CHAPTER 1. Learning Objectives
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1 CHTE sitin and Orientatin Definitins and Transfrmatins Marcel H. ng Jr., ug 26 Learning Objectives Describe psitin and rientatin f rigid bdies relative t each ther Mathematically represent relative psitin and rientatin f rigid bdies e able t physically visualie these mathematical representatins Marcel H. ng Jr., ug 26 2
2 STIL DESCITION used t specify spatial attributes f varius bjects with which a manipulatin system deals universe crdinate frame is implicit use Cartesian crdinate frames (x, y and axes we will use i, j, k, respectively) Marcel H. ng Jr., ug 26 3 i, x OSITION attribute f a pint 3 y k, y ( x, y, ) Cartesian crdinates f pt. expressed in frame j, y x sitin vectr - (crds has magnitude & directin, rigin impt) Marcel H. ng Jr., ug 26 4 x i j k, x +, y +, 2
3 OIENTTION attribute f a bdy, relative t sme reference frame attach a Cartesian crdinate frame t the bdy (i,j,k axes) rientatin f each axis f the Cartesian crdinate frame free vectrs rientatin vectrs describe rientatin magnitude & rigin nt imprtant nly directin is imprtant Marcel H. ng Jr., ug 26 5 x y i, j, k are the x, y and axes OIENTTION x y ( i, j, k ) unit vectrs alng each axis f frame free vectrs, nly directin is relevant, expressed in ( i j k ) x x x 3 3 iy jy ky Marcel H. ng Jr., ug 26 6 i j k i j k 3
4 tatin Matrix Since each clumn f the rtatin matrix represents unit vectrs alng the x, y and directins f the Cartesian crdinates, then they shuld be rthgnal t each ther Each clumn beys unit length cnstraints Cartesian crd frame : det ( ) + right hand rule is a rper Orthgnal Matrix - T r T I Marcel H. ng Jr., ug 26 7 tatin Matrix ll pssible rientatins f a rigid bdy (i.e. crdinate frame attached t the bdy) can be uniquely specified by a rtatin matrix Orientatin has 3 degrees-ffreedm in 3D space, i.e., 3 independent parameters are nly needed Marcel H. ng Jr., ug
5 Examples Marcel H. ng Jr., ug 26 9 Learning Objectives Transfrmatin f psitin and rientatin represents Crdinates Expressed in Frame Transfrmed t be expressed in Frame New psitin and rientatin after mtin (i.e., rigid bdy mtin) Duality between transfrmatins and rigid bdy mtins Marcel H. ng Jr., ug 26 5
6 y x Crdinate Transfrmatins & igid dy Mtin x y + +, x i, y j, k ix, jx, kx, x, iy. jy, ky, y, i, j, k,,, x i +, y j +, k ix, jx, k x, x, iy, jy, ky, y, i, j, k,, Marcel H. ng Jr., ug 26 Crdinate Transfrmatin Given: sitin f O in Orientatin f O in sitin f O in is: + +, x i, y j, k i j k x, x, x, x, iy, jy, ky, y, i, j, k,, Marcel H. ng Jr., ug
7 Hw abut Orientatin f O in? i j k i x, x, x, x, i ix, i+ iy, j+ i, k iy, jy, ky, iy, i i, j, k, i, i j k j x, x, x, x, j i, y j, y k, y j, y j i, j, k, j, i j k k x, x, x, x, k i, y j, y k, y k, y k i, j, k, k, ix, jx, k x, i j k i, y j, y k, y i j k i, j, k, ( ) ( ) Marcel H. ng Jr., ug 26 3 The ther way arund: Given O in, Find O in y x y x Marcel H. ng Jr., ug
8 Crdinate Transfrmatins & igid dy Mtin Generaliing a pint, O, with crdinates in Frame can be transfrmed t crdinates in Frame if the relative rientatin between & ( ) is knwn Crdinates f O in Crdinates f O in Is this always true? Nte ssumptin : rigins f Frames & are cincident nly difference is in rientatin Marcel H. ng Jr., ug 26 5 Crdinate Transfrmatins & igid dy Mtin What if the tw rigins are nt cincident? i.e., translatin between frames & y x y x y frame parallel t (nly different rigins) x (frm befre) Marcel H. ng Jr., ug
9 Crdinate Transfrmatins & igid dy Mtin + can nly add tw psitin vectrs if they are expressed in Frames that are parallel t each ther is parallel t s that adding respective crdinates make sense Therefre + since because is // Transfrming O t a frame // t Then adding the relative displacement f & Marcel H. ng Jr., ug 26 7 Duality (f Crdinate Transfrmatin) with igid dy Mtin Imagine a igid dy (yramid) whse apex is at pt. O y x Frame is attached t the rigid bdy y y x x Marcel H. ng Jr., ug
10 Duality (f Crdinate Transfrmatin) with igid dy Mtin The bdy is initially at and dy underges a mtin such that the new sitin & Orientatin f are & y x x y y x New Crdinates f O in, O can be fund : + Marcel H. ng Jr., ug 26 9 Elementary (asic, Fundamental) Mtins x x θ y y Initially, dy is at underges a tatin abut x by θ & is nw at after the rtatin t (x,θ) sinθ csθ Marcel H. ng Jr., ug 26 2 csθ - sinθ Similarly, a tatin abut y by θ : csθ sinθ t (y,θ) - sinθ csθ
11 Elementary (asic, Fundamental) Mtins Similarly, a rtatin abut by θ : x csθ t (,θ) sinθ - sinθ csθ Marcel H. ng Jr., ug 26, 2 Translatin dy underges pure translatin (nly) y x y,x,y Crdinates f in Hmgeneus Transfrmatins created s that rtatins and translatins are treated unifrmly. (i.e. as matrix multiplicatins) vectrs have 4 cmpnents ~ scaling factr (free vectrs, unit vectrs alng axis, rientatin vectrs) ~ scaling factr (psitin vectrs) i x x ; similarly y iy i with j & k i φ Marcel H. ng Jr., ug 26 22
12 2 Marcel H. ng Jr., ug Hmgeneus Transfrmatins C C + hmgeneus transfrmatin matrix describes the psitin and rientatin f frame in frame Chain rule : T C T T C C C 4 4 T Marcel H. ng Jr., ug Inverse f a Hmgeneus Transfrmatin Matrix Given T, Find T T T I I I Y X X I X - T Y + Y - Y T
13 Inverse f a Hmgeneus Transfrmatin Matrix T - T - T Nte that ( n a ) T T T n a x, n nrmal y, pening, a apprach Marcel H. ng Jr., ug Example y c x cube f unit length x Given : Initial sitin & Orientatin f cube specified by T Find : new crdinates f pt. c (crner f cube) after the cube is rtated by θ abut Marcel H. ng Jr., ug
14 Example : C t (, 9 ) T C C since C is attached rigidly t Marcel H. ng Jr., ug Tw ules Given dies and, with T dy mves, new psitin and rientatin T with respect t : re-multiplicatin T Mtin (4x4) T With respect t (bdy itself): st multiplicatin T T Mtin (4x4) Marcel H. ng Jr., ug
15 Other Orientatin epresentatins elative rientatin between tw bdies and can always be described by a sequence f 3 rtatins abut adjacent axes that are rthgnal t each ther The 3 angles f rtatin are used t represent rientatin, tgether with their meanings (axes f rtatin) Marcel H. ng Jr., ug t( x, d) t( y, e) t( x, f ) t( x, a) t( y, btc ) (, ) t(, p) t(, y q) t(,) r Many pssibilities Marcel H. ng Jr., ug
16 Given a definitin r sequence f axes f rtatins, are the three angles unique? Marcel H. ng Jr., ug 26 3 OLL - ITCH - YW ngles y ( φ, θ, ϕ ) x t (, φ ) t ( y, θ ) t ( x, ϕ ) csφcsθ sin φcsθ -sin θ ll itch Yaw csφsin θsin ϕ-sin φcsϕ csφsin θcsϕ+ sin φsin ϕ sin φsin θsin ϕ+ csφcsϕ sin φsin θcsϕ-csφsin ϕ csθsin ϕ csθcsϕ Marcel H. ng Jr., ug
17 Inverse Transfrmatin If n φ TN2 ( n, n ) θ TN2 (-n, ± n + n ) ϕ TN2 (, a ) 2 2 y x x y ( 2 Slutins ) If n +, θ 27, ϕ + φ TN2 (- x, y) If n -, θ 9, ϕ- φ TN2 (, ) x y Mathematical Singularity ccurs when n ϕ & φ describe the same rtatin & cannt be cmputed separately Marcel H. ng Jr., ug OLL - ITCH - YW ngles y ( φ, θ, ϕ ) x t (, φ ) t ( y, θ ) t ( x, ϕ ) ll itch Yaw sin( φ ϕ) cs( φ ϕ) cs( φ ϕ) sin( φ ϕ) t pitch 9 deg ll and Yaw describes the same mtin Marcel H. ng Jr., ug
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