Rigid Body Transformations
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1 igid od Kinemtics igid od Trnsformtions Vij Kumr
2 igid od Kinemtics emrk out Nottion Vectors,,, u, v, p, q, Potentil for Confusion! Mtrices,, C, g, h,
3 igid od Kinemtics The vector nd its skew smmetric mtri counterprt For n vector +
4 igid od Displcement igid od Kinemtics igid od Trnsformtion g : O t O igid od Motion g () t : O 4
5 igid od Kinemtics Coordinte Trnsformtions nd Displcements Trnsformtions of points Trnsformtion (g) of points induces n ction (g * ) on vectors p v q g (p) g * (v) g (q) Wht re rigid od trnsformtions? Displcements? g preserves lengths g * preserves cross products 5
6 igid od Kinemtics igid od Trnsformtions in Cn show tht the most generl coordinte trnsformtion from {} to {} hs the following form {} ' r P P ' O r P r O O' ' position vector of P in {} is trnsformed to position vector of P in {} description of {} s seen from n oserver in {} P P O r r + r ottion of {} with respect to {} Trnsltion of the origin of {} with respect to origin of {} 6
7 igid od Kinemtics ottionl trnsformtions in Properties of rottion mtrices Trnspose is the inverse Determinnt is + ottions preserve cross products u v (u v) ottion of skew smmetric mtrices For n rottion mtri : w T ( w) 7
8 igid od Kinemtics Emple: ottion ottion out the -is through θ ot (, θ) cosθ sin θ sin θ cosθ θ Displcement 8
9 igid od Kinemtics Emple: ottion ottion out the -is through θ ottion out the -is through θ ot (, θ) cosθ sin θ sin θ cosθ ot (, θ) cosθ sin θ sin θ cosθ ' θ ' ' θ 9 '
10 ' igid od Kinemtics igid Motion in ' {} {} r P P ' {} r P r P P ' O r P r O ' O' P P O r r + r Coordinte trnsformtion from {} to {} r P P r O O ' O' P P O r r + r Displcement of od-fied frme from {} to {} The sme eqution cn hve two interprettions: It trnsforms the position vector of n point in {} to the position vector in {} It trnsforms the position vector of n point in the first position/orienttion (descried {}) to its new position vector in the second position orienttion (descried {}).
11 Moile oots igid od Kinemtics θ W W g W cosθ sin θ sin θ cosθ
12 igid od Kinemtics The Lie group SE() () I r r T T SE,,,
13 igid od Kinemtics SE() is Lie group SE() stisfies the four ioms tht must e stisfied the elements of n lgeric group: The set is closed under the inr opertion. In other words, if nd re n two mtrices in SE(), SE(). The inr opertion is ssocitive. In other words, if,, nd C re n three mtrices SE(), then () C (C). For ever element SE(), there is n identit element given the 4 4 identit mtri, I SE(), such tht I. For ever element SE(), there is n identit inverse, - SE(), such tht - I. SE() is continuous group. the inr opertion ove is continuous opertion the product of n two elements in SE() is continuous function of the two elements the inverse of n element in SE() is continuous function of tht element. In other words, SE() is differentile mnifold. group tht is differentile mnifold is clled Lie group[ Sophus Lie (84-899)].
14 Displcement from {} to {} Displcement from {} to {C} Displcement from {} to {C} C C igid od Kinemtics Composition of Displcements C C r r r r O' O O O,, O O C r + C r POSITION r O {} O ' ' {} O' '' POSITION POSITION O'' ' {C} Note X Y descries the displcement of the od-fied frme from {X} to {Y} in reference frme {X} '' '' 4
15 igid od Kinemtics Composition (continued) Composition of displcements Displcements re generll descried in od-fied frme Emple: C is the displcement of rigid od from to C reltive to the es of the first frme. Composition of trnsformtions Sme sic ide POSITION {} O ' ' {} O' POSITION O'' ' {C} '' C C Note tht our description of trnsformtions (e.g., C ) is reltive to the first frme (, the frme with the leding superscript). '' POSITION Note X Y descries the displcement of the od-fied frme from {X} to {Y} in reference frme {X} '' 5
16 igid od Kinemtics Sugroup Nottion Definition Significnce The group of rottions in three dimensions SO() SO The set of ll proper orthogonl mtrices. T T ( ) {, I} ll sphericl displcements. Or the set of ll displcements tht cn e generted sphericl joint (S-pir). Sugroups of SE() Specil Eucliden group in two dimensions The group of rottions in two dimensions SE() SO() The set of ll mtrices with the structure: cosθ sinθ r sinθ cosθ r where θ, r, nd r re rel numers. The set of ll proper orthogonl mtrices. The hve the structure: cosθ sinθ sinθ cosθ, ll plnr displcements. Or the set of displcements tht cn e generted plnr pir (E-pir). ll rottions in the plne, or the set of ll displcements tht cn e generted single revolute joint (-pir). where θ is rel numer. The group of trnsltions in n dimensions. The group of trnsltions in one dimension. T(n) T() The group of SO() T() clindricl displcements The group of screw displcements The set of ll n rel vectors with vector ddition s the inr opertion. The set of ll rel numers with ddition s the inr opertion. The Crtesin product of SO() nd T() ll trnsltions in n dimensions. n indictes plnr, n indictes sptil displcements. ll trnsltions prllel to one is, or the set of ll displcements tht cn e generted single prismtic joint (P-pir). ll rottions in the plne nd trnsltions long n is perpendiculr to the plne, or the set of ll displcements tht cn e generted clindricl joint (C-pir). H() one-prmeter sugroup of SE() ll displcements tht cn e generted helicl joint (H-pir). 6
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