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The Planar Motion with Two Bounded Controls - the Acceleration and the Derivative of the Curvature

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(1)The Planar Motion with Two Bounded Controls - the Acceleration and the Derivative of the Curvature Elena Degtiariova-Kostova. To cite this version: Elena Degtiariova-Kostova. The Planar Motion with Two Bounded Controls - the Acceleration and the Derivative of the Curvature. RR-3444, INRIA. 1998. �inria-00073246�. HAL Id: inria-00073246 https://hal.inria.fr/inria-00073246 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. The planar motion with two bounded controls – the acceleration and the derivative of the curvature Elena Degtiariova-Kostova. N˚ 3444 Juin 1998 THÈME 4. ISSN 0249-6399. apport de recherche.

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(151) ¶ · gtfdlqnÆ  ‰! ½ rw l? [] Ë \ \ ycis{k\ [¡gtf(pgjikWÝSisWZ\Kn!bmi~•–l?fmfdlj™-r-•–{klq[ ¥ ª ikWgji ;  Æ kËX¶ · Fb « \ « ;  Æ kË %  % ;   gtfdlqnZ‰ isWZ\ ™$Wlqfd\¡\*ycis{k\ [¡gtfžpgjikW « @ \ nx \q ;  Æ   Ë % ;   gtn u ™'\¡lq”Nigtbdn±isWgtiˆgtfdlqn‰ikWZ\š™$WZl?fm\ \*ycis{k\ [¡gtf pOgjisW]ikWZ\6rspF\ \ u bdrž\ bmisWZ\K{,bdnx*{k\Kg?rwbdnZ‰;l?{ u \Kx {s\Hgqrsbmn‰•–eZnx0ikbml?n!¥–bmi u \ pF\ n u r,lqn]ikWZ\~bdnZbmisbhgtfSgtn u`œ ngtf ¤ ¤ x lqn u b ikbml?nr Ob \ ! gqn u  Å   aclOikWZ\

(152) \ ycis{k\ [¡gtf(pgjikW bhrX{keZnobmnisbdª []\ «  « % Æ kÆ Ë  %     )-Ë , N(Æ.S Æ ; ¤  )  , Ë(.S<Æ ;   ËsÆ Ë kË% %      ) ,  ( .› ¤Æ ; «   Ë « aclO,bdn isWZ\°rseZ”Sx g?rw\o” gqn u bm• 4 ™,\olq”Zikgtbdn5isWZ\Ä{k\KrseZfmi`ikWgjišl?pNisbd[¡gtf$pgtisWr gt{k\gqfmf g u []bdrkrsbm”Zfd\Äx*eZ{k—q\Hr

(153) ™$bmisW ikª WZ\ÄfmbdnZ\Hgt{¡; gt ”rslqfdeN· ik\—jgqfmeZ\olq•1rspO\K\ u  Æ kË %  ¤  ) , +.S<Æ ;   Ë Å   gtn u ™$WbdxWogt{k\{seZnbdnikbm[]\  %         ¤ « Ò ! ̌ Ò ‘ bdnisWZ\ˆx gqrs\  ™'\lq”NigtbdnikWZ\2{k\KrseZfmi'ikWgji-bm•D 4  F NisWZ\KnisWZ\;pgq{wi-lq• ikWZ\]\*ycis{k\ [¡gqfpOgjisW±x*l?{s{k\KrspFlqn u bdª nZ‰ islisWZ\]bdnDis\ {k—jgtf Æ   ½   Ë bdr;gfdbmn· \]rs; \ ‰?[]\ nD· i1bdn¦isWZ\ u bd{s\Hx0ikbml?n ¤ µ gqn u ™$bmisW5isW\pZbd\Kx \ ™$bhrw\ ´IfdbdnZ\Kgq{`gq”rwl?fmeZis\ —jgqfmeZ\lt•1rwpF\ \ u ! bm• D  ; Æ k&Ë 4 % · Dis¤ W\  )n, (lq.›pNÆ ik; bm[¡ gtË fSÅ pgt is WNrXikWZgq\ {s\1vgtgtfd{kfS\6g {ku eZ[]nobdrkbdnrsbm”Zikfdbm\[]x*\ eZ {k—q\H% rž™$ bmisWis WZÆ  \1kË  fd¤bm%nZ«\Hgt {$gt)”O, rw+l?.SfmeN<Æ ; ik\~ Æ —jkgtËwËkfdeZ¹ \lt•rwpF% \ \ · u    I,l?nrwb u \K{XisWZ\2xKgqrs\1 ª0« ^6rwbdnZ‰  A   ¨   ¶ · N™'\lq”Nigtbdn •–{klq[ ¥L ª ;  Æ k˶ · NWZ\Knx*\ ; !  Æ k˶ · Zb « \ « ¥­•–{klq[3¥ ªsª D '*È+) Æ  ÆkË  µ Ë Å ; + ÆkË  ÆkË&% ·. ¥wT„ S ª {sl?[ ¥§† ª ™,\lq”Zikgtbdn ¥w„ ª ; = Æ kË ¨ ©tÆ kË Å ;  % .·  ^6rwbdnZ‰  A   ™'\Alq”Nigtbdn6•–{klq[ý¥ L ªS¨F© Æ k*Ë %  ¢ ) , +.S<Æ ;>=?Æ kËwË « @ \ nx \ ! Æ k*Ë %  ¢ )-, (.SÆ ;>=qÆ kËwË •–l?{  º Æ   ½    Ë ¨S© » ·% 3½ $· À « {sl?[ ¥}„ ª •–lqfdfdlj™-ržikWgji-™'\[

(154) erwi~x*l?nrsb u \K{'i}™,l]pFl?rkrwbd”Zbdfmbmisbd\KrK³ g  ” ªª ;;  % 4 ·· « I,l?nrwb u \K{6ikWZ\¡rwe”FxKgqrs\ˆg ’ \ˆWOgT—q\ ›WZ\Knx*\?S•–{sl?[8¥}„ ™'\`lq”Nigtbdn VX! W\ n ;>=tÆ kËo¶ · ¥–”F\KxKgters\ ª*¨S« © Æ kË %8· ª -; b « \ %« · ;>! =?Æ kËo¶ · $WZ\Knx*\ ª ¥­•–{klq[.¥ ªwª ;; =+ ÆÆ kkËËo¨ ¶©tÆ kË· ~%b « ·\ «« ;VX+WcÆ ekrKË'N¶ ™'\1· FWOWgT—q\ \nx D \% ¥­•–{k· lq[ gqn u¥ ;ªwª ÆD k*Ë) , %C.ÃÆ ; Æ© kÆ Ë kË  %0µ ; Ë  % Æ k·Ë « %0{; sl?[Æ k*Ë ¥}%C„TS ;>ª =t™'Æ k\

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(156)   ¯ÄgjyNbd[

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(158) ikWZbhr$pZbd\Kx*\ « I,l?nrwb u \K{$isWZ\ˆrweZ”Sx g?rw\;” ’ \2WOgT—q\ gqn u Æ k&Ë %  ¢ )-, (.›Æ ;>=tÆ kËsË FWZ! \ nOx*\q•–{sl?[3¥w„ ª 4 ™'\lq”Nigtbdn  ;>=tÆ kË +%C;   ¢ -« ª0@ « \ nx \q ;7! =qÆ k; Ë ¶ · · gtn u ¥–•–¨F{sl?© [3 ¥ Zb \ ZWZ\Knx*\ ¥–•–{sl?[ ¥ ªwª D )-,/.›Æ  Æ kË  µ Ë %¸· « {sl?[ ¥}7„ S ª ™,\]l?”Nikgqbmn D ' (È ªwª) Æ ;  + Æ kÆ kË Ë ¶ µ · Ë %¸« «· « ; + VXÆ WDkËeO¶r (·™,\]WgT—?\ 

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(170) ¶“¶· · gqfml?nZ‰ÄisW\ ™$WZl?fm\\*ycis{k\ [¡gtf,pgjik! W « U'eNi`b • D % · gtn u ; + Æ kË

(171) ¶ · gtfdlqnÆ ‰! ½ rw l? [] Ë \ \ ycis{k\ [¡gtf(pgjikWÝSisWZ\Kn!bmi~•–l?fmfdlj™-r-•–{klq[ ¥ ª ikWgji ; = Æ kËX¶ · Fb « \ « ; = Æ kË %  % ; =  gtfdlqnZ‰ isWZ\ ™$Wlqfd\

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