igid od Kinemtics igid od Trnsformtions Vij Kumr
igid od Kinemtics emrk out Nottion Vectors,,, u, v, p, q, Potentil for Confusion! Mtrices,, C, g, h,
igid od Kinemtics The vector nd its skew smmetric mtri counterprt For n vector +
igid od Displcement igid od Kinemtics igid od Trnsformtion g : O t O igid od Motion g () t : O 4
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
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
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
igid od Kinemtics Emple: ottion ottion out the -is through θ ot (, θ) cosθ sin θ sin θ cosθ θ Displcement 8
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 '
' 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 {}).
Moile oots igid od Kinemtics θ W W g W cosθ sin θ sin θ cosθ
igid od Kinemtics The Lie group SE() () I r r T T SE,,, http://www.ses.upenn.edu/~mem5/notes/igidodmotion.pdf
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)].
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
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
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