Solving Triangle Position Analysis with Law of Sines and Cosines
Explore how to solve position analysis for a triangle using the constraints of the Law of Sines and Law of Cosines. Understand concepts like finding motor angles, adding servo motors, defining motor angles, and determining point coordinates in a triangle.
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Presentation Transcript
Solving position analysis for a triangle constraint Law of Sines ? sin ? ? c b = = ? sin ? sin ? c ? a ? Law of cosines ?2= ?2+ ?2 2??cos ? 1
?2 ?2 ?2 ?3 ?1 ?1 ?3 ?1 ?3 2
Adding Servo Motors ?2 ?2 ?2 ?3 ?1 ?1 ?3 ?1 ?3 3
Define Motor Angles ?2 ?2 ?2 ?2 ?3 ?1 ?1 ?1 ?3 ?1 ?3 ?3 4
How Do We Find the Motor Angles Problem: given points a1, ?2 a2, a3 and b1 b2 b3 and ?2 the link lengths calculate ?2 q1 q2 q3 ?2 ?3 ?1 ?1 ?1 ?3 ?1 ?3 ?3 5
How Do We Find the Motor Angles b ?2 ? ?2 ? c ? a ?2 Law of cosines ?2= ?2+ ?2 2??cos ? ?2 ?3 ?1 ?1 ?1 ?3 ?1 ?3 ?3 6
Law of cosines ?2= ?2+ ?2 2?? cos ? b cos ? =?2+ ?2 ?2 ? ? c 2?? ? a ?2+ ?2 ?2 2?? ? = ?????? ?1 ?1 ?1 ?1 ?1 How do we find the length ???1 ? 7
Length of a segment defined by two points 2 2+ ?? ?? ?1= ??,?? ???1= ? = ?? ?? ?1= ??,?? ?? ?? ?? ?? ?1 ?1= ??,?? ?1 ?1 ?1 ?1= ??,?? 8
Law of cosines ?2= ?2+ ?2 2?? cos ? b cos ? =?2+ ?2 ?2 ? ? c 2?? ? a ?1 ?1 ?1 ?1 2 ?2 2+ ?1?1 2 ?1?1 ?1 2 ?1 ? = ?????? How do we find the point ?1 so we can calculate the length ?? 9
How Do We Find the Motor Angles Problem: given point p, angle ? and the size of the ?2 triangle find points ?1,?2,?3 ?2 ?2 ?2 ? ?3 ?1 ? ?1 ?1 ?3 ?1 ?3 ?3 10
Finding points ?1,?2 and ?3 ?1?= ?x+ rcos ? ? ?1?= ?y+ r ??? ? ? 6 6 ?2 ?2 ?2?= ?x+ rcos ? +? ?2?= ?y+ r sin ? +? 2 ?2 2 ?3?= ?x+ rcos ? +7? ?3?= ?y+ r sin ? +7? ?? 6 ?2 ? 6 ?3 ?1 ? ?1 ?1 ?3 ?1 ?3 ?3 11
Finding points ?1,?2 and ?3 ?1?= rb cos ? ?1?= r? ??? ? 3 = ?? = ?? 6 2 ?2 6 2 ?2 ? 2 ? 2 ?? ?2?= rb cos = 0 ?2?= rb sin = ? ?2 ?2 7? 6 7? 6 3 ?3?= rb cos = ?? = ?? 2 ? ?3?= rb ??? 2 ?3 ?1 ?1 ?1 ?3 ?1 ?3 ?3 12
Finding the motor angles We will make sure that atan2 returns angles between 0 and 360 degrees ?2 = ????2(?2? ?2?,?2? ?2?) ?2 ?2 ?2 ?2= ?2 ?2 2? 3 ?2 ?2 ?1= ????2(?1? ?1?,?1? ?1?) ?2 ?1= ?1 ?1 ? ?3 ?1 ? ?1 ?1 ?3 ?1 ?1 ?3 ?3 ?3 ?1 ?3 ?3= ?3 ?3 4? ?3= ????2(?3? ?3?,?3? ?3?) 14 3
Numerical values for our robots ?_?=0.04 ? ?_?=0.1 ? L1 = 0.0500 L2 = 0.0500 b1x = 0.0866 b1y = -0.0500 b2x = 0 b2y = 0.1000 b3x = -0.0500 b3y = -0.0866 theta = -0.5236 px = 0 py = 0 a1x = 0.0346 a1y = -0.0200 a2x = 0 a2y = 0.0400 a3x = -0.0346 a3y = -0.0200 15
