Dipartimento di Elettronica e Informazione e Bioingegneria Robotics

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Aubrey Ward
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1 Diartimeto di Elettroica e Iformaioe e Bioigegeria Robotics arm iverse 5

2 IK ad robot rogrammig amera Tool gras referece sstem o the object the had has to reach the gras referece: T gras IK to solve the arm ( fid the joit values) Goal Base

3 IK: the roblem Give a referece (ositio ad orietatio) i artesia sace, fid the joit values to reach it with the had. ubroblems: eistece uicit solutio methods real-time issues

4 eistece Goal goal is outside the workig sace a oit ca be out of reach eve if iside the geometric volume of the workig sace due to hsical costraits a oit ca be out of reach i trajector followig sigular oit - whe acceleratio is ecessar to follow the trajector

Eistece - If the target oit is iside the workig sace ad the robot has at least 6 dof, the a solutio eists (the robot ca be solved) If the robot has less tha 6 dof, a solutio

5 Eistece - If the target oit is iside the workig sace ad the robot has at least 6 dof, the a solutio eists (the robot ca be solved) If the robot has less tha 6 dof, a solutio is ot guarated; we ca reach, however, a oit ear to the target ear i terms of orietatio Eamle: verif if the structure ca reach the give artesia coordiates.,, f, with Rot(, f ) ad

6 Multile solutios Goal the laar RR has solutios i geeral the umber of solutios icreases with umber of dof

7 Multile solutios - The umber of solutios is correlated with the umber of D-H arameters ma 6 for 6R If there are multile solutios, all should be roduced to choose the best to actuate Es: the best movemet of the robot cosiders the startig oit. Joits dislacemets are weighted (weights icreasig from ti to base) ad summed u the miimum combiatio is actuated Multile solutios are eloited for obstacle avoidace

8 Multile solutios - 3 Ifiite umber of solutios - as i the RRR laar i ( ) Redudat robot ommo situatio i ver deterous arms ad i legged robots

9 Direct kiematics fuctio? θ θ θ 3 Ed Effector Base f(θ)

10 Iverse kiematics fuctio θ θ θ3 Ed Effector Base θ f ( )

11 T matri Betwee a referece sstems there is a kiematic relatio of rototraslatio This relatio is rereseted as a matri i homogeeous coordiates relatioshi betwee the basic referece sstem ad the had referece sstem is eressed b the T matri

12 olutio methods from T matri T 6 A A ( θ, θ, θ, θ, θ θ ) 5,... 5,6 T 3 4 5, 6 losed form solutios ofte use T to fid eressios for each joit Geometric Algebric Iterative solutios - ca use other formulatios, as the Jacobia matri

13 Equatios T 6 A A ( θ, θ, θ, θ, θ θ ) 5,... 5,6 T 3 4 5, 6 We kow T ad must comute the θ joits values T cotais the 6 kow cartesia coordiates T is also the roduct of the A matrices that cotai the joit variables (the ukow) We work o equatios i 6 ukow variables (θ to θ6) osiderig that the 9 equatios eressig R are ot ideedet, we select 6 equatios i 6 ukow

losed form methods Eistece coditios for closed form solutios Methods: Paul method: re-multilicatio or ost-multilicatio of the A matrices Pieer method: defies ad

14 losed form methods Eistece coditios for closed form solutios Methods: Paul method: re-multilicatio or ost-multilicatio of the A matrices Pieer method: defies ad solves equatios for robots that satisf the Pieer coditios - ssufficiet coditios to have closed form solutios Others: geometrical, usig quaterio algebra, iterative methods

15 Eistece of closed form solutios Pieer coditios ufficiet oditios to fid a closed form solutio b Pieer (ad Ag) For 6 dof robot with 3 T joits 3 R joits with aes that itersect i oe oit (see PUMA) T joits ortogoal to aother T joit T joit ortogoal to joits with arallel aes 3 R joits with arallel aes (see ARA)

16 Numerical solutios If the robot has a comle geometr, the closed form solutio ma be imossible to fid. iterative solutios tart with a iitial estimate of joits values Loo: omute T usig the estimate omute the error i cartesia sace Produce a ew estimate to reduce the error (i geeral usig the Jacobia of the maiulator) Eecutio time idefied or lower boud of error idefied learig methods (eural ets)

17 Paul method - Take the matri equatio T 6 A A ( θ, θ, θ, θ, θ θ ) 5,... 5,6 T 3 4 5, A matrices cotai the θ variables 6 olve for the θi We ca use the re/ost multilicatio of the iverse of first/last Ai

18 Paul method - Equate T with the roduct of Ai for the first lik if i the secod term there are either: elemets cotaiig ol joit variable airs of elemets to reduce to a uique joit variable the solve those simle equatios for oe joit. else use remultilicatio for the iverse of the first A matri Reeat for all the liks Postmultilicatio for the iverse of the last A matri is also ossible

19 useful formulas siθ a, a [-, ] cosθ b, b [-, ] θ ata(a,b) gives a uique solutio bsiθ + acosθ θ ata(a,-b) ata(-a,b) we obtai solutios,

20 eamle IK (Paul): RR se l l ta ) cos( ) ( ) cos( ) cos( θ θ θ θ θ + a o a o a o l d l l T Pre-multilicatio iverse A / (+)/(-d) >θ ta- [(-d)/ (+)] ( ) ,, l l a a o o d a o a a o o A T A

21 Eamle IK (Paul): RT a o a o a o d d d T From, ad, elemets ta ) cos( ) se( θ θ θ ) se( ) se( θ θ d d From,4

22 Eamle IK (Paul): TR From, ad 3, elemets From 3,4 + a o a o a o d l l T ta ) cos( ) se( θ θ θ ) se( ) se( θ θ + l d d l

23 Eamle IK: TT d a o a o a o d d T d

24 eamle IK: RR o Paul solutios for θ + sig for elbow dow - sig for elbow u ϑ cos ta ( + l l ) (- ( + ) + (l + l ) ) (( + ) (l l ) ) ϑ ta cosϑ ( ϑ / ) + cosϑ ( t/ t) with + or sig l l t/ t l θ α θ l θ θ (, ) Δθ ϑ ta ta l l + l

25 taford arm DH arameters lik theta d a alha - L 9 - L L6

26 IK taford arm Obtaied b Paul: T A A A 3 A 4 A 5 A 6 Usig all the remultilicatios T 6 A - T A A 3 A 4 A 5 A 6 T 6 A - A - T A 3 A 4 A 5 A 6 T 3 6 A 3 - A - A - T A 4 A 5 A 6 T 4 6 A 4 - A 3 - A - A - T A 5 A 6 T5 6 A 5 - A 4 - A 3 - A - A - T A 6

27 Puma 56-6gdl DH arameters Lik a alha d theta

28 PUMA T matrices

29 IK Puma: algebraic solutio olved b Paul method ad geometric method 8 solutios: Right or left arm Elbow u or dow Wrist flied

30 Geometric stud of PUMA the first 3 liks The wrist Not aliged o (R or L)

31 stes IK comute the ositio c of the wrist to get the ed effector i the give artesia ositio c is obtaied from the had ositio traslated alog - 6. olve first 3 dof (ositio), the reformulate the roblem to solve the last 3 (orietatio) c deeds ol o the first 3 joits; 3 equatios to solve for θ, θ., θ 3. 4 solutios The orietatio deeds o θ 4, θ 5, θ 6.. solutios

32 olutio for st joit l3 Alig l ad l3. From to view, we solve for θ i the triagle usig the ad wrist comoets. ( solutios L or R arm) l Eamle with right arm

33 olutio for 3rd joit stud lik ad 3 as a laar RR. fid θ3 ( solutios elbow u or dow asθ i RR).

34 olutio d joit θ as the solutio for θ i RR The square roots i θ ad θ3, each with solutios, give a total of 4 solutios to get the wrist at a give ositio

35 IK Puma geometric algorithm. Fid the ceter oit of wrist c : c 6 (last colum i T) d 6 (tool offset legth) * 6 (3rd colum i T). From the equatio c 4 colum A 3 (θ, θ, θ 3 ) solve θ, θ, θ 3 3. Use θ, θ, θ 3 to comute R 3 ; 3 R 6 R 3 -. R 6 Use the equatios 3 R 6 3 R 6 (θ 4, θ 5, θ 6 ) to solve for θ 4, θ 5, θ 6.

36 The PUMA wrist solutio d6 Pc solutios θ4, π + θ4 θ5, π θ5 θ6, π θ6,,

37 oclusios OPERATOR ONTROL PLANNING the oerators uses artesia refereces artesia sace X, Y, Z, ψ, θ, ϕ IK Joit sace θ, θ, θ 3, θ 4, θ 5, θ 6 IK is cocetuall art of the cotrol sub sstem. However it is ofte maaged b the rogrammio sstem oversio Actuator sace (umber of stes, )

PUTNAM TRAINING PROBABILITY

PUTNAM TRAINING PROBABILITY (Last udated: December, 207) Remark. This is a list of exercises o robability. Miguel A. Lerma Exercises. Prove that the umber of subsets of {, 2,..., } with odd cardiality