Introduction To Robotics (Kinematics, Dynamics, and Design)

Transcription

1 Intoduction o obotics Kinematics Dnamics and Design EION # 9: satial Descitions & ansfomations li Meghdai ofesso chool of Mechanical Engineeing haif Univesit of echnolog ehan IN Homeage: htt://meghdai.shaif.edu

2 ansfomation ithmetic: Multilication of ansfoms omound ansfomations: Given fames {} {} {} and vecto : Find: We can wite: تبديل مركب whee : X Z {} {} Y {}

3 ansfomation ithmetic: Invesion of ansfoms: One ma invet the matices b standad techniques too long. Howeve a simle method eists: Given: Find:? OG a o n a o n a o n... a a a O o o o N n n n OG O N

4 Eamle: Invesion of ansfoms: Given: Find: a o n a o n a o n a a a O o o o N n n n

5 ansfom Equation: onside the figue shown: Given: We can wite two eessions fo : G W G W Fied {} {W} {G} {} {} G W G W G W G W ansfom Equation

6 ansfom Equation: E: If the ansfom is unnown? We can wite the following eession fo the oientation of the eg at insetion: W G G W {} {W} {G} {}? {} {} W W G G W W G G ansfom Equation

7 Moe on eesentation of osition & Oientation: lindical oodinates: o define atesian coodinates of a oint in tems of the lindical coodinates stat b a coodinate coincident on {} and:. anslate b along X-ais of the fame {}. otate b an angle about the Z-ais of {} 3. anslate b veticall along Z-ai of {}. ansfomations ae all along the Oiginal/Old {} fame then: EMULILY: Z ê ê ê ans Z ot Z ans X θ Y {} X

8 Moe on eesentation of osition & Oientation: Oeato : X ans Z ot Z ans in os in os in os in os os in in os

9 Moe on eesentation of osition & Oientation: lindical oodinates: o define atesian coodinates of a oint in tems of the lindical coodinates stat b a coodinate coincident on {} and: nothe oach:. anslate along the Z-ais of the fame {} b. otate about the New Z-ais b an angle 3. anslate along the New X-ais b. Z ê ê ê ansfomations ae all along the New fames then: OMULILY: ans Z ot Z ans X θ Y {} X

10 ight-to-left e-multil vs. Left-to-ight ost-multil: Eamle:. otate 3 about X-ais. otate 9 about the New tansfomed Y-ais 3. anslate 3 along the Old fied Z-ais 4. otate 3 about the New tansfomed X-ais. o wite the coesonding tansfom eession Just emembe: {FiedOld on the Left} and {Newansfomed on the ight}. heefoe the st tansfom is: os3 in3 in3 os3 he nd tansfom is: New-ight os3 in3 in3 os3

11 ight-to-left e-multil vs. Left-to-ight ost-multil: Eamle:. otate 3 about X-ais. otate 9 about the New tansfomed Y-ais 3. anslate 3 along the Old fied Z-ais 4. otate 3 about the New tansfomed X-ais. he 3 d tansfom is: Old-Left he 4 th tansfom is: New-ight os in in os os in in os os in in os

12 Moe on eesentation of osition & Oientation : o find lindical coodinates fom atesian oodinates: tant : is a wo-gument ac tangent function. It comutes tan - / but uses the signs of both and to detemine the quadant in which the esulting angle lies. E: tan=tan - /=tan-- = -35 tan = 45 tan- = -45 tan- = 35 tan = Undefined tan in os

13 Moe on eesentation of Oientation: o fa we intoduced a 3 3 otation Mati to define oientation such that: X Y Z X Y Z and 6-Deendencies X Y X Z Y Z {} o secif the desied oientation of a obot hand it is difficult to inut a nine-element mati with othogonal columns. heefoe we need: moe efficient wa to secif oientation {} eveal methods ae esent. 9-Quantities

14 Moe on eesentation of Oientation: oll itch and Yaw Fied ngles about Fied aes Y: o descibe oientation of {} elative to a fied nown fame {} stat with a fame coincident with {} and: X: oll. otx : oll. oty : itch 3. otz : Yaw Z Z Z Y Z Y Z Y: itch Z: Yaw Z Y X Y Y Y X X X haif Univesit of echnolog - ED X X

15 Moe on eesentation of Oientation: ince all otations ae about the oiginal/fied fame {} then e-multil to find the Y-Oeato as: X ot Y ot Z ot

16 Moe on eesentation of Oientation: Invese of this oblem is to comute the oll itch and Yaw angles fo a given otation Mati: Deendencies Unnowns Equations

17 Moe on eesentation of Oientation: heefoe with 3-indeendent equations one can find the 3- unnowns oll itch and Yaw angles as: tan tan tan 3 / 3 / 33 / / as long as ead the detailed discussion of the solution in ou boo.

18 Moe on eesentation of Oientation: Eule ngles about Moving aes: nothe method to eesent oientation. Z-Y-X Eule ngles: o descibe oientation of {} elative to a fied nown fame {} stat with a fame coincident with {} and:. otz. oty X 3. otx Z Z X Y Y Z X X Z Y Y Z X X Z Y Y

19 Moe on eesentation of Oientation: ince all otations ae about the Moving/New fame {} then ost-multil to find the Eule-Oeato as same esult as Y: X ot Y ot Z ot

20 Moe on eesentation of Oientation: imle Geneal otation: otation of a igid bod fame about a geneal fied ais in sace. Elementa otation: otation of a igid bod fame about one of the coodinate aes. Eule s heoem: n change of oientation about an abita ais fo a igid bod with a fied bod oint can be accomlished though a simle otation. he igid bod otation can be esolved into thee elementa otations whee the angles of these otations ae called the Eue s ngles. he 3-indeendent Euleian ngles and the Fied ngles conventions ma be selected in a vaiet of was and sequences. total of 4 conventions eist of which onl sets ae unique see ages

21 Eamle: he Unimation UM 56 Eule ngles onvention: Descition of oientation of the ool Fame {} elative to the fied Univesal fame {U}:. otz U o: Oientation. oty a: oach 3. otz t: wist o Z Z U Y Y U Z a Z Y Y Y t Z Z X Y X U X X X X

22 he Unimation UM 56 Eule ngles onvention: Fo UM-56 the ool Fame {} is not coincident with the Univesal fame {U}. heefoe the Zeo oientation of {} is: Z U U initial {} {U} U U ot Z o initialot Y a ot Z t oat ot oat ot at oat oat at ot ot oa oa a Z Y X U X Y U

23 Equivalent ngle-is eesentation Eule s heoem on otation: n oientation of a igid bod fame can be obtained though a oe is and ngle selection. imle Geneal otation Oeato = ot K : otation of a igid bod fame about a geneal fied ais K in sace. Oiginall {} is coincident with {} then aling ot K b ight- Hand-ule we can define: whee: v v v v v v v v v K os v j i K X Y Z X Y Z K {} {}

24 Equivalent ngle-is eesentation Eule s heoem on otation: When the ais of otation is chosen as one of the incial aes of {} then the Equivalent Geneal otation Mati tae on the familia fom of lana Elementa otations: os in in os Z ot os in in os Y ot os in in os X ot Z Y X

25 Equivalent ngle-is eesentation Eule s heoem on otation: o obtain K fom a given otation mati oientation: in os K 3 3 in K 3 3 in tan os

26 n combination of otations is alwas equivalent to a single otation about some ais K b an angle : Eamle: onside the following combined otation oeatos and obtain its coesonding equivalent angle-ais eesentation? ot Y9 ot Z9 3 tan K ot Y9 ot Z9 ot K in os i 3 3 j 3 3

27 ansfomations of Fee and Line Vectos: In mechanics we mae a distinction between the equalit and the equivalence of vectos. - wo vectos ae equal if the have the same dimensions magnitude and diection. - wo equal vectos ma have diffeent lines of actions. E. Velocit vectos shown. - wo vectos ae equivalent in a cetain caacit if V V each oduces the ve same effect in this caacit. * If the citeion in this E. is distance taveled all thee vectos give the same esult and ae thus equivalent in this sense. * If the citeion in this E. is height above the XY-lane then the vectos ae not equivalent desite thei equalit. {} Z Y V 3 X

28 ansfomations of Fee and Line Vectos: Line-Vecto : vecto which along with diection and magnitude is also deendent on its line-of-action o oint-of-action as fa as detemining its effects is concened. E: foce vecto osition vecto. Fee-Vecto : vecto which ma be ositioned anwhee in sace without loss o change of meaning ovided that magnitude and diection ae eseved. E: ue moment vecto velocit vecto. heefoe in tansfoming fee vectos fom one fame to anothe fame onl the otation mati elating the two fames is used. {} Z V Y V V 3 V V and not V V X

29 omutational onsideations: Efficienc in comuting methods is an imotant issue in obotics. Eamle: onside the following tansfomations: st oach: D D D 54Mul.+36dd. 9Mul.+6dd. nd oach: D 63Mul. 4dd. D D 7Mul. 8dd. *** he nd oach is moe efficient. ***.