Optimization Criterion. Minimum Distance Minimum Time Minimum Acceleration Change Minimum Torque Change Minimum End Point Variance

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1 Lecture XIV Trajectory Formatio through Otimizatio Cotets: Otimizatio Criterio Miimum istace Miimum Time Miimum Acceleratio Chage Miimum Torque Chage Miimum Ed Poit Variace Usig Motor Redudacies Efficietly Lecture XIV: MLSC - r. Sethu Vijayakumar 1

2 Trajectory Plaig Phases Trajectory geeratio Ivolves comutatio of the best trajectory for the object Force istributio Ivolves determiig the force distributio betwee differet actuators a.k.a. resolvig actuator redudacy Some of the aroaches solve the trajectory geeratio ad force distributio roblems searately i two hases. It has bee argued that solvig the two issues simultaeously as a global otimizatio roblem is suerior i may cases. Kiematics: refers to geometrical ad time-based roerties of motio; the variables of iterest are ositios e.g. joit agles or had Cartesia coordiates ad their corresodig velocities, acceleratios ad higher derivatives. yamics: refers to the forces required to roduce motio ad is therefore itimately liked to the roerties of the object such as it s mass, iertia ad stiffess. Lecture XIV: MLSC - r. Sethu Vijayakumar 2

3 Some Simle Cost Fuctios Shortest istace Refer: F.C.Park ad R.W.Brockett, Kiematic exterity of robot mechaisms, It. Joural of Robotic Research, 1391,. 1-15, 1994 Miimum Acceleratio Refer: L. Noakes, G. Heiziger, ad B. ade, Cubic slies o curved surfaces, IMA Joural of Mathematical Cotrol & Iformatio, 6, , Miimum Time Bag-Bag Cotrol Refer: Z. Shiller, Time Eergy otimal cotrol of articulated systems with geometric ath costraits, I. Proc. of 1994 Itl. Coferece o Robotics ad Automatio, , Sa iego, CA, x ẋ.. x t t t Lecture XIV: MLSC - r. Sethu Vijayakumar 3

4 Miimum Jerk Trajectory Plaig Proosed by Flash & Hoga 1985 Otimizatio Criterio miimizes the jerk i the trajectory T d x d y CJ dt dt dt For movemet i x-y lae The miimum-jerk solutio ca be writte as: x t y t where x y x 0 + y 0 tˆ t / t y f x f 15ˆ t f ad 15ˆ t x 4 0 4, 6ˆ t 6ˆ t y ˆ t 3 10ˆ t 3 are iitial coordiate s at t 0. eeds oly o the kiematics of the task ad is ideedet of the hysical structure or dyamics of the lat Predicts bell shaed velocity rofiles Lecture XIV: MLSC - r. Sethu Vijayakumar 4

5 Miimum Torque Chage Plaig Proosed by Uo, Kawato & Suzuki 1989 Otimizatio Criterio miimizes the chage of torque T d d CT dt dt dt τ1 τ For two joit arm or robot The Mi. Jerk ad Mi. Torque chage cost fuctios are closely related sice acceleratio is roortioal to torque at zero seed. No Aalytical solutio ossible for Mi. Torque chage criterio but iterative solutio is ossible. Like Mi. Jerk, redicts bell shaed velocity rofiles. But also redicts that the form of the trajectory should vary across the arm s worksace. Lecture XIV: MLSC - r. Sethu Vijayakumar 5

6 Mi. Jerk vs Mi. Torque Chage Oe way of resolvig how humas la their movemet is by settig u a exerimet which ca distiguish betwee the kiematic vs dyamic las Lecture XIV: MLSC - r. Sethu Vijayakumar 6

7 Mi. Jerk vs Mi. Torque Chage II No erturbatios at the start & ed. Most studies suggest that trajectories are laed i visually-based kiematic coordiates Lecture XIV: MLSC - r. Sethu Vijayakumar 7

8 Miimum Edoit Variace Plaig Proosed by Harris & Wolert 1998 Also called TOPS Trajectory Otimizatio i the Presece of Sigal deedet oise Basic Theory: Sigle hysiological assumtio that eural sigals are corruted by oise whose variace icreases with the size of the cotrol sigal. I the resece of such oise, the shae of the trajectory is selected to miimize the variace of the fial ed-oit ositio. Biologically more lausible sice we do have access to ed oit errors as oosed to comlex otimizatio rocesses mi. jerk ad mi. torque chage itegrated over the etire movemet that other otimizatio criterio suggest the brai has to solve. Refer: Harris & Wolert, Sigal-deedet oise determies motor laig, Nature, vol. 394, Lecture XIV: MLSC - r. Sethu Vijayakumar 8

9 Testig the Iteral Model Learig Hyothesis Lecture XIV: MLSC - r. Sethu Vijayakumar 9

10 Learig of Iteral Models Usig the force maiuladum, oe ca create a iterestig exerimet i which you modify the dyamics of your arm movemet i oly the x-y lae. Humas are very adet at learig these chaged dyamic fields ad adat at a relatively short time scale. Whe we remove these effects, after effects of learig ca be felt for some time before re-adatatio, rovidig evidece that we lear iteral models ad use them i a redictive feedforward fashio Lecture XIV: MLSC - r. Sethu Vijayakumar 10

11 Multile Model Hyothesis Lecture XIV: MLSC - r. Sethu Vijayakumar 11

12 Lecture XIV: MLSC - r. Sethu Vijayakumar 12 Recursive Bayes Estimatio d k k 1 x If we exlicitly idex the idividual exerieces by, i.e., Ð {x 1,,x }, 1 x where d x x We get Py x Px ypy Px Usig

13 Multile model hyothesis II Multile Paired Forward-Iverse Models MPFIM Lecture XIV: MLSC - r. Sethu Vijayakumar 13

14 Resolvig Motor Redudacies Iverse Kiematics 4 7 x f eye +head : R R x x x y y T R L R L Head1 Head 2 Head 3 Retial islacemet T EyeLP EyeLT EyeRP EyeRT Eye $ Head dislacemets Lecture XIV: MLSC - r. Sethu Vijayakumar 14

15 Resolvig kiematics with RMRC Resolved Motio Rate Cotrol with locality i joit ositios x J Itegrate over small icremets where liearity holds to get comlete trajectory Based o collected traiig data forward kiematics i velocity sace was almost comletely liear, irresective of joit ositio x & J & Hece, i this case, we ca use seudo-iversio with the costat Jacobia!! Lecture XIV: MLSC - r. Sethu Vijayakumar 15

16 Pseudoiverse & Null Sace maiulatio Give x & J &, what is? & & J k L # # J x& + I - J J ull, i J T JJ T 1 2 L ot curret, i 1 # k ull : Pseudoiverse wi curret, i default, i Otimizatio based o gradiet descet i Null sace 2 ot mi w i curret, i default, i Otimizatio criterio Lecture XIV: MLSC - r. Sethu Vijayakumar 16 Iertial Weightig for each joit