rgt Wmwk2T7r lt at In gt we 1172 weala zz 7172T 2T In wl7r Wk2T71 zlkr ilkr klkr B Rutanumlwmnun Quetron Grven a frame or r wame and a pornt p the coordrnate of p r p We rotate the pornt p around the ongln or wrth a rotatron mater I r e we bulld a coordrnate frame f1 ma the axe er 9 an 21 are parallel wrth axe em on and an Attache pornt p wrth frame fr Rotate thr frame wrth the rotatron mater 2E The pornt p wlll be at rt new portron Q The coordlnate of Q rn the frame or 7 mm r p What r p 7 mm the axe e m and z are parallel wrth axe en meoaremzrrw t the pornt p wrth the frame on 7 mm For convemenL we denote thr pornt a A When we rotate the pornt p around the ongrn or wrth a rotatron mater a the frame or 7 mm Wlll 2 h t H th n tntAwlll be colncldent wrth pornt Q For pornt A rt coordlnate rn the frame or 7 mm r p and rt Rammg Veda mm mm 50 coordrnate 1n the frame on 7 mm n put Accordmg to the rank 1n the Ian SubSecuon 1 p 3p 13 Example 1 The vector 1 wrth coordmate or 0 1 1T r rotated about or by 7r2 a Show rn Fxgure 339 The reSultmg vector 1 ha coordmate 31 en by 0 0 1 1Rw2mn 0 1 0 1 1 71 0 0 1 0 Ln Summary Three purpOSeS of a rotatron matrrx t b et The rotatron mamx can be uSed to tramform the coordmate of a pornt from one frame to another frame It 1 an operator takmg a vector and rotaung rt to a new vector rn the Same coordrnate Xy amv IL REPRESENTATIONS OF A ROTATION m Two FRAMES QueSuon Grven two coordrnate frame on r u yu u and on 7 mm the rotatron mamx IE 1 known Grven a rotatron thr rotatron r repreSented by A rn coordrnate fram and r repreSented by B rn coordrnate frame o 7 a A and B e on 7 mm mr We want to nd the relauon between Solutro Grven any pornt Pt 1t coordrnate m frame on r u yuzu r i We rotate P around on wrth rotatron matr t new pOSmon r denoted by Q The coordrnate of Q rn fra our mm 1 sf 5 o 14


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