Manipulation Planning with Probabilistic Roadmaps

TLDR: A general manipulation planning approach capable of addressing continuous sets for modeling both the possible grasps and the stable placements of the movable object, rather than discrete sets generally assumed by the previous approaches.

Abstract: This paper deals with motion planning for robots manipulating movable objects among obstacles. We propose a general manipulation planning approach capable of addressing continuous sets for modeling both the possible grasps and the stable placements of the movable object, rather than discrete sets generally assumed by the previous approaches. The proposed algorithm relies on a topological property that characterizes the existence of solutions in the subspace of configurations where the robot grasps the object placed at a stable position. It allows us to devise a manipulation planner that captures in a probabilistic roadmap the connectivity of sub-dimensional manifolds of the composite configuration space. Experiments conducted with the planner in simulated environments demonstrate its efficacy to solve complex manipulation problems.

Full text

HAL Id: hal-01987879
https://hal.laas.fr/hal-01987879
Submitted on 21 Jan 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic 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 diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Manipulation Planning with Probabilistic Roadmaps
Thierry Simeon, Jean-Paul Laumond, Juan Cortés, Anis Sahbani
To cite this version:
Thierry Simeon, Jean-Paul Laumond, Juan Cortés, Anis Sahbani. Manipulation Planning with Prob-
abilistic Roadmaps. The International Journal of Robotics Research, SAGE Publications, 2004, 23
(7-8), pp.729-746. �hal-01987879�

 ! "#$%&'()"
*&+-,/.1003254,/687.19;:=<>?.A@B:DCFE@AG-HJIK@AG-6L9;:NMO<N>?GN@A:QP9?03RS7.ATU<NVW:-,XTW4@B+YN@B:S,
IKVV!4?C3P[Z\]4
*=9?G-H/9?GNT.<N^-0_@B:a`b.
cedNfgh_i;jkg
l=mAnpoq;r_q?stuBs3r_vpow=npxym{z}|bxynp|U~qAvrk~A~Anp~AW/|Utt|U?|UxyoJzr_~BnqBAvr_xynp~A&z}|_UrkAvps|UUsxyo
rkz}|U~BN|UAoxrkvpso3DsNqAt|Uq?|UoysOrOUs~Astr_v1zr_~AnpqAAvrkxynp|_~=qAvr_~A~Bn~BrkqAqAty|1r_m3rkq;r_Bvs-x|r_u1
uAtsoyo=|U~1xynp~1A|UBoNoysxoO/|Utz}|1uAsvpnp~A?|Uxm}xmAsq?|UooynpAvpsbtr_oqAo=r_~AuexmAsoyxrkAvps=qAvr_sz}s~1xyo
|kxmAszW|_Ur_Avps-|UUsx31tr_xymBstKxm;r_~euBnotysxysoysxoKbs~Astr_vpvprkoyoAz}sue1xmAsqBtys1np|UAoOr_q1
qAt|rkmBso3Ol=mAsqBty|Uq?|Uosurkvb|UtnxmAztsvpnso|U~r&x|Uq?|Uvp|Ubnr_v?qAty|bq;stxxym;rkxm;r_tr_xstynpso
xmAss1npoyxs~As|ko|UvpAxynp|b~AoNnp~exymAsoyABoyq;rks|k|U~1?UAtr_xynp|b~AoNw=mAstsxymAs=t|U?|UxOUtrkoyqAoxymAs
|bUsx=qAvrksur_xr&oxr_Bvsq?|Uonxnp|U~Dxr_vpvp|_w=oNAoxy|&uAs1npoys[rJzrk~AnpqAAvr_xn|b~[qBvrk~A~Ast=xm;r_x
3rkqAxyBtysonp~rqAty|b;r_Anpvpnpoyxnty|1r_uAzr_qexymBs|U~B~Asxn1npx|_NoyB1XuBnzWs~Aon|b~;r_v?zrk~An/|UvpuAo|_
xmAs|UzWq;|boynpxys=|b~1;bAtrkxynp|U~oyq?r_sUO¡-1q;stynpz}s~1xoK|b~AuAAxysuw=npxm}xymBsqBvrk~A~Ast=np~Woynpze1
vr_xsu}s~1npty|b~Az}s~xoKuBsz}|b~Aoyxtr_xsnxo=s¢}3rk}x|eoy|Uvpbs|bz}qAvpsezr_~AnpqAAvrkxynp|U~qBty|UBvsz}o3
£ ¤D¥¦§N¨W©ª}«O¦-¬¨e¥
¯®B°?±²³?´®1µ±¶?°²?´®A°°?±°?·W¸_¶;°?¸_¹Uº°?»=µ3¼?¹®A³?µ¶;½®¾µ±p¸=·A¹U°?¹Uº®1µ±p¶;°¶1¿µ3¼?¹»¹_À³?¹U°?¸_¹¶1¿SºF¶;Á¶AµO½¶Aµ±p¶;°?»
®1´p´¶1Â±°?·Wµ¶½!®A°?±²³?´®¾µ¹}ÃÄAÅUÆ?Ç_ÈpÉ[Ä?ÇËÊ_É3ÌbÍËÎK®A½¶;°?·e¶;Á?»µ3®1¸_´p¹_»UÏ=Ð¼?¹²º¹_»¹U°?¸_¹¶¾¿S½¶1Ñ®AÁ?´p¹¶;ÁÒ¹_¸_µ»bÓ
±XÏÔ¹AÏÕ¶;ÁÒ¹_¸_µ»Wµ3¼®1µW¸U®A°Ö¶?°?´×ؽ¶1ÑA¹Â[¼?¹U°5·;º®1»²¹_ÙÚÁA×®Ûº¶;ÁܶAµUÓO´p¹U®1Ù»µ¶Ö®Õ½¶;ºF¹·A¹U°?¹Uº®1´®A°?Ù
¸_¶;½²³?µ3®1µ±¶?°®1´p´×e¸_¶?½²?´p¹_Ý}ÑA¹Uº»±¶?°e¶1¿µ3¼?¹=¸_´®1»»±p¸U®1´½¶Aµ±¶?°}²?´®A°°?±°?·}²º¶;Á?´p¹U½ßÞàUákâÏKã°?Ù¹_¹_ÙNÓAµ3¼?¹
º¶;Á¶Bµä¼®¾»µ3¼?¹}®AÁ?±p´p±pµ×µ¶]½]¶;Ù±¿×µ3¼?¹e»µ3º³?¸_µ3³º¹&¶1¿=±pµ»¸_¶;°Bå·;³º®1µ±p¶;°!»3²®1¸_¹}Ù¹U²Ü¹U°?Ù±°?·¶;°¼?¶1Â
µ3¼?¹}½¶1Ñ®AÁ?´p¹e¶;ÁÒ¹_¸_µ±p»·?º®1»3²Ü¹_ÙÕ®A°?ÙÂ[¼?¹Uº¹e±pµ±p»[º¹_´p¹U®1»¹_ÙÕ±°{µ3¼?¹e¹U°BÑ;±º¶;°½]¹U°AµUÏçæ[´p»¶ÓD½¶1Ñ®AÁ?´p¹
¶;Á?Ò¹_¸_µ»[¸U®B°Õ°?¶Aµ}½¶1ÑA¹}ÁB×Õµ¼?¹U½»¹_´pÑA¹_»Uè-¹_±pµ3¼?¹Uº&µ3¼?¹_×L®Aº¹Wµ3º®A°?»3²Ü¶;ºµ¹_ÙLÁA×ÕºF¶;Á¶Aµ»[¶;ºJµ3¼?¹_×Õ½³?»µ
º¹_»µe®1µ»¶;½]¹e»µ3®AÁ?´p¹²?´®1¸_¹U½¹U°BµUÏL¶Aµ±p¶;°Õ²?´®A°°?±°?·Û±°éµ3¼?±p»e¸_¶;°Bµ¹_Ý;µ&®A²²¹U®AºF»®1»&®¸_¶;°?»µ3º®1±°?¹_Ù
±°?»µ3®A°?¸_¹Õ¶¾¿[µ3¼?¹Õ¸_¶?¶;ºÙ±°®1µ¹_Ùê½¶Aµ±p¶;°Ú²?´®A°°?±°?·ê²º¶;Á?´p¹U½ÏQÐ¼?¹»¶A´³?µ±p¶;°ë¶1¿®Ö½®B°?±²³?´®1µ±¶?°
²?´®B°°?±°?·Ú²º¶;Á?´p¹U½íìX»¹_¹¹AÏî·ÏÚÞðï?Óàbá_âñ}¸_¶?°?»±p»µ»±°Ú®Û»¹_À³?¹U°?¸_¹Õ¶1¿[»³ÁBò²®1µ3¼?»»3®¾µ±p»¿X×?±°?·Lµ3¼?¹_»¹
½¶Aµ±p¶;°ÖºF¹_»µ3º±p¸_µ±p¶;°?»UÏ!L¶Aµ±p¶;°?»}¶¾¿µ3¼?¹ºF¶;Á¶Aµ¼?¶A´pÙ±°?·óµ¼?¹¶;Á?Ò¹_¸_µ®1µ}®!åÝ?¹_ÙØ·;º®1»3²L®AºF¹¸U®1´p´p¹_Ù
ÍôÆBõÎËö÷ÉbôøÆAÍËùAÎ_Ó-®A°?Ùó½¶Bµ±p¶;°?»[¶¾¿µ3¼?¹º¶;ÁܶAµÂ[¼?±p´p¹µ3¼?¹}¶;ÁÒ¹_¸_µ&»µ3®b×?»ä®¾µ®»µ3®AÁ?´p¹²?´®1¸_¹U½¹U°Bµ}®Aº¹
¸U®1´p´¹_ÙÚÍôÆAõÜÎ÷úËÍDøÆAÍËùBÎ÷Ï
û
¶;°?»±Ù¹Uºçµ3¼?¹½®A°?±²³?´®1µ±p¶;°{²?´®A°°?±°?·¹_Ý®A½²?´p¹&±p´´³?»µº®1µ¹_ÙÁA×Wü=±p·;³º¹}àAÏOÐ¼?¹[½®B°?±²³?´®1µ¶;º
®Aº½8¼®1»µ¶e·A¹_µ®}½]¶Ñ1®AÁ?´p¹¶;Á?Ò¹_¸_µìXµ3¼?¹Á®AºñO¶;³?µ¶1¿Sµ3¼?¹J¸U®1·A¹AÓ;®B°?Ùµ¶}²?´®1¸_¹&±µ¶;°Wµ3¼?¹¶Aµ¼?¹Uº=»±pÙ¹
¶1¿µ¼?¹e¹U°AÑ?±º¶;°½¹U°BµUÏý?¶A´pÑ?±°?·µ3¼?±p»e²º¶;Á?´p¹U½þº¹_À³?±º¹_»Jµ¶®A³?µ¶?½®1µ±p¸U®1´p´p×ÿ²ºF¶;Ù³?¸_¹Wµ3¼?¹e»¹_À³?¹U°?¸_¹
¶1¿µº®A°?»¿¹Uº

1µ3º®A°?»±pµ²®1µ3¼?»!»¹U²®Aº®1µ¹_Ù ÁB×·;º®1»3²?»

A³°?·?º®1»3²?»¶?²¹Uº®1µ±¶?°?»UÓ®1´p´p¶ÂJ±°?·Øµ¶Ö·B¹_µ¶;°?¹
¹_Ý;µ3ºF¹U½±pµ×!¶1¿µ¼?¹Á®Aºä¶;³?µ¶1¿Kµ3¼?¹e¸U®1·B¹Aè;µ¼?¹½®B°?±²³?´®1µ¶;º¸U®B°µ3¼?¹U°ºF¹òy·;º®1»3²!µ3¼?¹e¶;ÁÒ¹_¸_µÁB×µ3¼?¹
¹_Ý;µ3ºF¹U½±pµ×Ûµ3¼®1µ&Â®1»½!®1Ù¹®1¸_¸_¹_»»±Á?´p¹ÁB×Õµ3¼?¹²º¹_Ñ?±p¶;³?»½¶Bµ±p¶;°?»UÓ-²¹Uº¿¶;º½ ®{µ3º®A°?»¿¹Uº[²®1µ¼µ¶
¹_Ý;µ3º®1¸_µµ3¼?¹Á®Aºç¿yº¶;½µ¼?¹e¸U®1·A¹AÓ®A°?Ù{åS°®¾´´p×Õº¹U®¾¸÷¼{µ3¼?¹W»3²¹_¸_±å¹_ÙL·A¶;®¾´a²Ü¶A»±pµ±p¶;°-Ïãy°²®AºFµ±p¸U³?´®Bº_Ó
µ3¼?¹½¶Aµ±¶?°Õ»¼?¶Â&°Õ¶?°Aµ¶µ¼?¹»¹_¸_¶;°?ÙÖ±½®¾·A¹±´p´³?»µ3º®1µ±°?·Öµ3¼?¹»¶A´³?µ±p¶;°Øº¹_À³?±º¹_»}±pµ»¹_´¿¿¶;³º}º¹ò
·;º®1»3²?±°?·{¶;²¹Uº®1µ±p¶;°?»eµ¶{¶;Á?µ3®1±°Û®{»3³

¸_±p¹U°Aµe»´±pÙ±°?·ó½]¶Aµ±p¶;°¶¾¿µ3¼?¹Á®Aº_Ï[ÐJ¼?±»e¹_Ý®A½²?´p¹»3¼?¶1Â»
à

ü=±·?³º¹óà

Ä

8ÍÄóÃÆAõDúîø

;ÈpÆAÍÉÛÍËù;ÉÕÇÆBôö÷ôÄAà ú/ÍËÎ{ú/õDú/ÍúËÆAÈDøľÎ÷úËÍú/ÄBõ

yÍÄø

XÈpÉö_Í

ÕÍÄÖÍËù;É

?ÄUÆAÈ

ÇÄAÍ

ÍÄAÃ

XÈpÉö_Í

Où?É}Î_ÄAÈ

?Íú/ÄAõ

ô3ú

Aù?Í

]ôÉ

?ú/ôÉ_ÎeÎ_ÉUÅbÉUôÆAÈBøSúËÌ

ÆAõ

eøSÈpÆAÌÉÄøÉbôÆAÍú/ÄBõÎ
µ3¼®1µ=®e½®A°?±²³?´®1µ±p¶;°µ3®¾»
!
e²¶A»»±Á?´×!´¹U®¾Ù»=µ¶}®¸_¶?½²?´p¹_Ý»¹_À³?¹U°?¸_¹¶¾¿S½¶Aµ±p¶;°?»=±°?¸_´³?Ù±°?·»¹_ÑA¹Uº®1´
º¹òy·;º®1»²?±°?·{¶;²Ü¹Uº®1µ±p¶;°?»UÏ&æ8¸_¼®1´p´p¹U°?·A±°?·ó®1»3²Ü¹_¸_µ[¶¾¿½®A°?±²³?´®1µ±p¶;°Ö²?´®A°°?±°?·Õ±p»eµ¶!¸_¶;°?»±pÙ¹Uºeµ3¼?¹
®A³?µ¶;½®¾µ±p¸eµ3®1»
!
Ù¹_¸_¶?½²¶A»±µ±¶?°Õ±°Bµ¶»3³?¸_¼{¹_´p¹U½¹U°Bµ3®Aº×¸_¶A´p´p±p»±p¶;°?»ò¿yº¹_¹½¶Aµ±p¶;°?»bÏ
L¶A»µW¶1¿¹_Ý?±p»µ±°?·®1´p·A¶?º±pµ3¼½»óì˹AÏÔ·ÏêÞà1Óï?Ó
#"
?Óà
%$
;Ó
'&
AïkâXñ®1»»3³½¹µ¼®1µWåS°?±µ¹»¹_µ»¶1¿&»µ3®AÁ?´p¹
²?´®¾¸_¹U½¹U°Aµ»®A°?ÙÚ²¶A»»±Á?´p¹·;º®1»3²?»¶1¿µ3¼?¹½¶1Ñ®AÁ?´p¹{¶;Á?Ò¹_¸_µ]®Bº¹·A±pÑA¹U°Ø±°5µ3¼?¹Ù¹åS°?±pµ±p¶;°ë¶1¿[µ3¼?¹
²º¶;Á?´p¹U½Ï
û
¶?°?»¹_À³?¹U°Bµ´p×AÓ®e²®Aºµ=¶1¿Nµ3¼?¹Jµ3®1»
!
Ù¹_¸_¶;½²¶B»±pµ±p¶;°±p»µ3¼;³?»º¹_»¶A´pÑA¹_Ù{ÁA×µ3¼?¹[³?»¹UºO»±°?¸_¹
µ3¼?¹±°?±µ±®¾´
(!
?°?¶1Â´p¹_Ù·A¹Õ²º¶1Ñ;±pÙ¹_ÙQÂ±pµ3¼Øµ3¼?¹_»¹{åS°?±pµ¹»¹_µ»¼®1»µ¶L¸_¶;°Bµ3®1±°Øµ3¼?¹·?º®1»3²?»®A°?ÙÚµ3¼?¹
±°Bµ¹Uº½]¹_Ù±®¾µ¹ÿ²?´®1¸_¹U½]¹U°Aµ»º¹_À³?±º¹_ÙÚµ¶»¶B´ÑB¹µ3¼?¹{²º¶;Á?´p¹U½Ï
*)
¹¿X¹Uºº±°?·¯Á®¾¸
+!
Ûµ¶Ûµ3¼?¹¹_Ý®A½²?´p¹AÓ
·A¹_µµ±°?·µ3¼?¹eÁ®Aº¶?³?µ¶¾¿µ3¼?¹&¸U®1·A¹eÂ¶;³?´pÙÕº¹_À³?±º¹}®´®Aº·B¹}°;³½Á¹Uºä¶1¿K·;º®¾»3²?»®A°?ÙÕ²?´®1¸_¹U½¹U°Bµ»µ¶
Á¹}·B±ÑB¹U°Õ®1»±°²³?µeÙ®1µ3®Ï
ãy°¯µ3¼?±p»²®A²Ü¹Uº_ÓNÂ¹Ù¹_»¸UºF±Áܹÿ®{·B¹U°?¹Uº®1´®A²²º¶?®1¸÷¼LÁ®1»¹_ÙÖ¶;°Bµ¶Õº¹_¸_¹U°Bµº¹_»3³?´pµ»²º¹_»¹U°Bµ¹_ÙÖ±°
Þ
,&
Aá?Ó
-&.
bâÏ[ÐJ¼?¹]½!®1±°Õ¸_¶?°Aµ3ºF±Á³?µ±¶?°Õ±p»eµ¶{Ù¹U®1´OÂ±pµ3¼L®¸_¶?°Aµ±°?³?¶;³?»e»¹_µµ±°?·{¶1¿µ3¼?¹½®B°?±²³?´®1µ±¶?°
²º¶;Á?´p¹U½þÂ[¼?±p´p¹¸_¶1ÑA¹UºF±°?·{µ3¼?¹e»¸_¶?²¹¶1¿=µ3¼?¹²º¹_Ñ?±p¶;³?»e²º¶;²¶B»¹_ÙÖ®A²²º¶;®1¸_¼?¹_»ìý?¹_¸_µ±¶?°
/&
Añ_Ïãµe®1´ò
´¶1Â»&³?»µ¶Ù¹_Ñ?±p»¹®½!®A°?±²³?´®¾µ±p¶;°Õ²?´®A°°?¹UºJµ3¼®1µ[®B³?µ¶;½®1µ±¸U®¾´´p×{·A¹U°?¹Uº®1µ¹_»[®A½¶;°?·¸_¶;°Bµ±°?³?¶;³?»
»¹_µ»µ3¼?¹W·;º®1»²?»[®A°?Ùµ3¼?¹W±°Bµ¹Uº½]¹_Ù±®¾µ¹²?´®1¸_¹U½¹U°Bµ»eº¹_À³?±ºF¹_Ùóµ¶»¶A´pÑA¹e¸_¶;½!²?´±p¸U®1µ¹_Ù¯½!®A°?±²³?´®ò
µ±p¶;°²º¶;Á?´p¹U½»&´p±
0!
¾¹}µ3¼?¹e¶;°?¹e±p´p´³?»µ3º®¾µ¹_ÙÕ¶;°Bµ¶ü=±p·;³º¹{àAÏ=Ð¼?¹}®A²²º¶;®¾¸÷¼º¹_´p±p¹_»&¶;°®µ¶;²Ü¶A´p¶A·A±p¸U®1´
²º¶;²¹UºFµ×éåSº»µ&¹_»µ3®AÁ?´p±p»3¼?¹_Ùر°Þ
21
1âç®B°?Ù¯ºF¹_¸U®1´p´¹_Ù5±°5ý?¹_¸_µ±p¶;°ÖïÏeÐ¼?±p»²ºF¶;²¹Uºµ×Õ®1´p´p¶1Â»}³?»eµ¶Õº¹ò
Ù³?¸_¹µ3¼?¹{²º¶;Á?´p¹U½)ÁA×L¸_¼®Aº®1¸_µ¹UºF±
43
_±°?·Ûµ3¼?¹¹_Ý?±p»µ¹U°?¸_¹{¶1¿ä®é»¶A´³?µ±p¶;°5±°Öµ3¼?¹!´¶1Â¹UºeÙ±½¹U°?»±¶?°®1´
&

»3³Á?»3²®1¸_¹¶¾¿ä¸_¶?°Bå·;³º®1µ±p¶;°?»eÂ&¼?¹Uº¹µ3¼?¹ºF¶;Á¶Aµ
5
?ôÆ1ÎyøDÎ[µ3¼?¹½]¶Ñ1®AÁ?´p¹¶;Á?Ò¹_¸_µ&øSÈpÆAÌÉ

®¾µ}®{»µ3®AÁ?´p¹
²¶A»±pµ±p¶;°-Ïý?¹_¸_µ±p¶;°
61
eÙ¹_»¸Uº±Á¹_»µ3¼?¹&²º¶;²¶B»¹_Ù®A²²º¶?®1¸÷¼®A°?Ù!»3¼?¶1Â»¼?¶1ÂÚµ3¼?¹¸_¶?°°?¹_¸_µ¹_Ù¸_¶?½²¶1ò
°?¹U°Aµ»J¶1¿Kµ3¼?±p»&»3³Á?»3²®1¸_¹e¸U®A°Áܹ}¸U®A²?µ³º¹_Ù±°Õ®²º¶;Á®AÁ?±p´p±p»µ±p¸º¶?®1Ù½®A²!¸_¶;½²³?µ¹_Ù{¿¶;º®Ñ?±ºµ3³®1´
¸_´¶B»¹_Ù?òy¸÷¼®¾±°ë»×?»µ¹U½ÏLý?¹_¸_µ±p¶;°
$
Ù¹_µ®1±p´»!µ3¼?¹ÿ²?´®A°°?±°?·Øµ¹_¸_¼°?±À³?¹_»Ù¹_ÑA¹_´p¶;²Ü¹_Ùêµ¶Û±½²?´p¹U½¹U°Aµ
µ3¼?¹}®A²²º¶;®¾¸÷¼-Ï
87
J»±°?·µ3¼?¹
59
&±»±Á?±p´p±µ×Bò
;:#)
®1´p·A¶;º±pµ3¼½þÞ
,&$
_âO¹_Ý;µ¹U°?Ù¹_Ùµ¶Ù¹U®¾´NÂ±pµ3¼{»3³?¸_¼{¸_´p¶A»¹_Ù
»×;»µ¹U½»Þ
,<
kâÓOÂ¹åSº»µ¸U®B²?µ3³º¹{µ3¼?¹¸_¶?°°?¹_¸_µ±pÑ;±pµ×¶¾¿[µ3¼?¹{»¹U®Aº¸_¼Ö»²®1¸_¹±°Bµ¶Ö®Û»3½®1´p´ºF¶;®1Ù½®A²
¸_¶;½²¶B»¹_Ù¶1¿®W´p¶1Â°;³½ÁܹUº¶1¿K¸_¶;°°?¹_¸_µ¹_Ù¸_¶;½²¶?°?¹U°Aµ»ìý?¹_¸_µ±p¶;°
=$
Ïàbñ_Ï
û
¶?°°?¹_¸_µ±p¶;°?»[Á¹_µÂ¹_¹U°
µ3¼?¹_»¹¸_¶;½²Ü¶;°?¹U°Aµ»³?»±°?·}µº®A°?»±pµ=¶;ºµ3º®A°?»F¿X¹Uº=½¶Bµ±p¶;°?»®Aº¹µ3¼?¹U°W¸_¶;½²³?µ¹_ÙÁA×W»¶A´pÑ?±°?·®&´p±½]±µ¹_Ù
°;³½Á¹UºJ¶1¿²Ü¶A±°Aµòyµ¶1ò²¶A±°Bµ}²®1µ3¼Û²?´®B°°?±°?·L²º¶;Á?´p¹U½»ìý?¹_¸_µ±p¶;°
*$
Ï
,&
Bñ_ÏÐJ¼?¹}Ù¹_µ3®1±p´p»e¶1¿®A°é±½²?´p¹ò
½¹U°Aµ¹_Ù¯²?´®A°°?¹Uº±°Aµ¹UºF´¹U®bÑ?±°?·óÁ¶Bµ3¼Û»µ3®1·A¹_»&±°Ö®A°Û¹

¸_±p¹U°Bµ}Â®b×Õ®Aº¹}Ù¹_»¸Uº±Á¹_Ùر°5ý?¹_¸_µ±p¶;°
>$
ÏðïÜÏ
ü=±°®¾´´p×AÓý?¹_¸_µ±p¶;°êáÛ²º¹_»¹U°Bµ»»¶;½¹{¹_Ý²¹Uº±½¹U°Bµ»®A°?ÙÚ¸_¶?½½¹U°Bµ»}¶;°5µ3¼?¹Õ²Ü¹Uº¿X¶?º½®A°?¸_¹{¶1¿[µ3¼?¹
²?´®B°°?¹Uº_Ï
? @BA#CED
¦
-A
©
GF
¨§
H
I
°?¹e¶1¿Kµ3¼?¹e¸_¼®1´p´p¹U°?·A±°?·±p»»3³?¹_»J¶1¿½®A°?±²³?´®1µ±p¶;°Û²?´®A°°?±°?·±»Jµ¶}±°Bµ¹_·;º®1µ¹[µ3¼?¹}®¾ÙÙ±µ±¶?°®1´NÙ±

ò
¸U³?´µ×Û¶1¿²?´®A°°?±°?·µ3¼?¹·;º®1»3²?±°?·®A°?ÙÖº¹òy·;º®¾»3²?±°?·¶;²Ü¹Uº®1µ±p¶;°?»&µ¶{µ3¼?¹²®¾µ3¼Õ²?´®A°°?±°?·ó²ºF¶;Á?´p¹U½Ï
Ð¼?±p»e±°Bµ¹UºÙ¹U²Ü¹U°?Ù¹U°?¸_×Á¹_µÂ¹_¹U°Õ²®¾µ3¼Õ²?´®A°°?±°?·ó®B°?Ùó·?º®1»3²²?´®A°°?±°?·ÕÂ®1»JåSº»µJµ¶;³?¸_¼?¹_Ù¯³²Ü¶;°
ÁA×WÂ¶?º
!
&Ù¶;°?¹±°µ3¼?¹
J.LKNM
î»¿¶;ºNµ¼?¹Ù¹_ÑB¹_´¶?²½¹U°Aµ¶1¿a®B³?µ¶;½®1µ±¸º¶;ÁܶAµ²ºF¶A·;º®A½!½±°?·e»×;»µ¹U½»UÏOãy°
²®Aºµ±p¸U³?´®Aº_ÓBµ3¼?¹
PO
&®A°?Ù¹_×»×?»µ¹U½ßÞ
,&
àâ±°Bµ¹_·;º®¾µ¹_ÙÁ¶Aµ3¼W²?´®A°°?±°?·´p¹_ÑA¹_´p»®A°?Ù}Â®1»O¸U®A²®AÁ?´p¹µ¶e²?´®A°
»±½²?´p¹
Q:
=±p¸
!
®A°?Ù
R:
ç´®¾¸_¹¶?²¹Uº®1µ±¶?°?»[±°?¸_´³?Ù±°?·ó»¶;½¹}º¹òy·;º®1»3²?±°?·¸U®A²®AÁ?±p´p±pµ±p¹_»UÏeÐ¼?¹·A¹_¶;½¹_µ3ºF±¸
¿X¶;º½³?´®1µ±¶?°¶1¿a½!®A°?±²³?´®¾µ±p¶;°²?´®A°°?±°?·ÞðïÓàbá_âÓA»¹_¹U°®¾»®A°±°?»µ3®A°?¸_¹J¶1¿K½¶Aµ±p¶;°²?´®A°°?±°?·²º¶;ÁBò
´¹U½þ¹_Ý?µ¹U°?Ù¹_ÙÖÁB×µ3¼?¹²º¹_»¹U°?¸_¹¶1¿½¶1Ñ®AÁ?´p¹¶;Á?Ò¹_¸_µ»UÓ²ºF¶Ñ?±pÙ¹_ÙÖ®³°?±å¹_ÙL¿yº®A½¹_Â¶;º
E!
®1´p´p¶1Â±°?·
µ¶Á¹_µµ¹Uºµ3®1¸
!
B´¹eµ¼?¹e±°Bµ¹UºÙ¹U²Ü¹U°?Ù¹U°?¸_×L±»»3³?¹_»Áܹ_µÂ¹_¹U°ÛÁ¶Aµ¼²?´®B°°?±°?·´p¹_ÑA¹_´p»UÏ
L¶Aµ±p¶;°L²?´®A°°?±°?·Õ±°Ø²º¹_»¹U°?¸_¹¶1¿½¶1Ñ®AÁ?´p¹¶?Á?Ò¹_¸_µ»[±p»eåSºF»µ}®1ÙÙº¹_»»¹_Ù®1»e»3³?¸_¼Û±°ëÞ
,&L<
kâÏãy°
µ3¼?±p»eÂ¶?º
!
Ó-®A°é¹_Ý®1¸_µe¸_¹_´p´=Ù¹_¸_¶;½!²¶A»±pµ±p¶;°5®1´p·A¶;º±pµ3¼½þ±p»}²º¶;²Ü¶A»¹_ÙÛ¿¶;ºµ¼?¹]²®Bºµ±p¸U³?´®Aº&¸U®1»¹¶1¿®
²¶A´p×?·A¶;°®1´º¶?Á¶Aµ®A°?Ù¶1¿¶?°?¹[½¶1Ñ®AÁ?´p¹¶;Á?Ò¹_¸_µ=µ3º®B°?»´®1µ±°?·e±°®e²¶A´p×?·A¶;°®1´Â¶;º
E!
A»3²®¾¸_¹AÓ®1»»3³½±°?·
®WåS°?±pµ¹}·;º®1»3²{»¹_µ¶¾¿Kµ3¼?¹½]¶Ñ1®AÁ?´p¹e¶;Á?Ò¹_¸_µUÏ
Ð¼?¹Õ½®B°?±²³?´®1µ±¶?°ê·?º®A²¼Ø¸_¶;°?¸_¹U²?µ±p»{±°Aµ3ºF¶;Ù³?¸_¹_Ù%±°Þðïkâ¿¶;ºWµ3¼?¹Û¸U®1»¹¶1¿e¶;°?¹Õº¶;ÁܶAµ®A°?Ù
»¹_ÑA¹Uº®1´½]¶Ñ1®AÁ?´p¹¶?Á?Ò¹_¸_µ»½®A°?±²³?´®1µ¹_ÙØÂ±pµ3¼5Ù±p»¸Uº¹_µ¹{·;º®1»²?»}®A°?Ù²?´®1¸_¹U½¹U°Bµ»UÏÕã°Lµ3¼?±p»¸U®1»¹AÓ
µ3¼?¹°?¶?Ù¹_»e¶1¿=µ3¼?¹½®B°?±²³?´®1µ±¶?°Õ·?º®A²¼¸_¶?ºº¹_»3²Ü¶;°?ÙÕµ¶{Ù±p»¸Uº¹_µ¹W¸_¶;°Bå·;³º®¾µ±p¶;°?»®B°?ÙÕµ3¼?¹W¹_Ù·A¹_»
®Aº¹¸_¶;°?»µº³?¸_µ¹_Ù ÁB×Ø»¹U®Aº¸_¼?±°?·Ö¿¶;ºµ3º®A°?»¿¹UºÿìX¶;ºµ3º®B°?»±pµ3ñ²®1µ3¼?»{Á¹_µÂ¹_¹U°ê°?¶?Ù¹_»»3¼®Bº±°?·µ3¼?¹
»3®A½¹·?º®1»3²êìX¶;º}²?´®1¸_¹U½¹U°Bµ3ñW¶1¿µ3¼?¹½¶1Ñ®AÁ?´p¹{¶;Á?Ò¹_¸_µìË»3ñkÏü¶B´´p¶1Â±°?·Lµ3¼?±p»·A¹U°?¹Uº®1´ç¿ º®B½¹_Â¶;º
!
DÓ
µ3¼?¹®A²²ºF¶;®1¸_¼&Â䮾»S±½²?´p¹U½¹U°Bµ¹_ÙW¿¶;ºN®µ3º®A°?»´®1µ±°?·[²Ü¶A´p×;·A¶?°Þðïbâ®A°?Ù}®ï
JSUTWV
²?´®A°®BºK½®B°?±²³?´®1µ¶;º
Þà
.
kâÏ=æe°{¹_Ý®¾¸_µ¸_¹_´p´KÙ¹_¸_¶;½!²¶A»±pµ±p¶;°L®1´p·A¶?º±pµ3¼½±p»®¾´»¶²º¶;²Ü¶A»¹_Ù±°ØÞ
X1
bâN¿X¶?ºµ¼?¹}»3²Ü¹_¸_±ðå¸¸U®1»¹e¶1¿
®eµ3º®B°?»´®1µ±°?·²¶A´p×?·A¶;°®1´-º¶;Á¶Bµ¸U®A²®AÁ?´p¹&µ¶½!®A°?±²³?´®¾µ¹e¶;°?¹e½¶1Ñ®AÁ?´p¹e²¶A´p×?·A¶;°!Â±pµ3¼®A°!±°BåS°?±µ¹
»¹_µ¶1¿=·;º®1»²?»UÏ
Ð¼?¹}½®B°?±²³?´®1µ±¶?°ó²?´®A°°?±°?·{¿yº®A½¹_Â¶;º
E!
}±p»&¹_Ý;µ¹U°?Ù¹_ÙÕ±°ÖÞà
Y1
Óà
$
kâKµ¶½³?´pµ±ò®Aº½ ½!®A°?±²³?´®ò
µ±p¶;°{Â[¼?¹Uº¹e»¹_ÑB¹Uº®1´Nº¶;ÁܶAµ»¸_¶?¶;²¹Uº®1µ¹eµ¶¸U®AººF×®»±°?·A´p¹½]¶Ñ1®AÁ?´p¹}¶;ÁÒ¹_¸_µ®A½±pÙ»µJ¶;Á?»µ3®1¸_´p¹_»UÏãy°
µ3¼?±p»eÂ¶?º
!
ÓDµ3¼?¹°?³½ÁܹUº&¶1¿´p¹_·;®1´O·;º®1»²?»[¶1¿µ¼?¹¶?Á?Ò¹_¸_µ»[±p»&åS°?±pµ¹®A°?ÙLµ3¼?¹½¶1Ñ1®AÁ?´p¹¶?Á?Ò¹_¸_µe¼®1»
µ¶óÁܹÿ¼?¹_´pÙê®1µW´¹U®¾»µÁB×Ö¶;°?¹ºF¶;Á¶Aµ®1µ®B°A×Lµ±½¹{Ù³º±°?·®ÕºF¹òy·;º®1»3²¯¶;²¹Uº®¾µ±p¶;°-ÏÕÐ¼?¹²?´®A°°?¹Uº
²º¶;²¶B»¹_Ù±°LÞà
$
kâåSº»µä²?´®A°?»Jµ3¼?¹½]¶Aµ±p¶;°?»¶1¿Kµ3¼?¹e½¶1Ñ®BÁ?´¹e¶?Á?Ò¹_¸_µ³?»±°?·®A°®1Ù®B²?µ¹_ÙÑB¹Uº»±p¶;°!¶1¿
®eº®A°?Ù¶;½±
3
_¹_Ù²¶Bµ¹U°Aµ±®¾´?å¹_´Ù{²?´®B°°?¹UºÞ
,$
bâÓ®A°?Ù!µ3¼?¹U°åS°?Ù»=µ3¼?¹»¹_À³?¹U°?¸_¹&¶1¿Sº¹òy·?º®1»3²W¶;²Ü¹Uº®1µ±p¶;°?»
¶1¿µ3¼?¹®Aº½»&µ¶½¶1ÑA¹}µ¼?¹¶?Á?Ò¹_¸_µe®1´p¶;°?·µ3¼?¹¸_¶;½!²³?µ¹_ÙÖ²®1µ3¼-Ï}Ð¼?±p»}²?´®A°°?¹UºWº¹_´p±¹_»W¶;°»¹_ÑA¹Uº®1´
»±½²?´p±ðå¸U®¾µ±p¶;°?»UÓNÁ³?µ&±µJ¸U®A°{Ù¹U®1´NÂ±pµ3¼é¸_¶;½²?´p¹_ÝÕ®A°?ÙÕº¹U®¾´±p»µ±¸²º¶?Á?´¹U½]»UÏ
ï

æe°?¶Aµ3¼?¹Uº¼?¹U³º±p»µ±¸²?´®A°°?±°?·Õ®A²²º¶;®1¸_¼²º¶;²Ü¶A»¹_ÙÛ±°5Þ
,"
kâK±p»[µ¶±pµ¹Uº®1µ±pÑA¹_´p×Ù¹¿X¶?º½ ®¸_¶?¶;ºÙ±ò
°®1µ¹_Ù²®¾µ3¼JåSº»µO·A¹U°?¹Uº®1µ¹_Ùe±°Wµ3¼?¹¸_¶;½²Ü¶A»±pµ¹¸_¶?°Bå·;³º®1µ±p¶;°&»²®1¸_¹³?»±°?·}®Ñ1®Aº±®1µ±p¶;°®¾´;Ù×°®A½±p¸
²º¶A·;º®B½½±°?·µ¹_¸_¼°?±pÀ³?¹Wµ3¼®1µ[²º¶B·;º¹_»»±pÑA¹_´p×{¹U°B¿X¶?º¸_¹_»[µ¼?¹}½®A°?±²³?´®1µ±p¶;°Û¸_¶;°?»µ3º®1±°Aµ»bÏ
9
®Aº±®A°Bµ»¶1¿Nµ3¼?¹}½®B°?±²³?´®1µ±¶?°Õ²?´®A°°?±°?·²º¶;Á?´p¹U½ ¼®bÑA¹eÁ¹_¹U°é±°AÑA¹_»µ±p·;®1µ¹_ÙNÏã°ÕÞ
&L&
kâÓ?·;º®1»3²Bò
±°?·Ø±p»º¹U²?´®1¸_¹_ÙÁB× ²³?»3¼?±°?·Ú®A°?ÙÚµ3¼?¹»3²®¾¸_¹Õ¶1¿[»µ3®AÁ?´p¹ó²³?»3¼?±°?·ØÙ±º¹_¸_µ±p¶;°?»±½²¶B»¹_»®L»¹_µ!¶1¿
°?¶;°¼?¶A´p¶;°?¶;½±p¸¸_¶;°?»µº®1±°Aµ»[µ3¼®¾µ±°Aµ3ºF¶;Ù³?¸_¹»¶;½]¹}¸_¶;°Bµ3º¶A´p´®AÁ?±p´p±pµ×Û±p»»3³?¹_»eµ¶µ¼?¹²º¶;Á?´p¹U½Ï&Ð¼?¹
¼?¹U³º±p»µ±p¸ó®1´p·A¶?º±pµ3¼½)Ù¹_»¸Uº±Áܹ_Ùê±°Þ
,.
kâ¸_¶;°?»±pÙ¹Uº»®L²º¶;Á?´p¹U½)Â[¼?¹UºF¹Õ®1´p´µ3¼?¹{¶;Á?»µ3®1¸_´p¹_»¸U®B°ÚÁ¹
½¶1ÑA¹_ÙÕÁA×®¸_±º¸U³?´®Aº&º¶;ÁܶAµ±°¶;ºFÙ¹Uºµ¶WåS°?ÙÕ±pµ»&Â®U×!µ¶µ3¼?¹}·A¶?®1´Ï
ÐÂ¶}¶Bµ3¼?¹Uº¸_¶;°Aµº±Á³?µ±p¶;°?»¹_Ý?µ¹U°?ÙÕº¹_¸_¹U°Bµ[²?´®A°°?±°?·µ¹_¸_¼°?±pÀ³?¹_»&µ¶½®A°?±²³?´®1µ±p¶;°Õ²?´®A°°?±°?·Ï
ãy°Þ àâÓUµ3¼?¹
#Z
ô3úËÆAõÉ
[
Î
J\
ÈÉ
%
®1´p·A¶;º±pµ3¼½ Þðábâ;±p»=®A²²?´p±¹_Ùµ¶&®º¹_Ù³°?Ù®A°BµKº¶;ÁܶAµK½®B°?±²³?´®1µ±°?·}®ä»±°?·A´p¹
¶;Á?Ò¹_¸_µ=±°!®eï
W]
%Â¶?º
!
A»²®1¸_¹AÏ=Ð¼?¹½¹_µ3¼?¶?Ù®1»»3³½]¹_»=Ù±»¸Uº¹_µ¹&·;º®1»3²?»O¶1¿-µ3¼?¹[½¶1Ñ®BÁ?´¹J¶;Á?Ò¹_¸_µUè1±pµ±p»
¼?¶Â¹_ÑA¹Uºä¸U®A²®AÁ?´p¹µ¶Ù¹U®1´N±°Lº¹U®1´p±p»µ±p¸»±pµ3³®1µ±¶?°?»[Â±pµ3¼Ûº¹_Ù³°?Ù®A°Bµ½!®A°?±²³?´®¾µ¶;º»Þ
,&
kâK¿¶;ºÂ&¼?±¸_¼
¹U®1¸÷¼W·;º®1»3²²¶A»»±Á?´×¸_¶;ºº¹_»3²¶?°?Ù»µ¶e®A°W±°BåS°?±µ¹°?³½Á¹Uº=¶¾¿Sº¶;Á¶Bµ=¸_¶;°Bå·;³º®1µ±¶?°?»UÏOü=±°®¾´´p×AÓ-Þ
,&
Aïkâ
²º¶;²¶B»¹_»®&²º®1¸_µ±¸U®¾´½®B°?±²³?´®1µ±¶?°²?´®A°°?¹UºÁ®1»¹_ÙW¶;°Aµ¶µ3¼?¹¹_Ý?µ¹U°?»±p¶;°W¶1¿µ¼?¹
P:#)
ä¿ º®A½¹_Â¶;º
!
ÞàUï?Ó
^&W1
¾âÏÕÐ¼?¹²?´®A°°?¹UºW¸_¶;°?»µ3º³?¸_µ»]®Û½®A°?±²³?´®1µ±p¶;°Ø·;º®A²¼ÖÁ¹_µÂ¹_¹U°5Ù±»¸Uº¹_µ¹¸_¶;°Bå·;³º®1µ±p¶;°?»Uè
¸_¶;°°?¹_¸_µ±p¶;°?»}®Aº¹e¸_¶?½²³?µ¹_Ùó³?»±°?·®
`_abbcd#egf
²?´®A°°?¹UºJµ3¼®1µ[Á³?±p´pÙ»®ºF¶;®1Ù½®A²{ÂJ±µ¼{¹_Ù·A¹_»
®A°°?¶Aµ3®1µ¹_ÙÕÁA×Õ®{²º¶;Á®AÁ?±p´p±pµ×Õ¶1¿¸_¶A´p´p±p»±p¶;°Bò¿yº¹_¹U°?¹_»»UÏ
û
¶;½²³?µ±°?·{»3³?¸÷¼Lº¶?®1Ù½®A²?»±½²ºF¶ÑB¹_»[µ3¼?¹
¹

¸_±p¹U°?¸_×¶1¿Kµ3¼?¹e²?´®A°°?¹Uº¿¶;º»¶A´pÑ;±°?·µ¼?¹²Ü¶A»»±Á?´p×Õ¼?±p·;¼°;³½ÁܹUº¶¾¿²®1µ3¼{²?´®A°°?±°?·À³?¹Uº±p¹_»ì˱°
¸÷¼®A°?·B±°?·{¹U°BÑ;±º¶;°½]¹U°Aµ»ñº¹_À³?±ºF¹_ÙÕµ¶¸_¶?½²³?µ¹eµ3¼?¹e¸_¶;°°?¹_¸_µ±¶?°?»UÏ
hjiakmlnWop8qloriak8s
Ð¼?¹½®A°?±²³?´®1µ±p¶;°²?´®A°°?±°?·eµ¹_¸_¼°?±pÀ³?¹_»®AÁ¶1ÑA¹½¶B»µ´p×}®1ÙÙº¹_»»Oµ3¼?¹Ù±»¸Uº¹_µ¹
±°?»µ3®A°?¸_¹e¶1¿Nµ3¼?¹e²º¶;Á?´p¹U½Ï
I
°?´p×{µ3¼?¹e®1´p·A¶;º±pµ3¼½»±°LÞ
X1
ÓSàâN¸_¶;°?»±pÙ¹Uº½¶;ºF¹[Ù±

¸U³?´pµ±°?»µ3®B°?¸_¹_»¿¶;º
Â[¼?±p¸÷¼!µ3¼?¹[°?¶?Ù¹_»¶1¿Nµ3¼?¹[½®A°?±²³?´®1µ±p¶;°{·?º®A²¼Õì˱Ïî¹AÏOµ3¼?¹&²?´®¾¸_¹_»Â&¼?¹Uº¹[µ¼?¹¸_¶;°°?¹_¸_µ±p¶;°?»Á¹_µÂ¹_¹U°
µ3¼?¹W¿X¹U®1»±Á?´p¹µº®A°?»±pµe®A°?ÙÕµ3º®A°?»¿¹Uº[²®1µ¼?»¼®bÑA¹Wµ¶ÿÁܹ»¹U®Aº¸_¼?¹_Ùñ&¸_¶;ºº¹_»²¶;°?ÙLµ¶®¸_¶B´´p¹_¸_µ±p¶;°¯¶1¿
»3³ÁBò½®A°?±¿¶A´pÙ»¶1¿Nµ3¼?¹&¸_¶;½²¶B»±pµ¹[¸_¶?°Bå·;³º®1µ±p¶;°!»3²®1¸_¹AÓ?®1»¶?²²¶A»¹_Ùµ¶}Ù±p»¸Uº¹_µ¹&¸_¶;°Bå·;³º®1µ±p¶;°?»UÏ
ý³?¸÷¼Ø½®A°?±¿¶A´pÙ»]®Bº±p»¹Â&¼?¹U°Ö¸_¶;°?»±Ù¹UºF±°?·5±°BåS°?±pµ¹·?º®1»3²?»}®A°?Ùظ_¶;°Bµ±°;³?¶?³?»]²?´®1¸_¹U½]¹U°Aµ»¶1¿µ3¼?¹
¶;Á?Ò¹_¸_µUÏKÐ¼?±p»¸_¶;°Aµ±°?³?¶;³?»»¹_µµ±°?·}±p»=¶;°?´p×®1ÙÙº¹_»»¹_Ù±°Þ
X1
1â¿¶;ºOµ3¼?¹»3²Ü¹_¸_±å¸[¸U®1»¹¶1¿a®&µº®A°?»´®1µ±°?·
º¶;Á¶Bµ[±°Ö®{²¶A´p×?·A¶;°®1´=Â¶;ºF´ÙNÏ&Ö®A°?±¿¶A´pÙ»}®1´p»¶Õ®Aº±p»¹±°ØÞàâÁ¹_¸U®A³?»¹¶1¿µ3¼?¹º¹_Ù³°?Ù®A°?¸_×Û¶1¿µ3¼?¹
º¶;Á¶Bµ[®1´pµ3¼?¶;³?·;¼{µ¼?¹}²?´®B°°?¹Uºe®1»»3³½¹_»&®}»¹_µ¶1¿²º¹òyÙ¹åS°?¹_ÙLÙ±»¸Uº¹_µ¹}·?º®1»3²?»UÏ
ãµµ3¼?±p»²®A²Ü¹Uº_ÓÂ¹[²º¶;²Ü¶A»¹}®·A¹U°?¹Uº®1´N®A²²º¶?®1¸÷¼]¿¶;º=Ù¹U®1´p±°?·Â±pµ3¼{»³?¸÷¼!¸_¶;°Bµ±°;³?¶;³?»J»¹_µµ±°?·A»
¶1¿µ3¼?¹½®A°?±²³?´®1µ±p¶;°²?´®A°°?±°?·²º¶;Á?´p¹U½Ï
I
³º=²?´®A°°?±°?·®A²²º¶;®¾¸÷¼¸_¶;°?»±pÙ¹Uº»¸_¶?°Aµ±°?³?¶;³?»²?´®1¸_¹ò
½¹U°Aµ»®A°?Ù·?º®1»3²?»UÓA®B°?Ù±pµ±p»®1´p»¶®AÁ?´p¹µ¶¼®A°?Ù´p¹eº¹_Ù³°?Ù®A°Bµº¶;ÁܶAµ»UÏ=ã µº¹_´p±p¹_»¶;°®e»µ3º³?¸_µ3³º±°?·
¶1¿Nµ3¼?¹[»¹U®Aº¸_¼»3²®¾¸_¹®1´p´p¶1Â±°?·³?»µ¶}¹

{¸_±¹U°Bµ´p×{¸U®A²?µ3³º¹&µ3¼?¹&¸_¶;°°?¹_¸_µ±pÑ;±pµ×{¶1¿Nµ3¼?¹[»³ÁBò½®A°?±¿X¶B´Ù»
±°L®{²º¶;Á®AÁ?±p´p±p»µ±p¸º¶;®1Ù½®B²¸_¶;½²³?µ¹_ÙÕ¿¶;ºÑ?±ºµ3³®1´=¸_´p¶A»¹_Ù?òy¸÷¼®1±°Ö½]¹_¸÷¼®A°?±p»3½]»UÏeÐ¼?¹º¹_»3³?´pµ±°?·
²?´®B°°?¹UºO±»=·B¹U°?¹Uº®1´®A°?Ù²º®1¸_µ±p¸U®1´B¿X¶?º-»¶A´pÑ?±°?·e¸_¶?½²?´p±¸U®¾µ¹_Ù½®A°?±²³?´®1µ±p¶;°!²?´®B°°?±°?·²º¶;Á?´p¹U½»=±°
¸_¶;°?»µ3º®¾±°?¹_ÙÕïòyÙ±½]¹U°?»±p¶;°®1´N¹U°BÑ;±º¶?°½¹U°Aµ»UÏ8ü¶?º¹_Ý®A½²?´p¹AÓ¶?°?¹[¸U®A°{Ù¹_»¸Uº±Á¹}µ¼?¹[»¹_µ¶1¿O»µ3®AÁ?´p¹
²?´®¾¸_¹U½¹U°Aµ»ÁA×e¸_¶;°?»µ3º®1±°?±°?·}µ3¼?¹½¶1Ñ®AÁ?´p¹¶;Á?Ò¹_¸_µOµ¶Áܹ²?´®¾¸_¹_Ù¶;°eµ¶;²e¶1¿»¶;½¹¼?¶?º±
3
_¶;°Aµ®1´A¿y®1¸_¹_»
¶1¿Nµ3¼?¹[»µ3®1µ±p¸&¶;Á?»µ3®1¸_´p¹_»UÏ=ý³?¸_¼²?´®1¸_¹U½¹U°Bµ¸_¶?°?»µ3º®1±°Bµ»Ù¹åS°?¹}®WïbòyÙ±½¹U°?»±p¶;°®1´»3³ÁBò½®A°?±¿¶A´pÙ¶1¿
µ3¼?¹[¶?Á?Ò¹_¸_µ
%M
î»=¸_¶;°Bå·;³º®¾µ±p¶;°»²®1¸_¹]ìXµÂ¶&µ3º®A°?»´®¾µ±p¶;°?»±°!µ3¼?¹e¼?¶;º±
3
_¶;°Bµ3®1´-²?´®B°?¹®A°?Ù{¶;°?¹[ºF¶Aµ3®1µ±p¶;°
®Aº¶;³°?ÙQµ3¼?¹ÑA¹Uºµ±p¸U®1´&®1Ý;±p»3ñkÏêæ&´p»¶Ó¶;°?¹¸U®A°ë¸_¶;°?»±pÙ¹Uº{»¹_µ»!¶1¿¸_¶?°Aµ±°?³?¶;³?»{·;º®1»3²?±°?·ÖÙ¶?½®1±°?»
»3³?¸÷¼{µ¼®1µµ3¼?¹Ò®bÂ»¶1¿®²®Aº®1´p´¹_´·;º±²²¹Uº&¼®bÑA¹®]¸_¶;°Aµ®1¸_µÂ±pµ3¼{µÂ¶·A±pÑA¹U°!¿y®1¸_¹_»¶¾¿µ3¼?¹e¶?Á?Ò¹_¸_µbÏ
ý³?¸÷¼Ø·;º®1»3²5¸_¶;°?»µ3º®¾±°Bµ»®1´p»¶ÖÙ¹åS°?¹Õ®LïbòyÙ±½¹U°?»±p¶;°®1´JÙ¶;½®1±°ì˵Â¶µ3º®A°?»´®¾µ±p¶;°?»²®Aº®¾´´p¹_´µ¶
µ3¼?¹e·;º®1»²¹_Ù¿y®1¸_¹_»[®B°?Ù¶;°?¹}º¶Aµ3®¾µ±p¶;°®Aº¶;³°?Ùµ¼?¹}®1Ý;±p»e²¹Uº²Ü¹U°?Ù±p¸U³?´®Bºeµ¶µ3¼?¹¿ ®1¸_¹_»ñ_Ï
1

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Manipulation planning with probabilistic roadmaps" ?

HAL this paper is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.