Real-time obstacle avoidance for manipulators and mobile robots..pdf

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REAL-TIME OBSTACLE AVOIDANCE FOR M.ANIPULATORS AND MQUIEE ROBOTS 0. Khatib Artificial Intciligence Laboratory Stanford University, Stamford, 94305 7'(z, &) = -k*A(z)k; 1 2 ~H2152-7/85/0008/0500$01 .OQ 0 1985 lEEE 500

i n Fiqtrrc 1, lhc vclocity vcctor i Wilt1 lhis schcmc, shown its is conlrollcd l o be poinl.rd lowarti Irlagniludc is Iirrrilctl to V,,,. The c!nd-cflrct.or will t h e n 1r:tvcl a1 1,hal spccd, in n slraight linc, cxccpl, during thc accclcr:llion and dcct!lcralion scgmIcn1.s or whcn it is inside lhc rcpulsive potcntial Iicld regions of inllucncc. lhc goal posilion whilc f IRAS potcnlial ficld Uavt (7) can be written as: F* = FHd + F i ; (9) with: Fkd is an attractive force allowing lhc point z of Lhc end- effector l o reach the goal position Z d , and F> rcprcscnts a Ig'orce Inducing an ArtiJicial Ilepulsion Jrorn the Surface of the obstxlc (FIIiAS, I'rorn Lhc lhxtch), crcalcd by lhc potcntial firld UO(Z). Fz. corrcsponds to l h c proportional tcrm, Le. - k ( z - Z d ) , in it convcnliond 1'11 scrvo, whcrc k is thc position gain. Thc altrsctivc polcnlial ficld b's,c(z) is simply: FIRAS Function field (T/o(z) shoultl bc designed to rncet Tho arlificial polcnl,ial the rn:mipul:ltor slabilily condition and l o crcatc at cach point on Ihc! obslaclc's surl'acc a poknlial barricr which bccorrrcs ncg- lhat surfxc. Spccific;tlIy, UO(Z) sho111d be a ligiblc bcyond non-ncgativc cont,inuous and dill'crcnliat)lc funclion whose value lcnds to inlini1.y as Lhc end-oiTretor approaches thc obslnclc's surl'acc. In ordcr lo avoid ur~dcsir:~hlc pcrl.urbing lbrccs hcyorltl thc obs1:tclc's vicinily, Lhc illllucrrcc of this potcntial Rcld must bc lirnitcd to a givcn region surrounding thc obstaelc. 50 I

502

(3%) Joint Limit Avoidance field approach can be rlscd The polcnlial L c l , q, and vi be rcspec- l o satisry thc r~~:~~~ip~rl;~Lor internal joint conshainks. tivcly i.hc rninirnnl and maximal bounds of l h c ith joint eoor- dinate 9;. q; can be kept within t.hcsc boundarics by creating barriers or polcnt,ial a1 each of L h c hypcrplarlcs (q; = 2;) and (qi = qi). The corresponding join1 I'orces arc: and: 503

X i Figure 4. Operatio.rsts1 Spucc Control S?/stem Architccture 504

Appendix 111: Link Distance to rt Cone In this (::EM:, I,hc I'rarrlc: of rdcr(:n(:(! Irl is c:hoscn such th:d, i1.s z-axis is I,ho COIIO :%xis of' syrrlrrlclry arid il,s origill is I,hc (:enter of the conc circular base. r , h , : ~ r l t i p rvprcscnt, rcspcclivcly, the conc base r:tdius, hcighl ;md half angle. Distance t,o the Cone-Shaped Surface The problcrn ol' locnting m ( . c , yl 2) is id(wtica1 to that for the cylinder c:asc. The distancc can be written as: p = z sin(/?) + ( d m - l)Cf>8(/3). (A3 - 1) 'I'hc partial derivatives come rrom t h e cquntion: z2 + y2 = rz; (A3 - 2) ( A I - 3) where: r, = tun(P)[h + p sin(/?) - 21. (A3 - 3) 'I'hcy arc: Distance to an Edge Ily :t projcclion i l l L11c pl:mo ~)c~rl)c~r~dic.~tl:v t h c c:ortsid(:rcd or ~ ( I Z ) , Lhis prohIcm c:~n b(! rcduc:ctI ~ , o I,II:IL ctigc: (zoy, ~ O Z , of finding the distance to a vcrlcx in the plane. This leads to cxprcssions similar to I,hosc of (AI-I)-(hl-3) with a mro partial dcriv;ll,ivc of the distancc: w.7.t. the :mis parallel lo thc cdgc. Distance to a Face 1.0 In this c;LscI the distance can be directly obtained by comparing lhc absolute values of thc coordinxlcs of' rnl and m 2 along the axis pcrpcndicular to the idcrllical to the unit r~orn~al vector of this face. face. Tllc partial dcrivativc vector is Appendix 11: Link Distance to a Cylinder The frarnr of rcfcrcnc:c X is chosen such that its z-axis is the cylinder axis of symmetry a r t d its origin is the cylinder center of artd mass. r and h designato, respectively, lhc cylinder radius hcight. Distance to the Circular Surface The closcst point of the link (27) to the circular surface of the cylinder can be dcduccd from the dislancc to a vcrtcx considered i n the zoy platlc ; ~ n d by :dowing Tor the radius r . Distance to the Circular Edges lo the cylintior circular cdgc can be ob- The closest distance hincd from that or lhc circular s u r f x c by laking i n h account the rvI:Ltivc z-coordinaI,c: 01' m to thc circ:ular cdgc i.e. ( z + h/2) for thc base nnd (z - h/2) for tho top. 'I'hc distance parlial dcrivativc vector rcsulls I'rorn the torus cqu;llion: [z2 + y2 + ( z f h/2I2 - r2 - p212 = 4r2[p2 - ( z f /&/2)']. 'This vector is: with: (A2 - I) (A2 - 2) 505

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