Path Planning for Robust Image-Based Visual Servoing

38
HAL Id: inria-00072534 https://hal.inria.fr/inria-00072534 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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. Path Planning for Robust Image-Based Visual Servoing Youcef Mezouar, François Chaumette To cite this version: Youcef Mezouar, François Chaumette. Path Planning for Robust Image-Based Visual Servoing. [Re- search Report] RR-4097, INRIA. 2001. inria-00072534

Transcript of Path Planning for Robust Image-Based Visual Servoing

Page 1: Path Planning for Robust Image-Based Visual Servoing

HAL Id: inria-00072534https://hal.inria.fr/inria-00072534

Submitted on 24 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Path Planning for Robust Image-Based Visual ServoingYoucef Mezouar, François Chaumette

To cite this version:Youcef Mezouar, François Chaumette. Path Planning for Robust Image-Based Visual Servoing. [Re-search Report] RR-4097, INRIA. 2001. �inria-00072534�

Page 2: Path Planning for Robust Image-Based Visual Servoing

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--40

97--

FR

+E

NG

ap por t de r ech er ch e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Path Planningfor Robust Image-based Visual Servoing

Youcef Mezouar, François Chaumette

N˚4097

Janvier 2001

THÈME 3

Page 3: Path Planning for Robust Image-Based Visual Servoing
Page 4: Path Planning for Robust Image-Based Visual Servoing

������������� ������������������ � � ��� �!����"$# � � � "&%(')� �*� ���,+-" �/.�� �� �

02143658789�:;7=<8143?>A@8BDCD@E>AF6G814HJILKNM?>O36PQ7SRERE7

TVUXW�Y[Z�\2]_^a`=bcZedgf8hibgjlk8`LUXk8YNY[Z�mJY-fShnUojl`XZ8pjqY-fsr8ZetepvuXkS`X`XweZetepXhk8`X`ofsjxtgtcfS`ohZEt

yDdgksz{Z�b&|}jxt{bnf

~�fs�o�Ok8d�b�u�Z�dgZehnUXZedghnUoZ`2�s�=�S�=��](�8fS`=�vjqZ�d��s�8�X��]�\s�[�ofSrSZet

�L�/�e�����X�S�S� |}jxt�jqkS`[��ZeZeu��OfShn��h�kS`=bcdgkS�o�lkvkS�NbgZehnUX`Xjx�=�XZet$fsdgZ&Z��hjqZ�`=bD��k8d$f��xfsdgrSZ�h��qf8tct?kS� fs�X�X�qjxh�f�mbgjlk8`ot��o��b}bgUXZ�¡¢hk8YNZ2�X��fsr=fsjq`ot{b}uXj£��h�o�£bgjlZEt}¤&UXZe`¢bcUXZ�jl`oj£bgjqfS��fs`Ou¥u�ZetcjldgZeu¢dgkS�Aksb��Ak8tcjlbcjqkS`ot¦fsdgZuXjqt�bgfS`8bE§V¨$�xfStgt�jxh�fS� fs�X�odck=fShnUXZEt&fsdgZ©�OfStcZeu kS` bcUoZ�dcZerS�X�xf�bgjlk8`Lbgk�ª�Zedck[kS�?fS`¢Z�dgdck8dV���X`ohibgjlk8`�hk8Y�m�o��bcZEu ��dck8Y«bgUXZ2h�odcdgZ�`=b�Y[Zef8t��XdgZ�Y[Ze`8b}fS`ou¢f�hk8`ot�bgfs`=b}u�ZEt�jqdgZeu kS`XZ8§­¬­¡ �otcjl`Xr�tc�ohnU�fS�X�Xdgk8fShnU*pjlb�jqt©`XkSb©k8�v�=jqkS�Ot�bgk�jq`=bcdgkvuX�ohZ[fs`v¡�hk8`ot�bcdnfsjq`8b©jl`®bcUoZNdgZefS�ljqª�ZEu¢bcdnf�z{ZEhibgkSdgjlZEt}fs`Ou�bgk�Ze`ot��odcZ�bcUXZh�kS`v�SZedcr8Z�`oh�Z©��kSd¦fS�l�*bcUXZ�jl`oj£bgjqfS�/h�kS`�¯or8�Xdnf�bcjqkS`Ot�§V^a`¥bgUXjqt}�ofs�AZ�dEpA¤$Z��odck8�Ok=t�Z2f�`XZ�¤fs�o�Xdck=fShnU bckdgZetckS�q�SZ�bcUXZEt�Z©u�j£��h��X�£bgjlZEt­�=¡�h�kS�X�o�ljq`Xr2�Of�bcU��X�xfs`X`Xjq`Xr�jl`�jlY-fsr8Z¦tc�ofSh�Z¦fS`ou-jqY[fSrSZ�mJ�of8t�ZEu-hk8`8bgdck8�°§¨$k8`ot�bcdnfsjq`8bnt�tc�ohnU±bcUofsb2bgUXZLk8��z{Zehb2dcZeY-fsjq`ot�jq`²bcUoZ�h�fSY[Z�dnf ¯oZ��xu²ks���vjqZ�¤³kSd2bcUXZ�dck8�OkSb�f´�Sk8jquotjlbgt?z{kSjq`8b��qjqYNjlbgt}h�fs`¢�AZ�bgfS�SZ�`¢jq`=bck-fShehkS�o`8b�fsb&bcUXZ�bnfStc�L�X�xfs`X`ojl`Xr[�qZ��8Z��J§¶µX�XdcbcUoZ�dgYNk8dcZ8p��v¡��otcjl`XrbgUXjxt}fs�o�Xdck=fShnU pXh��XdgdcZe`8b�Y[ZEfStc�XdcZeY[Z�`=bgt�fS�l¤Vf´¡�tVdcZeY-fsjq`¢h��lk=t�Z�bgk-bcUoZ�jqd¦u�ZEt�jqdcZEu �´fS�l�oZ2fs`ou¢f�h�kS`�mbgdck8�6�=¡¥jqY-fsr8Zm·�ofStcZeu�tcZ�dg�Sk8jl`or�Z�`otc�XdgZet¦bcUoZ�dgkS�X�Ot{bg`XZetgt©¤&j£bgU®dgZetc�OZEhib©bck Y[kvuXZ��qjl`Xr¢ZedcdgkSdnt�§�TVUXZ�odck8�Ok=t�ZEu-Y[ZbcUokvu�jxt$�OfStcZeu�k8`�bgUXZ¦�AksbgZ�`=bcjxfs�A¯oZe�qu fs�o�Xdck=fShnU�fS`ou�jxt&fs�X�o�ljqZeuL¤&UXZ�`�kS��z{ZEhib&t�UofS�OZfS`ou�u�jqY[Z�`otcjqkS`ot�fsdgZ2�v`Xk�¤&`¸kSd©`XksbEp4fS`ou�¤&UoZ�`�bgUXZ[h�fs�qjq�XdgfsbcjqkS`��OfsdnfsY[ZbgZ�dnt�ks�?bgUXZ-h�fSYNZedgfLfsdgZ¤­Z��q�Xk8d?�ofSu��q¡2ZEt{bgjlY-fsbcZeu §6µ/jq`ofs�q�q¡SpSdgZefS�vbcjqY[Z�Z¹��AZ�dgjlY[Z�`=bnfs�XdgZetc�X�lbgt?�otcjq`Xr�fs`[Z�¡8Zm·jl`XmJUofS`ou�dck8�OkSbcjxhtc¡�t{bgZ�Y«fSdcZ��XdgZetcZ�`=bgZeu fs`ou hk8`�¯odgY�bcUXZ���fS�ljxu�jlb{¡�ks�6kS�Xd�fS�X�Xdgk8f8hnU §º¥»=¼4½g¾2¿ ��À/�S� y?fsbcU¢�X�xfs`X`Xjq`XrOp�|}jxt��ofS�*tcZ�dg�SkSjq`Xr

Áx ÃcÄ·ÅeÆVÃ&Ç/ÈqÄ·ÉJÊË

Unité de recherche INRIA RennesIRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France)

Téléphone : 02 99 84 71 00 - International : +33 2 99 84 71 00Télécopie : 02 99 84 71 71 - International : +33 2 99 84 71 71

Page 5: Path Planning for Robust Image-Based Visual Servoing

����� �����­����� � ��% " � � ��� "��¶� � � � " � % ��� � �������!����" �����«� �� �*� " ��. � ��� "��!"&��� . � �*� "&� �6��� � � �/"

� � ����� � � � ZEt¦bcZehnUo`Xjq�=�XZEt�u�� f8tctcZ�dg�vjqtgtcZ�Y[Z�`=b}�vjxt��XZe�Dt�k8`=b©Z��-hefSh�Zet¦�Ok8�Xd��X`XZ[rSdnfs`ouXZ�h�xfStgtcZu�� fS�X�X�qjqhef�bcjqkS`²Y-fSjqt�t�Z�UXZ��XdcbcZe`=b���u�ZEt�uXj£��h�o�£bgwetN�8�Ofs`ou²�qZ�uXw��X�xfSh�Z�Y[Z�`=b��¸Z��4Zehibg�XZ�dNZet�b�bgdck8�jqY[�Ok8d�bnfs`=be§ � Zet2fs�X�Xdgk�hnUXZet�h��qf8tctcjq�=�XZEt�t�k8`=b��ofStcw�ZEt�t��Xd��xf¢dgw�rS�o�qfsbcjqkS`���ªew�dgk�u�Z-��� ZedcdgZ��Xd�Ze`8bgdcZ�qZet2��fs�qZ��Xdnt�hk8�Xdnfs`=bcZ-Z�b�u�wetcjqdcweZ�u�� jq`���k8dcY-f�bgjlk8`²�vjqtc�XZ��q�qZet�tcw��qZehbcjqkS`X`XweZet�t�k8j£b2uXfS`ot���� jlY-fsr8ZSp6t�k8j£buofs`otV��� ZEt��OfShZ�\sm���§�^a� ZEt{b}fS�lk8dgt&u�jl��hjq�lZ�u�� jq`=bcdgkvuX�XjldgZ�u�ZEt&hkS`=bgdgfSjl`=bcZEt&t��od&�qfNbgdgfsz{ZehibgkSjqdcZ©dgwefs�qjxt�weZZ�bVu�� f8tctc�XdgZ�dD�xf2h�kS`v�SZedcr8Z�`oh�Z��8�oZ��q�lZ��=�XZ¦tckSjlb$�xf2�Ak8tcjlbcjqkS`�jq`Xjlbcjxfs�qZ¦u���dck8�OkSbe§��¦fS`ot­hZ}dnfs�o�Ok8d�bEp8k8`w�bc�ouXjlZ­�lZ­�XdgkS�X�qW�Y[Z­u�Z$�xf��o�qfS`Xj£¯Ah�f�bgjlk8`2u�Z$bcdnf�z{ZEhibck8jldgZ¶uXfS`ot�� � jlY-fSrSZ­Zb6� � kS`��odck8�Ok=t�Z$�X`XZ­YNw�bcUXk�u�Z�AZ�dgY[Zbcbgfs`=b�u�� f8tctc�XdgZ�d��xf�hk8`=�8Z�dgrSZe`ohZ-�=�XZ��q�qZL�=�XZLtckSjlb��lZ u�w��X�xfSh�Z�Y[Z�`=b!�¥dgwefS�ljxtcZ�d�Z�`QhkS�o�X�qfS`=br8w�`XwedgfsbcjqkS`±u�Z-bgdgfsz{ZehibgkSjqdcZ[Z�bNfStgtcZ�dg�=jxtgt�ZeYNZe`=b©�vjxt��XZe�$uXfS`ot���� jlY-fsr8ZS§ � f¢Y[wbgUXk�u�Z��XdgkS�Ak8tcw�Z-ZEt{b�OfStcw�ZVt��Xd?��� ��bcjq�ljxtgf�bcjqkS`-u�Z&hnUofSY[�ot?u�ZV�OkSbcZe`8bgjlZe�XZb?ZEt{b¶fS�X�X�qjq�=�XweZ­�qkSdntg�8�oZ$��� k8��z{ZbDh�kS`otcjquXw�dgw$ZEt{b¶u�Z��k8dcY[Z¦Zb�u�Z�u�jlY[Ze`ot�jqkS`Ot­h�kS`X`v�XZEt¶k8�Ljq`ohk8`X`v�XZet­Z�bVfS��h�f8t¶k#"L�qZet­�ofSdgfSYNW�bcdgZet­u�Z©�qfNh�fSYNwedgf�t�k8`=b�ojlZe`LkS��Y-fs�AZet�bcjqYNwEt�§ � Zet­�XdgkS�X�qW�Y[Zet­u�Z©�vjqtcjl�ojl�qj£bgw©uXZ¦��� k8��z{ZbVZb&u�� w��vjlbcZ�Y[Ze`8b&u�ZEt$�o��bcweZetVfsdcbcjxh��m�xfsjqdgZetDu��Ldck8�OkSb­tckS`=b¶w�r=fs�qZ�Y[Z�`=bDbgdgfSj£bgwete§/µ/jq`ofs�qZ�Y[Ze`8bEp=uXZet$dcwEt��X�lbgfsbgt¶Z¹��OwedcjqY[Z�`=bgfS��¹ p8k8��bcZe`v�otDZe`�Xbcjq�ljxtcfS`=b}�X`XZNh�fsY[wedgf[ZeY2�ofSdg�=�XweZ�t��Xd}�X`�dgkS�Aksb$��tcjl¹¢u�ZerSdgwet�uXZ��ljq�AZ�dcbcwSp4hk8`�¯odgY[Z�`=b}�xf-��fs�qjqu�jlbcwuXZ©`oksbcdgZ�fs�X�odck�hnUXZ8§% ¿ �E� ½ ��& � � y¶�qfS`Xj£¯Ah�f�bgjlk8` uXZ�bcdnf�z{Zehbck8jldgZetep('}tgt�Zedc�vjxtctcZ�Y[Z�`=bV�vjqtc�XZ��

Page 6: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� \

- � �2�/"&�2� �

.0/�1 �e� ¿ À �/�s�32 ¿ 1 4

576 � 1 À�� � » 1 �e� &°� 8�X§l�79©Ze`XZ�dnfs� uXZetghdgjl��bgjlk8` fS`ou�`Xksbnf�bcjqkS` §�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ :�X§ � y6kSbcZ�`=bgjqfS�4¯oZ��xu YNZ�bcUXk�u §�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ ;

<>= ���@?BA & � 1C1 2 1EDGF ¿ �L�IH 1 ¿v¾ 1 �E��� D » � J\o§l� ¨­fSY[Z�dnf�bgdgfsz{ZehibgkSdg¡-jl` bgUXZ�\�m � ¨­fsdcbcZEt�jxfs` t��of8hZ §�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �\o§ � K¦�Xz{Zehib&bgdgfsz{ZehbckSdg¡[jl` bgUXZ�jlY-fSrSZ�t��OfShZ §�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �E�\o§ \ T�dnf�z{Zehbck8dcjqZet­jq`LbgUXZVz{kSjq`=b}t��of8hZQ§�§©§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �E�\o§ � '­bcbcdnfShbcjq�SZ��OkSbcZe`8bgjqfS� fs`ou���k8dgh�Z §�§©§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �8�\o§ :ML�ZehnUofS`Xjqhefs� fS`ouL�vjxt�jq�Xjq�ljlb{¡Lh�kS`ot�bcdnfsjq`=bgt§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �´�\o§ ; N��XY[Y[fSdc¡ §©§�§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �,:

4O= ���@?BA & � 1C1 2 1EDGF ¿ �L� 1 � 1 H 1 ¿v¾ 1 �E��� D » � .�P�O§l�7NXh�fs�qZeuRQD�oh��ljxu�ZEfs`¢dcZEhk8`ot{bgdc�OhibcjqkS` §©§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �3;�O§ � NXh�fs�qZeu \�m � ¨­fSd�bgZetcjqfS`�bcdnf�z{ZEhibgkSdg¡ §©§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ ����O§ \ K¦�Xz{Zehib&bgdgfsz{ZehbckSdg¡[jl` bgUXZ�jlY-fSrSZ�t��OfShZ §�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �3S�O§ � ^a`UTo�XZ�`OhZ�ks�/Z�dgdck8dgt­k8`WVYX §�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �E��O§ : ^a`UTo�XZ�`OhZ�ks�/Z�dgdck8dgt­k8`LbgUXZ�jl`=bgdcjq`otcjqh©�ofSdgfSYNZ�bcZedgt®§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �E��O§ ; N��XY[Y[fSdc¡ §©§�§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �S�

8 = » � F ¿ � �Z2 1ED\[] 2 � � D » �e����^ » �S� ¿ ��2 » � 5_.

PM` ¿ 1 ��� ¿ &V�E�,? » � » 5�5

a7b�c A » ��2 � » 1 �e� &D� » ����&��E� 5 <��§l�79©Ze`XZ�dnfs� jq`=bcZedcZEt{bV��kSd&�Of�bcU¢�X�xfs`o`Xjl`or §�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �S\��§ �OQD¹v�AZ�dgjqYNZe`=bgfs�*dcZEt��X�lbgt­��k8d�fN�=`ok�¤&`��X�xfs`OfsdVkS��z{ZEhib §�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ �d:��§ \MQD¹v�AZ�dgjqYNZe`=bgfs�*dcZEt��X�lbgt­��k8d�fs` �X`X�v`Xk�¤&` �X�xfs`ofSd&kS��z{ZEhib±§©§�§�§�§�§�§�§�§�§�§�§�§©§�§ ��;��§ � QD¹v�AZ�dgjqYNZe`=bgfs�*dcZEt��X�lbgt­��k8d�fs` �X`X�v`Xk�¤&` `XkS`XmJ�X�xfs`OfsdVkS��z{ZEhib §�§�§�§�§�§�§�§�§�§©§�§ �S���§ : ^a`=bgdck�u��oh�jl`or¦z{k8jl`=b��qjlY[jlbgt�f´�8kSjxuXfs`OhZ2§�§�§�§�§�§�§�§�§�§�§©§�§�§�§�§�§�§�§�§�§�§�§©§�§)\o�

eM` ¿ 1 �#& �/�@2 ¿ 1 <f5

gCgihkj�l�m#n�o

Page 7: Path Planning for Robust Image-Based Visual Servoing

� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

� �A��� �6� % � �¶��� � �

¨$�xfStgtcjqhefs�*fS�X�Xdgk8fShnUoZet&�otcjq`XrL�vjxt��ofS�*jq`���kSdgY-f�bgjlk8`¥jl`¸��Z�ZeuX�ofShn�¢hk8`8bgdck8�*�qkvkS�ot¦fsdgZ��Ok8jl`=b¦bckL�AkSjq`=b�m�OfStcZeu p?jJ§ Z�bcUoZ�dgkS�AksbNY2�ot�bNdgZef8hnU f¸uXZetcjldgZeu²r8k8fS�­h�kS`�¯or8�Xdnf�bcjqkS` t�bgfSd�bgjl`or���dgkSY f¸r8jl�8Z�` jq`Xjlbcjxfs�h�kS`�¯OrS�Xdnf�bgjlk8` §��}Z�ZeuXZeu-jq`���kSdgY-f�bgjlk8`-jqt$dcZEu��oh�ZeuNbck2bcUXZ¦rSk=fs�Oh�kS`�¯OrS�Xdnf�bgjlk8`[fS`ou�f�r8�lk8�ofs�q�q¡Nt�bgfS�Xj£m�qjqª�jq`Xr���ZeZeu��of8hn�-jqt$dgZe�=�XjqdcZEu §���k�¤­Z��SZedep8j£�*bcUXZ�jq`Xj£bgjqfS�4Z�dgdck8d¶jxt¶�xfsdgrSZ8p=tc�ohnU�f�hk8`8bgdck8�OY-f´¡[�Xdgk�u��ohibfS`-Z�dgdgfsbcjxhV�OZeUof´�vjlk8d¶fs`Ou��´k8d?�qfSdcr8Z�hkS`=bgdck8�XZ��4kSdcbep=Zetc�OZEhjxfs�q�l¡�jl`�bcUXZ��odcZEt�Ze`ohZ�kS� YNk�u�Ze�ljq`Xr�ZedcdgkSdnte§µo�XdcbcUXZedcY[kSdgZSpv�8Z�Z��ojl`Xr-fS�l� bgUXZ�kS�Xz{Zehib&jq` bcUXZ�hefsY[Z�dnf�¯OZ��xu�kS�/�vjqZ�¤;dcZeY[fSjl`Ot&f�`oksb�kS�v�vjqkS�ot$bgf8t��4§|¦jqtc�ofS�$tcZ�dg�SkSjq`Xr�tghnUXZ�Y[ZEt2fsdgZ��qk�h�fs�$��Z�ZEu��of8hn� h�kS`=bcdgkS�¶tckS�q��bgjlk8`ot�§®TVUXZ�¡ bcUv�ot�dgZe�=�XjqdcZ-bgUXZ�uXZ¯Xm`oj£bgjlk8`®kS�¶jl`=bgZ�dgYNZEu�jxf�bcZ-tc�X�Xr8k8fs�xt¦jl`®bcUXZ-tcZ�`otckSd�tc�ofSh�ZNfsb¦bgUXZ[bgf8t��¥�X�xfs`o`Xjl`or��qZ��8Z��J§�TVUXjxt©�ofs�AZ�duXZefs�xt�¤&jlbcU®bcUXZ-�XdgkS�o�lZeY ks�­�of�bgU �o�qfS`X`Xjq`Xr¢jl` jlY-fSrSZ-t��OfShZ-fS`ou¸�odcZEt�Ze`8bnt�f h�kSY[�X�qZbgZ�t�k8�l��bgjlk8`bgk�tc�OZEhjl��¡�fs`ou¥bck�bcdnfShn� jlY-fsr8Z�bgdgfsz{ZehibgkSdgjlZEt�kS�¶fs`¸k8��z{Zehb¦k8�ot�Zedc�8Zeu¢�v¡¥fs`�Ze¡SZ�mJjq`�m·Uofs`ou�dck8�OkSbcjxhtc¡�t{bgZ�Y¢§

y6k8tcjlbcjqkS`�m·�ofStcZeu±fS`ou±jlY-fSrSZm·�of8t�ZEu®tcZ�dg�Sk8jl`Xr¥fSdcZ-`Xk�¤³¤­Z��q�¶�v`Xk�¤&`²fs�o�Xdck=fShnUXZEt��l�E���·p�� ;��·p�� S��J§TVUoZ$¯Odgt�bDkS`XZ�� �s���·p��q�����Jp8�ofStcZeu�kS`�bcUXZ¦hkSY[�X�Xbgf�bgjlk8`�kS�Af�\sm�� ¨­fSd�bgZetcjqfS`2Z�dgdgkSdEp�dcZE�8�ojldgZet6f��AZ�dc��Zehib¨ '$��m·Y[k�u�Z��4ks�*bcUXZ�kS�Xz{Zehib&fs`OuLf2hefs�qjl�odgfsbcZeuLh�fsY[Zedgf�bckNkS��bnfsjq`L�X`v�XjxfStcZeu-�Ak8tcZ©Zet�bcjqY[fsbcjqkS` §?TVUXZY-fSjl` f8u���fs`=bgfSrSZ-kS�&bcUXjxt�fS�X�Xdgk8f8hnU jxt2bcUofsbNjlbNh�kS`=bcdgkS�xt�u�jqdcZEhibg�l¡ bcUoZ hefsY[Z�dnf bcdnf�z{ZEhibck8dc¡®jl` bcUXZ¨­fSd�bgZetcjqfS`-t��of8hZ8§��}k�¤$Ze�SZedep�bgUXZ�dgZ}jxt$`Xk2hk8`=bcdgkS�ojq`-bgUXZ¦jqY-fsr8Z}fs`ouLt�k8YNZ}�ofsdcb$kS� bcUXZ¦kS��z{ZEhib$Y-f´¡r8Zb}k8��b¦kS�6bcUXZNh�fsY[ZedgfN¯oZe�qu�ks�?�vjlZe¤ u��Xdgjl`Xr�t�Zedc�8kSjq`Xro§ '&uXu�jlbcjqkS`ofS�l�q¡SpObcUoZ�hk8Y[�X��bgfsbcjqkS`�fsb¦Zef8hnUjlbcZedgfsbcjqkS`¢ks�6bgUXZ2hk8`=bcdgkS� �qk=k8�¥ks�Df-\�m � ZedcdgkSdVY-f´¡ �lZEfSu�bckLf-t�bcdgkS`or�t�Ze`ot�jlbcjq�vj£b{¡�¤&j£bgU�dgZetc�OZEhib&bckY[k�u�Ze�ljq`Xr¢Z�dgdck8dgt�fS`ou¸`okSjxt�Z[�AZ�dcbc�Xdg�of�bgjlk8`ot��q���J§-^a` bgUXZ�t�ZEhk8`ou®fS�X�Xdgk8f8hnU p4bcUXZ-ZedcdgkSd�jxt�u�jqdcZEhibc�q¡Y[ZEfStc�XdcZEu jl`¥bgUXZ2jqY-fsr8ZS§}TVUofsb}jqYN�o�ljqZet©t�k8Y[Z2u�Z�r8dcZeZet&kS�Ddck8�X�ot�bc`XZEtct�¤&j£bgU�dcZEt��AZehb}bgk�Y[kvuXZ��qjl`XrZedcdgkSdnt&fs`Ou¥`XkSjxtcZ��OZed�bg�Xdg�of�bgjlk8`ot�§�K¦`�bcUXZ�ksbcUoZ�d}UOfs`ou p4tc�ohnU�f�hk8`=bcdgkS�*Y-f´¡ �lZEfSu bgkLjq`ofSu�ZE�=�of�bgZhefsY[Z�dnf&Y[ksbgjlk8` ��`oksb6kS��bgjlY-fs�=kSd�`Xksb6�XUv¡�t�jxh�fS�l�q¡©�´fS�ljxu2hefsY[Z�dnfVbgdgfsz{ZehibgkSdg¡"!��l�#�J§6¬­¡�h�kSY[�Ak8tcjl`Xr}bcUXZZedcdgkSdV���o`ohibgjlk8`¸ks�D\sm��³¨­fsdcbcZetcjxfs`¢��Zefsbc�XdgZet¦fs`ou¢jqY-fsr8Z���Zefsbc�XdgZetep�L�fS�ljxt�Z�b ���o�XdgkS�Ak8tcZ�f-rS�qkS�Ofs�q�l¡t�bgfS�Xjq�ljqª�jq`XrLtckS�q��bcjqkS`¥hefs�q�lZEu%$'&�(�$ ) �=jxtc�ofs�/t�Zedc�8kSjq`XrN��kSd}r8Z�`XZedgfS�*tcZbc�o�*�l�E\��J§���k�¤­Z��SZed+$,&-(�$�)�vjxt��Ofs��tcZ�dg�Sk8jl`orNjxt¦dcZe�qfsbcjq�SZ��q¡�tcZ�`Ot�jlbcjq�SZ�bgkLY[Zef8t��odcZeYNZe`=b}�OZed�bg�Xdg�of�bgjlk8`�t�jq`ohZ�fs`�UXk8Y[kSrSdnfs�oU=¡Ljqth�kSY[�X��bgZeu²fsb�Zef8hnU²j£bgZ�dnf�bgjlk8` kS�&bcUXZ hk8`8bgdck8�D�qk=k8�±bck®Z¹vbcdnfShb�bcUXZ¥¨­fsdcbcZetcjxfs`±�ofsdcb�kS�VbcUXZ�Z�dgdck8d���o`ohibgjlk8` §^·b jqt�¤­Z��q�¦�v`Xk�¤&`,bgUof�b jlY-fsr8Zm·�ofStcZeu tcZ�dg�Sk8jl`or²jqt��lk�hefs�q�l¡,t{bnfs�X�qZ®fS`ou,dgkS�X�ot�b�¤&jlbcU dcZEt��AZehbLbckY[k�u�Ze�ljq`Xr ZedcdgkSdntLfS`ou�`okSjxt�Z®�OZed�bg�Xdc�Of�bcjqkS`Ot�§³TVUXZ �SZ�¡,jxu�ZefQks��k8�Xd ¤$k8dc�,jxt�bgU=�OtLbgkQ�Ot�Z®bcUXZ�qk�h�fS�$t�bgfs�ojl�qj£b{¡±fs`Ou²dck8�X�ot�bc`XZEtct2ks�&jqY[fSrSZ�mJ�of8t�ZEu±t�Zedc�8kSjq`Xr¥�v¡±tc�AZehjl��¡vjl`or¥bcdnf�z{ZEhibck8dcjqZet�bck���k8�l�qk�¤jq`QbcUXZ¢jqY[fSrSZ8§Q^a`ouXZ�Zeu pD��kSd�f¸bcdnf�z{ZEhibgkSdg¡®��k8�l�qk�¤&jq`Xrop­f¸�qk�h�fs��h�kS`=bcdgkS�VtckS�q��bcjqkS` ¤$k8dc��t��XdgkS�AZ�dg�l¡tcjq`ohZ�h�XdgdgZ�`=b­fS`ouLu�ZEt�jqdgZeu�h�kS`�¯or8�Xdnf�bcjqkS`OtDdgZ�Y-fsjq`Lh�qk8tcZS§kL¥k8dcZek��SZedep���k8d&fs�q�ObgUXZ©�XdcZe�vjlk8�ot��q¡-hjlbcZEuh�kS`=bcdgkS�XtghnUXZeYNZEt�pS�SZ�Ze�Xjq`Xr�bcUXZ&¤&UokS�qZ&kS��z{ZEhibDjq`[bcUXZ}hefsY[Z�dnf}¯oZ��xu[ks�4�vjlZe¤ fS`ou-f´�Sk8jqu�jq`Xr¦bcUXZ}dck8�OkSbz{k8jl`=b[�ljqY[j£bnt[u��Xdgjl`Xr t�Zedc�8kSjq`Xr�dgZ�Y-fsjq` f¸`Xksb[kS�v�vjlk8�ot�bnfStc�²fs`ouQh�fs` jl`v�Sk8�l�8Z-bcUoZ�� fSjl�q�XdgZLkS��bcUXZtcZ�dg�Sk8jl`oro§6TVUXZetcZ�Zetgt�Ze`8bgjqfS�4jqtgt��oZet&h�fS`¢�OZ�Zef8t�jq�q¡�jl`=bgdck�u��oh�Zeu f�b&bgUXZ�bgf8t����o�qfS`X`Xjq`Xr[�lZe�SZ��J§K¦`o�l¡���Ze¤ �ofS�OZedgt6u�ZEfs�v¤&j£bgU��Of�bcUN�X�xfs`X`ojl`Xr©jq`�jqY-fsrSZ­t��OfShZ8§/µokSd6f©�SZ�dg¡�t�jqY[�X�qZVhefStcZ���bcUXZ&¤­kSdg�qu�jqttc�X�X�Ak8tcZeu�bckN�OZ©�X�xfs`ofSd­fS`ou-bgUXZ�h�fsY[Zedgf2jqt­kS`oZ©u�jqY[Z�`otcjqkS`ofS�.!pX¨$k�¤Vfs`�fs`ou0/}k�u�jlbgtghnUXZ��[u�ZEtch�dcjq�OZjq`1� \��*f�r8�lk8�ofs�q�q¡[t�bgfS�Xjq�ljqª�jq`Xr[Y[ZbcUokvu �otcjq`Xr�`Of´�=jqr8fsbcjqkS`L���X`ohibgjlk8` §?^a`1� ���JpXf�bcdnf�z{ZEhibgkSdg¡Nr8Z�`XZedgfsbck8d�Ot�jq`Xr¸f®t{bgZ�dgZ�k¸tc¡�t{bgZ�Y jqt��XdgkS�Ak8tcZeu²fS`ou²fs�X�o�ljqZeu±bgk¸k8�ot{bnfSh��lZLf´�8kSjxuXfs`OhZS§ '�` fS�ljqrS`XY[Ze`8b�bgf8t���Ot�jq`Xr jl`=bcZedcY[ZEu�jqfsbcZN�vjlZe¤�t¦kS�$fs`®kS��z{ZEhib�tc¡=`=bgUXZetcjlªeZeu¥�v¡�jqY-fsr8Z2Y[kSdg�XUXjq`Xr jqt©�XdgZetcZ�`=bcZEu�jq`2�q�@S3�J§';�ofsbcUL�o�qfS`X`Xjq`Xr���k8d­fNt{bgdgfSjlr8U=b�m·�ljq`XZ�dgkS�Aksb$bcdnfs`otc�qfsbcjqkS`�k8�otcZ�dg�SZeu[�v¡-f�¤$ZEfs�v�l¡[h�fS�ljq�Xdnf�bgZeu�t�bcZedcZek

465Eg7498

Page 8: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� :

tc¡�t{bgZ�Y¢p jxt��AZ�dc��kSdgYNZEu¸jq` �l��� �·§���k�¤­Z��SZedep `okS`XZ-kS�¶bcUoZ�Y ¤­Z�dgZ-u�ZefS�ljq`Xr¢¤&jlbcU±dck8�X�ot�bc`XZEtct�jxtgt��XZEt�§^a` �XdgZ��vjqkS�otN¤­kSdg� �q�3:��Jp­¤$Z¢Uof´�8Z��odck8�Ok=t�ZEu f �OkSbcZ�`=bgjqfS�V¯oZ��xuvm·�ofStcZeu �of�bgUQ�X�xfs`o`Xjl`or r8Z�`XZedgfsbck8dbgUof�b[u�ZbgZ�dgY[jl`XZEt�bgUXZ�bcdnf�z{Zehbck8dcjqZet�jq` bgUXZLjqY-fsr8ZLks�}f®t�Z�b�kS���Ok8jl`=bgt��q¡=jq`Xr®kS` f¸�X�qfS`ofsd2bgfSdcr8Zbe§^a`�bcUXjxt©�ofs�AZ�dEp4¤$Z�rSZ�`oZ�dnfs�qjlªeZ�bcUXjxt©YNZ�bcUXk�u�bck�bgUXZ[h�fStcZ�kS�DkS��z{ZEhibgt¦ks�­fs`v¡��o`X�v`Xk�¤&`¸tcUofs�AZete§¦^·bjxt�f�dck8�X�ot�b�fS`ou t{bnfs�X�qZ[YNZ�bcUXk�u®Z��8Z�` j£�­jl`Xjlbcjxfs�$fs`ou u�ZetcjqdcZEu�dgkS�Aksb��Ok=t�jlbcjqkS`Ot©fSdcZ[u�jxt{bnfs`=be§�K¦�XdfS�X�Xdgk8f8hnUQhk8`otcjqt�bgt-kS��bcUXdgZ�Z¥�XUOfStcZete§;^a`,bgUXZ¥¯odnt{b�k8`XZSp­bgUXZ¸u�jxtch�dcZ�bcZ¥r8Z�kSY[Z�bcdgjqh¥hefsY[Z�dnf �of�bgU��bcUofsb�Z�`otc�XdgZet$bcUoZ��XUv¡vtcjxh�fs�4��fs�qjxu�j£b{¡�ks��bcUXZ�dgkS�Aksb­bcdnf�z{ZEhibck8dc¡"!¶jqtV�AZ�dc��kSdgY[Zeu fSt&f[tcZe�=�XZ�`OhZ�ks���jq`=bcZedcY[ZeuXjqfsbcZ}hefsY[Z�dnf��Ak8tcZetD¤&UojqhnULfS�X�Xdgk8f8hnUXZet?f8t¶Y��ohnU�fSt¶�Ok=tctcjl�o�lZ©f�t{bgdgfSjlr8U=b¶�ljq`XZ}bcdnfs`otc�qfsbcjqkS` §^a`�bcUXjxt*�oUofStcZSpEbcUoZ$Y[ZehnUOfs`Xjxh�fS�8fs`Ou��vjqtcjl�ojl�qj£b{¡�h�kS`ot�bcdnfsjq`=bgt fSdcZDjq`=bcdgk�u��ohZEu §*^a`2bcUXZ­t�ZEhk8`ou�kS`XZ8p´bcUXZuXjqtghdgZbgZ$r8Z�kSY[Z�bcdgjqh¶bcdnf�z{ZEhibck8dc¡�kS�obcUoZ$bnfsdgrSZ�b/jq`2bgUXZ&jqY[fSrSZVfs`ou�bcUXZ�uXjqtghdgZbgZ$r8Z�kSY[Z�bcdgjqh¶bcdnf�z{ZEhibck8dc¡kS�¦bgUXZ�dgkS�Aksb-jq`,bcUXZ[z{k8jl`=bLtc�ofSh�Z�fsdgZ k8��bgfSjl`oZeuQ��dck8Y bgUXZ�h�fSYNZedgf �ofsbcU § µ/jl`ofS�l�q¡Sp&h�kS`=bcjq`v�XkS�otfS`ou u�Zedcjq��fs�X�qZ rSZ�k8Y[Zbcdgjxh-�of�bgUotNjl` bcUXZ jqY[fSrSZ�¤&j£bgUQfs` fStgt�k�h�jqfsbcZeu bgjlY[jq`Xr �qf´¤ fSdcZ�rSZ�`oZ�dnf�bcZEufS`ou¸bcdnfShn�SZEu¥�otcjl`or¢fS`®jqY-fsr8Zm·�ofStcZeu�h�kS`=bcdgkS�?tghnUXZ�Y[Z8§N¬­¡��otcjq`Xr bcUXjxt�fs�X�odck=fShnU p h�odcdgZ�`=b��vjqtc�ofs���ZEf�bg�XdcZEt�fs�q¤­f´¡�t¦dcZeY-fsjq` h��lk=t�ZNbck¢bgUXZ�jqd�u�ZEt�jqdcZEu¸��fS�l�XZ�fS`ou f¥hkS`=bgdck8�/�v¡�jqY-fsr8Zm·�ofStcZeu®t�Zedc�8kSjq`XrZe`otc�XdcZEt�bgUXZ[dck8�X�ot�bc`XZEtct¦¤&jlbcU¸dgZetc�AZehib©bck Y[k�u�Z��qjl`or�ZedcdgkSdnt}fS`ou�`Xk8jqtcZN�OZed�bg�Xdc�Of�bcjqkS`Ot�§�¨$k8`8bgdgfSdc¡bgk[ksbcUoZ�d�fs�o�Xdck=fShnUXZEt¶Z�¹��X�lk8j£bgjl`or�bgUXZ�dgkS�Aksb&dcZEu��X`ouofs`oh�¡'� �3�Jp7�q�@;3�Jp�bgUXZ�Y[ZehnUofS`Xjqhefs� fS`ouL�vjxt�jq�Xjq�ljlb{¡h�kS`ot�bcdnfsjq`=bgt©h�fS`¸�AZNZe`otc�XdcZEu¥Z��8Z�`®j£�Vfs�q��bgUXZ[dck8�OkSb�uXZ�rSdgZ�ZEt}kS�D��dgZ�ZeuXkSY fsdgZN�ot�ZEu¥bgk�dgZefS�ljqª�Z�bcUXZbnfStc�4§kK¦�Xd­�ofsbcUL�o�qfS`X`Xjq`Xr�t�bcdnf�bgZ�rS¡[jxt¶�OfStcZeu-k8`�bcUoZ¦�AksbgZ�`=bcjxfs�O¯oZ��xu�Y[Z�bcUXk�u,� �3�·p �l�E��� fs`ouLh�fs`��OZfS�X�X�qjlZEu Z��SZe`�jl�6`XkS`XZ�\sm�� Y[k�u�Ze� jqt}f´�´fSjl�xfs�o�lZ�fs`OuLZe�SZ�`¢jl��bgUXZ�h�fSY[Z�dnf2jq`=bcdgjl`otcjxh��ofsdnfsY[ZbgZ�dntVfsdgZ�OfSu��q¡�Zet�bcjqY-f�bgZeu §TVUoZ-�ofs�AZ�d�jxt�k8dcr=fs`Xjqª�ZEu®f8t���kS�q�lk�¤�te§�^a` NvZEhibcjqkS` �Xp*¤­Z-dcZEh�fS�l�DtckSY[Z-�of8t�jxhN���X`ouXfSY[Z�`=bgfS�qte§�TVUXZY[Z�bcUXk�u¢ks�?�ofsbcU¥�X�xfs`o`Xjl`or-��kSd}f��=`ok�¤&` k8��z{Zehb}jxt��XdgZetcZ�`=bcZEu jl` NvZEhibgjlk8`�\�fs`ou¢Z�¹vbcZ�`Ou�Zeu bgk-bcUXZhefStcZ�¤&UoZ�dgZ¢f±\sm��)Y[k�u�Ze�&ks�©bcUXZ¥k8��z{Zehb[jxt[`XkSbLf´�´fSjl�xfs�o�lZ jl` NvZEhibgjlk8`Q�O§�¨­fSY[Z�dnf®hefs�qjl�Xdnf�bgjlk8`ZedcdgkSdnt}fSdcZ[fs�xtck�hk8`otcjqu�ZedcZEu §�^a`GNvZehbcjqkS`\:�p fLbcjqYNjq`Xr¢�xf´¤ jqt�f8tctck�hjxf�bcZEu¢bck bgUXZ�r8Z�k8YNZ�bcdgjqh��ofsbcU §^a`WNvZehbcjqkS`�;op8¤­Z¦tcUXk�¤,UXk�¤,bck��otcZ¦fs`-jqY-fsr8Zm·�ofStcZeu-h�kS`=bcdgkS�ofS�X�Xdgk8f8hnU�bgk�bgdgf8hn�2bgUXZ}bgdgfsz{ZehbckSdgjqZete§TVUoZ©Z�¹��OZedcjqY[Z�`=bgfS� dcZEt��X�lbgt&fSdcZ�r8jl�8Z�` jl` NvZehbcjqkS`¸�v§

� � � ��% �2�!"&������� �

^a`�bcUXjxt¦tcZehbcjqkS` pA¤­Z�jl`=bgdck�u��oh�Z�`XkSbgf�bgjlk8`ot}fS`ou�h�kS`oh�Z���bnt&bcUOf�b©fSdcZ2`XZeh�ZetgtcfSdc¡Lbck�uXZetcjlr8`¥kS�Xd¦�of�bgU�o�qfS`X`Xjq`Xr[rSZ�`oZ�dnf�bck8de§

���� ��� ���������������������� ��"!# $�� ��% �!� &�' ��!# � Zb)(�*-�AZ�f¥��dnfsY[ZLfsb�bnfShnUXZEu bck�bgUXZ�bgfsdgrSZ�bep�(�+{p�(�,�fs`ou-( X �OZ�bcUoZ���dnfsY[Zet�f�b�bnfShnUXZEu±bck¸bcUXZhefsY[Z�dnfNjl`¥jlbgt�jl`Xjlbcjxfs�JpOh��XdgdcZe`8b}fS`ou¢u�ZetcjqdcZEu��Ak8tcjlbcjqkS`ot&dgZetc�AZehibgjl�8Z��q¡ �°t�ZeZ�µ�jqrS�odcZ2� !i§/.$0�u�Ze`XksbgZetf \�m ��bgfSdcr8Zb[�Ok8jl`=b-¤&j£bgUQUokSY[kSr8Z�`XZekS�otNhkvkSdnu�jq`of�bgZet)12035476 8120:93<;>=?9@4A6 BC03�DE03GF/03/;H=I9KJLjq`( 3 � ¤&j£bgUMLONQPSR�T�UVT�W�TYX�Z�!i§[.$0¥jqtL�Xdgksz{ZehbcZEu,jl`,bgUXZ¸jqY-fsrSZ�fsb�bcjqY[Z\W kS`=bgk f±�Ok8jl`=bL¤&jlbcUUokSY[kSr8Z�`XZekS�ot2`XkSdgY-fs�qjlªeZeu h�k=k8dguXjl`ofsbcZEt^] 0 , 4_6 L 0 ,a` 0, ;H=I9¶§ TVUXZ¢hkSdgdgZetc�Ok8`ou�jq`Xr¸hkvk8dgu�jq`ofsbcZet�jq`�oj£¹�Z��xt­fsdgZ¦u�Ze`XksbgZeuGb 0 , 4K6 c 0 ,'d 0, ;H=I9e4@6 fb 0:9, ;H=I9e4Mgih 0 , pv¤&UXZ�dgZ�bgUXZ©Y[fsbcdgj£¹ig�jqtVf2`XkS`Ltcjq`XrS�X�xfsdjk�limSn hSoVp n"q psr n putsv�h qsw o qsn oVx�psr nyk�zI{ v|pst~} �

gCgihkj�l�m#n�o

Page 9: Path Planning for Robust Image-Based Visual Servoing

; ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

T

RO

BO

T

s

+

s*(t)

Trajectories Planning Control law

Features Extraction

desired image

initial image Constraints

µ/jlr8�XdcZ[���6¬­�qk�hn�Lu�jxfsrSdnfsY�kS�*bgUXZ��XdgkS�Ak8tcZeu�YNZ�bcUXk�u

Y-fsbcdgj£¹�h�kS`=bgfSjl`ojl`Xr�bcUoZ�h�fsY[Zedgf�jl`=bgZ�dg`ofs� �ofSdgfSY[ZbcZedgt��g 4

������ ����� �� R�������� c �� �������� U�������� d �� � ;

� 4

���!" !#�� c�� !#� d �� � ;

¤&UoZ�dgZ^c � fs`Ou d � fSdcZ2bcUXZN�Xjl¹�Z��xt¦h�k=k8dguXjl`ofsbcZEt�ks�D�odcjq`ohjq�ofS���AkSjq`=bep4�¶jxt�bgUXZ���kvhefs�/�lZe`XrsbgU p � fs`Ou� � fSdcZ�bgUXZ�Y-fsr8`Xjl¯Oh�fsbcjqkS`ot&dgZetc�AZehibgjl�8Z��q¡Ljq`¢bcUXZ c¸fs`Ou d u�jqdcZEhibgjlk8`ot}fS`ou$��jqt�bcUXZ�fs`Xr8�lZ��AZb{¤­Z�Ze`bgUXZetcZ�f�¹�Zete§TVUoZ�fsjqY ks�4kS�odD¤$k8dc��jxt?bgk�u�ZetcjqrS`[f©bcdnf�z{Zehbck8dcjqZet6r8Z�`XZedgfsbckSd6�AZ�dc��kSdgY[jl`Xr�f�h�kS`=bcjq`v�XkS�ot?fS`ou[u�Z�dgjq�8mfS�X�qZ­h��Xdc�8Z&%����'�y4@6 fb)(+*,.-�/102020 fb#34*,.-�/ =I9 �AZb{¤­Z�Z�`2bcUXZVjq`Xjlbcjxfs�Xhk8`�¯or8�XdgfsbcjqkS`5% +�4@6 fb�('*+ 0�020 fb634*+ =?9¢fs`ou2bcUXZuXZetcjldgZeu¢k8`XZ5% X 4 6 fb (+*X 0�020 fb 34*X = 9 §}µ/jldnt{bEpobgUXZ[u�jqtghdgZbgZ�rSZekSY[Zbgdcjxh�h�fSY[Z�dnf-�of�bgU�jxt��AZ�dc��kSdgYNZEu�fStf�t�ZE�=�XZ�`oh�Z}kS���)jl`=bgZ�dgYNZEu�jxf�bcZ©h�fSYNZedgf2�Ok=t�ZEt�7 4[P98 , � W N-; 020�0 � Z¦�Ot�jq`Xr�bgUXZ©�OkSbcZ�`=bgjqfS�O¯OZ��xut�bcdnf�bgZ�r8¡Nu�ZEtch�dcjq�AZeu�jq`�bgUXZ©`XZ¹vb&t��X�Ot�ZEhibcjqkS`*§6TVUoZ�` p=bcUoZ©u�jxtghdgZbcZ©kS�Xz{Zehib$bcdnf�z{ZEhibck8dc¡Njq`LbgUXZ¦jqY-fsr8Z: 4 P�% , � W N ; 020�0 � Z[fs`ou¢bgUXZ2dgkS�Aksb}bcdnf�z{Zehbck8dc¡ jl`�bcUXZ}z{kSjq`8b�tc�ofSh�Z<;K4 P>= , � WeN ; 0�020 � ZfSdcZ}kS��bnfsjq`XZeu[��dgkSY?7 §?µ/jl`Ofs�q�l¡8pvh�kS`=bcjq`v�XkS�Ot¶fs`Ou�u�Z�dgjq�´fS�X�qZ}rSZekSY[Zbgdcjxh&�ofsbcU�jq`�bgUXZ}jqY-fsr8Z}¤&jlbcULfS`f8tctck�hjxf�bgZeu-bgjlY[jq`Xr-�xf´¤@%����'�­jxtV�OZed���k8dcY[Zeu fs`Ou�bgdgf8hn�SZeu � tcZ�Z�µ/jqrS�XdgZ[� !i§���u� AC!� &�� ��"���CB ���:�ED � GF�!/�K¦�od&�of�bgU �o�qfS`X`Xjq`Xr-t{bgdgfsbcZ�r8¡-jqt��ofStcZeu�kS` bcUoZ��OkSbcZ�`=bgjqfS�4¯oZ��xu YNZ�bcUXk�u §$TVUXjqt�YNZ�bcUXk�u ¤VfStVkSdgjqrSjlm`Ofs�q�l¡LuXZ��SZe�lk8�OZEu���k8d�fs` kS`�m·�qjl`XZ�h�kS�q�ljxt�jqkS`¢f´�Sk8jquofs`oh�Z � �3�·p7�l�E���·§

` &°�X�e�32 �8� &­�YACA/� ¿ �o�3?^a`³bgUXjxt fs�o�Xdck=fShnU bcUoZ�dck8�OkSb Y[ksbgjlk8`ot fsdgZ �X`ouXZ�d bgUXZ jq`YTo�oZ�`oh�Z�ks��fs` fSd�bgj£¯Ahjxfs���AksbgZ�`=bcjxfs�¯OZ��xu �IH !¥uXZ¯o`XZEu fSt bcUoZ t��oY ks�-fs`fsb�bcdnfShbcjq�SZ±�AksbcZe`=bcjxfs� �IH J3!¢�o�X�l�qjq`Xr,bcUXZ dck8�OkSb bgk�¤­fSdguotbgUXZ¢rSk=fs�&hk8`�¯orS�odgfsbcjqkS` ��8 X ![fS`ouQf®dgZ��X�X�xtcjl�8Z��AksbgZ�`=bcjxfs� �IHLK�!��X�otcUXjl`or®bgUXZ¥dgkS�Aksb-f´¤­f´¡ ��dgkSYbgUXZ¸k8�ot�bgfSh��lZEt�§ L¥ksbgjlk8`��X�xfs`X`Xjq`Xr jxt��OZed���k8dcY[Zeu�jl` fS`�jlbcZ�dnf�bgjl�8Z¥� fStcUXjqkS` § '­b ZEfShnU jlbcZ�dnf�bgjlk8`fS` fsdcbcjl¯Ohjxfs�V��k8dgh�ZNMO�I8$�ip&¤&UXZedcZ bgUXZQPNR�;¥�8ZehbckSd<8 dgZ��XdgZetcZ�`=bgt[f±�ofSdgfSY[ZbcZedcjqªefsbcjqkS`Qks��bcUXZ

465Eg7498

Page 10: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �

dgkS�Aksb�¤$k8dc��tc�ofSh�Z�� �����vp6jxt2jq`ou��oh�Zeu²�v¡ bcUoZL�AksbcZe`=bcjxfs�D���o`ohibgjlk8` §®TVUXjxt2��kSdnhZ�jxt�uXZ¯o`XZEu fStMO��8$�#4 ���)9 H ¤&UXZedcZ �)9 �H u�Z�`oksbcZEt$bcUXZ©bcdnfs`Ot��Ak8tcZ}kS� bcUXZ�r8dgf8u�jlZe`=b$�SZEhibgkSd­ks�#H f�b�8¸§��}tcjl`XrbgUXZetcZ[hkS`v�8Z�`=bcjqkS`otep MO�I8$�©h�fs`®�OZ-u�ZEhk8YN�Ak8tcZeu¸fSt¦bcUXZ[tc�XY)ks�$b{¤$k��SZehbck8dgtepM JG�I8$� 4 � � 9 HJfS`ou M K ��8<�C4 � �)9 H K pV¤&UXjxhnU�fsdgZ¢dcZEt��AZehbcjq�SZe�l¡ hefs�q�lZEuQbcUXZ¸fsb�bgdgf8hibcjq�SZ¥fS`ou dcZe�X�X�xt�jq�SZ ��kSdnhZEt�§y?fsbcU rSZe`XZ�dnf�bgjlk8` �Xdgk�hZ�ZEuXt¥fs�qkS`or bgUXZ u�jqdcZEhibgjlk8`;kS� MO��8<� dgZ�r=fsdnu�Zeu;fSt bgUXZ Y[k8t�b¥�Xdck8Y[jqtcjl`oruXjldgZehbcjqkS`�kS�2Y[ksbcjqkS`*§ TVUv�otep}ZEfShnU tcZ�r8YNZe`=b�jxtLk8dcjqZ�`=bcZEu�fs�qkS`or bcUoZ®`XZer8fsbcZeu�rSdnfSu�jqZ�`=bLkS��bcUXZ�AksbgZ�`=bcjxfs�¶���X`ohbcjqkS` h�kSY[�X��bgZeu fsb�bcUXZ hk8`�¯orS�odgfsbcjqkS`²f�bcbgfsjq`XZEu±�=¡®bcUoZL�XdgZ��vjqkS�ot�tcZ�r8Y[Z�`=be§�TVUXZuXjqtghdgZbgZmJbcjqYNZ©bgdgfsz{ZehibgkSdg¡-jqtVr8jl�8Z�` �v¡-bcUXZ�bgdgfS`otcj£bgjlk8`�ZE�8�Of�bcjqkS` �

8 ,� ( 4 8 ,���� , MO�I8 , �� MO��8 , ����� �{� !

¤&UoZ�dgZ W-jxt$bcUXZ©jq`ohdgZ�Y[Z�`=bVjl`Ou�Z¹LfS`ou�� ,�jxtVf��Ak8tcj£bgjl�8Z©tgh�fS�ljq`Xr�� f8hibgkSdVu�Z�`XkSbcjq`Xr�bgUXZ��lZe`XrsbgULkS� bcUXZW -�� jl`oh�dcZeY[Z�`=be§% ¿ ÀC2�� » À 6 ¿ ��� » �¨$k8`otcjqu�Zed­bcUXZ��X`OhkS`Ot{bgdgfSjl`XZEuL�odck8�X�lZeY��

h U��CH �I8$� T 8 N�� �' h��qf8tctcjxh�fs�Xh�kS`=bcjq`v�XkS�Ot?rSdnfSu�jqZ�`=bDt�bcdnf�bcZerS¡���k8d?¯o`ouXjl`Xr�f�Y[jq`XjqY2�XY ks�1Hhk8`otcjqt�bgt6kS�AY-fs�vjq`Xr 8 ���'���fSdc¡�f8h�hk8dguXjl`Xr�bck[bgUXZ�Z��8kS�q��bcjqkS`�Ze�=�ofsbcjqkS`1��8 4 ����� � 9 H �°� !¤&UoZ�dgZ � jxt?f¦�Ok=t�jlbcjq�SZ&tgh�fs�xfsd6fs`Ou � jqt?f�h�kS`ot�bgfS`8b6�Ak8tcj£bgjl�8Z$Y-f�bgdcjl¹ §�y¶dcZeY2�X�lbcjq�X�q¡=jq`Xr �°� !��=¡ � H-p¤­Z�rSZ�b��

VV�� H ��8<� 4 ��� � H � � 9 H � � � \ !

TVUv�otOH�u�ZEhdgZefStcZet¦¤&jlbcU®bcjqYNZ-f8t¦�qkS`or f8t � H"!4 � p fS`ou®dcZeY[fSjl`Ot©h�kS`ot�bgfS`=b©¤&UXZe` � H$4 � §' hk8Y[YNk8`¸fs`Ou¸tcjlY[�X�qZNhnUokSjxhZ���k8d � jxt}bgUXZNjqu�Ze`=bcjlb{¡¥Y[fsbcdgj£¹$#�§2^a`�bcUojqt�h�f8t�Z8p 8 YNk��8Zet}jl`®bcUXZuXjldgZehbcjqkS`¢kS�X�Ak8tcjlbcZ�bgk-bcUXZ2rSdnfSu�jqZ�`=b}f�bO8�§�TVUojqt¦t{bgdgfsbcZ�r8¡Ljxt}f8u�kS��bgZeu jq`¥bcUXZ�h�xfStgt�jxh�fS�*fs�X�odck=fShnUuXZetghdgjl�AZeuQ�XdgZ��vjlk8�otc�l¡8pD¤&UXZ�dgZ M 4 � � 9 H[§ ¨$kS`otcjquXZ�d[`Xk�¤�f®�AksbcZe`=bcjxfs�V¯oZe�qu H&% 4 H �('���8$�'�¤&UoZ�dgZ)'¢jxt-u�j �AZedcZe`8bgjqfS�X�qZ Ze�SZedc¡v¤&UXZedcZ�jl`*� §QTVUXZ¢Z��8kS�q��bcjqkS`QZe�=�ofsbcjqkS` ks�+'8p$¤&UoZ�` 8 Y[k��SZetf8h�h�kSdnu�jl`or2bgk �J� !pXjqtVr8jl�8Z�` �v¡�

�' 4 ���-,/. '. 810 � � 9 HK4 ���+,). '. 820 �3,). '. 820 9 � 9% H ��� !

^a` kSdnu�Zed$bgUof�b�'�Y[k��8Zet$jl` bcUoZ�uXjldgZehbcjqkS` kS�X�Ak8tcj£bgZ}bgkNbcUXZ�r8dgf8u�jlZe`=b­kS�6H f�b�'8pvbcUXZ�Y-f�bgdcjl¹ � h�fS`�AZ�hnUXk=t�Ze`�fSuXZe�=�of�bgZ��q¡ � 4 � 4 � %/4 ,5. '. 820

,). '. 810 9

� : !687 r n { v|putu} � k:9 m n h oVp n"q pur n wSqun<;Sm o z } h�= n t qun oVx pur n�{ v|putu} � k?> } p�} q psr n } h�= n t qsn�{ v|pst~} �y} x k } q vCh o#h z?q } hA@ ;CB vVtqED�; vVt ny{ v|putu} �

gCgihkj�l�m#n�o

Page 11: Path Planning for Robust Image-Based Visual Servoing

S ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

�}ksbgZ$bgUof�b � jxt?f��Ak8tcj£bgjl�8Z­Y-fsbcdgj£¹�fS`ou2bcUv�ot6bgUXZ&dcZe�qfsbcjqkS` � \ !*jxt?�8Z�dgj£¯oZEu §�TVUoZ&Z��Sk8�l�XbcjqkS`NZe�=�of�bgjlk8`� � !­h�fS` bcUv�otV�AZ�dcZe¤&dcjlb�bgZ�`1� �' 4 ��� � 9% HTVUoZ�fSd�bgj£¯Ahjxfs�4��kSdnhZ�f8tctck�hjxf�bcZEu-bgk�bgUXZ��AksbcZe`=bcjxfs� ¯oZe�qu H % ��'��I8$�'��jqt­bgU=�Ot2�

M�%��I8$�#4 �+� � 9 H&%�4 � ,/. '. 810 � 9% H % � ; !

� UoZ�`�tcZ��SZedgfS� �OkSbcZe`8bgjqfS� ���X`OhibcjqkS`Ot}fsdgZ�hk8`otcjqu�ZedcZEu p�bgUXZ2u�k8YNjq`ofS`8b}fSd�bgj£¯Ahjxfs� ��k8dgh�Z�u�Z�dgjl�8Zeu���dgkSYbgUXZ��OkSbcZe`8bgjqfS� H %©hdgZefsbcZetVfNu�kSY[jq`ofs`=b&Y[ksbgjlk8`�kS� '�jq`LbgUXZ�u�jqdcZEhibcjqkS`�kS�X�Ak8tcjlbcZ©bckNbcUoZ©r8dgf8u�jqZ�`=b­kS�H&%�f�b+'8§&^a`¥�XdnfShbcjxhZSpO�v¡��otcjq`Xr�t��OhnU¥�Xdck�h�Zetgt�pXjlb¦jxt�Y[k8dcZ�ZEfStc¡LbgkLhk8`8bgdck8�4bcUXZ�dcZe�qfsbcjq�SZ�jq`YTo�XZe`ohZkS��ZEfShnU���kSdnhZ�fs`ouLbcUv�ot&bck-hk8`8bgdck8�4bcUXZ�hefsY[Z�dnfNkSd­bcUXZ�k8��z{Zehb&bcdnf�z{ZEhibck8dcjqZete§^a` kS�od�hefStcZ2bgUXZ�hk8`=bcdgkS�/kS��z{ZEhibcjq�SZ-hefs`¸�AZ[��kSdgY2�o�qfsbcZeu®fSt¦��kS�q�qk�¤O�}bgk�bgdgfS`ot���Z�d©bcUXZ�tc¡�t{bgZ�Y bgk¢fuXZetcjldgZeu��Ok8jl`=b&jq`�bgUXZ�tcZ�`otckSd&tc�ofSh�Z�tgf�bgjqt���¡vjl`or�bgUXZ���kS�q�qk�¤&jl`Xr[hk8`ot�bcdnfsjq`8bnt2�

�S§­bcUXZ�jqY-fsr8Z¦bgdgfsz{ZehbckSdgjqZetVhk8dcdgZetc�Ok8`ou-bgk-f���fS�ljxu dck8�OkSb­bgdgfsz{ZehbckSdg¡��§&fs�q� bcUoZ�hkS`Ot�jxu�Z�dgZeuLjlY-fSrSZ©��Zefsbc�XdgZetVdgZ�Y-fsjq` jl` bcUoZ�hefsY[Z�dnf�¯oZ��xu�ks�6�=jqZ�¤\X§­bcUXZ�dgkS�Aksb6z{k8jl`=b��Ak8tcj£bgjlk8`otVdcZeY-fsjq`��AZb{¤­Z�Ze`LbgUXZ�jqd��qjlY[jlbgt

T�k�u�ZefS�=¤&jlbcU�bcUoZ$¯Odgt�b6h�kS`ot�bcdnfsjq`=bep´bgUXZ$Y[kSbcjqkS`Njqt�¯odnt{bg�l¡��o�qfS`X`XZeu�jq`�bcUoZV\sm���¨­fSd�bgZetcjqfS`2t��OfShZVfs`OubgUXZ�`¥�odckSz{ZehibgZeu¢jq`¢bcUXZ2jlY-fsr8Z2tc�ofSh�ZS§VTVUXZ2f�bcbcdnfShbcjq�SZ��AksbcZe`=bcjxfs� �IH !&�o�X�l�qjq`Xr-bcUoZ�dck8�OkSb�bck�¤VfsdnubgUXZ�r8k8fs� h�kS`�¯OrS�Xdnf�bgjlk8` �I8 X !VjxtVbcUv�ot¦u�Z¯O`XZeu jq`¢bcUXZ2\#��m�¨­fsdcbcZEt�jxfs` t��OfShZ8§$TVUXZ�t�ZEhkS`Ou�fS`ou bcUXZbgUXjqdgu�h�kS`ot�bcdnfsjq`=bgt¦fsdgZ�jl`=bgdck�u��oh�Zeu¢bgUXdgkS�Xr8U�f�dgZ��X�X�xtcjl�8Z��OkSbcZe`8bgjqfS�6H���u�Z¯O`XZeu¸jl`¥bgUXZNjlY-fsr8Z2fs`OufNdgZ��X�o�qtcjl�8Z¦�AksbgZ�`=bcjxfs�#H�� u�Z¯o`oZeu jl` bgUXZVz{kSjq`=b&tc�of8hZS§?TVUXZ�bgksbnfs�4��kSdnhZ�jxtVrSjq�SZe`��v¡ �

M�4 M ��� M � ���#M � �J��!¤&UoZ�dgZ2bgUXZ-tchefs�qjl`Xr�� fShbck8dgt� fs`Ou� fs�q�lk�¤ �ot©bck¥fSu�z{�ot{b�bgUXZ[dcZe�qfsbcjq�SZNjl`YTO�XZ�`oh�Z[ks�¶bgUXZ-u�j �AZedcZe`8b��k8dgh�Zet�fs`Ou hefs`®bcUv�ot��AZ-�otcZeu®bck¢bnfs�8ZNk8��b�ks�V�OkSbcZe`8bgjqfS�D�qkvhefs�?Y[jq`XjlY-fo§-^·�Vf¢�qk�h�fs�?Y[jq`XjqY2�XY jqtdgZef8hnUXZeu p=f2YNkSbcjqkS`�jqt­Z¹�ZEh��bgZeu-bgk2bnfs�8Z}kS�Xb­kS�*j£bV�v¡[� f´�Sk8dcjq`Xr�bgUXZ¦dgZ��o�X�qtcjq�SZ���k8dgh�ZS§�'&h�h�kSdnu�jq`Xr�bck� ; !p�bcUXZ�fSd�bgj£¯Ahjxfs�A��kSdnhZEt&h�fs` �AZ�¤&dcjlb�bgZ�`¥fSt$��kS�q�qk�¤O�

������� �������

M 4 ����� � �� �)9 H 4 � �)9 �H M � 4 � ��� ���� ���� �� � 9� H � 4 � 1 �� �)9� H �M � 4 ��� � ���� ���� �� �)9� H � 4 � 1 � � =1� �)9� H �

� S !

¤&UoZ�dgZE1�jxt­bcUXZ��=fShk8�Xjxfs`LY-fsbcdgj£¹�bgUof�b&dgZ��xf�bgZet­bcUoZ©��fSdcjxf�bgjlk8`�kS�*bcUoZ�h�fsY[Zedgf2�SZ��qk�hjlb{¡"!�#­bgk[bcUXZ��fSdcjxf�bgjlk8`�kS��bgUXZ�hnUXk8tcZ�` �ofSdgfSYNZ�bcZedcjqªefsbcjqkS`�8Q�$!�# 4 1 �8¸§¶TVUXZ���k8dcY�ks�/bcUXZ�Y-fsbcdgj£¹C1 ¤&jq�l���OZr8jl�8Z�`®jl`®bcUXZ-tcZe�=�XZe����k8d�bcUXZ-hnUXk=t�Ze`¸¤­kSdg��t��of8hZ��OfsdnfsY[ZbgZ�dgjlªEf�bcjqkS`Ot�§�TVUXZ[Y-f�bgdcjl¹ � � =1��u�Ze`XksbgZetbgUXZ�dgkS�Aksb¶�=fSh�kS�Xjxfs`[fs`ou � jxt?bgUXZ�jq`=bcZ�dnfShbcjqkS`NY-f�bcdgjl¹NdgZ��xf�bgZeu2bck5% �°fs�xt�k�hefs�q�lZEuNjlY-fsr8Z��8f8hk8�XjqfS`�!i§

465Eg7498

Page 12: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �

^·b$�ljq`X��t?bgUXZ}��fsdgjxf�bcjqkS`[kS�AbgUXZ��vjxt��ofS�X��ZEf�bc�odcZEt?¤&j£bgU�dgZetc�OZEhibDbgk�bcUXZ¦h�fsY[Zedgf��8Z��qkvh�j£b{¡�!�#>���%�4 � !�#E§µokSd&fN�Ok8jl`=b�.$0©¤&jlbcU¥h�k=k8dguXjl`ofsbcZEt 6 BC0�D 0aF 0>=?9 jl` bgUXZ�h�XdgdcZe`=bVhefsY[Z�dnf���dnfsY[Z�fs`OuLhkvk8dgu�jq`ofsbcZetb 0 4 6 c 0 d 0 ;>= 9 � fS`ou56 L 0 ` 0 ;>= 9 4 g�� ( b 0 !ip bcUXZ[jq`8bgZ�dnfShbcjqkS`¸Y-fsbcdgj£¹�dgZ��xf�bgZeu¢bgk�%�4$6 c 0 d 0 = 9 jqtr8jl�8Z�`��=¡ �

� � b 0 TVF 0 �#4 � ��� � ;F 0 � L 0

F 0 L&0 ` 0 � �; � L�0 ] � ` 0� � ;

F 0` 0F 0 �"; � ` 0 ] � � L&0 ` 0 � L�0

� � ¤&UoZ�dgZ4� � 4�� ! ! ��� ! �� UoZ�` %�jxtNh�kSY[�Ak8tcZeu²kS�&bcUXZ jlY-fSrSZ�h�kvkSdnu�jl`Of�bcZEt�kS� ���Ok8jl`=bnt�p?bgUXZ�h�kSdgdcZEt��AkS`Ou�jl`or�jl`=bgZ�dnfShibgjlk8`Y-fsbcdgj£¹Ljxt�� � � % T�� �#4 � 9 � � ( TVF ( � 020�0 � 9 � � 3 TVF 3 ��� 9� ������� ������� �� � ����� ���-� ��� � ��� � ��"­�

^a`LbcUXjxt­�ofsdcbep=bgUXZ©hefs�qjl�odgfsbcjqkS`��ofSdgfSY[ZbcZedgt$fs`ou-bcUXZ�\�m �;YNk�u�Ze�4ks� bgUXZ}bnfsdgrSZ�b­fsdgZ¦tc�X�X�Ak8tcZeu�bckN�OZ�AZ�dc��Zehbc�q¡[�=`ok�¤&` §�/�`Xk�¤&jl`or�bcUXZ©hkvkSdnu�jq`of�bgZet 120 * p=Z¹��XdgZetgt�ZEu-jq`i( * pvks�*fsb$�qZef8t{b$��kS�Xd­�Ok8jl`=bnt .$0Spjlb�jxt&�Ok=tctcjq�X�lZ���dgkSY«bcUXZejld}�XdckSz{ZehbcjqkS`�bck�hk8Y[�X��bcZ�bgUXZ�jq`Xj£bgjqfS��fS`ou¢u�ZetcjldgZeu��Ak8tcZet&¤&jlbcU¢dgZetc�OZEhib&bck( * � �3�·p �l�8���·pSbgUof�b­jqtDbcUXZ¦dckSbgf�bgjlk8`[Y[fsbcdgj£¹ + � * ��dgZetc� § X�� * !¶fs`ouNbgUXZ�bgdgfS`ot��xf�bgjlk8`[�SZehbck8d +�� * ��dgZetc� §X � * !­�OZ�b{¤$ZeZ�` ( + fs`ou ( * ��dgZetc� §?�AZb{¤­Z�Ze` ( X fS`ou ( * !i§���� � �6D �����[ &����� ���Y &!y��� �: Q GF������! �\���& &�����"�� � � ������ ZbWX�� , fs`Ou X � , �AZ¥bcUXZ¥dgksbnf�bgjlk8`ofs�VY-f�bgdcjl¹ fS`ouQbcUXZ¢bgdgfS`ot��xf�bgjlk8`ofs�V�8ZehibgkSd-�AZb{¤­Z�Ze` ( , fs`Ou( X � tcZ�Z�µ/jlr8�XdgZ-� !i§ � Zb ��, fs`ou � , �OZ[bgUXZLfs¹vjxt�fs`Ou¸bgUXZ-dgksbgfsbcjqkS`±fs`Xr8�lZ-k8��bgfSjl`oZeu¸��dck8Y X�� , §� Z®hnUok=k=t�Z 8 , 4 6 X � 9 , � � ���"9, =?9 fStLf �ofsdnfsY[ZbgZ�dgjlªEf�bgjlk8` kS��bcUXZ®¤$k8dc��tc�ofSh�ZS§ � Z�bcUv�ot�Uof´�8Z8 + 4 6 X � 9+ � � ���9+ =?9 fs`ou 8 X 4#" (%$�& §LTVUXZ-jq`Xj£bgjqfS�¶h�fSY[Z�dnf��Ak8tcjlbcjqkS` 8 + jqt�kS��bnfsjq`XZeu¸��dck8Y X � +fS`ou X�� + � � � X�� + 4 X�� * + � 9 *

X � + 4 � X�� + +�� * � X � *'&hehk8dgu�jq`Xr[bgk-bcUXZ�bcdnfs`otcj£bgjlk8`¥Ze�=�ofsbcjqkS`*�{��!ipA¤&UXZ�dgZ�bcUXZNf�bcbcdnfShibgjl�8Z�fs`ou bgUXZ�dgZ��X�o�qtcjl�8Z���kSdnhZEt&¤&jq�l��AZ rSjq�SZ�` jl`QbcUXZ¢tc�X�otcZehibgjlk8` \X§ �±fs`ou \o§ :¸dgZetc�OZEhibcjq�SZe�l¡8p?¤$Z¢hk8`ot�bcdg�ohib[f¸�ofsbcU 7 4 P�8 , � W N; 02020 �-Z¦fSt?bcUXZ©t�ZE�8�oZ�`oh�Z�ks�*tc�oh�h�Zetgt�jq�SZ&�Of�bcU�tcZ�r8Y[Z�`=bgt$t{bnfsdcbcjq`Xr�f�b$bcUXZ�jq`Xjlbcjxfs�4hk8`�¯orS�odgfsbcjqkS` 8 + §µo�XdcbcUXZedcY[kSdgZSpSbcUXZ F/0nmahkvk8dgu�jq`ofsbcZ�jq`i( , ks� ZEfShnU-�AkSjq`=b/.$0spv¤&UXjxhnU�¤&jl�q�A�OZ¦�ot�ZEu-jq`�bcUoZ}dgZ��X�X�xtcjl�8Z��k8dgh�Z�fs`ou�jl` bgUXZ�hk8`8bgdck8�4�qf´¤�pXhefs` �OZ�ZEfStcjl�q¡�kS�Xbgfsjq`XZEu�fsbVZEfShnU j£bgZ�dnf�bgjlk8` tcjl`OhZ4�

81 0 , 4K6 B 0, D 0, F 0, = 9 4' X�� 9 , X�� * X�� ,G�dX � * � X � ,4� � 1 0 *

gCgihkj�l�m#n�o

Page 13: Path Planning for Robust Image-Based Visual Servoing

�E� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

k k

i

i i

oti

o

m

*

*mR

*k

o

m

j

R o

ot

tR

R t

ij

k j

ij

**

* *

F F

F

F

*

M

µ/jqrS�XdgZ�� �¶¨$kSY[�X�Xbcjq`Xr-jl`=bgZ�dgYNZEu�jxf�bcZ��vjqZ�¤�tVks��bcUXZ�bnfsdgrSZ�bV�AkSjq`=bgt'�hk8Y[�X�lZ�bcZ¢h�fSYNZedgf��of�bgU jq` bgUXZ \sm��)¨­fsdcbcZEt�jxfs`Qt��of8hZ�jqtNkS�Xbgfsjq`XZEu fSt2bcUoZ tcZe�=�XZe`ohZ<7 ks� �jq`=bcZedcY[ZeuXjqfsbcZ¶hefsY[Z�dnfV�Ok=t�ZEt�§ ' �Ak8tcj£bgjlk8`�mJ�OfStcZeu�hk8`8bgdck8�Shk8�X�xu©bcUv�ot��OZ$�ot�ZEu©bck&bgdgf8hn�}jlbe§���k�¤­Z��8Z�dEpjlb�jxt&Y[kSdgZ©jq`=bcZedcZEt{bgjl`XrNbgkN�AZ�dc��kSdgY ��ZEf�bg�XdcZEt­bcdnf�z{ZEhibck8dcjqZet­jq` bcUXZ�jqY-fsrSZ8§?^a`ou�ZeZeu pX¤­Z©bcUv�ot&Z�¹��X�lk8j£bf8t­¤$Ze�l� f8t$�Ak8tgtcjl�X�qZ©bcUXZ��qk�h�fs� t�bgfS�Xjq�ljlb{¡�fs`ou�dgkS�X�ot�bc`oZetgt$kS��jqY-fsrSZ�mJ�OfStcZeu�tcZ�dg�Sk8jl`XrO§/µo�XdcbcUXZedcY[kSdgZSpjlb}f´�Sk8jquot¶bgUXZ�hk8YN�o��bgfsbcjqkS`¢ks��bcUXZ�hefsY[Z�dnf2�Ak8tcZ�f�b�ZEfShnU j£bgZ�dnf�bcjqkS` ks��bcUXZ�tcZ�dg�Sk8jl`or2�Xdgk�hZetgte§

���u� ��� � ���Y 5 &����� ���Y &!y��� � $ GF�� �'D ���y� � �������T�k-�AZ�dc��kSdgY«jqY[fSrSZ�mJ�of8t�ZEu�h�kS`=bcdgkS�Jp�¤­Z��X�Xjq�qu bgUXZ�bgdgfsz{ZehibgkSdg¡�ks�/bcUoZ��Xdgksz{ZehbcjqkS`Cb�0�kS�6ZEfShnU¢�AkSjq`8b.$0¸k8`=bck bgUXZ¸jqY-fsr8Z¸�otcjl`or bcUoZ®�v`Xk�¤&`�h�k=k8dguXjl`ofsbcZEt 120 * ks� .$0¸jq` ( * §�TVUXZ¸bcdnf�z{ZEhibck8dc¡ jq`bgUXZ jlY-fsr8ZLjxtNkS��bnfsjq`XZeu²�Ot�jq`Xr®bcUXZ¢h�xfStgt�jxh�fS�$f8tctc�XY[��bcjqkS` bcUofsbNbgUXZ¢h�fsY[Zedgf¥�AZ�dc��kSdgY-t�f®�AZ�dc��Zehib�AZ�dntc�OZEhibcjq�SZ©bgdgfS`ot{��k8dcY-fsbcjqkS`L¤&jlbcU¢dgZetc�OZEhibVbck[bgUXZ�h�fSYNZedgfNk8��bcjxh�fS� hZ�`=bgZ�d ���ojl`XUokS�qZ©Y[k�u�Ze�.!��

� 0 , b 0 , 4K6 � 0 , c 0 , � 0 ,Sd 0 , � 0 , = 9 4�g\6 , � * , � * =I1 0 * � � !

b 0 , jqt&ZEfStcjl�q¡�k8��bgfSjl`XZEu���dgkSY � � !­�v¡LuXjl�vjxu�jl`or � 0 , b 0 , �v¡�j£bntV�qf8t{b}h�kSY[�Ok8`XZ�`=bE§

��� � �)����� ���Y &!y� �"��� �: GF�� � !#�: # � � �����T�k�fs`=bgjqh�jl�ofsbcZ[bgUXZL�Ak8tgt�jq�X�qZ-Z�`oh�kS�X`=bgZ�d�kS�Vf[z{k8jl`=b2�qjqYNjlb�fS`ou bck�f´�8kSjxu®jlbep/¤­Z�Uof´�SZNbgk�Zet�bcjqY-f�bcZbgUXZ¢bcdnf�z{ZEhibgkSdg¡²ks�©bcUXZ¥dgkS�Aksb-jq`QbgUXZ[z{kSjq`=bLt��OfShZ8§ ^a`ou�ZeZeu p$bcUXZ¥Y[ZEfStc�XdcZ¢kS�¦bgUXZ�h�XdgdgZ�`=b�z{kSjq`8bh�kvkSdnu�jl`Of�bcZEtVjqt}�otcZeu jl`¢bgUXZ2h�kSY[�X��bnf�bcjqkS`¥kS��bgUXZ2dgZ��o�X�qtcjq�SZ��OkSbcZe`8bgjqfS�*dgZ��xf�bgZeu�bck-bgUXZ&z{k8jl`=b}�qjlY[jlbgtf´�8kSjxuXfs`OhZS§�^·��bgUXZ�Y-fs`Xjq�X�X�xf�bgkSd&�Ak8tcj£bgjlk8`�jq` bcUXZ­z{k8jl`=b}tc�ofSh�Z©jxtVdgZ��XdgZetcZ�`=bcZEu��v¡ = 4[6 � ( 0�020 ���= 9 p¤­Z�Uof´�SZ�� . =. 8 4 . =.� .�. 8 4 � � =1�"1 ���e� !

465Eg7498

Page 14: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �8�

1 jxt&bgUXZ P R P��=fSh�kS�Xjxfs` Y-f�bcdgjl¹�bgUof�b}dgZ��xf�bgZetVbgUXZ���fsdgjxf�bcjqkS`¢kS�/bgUXZ2hefsY[Z�dnfN�SZ��qk�hjlb{¡ !�#}bgk-bcUXZ��fSdcjxf�bgjlk8`Lks� 8Q�

1 4 � X�� 9 , "�� $ �"�� $ � � � ,

TVUoZ�h�kSY[�X��bnf�bgjlk8` kS� � � , hefs` �OZ���k8�X`ou jq` �q�e\��I� ��

� , 4 # � $ � � � ,�tcjl`Oh ] , � ,

� 0 6 � , =�� � �; � tcjq`oh4� � , �'�>6 � , = ] �TVUoZ?bcdnf�z{ZEhibck8dc¡¦ks�vbcUoZ¶dgkS�Aksb�h�k=k8dguXjl`ofsbcZEt jl`�bgUXZ z{k8jl`=b6t��of8hZ?jxt bgUXZ�`2kS��bnfsjq`XZeu©��dgkSY;bcUXZ¶bcdnf�z{ZEhibck8dc¡kS�)8)�=¡ fN�ljq`XZefSdcjqªefsbcjqkS`�ks�����e� !VfsdgkS�X`ou<=�, �

= ,� ( 4 = , � � � = , ��1 �I8 ,� ( � 8 , �^a`¢bgUXZ2`oZ¹vb¦tc�X�otcZehibgjlk8`otepo¤­Z��XdgZetcZ�`=b�UXk�¤ bcUXZ2�OkSbcZ�`=bgjqfS� ���X`ohbcjqkS`ot©fs`Ou bcUXZ�jq`ou��OhZeu¥��kSdnhZet}fsdgZuXZ¯o`XZEu¢fs`ou h�fS�qh��X�qfsbcZEu §

����� � ������Y ���#�M� !� &�� ��"��� �� � �V!y�����TVUoZ�fsb�bgdgf8hibcjq�SZ©�AksbcZe`=bcjxfs�A¯oZ��xu H jxt&tcjlY[�X�q¡Lu�Z�¯o`XZeu fStVfN�ofSdgfS�Ok8�ljxh����o`ohibgjlk8`�jq`�k8dguXZ�d­bck[Y[jl`oj£mY[jqª�Z©bcUoZ�u�jqt�bgfS`ohZ��AZb{¤­Z�Ze`LbgUXZ�h�odcdgZ�`=b&�Ok=t�jlbcjqkS`¢fs`Ou�bgUXZ�u�ZetcjqdcZEuLk8`XZ4�

H �I8$�#4 ;�� 8 � 8 X � ] 4 ;

�� 8 � ]

TVUoZ����X`ohbcjqkS` H jqt¦�Ok=t�jlbcjq�SZ2kSd}`v�X�q�6fs`ou�fsb�bnfsjq`ot}jlbgt¦Y[jl`ojlY��XY)f�bO8 X ¤&UXZ�dgZ H �I8 X ��4 � §}^·br8Z�`XZedgfsbcZEt­f���k8dgh�Z M bgUof�b}h�kS`v�SZedcr8Zet$�ljq`XZEfsdg�l¡[bgk�¤­fSdguXt$bgUXZ�rSk=fs� hk8`�¯orS�odgfsbcjqkS` �°t�ZeZ � S ! ! �

M �I8$�y4 � � 9 H 4 � 8 ���S� !

� UoZ�`�bcUXZ�dgZ��X�o�qtcjl�8Z¦�AksbgZ�`=bcjxfs�xtVfsdgZ�`Xksb&`XZeZeu�ZEu pvbcUXZ©bgdgfS`ot�jlbcjqkS` Ze�=�ofsbcjqkS` h�fs` �AZ©¤&dgjlb�bcZe`1� dcZ���Z�dbgk'�{� !VfS`ou1�{�S��! !�� 8 ,� ( 4 , ; � � ,� 8 , � 0 8 ,TVUv�otep 8 , jqt �q¡=jq`Xr kS` bcUXZ²t�bcdnfsjqrSU=b��qjl`oZ®�of8tctcjq`Xr �v¡@8 +[fs`ou 8 X § '¦t f hk8`otcZe�=�XZ�`OhZSp}bcUXZbgdgfS`otc�qfsbcjqkS`L�AZ�dc��kSdgY[Zeu �v¡�bgUXZ�h�fSYNZedgf�jxt�f[dcZEfs� t�bcdnfsjqrSU=b&�ljq`XZ�tcjl`OhZ58 ,Njqt�uXZ¯o`XZEu ¤&j£bgU dgZetc�AZehibbgk�fLYNkSbcjqkS`X�qZetgt&��dnfsY[Z � bcUOf�b�jqt ( X !§��}k�¤$Ze�SZedep�bgUXZNkS��z{ZEhib©hefs`�r8Zb©kS��b©ks�?bcUoZNhefsY[Z�dnf[¯oZe�qu¥kS��vjqZ�¤fs`Ou bcUXZ�dgkS�Aksb©h�fs`�fsb�bnfsjq`¥j£bnt?z{kSjq`=b¦�qjlY[jlbgt¦fs�qkS`Xr[bgUXjqt}bcdnf�z{ZEhibgkSdg¡S§VT*kLf´�8kSjxu�bcUXjxt��AksbgZ�`=bcjxfs��odck8�X�lZeY-t�p}b{¤$kQdcZe�X�X�xt�jq�SZ®��kSdnhZEt�fSdcZ®jl`=bcdgk�u��oh�Zeu��v¡�u�Ze�vjqfsbcjq`Xr bgUXZ±h�fsY[Zedgf bcdnf�z{Zehbck8dc¡,¤&UXZe``oZ�ZeuXZeu §

��� ����� mSn hSoVp n"q pur n�q�� n�� q�� {�{�n putu}�� { v|pst~} � v qsq o��"} v|p nm pso pur n = n � pso t �

gCgihkj�l�m#n�o

Page 15: Path Planning for Robust Image-Based Visual Servoing

�´� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

����� �A���LF��� ������)�� �� ����� � ��:� � ��!# �� &����� ������� ¿ 2 1 � & 2 �Z2��e� ��� ¿ 2 À/� 1 � » � TVUXZ¥dgkS�Aksb�hk8`�¯orS�odgfsbcjqkS` =;jxt-h�fs�q�qZeu fSh�h�Z���bnfs�X�qZ¢j£��Zef8hnUQkS�jlbgtNhk8Y[�Ok8`XZ�`=bgt2jqtNtc����hjqZ�`=bc�q¡¸� fSdNf´¤Vf´¡¥��dgkSY jlbgtNhkSdgdgZetc�Ok8`ou�jq`Xr[z{kSjq`=bgt��qjqYNjlbgte§®TVUOf�b�jxtep�=,jqtf8h�h�Z���bnfs�X�qZ�j£�6��kSd�fS�l�op � 0 N 6 � 0

� + 3 ���?0� � 0� J 3 � �I0>=·p � 0

� + 3 fS`ou � 0� J 3[�AZ�jq`Xr[bcUoZ�YNjq`XjqY2�XY fs`OubgUXZ&Y-f�¹�jqY2�XY fs�q�lk�¤Vfs�X�qZ/z{kSjq`=bD�´fS�l�oZet6��k8d?bcUXZ� -�� z{k8jl`=b$fS`ou��I0&�OZejl`or�bcUXZ�u�jxt�bgfs`OhZ&ks� jq`YTo�XZe`ohZ&kS�bgUXZ� -�� z{kSjq`8b¦�ljqY[j£bE§ �®Z�u�Z�`XkSbcZ�� bcUXZ�t��X�Ot�Z�b¦ks�6bgUXZ&z{kSjq`=b©tc�ofSh�Z�ks�DfShehZe��bgfS�X�lZ�hkS`X¯orS�Xdnf�bgjlk8`ote§TVUoZ©dgZ��o�X�qtcjq�SZ��OkSbcZe`8bgjqfS� H���jqt}u�Z¯o`oZeu¢fSt+�°t�ZeZ�µ/jqrS�XdgZ�\ !��

H�� � =1� 4����� �����

� d ]� �lk8r �� ��0��#( �; � � 0

� 0� J 3 �2�"; �� 0� 0� + 3 ���� jl�6= �N��

� jl�6=\N�� ���E� !

^a`¥k8dguXZ�dVbckLkS��bnfsjq`¢f-h�kS`=bcjq`v�XkS�Ot�fs`ou¢u�Zedcjq��fs�X�qZ��AksbcZe`=bcjxfs� ¯OZ��xuNH � p d � jqt¦hnUXk8tcZ�`¢fSt�f��Ok8�X`ou�ZEu���o`ohibgjlk8` ¤&jlbcU¢`v�X�q� ��fs�q�XZ�jl` bgUXZ��Ok8�X`ouXfSdc¡Lks��� �d � � =1�#4 3�

0��"( � � 0 � � � 0� J 3 �2� � 0 � � � 0� + 3 �¤&UoZ�dgZ � � 0

� + 3 4 � 0� + 3 �!�I0NfS`ou � � 0

� J 3 4 � 0� J 3 � �?0S§�TVUXZN�OkSbcZ�`=bgjqfS�)H � jxt©�Ok=t�jlbcjq�SZ�kSd�`v�X�l�Jp4bcZe`ouXt

bgkLjq`�¯o`Xjlb{¡¥fSt =±rSZbnt¦h��lk=t�Zed&bck f�z{k8jl`=b¦�qjqYNjlbep fS`ou¥jlb¦jxt}`v�X�q��¤&UoZ�`¥bgUXZ�u�jxt�bgfs`OhZ2�AZb{¤­Z�Ze`Q=²fs`OubgUXZ�dgkS�Aksb�Y[ZehnUOfs`Xjxh�fS� �ljqY[j£bnt}jxt&Y[kSdgZ�bgUofs`"�?0�§&TVUXZ�fsdcbcjl¯Ohjxfs� dgZ��o�X�qtcjq�SZ���kSdnhZ�uXZ�dgjl�vjq`Xr[��dck8Y H �jxt+��dgZ��ZedVbck,� S ! !�� M �G4 � 1 � � 9� H��

q qq q maxmin min maxjl l j

q j

Vq

l ljj

µ/jqrS�XdgZ�\G�D~&Ze�X�X�xt�jq�SZ��AksbgZ�`=bcjxfs�4��k8d6z{kSjq`8bnt&�qjlY[jlbgt�f´�8kSjxuXfs`oh�Z

465Eg7498

Page 16: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �E\

¤&UXZedcZ��

� 9� H � 4

��������������� ���������������

�������� d (�§§§ d +�§§§ d ��

� ������ d ��� � � d ]������������ (�������� �"; � � ��������� � � ( � (� ���� �; � � �� ��� � � (§§§� (� ����� �"; � � � ����� � � ( � (� ��� �; � � � �� � � (§§§� (� ������ �"; � � �� ������ � � ( � (� ���� �; � � �� ��� � � (

� �������� jl�)= �N��

� jl�)=eN��¤&jlbcU ����������� �����������

� � 4��qkSr�� ��0��#( �; � � 0

� 0� J 3 �2�"; �� 0� 0� + 3 � ��

d +� 4 � � � + � � � +� J 3 � � � +� + 3 � h� 4K;5!4MU � � 0� � � 0� J 3��2� � 0 � � � 0� + 3 �

� ���W2 �32��!2 & 2�� ¼ � ¿ 1 �e�����U2 1 ��� ' �Ok8jl`=b .$0Sp ¤&UXjqhnU®�Xdgksz{Zehbgt}kS`=bck�bcUXZNjqY[fSrSZ��X�qfS`XZ[f�b©f��AkSjq`8b¤&jlbcUNjqY[fSrSZVhkvkSdnu�jq`of�bgZet'b�0 4@6 c�0 d 0y;>=?9Dpsjxt6�v`Xk�¤&`�f8t/�=jxtcjl�X�qZVj£�Yc 0 N 6 c � c��G=ofS`ou d 0 N 6 d � d � =Jp¤&UoZ�dgZ c � p&c � p d � p d � fsdgZ}bgUXZ��qjlY[jlbgtVks�/bcUoZ©jqY-fsr8ZS§?TVUXZ��SZEhibck8dVks��jqY-fsr8Z¦��ZEf�bg�XdcZEt %�jqt&hefs�q�lZEuf8h�h�Z���bnfs�X�qZ&j£�4��k8d¶fs�q� iNeP ; 020�0 ��Z8p c�0EN-6 c � � ! /c � � ! =AfS`ou d 0EN-6 d � � ! d � � ! =Jp8¤&UXZ�dgZ !jxt�f[�Ak8tcj£bgjl�8Z�hk8`ot{bnfs`=b}u�Ze`Xksbgjl`Xr[bgUXZ2u�jxt�bgfs`OhZ�ks�?jq`YTo�XZe`ohZ�kS�6bcUXZ�jqY-fsr8Z��Ok8�X`ouXfSdc¡ �°t�ZeZ2µ/jlr8�XdgZ� � f ! !i§ �®Z u�Z�`oksbcZ��,bcUXZ¢t�Z�b�kS�}fSh�h�Z���bnfs�X�qZ�jqY-fsrSZ���Zef�bg�XdgZete§GK¦`XZ�¤­f´¡¸bck®h�dcZEf�bgZLf��AksbgZ�`=bcjxfs��OfsdgdcjqZ�dVfsdgkS�X`Ou[bcUXZ�hefsY[Z�dnf�¯oZe�qu�ks�/�vjlZe¤�p=Ze`otc�Xdcjq`Xr�bcUofsb�fs�q�O��ZEf�bg�XdcZEtVfsdgZ©fS�l¤Vf´¡�tD�vjqtcjq�X�lZ�fS`ouLu�k`oksb­f �AZEhib$bcUXZ�h�fSYNZedgf�YNkSbcjqkS`�¤&UoZ�`-bgUXZ�¡�fSdcZ}tc����hjqZ�`=bg�l¡N� fsd­Z�`okS�Xr8U-��dck8Y bcUXZ¦jlY-fsr8Z}�qjlY[jlbgtepvjqtbgk-u�Z¯o`oZ�bcUXZ�dgZ��X�o�qtcjl�8Z¦�AksbgZ�`=bcjxfs�#H��9� %��VfSt+�°t�ZeZ�µ/jlr8�XdcZ©��� ��! !O�

H � � %��#4����� �����

� d ]��� ����� 3�0��#( �; � c�0

c�� ���; � c�0c �

�2�"; � d 0d � ���"; � d 0

d �� �� jl�)% �N��

� jl�)% N��

���e\ !

'¦tD��k8d¶bgUXZ}�odcZe�=jqkS�OtD�AksbcZe`=bcjxfs�O���X`ohibgjlk8` p d � jxt$hnUXk=t�Ze`�fSt­f��AkS�X`Ou�Zeu[���o`ohibgjlk8`L¤&jlbcU�`v�X�q�4�´fS�l�oZ}jq`bgUXZ��AkS�X`ouofsdg¡-ks���6�d � � %>�y4 3�

0��"( � c 0 � c��� �2� c 0 � c��� �2� d 0 � d �� ��� d 0 � d �� �gCgihkj�l�m#n�o

Page 17: Path Planning for Robust Image-Based Visual Servoing

�e� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

¤&UoZ�dgZ c ��4Mc � � ! p�c �� 4 c�� � ! p d �� 4 d � � ! fs`ou d �� 4 d � � ! §

α

α

um

uM

mV VM

C

Vm

Um

U M

V M

Vr

Vm

Um

U M

V M

Vr

�°f ! � ��!

µ/jlr8�XdgZ©� � � f !­^aY-fSrSZ��ljqY[j£bnt�p�����!V~&Ze�X�X�xt�jq�SZ��AksbgZ�`=bcjxfs� ��k8d&�vjqtcjl�ojl�qj£b{¡�hkS`Ot{bgdgfSjl`=bTVUoZ ���X`ohbcjqkS`?H��±jxt¸�Ak8tcj£bgjl�8Z k8d¸`v�X�q�°p�bcZe`ouXt¸bck jq`�¯o`oj£b{¡¤&UXZe` f�b �lZEtct®kS`oZQtcZ��qZehibgZeujqY-fsr8Z��ZEf�bg�XdcZEtDr8Zbgt$h�qk8tcZ�d?bgk�bgUXZ}jqY-fsrSZ&�qjqYNjlbgtep�fs`ou[jlb$jxt$`=�o�l�A¤&UXZ�`�fS�l�AjqY[fSrSZ&��ZEf�bg�XdcZEtDfsdgZ}tc����hjqZ�`=bc�q¡� fSd�f´¤­f´¡N��dgkSY«bcUXZ�jqY[fSrSZ��qjlY[jlbgte§¶TVUXZ�fsdcbcjl¯Ohjxfs� dgZ��o�X�qtcjq�SZ¦��kSdnhZ�u�Z�dgjq�=jq`XrN��dgkSY H��}jxt2�

M � �I8$�#4 � 1 � � 9� H � ����� !

¤&UoZ�dgZ �)9� H��¦jqt&ZEfStcjl�q¡-kS�Xbgfsjq`XZEuL��dgkSY �{�e\ ! �

� 9� H � 4

������������������ ������������������

������������

d ��§§§ d ] +� d ] +� (� §§§ d ] 3 � (�

� ���������� d ��� � � d ]�

�������������

� (�� �; � ��� � � ( � ( � �"; � � � � � (§§§� (� � �; � � � � � � ( � (� � �"; � � � � � � (� (�� �; � �� � � ( � ( � �"; � � � � (§§§� (� � �; � � � � � � ( � (� � �"; � � � � � � (

� ����������� jl�6% �N �

� jl�6% N �

465Eg7498

Page 18: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �,:

¤&jlbcU�������������������� ��������������������

� ��4 � � ��� 3�0��"( � �"; � �� � ���; � �� � �2�"; � ������ ���; � ���� � � � ��

d ] +� 4 � � c + � c �� � c �� ��� d + � d �� ��� d + � d �� ��� 4K;5!4MU � c�0

� c �� �2� c�0 � c �� ��� d 0 � d �� ��� d 0 � d �� �

d ] +� (� 4 � � d + � d �� � d �� ���~c + � c �� �2� c + � c �� ��� 4 ;5!4 U �~c�0

� c �� ���~c�0 � c �� ��� d 0 � d �� �2� d 0 � d �� �

����� ����D?D �����¬­¡��ot�jq`Xr�fNbnfsdgrSZb&Y[k�u�Z��/fs`ou¢f-hefs�qjl�Xdnf�bgZeu h�fSYNZedgfopvbcUXZ�bcdnf�z{Zehbck8dc¡�kS�6f-tcZb�kS�#�±�AkSjq`=bgt&jq`¢bcUXZjqY-fsr8Z�UofSt¶�OZeZ�`�k8��bgfSjl`oZeu-fSt$f�tcZe�=�XZ�`OhZ�kS��� �SZehbck8dgt : 4 P>% , � W N ; 020�0 �-Z8§ '��o�lk�hn�-u�jxfsrSdnfsYkS�¶bgUXZ��ofsbcU �X�qfS`X`Xjq`Xr�tghnUXZ�Y[Z[jqt�rSjq�SZ�`®jl`²µ/jqrS�XdgZ :�§�TVUXZ-k8��bgfSjl`XZEu�bgdgfsz{ZehbckSdg¡¥�Xdgk��vjqu�ZEt�t�k8Y[Zr8kvkvu Z�¹v�AZehbcZEu²�XdgkS�AZ�dcbcjqZet���fs�qkS`or¥bcUXjxt�bgdgfsz{ZehbckSdg¡�bgUXZ-bnfsdgrSZb�dcZeY[fSjl`Ot�jl` bcUXZ h�fsY[Zedgf�¯oZe�qu²kS��vjqZ�¤�pAbcUoZNh�kSdgdcZEt��AkS`ouXjl`XrLdck8�OkSb¦Y[ksbgjlk8`�jqt©�XUv¡�tcjqhefs�q�l¡¢dgZefs�qjqªefs�o�lZ2fS`ou¥bgUXZ[h�fsY[Zedgf[bgdgfsz{ZehbckSdg¡ jqtf-t�bcdnfsjqrSU=b&�qjl`XZ�k8��bgtcjxu�Z�bcUXZ�fsdgZefN¤&UXZ�dgZ©bcUXZ2dcZe�X�X�xt�jq�SZEt­��kSdnhZet&fSdcZ�`oZ�ZeuXZeu §­TVUXZ�t�Z�b� 4KP � , 46 F ( 02020 F 3 = � W N\; 02020 �-Z8p�¤&UXjxhnU¢¤&jl�q� �AZ��ot�ZEu jl` bcUoZ�hkS`=bgdck8�A�xf´¤�p�Uof8tV�OZeZ�`¥fs�xt�k[u�Z�bcZ�dgY[jl`oZeu §^a`LbcUXZ¦`XZ¹vbV�ofsdcbepv¤­Z}Z�¹=bgZ�`ou�bcUXjxt$YNZ�bcUXk�u�bck�bcUXZ�h�f8t�Z�¤&UoZ�dgZ�bcUoZ¦k8��z{ZehbVtcUofs�AZ¦fS`ou�u�jqY[Z�`otcjqkS`otfSdcZ©�X`X�v`Xk�¤&`¢fs`Ou�¤&UoZ�dgZ¦bgUXZ�h�fS�ljq�Xdnf�bgjlk8`LjxtV¤$Ze�l� k8dV�ofSu��q¡LZEt{bgjlY-fsbcZeu §

M L

Fk

γk

sZk k γ k k γk+1

If k>1

If k=1

Initial image

Desired image k=1γPose

Transition equation

Initialization

Estimation

kq

Target and camera models

µ/jlr8�XdcZ :G�6¬­�qk�hn�Lu�jxfsrSdnfsY�kS�*bgUXZ��of�bgU¢�X�qfS`X`Xjq`XrN��kSd�fN�v`Xk�¤&` kS��z{ZEhib

gCgihkj�l�m#n�o

Page 19: Path Planning for Robust Image-Based Visual Servoing

�3; ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

� ������� ������� �� � ����� �2� � � �-� ��� � ��� � ��"­�

^a`¢bgUXjxt¦tcZehbcjqkS` pA¤$Z�hkS`Ot�jxu�Z�d�bcUofsb}bcUoZ�bgfSdcr8Zb�Y[k�u�Z��/jxt�`XkSb©f´��fsjq�qfS�X�qZS§V^a`¢bcUXjxt©h�fStcZ�bgUXZ2h�fSY[Z�dnf�Ak8tcZ¥h�fS` `oksb��AZ¥Zet�bcjqY[fsbcZEu § K¦`X�q¡Qf±tchefs�qZeu QD�oh��ljxu�ZEfs`,dcZEhk8`ot{bgdc�OhibcjqkS` h�fS`Q�AZ�kS�Xbgfsjq`XZEu �v¡�AZ�dc��kSdgY[jl`orQfQ�ofsdcbcjxfs���Ak8tcZ®ZEt{bgjlY-f�bgjlk8`;f8t�u�ZEtch�dcjq�AZeu�jq`;bcUXZ `XZ¹vb�tc�X�otcZehibgjlk8` §«TVUXjxt �ofsdcbcjxfs��Ak8tcZ�Zet�bcjqY-f�bgjlk8`¢fS`ou bcUoZ�dcZe�qfsbcjqkS`otV�qjq`X�vjl`Xr�b{¤$k-�vjqZ�¤�t&kS�?f-t�bgfsbcjxh�kS��z{ZEhib¦fSdcZ©bgUXZ�`�Z�¹��X�lk8j£bgZeu�bckuXZetcjlr8`�fN�of�bgU�kS�*bgUXZ�kS��z{ZEhib&jq`LbgUXZ�jlY-fsr8Z�t��of8hZ8§6µ/jqdgt�bep�¤­Z©�XdcZEt�Ze`=b­bgUXZ�Y[ZbcUokvu ¤&jlbcU¢fSheh�Xdnf�bgZhefs�qjl�odgfsbcjqkS`L�OfsdnfsY[ZbgZ�dnt­fS`ouLbcUXZe` p�bcUoZ�dck8�X�ot�bc`XZEtctV¤&jlbcU¢dgZetc�OZEhibVbck[Y[k�u�Z��qjl`or-Z�dgdck8dgt$jqt}t{bg�ou�jqZeu §µokSd2h�kS`v�SZe`Xjxfs`oh�ZSp bgUXZ�dgksbnf�bcjqkS`±Y-f�bgdcjl¹\X � , fS`ou bcUoZ-bcdnfs`otc�xf�bcjqkS`±�8ZehibgkSd X � , fSdcZ-u�Ze`XksbgZeu�� ,fS`ou � , jl` bgUXZ�t�ZE�=�XZ��J§���� �������"����� ��� �:������� ���'��!# �� &� ���Y ��!# ¨$k8`otcjqu�Zed&f�\sm�� dgZ��ZedcZe`ohZ©�X�xfs`XZ�� rSjq�SZe`Ljq`LbgUXZ�u�ZetcjqdcZEuLh�fSY[Z�dnf���dnfsY[Z ( X �=¡�bcUXZ��SZEhibgkSd���9e46 �kX � V�X =·pO¤&UXZedcZ��kX�jxt�jlbgt��X`Xjlbgfsdg¡ `XkSdgY-fs� jq` ( X fs`Ou V�X�jqtVbgUXZ�uXjqt�bgfS`ohZ���dgkSY�� bck-bgUXZ�k8dcjqrSjq`kS� ( X � tcZ�Z�µ/jqrS�XdgZ ; !§L^·b2jxt�¤­Z��q�¶�v`Xk�¤&`®bcUofsb�bcUXZedcZ-jxt2f¢�odckSz{Zehibgjl�8Z[UXkSY[kSr8dgfS�XUv¡¥Y-f�bcdgjl¹C,vpdgZ��xf�bgjl`or2bgUXZ�jlY-fSrSZ��Ok8jl`=bnt&jl`�bcUoZ�h�XdgdcZe`=b&fS`ouLbcUXZ�u�ZEt�jqdgZeu�jlY-fsr8ZetepXt��OhnULbgUof�b � : ���

! 0 , b 0 , 4� , b 0 X � � 0�� , ¤&jlbcU � , 4 � g/� 9 , � , ���3: !

¤&UoZ�dgZ ! 0 , jxt�f¢�Ak8tcj£bgjl�8Z�tchefs�qjl`Xr¢� f8hibgkSd2fS`ou���0�jqt�f¥hkS`Ot{bnfs`=b2tchefs�qjl`or�� f8hibgkSd�`v�X�q�¶jl�VbcUXZ-bnfsdgrSZ�b�AkSjq`=b}�AZ��qkS`Xr=t&bck���§�L¥k8dcZ2�XdcZEhjxt�Ze�l¡8pojl�D¤$Z�u�Z¯o`oZ�bcUoZ�tcjlr8`XZeu¥u�jxt�bgfs`OhZ V�~.$0 T�� ��4�� 1 0 X pA¤$ZUOf´�SZ4�

� 0 4 � V�~. 0 T�� �F 0X V X

���@; !

9©jq�SZe`Lf�bV�qZefSt�b¶��k8�Xd­Y[fsbghnUXZEu��AkSjq`=bgt$�OZe�lk8`XrSjq`Xr�bgk��2p� , h�fS`��AZ¦ZEt{bgjlY-f�bgZeuL�=¡-tckS�q�vjl`Xr�f2�qjl`oZefsdtc¡�t{bgZ�Y¢§*^·�ObcUXZ­�X�xfs`XZ��Qjqt6u�Z�¯o`XZeu��v¡�\}�AkSjq`=bgtepsf�b6�lZEfSt�b�¯o�8Z­tc�X�X�o�lZeYNZe`=bgfsdg¡��Ok8jl`=bnt/fSdcZ$`XZeh�ZetgtcfSdc¡bgk�Zet�bcjqY-f�bgZ�bcUoZ2UXk8YNk8rSdnfs�XUv¡�Y-fsbcdgj£¹¢�v¡��otcjq`Xr-��kSd¦Z¹XfsY[�X�qZ�bgUXZ2�qjl`oZefsd©fs�qrSk8dcjlbcUXY��XdgkS�Ak8tcZeu¢jq`�q�E���J§ '}tgt��XY[jq`Xr bcUofsb[bcUXZ¥h�fSY[Z�dnf¸h�fS�ljq�Xdnf�bgjlk8` jxt[�=`ok�¤&` p?bcUoZWQ¶�oh�qjquXZefs` UXk8YNk8rSdnfs�XUv¡� , kS��o�qfS`XZ�� jqt&ZEt{bgjlY-fsbcZeu fSt­��k8�l�qk�¤O�

� , 4Mg , g ���´��!fS`ou�j£b}hefs` �OZ�u�ZEhk8YN�Ak8tcZeu�jq`8bgk�fNdckSbgf�bgjlk8`LY-f�bgdcjl¹�fS`ou�f[dnfs`o�¥�©Y-fsbcdgj£¹ � :����

� , 4 � 9 , � � 9 , ��� , � X:9 ¤&UXZedcZ ��� , 4 � ,V X ���@S !

µodck8Y�� , p�j£b?jxt�bcUv�ot/�Ok=tctcjq�X�lZ¶bck�u�ZbgZ�dgY[jl`XZ$bcUoZVhefsY[Z�dnf&Y[ksbgjlk8`2�ofSdgfSY[ZbcZedgt � bgUof�b?jxt*bgUXZ­dckSbgf�bgjlk8`� , fs`Ou[bcUXZ�tgh�fs�qZeu-bcdnfs`otc�qfsbcjqkS` ��� , !¶fS`ou-bgUXZ¦t�bcdg�ohibg�XdgZ}ks�*bcUXZ¦kS�otcZ�dg�SZEu-tch�Z�`XZ�� ��kSdVZ�¹�fSY[�X�lZ}bcUXZ�8ZehbckSd��kX ! � :��·§DTVUXZ�dnf�bcjqk�� 0 , 4 F 0, � V�X©hefs`¢Zef8t�jq�l¡-�AZ�Zet�bcjqY-f�bgZeu1�l�E\����

����� ������ 0 , 4 F 0,

V X 4; � � X:9 � ] 9, ��� ,� X9 � ,S] 0 , jl� .$0EN��

� 0 , 4 F 0,V X 4 � 0 ,

� � 9 , ��� , �� � 0 , ] 0 , � � 9 , ] 0 X � jl� .$0 �N�����e� !

465Eg7498

Page 20: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� ���

¤&UoZ�dgZ � 0 , 4 � 6 � 9 , ��� , =�� � 9 , ] 0 X �� 6 � 9 , ��� , =�� ] 0 , � §DTVUoZetcZ��ofsdnfsY[ZbgZ�dnt$fSdcZ�jqY[�Ok8d�bnfs`=b�t�jq`ohZ�bgUXZ�¡�fsdgZ©�Ot�ZEu�jq` bcUXZ�Of�bcU �X�xfs`X`ojl`Xr-r8Z�`XZedgfsbck8d­fS`ouLjq` bcUXZ�h�kS`=bcdgkS� tghnUXZ�Y[Z8§

n*

d*

jm

m j

*

m*j

F

F

i

k

Rk kt

M

jd(M ,Π)

Π

k

i

j

FR i ti

µ/jqrS�XdgZ ;G� N�hefs�qZeu \�m � ¨­fsdcbcZEt�jxfs`Lbcdnf�z{Zehbck8dc¡

���u� �������"���'� � � �a�� ��'� ��� &����� ���Y &!y� �� Z�`Xk�¤ hnUXkvk8tcZO8 , 4@6 � 9 � , � � ���9, =?9 f8tV�ofsdnfsY[ZbgZ�dgjlªEf�bgjlk8`�ks�/bcUXZ�¤­kSdg��t��OfShZ8p�t�jq`ohZ � � ,2jxt&f´��fsjq�£mfS�X�qZ�¤&UXjq�lZ � , jxt2`oksbe§ �®ZLbcUv�ot�Uof´�SZ�8 9+ 4 6 � 9 � + � � ���9+ =&fs`ou 8 X 4" &!$ ( § µXdgkSY jl`Xjlbcjxfs�Vfs`OuuXZetcjldgZeu[jqY-fsrSZEt�p=jlb­jxt¶�Ak8tgtcjl�X�qZ�bgk�h�kSY[�X��bgZ�bcUoZ}UXk8YNk8rSdnfs�XUv¡�� + fS`ou[bcUoZ�`�bgk2k8��bgfSjl` � + p ��� + p �!XfS`ouLbcUv�ot�8 + § '¦tVjl`�bcUXZ��odcZe�=jqkS�Ot$tcZehbcjqkS` p�¤­Z�hkS`Ot{bgdc�ohbVfN�ofsbcU$7�t�bgfSd�bgjl`Xr[f�b 8 + fs`OuLk8dcjqZ�`=bcZEufS�lk8`Xr�bcUXZ�jq`ou��oh�Zeu���kSdnhZEtVrSjq�SZe`Ljq`�bgUXjxt�h�f8t�Z��v¡ �

����� �����M �I8$� 4 � 8M ����8$� 4 � 1 �� � 9� H��M � ��8$� 4 � 1 � �)9� H��

TVUoZ��=fSh�kS�Xjxfs`�Y-f�bcdgjl¹Lks�6�ofsdnfsY[Z�bcZ�dgjqªef�bgjlk8` 1 � V�X>�­jxtV`Xk�¤ r8jl�8Z�` �=¡ �1 � V X �y4 � VYX�� 9 , " � $ �" � $ � �

� , �J�s� !

�}ksbgZ¦bgUof�b�1 `Xk�¤;u�Ze�OZe`ouXtVkS�*bcUoZ©�o`X�v`Xk�¤&`L�ofSdgfSY[ZbcZed V�Xs§ ��k�¤­Z��8Z�dEp=f8t­tcUXk�¤&`�jl`�bcUXZ�tcZe�=�XZe�°pbgUXjxt��ofsdnfsY[Z�bcZ�d�uXk=ZEt�`XkSb�Uof´�SZ-fs`v¡®jl`YTO�XZ�`oh�Z-kS` bcUoZ��of�bgU±�X�qfS`X`Xjq`Xr¥�Xdgk�hZetgt�jl�Vk8`X�q¡�bcUoZ��vjqtcj£m�ojl�qj£b{¡ hk8`ot{bgdgfSjl`=b�jxt2hk8`otcjqu�ZedcZEu §Lµo�Xd�bgUXZ�dgY[kSdgZSp*bcUXZ-jq`=bcZ�dnfShbcjqkS`±Y-f�bgdcjl¹ u�Ze�OZe`ouXt�kS�VbcUXZ�u�Ze��bcU

gCgihkj�l�m#n�o

Page 21: Path Planning for Robust Image-Based Visual Servoing

�3S ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

�8ZehbckSd ��§-¨$k8`8bgdgfSdc¡ bgk�bgUXZ[�XdgZ��vjlk8�ot�h�f8t�Z8p j£b�h�fS`¸`XkSb��OZ-hk8Y[�X��bcZEu®uXjldgZehbc�q¡¥��dck8Y bcUXZ�tgh�fS�lZEu�OfsdnfsY[ZbgZ�dgjlªEf�bcjqkS` 8¸§ ¬$�XbNbgUXZ dgfsbcjqk � 0 , 4 F 0, � V�X hefs`QZefStcjq�l¡±�AZ ZEt{bgjlY-fsbcZeu ��dgkSY bcUoZ¢tgh�fS�lZEu�OfsdnfsY[ZbgZ�dgjlªEf�bcjqkS`-fS`ou-bgUXZ¦jqY-fsr8Z���Zefsbc�XdgZet­fShehk8dgu�jq`Xr�bgk ���e� !i§?TVUv�ot¶¤­Z©dcZe¤&dcjlbcZ}bcUXZ©jl`=bgZ�dnfShibgjlk8`Y-fsbcdgj£¹ � � % T�� �VfSt­��k8�l�qk�¤O� � � % T���T V X �y4�� ;

V X�� ��� �J��� !¤&UoZ�dgZ�� 4 �G(, 02020 ��3 , ��p � 4K6 � ("9 0�020 � 3 9�=I9 fs`ou � 4K6 � (9 0�020 � 3 9 =?9 fSdcZ¦b{¤$k � � R[Y-f�bcdgjxhZetjq`ou�Ze�OZe`ou�Ze`8b�kS�!V X �

������������ ������������

� 0 4 ������ (� � � � 3 � �

� � �� � (� � � ��� � �

� ��

� 0 4 � ��� L 0 , ` 0 , � ; � L 0 , ] ` 0,

; � ` 0 , ] � L 0 ,S` 0, � L 0 ,� �

TVUoZ2�ofSd�bgjqfS���Ak8tcZ2kS�DbcUXZ[h�fSY[Z�dnf<8 , jxt�bgU=�Ot©h�kSY[�X��bgZeu®f�b¦ZEfShnU�jlbcZedgfsbcjqkS` §�TVUXZe` p � , fS`ou ��� ,fSdcZ�uXjldgZehbc�q¡¸kS�Xbgfsjq`XZEu ��dck8Y 8 ,v§ '&h�h�kSdnu�jl`or¢bck ���e� !p*bgUXZ��8ZehibgkSd�� 4 �G(, 02020 ��3 , � jxt�kS�Xbgfsjq`XZEu��dgkSY 8 ,�§ ��ksbgZ�bcUofsb&jl` bcUojqt�hefStcZ©bcUXZ��ofsbcU ks�/bcUXZ�dgkS�Aksb&jq` bcUXZVz{k8jl`=b�t��OfShZ�jxt&rSjq�SZe`L�v¡ �

=�,� ( 4 =�,�� � � =�,4��1@, � V X � ��8 ,� ( � 8 ,��µ/jq`ofs�q�q¡SpvbcUoZ©jqY-fsr8Z}��ZEf�bc�odcZEt­fsb­jlbcZedgfsbcjqkS`CWLfSdcZ©hkSY[�X�XbcZeu fStVu�ZEtch�dcjq�AZeu�jl`LbcUXZ�`XZ�¹vbVtc�X�otcZehbcjqkS` §

��� � ��� � ���Y 5 &����� ���Y &!y��� � $ GF�� �'D ���y� � �������TVUoZ¥UXkSY[kSr8dgfS�XUv¡²Y-f�bcdgjl¹ , ks���X�xfs`XZ � dcZe�qfsbcjq`Xr±bcUXZ�h��XdgdcZe`8bLfS`ou,u�ZetcjqdcZEuQjqY-fsr8Zet-h�fS` �OZh�kSY[�X��bgZeu���dgkSYE8 , �=¡��otcjl`Xr'���´��!­fS`ou%�{�@S ! �

, 4Mg ��� 9 , � � 9 , ��� , � X:9 �g �J�S� !'&hehk8dgu�jq`Xr�bck'�{�,: !$bgUXZ�jlY-fSrSZ�hkvkSdnu�jq`of�bgZet­ks�/bcUXZ��AkSjq`=bgt�. 0 f�bVbcjqY[Z^W fsdgZ¦r8jl�8Z�` �=¡ �

� 0 , b 0 , 4[6 � 0 , c 0 , � 0 ,Sd 0, � 0 , = 4�C,Sb 0 X � � 0 � , �J�s\ !¤&UoZ�dgZ�� dcZ���Z�dVbgk ���3: !Vfs`ou1�{�3; ! !��

� 0�� ,E4 V �u.$0 T�� �F 0X V X

g � 9 , � ,E4 V �u.$0 T�� �F 0X

g/� 9 , � � ,�¦t�jq`XrNbcUoZ��XdcZe�vjlk8�ot­dcZe�qfsbcjqkS` p7�J�s\ !­hefs` �OZ�dgZ�¤&dgj£bcbcZe`1�

� 0 , b 0 , 4 C,Sb 0 X � V�~.$0 T�� �F 0X

g � 9 , � � , �J��� !

465Eg7498

Page 22: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �E�

µo�XdcbcUXZedcY[kSdgZSp}j£��bcUXZ±dgZ��xf�bcjqkS` �°��� !�jxt¢fs�X�o�ljqZeu;�OZ�b{¤$ZeZ�` bcUXZ±u�ZetcjqdcZEu fs`Ou bcUoZ jq`Xjlbcjxfs��h�fSY[Z�dnf�Ak8tcjlbcjqkS`otep�¤$Z�k8��bgfSjl` Zef8t�jq�l¡��4�

V�~. 0 T�� �F 0X

4 t�jqrS`�� � � 0 + b 0 + � +~b 0 X � (�~g/� + ��� + � ( �� � +ub 0 X�� b 0 + �� g/� + � � + � b 0 + � �J��: !

TVUoZ&dcZe�qfsbcjqkS`ot��°�8� !p��°�s� !/fs`Ou �J��: !�fS�l�qk�¤ bgk�hk8YN�o��bcZ � 0 , b 0 , ��dck8Y?8 ,©fs`ou�bcUoZ&jl`Xjlbcjxfs�Ofs`ou[u�ZEt�jqdcZEu�vjxt��Ofs�X��ZEf�bg�XdcZEt�§?TVUXZ¦jlY-fSrSZ¦h�kvkSdnu�jl`Of�bcZEtyb 0 , fsdgZ�¯O`ofs�q�l¡-h�kSY[�X��bgZeu-�v¡-u�jq�=jxu�jq`Xr � 0 , b 0 , �=¡[jlbgt$�qf8t{bh�kSY[�AkS`XZe`8bE§^a`�bcUXZ©`XZ¹vb&t��o�ot�ZEhibgjlk8`ot�p=¤­Z¦hk8`otcjqu�ZedDbgUof�b­bcUXZ©dgZ��X�X�xtcjl�8Z&��kSdnhZ¦dcZe�qfsbcZeuNbgk�bgUXZ¶z{k8jl`=bV�ljqY[j£b&f´�Sk8jqu�mfS`ohZ¦jqt$`XkSbVf8hibcjq��f�bgZeu � j°§ Z �a,E4 � !¶fS`ou[bcUoZ¦Z �AZEhibgt­kS�*Z�dgdck8dDkS`LbcUXZ©jq`8bgdcjq`otcjqh}�ofsdnfsY[Z�bcZ�dnt¶fS`ou-k8`bgUXZ�u�Ze��bcU VYX�fsdgZ�t{bg�ou�jqZeu §

����� � �� ���� ���2! � ������!y��� !# ��^·�$bcUXZ©z{k8jl`=b��ljqY[j£bnt�f´�Sk8jquofs`oh�Z�jqt�`oksb�hk8`ot�jxu�ZedcZEu p4bcUXZ[�ofSdgfSY[ZbcZed V X fS�X�OZEfsdnt}k8`X�q¡�jl` dgZ��X�X�xtcjl�8Z��k8dgh�Z�bcUXdgkS�orSU bcUXZ2Y[fsbcdgj£¹�� u�Z¯o`oZeu¥fSt&bcUXZ��odck�u��ohb�ks�y1 � V�X>�&fS`ou � � % T���T VYX��i§ ��k�¤­Z��8Z�dEpf8h�h�kSdnu�jl`or2bgk �J�s� !$fS`ou%�°�X� !ip�¤­Z�Uof´�SZ�f�b&bgjlY[Z^W �

� �I8 , T�� , T % , �#4 � � � % , T � , T V X �"1 , � V X �+� 4 � � � % , T � , � � 9 ,� � % , � � � ,

� jxt�jq`ou�Z��AZ�`Ou�Z�`=b}k8`�bgUXZ��ofSdgfSY[ZbcZed V�XS§$TVUv�otepXbgUXZ�bcdnf�z{ZEhibgkSdgjlZEtVjl` bcUoZ�dcZ�bcjq`ofs��t��of8hZ�fSdcZ�fs�xtckjq`ou�Ze�OZe`ou�Ze`8b�k8` VYX¢fs`OuQfSdcZ `Xksb�f �AZEhibcZEu �v¡²�Ak8tgt�jq�X�qZ¢Z�dgdck8dgt�kS` V�Xs§ � Zb-�ot[`oksbcZ bgUof�b�jlb-jqt�o`���kSdcbc�X`Of�bcZe�l¡L`Xksb&bgUXZ�h�f8t�Z©j£�Az{kSjq`=bgt&�qjqYNjlbgt}f´�SkSjxuXfS`ohZ¦jqt�fS�qtck[hkS`Ot�jxu�Z�dgZeu §

����� � �� ���� ���2! � ������!y��� !# $ GF��[�: # &���: ����� � �a���6D � &��� �^·�$bcUXZ[h�fSY[Z�dnf�jxt¦`oksb��OZed���ZEhibg�l¡�hefs�qjl�Xdnf�bgZeu�fS`ou� g jxt©�ot�ZEu¸jq`ot�bcZEfSu�kS� g�p bcUXZNZet�bcjqY-f�bcZEu¸jq`Xjlbcjxfs�UokSY[kSr8dgfS�XUv¡NY-fsbcdgj£¹ � +6jxt��

� + 4 g g � + g g 4�� g � + � g �J��; !

¤&UoZ�dgZ�� g 4� g g¸§ � Z�b��otVY-fs�SZ©bgUXZ���kS�q�lk�¤&jq`Xr[fStgt��oYN�XbcjqkS`1� � +a4�� g � +�� g 4� � , 4�� g � ,�� g �J�8��!

^·b­Y[Zefs`OtDbgUof�b$bcUXZ©jl`oj£bgjqfS�OZedcdgkSdDk8`-bcUoZ¦ZEt{bgjlY-f�bgZeu-UokSY[kSr8dgfS�XUv¡�jqt­�XdgkS�Ofsr8fsbcZEuNfS�lk8`Xr�bgUXZ�bgdgfsz{ZehimbgkSdg¡�fS`ou¥jqYN�o�ljqZet�bcUofsb}bcUoZ2Zet�bcjqY-f�bgZeu¥UXk8Y[kSrSdnfs�oU=¡�f�b�bgjlY[Z)W¢hefs`��AZ2u�ZEhk8YN�Ak8tcZeu¢jq`¥bgUXZ�tc�XYkS�/fNY-f�bgdcjl¹�tcjlY[jq�qfSdVbck�fNdckSbgfsbcjqkS`LY-fsbcdgj£¹�fS`ou¢fNdgfS`X�¥��Y-f�bcdgjl¹ �

� , 4 ��� , � ! , � X:9 �J��S !�������! } q pur n#"%$'& �"o {�w o#h n h>p q oVx � v�h m)(+*-,/. �102� mSn hSoVp n"q pur n#3uq } @#h%4�oVx 065

gCgihkj�l�m#n�o

Page 23: Path Planning for Robust Image-Based Visual Servoing

�S� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

¤&UoZ�dgZ ��� ,�4 � g � 9 , � g p �!X:9 4 ��� *����� � � * ��� � fs`ou ! ,)4 � �kX9 � g � � g ! ,[¤&j£bgU !^,G4 � 9 , � � ,

�q�e\3�J§ K¦�v�vjlk8�ot��q¡Sp +�p�b 0 X fS`ouGb 0+ fsdgZ�`XkSbVf �AZEhibcZEu-�v¡[Z�dgdck8dgt6k8`�jl`=bgdcjq`otcjqh}�ofsdnfsY[ZbgZ�dnt�p=t�jq`oh�Z�bcUoZ�¡fSdcZ�Z�¹=bgdgf8hibgZeu kSd}h�kSY[�X��bgZeu¢u�jqdcZEhibg�l¡���dck8Y«jqY-fsrSZ�uXfsbgfo§ '&hehkSdnu�jq`XrNbck,�J��� !p��J��: !VfS`ou%�°��S !ipobcUXZk8��bgfSjl`oZeu UXkSY[k8rSZ�`oZ�kS�Ot$jqY-fsr8Z�hkvkSdnu�jq`of�bgZet b 0 , jq`¢bcUoZ��XdcZEt�Ze`ohZ�kS�6hefs�qjl�odgfsbcjqkS` Z�dgdck8dgtVfsdgZ�rSjq�SZe`�v¡ �

� 0 , b 0 , 4 g � , g b 0 X � � U ��� �� � � 0 + b 0 + � +ub 0 X � (� g ! + � (�� � + b 0 X � b 0 + �� g ! + � b 0 + � g ! , �J�s� !

N�jl`oh�Z g ! , 4 � �kX:9#� g � g ! , pX¤­Z�Uof´�SZ¦bcUXZ���k8�l�qk�¤&jq`Xr[Ze�=�ofs�qjlbcjqZet���������� ��������� U � � �� � � 0 + b 0 + � + b 0 X � (� g ! + � (

�� 4 � U � � �� � � 0 + b 0 + � +~b 0 X � (�sg ! + � ( ��� + b 0 X � b 0 + �� g ! + � b 0 + � g !^, 4 � + b 0 X � b 0 + �� g ! + � b 0 + � g !^,

�°\S� !

µo�XdcbcUXZedcY[kSdgZSpv¤­Z�fs�xt�k[Uof´�8Z4� g� � , g 4 g�� g � , � g g 4Mg � , g �°\X� !

¬­¡�jq`�z{ZEhibcjq`Xr,�°\S� !­fS`ou%� \X��!$jl`*�°�S� !p�¤$Z©¯o`ofS�l�q¡�k8��bgfSjl`1� � 0 , b 0 , 4 � 0 , b 0 ,

TVUoZ�dgZ��kSdgZSpv�X`Ou�Z�d�f8tctc�XY[��bcjqkS`1�°�=��!ipvbgUXZ�bcdnf�z{ZEhibgkSdgjlZEt¶jq`LbgUXZ�jqY[fSrSZ�fsdgZ¦`oksb�u�jxt{bg�Xdc�AZeu��v¡�Z�dgdgkSdntk8`�jq`=bcdgjl`Ot�jxh��ofsdnfsY[Z�bcZ�dnte§ � Z�¤&jq�l��hnUXZehn�-bgUXjxt&`XjxhZ��XdgkS�AZ�dcb{¡�kS` bgUXZ�Z¹��AZ�dgjlY[Z�`=bnfs� dgZetc�X�£bnt&rSjq�SZe`jq` NvZehbcjqkS`��v§

����� ����D?D �����^a` bcUXjxt �ofsdcbep}bcdnf�z{ZEhibck8dcjqZet�ks�Nf tcZb¢ks� � �Ok8jl`=bgt jq` dgZbgjl`ofS�©tc�of8hZ Uof´�SZ®�OZeZ�` k8��bgfSjl`oZeu f8t�ftcZe�=�XZe`ohZ¥kS�E���8ZehibgkSdnt : 47P�% , � W[NQ; 02020 � Z�¤&j£bgUXkS�Xb�fs`v¡ Y[k�u�Ze��ks��bcUoZ�tch�Z�`XZ8§;TVUXZ¸tcZb� 4AP � , � W2N ; 0�020 � Z8p¶¤&UojqhnU,¤&jq�l�&�AZ¥�otcZeuQjl` bcUoZ¢h�kS`=bcdgkS�V�xf´¤�p¶Uof8t[�OZeZ�` fs�xtck h�kSY[�X��bgZeu §L�kSdgZ�k��SZedepO¤$ZNUof´�8ZNtcZ�Ze`�bcUOf�b©bcUXZ[�X�xfs`X`oZeu��Of�bcU jxt¦jq`ou�Ze�OZe`ou�Ze`8b�kS�­h�fSY[Z�dnfLh�fS�ljq�Xdnf�bgjlk8`�j£�$bcUXZz{k8jl`=b��qjlY[jlb©f´�8kSjxuXfs`oh�Z©h�kS`ot�bcdnfsjq`=b&jqt&`oksb¦h�kS`otcjquXZ�dgZeu § ' �X�lk�hn� u�jqfSrSdnfsY kS�6bcUXZ��ofsbcU¥�X�xfs`o`Xjl`or-jqtr8jl�8Z�`�jl`¥µ/jlr8�XdgZ���§

465Eg7498

Page 24: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �X�

Fk

γk

γk+1

If k>1

Initial image

Desired image

Initialization

Transition equation

ρk

If k=1k=1

γProjective

reconstruction

Hk s k

qk

Wk

µ/jlr8�XdgZ�� �D¬$�qk�hn�Lu�jxfsr8dgfSY�ks�*bgUXZ��ofsbcU¢�X�xfs`X`Xjq`XrN��kSd�fS`��o`X�v`Xk�¤&`�k8��z{Zehb

� ��" �6����� �!������� � ���!���2" � � �$� "��D� ��� �" �

^a` bcUXZ±�XdgZ��vjqkS�ot tc�X�otcZehibgjlk8`otep}¤­Z®Uof´�8Z¸k8��bgfSjl`XZEu u�jxtghdgZbcZ¸bcdnf�z{ZEhibck8dcjqZete§³^a`;k8dguXZ�d�bgk uXZetcjlr8`h�kS`=bcjq`v�XkS�Ot}fs`Ou�u�Z�dgjq�´fS�X�qZ�h�odc�8Zet}fS`ou¢bgU=�Ot}bgkLjqY[�Xdck��8Z�bcUoZNu�¡v`ofSY[jqh2�OZeUof´�vjlk8d}ks�?bgUXZ[t�¡�t�bcZ�Y¢p¤­Z2�Ot�ZNh�X�XjxhN¬¶matc�X�ljq`XZ�jl`=bcZedc�AkS�xf�bgjlk8` §¦TVUoZ�tc�X�ljq`XZNjl`=bgZ�dg�Ok8�qfsbcjqkS`¥�XdgkS�X�qZ�Y jxt}�Ot��ofS�l�q¡¥t{bnf�bgZeu¸f8t2�r8jl�8Z�`Luof�bgf2�Ok8jl`=bnt : 4[P�%S, � W N\; 02020 � Z�fS`ou�f�t�Z�b­kS�*�ofsdnfsY[Z�bcZ�d$��fs�q�XZet��@42P>�, � W N\; 02020 � Z8p¤­Z�¤­fS`=b}fNh��X�Xjxh�¬$m·tc�X�qjl`XZ�h��Xdg�SZO%�� �'�Vt��OhnU�bgUof�b�%����,4�#4 %S,=§¶^a`¢�XdnfShbcjxhZ8p��ofsdnfsY[ZbgZ�dV��fs�q�XZet}fsdgZdnfsdgZ��q¡¸r8jl�8Z�` § ^a` k8�XdNhefStcZSp6¤­Z�h�fS` fSu�z{�ot{b�bcUXZeY bck®bcUXZ u�jxt{bgdcjq�X��bgjlk8` ks�&bgUXZL�8ZehbckSd�ks�}jqY-fsr8Z��ZEf�bg�XdcZEt % ,¸kSd[�otcjl`or¸bgUXZ¥u�jqt�bcdgjq�X��bcjqkS` ks�¦bcUXZ�hefsY[Z�dnf��Ak8tcjlbcjqkS`ot 8 ,�§ TVUXZ¥u�jqt�bgfS`ohZ¢�AZb{¤­Z�Ze`b{¤­k±t��ohehZEtctcjl�8Z�hefsY[Z�dnf¸�Ak8tcjlbcjqkS`ot 8 , fs`ou 8 ,� ( h�fS` �AZ¥hnUXk8tcZ�` hkS`Ot{bnfs`=b-�=¡±¯o¹vjq`Xr � ,¸fsb[fh�kS`ot�bgfS`=b¦��fs�q�XZS§ � Z�b©�Ot}`XkSbcZ�bcUofsb}bgUXZNuXjqt�bgfS`ohZNjl`�bcUXZNjlY-fSrSZ2�AZb{¤­Z�Ze`¸f��Ok8jl`=b©fsb¦bgjlY[ZGW¥fs`OuW �2;Njqt�`XkSb�`XZeh�ZetgtcfSdcjq�l¡¥h�kS`ot�bgfS`8b��otcjq`Xr f¢h�kS`ot�bgfS`8b���fs�q�XZN��kSd-� , �°t�ZeZNµ/jqrS�XdgZ$S !§2^a`±kSdnu�Z�d¦bckh�kS`=bcdgkS�4Z��-h�jlZe`=bc�q¡�bcUoZ2h�fSYNZedgf��SZe�lk�h�j£b{¡8p�bcUXZ�bgjlY[Z���fs�q�XZEt�fsdgZ�hnUXk8tcZ�`¢t��OfShjq`Xr[�XdgkS�AkSdcbcjqkS`ofS�l�q¡[bckbgUXZ2u�jxt�bgfs`OhZet��OZ�b{¤$ZeZ�`�h�fSY[Z�dnf��Ak8tcjlbcjqkS`ote§­TVUv�otepXbgUXZ�bgjlY[Z2�AZb{¤­Z�Ze` b{¤$k�h�kS`otcZeh���bcjq�SZ���dnfsY[Zet�jqth�kS`ot�bgfS`=b���� � ,� ( 4 � ,� ( � � , 4���p�¤&UoZ�dgZ� h�fS`��AZ�hnUXk=t�Ze`L��k8d&Z¹XfsY[�X�qZ�fSt­bcUoZ��=jxu�ZekNdnf�bgZS§9©jq�SZe`-bcUoZ©uXfsbgf2�SZehbck8dgt % , fS`ou[bgUXZ¦�OfsdnfsY[ZbgZ�dnt?��fs�q�XZet � , p=bcUXZ©jqY[fSrSZ©uXf�bnf2h�fS`��AZ¦jq`=bcZ�dg�AkS�xf�bcZEu�v¡[�otcjl`or�f�`of�bg�XdgfS�Oh��X�Xjxh¦¬$mat��X�qjq`XZ}jq`=bcZ�dg�AkS�xf�bcjqkS`�fs`ou-¤­Z©kS��bnfsjq`Lf� ] ���X`ohbcjqkS`�%�� �'�¶u�Z�¯o`XZEu[��kSd�sW � ;���� � � � W � �v¡ � %����'�y4 g ,9��� ����,�� ] ���),�� ���C, �°\8� !¤&UoZ�dgZ¢bcUXZN� R � uXjqfSrSkS`Ofs�&Y-f�bgdcjxhZEtig ,vp���,vp��),vp��C,±fsdgZ¥kS��bnfsjq`XZeuQ��dgkSY : fs`ou��-§TVUXZuXZ���bgU F ��dgZetc� §?bcUoZ�dgfsbcjqk�� !Vjq` bcUXZ�hefStcZ©kS�6f[�v`Xk�¤&` kS�Xz{Zehib � dcZEt��*§Djl` bgUXZ2h�f8t�Z�kS�6fS`��X`o�=`ok�¤&`k8��z{Zehb-!�fs�X�AZefSdgtNjq` bgUXZ¥hk8`8bgdck8�$�xf´¤�§ ¬­¡ �otcjq`Xr¸bgUXZ¥tcfSY[Z��Xdgk�hZEtctep �����'� 4 F (����'� 0�020 F�3 ���'� �� dcZEt��*§ ��� �'��4 �G(9���'� 02020 ��3 � �'��� !�jqt©hk8Y[�X��bcZEu¢��dgkSY �Q4 P � , � W N-; 020�0 �-Z�fS`ou�� ��dgZetc� §&��dgkSY� 4[P � , � W N\; 0�020 � Z�fS`ou�� !i§

gCgihkj�l�m#n�o

Page 25: Path Planning for Robust Image-Based Visual Servoing

�8� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

∆ t k+1

s k+1 k+2

γ

k s

s

k k+1

tk

γ γk+2

= T = T

����

����

����

µ/jlr8�XdcZ SL�$¨$kS`=bcdgkS�q�ljq`Xr�bcUXZ�bgjlY[Z�fS�lk8`Xr2bgUXZ�h�fSY[Z�dnf�bgdgfsz{ZehibgkSdg¡� - � ��� �6� � � �6� "��!"

T�k-bgdgf8hn�LbgUXZ�jqY-fsr8Z�bcdnf�z{ZEhibck8dcjqZet��ot�jq`Xr fs`¥jqY[fSrSZ�mJ�of8t�ZEu¢hkS`=bgdck8��tghnUXZ�Y[ZSpA¤­Z2�otcZ�bgUXZ���k8�l�qk�¤&jl`or�vjxt�jqkS`XmJ�of8t�ZEu-bnfStc�-���X`ohbcjqkS` � bck-�AZ�dgZ�rS�o�qfsbcZeu�bck " �

� 4 � � %�� ���'�+� � % X ���'�'�TVUoZ$bgjlY[ZV��fsdg¡vjl`Xr©�SZEhibgkSd)%,X4���'�/jxt/bcUoZVuXZetcjldgZeu2bcdnf�z{ZEhibck8dc¡�kS�%&hk8Y[�X��bcZEu�f8t6�XdgZ��vjlk8�otc�l¡�Z¹��X�xfsjq`XZEufS`ou©bcUoZ¶Y-f�bgdcjl¹ � jxt bcUoZ$�otcZ��Ou�ksm·jl`v�8Z�dnt�Z6ks�of}hnUXk8tcZ�`�Y[k�u�Z��=ks� � §/TVUXZ$��fs�q�XZDks� � f�b�bcUXZ­h�XdgdcZe`=buXZetcjldgZeu��Ok=t�jlbcjqkS` jxt&�otcZeuL��kSd � §!L�kSdgZ©�odcZEhjxt�Ze�l¡ �

� j£�/bcUoZ�bgfsdgrSZ�b&jqtV�v`Xk�¤&` � 4 � � %�X����'� T � X�� �'�'�� Z��xt�Z � 4 � � %,X4���'�>T � X4� �'� T V�X>�ip V�X©�AZ�jq`Xr�fS`�ZEt{bgjlY-f�bgZeu ��fs�q�XZ�ks�kV�X � tcZ�Z �°�X� ! !��� � % X � �'� T�� X ���'�>T V X �y4 � ;

V X � � %�X4���'� T��X����'�+� � � %,X����'�+� �TVUoZ�Z�¹v�AkS`oZ�`=bcjxfs�Vu�Zehef´¡±ks� � bgk�¤­fSdgu " hefs` �AZ¢kS��bnfsjq`XZeu²�v¡²jqY[�Ok=t�jq`Xr �� 4 �� � � � �AZ�jq`Xr f�odck8�Ok8d�bgjlk8`ofs�Ar8fSjl`�!p�bcUoZ�hkSdgdgZetc�Ok8`ou�jq`Xr�h�kS`=bcdgkS�4�xf´¤ jxt2�

! # 4 �� � � . �. � �°\S\ !¤&UoZ�dgZ�! # jxt¦bcUXZ-hefsY[Z�dnf��8Z��qkvh�j£b{¡¥tcZ�`=b©bck�bcUXZ[dgkS�Aksb�hkS`=bgdck8�l�qZ�dE§�^·�¶bcUoZ2bnfsdgrSZb�jxt¦�v`Xk�¤&`�bck¢�OZY[kSbcjqkS`X�qZetgt�p�¤­Z�Uof´�SZ . �. � 4 � � . % X. � fS`ou�bcUXZ�hefsY[Z�dnf��SZ��qk�hjlb{¡Lh�fS`��AZ�dgZ�¤&dgj£bcbcZ�` �

!�# 4 �� � � � . %,X. �¤&UoZ�dgZ-bcUXZLbcZedcY � � � �� - fs�q�lk�¤�t�bck¸h�kSY[�AZ�`otgf�bgZ-bcUXZLbcdnfShn�vjq`Xr¥Z�dgdck8de§ L¥k8dcZ��odcZEhjxt�Ze�l¡8p�¤­Z�Uof´�8Z��dgkSY � \8� ! � . % X. � 4 g , � ] � � � , � � � , ��kSd �uW � ;���� � � � W �'¦t�¤&jq�l�$�OZLtcUXk�¤&`±jq`±bcUXZ�`oZ¹vb�tcZehibgjlk8` p�bcUXjxt2h�kS`=bcdgkS�?�xf´¤ jxt�dgkS�X�Ot{b�¤&jlbcU²dgZetc�OZEhib�bgk¥Y[kvuXZ��qjl`XrZedcdgkSdnt}fS`ou�`okSjxt�Z��OZed�bg�Xdc�Of�bcjqkS`Ot¦tcjl`OhZ�bcUXZ[Z�dgdgkSd����o`ohibgjlk8`®�otcZeu®fSt¦jl`o�X��b�dcZeY-fsjq`ot¦tcY-fs�q�?fs`ou¸jqtuXjldgZehbc�q¡Lhk8YN�o��bcZEuL��dgkSY«�=jxtc�ofs�4��ZEf�bc�odcZEt�§

465Eg7498

Page 26: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �S\

Desired configuration Initial configuration

Initial imageDesired image

µ/jlr8�XdcZ��L�EQD¹v�AZ�dgjqYNZe`=bgfs��t�Z�bc�X�� ��� ®" � ���!"&������� � " �*� �n� �

TVUoZ��XdgkS�Ak8tcZeu¢Y[Z�bcUXk�uXt©Uof´�8Z2�AZ�Z�`¸bcZEt{bgZeu�k8`¸f t�jl¹�uvmJkSm°�DZe¡SZ�mJjq`�m·Uofs`ou¸t�¡�t�bcZ�Y¢§ Nvjq`ohZN¤­Z2¤­Z�dgZ`oksb[jl`=bcZedcZEt{bgZeuQjl` jqY[fSrSZ��Xdck�h�Zetgt�jq`Xr jqtgt��oZetNjl` bcUXjxt[�ofS�OZedep¶bcUXZ bgfSdcr8Zb[jqt[h�kSY[�Ok=t�ZEu �v¡ ¤&UXjlbcZY-fSdc��t�� tcZ�Z�µ/jlr8�XdcZ�� !i§?TVUXZ�Z¹vbcdnfShbcZEu��vjqtc�ofS�O��ZEf�bg�XdcZEt&fsdgZ}bgUXZ�jlY-fSrSZ�hkvkSdnu�jq`of�bgZet$ks�*bgUXZ�hZe`=bcZ�dkS�/r8dgf´�vjlb{¡-ks�6Zef8hnU Y-fSdc�4§¶TVUXjxt�tcZehbcjqkS`¢jqt�kSdgr8fs`ojlªeZeuLf8tV��kS�q�qk�¤�t�§D^a` Nv�o�ot�ZEhibgjlk8`���§l�8pX¤­Z��XdcZEt�Ze`=bZ�¹��OZedcjqY[Z�`=bgt$¤&UXjxhnULhk8`�¯odgY bgUof�bVjq`8bgdck�u��Ohjq`Xr�f2�of�bgU��X�xfs`X`Xjq`Xr[t{bgZ���jqY[�Xdgk��SZet$t�jqrS`oj£¯Ohefs`=bc�q¡NbcUXZdgkS�o�ot{bg`XZetgt�¤&jlbcU²dgZetc�OZEhib�bck¸YNk�u�Ze�ljq`Xr¥ZedcdgkSdnt�§-^a`±bgUXZRNv�o�ot�ZEhibgjlk8`ot��v§ ��p?�v§ \¥fs`ou ��§ �Op�k8`X�l¡¸bcUXZdgZ��o�X�qtcjq�SZ��AksbcZe`=bcjxfs��f8tctck�hjxf�bgZeuLbck-bgUXZ2�vjxt�jq�Xjq�ljlb{¡ h�kS`ot�bcdnfsjq`=b�jqt¦fShbcjq�´fsbcZEu §&TVUXZ�dgZetc�X�£bnt&kS�Xbgfsjq`XZEu¤&jlbcU�f��o�qfS`ofsd6bnfsdgrSZb¶fS`ou[fs`-�X`o�=`ok�¤&`N`okS`�m·�X�xfs`ofSd�bnfsdgrSZ�b¶fsdgZ�u�jqtgh�OtctcZeu §/^a` Nv�X�otcZehbcjqkS`���§ :Xps¤$Zh�kSY[Y[Z�`=b¦Z�¹��OZedcjqY[Z�`=bgt¦kS��bnfsjq`XZeu¸¤&UXZ�`¸�OkSbcU�z{k8jl`=b��ljqY[j£bnt�f´�SkSjxuXfS`ohZ2fS`ou¥�vjxt�jq�Xjq�ljlb{¡¥hkS`Ot{bgdgfSjl`=bgtfSdcZ¦bgfS�SZe`�jq`=bck�fShehk8�X`=be§

� �� ��� ���������� &������� �V!y� ���a GF � ��� ��: �� Z�¯odgt�b©tcUXk�¤ bcUOf�b�j£b�jqtep4jl`¸rSZ�`oZ�dnfs�JpOY[kSdgZ�jq`=bcZ�dgZet�bcjq`XrLbckL�o�qfS`�bcUoZ2jqY[fSrSZ2bcdnf�z{Zehbck8dc¡$%,X4���'�}¤$Z¤Vfs`=bDbgk�f8hnUXjlZe�SZ�dnf�bgUXZ�d¶bcUofS`-bck��otcZ¦f2h�kS`ot�bgfS`=b¶dcZ���Z�dgZ�`oh�Z�%�X�§?T�k�fShnUojlZe�SZ}fS`-Z¹��Ok8`XZ�`=bgjqfS�Au�ZEh�f´¡kS�¶bgUXZ�t�Ze`otckSd�t�jqrS`Ofs�/��dck8Y jlbgt�jq`Xjlbcjxfs�?��fs�q�XZ % + bgk¥j£bnt�u�ZetcjqdcZEu¸k8`XZ %,X�¤­Z-h�fs` �otcZNbcUXZ[��k8�l�qk�¤&jl`orh��qf8tctcjxh�fs� h�kS`=bcdgkS�4�xf´¤O� !�# 4 �� � � %�� �'� � % X �¤&UoZ�dgZ � 4 � � %�� �'� T � � �'�'�i§ � jlbcU¸bcUXjxt¦tghnUXZeYNZ8pO¤­Z2Z�¹��OZEhib}bgUof�b©Zef8hnU��Ok8jl`=b©YNk��8Zet&jq`¥bgUXZ�jqY-fsr8Zf8h�h�kSdnu�jl`or2bgk[f-t�bcdnfsjqrSU=bV�ljq`XZ8pXZ��8Z�`¢j£�6¤­Z©�v`Xk�¤ bcUofsb�j£b�jxt&`XkSb��Ok=tctcjq�X�lZ�jq`¥fs�q�*h�f8t�ZEt��q�#�·§?^a`¢kSdnu�Z�dbgk-hnUXZehn�-bgUXZ�dgkS�X�ot�bc`oZetgt$¤&jlbcU¢dgZetc�OZEhib&bgkNY[k�u�Ze�ljq`Xr[Z�dgdck8dgtepv¤$Z�hefsdgdcjqZeuLkS��b&bgUXdcZeZ�Z¹��OZedcjqY[Z�`=bgt��.�� bcUXZ�h�kSdgdcZEhibVjl`=bcdgjq`ot�jxh©�ofsdnfsY[Z�bcZ�dntVfsdgZ��ot�ZEu §DTVUXZ�dgZ��Z�dgZ�`OhZO% X UOfStV�AZ�Z�` bnfs�SZe` h�kS`ot�bgfS`=be§

gCgihkj�l�m#n�o

Page 27: Path Planning for Robust Image-Based Visual Servoing

�s� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

5 � '�`¥Z�dgdgkSd&ks��� ��� jxt&jq`=bcdgkvuX�ohZEu¢kS` bcUXZ2jl`=bcdgjq`ot�jxh��ofSdgfSY[ZbcZedgte§ '¦t�jq`¢bcUXZ��odcZe�=jqkS�Ot&h�fStcZSpobcUXZdcZ���Z�dgZ�`oh�ZO%,X¦Uof8tV�OZeZ�` bgfS�SZe` h�kS`ot�bgfS`8bE§

< � TVUXZ-tcfSYNZNZ�dgdgkSd©ks��� ��� jxt�jl`=bcdgk�u��oh�Zeu�k8`¸bgUXZ[jl`=bgdcjq`otcjqh��ofsdnfsY[Z�bcZ�dnte§ ' ��fSdcjxfs�X�qZNdcZ���Z�dgZ�`oh�Z% X ���'�¶jxtD�otcZeu�tc�ohnU-bgUof�b$bcUXZ¦Z¹��AZehibgZeu[bgdgfsz{ZehbckSdgjqZetDfSdcZ}t�bcdnfsjqrSU=b¶�qjq`XZet¶jl`LbcUXZ¦jlY-fsr8ZS§ � Z�b$�otdcZEh�fS�l� bgUof�b©t��ohnU bgdgfsz{ZehibgkSdgjlZEt&fsdgZ�`Xksb©fs�q¤­f´¡�t­�XUv¡�t�jxh�fS�l�q¡L��fs�qjxu%�l���·§�Nv�ohnU¥tcjlY[�X�qZ��X�xfs`X`ojl`Xrfs�X�odck=fShnU-h�fS`[bcUv�otD`oksb$�AZ�r8Z�`XZedgfS�ljqª�ZEu §�^a`[bgUXjxt¶Z�¹v�AZ�dgjqYNZe`=bep=¤$Z�UOf´�SZ�hnUXk=t�Ze`[f��ofsdcbcjxh�X�xfsdfs`ou���fs�qjqu�hk8`�¯orS�odgfsbcjqkS`2tc�ohnU�bgUof�b/bcUoZ­dgZefS�ljqª�ZEu�bcdnf�z{ZEhibgkSdgjlZEt�fSdcZ¶bcUXZ&tgfsY[Z¶��k8d6�OkSbcU�h�kS`=bcdgkS�tchnUoZ�Y[ZetV¤&UXZe`¥hkSdgdgZehib&h�fS�ljq�Xdnf�bgjlk8`Ljxt&�otcZeu §

TVUoZ�jqY-fsrSZEt¦h�kSdgdcZEt��AkS`Ou�jl`or�bck�bcUoZ[uXZetcjldgZeu�fS`ou�jq`Xjlbcjxfs�DhefsY[Z�dnf��Ak8tcj£bgjlk8`ot�fsdgZ2r8jl�8Z�`�jq` µ/jqrS�XdgZet�E���°f !¦fs`ouQ�e������!¦dgZetc�OZEhibcjq�SZe�l¡8§NTVUoZNk8��bgfSjl`oZeu¸bcdnf�z{Zehbck8dcjqZet©jq`¸bgUXZ-jlY-fSrSZ���kSd�bcUoZ[bcUXdgZ�Z[Z¹��AZ�dgj£mY[Ze`8bnt&fsdgZ�rSjq�SZ�` jq`LbgUXZ�µ/jlr8�XdcZEt��S� � f !p �S� ����!­fs`Ou®�8� �°h !$dcZEt��AZehbcjq�SZe�l¡8§'¦t[Z¹��AZehibgZeu p$bcUoZ�bgdgfsz{ZehbckSdg¡²ks�©Zef8hnU �AkSjq`8b-jq` bcUXZ¢jqY[fSrSZ jqt-f±t�bcdnfsjqrSU=b[�ljq`XZ¢¤&UXZ�` f hk8dcdgZehbhefs�qjl�odgfsbcjqkS`Ljxt&�otcZeu1� tcZ�Z�µ/jqrS�XdgZ[�S� � f ! !i§?¬­��b�jq` �odcZEt�Ze`ohZ�ks�6Y[k�u�Z��qjl`orNZedcdgkSdntep=¤­Z�`XksbgZ�bcUofsb&bcUXZbgdgfsz{ZehbckSdgjqZetDfSdcZ¦�SZ�dg¡�� fsdVf´¤­f´¡2��dck8Y bgUXZ©Z¹��OZEhibgZeu�k8`XZet�� tcZ�Z�µ/jlr8�XdcZ2�S� � ��! !§DTVUXZ}Y[kSbcjqkS`�kS� bcUXZ�AkSjq`=bgtVjq` bcUXZ�jqY-fsr8Z�fSdcZ¦bcUv�ot��X`X�XdgZeu�jxhibnfs�X�qZ�fs`Ou�f[�ofSd�b&kS�*bcUoZ�jlY-fSrSZ©��Zefsbc�XdgZetVY-f´¡�r8Zb&kS��b�kS�bgUXZ2hefsY[Z�dnf�¯oZ��xu ks�?�vjlZe¤�§ '}t¦h�fS` �AZ2tcZ�Ze` jq`�µ/jlr8�XdgZ-�S� �°h !pX�v¡L�otcjl`or�f[�´fSdcjxfs�o�lZ�dgZ��ZedcZe`ohZ8p�bcUXZZ�¹��OZEhibcZEu bcdnf�z{ZEhibck8dcjqZet&fSdcZ�dgZefS�l�q¡Lk8��bgfSjl`oZeu¢Z��8Z�`¢jl�DjlY[�AkSdcbgfs`=b}Y[k�u�Ze�ljq`Xr-Z�dgdgkSdnt&fsdgZ�jl`=bcdgk�u��oh�Zeu §TVUoZ¦Y[ksbgjlk8`�ks��ZefShnUL�Ok8jl`=b&jl`LbcUXZ©jqY[fSrSZ©fsdgZ�bcUv�ot­�AZ�dc��Zehbc�q¡-�XdcZEu�jxhibgfS�X�qZS§?TVUXjxt$Z�¹��OZedcjqY[Z�`=bVh�kS`�m¯OdcY-t/bcUOf�bDf¦bcdnf�z{Zehbck8dc¡���k8�l�qk�¤&jl`or¦jq`2bgUXZ&jqY[fSrSZVt��OfShZVr8jl�8Zet/�OZ�b�bgZ�dDdgZetc�X�lbgt�bcUofS`-f¦�AkSjq`=b�mJbckSmJ�AkSjq`=bY[kSbcjqkS`¢fs`ou�jlY[�Xdgk��SZEtVt�jqrS`Xjl¯Ohefs`=bc�q¡-bcUoZ�dck8�X�ot�bc`XZEtct­¤&jlbcU¢dcZEt��AZehb&bck[Y[k�u�Z��qjl`orNZedcdgkSdnte§

50 100 150 200 250 300 350

50

100

150

200

250

1

2

3

4

50 100 150 200 250 300

50

100

150

200

250

1 2

4 3

�°f ! � ��!

µ�jqrS�odcZN�e�L���°f !­^a`Xjlbcjxfs�Jp � ��!Vu�ZetcjqdcZEuLjqY-fsrSZEtVks��bcUXZ�bnfsdgrSZb

465Eg7498

Page 28: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �d:

�°f !50 100 150 200 250 300 350

50

100

150

200

250����!

50 100 150 200 250 300 350

50

100

150

200

250� h !

50 100 150 200 250 300 350

50

100

150

200

250

µ/jqrS�XdgZ��8�4��T�dnf�z{Zehbck8dcjqZet?jq`-bgUXZ}jqY-fsrSZ}¤&UXZ�`1� �°f !Df2hkS`Ot{bnfs`=b¶dgZ��Z�dgZ�`OhZ�%�X&fS`ou�f��AZ�dc��Zehb­h�fSY[Z�dnfhefs�qjl�odgfsbcjqkS`²fsdgZ-�otcZeu�� � ��!�f�h�kS`ot�bgfS`=b�dcZ���Z�dgZ�`oh�Z %,X-jqt2�ot�ZEu²fs`ou²f��8���)Z�dgdck8d�kS`±bgUXZ�jq`8bgdcjq`otcjqh�OfsdnfsY[ZbgZ�dnt[jqtLjl`=bcdgk�u��oh�Zeu���� h !�f²��fsdgjxfs�X�qZ¥dcZ���Z�dgZ�`oh�ZQ%�X9���'��jqtL�otcZeu�fS`ou f²�8��� ZedcdgkSd�k8`,bcUXZjq`=bcdgjl`Ot�jxh©�ofsdnfsY[ZbgZ�dnt­jqtVjq`=bcdgkvuX�ohZEu

� �u� ����� �����+D �� ����^����� ���" &� �|!#� ���/ �! � �"�� ��a� ! � � ���Y � Z¢fsr8fSjl` h�kS`otcjxu�Z�d[f��X�xfs`OfsdNkS��z{ZEhib�¤&jlbcUQ��kS�od�¤&UXjlbcZ Y-fsdg��t�§±TVUXZ jlY-fsr8Zet�hkSdgdgZetc�Ok8`ou�jq`Xr¥bckbgUXZ-u�ZetcjqdcZEu®fS`ou¸jq`Xjlbcjxfs�DhefsY[Z�dnf��Ak8tcj£bgjlk8`ot�fsdgZNrSjq�SZ�`®jl`²µ/jqrS�XdgZetN�E��� f !}fS`ou �´�"� ��!¦dcZEt��AZehbcjq�SZ��q¡S§TVUoZ�hkSdgdgZetc�Ok8`ou�jq`XrNhefsY[Z�dnf[u�jxt��X�xfSh�Z�Y[Z�`=b��AZb{¤­Z�Ze`�bgUXZ�u�ZEt�jqdcZEu¢fs`ou�bcUoZ�jl`oj£bgjqfS�*h�fSYNZedgf���dnfsY[Zetjxt��8Z�dg¡�jlY[�AkSdcbgfS`8b�� � 3 4 ��� Y[Y¢p1� 4�� � � Y[Y¢p ��� 4 ; ��� YNY ���G��� 3 4 � � u�rOp ��� ��� 4�� � u�rop� �G�����C4��e�v��uXr !�fs`Ou p6jl`±bgUXjqtNh�f8t�Z8p/h��qf8tctcjxh�fs�¶jlY-fsr8Zm·�ofStcZeu±fS`ou±�Ok=t�jlbcjqkS`XmJ�of8t�ZEu �=jxtc�ofs�­t�Zedc�8kSjq`Xr� fSjl�J§V^a`¥kSdnu�Z�d�bck-Z�Y[�XUOfStcjlªeZ�bcUXZ2jlY[�AkSdcbgfs`OhZ�kS�/bgUXZ�jq`=bcdgkvuX�ohZEu h�kS`ot�bcdnfsjq`=b�jl`¢bgUXZ�bgdgfsz{ZehbckSdgjqZetep¤­Z2�AZ�dc��kSdgY�bcUXZ��of�bgU¸�X�xfs`o`Xjl`orLks�?bgUXZ�bgfsdgrSZ�b¦¤&jlbcUXk8��b¦dgZ��o�X�qtcjq�SZ2�AksbgZ�`=bcjxfs��� tcZ�ZNµ/jlr8�XdcZ��E��� h ! !i§� Z�kS�Ot�Zedc�8Z}bgUof�bVbcUoZ��=jxtc�ofs�4��ZEf�bc�odcZEtV�qfSdcr8Z��q¡-rSZb�k8��b&ks�/bcUXZ�hefsY[Z�dnf2¯oZ��xu ks���vjqZ�¤�§

�°f !50 100 150 200 250 300

50

100

150

200

2501

2

3 4

����!50 100 150 200 250 300

50

100

150

200

250

1 2

4 3

� h !−150 −100 −50 0 50 100 150 200 250 300 350 400

0

100

200

300

400

500

600

700

µ/jqrS�XdgZ��´� �'� f !�^a`Xjlbcjxfs�Jp ����!�u�ZEt�jqdcZEu±jlY-fSrSZet2ks�&bgUXZ�bnfsdgrSZbEp � h !�bcdnf�z{ZEhibgkSdgjlZEt�¤&j£bgUXkS�Xb�dgZ��X�X�xtcjl�8Z�AksbgZ�`=bcjxfs�

^a`[bgUXZ}Z�¹��OZedcjqY[Z�`=b¶¤&UXk=t�Z�dgZetc�X�lbgt$fsdgZ&dcZe�Ok8d�bgZeu[jq`�µ�jqrS�odcZ��E\XpSbcUXZ¦jl`=bcdgjq`ot�jxh&�ofSdgfSY[ZbcZedgtDrSjq�SZ�`[�v¡bgUXZ-h�fSYNZedgf�Y-fs`v��� f8hibc�odcZed©fSdcZN�otcZeu §[TVUXZ-\�m �³�ofsdnfsY[Z�bcZ�dnt¦�ot�ZEu®f8t©jl`Xjlbcjxfs�qjqªef�bgjlk8`®kS�¶bgUXZ[�of�bgU�o�qfS`X`Xjq`Xr fS�lr8kSdgj£bgUXY � bcUOf�b-jqt�bgUXZ¢jl`Xjlbcjxfs�&fS`ou u�ZEt�jqdgZeu hefsY[Z�dnf¥�Ak8tcZet-!�UOf´�SZL�AZ�Ze` k8��bgfSjl`XZEu �v¡h�kS�X�o�ljq`Xr�bcUXZ �¦Z�Y[Z�`=bcUokS`±fs�qrSkSdgjlbcUXY � �3�¶fs`Ou�bgUXZ � k�¤­Z�fs�qrSk8dcjlbcUoY �l�8���J§NyD�xfs`X`oZeu fs`ou�bcdnfShn�SZEubgdgfsz{ZehbckSdgjqZetNfsdgZ��X�qksbcbcZEu jq` µ�jqrS�odcZEt��E\��°f !�fs`ou;�e\�����!�dgZetc�OZEhibcjq�SZe�l¡8§ � Z�h�fS` `XkSbcjxhZ bcUofsb-bcUXZbgdgf8hn�SZEu}bgdgfsz{ZehbckSdgjqZet fS`ou©bcUXZ$�X�xfs`X`XZEu�bcdnf�z{ZEhibgkSdgjlZEt fsdgZDfs�qY[k8t�b/tcjlY[jq�qfSde§*TVUojqt/t�Uok�¤�t4bcUXZ$Z��hjqZ�`Oh¡

gCgihkj�l�m#n�o

Page 29: Path Planning for Robust Image-Based Visual Servoing

��; ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

kS�­k8�Xd�hk8`=bcdgkS�DtghnUXZ�Y[ZS§�TVUoZ[bcdnfShn�vjl`or ZedcdgkSd � %�� �'� � %,X4���'� !�jxt��X�qksb�bgZeu jq`±µ/jqrS�XdgZ¥�e\���Z�!ip�fS`ou j£bh�kS`�¯OdcY-t©bcUoZ-�XdcZe�vjlk8�ot�hkSY[Y[Z�`=b2tcjl`OhZNbcUXZ-Y-fs¹vjqY-fs�?ZedcdgkSd�jxt�fs�q¤­f´¡�t©�lZEtct©bgUofs` ���Xjl¹�Z��xt�§�TVUXZZedcdgkSd­kS`�bcUoZ�h�kvkSdnu�jl`Of�bcZEt­ks��ZEfShnULbgfsdgrSZ�b­�AkSjq`=b&�OZ�b{¤$ZeZ�` j£bnt�h�XdgdgZ�`=bVfS`ou�j£bnt&u�ZetcjldgZeuL�lk�hef�bcjqkS` jq`bgUXZ�jqY-fsrSZ0� %����'� � %�X !�jqt}rSjq�SZe` jq`¸µ/jqrS�XdgZ-�e\�� u�!§�TVUXZ2hk8`v�SZ�dgrSZe`ohZ�kS�/bgUXZ�h�kvkSdnu�jl`Of�bcZEtVbck�bcUXZejlduXZetcjldgZeuL�´fS�l�oZ�uXZ�Y[kS`ot�bcdnf�bgZet¶bcUXZ�h�kSdgdcZEhibVdcZEfs�qjlªEf�bcjqkS`Lks�*bgUXZ©bgf8t��4§?TVUXZ�hk8YN�o��bcZEu�h�kS`=bcdgkS�A�qf´¤ jqtr8jl�8Z�`®jl`²µ/jlr8�XdgZ �E\�����!i§ � Z-h�fs` `XkSbcZ-jlbgt�tgf�bcjxt�� fShibgkSdg¡¥��fsdgjqfsbcjqkS`ot�u��XZ[bgk bcUXZ-dgZ�r8�X�qfSdcjlb{¡¢ks�­bcUXZZedcdgkSd$���X`ohbcjqkS` §$TVUXZ�\�m � h�fSYNZedgf2bcdnf�z{ZEhibgkSdg¡-jqtV�X�qksbcbcZEu�jq`¥µ/jlr8�XdcZ[�E\�� h !i§

� f !50 100 150 200 250 300

50

100

150

200

250 ����!50 100 150 200 250 300

50

100

150

200

250 � h !

�°u�!0 100 200 300 400 500 600 700 800

−50

0

50

100

150

200

250

� Z !0 100 200 300 400 500 600 700 800

−10

−8

−6

−4

−2

0

2

4

6

8

10

iteration number

tra

ckin

g e

rro

r (p

ixe

l)

����!0 100 200 300 400 500 600 700 800

−8

−6

−4

−2

0

2

4

6

translations

rotations

µ/jqrS�XdgZ �e\G� Q?¹��AZ�dgjlY[Ze`8b��Ot�jq`Xr bcUXZNbnfsdgrSZb�Y[k�u�Z��I� � f !©�X�qfS`X`XZEu�bgdgfsz{ZehibgkSdgjlZEt�p � ��!¦��kS�q�qk�¤$ZEu¥bcdnf�mz{ZEhibgkSdgjlZEt�p � h !}h�fSYNZedgfNbgdgfsz{ZehbckSdg¡Sp�� u�!}Z�dgdck8d&jl`¸jlY-fsr8Z��AkSjq`8bnt}h�k=k8dguXjl`ofsbcZEt ���Xjl¹�Z��xt�!p � Z !�bcdnfShn�vjq`XrZedcdgkSd����oj£¹�Z��xt-!Vfs`ou%� �-!V�SZe�lk�hjlbcjqZet+�°hY ��t&fS`ou u�r �st�!� � � ����� �����+D �� ����^����� ���" &� �|!#� �� �� �� �! %� ��� ���� ! � � ���Y � Z¶`ok�¤®�XdgZetcZ�`=b bgUXZ¶Z�¹��OZedcjqY[Z�`=bgfS�SdgZetc�X�£bnt kS�Xbgfsjq`XZEu�¤&j£bgUXkS��b�bgfS�=jq`Xr�jq`=bck}f8h�h�kS�X`=b bgUXZD�v`Xk�¤&�qZeu�r8ZkS�$tcUofs�AZ�fS`ou¸u�jqY[Z�`otcjqkS`¸kS�DbcUoZ�k8��z{Zehbe§�TVUXZNjl`Xjlbcjxfs�?fS`ou¸u�ZEt�jqdgZeu�h�kS`�¯or8�Xdnf�bcjqkS`Ot}fsdgZ�bgUXZ-tcfSY[Zk8`XZet�fSt�bcUXk=t�Z®�ot�ZEu jq` bcUXZ®�XdgZ��vjlk8�ot�Z�¹��OZedcjqY[Z�`=be§ TVUoZ¸�ofSd�bgjqfS�}Y[ksbgjlk8`��OZ�b{¤$ZeZ�`,bgUXZ®jq`Xjlbcjxfs�fS`ou u�ZetcjldgZeu h�fSY[Z�dnf®�Ak8tcjlbcjqkS`otep$�Ot�ZEu bgk²jl`Xjlbcjxfs�qjqª�Z¢bcUoZ¥�of�bgU �o�qfS`X`Xjq`Xr²fs�qrSkSdgjlbcUXY jxt[kS�Xbgfsjq`XZEu��dgkSY)f��Xdgksz{Zehbcjq�SZ�dgZeh�kS`ot�bcdg�ohbcjqkS` § '}t¦� fsd�fSt©h�fS�ljq�Xdnf�bgjlk8`�jqt©hk8`ohZedc`oZeu pAbcUodcZeZNu�j �4Z�dgZ�`=b©tcZbnt¦kS��OfsdnfsY[ZbgZ�dnt$UOf´�SZ©�OZeZ�`¢�otcZeu �.�� � ���#� %-��������� � ��� ��� � ��� �

bgUXZ�hk8dcdgZehb¶jl`=bgdcjq`otcjqh¦�ofsdnfsY[Z�bcZ�dnt­fs`ouL�´fS�l�oZ¦kS�EVYX�� bcUOf�b&jqtV\�:�hY�!$Uof´�8Z�OZeZ�`¢�otcZeu%� tcZ�Z�µ/jqrS�XdgZ[��� ! �

5 � � �3������% � ��� � ��� ��� � ��� �fS`LZ�dgdgkSd¶kS�/�s��� UofSt$�OZeZ�`¢fSuXuXZeu-k8`�bcUoZ¦jq`=bcdgjl`otcjxh}�ofSdgfSY[ZbcZedgtepS¤&Uojl�qZ VYX

UofStV�AZ�Ze`¥t�Z�b&bck��s�[hY � tcZ�Z�µ/jlr8�XdgZN�,: ! �< � � �d( ����� � ��� ��� � ��� �

fS` ZedcdgkSd�kS� :S����Uof8t��OZeZ�` f8uXu�Zeu®kS` bgUXZ-jl`=bgdcjq`otcjqh[�ofSdgfSY[ZbcZedgtep ¤&UXjq�lZ V XUofStV�AZ�Ze`¥t�Z�b&bckL���[hY � tcZ�Z�µ/jlr8�XdgZN�3; !

465Eg7498

Page 30: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �=�

&�� ���#� %���� ����� � ��� ��� � ����µ/jldnt�bep ¤­Z2`XkSbcZ�bcUofsb}bcUoZ2bgdgfsz{ZehibgkSdgjlZEt�k8��bgfSjl`XZEu¥¤&jlbcU kSd}¤&jlbcUokS��b��otcjl`Xr

bgUXZ�k8��z{Zehb¶Y[kvuXZ��OfSdcZ�fS�lY[k=t{b$tcjqYNjq�xfsd � dcZ���Z�d?bgk2µ/jlr8�XdcZEt&�e\2fs`ou¢�e� !i§ K¦`oh�Z�fsr8fSjl`*p8fSt¶h�fs`-�AZ}tcZ�Ze`jq`�µ/jqrS�XdgZet}��� � f !¶fs`ou¸��� ����!?bcUXZ©�X�xfs`X`XZEuLfS`ou[bcUoZ}bgdgf8hn�SZeuNbgdgfsz{ZehbckSdgjqZet¶fSdcZ©fs�xt�k�t�jqY[jl�xfsdVfs`ou�bcUXZbgdgf8hn�vjl`Xr2Z�dgdck8dep=rSjq�SZe`�jl`¢µ/jlr8�XdgZ2��� ��Z�!ipvdcZeY-fsjq`otVt�Y-fs�q�4u��Xdgjl`Xr�bcUoZ�tcZ�dg�Sk8jl`or ���qZetgt$bcUofS`��2�Xj£¹�Ze�qt-!i§TVUv�otep=bcUXZ©�XdgZ��vjqkS�ot$hkSY[Y[Z�`=bnt�pvjq`�bcUoZ©hefStcZ}ks�*f2�v`Xk�¤&`�Y[k�u�Z��AbgfSdcr8ZbEp=fSdcZ©fs�xt�k��´fS�ljxu §?TVUXZ¦bgf8t��jxt�hk8dcdgZehbc�q¡ dgZefS�ljqª�ZEu §�TVUXjxt©jqt�tcUXk�¤&`��v¡¢bcUXZ-h�kS`v�SZedcr8Z�`oh�Z�ks�DbgUXZ[jlY-fSrSZ2�AkSjq`=bgt�hkvk8dgu�jq`ofsbcZet}bckbgUXZ�jqd�u�ZEt�jqdcZEuL��fS�l�XZ��°t�ZeZ�µ�jqrS�odcZ[��� � u�! !i§$��$���f(����0 �@������% ���*(�� �d( ��� � ��� ��� � ��� �

�� Z}`Xk�¤ bgZet�b¶bcUoZ¦dgkS�X�Ot{bg`XZetgt?ks� k8�XdVfs�X�Xdgk8f8hnU�¤&jlbcU

dgZetc�AZehib�bck±h�fS�ljq�Xdnf�bcjqkS` Z�dgdck8dgt�fs`ou Z�dgdck8dgt�k8` bgUXZ¢�ofsdnfsY[ZbgZ�d V�Xs§ '}t[hefs` �AZ¥t�ZeZ�` jq`Qµ/jqrS�XdgZet�e���°f !ipO��� ����!6fS`ou[jl`�µ/jqrS�XdgZet��3:"�°f !po�3:�����!psbgUXZ��X�xfs`X`oZeu-fs`OuNbcdnfShn�SZEu�bgdgfsz{ZehbckSdgjqZet6jq`[bcUXZ}hefStcZet?kS�f[h�kSdgdcZEhib�fs`ou hk=fsdnt�Z�h�fS�ljq�Xdnf�bgjlk8` fSdcZ�tcjlY[jq�qfSde§DTVUXZ�bgdgfsz{ZehbckSdgjqZet­kS��bnfsjq`XZeu�¤&jlbcU¥�of8u�hefs�qjl�Xdnf�bgjlk8`fSdcZ©h�qk8tcZ�bgk2bgUXk8tcZ¦k8��bgfSjl`oZeu-jq`LbgUXZ©ksbcUoZ�d&h�f8t�ZEt � tcZ�Z�µ/jlr8�XdgZ��3; !i§?TVUofsbVh�kS`�¯OdcY-t¶bcUXZ©dck8�X�ot�bc`XZEtctkS� bcUoZ¦�ofsbcU��X�qfS`X`Xjq`Xr[fs`ouLks� bgUXZ©h�kS`=bcdgkS�4tghnUXZ�Y[Z}¤&jlbcU�dcZEt��AZehb$bgkNhefs�qjl�odgfsbcjqkS`�ZedcdgkSdntDfs`Ou�Z�dgdgkSdntk8` VYXs§�TVUXZ�bcdnfShn�vjq`Xr�ZedcdgkSd �°µ/jlr8�XdcZ �3:�� h ! !&dgZ�Y-fSjl`ot©tcY[fS�l�/jq`�bcUoZNh�k8fsdntcZ2h�fS�ljq�Xdnf�bcjqkS`�hefStcZ0� �lZEtctbgUofs`Z: �oj£¹�Z��xt-!i§ ^·b2dcZeY-fsjq`ot�tgf�bcjxt�� fShibgkSdg¡�jq`±bcUXZ��OfSu²h�fS�ljq�Xdnf�bgjlk8` hefStcZSp�tcjq`ohZLfS�l¤Vf´¡�t��lZEtct�bcUofS`�E�¥�Xjl¹�Z��xt � µ/jqrS�XdgZ¸�3;�� Z ! !§¸^a` fS�l�¶bcUXZ¢hk8`�¯or8�XdgfsbcjqkS`otep�bcUXZLbcdnfShn�vjl`or¥Z�dgdck8d�dcZeY-fsjq`ot2tc����hjqZ�`=bc�q¡�qjlb�bc�qZNbck¢Z�`Ot��XdgZ[fLr8k=k�u¸�OZeUof´�vjlk8d¦kS�¶bcUoZ-��m��³h�kS`=bcdgkS�?tghnUXZ�Y[ZS§��®Z[`XkSbcZNbcUXZ-t�bgfS�Xjl�qjlb{¡�fs`ou¸bcUXZdgkS�o�ot{bg`XZetgt­ks�/bcUXZ�h�kS`=bcdgkS�4�xf´¤ �°t�ZeZ�µ�jqrS�odcZEt¦�e������!ip��3:"����!Vfs`Ou¸�3;�� �-! !i§$µ�jq`ofS�l�q¡SpX¤­Z�`XksbgZ�bcUofsb&bcUXZbnfStc�¸jxt2h�kSdgdcZEhibc�q¡�dgZefS�ljqª�ZEu f8t�¤­Z��q�D��kSd2bcUXZ�hk8fSdgtcZ-h�fS�ljq�Xdnf�bcjqkS`²hefStcZ�fSt���k8d�bcUXZ��ofSu±hefs�qjl�Xdnf�bgjlk8`hefStcZ�� dcZ���Z�d&bgk-µ�jqrS�odcZEt��3:"�°u�!­fs`ou®�@;�� u�! !i§

� f !50 100 150 200 250 300

50

100

150

200

250����!

50 100 150 200 250 300

50

100

150

200

250� h !

�°u�!0 100 200 300 400 500 600 700 800

−50

0

50

100

150

200

250

� Z !0 100 200 300 400 500 600 700 800

−10

−8

−6

−4

−2

0

2

4

6

8

10

iteration number

tra

ckin

g e

rro

r (p

ixe

l)

����!0 100 200 300 400 500 600 700 800

−8

−6

−4

−2

0

2

4

6

translations

rotations

µ/jqrS�XdgZ-��� � NXfsY[Z�Z¹��AZ�dgjlY[Ze`8b¦¤&j£bgUXkS��b©�otcjl`Xr�bcUXZ�bnfsdgrSZ�b}Y[kvuXZ���fS`ou¢�otcjl`orLhk8dcdgZehb}h�fs�qjq�XdgfsbcjqkS`1��°f !­�X�xfs`o`XZeu bgdgfsz{ZehibgkSdgjlZEt�p7� ��!­��k8�l�qk�¤$ZEu�bgdgfsz{ZehbckSdgjqZetep7� h !�h�fsY[Zedgf�bcdnf�z{ZEhibgkSdg¡Sp7� u�!&Z�dgdgkSdVjl`¥jqY-fsr8Z�AkSjq`=bgt&h�k=k8dguXjl`ofsbcZEt � �Xjl¹vZe�qt-!ip���Z !$bgdgf8hn�vjl`XrNZ�dgdgkSd+���Xjl¹�Z��xt�!VfS`ou1� ��!V�8Z��qk�hjlbcjqZet+� h�Y ��t&fs`Ou uXr ��t-!

gCgihkj�l�m#n�o

Page 31: Path Planning for Robust Image-Based Visual Servoing

��S ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

�°f !50 100 150 200 250 300

50

100

150

200

250� ��!

50 100 150 200 250 300

50

100

150

200

250�°h !

� u�!0 100 200 300 400 500 600 700 800 900

−50

0

50

100

150

200

250

� Z !0 100 200 300 400 500 600 700 800 900

−10

−8

−6

−4

−2

0

2

4

6

8

10

iteration number

trac

king

err

or (

pixe

l)

� �-!0 100 200 300 400 500 600 700 800 900

−10

−8

−6

−4

−2

0

2

4

translations

rotations

µ/jqrS�XdgZN�3: � N�fSY[Z©Z�¹��OZedcjqY[Z�`=b&�ot�jq`Xr-hk=fsdnt�Z�h�fS�ljq�Xdnf�bgjlk8`1���°f !$�X�xfs`X`oZeu�bcdnf�z{ZEhibck8dcjqZetep ����!$��k8�l�qk�¤$ZEubgdgfsz{ZehbckSdgjqZetep�� h !Dh�fSY[Z�dnf}bgdgfsz{ZehbckSdg¡Sp � u�!{Z�dgdck8dDjq`-jlY-fsr8Z&�Ok8jl`=bnt$h�k=k8dguXjl`ofsbcZEt ���Xjl¹�Z��xt�! ��Z !?bcdnfShn�vjq`XrZedcdgkSd����oj£¹�Z��xt-!Vfs`ou%� �-!V�SZe�lk�hjlbcjqZet+�°hY ��t&fS`ou u�r �st�!

� f !50 100 150 200 250 300

50

100

150

200

250����!

50 100 150 200 250 300

50

100

150

200

250� h !

�°u�!0 200 400 600 800 1000

−100

0

100

200

300

� Z !0 200 400 600 800 1000

−15

−10

−5

0

5

10

15

20

����!0 200 400 600 800 1000

−8

−6

−4

−2

0

2

4 translation rotation

µ/jqrS�XdgZ¥�@;L� N�fSY[Z�Z¹��AZ�dgjlY[Z�`=b2�Ot�jq`Xr¥�of8u²h�fs�qjq�XdgfsbcjqkS`1�0� f !��X�xfs`X`XZEu bcdnf�z{ZEhibck8dcjqZetep � ��!���k8�l�qk�¤$ZEubgdgfsz{ZehbckSdgjqZetep � h !*hefsY[Z�dnfVbgdgfsz{ZehibgkSdg¡Sp � u�!*Z�dgdck8d jq`�jlY-fSrSZ¶�AkSjq`=bgt/hkvkSdnu�jq`of�bgZet � �Xjl¹vZe�qt-!ip"� Z ! bcdnfShn�vjq`XrZedcdgkSd����oj£¹�Z��xt-!Vfs`ou%� �-!V�SZe�lk�hjlbcjqZet+�°hY ��t&fS`ou u�r �st�!

465Eg7498

Page 32: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� �S�

� ��� ����� �����+D �� ����^����� ���" &� �|!#� �� �� �� �! �! � � �"�� ���� ! � � ���Y TVUoZ¦bnfsdgrSZ�b­jxtV`Xk�¤ h�kSY[�Ok=t�ZEu�kS��`Xjq`XZ�¤&UXjlbcZ�Y-fsdg��t$�q¡vjl`orNk8`�bgUXdcZeZ�u�j �4Z�dgZ�`=b&�X�xfs`XZEt+� tcZ�Z�µ/jlr8�XdgZ����!§�^a`¸bgUXjqt�Z�¹��OZedcjqY[Z�`=bep4bgUXZ-h�fsY[Zedgf�u�jqtc�X�xfSh�Z�Y[Z�`=b�jqt©�SZedc¡¢jqY[�Ok8d�bnfs`=b�� � 3 4 � P�� � h h®p� 4� ; � P � h h®p"� � 4 ��P � h h®p ���L��� 3 4 ��� P�V �4p&� �G��� 4 �4P�V �Ap ��� ��� � 4 ; ���V � !§ '}t��XdcZe�vjlk8�ot��q¡SpbgUXZ��ofsbcU¢�X�xfs`X`Xjq`Xr�fS�lr8kSdgj£bgUXY jxtVjl`Xjlbcjxfs�qjqª�Zeu �v¡L�Ot�jq`Xr�fN�Xdgksz{Zehbcjq�SZ�dgZehk8`ot�bcdg�ohibgjlk8`�fs`Ou f-tgh�fS�lZEuQ¶�oh�qjquXZefs` dgZehk8`ot�bcdg�ohibgjlk8` §QTVUXZ¥tgfsY[Z¢��fs�q�XZet[kS�¦jq`8bgdcjq`otcjqh��ofSdgfSYNZ�bcZedgt�bcUOf�b-jl` bgUXZ¥h�fStcZ ks�©f�o�qfS`ofsdNkS��z{ZEhibNUof´�SZ��OZeZ�` �Ot�ZEu § TVUXZ¢u�Z���bgU VYXLUofStN�AZ�Ze`QtcZbNbck ���Sh�Y¢p :s�Sh�Y fs`ou �e�S�8hY jq`bgUXZ-h�f8t�ZEt¦kS�­h�kSdgdcZEhibep h�k8fsdntcZ�fs`Ou¸�of8u®hefs�qjl�odgfsbcjqkS`�dgZetc�OZEhibgjl�8Z��q¡S§2^a`®bcUojqt��ofSd�bEp ¤­Z�k8`X�l¡¸Z¹�UXjq�Xj£bbgUXZ�dcZEt��o�£bnt&t�jq`ohZ©bgUXZ�hkSY[Y[Z�`=bntVjl`�bcUXZ�hefStcZ¦kS�/fS`��X`o�=`ok�¤&`L�X�xfs`ofSdVkS��z{ZEhib�fsdgZ©fS�qtckN�´fS�ljxu�UXZ�dgZS§�}k�¤$Ze�SZedep=¤$Z�`oksbcZ8p�kS`oh�Z�fsr8fSjl`*pvbcUofsb­bgUXZ�dgZetc�X�£bnt&fsdgZ�tcfsbcjxt{� f8hibck8dc¡-jq` bcUXZ©bgUXdcZeZ�h�f8t�ZEt�§

�°f !100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

450

500����!

100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

450

500

µ�jqrS�odcZN�´� ���°f !­^a`Xjlbcjxfs�Jp � ��!Vu�ZetcjqdcZEuLjqY-fsrSZEtVks��bcUXZ�bnfsdgrSZb

� f !100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

450

500����!

100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

450

500� h !

�°u�!0 200 400 600 800 1000

−300

−150

0

150

300

450

600

iteration number

erro

r be

etw

een

curr

ent a

nd d

esire

d co

ordi

nate

s (p

ixel

)

� Z !0 200 400 600 800 1000

−15

−10

−5

0

5

10

15

����!0 200 400 600 800 1000

−1

−0.5

0

0.5

1

iteration number

velo

city

translations

rotations

µ/jqrS�XdgZ[�@SL� Q?¹��OZedcjqY[Z�`=b}¤&jlbcU¥f[`Xk8`�mJ�o�qfS`ofsd&k8��z{Zehb�¤&j£bgUXkS��b¦�ot�jq`Xr[bgUXZ�bgfSdcr8Zb�Y[k�u�Z��*fS`ou �otcjl`Xrh�kSdgdcZEhibDh�fS�ljq�Xdnf�bgjlk8`1� �°f !6�X�xfs`o`XZeu�bcdnf�z{ZEhibgkSdgjlZEt�p�� ��!6��k8�l�qk�¤­Zeu2bcdnf�z{Zehbck8dcjqZetep�� h !?h�fSY[Z�dnf©bcdnf�z{Zehbck8dc¡8p�°u�!¦Z�dgdck8d}jq`¸jqY-fsrSZN�AkSjq`8bnt�hkvkSdnu�jq`of�bgZet����Xjl¹�Z��xt�!p ��Z !}bgdgf8hn�=jq`Xr�ZedcdgkSd����oj£¹�Z��xt-!}fS`ou ����!©�SZe�lk�hjlbcjqZet�°hY ��t&fS`ou u�r �st�!

gCgihkj�l�m#n�o

Page 33: Path Planning for Robust Image-Based Visual Servoing

\8� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

� f !100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

450

500����!

100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

450

500� h !

�°u�!0 200 400 600 800 1000

−300

−150

0

150

300

450

600

� Z !0 200 400 600 800 1000

−15

−10

−5

0

5

10

15

����!0 200 400 600 800 1000

−1

−0.5

0

0.5

1

translations

rotations

µ/jqrS�XdgZN�e�G� N�fSY[Z©Z�¹��OZedcjqY[Z�`=b&�ot�jq`Xr-hk=fsdnt�Z�h�fS�ljq�Xdnf�bgjlk8`1���°f !$�X�xfs`X`oZeu�bcdnf�z{ZEhibck8dcjqZetep ����!$��k8�l�qk�¤$ZEubgdgfsz{ZehbckSdgjqZetep � h !*hefsY[Z�dnfVbgdgfsz{ZehibgkSdg¡Sp � u�!*Z�dgdck8d jq`�jlY-fSrSZ¶�AkSjq`=bgt/hkvkSdnu�jq`of�bgZet � �Xjl¹vZe�qt-!ip"� Z ! bcdnfShn�vjq`XrZedcdgkSd����oj£¹�Z��xt-!Vfs`ou%� �-!V�SZe�lk�hjlbcjqZet+�°hY ��t&fS`ou u�r �st�!

� f !100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

450

500����!

100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

450

500� h !

�°u�!0 200 400 600 800 1000

−300

−150

0

150

300

450

600

� Z !0 200 400 600 800 1000

−20

−15

−10

−5

0

5

10

15

20

����!0 200 400 600 800 1000

−1.5

−1

−0.5

0

0.5

1

1.5

translations

rotations

µ/jqrS�XdgZL�s�L� N�fSY[Z�Z¹��AZ�dgjlY[Z�`=b2�Ot�jq`Xr¥�of8u²h�fs�qjq�XdgfsbcjqkS`1�0� f !��X�xfs`X`XZEu bcdnf�z{ZEhibck8dcjqZetep � ��!���k8�l�qk�¤$ZEubgdgfsz{ZehbckSdgjqZetep � h !*hefsY[Z�dnfVbgdgfsz{ZehibgkSdg¡Sp � u�!*Z�dgdck8d jq`�jlY-fSrSZ¶�AkSjq`=bgt/hkvkSdnu�jq`of�bgZet � �Xjl¹vZe�qt-!ip"� Z ! bcdnfShn�vjq`XrZedcdgkSd����oj£¹�Z��xt-!Vfs`ou%� �-!V�SZe�lk�hjlbcjqZet+�°hY ��t&fS`ou u�r �st�!

465Eg7498

Page 34: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� \o�

� ��� � ���!/� ��� �: � � !#�: ��+D �" ��e� !#����� ���TVUoZ�jqY-fsrSZEt¦h�kSdgdcZEt��AkS`Ou�jl`or�bck�bcUoZ[uXZetcjldgZeu�fS`ou�jq`Xjlbcjxfs�DhefsY[Z�dnf��Ak8tcj£bgjlk8`ot�fsdgZ2r8jl�8Z�`�jq` µ/jqrS�XdgZet�X� �°f !-fS`ou ��� ����!�dcZEt��AZehbcjq�SZe�l¡8§ K¦` fs�q�¦bcUXZ¸��kS�q�lk�¤&jq`Xr �X�qksbnt�pDz{kSjq`8b �Ok=t�jlbcjqkS`Ot�fSdcZ�`okSdgY[fS�ljqª�ZEu�AZb{¤­Z�Ze` � mn� �q�#�·p�¤&UXZ�dgZ©mg��fs`Ou¸��dgZ��XdgZetcZ�`=bVbcUXZ­z{k8jl`=b��qjlY[jlbgte§� Z�¯Odgt�b2�AZ�dc��kSdgY bgUXZL�ofsbcU²�X�xfs`o`Xjl`or�¤&j£bgUXkS�Xb�dgZ��X�X�xtcjl�8Z��OkSbcZe`8bgjqfS�°§�TVUoZLdgZetc�X�£bnt2fSdcZ-r8jl�8Z�`²jq`µ/jqrS�XdgZet¦��� �°h !&fS`ou���� � u�!§ �®ZNh�fS`�t�ZeZ�bcUofsb�bcUoZ2�vjqtc�ofS� ��Zefsbc�XdgZet�rSZb¦kS��b©�qfSdcr8Z��q¡�ks�?bgUXZ2h�fSY[Z�dnf¯OZ��xu ks���vjqZ�¤ fs`Ou,bcUXZ f�¹�jxt �

�f�bcbgfSjl`otLj£bnt�z{kSjq`8b �ljqY[j£bE§ TVUoZ�` p�k8`X�q¡QbgUXZ¸dgZ��o�X�qtcjq�SZ��AksbgZ�`=bcjxfs�

f8tctck�hjxf�bgZeu-bgk[bcUXZ��vjxt�jq�Xjq�ljlb{¡Lh�kS`ot�bcdnfsjq`=bVUOfStV�AZ�Z�`¥f8hibcjq��f�bgZeu §?^a` bcUOf�b}h�f8t�Z8p�Z��8Z�` j£�/bcUXZ��vjxt�jq�Xjq�ljlb{¡h�kS`ot�bcdnfsjq`=b?jqt¶Z�`otc�XdgZeu � µ/jlrN��� � Z ! !/bcUXZ�tcZ�dg�Sk8jl`or�hefs`[`XkSb¶�OZ&dgZefS�ljqª�ZEuN�OZEh�fs�Ot�ZVbgUXZ}f�¹�jxt �

�dcZEfShnUXZEt

jlbgt¦z{k8jl`=b[�ljqY[j£b'� µ/jqr±�X� � �-! !i§²^a` µ/jlr8�XdcZ¢�S�Xp6bcUXZ�b{¤$k®dcZe�X�X�xt�jq�SZ��OkSbcZe`8bgjqfS�qtNfsdgZ�f8hibgjl��f�bgZeu §²TVUXZbnfsdgrSZ�b¦dgZ�Y-fsjq`ot¦jl`®bcUXZ-hefsY[Z�dnf-¯oZe�qu¸ks�¶�vjqZ�¤ � tcZ�Z[µ�jqrS�odcZ[�S�"�°f !¦fs`ou �S��� h ! !}fS`ou¸fS�l�?fs¹vZEt¦f´�8kSjxubgUXZ�jqd/z{kSjq`=b­�qjlY[jlbgt+�°t�ZeZ©µ/jqrS�XdgZ��8�"� ��!¶fS`ou��8�"�°u�! !§ K¦`OhZ�fsr8fSjl`*pSbgUXZ��Ok=t�jlbcjqkS`Xjq`Xr�bgf8t��[jxt&hkSdgdgZehibg�l¡dgZefS�ljqª�ZEu §

�°f !50 100 150 200 250 300 350

50

100

150

200

250

1 2

3 4

����!50 100 150 200 250 300 350

50

100

150

200

250

1 2

3 4

� h !0 100 200 300 400 500 600

0

100

200

300

400

500

� u�!0 100 200 300 400 500 600

−1

−0.5

0

0.5

1

��Z !50 100 150 200 250 300 350

50

100

150

200

250� �-!

0 100 200 300 400 500 600

−1

−0.5

0

0.5

1

µ/jqrS�XdgZ������?^a`Xj£bgjqfS� � f !Vfs`ou¢u�ZEt�jqdgZeu%����!­jqY-fsr8Zet �X�X�xfs`X`XZEu�bcdnf�z{Zehbck8dcjqZet­¤&jlbcUXk8��b�dcZe�X�X�xt�jq�SZ��AksbgZ�`�mbgjqfS� ��� f !/jl`[bgUXZ&jqY[fSrSZ8p���� !6jl`[bgUXZ?z{k8jl`=b¶tc�ofSh�Z �S�o�qfS`X`XZeu�bgdgfsz{ZehibgkSdgjlZEt6¤&j£bgUXkS�Xb¶dcZe�X�X�xt�jq�SZV�AksbgZ�`=bcjxfs�f8tctck�hjxf�bgZeu-bgkNbcUXZVz{k8jl`=b��qjlY[jlbgt�f´�8kSjxuXfs`oh�Z4��� h !­jl` bcUoZ�jlY-fsr8ZSp � u�!Vjl` bgUXZVz{kSjq`=b&tc�of8hZ

gCgihkj�l�m#n�o

Page 35: Path Planning for Robust Image-Based Visual Servoing

\=� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

�°f !50 100 150 200 250 300 350

50

100

150

200

250� ��!

0 100 200 300 400 500 600

−1

−0.5

0

0.5

1

�°h !50 100 150 200 250 300 350

50

100

150

200

250� u�!

0 100 200 300 400 500 600

−1

−0.5

0

0.5

1

��Z�!0 100 200 300 400 500 600

−6

−4

−2

0

2

4

6

Translation Rotation

����!0 100 200 300 400 500 600 700

−150

−100

−50

0

50

100

150

200

250

µ/jqrS�XdgZ��8� �VyD�xfs`X`XZEu¥bgdgfsz{ZehibgkSdgjlZEt&¤&jlbcU��AksbgU¸dgZ��X�X�xtcjl�8Z��OkSbcZe`8bgjqfS� �+�°f !}jl`¥bgUXZ2jqY-fsr8ZSp�����!}jl`¸bcUXZz{k8jl`=b6t��of8hZ ��dcZEfs�qjlªeZeu�bgdgfsz{ZehibgkSdgjlZEt2���°h !*jl`�bgUXZ­jlY-fsr8ZSp � u�! jq`�bgUXZ z{k8jl`=b?t��of8hZ �"��Z !�h�fSY[Z�dnf&�SZe�lk�hjlbcjqZet�°u�r �st&fs`ou hY �st�!$�SZedgtc�ot­jlbcZ�dnf�bgjlk8`�`v�XY��OZedep�� ��!VZedcdgkSd­jq` jqY-fsr8Z©�AkSjq`=b}hkvkSdnu�jq`of�bgZet����oj£¹�Z��xt-!

� - � ���$� � � � � �

^a` bgUXjqt2�ofs�AZ�dEp*¤­Z-Uof´�8ZN�odcZEt�Ze`8bgZeu f¢dck8�X�ot�b�Y[ZbcUokvu Ze`ot��odcjq`Xr¢bcUoZ�hkS`v�8Z�dgrSZ�`OhZ���kSd2fS�l�Djq`Xjlbcjxfs�hefsY[Z�dnf��Ak8tcjlbcjqkS`ote§ ¬­¡ h�kS�X�o�ljq`Xr²fs`QjlY-fsr8Zm·�ofStcZeu bcdnf�z{ZEhibck8dc¡±r8Z�`XZedgfsbckSd[fs`OuQfS`QjqY-fsrSZ�mJ�OfStcZeutcZ�dg�Sk8jl`orop4bcUoZ-�Xdck8�Ok=t�ZEu¸Y[Z�bcUXk�u Z¹vbcZe`ouXt�bgUXZ-¤­Z��q�£m·�v`Xk�¤&`¸dgkS�X�Ot{bg`XZetgt�fs`ou±t�bgfS�Xjl�qjlb{¡�ks�VjlY-fSrSZm�OfStcZeu²tcZ�dg�SkSjq`Xr¢¤&UXZe`²jl`Xjlbcjxfs�Vfs`Ou²u�ZetcjldgZeu²hefsY[Z�dnf �lk�hef�bcjqkS` fSdcZ�h��lk=t�Z�bck¥bgUXZ�hefStcZ-¤&UXZ�dgZ-bcUoZ�¡fSdcZ�uXjqt�bgfS`8bE§}TVUXZ�k8��bgfSjl`oZeu¢bgdgfsz{ZehibgkSdgjlZEtV�Xdgk��=jxu�Z�tckSY[Z�r8k=k�u¢Z�¹v�AZehbcZEu¥�XdgkS�AZ�dcbcjqZet��­fS�lk8`Xr[bcUXZEt�Zbgdgfsz{ZehbckSdgjqZetepsbcUoZ¦bnfsdgrSZ�b$dgZ�Y-fsjq`ot$jl`�bcUXZ�h�fSY[Z�dnf�¯oZe�quLks�*�vjqZ�¤�p=bcUXZ�h�kSdgdcZEt��AkS`ouXjl`Xr2dck8�OkSb$Y[ksbgjlk8`jxt©�XUv¡�t�jxh�fS�l�q¡¢dcZEfs�qjlªEfs�X�qZ�fS`ou¥bgUXZ-h�fsY[Zedgf�bcdnf�z{Zehbck8dc¡ jqt�f t{bgdgfSjlr8U=b¦�qjl`XZ[k8��bgtcjquXZ2bgUXZ-fsdgZef�¤&UoZ�dgZbgUXZ©dcZe�X�X�xt�jq�SZ}��kSdnhZet$fsdgZ}`XZeZeu�ZEu §!Q?¹��OZedcjqY[Z�`=bgfS�4dcZEt��X�lbgt­tcUXk�¤,bcUXZ©��fs�qjquXj£b{¡-kS� kS�XdVfs�X�odck=fShnU�fs`Oujlbgt­dgkS�X�Ot{bg`XZetgt¶¤&jlbcU dgZetc�OZEhib­bck[Y[kvuXZ��qjl`XrNZ�dgdgkSdnt�§/µX��bg�XdcZ©¤­kSdg�N¤&jq�q� �OZ�u�Ze�SksbgZeu[bgk[rSZ�`oZ�dnf�bcZ}bcUXZbgdgfsz{ZehbckSdgjqZet©jl`²jqY-fsr8Z�t��of8hZ[ks�&Y[kSdgZ-hk8YN�o�lZ�¹¸��Zefsbc�XdgZet�bcUofS` �Q�AkSjq`=bgt�jq`²kSdnu�Z�d�bgk¸fs�o�X�l¡®kS�XdY[Z�bcUXk�u�bck[`ofsbc�Xdnfs� k8��z{Zehbgte§

465Eg7498

Page 36: Path Planning for Robust Image-Based Visual Servoing

������������� �������������������������! #"$����%�&'������%�($)*%��#+,������� \8\

� " � " � "&� �­" �

�l� ��µ­§­¨$UOfs�XY[ZbcbcZ8§ y/kSbcZe`8bgjqfS�&�XdgkS�X�qZ�Y-t[ks��t{bnfs�Xjq�qj£b{¡ fS`ou,hk8`=�8Z�dgrSZe`ohZ�jl` jlY-fSrSZm·�of8t�ZEu fs`Ou�Ak8tcjlbcjqkS`�m·�ofStcZeu¸�vjqtc�ofs�?tcZ�dg�Sk8jl`XrO§��*�U% ������ %�� ��% �'��� � �#� ��� ���f( k��� � � ����� )�� �#� % ��"$����� ��� ����%���������� ������%�� %�(�� ������� �) )*%��#� %#���k)��*�#������%�����%��#� ���SpO�s\v� � ;�;��o��SopX�e�8��SX§

� �3��µ­§ ¨$Uofs�XY[Z�b�bcZ�fs`ouWQ�§fL�fsdnhnUofs`Ou § '`XZe¤;dgZeuX�X`ouXfS`oh¡=m·�ofStcZeu�j£bgZ�dnf�bgjl�8Z�tghnUXZeYNZ���k8d}f´�Sk8jqu�mjq`Xr-z{k8jl`=b��qjlY[jlbgt���'}�X�X�qjqhef�bcjqkS`±bgk¥�vjqtc�ofs�$t�Zedc�8kSjq`Xro§-^a`Z �� � � �� �!�� k���@�"�����I��������� � ��� ���f(���Y� ��"$��� � ���OpX�Sk8�l�oYNZ��Xp��ofsr8Zet©�´�S�s���4�´�S��:�p�N�fs`¥µXdnfs`Ohjxtch�kopo¨­fs�qj£��k8dc`ojqfop '}�Xdgjl���S�S�8�X§

� \��+�2§ �O§4¨$k�¤Vfs`¥fS`ou �2§ Q�§ /}k�u�jlbgtghnUXZe�A§$yD�xfs`ofSd�jlY-fSrSZ��of8t�ZEu¢�=jxtc�ofs�*tcZ�dg�Sk8jl`or[f8t�f-`of´�vjqr8fsbcjqkS`�odck8�X�lZeY¢§ ^a` �� � � �� �!� ���3�"� ���Z� �d����� � �#� ���f(#���Y� ��" ��� � ���opD�ofSrSZEt ;o�S�$�Y;X���vp��}Z�bcdgkSjlbepL�jqhnUXjqr8fS` pUL�f´¡��E�S�8�X§

� � � ��§ �¦Z�Y[Z�`=bcUokS`[fs`ou � § N4§ �¦f´�vjxt�§YL¥k�u�Ze�£m·�of8t�ZEu�kS�Xz{Zehib6�Ak8tcZ$jq`[�d:��qjq`XZet/ks�4hk�u�Z8§Y �� �!�&%����Y�#�f����'� ��"'�f�Y� %��(�C� �#� ���)��&+*,� &�(�$��

�&�$ ��& &.-�&spA�8�X`XZ-�E�S�d:X§

� :3� K�§XµOfs�Xr8Z�dnfSt­fs`ou�µ­§ � �ot�bcY-fs`*§EL¥kSbcjqkS` fs`ou t{bgdc�Ohibc�odcZ¦��dck8Y�Y[ksbcjqkS` jq` f��XjqZehZe¤&jqtcZ©�X�qfS`ofsdZe`v�=jqdgkS`XY[Z�`=bE§ #� �!��%Y���Y�#�*��� � � � ��� � %��#�W��%-� � ��� ��� � ���I���f(/���#� � 0���� ���� #��� %���� � ��%�� � %�pA��� \ ! � �dSd:��:S��Sop �e��SdSX§

� ;��+/[§��}fStcUXjqY[ksbckO§1�C� �#� ���k)_%��#+���������� %����2� ��"$% k��� � � ���� � � �d����� � ��� � �f�Y� ��� ����� � ����% (I���

�C� �#� ���_)_%�� �����$3 %�%�(���� ��4s§ � kSdg�qu N�hjqZ�`=bgj£¯Oh N�Z�dgjlZEtVjl`¥~&k8�OkSbcjxh�t&fs`Ou '}��bck8Y[fsbcZEu N�¡vt�bcZeY-t�p|?k8�/�vp � kSdg�qu N�hjqZ�`=bgj£¯Oh�y¶dcZEtctep Nvjq`Xr8fS�Ok8dep4�e�8�S\o§

� ���+/[§���k8tck�uXfXp /[§ N�fS��fsY[ksbgkopXfs`OuRL²§ '}tgfSuXfo§/T�dgfsz{ZehbckSdg¡-rSZe`XZ�dnf�bgjlk8`���k8d&kS�ot�bgf8h�qZ©f´�8kSjxuXfs`oh�ZkS� �X`ohefs�qjl�Xdnf�bgZeu�t�bcZedcZek��vjqtc�ofS�OtcZ�dg�SkSjq`Xr�¤&jlbcUXk8��b&\Su-dgZeh�kS`ot�bcdg�ohibgjlk8` § 5� �6��(�� )7% #���8� ���@�"����W �� � %���� � ��%�� �k� ������� � ���*($)&3,��� %�" �ip � �°\ !�� �S���v\S�op '��Xr8�ot�b��E�S��:�§

� S�� N § �}��bghnUojl`otckS`*p 9N§ ��§ �}fsr8Z�dEp fs`ou¥y6§ ^i§�¨$kSdg�SZ8§$' bc��bgkSdgjqfS�/k8`��vjqtc�ofS�/tcZ�dg�SkLh�kS`=bcdgkS�J§ 5� � ��U� ���U�.�k��� � ������� � ��� ���f(/���Y� ��" ��� � ���op �´�"� : ! � ;d:X�$�Y;=���op�K©hibgkS�AZ�d��E�S��;o§

� ��� K�§ /�Uof�bgjl� § ~&ZEfs�VbgjlY[Z¥kS�Ot{bnfSh�qZ¥f´�Sk8jquXfS`ohZL��kSd-Y-fs`ojl�X�o�qfsbckSdnt-fs`Ou Y[kS�ojl�qZ¥dck8�OkSbgte§ �� �!�%Y���Y�#�f���E� � ��������� � �#��� %���%���� ���vp�:��{��! � �S������SopO�E��S�;o§

�q�e�����O§A¨¦§ � f�bck8Y2�AZS§ ��������� � ��� � ��� � ������ �����S§ /��q�=¤­Z�d�'&hefSu�ZeYNjxh©yD�X�X�qjxt�UXZedgtep �E�S�X�8§�q�S� � ��§C9N§ � k�¤­ZS§�µ�jlb�bgjl`or �OfsdnfsY[ZbgZ�dgjlªeZeu¢bgUXdcZeZmau�jqYNZe`otcjlk8`ofs�/Y[kvuXZ��xt©bck jlY-fsr8ZS§ �� � �9�U� ���U�.�

��� � ��� � %��#�:���f����3,�#� � ���f( � � �������f% �� � %���� � ��%�� ��%p��e\�� : !�� �8�o�"�=��:S�Xp�L¸f´¡��e�8�X�S§�q�E�3� Q}§*L¸fs�qjqt¦fs`ou�µ­§ ¨$Uofs�oYNZ�b�bgZS§¦�����s�[u��=jxtc�ofs�/t�Zedc�8kSjq`Xr-¤&j£bgU�dgZetc�OZEhib}bgk��X`X�v`Xk�¤&`¥k8��z{Zehbgt

bgUXdgkS�Xr8U f `XZe¤ Zet�bcjqY-f�bgjlk8`±tghnUXZeYNZNks�&h�fSY[Z�dnf�u�jxtc�X�qf8hZeYNZe`=be§ #���8�;%Y���Y�#�f��� � � ��"'�f�Y� %���C� �#� ���opA\=�"�{� !�� �s�����=�vp=�S�o`XZ2�s�8�S�X§

�q�e\�� Q}§ L�fs�qjxt�p­µ$§V¨$Uofs�XY[Z�b�bcZ8p­fS`ou N4§D¬­kS�ou�Z�be§ �Q���s�®uQ�=jxtc�ofs�&tcZ�dg�SkSjq`XrO§ 5�6� �<�Y� ���U�.� ������������ � ��� ���f(=���Y� ��" ��� � ���op �,:"�°� ! � �s\dS����d:s�op�'��Xdgjq�D�e�8�S�X§

gCgihkj�l�m#n�o

Page 37: Path Planning for Robust Image-Based Visual Servoing

\S� ������� % ��� %��3��� �����_� �����,��� �� �U���Y" %�� � %

�q��� �©y6§kL�fSd�bgjl`XZ�bep���§��¦fS�ohnUXZedep?�O§E9�fs�q�qjqh�ZSp?fs`ou\L §��¦UXkSY[Z8§ � �������+����� � � ��� ��������� "$���*� ���Y����� �,��% %��#� ��"$��� � ���O§ � kSdntcUXkS�±kS` �}Z�¤�T�dcZe`ouXt�jq`²^aY-fsrSZ�mJ�OfStcZeu±~&kS�Aksb N�Z�dg�SkSjq`XrOp*^{~�K N � �=�vp9©dgZ�`okS�X�qZSp �E�S�=��§

�q�3:3���N§�L¥Z�ªekS�ofSd/fS`ou2µ­§S¨$UofS�XY[Zb�bgZS§�y?f�bgU2�X�xfs`o`Xjl`or¦jq`2jqY-fsrSZVtc�ofSh�ZD��kSd6dgkS�X�Ot{b/�=jxtc�ofs��tcZ�dg�SkSjq`XrO§ �� � � #� �!� ���@�"� ���W��������� � ��� ���f(/���Y� ��" ��� � ���opo\L� �=��:s� ���=��;s�Op '��Xdgjq���s�8�S�X§

�q�@;���¬©§ �o§ ��Ze�qtckS`�fs`ou-y6§ /[§ /�UXk8tc�qfo§ N=bcdnf�bgZ�r8jlZEt?��kSd­jq`ohdgZef8t�jq`Xr�bcUoZ}bgdgf8hn�=jq`Xr�dgZ�r8jlk8`-ks�*fS`�Z�¡8Zm·jl`�mUOfs`ou�tc¡�t{bgZ�Y �=¡©t�jq`Xr8�X�qfSdcjlb{¡¦fS`ou$z{kSjq`=b*�ljqY[j£bnt�f´�8kSjxuXfs`OhZS§� #���8�)%����Y�#�f�����'�C� �d����� � �#�E��%#��%������ � �&.-�� ���

�$ *+*�& $����Sp �S�X`oZN�E�S��:�§

�q�´��� '2§A~����¶fs`ou¥~�§ �}kSdnfs�ou §V|¦jqtc�ofs�*bcdnf�z{ZEhibck8dcjqZet­��dgkSY��X`ohefs�qjl�odgfsbcZeu¥t�bcZedcZeko§ 5� �6� �� �!�� ���@�"����W �� � %���� � ��%�� �k� ������� � ���*($)&3,��� %�" �ip��ofSrSZet S8\��v�o�SpA�e�8�=�v§

�q�@S���~�§*Nvjq`Xr8U pA~�§�L²§A|?k�¡v�lZ8p ��§ � j£bcbgfs�*pOfs`Ou,��§Oy/§Ay?fs�ofS`Xjq�SkS�qkS�AkS�o�lk=t�§ '��qjlr8`XZ�Y[Z�`=b©ks�Dfs`¥Ze¡SZmjq`�m·Uofs`Ou tc¡�t{bgZ�Y bck¥dgZefs�?k8��z{Zehbgt��Ot�jq`Xr¢�vjldcbc�ofS�DjqY[fSrSZEt�§�� ���54��'� ��� ���G� �������#� �C� ��� ���R������C� �#� ��� & ���,��%�( k��� � � ���� � � ��� � ���)� 5� �6� #���8�� ���3�"� ���I��������� � ���$���f(����Y� ��"$��� � ���Op L�f´¡�E�S�dSX§

�q�e��� � § Q �®Zejqtgtep '2§ ¨ N�fs`ouXZ�dnt�k8` pOfS`ou ¨¦§ y ��Ze�XY-fs` § �¦¡=`OfsY[jqhNtcZ�`otckSdcmJ�OfStcZeu¢hk8`8bgdck8� ks�$dck8�OkSb¤&jlbcU��vjqtc�ofs�8��Z�ZEu��of8hn�A§Y 5�6� � %Y���Y�#�f���U� � � �d����� � �#� ���f( ���Y� ��" ��� � ���ops\�� : !�� �=�s� �v�o���vp3K©hbckS�AZ�d�E��Sv�v§

� �s��� �§ ��� jq�qtckS`*pV¨¦§ ¨ � jq�l�qjxfsY-t ���X�q�xt�pVfs`ou 9N§ NQ¬­Ze�l�J§ ~�Z��xf�bcjq�SZ¢Ze`ouvm·Z��4ZehibgkSd-hk8`=bcdgkS�V�otcjl`XrhefsdcbcZEt�jxfs` �Ak8tcjlbcjqkS`�m·�ofStcZeu �=jxtc�ofs�&tcZ�dg�SkSjq`XrO§\ �� � � �U� ��� �"� ���Z��������� � ��� ���*( ���Y� ��" ��� � ���op�´�"� : ! � ;�SS� �Y;S��;opo�E�S��;o§

465Eg7498

Page 38: Path Planning for Robust Image-Based Visual Servoing

Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique,615 rue du Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY

Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES CedexUnité de recherche INRIA Rhône-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN

Unité de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY CedexUnité de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex

ÉditeurINRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)��������� ���� ���������� ��� ���

ISSN 0249-6399