Advanced Mechanical Elements

Transcription

1 May 3, 08 Advanced Mechancal Elements (Lecture 7) Knematc analyss and moton control of underactuated mechansms wth elastc elements - Moton control of underactuated mechansms constraned by elastc elements or gravtatonal force- Tokyo Insttute of Technology Dept. of Mechancal Engneerng School of Engneerng Prof. Nobuyuk Iwatsuk

2 . nderactuated Mechansms Ordnary mechansm n whch a number of actuators s equal to DO. Output moton can be determned by gvng two crank nputs. P (x P,y P ) M θ M θ Planar 5-bar lnk mechansm wth DOand actuators

3 Overactuator mechansm n whch a number of actuators s more than to DO. All actuators should be drven dependently to each other. P (x P,y P ) θ 3 Explaned n last lecture M M θ M θ Planar 5-bar lnk mechansm wth DOand 3 actuators

4 nderactuated mechansm n whch a number of actuators s less than DO. Output moton cannot be determned by gvng only one crank nput. P (x P,y P ) θ M θ Planar 5-bar lnk mechansm wth DOand actuator

5 nderactuated mechansm constraned wth an elastc element Dependent output motons are determned so as to mnmze potental energy. P (x P,y P ) θ M θ Planar 5-bar lnk mechansm wth DOand actuator

6 nderactuated mechansm constraned wth an elastc element Dependent output motons are determned so as to mnmze potental energy. P (x P,y P ) Mechansm wll be drven so that the added sprng becomes ts natural length. θ M θ Planar 5-bar lnk mechansm wth DOand actuator

7 nderactuated mechansm constraned wth an elastc element Dependent output motons are determned so as to mnmze potental energy. P (x P,y P ) In case where the mechansm s constraned wth more complcated combnaton of sprngs. θ M θ? Planar 5-bar lnk mechansm wth DOand actuator

8 . Knetostatcs Analyss of nderactuated Mechansms wth Elastc Elements. orce and moment applyng to mechansm Let assume jont forces and a drvng torque and a vrtual torque and sprng forces J S3 3 J 3 4 J S4 τ 4 θ τ P θ Planar 5-bar lnk mechansm wth DOand actuator M P (x P,y P ) J S - S S J 4 J S - S J S J

9 . Statcs equaton S S S S τ τ S S S P S P S P S P S S J J J 0 J J P 0 J P J 0 J J J 0 Statcs equaton for each lnk where poston vectors, P, J 3, J 4, J S3, J S4 and sprng forces, S, S are gven as functons wth respect to jont angle θ

10 .3 orward analyss [ ] [ ] ) ( ) ( θ P b A τ τ θ [ ] [ ] ) ( ) ( θ P b A θ τ τ Inverse statcs analyss The assumed vrtual torque τ 4 should be zero. Nonlnear equaton wth respect to θ, τ 4 (θ )0, can be solved wth numercal method.

11 .4 Inverse analyss [ ] [ ] ) ( ) ( θ P b A τ τ θ [ ] [ ] ) ( ) ( θ P b A θ τ τ Inverse statcs analyss Nonlnear equaton wth respect to θ, τ 4 (θ )0, can be solved wth numercal method. After specfyng desred output moton, θ, poston vectors, P, J 3, J 4, J S3, J S4 and sprng forces, S, S are gven as functons wth respect to jont angle θ

12 Parallel manpulator wth Remote Center Complance 3. Applcaton to Spatal Parallel Mechansms 3. Expected applcaton utlzng flexblty Postonng wth remote center complance M M M nderactuated parallel mechansm constraned wth elastc elements

13 3. Example Amng to develop a novel flexble jonts wth mult-do, underactuated spatal parallel mechansms constraned wth several elastc elements are proposed. ()Inverse knematcs analyss based on statcs analyss wth vrtual forces s proposed and formulated. ()or an example, 6-RS spatal parallel mechansm wth 3 rotary actuators and 3 torsonal sprngs s analyzed. R X x S θ P Z z O xed platform Movng platform y Y Rotary actuator Torsonal sprng

14 3.3 Inverse Knetostatcs analyss of 6-RS spatal parallel mechansm Inverse knematcs based on RS chan S Z z Movng platform Poston and posture of movng platform are gven. x P y Postons of sphercal pars can be calculated. R O xed platform Input angles of revolute pars and angular dsplacements of unversal pars can be determned wth knematcs of RS lnk chans. Y X θ Rotary actuator 6-RS spatal parallel mechansm wth 6 DO

15 //K R Relatve poston of sphercal par from unversal par s specfed. Q Poston s specfed. S Sphercal par R R R S x H z R S nversal par Z u R Relatve poston of unversal par from revolute par s specfed. H R x R K R H y H Poston and posture are specfed. R Revolute par y R X O Y RS two adjacent lnk chan

16 A crcle Q passng through //K R S Sphercal par S A plane passng Z through perpendcular to axs of R Sphere nversal par R R R S x H z R u R H can be calculated. O Y X RS two adjacent lnk chan R x R K R H A crcle passng through y H Cylnder R Revolute par Resultantly postons of all pars y R nversal par must be located at the ntersecton between sphere and crcle

17 Inverse statcs of mechansm wth sprngs orce and moment balance S Z z Movng platform Vrtual force to balance sprng forces V x P orce and moment actng on passve jonts y J, N J R X θ O xed platform m g Y Rotary actuator Torsonal sprng Gravtatonal force orce and moment actng on actve jonts A, N A 6-RS spatal parallel mechansm constraned wth sprngs

18 Statcs equaton: 0 G S 0, g 6, 6, ) ( JS P P V JS m Movng plate: 6) ( ) ( ) (,,,,,,,, ~ m J J JS J JS 0 N G G S - 0, g - pper lnks:,3,5) ( - ) ( ) (,,,,,,, m A, JR, J, JR L J L L JR J 0 N N N G R G - 0, g - Lower lnks wth actuator: [ ],4,6) ( ) ( ) ( T,,,,,,, K T m R, JR, J, JR L J L L JR J 0 N N G R G - 0, g - θ Lower lnks wth sprng: [ ],4,6), ( 0 0 0, T T A R, N Constrants:

19 A system of 78 lnear equatons wth respect to 78 unknowns of jont forces and moments, drvng torques and vrtual forces: S,,X N S,,Y S,,Z,,X [ A] [ b],4,x V,X V,Y V,Z Vrtual forces: V,X, V,Y, V,Z can be calculated wth Gauss-Jordan method.

20 Inverse knetostatcs analyss of 6-RS spatal parallel mechansm wth sprngs The assumed vrtual forces, V,X, V,Y, V,Z should be zeros. Procedure to obtan the balanced confguraton of the mechansm wth sprngs: ()Specfy translatonal moton of movng platform, X, Y, Z ()Assume angular moton of movng platform, α, β, γ, as varables (3)Inverse statcs analyss to calculate vrtual forces, V,X, V,Y, V,Z

21 Example of nverse knetostatcs analyss of 6-RS spatal parallel mechansm wth sprngs S Z z γ Movng platform Gven translatonal moton Y X x α P (X,Y,Z) y β Z R O xed platform Y Determned angular moton α β γ X θθ Rotary actuator Torsonal sprng Determned nput moton θ 4 θ 5 θ Smooth dsplacement can be obtaned. θ θ 3 θ 6

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23 4. Knetostatcs Analyss of Wre-drven nderactuated Mechansms Constraned by Gravtatonal orce Moton Control of Maronette

24 4. Wre-drven mechansms Conventonal Wre-drven Mechansms Q P Q P J Pullng force O (X,Y) j J θ J 3 J 4 Output lnk 4 3 P Q 4 3 Q 4 P 3 A planar wre-drven mechansm Wre Motor and pulley

25 Inverse knematcs: 0 ) ( ) ( cos sn sn cos Y X l P P P Q J j J J θ θ θ θ Inverse statcs: 0 0 ) ( > Y X J J J 0 J J P P P P Wre cannot generate pushng force! [ ] b A [ ] [ ] [ ] k A A I b A ) ( 3 4 # # The both of knematcs and statcs should be taken nto account. (Wre tenson should be controlled.) Number of wres should be more than DO because wre cannot generate pushng force.

26 nderactuated Wre-drven Mechansms 6 DO Controller 6 DO Control lnk chan (4 DO) Output lnk chan (5 DO) Wres 6 DO A general wre-drven mechansm (6 DO crane) Poston and posture control of movng table by adjustng wre lengths and wre wngng poston Puppet Output An example lnk chan of underactuated can 7 be DO drven wth wres wre-drven wth constant mechansm length Maronette nderactuated as a typcal mechansm mechansm

27 Maronette: An underactuated wre-drven mechansm Manpulated by a puppeteer wth a good skll to express human or anmal s moton It s dffcult to determne the optmum moton of controller for desred moton of puppet. Reasons: Lack of controllable DO Wre-drven Not only knematcs but also statcs due to gravtatonal force Controller 6 DO Puppet Wres 7 DO

28 Some researchers deal wth a maronette wth controller wth enough DO or drectly control wre length. It s thus expected to establsh the control method for underactuated maronette or the desgn method for optmum controller. Objectves To establsh the general control method to generate the desred moton of planar and spatal underactuated wre-drven lnk mechansms based on knetostatcs analyses Controller 6 DO Puppet Wres 7 DO

29 4. Smple example of planar wre-drven underactuated lnk chan Wre-drven three lnks chan Two lnks controller wth 4DO Poston of revolute jont C (DO) Posture angles λ, λ (DO) Three lnks output chan wth 5DO Poston of pont J (DO) Posture angles φ, φ, φ 3 (3DO) φ φ 3 φ The smplest underacutuated wre-drven mechansm φ A wre-drven three lnks chan

30 orward analyss Confguraton of controller: C, λ, λ Confguraton of lnk chan: J, φ, φ, φ 3 To obtan confguraton of output lnk chan, whch has the mnmum potental energy φ φ 3 φ φ A wre-drven three lnks chan

31 Procedure: ()Specfy confguraton of controller C, λ, λ ()Obtan wre connectng ponts S, S Wres can be assumed rgd and massless (3)Assume posture angles θ, θ (4)Obtan wre connectng ponts J, J 4 (5)Assume lnk angle φ (6)Jont J 3 can be calculated (7)Jont J can be calculated because J s located at center of adjacent lnks Confguraton of output lnk chan can be determned wth respect to θ, θ, φ. J J A wre-drven three lnks chan J 3 J 4 φ

32 (8)Calculate center of gravty of lnks, G (θ,θ,φ) (9)Calculate of y-coordnate of center of gravty of output lnk chan m yg, ( θ, θ, φ) Φ θ θ φ y Objectve functon: (,, ) G Desgn varables m Optmzaton wth the gradent method Mnmzaton of potental energy φ Confguraton of underactuated wre-drven mechansm can be determned wth the obtaned θ,θ,φ A wre-drven three lnks chan

33 Example of forward analyss In case where C moves along a straght lne and control lnks becomes open Moton of output lnk chan can be calculated. 0

34 Inverse analyss Confguraton of controller: Confguraton of lnk chan: J, φ, φ, φ 3 C, λ, λ orce balance (3)Solve statc equatons, τ ) ( ) ( 0 ) ( ) (, 0 ) ( ) (, G J G J g G J G J g G J G J g m m m Procedure: ()Specfy confguraton of lnk chan J, φ, φ, φ 3 φ φ φ 3 ()Assume jont forces, gravtatonal forces m g, vrtual torque τ

35 Wre connectng ponts S*, S* should be located n drecton of forces, 4 (4) S*, S* can be calculated as w * w J, S 4 J 4 4 * S (5) Confguraton of control lnk, C, λ, λ, can be calculated wth S*, S* However, there exsts mpossble Confguraton: )Control lnk cannot reach S*, S* )Vrtual torque s not equal to zero

36 (6)Modfy confguraton of output lnk Modfyng wth optmzaton Objectve functon : * Φ ( ψ * ) p S S p τ j Desgn varable To make controller reach to wre connectng ponts j To make modfcaton small max To make vrtual torques zero Set the lnk angle whch has maxmum senstvty Φ φ as desgn varable φ ψ φ φ 3 Possble confguraton of output lnk chan can be obtaned wth the gradent method φ

37 Example of nverse analyss Specfed Modfed Expected confguraton of output lnk chan can be calculated wth nverse analyss.

38 Wre angles Drectons of wre agree wth those of forces Vrtual torque before and after orce modfcaton angles Confguraton can be correctly modfed

39 4.3 Moton Plannng of Human Type Maronettes A planar maronette Controller A controller wth 5 DO: Poston of trple revolute jont: C (DO) Posture angles: λ, λ, λ 3 (3DO) Wre A puppet wth DO: Left hand and rght foot are connected wth a control bar. Rght hand and left foot are connected wth a control bar. Puppet It s just an underactuated mechansm however does not requre vrtual torque.

40 orward analyss Confguraton of controller: C, λ, λ, λ 3 Same procedure as two-lnk chan : ()Specfy confguraton of controller: C, λ, λ, λ 3 ()Obtan wre connectng ponts: S,,S 6 (3)Assume posture angles of massless wres, θ,,θ 6 (4)Obtan wre connectng ponts: J, J 4, J 6, J 8, J 0, J (5)Obtan confguraton of puppet based on knematcs (6)Obtan center of gravty of lnks (7)Solve the optmum wre angles whch mnmze potental energy of whole puppet (8)orward analyss s completed Confguraton of puppet: J,, J Two lnks chans are sequentally determned θ 5 θ θ 4 θ 3 θ 6 θ

41 Inverse analyss Confguraton of puppet: Confguraton of controller: J,, J C, λ, λ, λ 3 Same procedure as two-lnk chan : ()Specfy confguraton of puppet J,, J ()Inverse statcs analyss 30 varables (3)Determne drecton of wres (4)Obtan wre connectng ponts on a controller: S *,.., S 6 *

42 (5)Determne confguraton of a controller Generally no soluton due to lack of DO (6)Optmze the rotatng angle of lnk chan n the puppet to satsfy the constrant Desgn varables: Rotaton angles of lnk chans about J, J : ψ (,,3,4) ψ ψ J J ψ 4 ψ 4 Objectve functon: Summaton of dstances between S* k and S k : Φ ( ψ, ψ, ψ, ψ S * I 3 4) k k S k It should be zero.

43 Example of nverse analyss of maronette Specfcatons of planar maronette 0

44 Squattng Runnng Specfed Modfed An example of nverse analyss of a planar human type maronette

45 Experments Lnks are made of acrylc plate wth 4mm n thckness. Revolute jonts are metal shaft wth screws at both ends and two bolts as flange. Wres are cotton thread.

46 Maxmum error jont angle of 5.3 The prototype maronette can generate the desred confguratons calculated wth the proposed nverse analyss 6

47 5. Concludng remarks Amng to control underactuated lnk mechansms constraned wth elastc elements or gravtatonal force, forward/nverse knetostatcs analyses are establshed. ()Inverse knetostatc analyss based on statcs analyss takng account of vrtual torque and soluton of nonlnear equatons wth respect to unknown jont varables s proposed. ()A planar 5-bar mechansm wth DO and actuator and a spatal 6RS parallel manpulator wth 6 DO and 3 actuators are analyzed. (3)The confguraton of lnk chan hung wth several wres can be calculated wth the optmzaton to mnmze the vertcal poston of center of gravty of the chan.

48 (4)The confguraton of controller can be calculated wth wre drectons based on the nverse analyss and the optmzaton of the modfyng angles of lnk chans. (5)Moton plannng of planar human type maronette can be acheved wth the proposed nverse knetostatcs. (6)The proposed method was expermentally valdated wth prototypes of the human type maronettes composed of lnks of acrylc bars and cotton threads.

49 Concludng remarks for whole lecture Through ths lecture, you are expected to be able to: () Explan moblty of mechansm and relaton between nput/output moton of mechansm () Analyze dsplacement, velocty and acceleraton of planar/spatal closed-loop lnk mechansm wth the systematc knematc analyss method (3) Analyze the dynamcs of planar/spatal closed-loop lnk mechansm utlzng the systematc knematc analyss method (4) Explan the optmum moton control of redundant lnk mechansms (5) Explan moton control of overactuated or underactuated mechansm wth elastc elements

50 Important ssues explaned n ths lecture are as follows: ()Knematc analyses of planar/spatal lnk mechansm wth the systematc knematc analyss method Dsplacement, velocty and acceleraton analyses of planar/spatal closed-loop lnk mechansm can be easly acheved. ()Dynamc analyses of planar/spatal lnk mechansm Drvng forces and jont forces can be analyzed usng the systematc knematc analyss. (3)Optmum moton control of redundant lnk mechansms Dexterty can be maxmzed by utlzng redundancy.

51 (4)Moton control of overactuated lnk mechansms Overactuated mechansms can be controlled wth the relaxaton of nterference by addtonal elastc elements. (5)Moton control of underactuated lnk mechansms nderactuated mechansms constraned wth elastc elements can be controlled by takng account of knematcs and statcs. (6)Moton control of wre-drven underactuated lnk mechansms Maronette can be theoretcally controlled.

52 Subject of fnal report Calculate the desred nput motons for the target output moton, (X,Y), of the planar sx lnk mechansm wth only revolute pars shown n the next page under the followng condtons. ()You can locate rotary actuators at revolute pars. Note: The mechansm wll become a redundant mechansm. Therefore you have to acheve the optmum nverse knematcs wth a certan objectve functon based on nematc performance or knetc performance. ()You can determne lnk dmensons arbtrarly. (3)The target trajectory should be a straght lne. You can determne ts parameters a and b and ther tme hstory. (4)You wll show tme hstores of output/nput motons. The report wll be summarzed n A4 sze PD wth less than 0 pages and sent to nob@mep.ttech.ac.jp by June 6, 08.

53 Target trajectory YaXb Output pont (X,Y) R R Y R R O X R R A planar 6 lnk mechansm wth only revolute pars