Recap: rigid motions. [L7] Robotics (ME671): Forward Kinematics. Recap: homogeneous transforms. Robot Kinematics Suril Shah IIT Jodhpur

Transcription

1 --6 Rep: rgd motons [L7] Robots (ME67): Forwrd Knemts Rgd moton s ombnton of rotton nd trnslton It n be represented usng homogeneous trnsform R d H Surl Shh IIT Jodhpur Inverse trnsforms: T T R R d H Rep: homogeneous trnsforms Bs trnsforms: Three pure trnslton, three pure rotton Trns x, b Trns y, b Trns z, Rot Rot Rot x, y, z, γ s s s s γ sγ sγ γ 3 Robot Knemts Study of moton of robot wthout onsderton of fore tht use t Forwrd nd nverse knemts θ θ Jont ngles (θ, θ ) Forwrd knemts Inverse knemts e, Q e End-effetor motons ( e, R e )

2 --6 Forwrd knemts ntroduton Inputs: Jont prmeters of mnpultor Output: Fnd the poston nd orentton (Pose) of the tool frme wrt Inertl frme Mppng between the tool nd the nertl frmes Funton of ll jont prmeters nd the physl geometry Purely geometr: We do not worry bout jont torques or dynms Conventon A n-dofmnpultor wll hve n jonts (ether revolute or prsmt) nd n+ lnks (sne eh jont onnets two lnks) Eh jont only hs one DOF (wthout loss of generlty) The o -x y z frme s the nertl frme nd o n -x n y n z n s the tool frme Jont onnets lnks - nd The o s onneted to lnk Jont vrbles, q θ q d f jont s revolute f jont s prsmt 5 6 Conventon Conventon Pose of o j wth respet to o usng Homogeneous trnsformton An ntermedte step: Determne the trnsformton mtrx tht gves pose of o wth respet to o - : A Next, we n defne the trnsformton o j to o s: A + A +... Aj Aj f < j Tj I f j j f j ( T > ) The pose of the tool frme wth respet to the nertl frme s gven by For n-dof mnpultor Rn on H ( ) ( ) ( ) T A q A q A q n n n 7 8

3 --6 How to fnd A We n fnd A usng 6 bs trnsformtons Three ngles (Euler ngles, for exmple) nd 3x poston vetor R o A z - x - Cn we represent t usng prmeters n se of -DOF jonts? y - Jont xs The unque prmeters n robots re the jont xs. For rotry jonts, the entre lne of the jont defnes ths. For trnsltng xs, the lne s somewht more dffult to spefy 9 Lnk prmeters Jonts n be hrterzed by two lnes n 3D Lnes geometrlly relted by two unque prmeters. Unque ommon norml between the two lnes lled lnk length. Angle between the two lnes, lled the twst ngle. Jont Prmeters A revolute jont s hrterzed by the jont xs z -, nd the two ommon norml, N - nd N N O d O Z N θ N ' Length d whh seprtes the two ponts o - nd o t whh the norml N - nd N nterset the lne z - s lled the jont offset. On trnsltng the norml N - (keepng orentton sme) long the z - xs, we get N -. The ngle between N - nd N, θ s lled the jont ngle. 3

4 --6 The prmeters,,, d nd θ re lled DH prmeters. The Denvt-Hrtenberg(DH) Conventon z z Indvdul homogeneous trnsformton s the produt of four bs trnsformtons: d O Lnk θ y x x y z O x 3 A Rot Trns Trns Rot z, θ z, d x, x, θ s θ s s θ θ d s θ s s s θ θ θ sθ θ θ s sθ s d Assgnng oordnte frmes. Choose z s xs of rotton for jont + z s xs of rotton for jont, z s xs of rotton for jont, et. If jont + s prsmt, z s the xs of trnslton for jont +. Assgn bse frme Cn be ny pont long z 3. Chose x, y to follow the rght-hnded onventon Assgnng oordnte frmes. Now strt n tertve proess to defne frme wth respet to frme - Consder three ses for the reltonshp of z - nd z :. z - nd z re non-oplnr. z - nd z nterset z - nd z re oplnr.z - nd z re prllel 5 6

5 --6 Assgnng oordnte frmes. z - nd z re non-oplnr There s unque shortest dstne between the two xes Choose ths lne segment to be x. z - nd z nterset Choose x to be norml to the plne defned by z nd z -. z - nd z re prllel Infntely mny normls of equl length between z nd z - Choose x to be long the norml tht ntersets t o -, the resultng d wll be zero Fnd DH prmeters θ (jont ngle): ngle from x - nd x bout z - d (lnk offset) : dstne from x - nd x long z - (lnk length): dstne from z - nd z long x (lnk twst) : ngle from z - nd z round x 5. o s t the nterseton of z nd x 6. Choose y by rght-hnded onventon 7. Ple the tool frme: Assgn z n long z n- 7 8 The Denvt-Hrtenberg(DH) Conventon Indvdul homogeneous trnsformton s the produt of four bs trnsformtons: A Rot Trns Trns Rot z, θ z, d x, x, θ s θ s s θ θ d s θ s s s θ θ θ sθ θ θ s sθ s d 9 Rep: Forwrd knemts Step : Frme ssgnment. Lbel jont xes s z,, z n- (xs z s jont xs for jont +). Choose bse frme: set o on z nd hoose x nd y usng rght-hnded onventon 3. Ple o where the norml to z nd z - ntersets z. If z ntersets z -, put o t nterseton. If z nd z - re prllel, ple o long z suh tht d. x s the ommon norml through o (y usng rght-hnded onventon) 5. Ple the tool frme: set z n prllel to z n- Step : Assgn DH prmeters θ (jont ngle): ngle from x - nd x bout z - d (lnk offset) : dstne from x - nd x long z - (lnk length): dstne from z - nd z long x (lnk twst) : ngle from z - nd z round x Step 3: To perform forwrd knemts Get homogeneous trnsformton mtres, A, A. A n Crete T n tht gves the poston nd orentton of the end-effetor n the nertl frme n T A A... A n 5

6 --6 Two-lnk plnr mnpultor SCARA mnpultor lnk d θ θ θ DOF: need to ssgn fve oordnte frmes:. Choose z xs (xs of rotton for jont, bse frme). Choose z -z 3 xes (xes of rotton/trnslton for jonts -) 3. Choose x xes. Choose tool frme 5. Fll n tble of DH prmeters: T A T s s s s s s A, A s + s s + s A A lnk d θ θ 8 θ 3 d 3 d θ SCARA mnpultor Now determne the ndvdul homogeneous trnsformtons: s s s s s s s s A, A, A3, A d d 3 Homework: Stnford Arm Assgn DH frmes Tbulte DH prmeters T A A + s s s + s + s s s s s + s d3 d 3 6

7 --6 THANK YOU 5 7