Manipulator Dynamics. Amirkabir University of Technology Computer Engineering & Information Technology Department

Transcription

1 Mapulator Dyamcs mrkabr Uversty of echology omputer Egeerg formato echology Departmet

2 troducto obot arm dyamcs deals wth the mathematcal formulatos of the equatos of robot arm moto. hey are useful as: sght to the structure of the robot system. bass for model based cotrol systems. bass for computer smulatos.

3 Equatos of Moto he way whch the moto of the mapulator arses from torques appled by the actuators or from exteral forces appled to the mapulator.

4 Forward ad verse Dyamcs Gve a trajectory pot ad fd the requred vectors of jot torques τ. τ Gve a torque vector calculate the resultg moto of the mapulator ad. : problem of cotrollg the mapulator : problem of smulatg the mapulator

5 wo pproaches Eergy based: Lagrage-Euler. Smple ad symmetrc. Mometum/force approach:newto- Euler. Effcet dervato s smple but messy volves vector cross product. llow real tme cotrol.

6 Newto-Euler lgorthm Newto-Euler method s descrbed brefly below. he goal s to provde a bg pcture uderstadg of these methods wthout gettg lost the detals.

7 Newto-Euler lgorthm Newto-Euler formulatos makes two passes over the lks of mapulator eloctes cceleratos Gravty Forces momets

8 Newto-Euler lgorthm Forward computato Frst compute the agular velocty agular accelerato lear velocty lear accelerato of each lk terms of ts precedg lk. hese values ca be computed recursve maer startg from the frst movg lk ad edg at the ed-effector lk. he tal codtos for the base lk wll make the tal velocty ad accelerato values to zero.

9 Newto-Euler lgorthm ackward computato Oce the veloctes ad acceleratos of the lks are foud the jot forces ca be computed oe lk at a tme startg from the ed-effector lk ad edg at the base lk.

10 ccelerato of a gd ody Lear ad agular acceleratos:. lm lm 0 0 t t t t dt d t t t t dt d t t

11 Lear ccelerato... dt d dt d dt d : orgs are cocdet. : re-wrte t as. : by dfferetatg.

12 : the case whch the orgs are ot cocdet. 0.. OG OG : whe s costat : the lear accelerato of the lks of a mapulator wth rotatoal jots.

13 gular ccelerato s rotato relatve to ad s rotatg relatve to... dt d : the agular accelerato of the lks of a mapulator.

14 Mass Dstrbuto erta tesor- a geeralzato of the scalar momet of erta of a object

15 Momet of erta he momet of erta of a sold body wth desty w.r.t. a gve axs s defed by the volume tegral ρr ρ r r dv where r s the perpedcular dstace from the axs of rotato. hs ca be broke to compoets as

16 Momet of erta. dxdydz y x yz xz yz x z xy xz xy z y z y x d x x r r x x r m k j jk jk k j jk jk ρ δ ρ δ for a dscrete dstrbuto of mass for a cotuous dstrbuto of mass

17 Momet of erta Kroecker delta: δ j 0 for for j j. he dscrete verso of the delta fucto

18 Momet of erta he erta tesor relatve to frame {}: xx xy xz xy yy yz xz yz zz Mass momets of erta xx yy zz y z x z x y ρdv ρdv ρdv Mass products of erta xy xyρdv xz xzρdv yz yzρdv.

19 Momet of erta f we are free to choose the oretato of the referece frame t s possble to cause the products of erta to be zero. Prcpal axes. Prcpal momets of erta.

20 Example w l m h w m l h m w l m hl m hw m hl m h w m wl m hw m wl m h l m {}

21 Parallel xs heorem elates the erta tesor a frame wth org at the ceter of mass to the erta tesor w.r.t. aother referece frame. M zz xy zz xy y m x mx c c y c [ m P P P P ]. 3 c

22 Measurg the Momet of erta of a Lk Most mapulators have lks whose geometry ad composto are somewhat complex. pragmatc opto s to measure the momet of erta of each lk usg a erta pedulum. f a body suspeded by a rod s gve a small twst about the axs of suspeso t wll oscllate wth agular harmoc moto the perod of whch s gve by. π k where k s the torso costat of the suspedg rod.e. the costat rato betwee the restorg torque ad the agular dsplacemet.

23 Force causg the accelerato Newto s Equato F mv

24 Mometum causg the rotato Euler s Equato N ω ω ω

25 teratve Newto-Euler Dyamc Formulato Outward teratos to compute veloctes ad acceleratos he force ad torque actg o a lk ward teratos to compute forces ad torques

26 he Force alace for a Lk F f f

27 he orque alace for a Lk N P f P P f

28 Force alace Usg result of force ad torque balace: f P F P N teratve form: f P F P N f F f

29 he teratve Newto-Euler Dyamcs lgorthm st step: Lk veloctes ad acceleratos are teratvely computed from lk out to lk ad the Newto- Euler equatos are appled to each lk. d step: Forces ad torques of terato ad jot actuator torques are computed recursvely from lk back to lk.

30 Outward teratos. ˆ ˆ ˆ 5 0 : N v m F v P P v v P P v Z Z Z ω ω ω ω ω ω ω ω ω θ θ ω ω ω θ ω ω

31 ward teratos. ˆ 6 : Z f P F P N F f f τ

32 cluso of Gravty Forces he effect of gravty loadg o the lks ca be cluded by settg 0 v G where G s the 0 gravty vector.

33 he Structure of the Mapulator Dyamc Equatos τ M G : state space equato M : : mass matrx : : cetrfugal ad orols terms G : : gravty terms τ M : / [ ] [ ] [ ]: / [ θ θ θ θ L θ θ ] : [ ] [ ] : θ θ L θ G : cofgurato space : matrx of orols coeffcets 3 : cetrfugal coeffcets

34 orols Force fcttous force exerted o a body whe t moves a rotatg referece frame. F orols m v

35 Lagraga Formulato of Mapulator Dyamcs eergy-based approach N-E: a force balace approach N-E ad Lagraga formulato wll gve the same equatos of moto.

36 Ketc ad Potetal Eergy of a Mapulator ref u u u P g m u k k v m v k ω ω otal ketc eergy of a mapulator otal potetal eergy of a mapulator

37 Lagraga s the dfferece betwee the ketc ad potetal eergy of a mechacal system L k u.

38 he equatos of moto for the mapulator d dt L L τ d dt k k u τ vector of actuator torque

39 Example 6.5 : varable he ceter of mass of lk ad lk

40 Mapulator Dyamcs artesa Space. G M M G M G M G M G M x x x X F X X X F X F τ τ ot space formulato artesa space formulato

41 Expressos for the terms the artesa dyamcs:. G G M M M x x x

42 he artesa cofgurato space torque equato: [ ] [ ] [ ] [ ] [ ] [ ]. : : / : / : 3 x x x x x x x x G M G M θ θ θ θ θ θ θ θ θ τ τ L L X X :orols coeffcets :etrfugal coeffcets

43 Dyamc Smulato: Euler tegrato τ M G F M 0 0 Norgd body effects: frcto [ τ G F ] 0 : Gve tal codtos t t t t t t t t t t t t.

44 Next ourse: rajectory Geerato mrkabr Uversty of echology omputer Egeerg formato echology Departmet