Orientation. Orientation 10/28/15

Transcription

1 Orietatio Orietatio We will defie orietatio to mea a object s istataeous rotatioal cofiguratio Thik of it as the rotatioal equivalet of positio 1

2 Represetig Positios Cartesia coordiates (x,y,z) are a easy ad atural meas of represetig a positio i 3D space There are may other alteratives such as polar otatio (r,θ,φ) ad you ca ivet others if you wat to Represetig Orietatios Is there a simple meas of represetig a 3D orietatio? (aalogous to Cartesia coordiates?) Not really. There are several popular optios though: q Euler agles q Rotatio vectors (axis/agle) q 3x3 matrices q Quaterios q ad more 2

3 Euler s Theorem Euler s Theorem: Ay two idepedet orthoormal coordiate frames ca be related by a sequece of rotatios (ot more tha three) about coordiate axes, where o two successive rotatios may be about the same axis. Not to be cofused with Euler agles, Euler itegratio, Newto-Euler dyamics, iviscid Euler equatios, Euler characteristic Leoard Euler ( ) Euler Agles This meas that we ca represet a orietatio with 3 umbers A sequece of rotatios aroud pricipal axes is called a Euler Agle Sequece Assumig we limit ourselves to 3 rotatios without successive rotatios about the same axis, we could use ay of the followig 12 sequeces: XYZ XZY XYX XZX YXZ YZX YXY YZY ZXY ZYX ZXZ ZYZ 3

4 Euler Agles This gives us 12 redudat ways to store a orietatio usig Euler agles Differet idustries use differet covetios for hadlig Euler agles (or o covetios) Usig Euler Agles To use Euler agles, oe must choose which of the 12 represetatios they wat There may be some practical differeces betwee them ad the best sequece may deped o what exactly you are tryig to accomplish 4

5 Vehicle Orietatio Geerally, for vehicles, it is most coveiet to rotate i roll (z), pitch (x), ad the yaw (y) I situatios where there is a defiite groud plae, Euler agles ca actually be a ituitive represetatio z y frot of vehicle x Gimbal Lock Oe potetial problem that they ca suffer from is gimbal lock This results whe two axes effectively lie up, resultig i a temporary loss of a degree of freedom This is related to the sigularities i logitude that you get at the orth ad south poles 5

6 Iterpolatig Euler Agles Oe ca simply iterpolate betwee the three values idepedetly This will result i the iterpolatio followig a differet path depedig o which of the 12 schemes you choose This may or may ot be a problem, depedig o your situatio Iterpolatig ear the poles ca be problematic Note: whe iterpolatig agles, remember to check for crossig the +180/-180 degree boudaries Euler Agles Euler agles are used i a lot of applicatios, but they ted to require some rather arbitrary decisios They also do ot iterpolate i a cosistet way (but this is t always bad) They ca suffer from Gimbal lock ad related problems There is o simple way to cocateate rotatios Coversio to/from a matrix requires several trigoometry operatios They are compact (requirig oly 3 umbers) 6

7 Gimbal Lock Solutio? Quaterios Discovered by Sir Hamilto i 1843 Preferred rotatio operator i chemistry, robotics, space shuttle cotrols, 3D games, VR, etc Advatages q Represets pure attitude or rotatio q No mathematical sigularities q Operatios are computatioally easy q Smooth ad easy iterpolatio à great for aimatio Disadvatages q Completely uituitive à fortuately ot a real issue 10/28/

8 Solutio? Quaterios 10/28/15 15 Solutio? Quaterios 10/28/

9 Quaterios & Iterpolatio Quaterios are very suitable for aimatig attitude q Liear iterpolatio q SLERP iterpolatio Equally suitable for aimatig a camera (or more camera s) q Camera is just aother object Positio q Liear iterpolatio q Splies Other aspects q Scalig q Morphig 10/28/15 17 Motio Fusio: MPU