12

KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

Embed Size (px)

Citation preview

Page 1: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

������������ ������������������������! ����#"����#���$�� %�� '&(�$")�*,+-�$���# ��!�.�� ��/�102���3�4�#���$�)�5+-�# ��!�.���$�#�765�8���9�:�:����$���#���$�� ;���;<=���#�4���3�4�������1�>����"��?�����#�$�@�A�B,CDB,E5F�G�H�I4J�I3KLKMB NOB,E�E�B,A�P�Q�G/FSR�TUR�E�B,V @�B,A�PXWYG�Z\[�A^]�P`_aE�bcR�EKdIeC!I3KLKaB!fg],b,Gh[`_a[�]!GhBei`[ j7RDT�R�E�B,V#fg],b,Gh[`_a[�]!GhBei`[ bcB,A�P�fg],bkGh[`_a[�]!GhBei`[

l P�Bnm�Be]�o`jpI!ESq$E�bcAr_MAU[�A�BI!E�is B,t�AeGR4uwvxKaBe]^A�Tc_d],I3K�I!E�izy�R�j7t�[�A�B,T\v{E�|!_aE�B,B,Tc_aE�|} E`_MCDB,T~b�_aAcVXR!uwq8KLK�_aE�R!_db} TU��I!E�I���qcH����3������������^�,���,�

�1�>���U���d�c�,��c�,�����3�8�������,�( ��L��¡p¢��,£e�r¤��d�,�S�4��¤c�8�U���d�k�'¥e�U�!¤��L�.�d�r¤n¡4��¤7�¦�c�c���!�.�§£��¦�c�c�U�a¨��c¥��O���c¡©���d�$���4���d�k��ªc�^��¡�«�-��¡4�¬¤c�$�^�.�L¤c���d�r¤p�,��¥?�¦�k���`�3�8�U�p¨^�L¤��d�k�­�c�,�O�O�>�D�L�����c¤�®�¯#¡>�M¤n�4�r�4�U�X�c°c���L�,�¦�r¤7��¡4�X�c�,�����3�$�^�.�d�,�±�8²p���,�8³��«���,�¥k�L¤c���U�dª��3�.�d�,�4¤/²~�k�\�´�^¥e�U�h¤1��¡4���¡4�,¨,�Oª¦�c�U�p )�d¥e�U�§£n�c�,�4¤r��¥e�U�¦�c¥��a�X�c�,�����3�$���\¨^�L¤��d�k��®n¯#¡4�r¤~�g�c�k���`�e�$�^�.�d�k�!¤´�c�,�Xªc��3¤~�c¥µ�8�O�^¤c¤~�r¤¦¤�¡4�k�´�r³k�U���U�L�.£²~�,�{¤U�8³,�´�U�!�$�^�.�d�,�4¶)�k�·�c�,�7ªc�1�3¤~�c¥x²~�,���´�^¥e�U�`¤~���L�¦�r�.�d�,��®x¸��O���,�c�.�d�U�>�L�,�c¶� =�1¥,�L¤~�U�3¤¦¤�c�k���`�e�$�^�.�d�k���´�r�d¡��~¥�¤=��¡4���¹�r�e���º£��8�\�·»¬�k�c¼e�,¨��c�,��¥3�,��½=�U��¥�²^�,���O�!���^�.�d�k�!¶¾�«���`�§�d�U�L�����,�§£,�`�,�O�d�,�!¤��!�a²~�3�c����~¥3���¿¤¦¶�k��¥1���,�c�,���c�.���d�w�4�k�§£,���k�g�d�k��¤��!�a²~�3�¦�µ���~¥3���¿¤¦¶=�k��¥µ���¦�r¤~���4�=¤~�,���O¥e�U�´�k�!¤c���c�^�.�a¨��µ�c°c�4�U���a���U�!�d¤�®À Á�ÂOÃ�Ä�ÅnÆ�ÇYÈÃxÉ~Å�ÂÊ\E�B\iD_�ËX]^[`KMAcVYA�P�I!A¹_«bxR4u�A�B,E2B,E�]^R�[�E`A�B,T�BeiYÌ\P�B,E7I!t�t`KaVD_aE�|nm·I3V�Beb¦_«I4EÍI4E�I3KaV¾b�_db)_dbxA�P�B�B^ξt#B,E�b�_aC�Bg]^R�j7t�[�A^I!Ac_aR�EYA�P�I!A�_db_aE�]^[�T�T�Bei�G�q$EpI2bcA~I!E�i�I!T^i7I!t�t`KL_d],I!Ac_aR�EXR!u=m1I3V�Beb,Ï!T�[`KaB���I!E¬_ME`A�B,|�T^I3K�Ð$R�T1bc[�j7jpI!Ar_MRDE�Ñ�_«b�T�BeÒD[`_aT�Bei�A�R�jpI4T�|!_aE�I3KL_aÓ,B�R�E�BbcB,A\R!uA�P�B�T~I4E�i`R�j�C>I4Tc_dI!�`KaBebOÌ1_aA�PXT�Bebct#Be]^AgA�R�I!E�R�A�P�B,TeG l P`_dbO],I!E5�#B']^R�brAcKaVpÌ\P�B,ESA�P�BniD_aj7B,E�b�_aR�E�R!uA�P�B�T~I!E�i`RDjC>I!Tr_«I4�`KMBebÍ_dbYP`_a|�P��1I�b2_dbYR!u.A�B,E(A�P�B5],I�bcBXÌ1_aA�P±brA~I!Ac_dbcAc_d],I3K1_ajpI!|DBXj7R¾i`B^Kdb,G(q�EÔA�P`_dbYt�I!t�B,T�Ì1Bpt�TUBebcB,E�A¬E�[�j7B,Tr_«],I>Kj7B,A�P�R�i�bOu.R�T�B^ËÕ]U_aB,E`AcKaV-B,C>I3Ka[�I!Ar_ME�|�A�P�BYjpI4T�|!_aE�I3KL_aÓ^_ME�|z_ME`A�B,|�T^I3KdbOu.R�T�t�R�t�[`KdI!TYbcA~I!Ar_«brAc_d],I3K_ajpI!|�B7j¬R¾i`B^Kdb,�iD_db�]^[�b�bI!t�t`KL_d],I!Ac_aR�E�b,��I!E�iXt�TUBebcB,E�AObcR�j7B�B^Î�t�B,Tc_aj7B,E`A~I3KxT�Bebc[`KaA~bµA�P�I!AgiD_aT�Be]^AcKaVX[�bcB2A�P�B�j7B,A�P�R�i�b,GÖ [�j¬B,Tc_d],I3K=]^R�j7t�[�A~I!Ar_MRDE7R�E7A�P�BOt�R`bcA�B,Tc_aR�T1bct�I�]^BOA�P�I!A�TUBebc[`KaA~b{u�T�RDj×Inm1I3V�Beb�_dI!EXI!E�I3KaV¾b¦_«b�P�I�b{��B,B,E5I2bc[���ØcBe]^A�R!uI�]^Ac_aC�B�_aE�AUB,T�BebcAx_aE7A�P�B\bcA~I4Ac_dbcAc_d],b]^R�j7jY[�E`_aAcV�G l P�B·AcV`t`_d],I3K�P`_a|�PpiD_aj7B,E�b�_aR�E�I3KL_aAcV7R!u�t#R�bcA�B,Tc_aR�T^b)_ME�m·I3V�Beb¦_«I4EÍI4E�I3KaV¾b�_dbP�I�b�KaBeiXAUR7T�Be]^B,E`Ag]^R�j7t�[�A~I4Ac_aR�E5A�Be]UP�E`_dÒ�[�BebµA�P�I!AµP�IeCDBO_ME�]^TUBeI�bcBei�_aA~bµI!t�t`KL_«],I4�`_�KL_dAcV�G�Ù\_a����bgbUI!j7t`KL_aE�|7_dbOI7F�I!T�o�R!C]�P�I>_ME�Ú.��IDbcBei2A�Be]�P�E`_dÒD[�B\A�P�I!AxI3KLKMR!Ìgbxb�I!j¬t`K�_aE�|Ou.T�R�j%A�P�B�ØcR!_aE`A=t#R�bcA�B,Tc_aR�TiD_dbcA�Tc_a��[�Ac_aR�EYR!u/I4EnB^Î�t�R�E�B,E`Ac_dI3K`u8I!jÍ_�KaV��`I!E�iP�I�b1t�T�R!C�B,E�bc[�],]^Beb�b�u.[`Kx_MEÛ_ajpI!|�B2t�T�R¾]^BebUb�_aE�|ÕI!t�t`KL_d],I!Ac_aR�E�b¬Üa���^Ý$GWÞTUBe]^B,E�A�iD_db�]^[�b�b�_aR�E-I!E�iS]^R�j7t�I!Tc_dbcRDEXR!u·F�I!T�o�R!C]�P�I>_ME¬j7B,A�P�R�i�bxA�P�I!A[�bcBgFSR�E`A�B,Úcy·I4TcKaR´b�_ajY[`KdI!Ac_aR�E��`Ì\P`_d]UPY_aE�]UKM[�i`Beb�A�P�B�Ù\_a����b�b�I4j7t`KaB,Te��I!t�t#BeI!T~b)_aE5ܧß`��à!áDÝ$G@�jY_aA�PP�I�b�t�TUR>CD_di`Bei-IXj¬R�T�BY|�B,E�B,T~I3K·br[�T�C�B,V�R!uâm1I3V�Beb�_dI!E(]^R�j7t�[�A~I!Ar_MRDE-j7B,A�P�R�i�b,��_aE�]UKa[�iD_aE�|­I4E�I3KaV�Ac_d]7I4t�t�T�R3Î�_ajpI!Ar_MRDE�bA�R7A�P�BO_aE`A�B,|�T~I>K«bk��t�I!T~I!j¬B,A�Tc_aÓeI!Ac_aR�E�b\I4E�iXÒ�[�I�i`T^I!A�[�T�B�T�[`KaBeb,��I!E�iXbcRDj7BnI�i�I!t�Ac_aC�BnbUI!j7t`KL_aE�|XA�Be]�P�E`_dÒD[�Bebnܺà�ß3Ý$Gã R�T�RD[�Tn|�B,E�B,T~I>K·m·I3V�Beb¦_«I4E�I!E�I3KaV�b�_db,�xÌ·B�]^R�E�b�_di`B,TYA¦Ì1R�C�Be]^A�R�T^b�R!u1T~I!E�i`R�jäC>I!Tc_dI!�`KaBeb,�åæI4E�i-ç(G l P�BpCDBe]^A�R�TåèT�B,t�T�BebrB,E�A~b2I5]^R�E`Ac_aE`[�R�[�b�t�I!T^I!j7B,A�B,TYbct�I�]^B��=I4E�i-çæTUB,t�T�BebcB,E`A~b2A�P�B7R���bcB,TUC>I!Ac_aR�E�bkG l P�B�R���bcB,T�C!I!Ac_aR�E�b2],I!E-��BA�P�B2_ajpI!|�B7i�I!A~I¬R�TnbcRDj7B'bcA^I!Ac_dbcAc_d],bOR!u�A�P�B2_ajpI!|DBÍi�I4A~I�G l P�B,TUBn_db�I�E�R!_dbcBYj7R�i`B^K$��é�Ð�êâëºìxÑ~��̵P`_«]UP-T�B,t�TUBebcB,E�A^bOA�P�BI!E`Ac_d]U_Mt�I4A�BeiXR���brB,T�C>I!Ar_MRDEXu.R�T\IY|!_aC�B,E5t�I4T~I!j7B,A�B,T1C!I3Ka[�B��/I!E�iSIYt�Tc_aR�Tgi`B,E�b¦_MAcVXR�EXAUP�B�t�I!T~I!j7B,AUB,Tgbct�I�]^B���é¹Ð�ì=Ñ^Gí B�Ì1_LKLKwE�B^Î�Agi`Beb�]^Tr_M�#BYA�P�T�B,BO_aj7t�RDT�A~I!E`Agm1I3V�Beb�_dI!E5]^R�E`A�B^Î�A~b�_aE5Ì\P`_d]UP5A�P`_dbµB^ξt�TUBeb�b�_aR�E5R�],]^[�T~b,î

ï é�Ð$ê\ëºìxÑdé�Ð�ìxѦð{ì¹ñ Ð~�!Ñl P`_db\B^Î�t�T�Beb�b¦_MRDE©TUB,t�T�BebcB,E`A~b1A�P�B2jpI!T�|!_aE�I3KL_aÓeI!Ac_aR�ESR!uê�Ì1_aA�PST�Bebct#Be]^AgA�RÛìw��I4E�iXA�P�BY]^R�j7t�[�A~I3_aA�RDEXR!u=_aA~bµC>I>KM[�B2u.R�TbcA~I!Ar_«brAc_d],I3K�_ajpI!|�B�j¬R¾i`B^Kdbâ_«bµA�P�B�t�Tc_ajpI4T�VÕ]^RDE�]^B,T�E5R!uA�P`_db1t�I!t#B,TeGFSR¾i`B^K�R�T^i`B,T�bcB^KaBe]^Ac_aR�EpP�IDb��B,B,E�Inbc[���ØcBe]^AR!u#_ME`A�B,T�BebrA_aE7A�P�Bg]^RDj7t�[�A�B,TCD_«b¦_MRDEÕ]^R�j¬j'[�E`_aAcVpÜ¿á����eò3Ý$��t�I4T�Ac_d]^[`KdI!TcKaVu�RDTwI!t�t`KL_d],I!Ac_aR�E¬A�RgbcB,|�j¬B,E�A~I4Ac_aR�E�G l P`_dbI�i�i`TUBeb�bcBeb)A�P�B1t�T�R��`KaB,jÞR!u/i`Be]U_diD_aE�|nÌ\P`_d]UP'j¬R¾i`B^K$��å�óDR�T=å��3_dbI!t�t�T�R�t�Tr_«I4A�B

Page 2: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

u�RDTâIÍ|!_aC�B,E©i�I!A^I'brB,Ae��Ì\P�B,EXA�P�B�j¬R¾i`B^KdbOI!T�B�E�BebcA�Bei��#å ó�� å�G ã R�T\B^Î�I!j7t`KaB���å ó ]^R�[`KdiÛT�B,t�T�BebcB,E`AgI2KL_aE�BeI!TOj7R¾i`B^K$�I!E�iXå IYÒD[�I�i`T~I!Ar_«]�j7R�i`B^K8G ã R�T\AcÌ1RYE�BebcA�Bei�t�I!T^I!j7B,A�B,Tgbrt�I�]^Beb,�`A�P�B�u.R!KLKaR!Ì1_aE�|pT~I4Ac_aR'R4uwjpI4T�|!_aE�I3Kdb1P�I�b1�#B,B,E5[�bcBeiB^ξAUB,E�b�_aC�B^KaVXu.R�T1j7R¾i`B^KbcB^KaBe]^Ac_aR�ES_ME�I7m1I3V�Beb�_dI!E�u.T~I!j7B,Ì1R�T�oSÜa�!������à��!Ý$îï é�Ð$ê\ëºì ó Ñdé�Ð�ì ó Ñcðì óï é�Ð$ê\ëºìxÑdé�Ð�ìxѦð{ì ñ Цà`Ñ

W´b´Ð¦à`Ñx_aE�]^T�BeI�bcBeb,��]^R�E��/i`B,E�]^B2_aE©å�ó�I3KdbcRY_aE�]^T�BeIDbcBeb,� u8I3C�R�Tr_ME�|YA�P�Bnb¦_Mj¬t`KMB,T�j¬R¾i`B^K$Gq$E�I(m·I3V�Beb¦_«I4E BebrAc_ajpI!Ac_aR�E�]^R�E`A�B^Î�Ae�1R�E�BS_db¬_ME`A�B,T�BebrA�Bei _ME%bcB^KaBe]^Ac_aE�|±A�P�B(ì�A�P�I!A�jpI3Î`_ajY_aÓ,BebXA�P�BÛKL_MoDB^K�_aP�R`R¾i��é�Ð$ê\ëºìxÑdé�Ð�ìxÑ~G l P`_dbâ_«bOi`R�E�Bnb�_aE�]^BYA�P�BnI!t�t`KL_d],I!Ac_aR�E�R!u�m1I3V�Beb,Ï`T�[`KaB��é¹Ð�ì{ë ê�Ñ�� é�Ð$ê\ëºìxÑdé�Ð�ìxÑï é�Ð$ê\ëºìxÑdé�Ð�ìxѦð{ì� Цß`Ñ

_aj7t`KL_MBeb�A�P�I!Aé�Ð�ì�ë ê�Ñx_«bµjpI3Î`_ajY_aÓ,Bei�G\m�V5]^R�j7t�[�Ac_aE�|�A�P�Bni`B,E�R�jY_aE�I!A�RDTgR!u1Цß`Ñ^��RDE�B'],I!EX[�bcB2A�P�BnBeÒD[�I!Ac_aR�E�iD_aT�Be]^AcKaVA�R�R���A~I3_aE-I7E�RDT�jpI3KL_aÓ,Bei�t/iDu1C>I3Ka[�BYu.R�T�I7t�I4T~I!j7B,A�B,T3��|!_aC�B,E�A�P�BYR���brB,T�C>I!Ar_MRDE�b,��é�Ð�ì�ë ê�Ñ~G } b¦_ME�|XAUP`_«bk��]^R�j7t�I!Tc_dbcRDE�b],I!EX��B2jpI�i`B�A�R¬A�P�B�t�Tc_aR�TOt/iDuC>I3Ka[�Beb,�`é�Ð�ìxÑ~Gl P�BOA�P`_aT~ipI4t�t`K�_d],I!Ar_MRDE5R!u=AUP`_«b�jpI4T�|!_aE�I3K¹]^RDj7t�[�A~I!Ac_aR�E¬_db{u�R�T1A�P�B�I�b�brBeb�bcj7B,E`A�R!u�T�B,|4_MRDEXP�R�j7R�|�B,E�B^_aAcVYu.R�T_ajpI4|�BbcB,|�j7B,E`A~I!Ar_MRDE�G ã R�T\AcÌ1R7bc[���bcB,A~b,�� �gI!E�i� �e� R!uwI!E�_ajpI!|DB��`_aAgP�I�bâ��B,B,E©brP�R>̵EXA�P�I!A\A�P�B�T^I!Ac_aR��� ï é�Ð$ê�� ëºì��`Ñdé�Ð�ì���Ѧð{ì���� � ï é�Ð�ê���ëºì���Ñ�é�Ð�ì���Ѧð{ì����

ï é�Ð$ê � ëºì ��� Ñdé�Ð$ê � ëºì ��� Ñdé�ÐUì ��� Ѧðì ��� � Ð$á�Ñ

],I!E-�#Bp[�bcBei(I3KaR�E�|SÌ1_aA�P­m1I3V�Beb,Ï�TU[`KMB�A�RSR���A~I3_aE-A�P�B�t�T�R���I4�`_�KL_aAcV-A�P�I!A2A�P�Bpi�I!A^I�_aE� � I!E�i� � Ì1B,T�B7|�B,E�B,T~I4A�Bei��V�A�P�BYb�I!j7BOt�I!T~I!j¬B,A�B,T\C>I>KM[�B¬Ð.u.R�TgbcR�j¬B�|!_aC�B,EXt�I!T~I!j¬B,A�B,Tgbct�I�]^B!ÑOÜa���3Ý$G l P�B�C!I!Tc_dI!�`KaB�ì ��� T�B^u.B,T~b1A�RpI¬]^R�jY�`_ME�Beit�I!T~I!j¬B,A�B,T\bct�I�]^BnAUP�I!A�_«bOI�b�brR¾]U_dI!A�Bei�Ìâ_MAUP5��RDA�P� ��I!E�i� �3Gl P�B�AcÌ·R(T~I!Ar_MR`b,�2Цà`Ñ7I!E�i%Ð$á�Ñ^�OЦI!E�i�b�_ajY_LKdI!TÛu�R�TUjpb^ÑYP�IeCDBzI4t�t�BeI!T�Bei T�Be]^B,E�ArKMV _aE%Ì·RDT�o(u�TUR�j A�P�B-bcA~I!Ar_«brAc_d],bKL_MAUB,T~I!A�[�T�B��`I!E�iYI!T�B1A�B,T�j¬Bei��! �"�#�$&%' �(�)+*�,-$,G�@�jY_aA�P7I!E�i¬@�t�B^_a|�B^KaP�I3KaA�B,T\[�brBeinAUP`_«b{T~I!Ac_aRµu�RDT=j7R�i`B^K/bcB^KaBe]^Ac_aR�E���B,AcÌ1B,B,EE�BebcA�Bei KL_aE�BeI!TSt�I!T~I!j7B,AUTc_d]5j¬R¾i`B^Kdb-ܺà��3Ý$GÞWµ_MAUo�B,E P�I�bXi`B,C�B^KaR�t�Bei%IÔm1I3V�BebYu$I�]^A�RDT7u.R�TXj7R�i`B^K']^R�j7t�I!Tr_«brR�E A�P�I!A]^R�E�iD_aAc_aR�E�b2A�P�B7t�Tc_aR�T2j7R�i`B^K�R�E-A�P�Bpi�I4A~I�Üa�^Ý$G7N�I�b�b2I!E�i�J�I3_di`V�I4E�I!A�P�I!E-t�TUBebcB,E�AnI4E�i-iD_db�]^[�b�bYbcR�j7B7I�brV�j7t�AUR�Ac_d]I!t�t�T�R3Î`_Mj�I!Ac_aR�E�bYI!E�i-bcB,E�b¦_MAr_MCD_aAcV�A�RSC>I!TUV�_aE�|�t�Tc_aR�T~b2R!u\A�P�BXm1IeVDBebOu$I�]^A�R�T�Üa�,á!Ý$G�.xB,Ac_aAYI3KdbcR-iD_«bU]^[�b�bcBebYt�Tc_aR�T^b,�x��[�AÌ1_aA�P-]^R�E�]^B,T�ESu.R�TgTUR���[�bcA�E�Beb�bOÌâ_MAUP©TUBebct�Be]^A�A�R�R�[�AcKL_aB,T~bYܧà �~Ý�G l P�BÍm1I3V�Bebâu8ID]^A�R�T\P�I�b�I>K«brRp�#B,B,E�],I!T�B^u.[`KLKMV-bcAU[�iD_MBeiu�RDT\B,C�_di`B,E�]^BYB,C>I>KM[�I4Ac_aR�EX_aE©I'u�R�TUB,E�b�_d]nb�]U_aB,E�]^B7]^R�E`A�B^ξA�ܧà �¾ò4Ý$G@�Be]^Ac_aR�EXàniD_db�]^[�b�bcBebµbcR�j7B\t#R�t�[`KdI!TgbcA^I!Ac_dbcAc_d],I3K#_Mj�I!|�B\j7R�i`B^Kdb1A�R2Ì\P`_d]�P�A�P`_db{u�R�TUj×R!u=m1I3V�Beb�_dI!EpI!E�I>KMV�b�_db\I!t�t`KL_aBeb,G@�Be]^Ac_aR�EÔßXi`Beb�]^Tc_a��Beb7I4E�_aE�A�B,|DT~I!Ac_aR�E-j¬B,A�P�R¾izA�P�I!A�t�B,TUA~I3_aE�b2A�RXj7R�i`B^K«b�_aE(Ì\P`_d]�P�é�Ð$ê\ëºìxÑdé�Ð�ìxÑ_dbnIÕu�[�E�]^Ac_aR�E(R!u\IÒ�[�IDi`T~I!Ac_d]�_aEXì��I!E�iÍA�P�B\T�B,|!_aR�E¬R!u�_aE`A�B,|�T~I!Ar_MRDEY_db0/21YG�@�Be]^Ac_aR�E7á'I>KMAUB,T�E�I!Ac_aC�B^KaV7i`Beb�]^Tc_a��Beb1I�j7R�T�B1|�B,E�B,T~I>K8�D��[�AKaBeb�bB^ËÕ]U_aB,E�A3�¾FSR�E`A�B�y1I!TcKaR�Ú.��I�bcBei'_ME`A�B,|�T^I!Ac_aR�E�j7B,A�P�R�i�G l P�Bgj7B,AUP�R¾i¬R!u@�Be]^Ac_aR�EXßnI!t�t`KL_aBeb1A�R2t�I!T~I!j7B,AUTc_d]1t�R!KaV`E�R�jY_dI3Kj7R�i`B^KwÐU@�Be]^Ac_aR�E5à�Ga�!ÑI!E�i�I'F�I!TUo�R!CnT~I!E�i`RDj3��B^KdiÛj7R¾i`B^KÐ�@�Be]^Ar_MRDEÕà�Gºà`Ñ~��I4E�i�A�P�Bgj¬B,A�P�R¾i¬R!u�@�Be]^Ar_MRDEXáÍI4t�t`K�_aBebµA�RA�P�B1_aj7t`KL_d]U_aA´bc[�Tru8I�]^Bµj7R¾i`B^KxÐ�@�Be]^Ac_aR�Epà G§ß`Ñ~G)q$ES@�Be]^Ac_aR�E4��Ì1BgbcP�R!̱bcR�j7BObcB,|�j7B,E`A~I!Ac_aR�E¬T�Bebc[`KaA~bAUP�I!AÌ1B,T�B\R���A~I>_ME�Bei[�b�_aE�|pAUP�B�j7R¾i`B^Kdbâ_aE-@�Be]^Ac_aR�E�à�G ã _aE�I3KLKaV��/bcRDj7Bn]^R�E�]UKa[�b�_aR�E�b�I!T�B�t�TUBebcB,E�AUBei�_aE-@�Be]^Ac_aR�E���G5 687�91È1É-:�È<; Å2Æ�9>=�?@7A7B=,É~È0C1ÃxÉ~Å�ÂEDí BOÌ1_LK�Kxt�T�BebcB,E`A·AUP�B�|�B,E�B,T~I3K�j7B,A�P�R�i�b1R!u=E`[�j7B,Tc_d],I3K�_aE�AUB,|�T~I!Ac_aR�E¬_aE�@�Be]^Ac_aR�E5ßYI!E�i�á��`I!E�i¬_aEXA�P`_db\bcBe]^Ac_aR�E�Ì1B!��T~brAi`Beb�]^Tc_a��B¬_ajpI!|�BYj7R�i`B^KdbOA�RXÌ\P`_d]UP©AUP�BebcBYj7B,A�P�R�i�bnI!t�t`KaV�G ã R�TOBeI�]UPzI4t�t`K�_d],I!Ar_MRDE��=bc[`ËÕ]U_aB,E`A�_aE`u.R�T�jpI!Ar_MRDE5_dbO|!_aC�B,EA�R�u.R�T�j A�P�BY_aE�AUB,|�T~I!E�i�R4ugÐ~�!Ñ^��é�Ð�êâëºìxÑdé�ÐUì=Ñ~GOq�E-BeI�]UP�bcBe]^Ac_aR�E-Ì1BYT�B^u.B,T�A�R5IXbcB,A�R!u�_ajpI!|�B7B^KaB,j7B,E`A~b2I�b Y��Ì\P`_d]�P]^R�[`KdiX�#B'I7bcB,AµR!u=_aE`A�B,E�b�_aAc_aBebOR�T1T~I!E�|DBn]^R�R�T^iD_ME�I4A�Beb,��i`B,t#B,E�iD_ME�|SR�EXA�P�BO_ajpI!|DB�A¦V`t�B�G

Page 3: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

����� 0n��������� �����$������:��$�����.���'02���� � ������$���\� �������l P�B�|�B,E�B,T~I>K�u�RDT�j�R!u=AUP�Bnt�I!T~I4j7B,A�Tc_d]Ot�R4KMV`E�R�jY_dI3Kj7R�i`B^K�_db

�µÐ�ì���� ��� Ñ ������� 1���� � � � ����� �! � � Ð �`Ñ_aEXÌ\P`_d]�P#" � I!E�i%$ � I!T�BOt�R�b�_aAc_aC�BO_aE`A�B,|�B,T~bkG l P�B'&�#)( , # #1R!u�A�P�B�j¬R¾i`B^K�_db1A�P�BOjpI3Î`_aj'[�j R!C�B,T+* R!u�" � �,$ � G l P�Bt�I!T~I!j¬B,A�B,T\bct�I�]^B�_dbµA�P`[�b\bct�I!E�E�BeiÛ��V�A�P�Bn]^R`B^ËÕ]U_aB,E�A~bµA�P�I!AOI!T�BO[�bc[�I3KLKaV©bcB^KaBe]^A�BeiÕu�R�TObc[�Tcu$I�]^B�BebcAc_ajpI!Ar_MRDE�Gl P�BOR���bcB,T�C!I!Ac_aR�E�b,�`ç��`I!T�B\TUB,t�T�BebcB,E`A�Bei7�`VpI�C�Be]^A�R�T�R!u=t�R4_ME`A�Ú.A�RDÚ$bc[�Tcu$I�]^B´iD_dbct`KdI�]^B,j¬B,E�A~bâR!u�A�P�B\_aE`A�B,E�b�_aAc_aBeb�_aE�Y�|!_aC�B,E©IÍt�I!T~I!j7B,AUB,T·C!I3Ka[�B7ìG í BYi`B,E�R�A�BnI7b�_aE�|!KaBYiD_«brt`K«ID]^B,j7B,E�AnIDb

-DÐ).)Ü/� ��� Ý � ì=Ñ ��.)Ü/� ��� Ý102�µÐ�ì���� ��� Ñ Ð¦�`Ñ_aE�Ì\P`_d]UP3.�Ü4� ��� Ý�_db�A�P�BY_ajpI!|�BÍ_ME`A�B,E�b�_aAcV-I!AnAUP�B5�7698�T�R!Ì×I!E�i � 698X]^R!Ka[�j7E�G l P�BpiD_aj7B,E�b�_aR�E(R!u\çè_db�BeÒD[�I3K·AURXA�P�BE�[�jY�#B,TgR!uxt`_�Î�B^Kdbµ_aE YGí BpI�bUbc[�j7B7I!E�I�i�iD_aAc_aC�BSÙ´I4[�b�b�_dI!ES_L_«i(Ó,B,TUR�Ú.j7BeI!E�E�R!_dbcB�j7R�i`B^K8�I�b�]^RDE�b�_di`B,T�Bei�_aE ܧà�à3Ý$G l P�B�ئR!_aE`Ani`B,E�b�_aAcV5_dbR���A~I3_aE�BeiÛ��V�A~I!oD_aE�|7A�P�B�t�T�R�i`[�]^AgR4uwA�P�BO_aE�iD_aCD_«i`[�I>K·iD_«brt`K«ID]^B,j7B,E�Ani`B,E�b¦_MAr_MBebkî

é�Ð$ê\ëºìxÑ���Ðcà;:=< � Ñ�> ?@>ACB .�éEDF G 0��à;< � �HJI KML N�OQPJR Ü/-DÐ).)Ü/� ��� Ý � ì=Ñ$Ý�@S TU ñ ÐWV�Ñ

í BXi`B ��E�BXA�P�Bpt�Tr_MRDT'j¬R¾i`B^K\�`VzIDb�b�_a|�E`_aE�|�IX[�E`_Lu.R�T�j i`B,E�b¦_MAcV-AUR©IS��R�[�E�i`Bei(t�I4T~I!j7B,A�B,TYbct�I�]^B�G ã R�TYT�B,|4_MRDE�bA�P�I!A�Ì·BµP�IeCDBg]^R�E�b�_di`B,T�Bei���I�T�Be]^A~I!E�|D[`K«I4T�t�RDT�Ac_aR�E�R!u�A�P�Bgt�I4T~I!j7B,A�B,T�brt�I�]^B´],I4EÕI3KaÌ\I3V¾b{��Bµ_di`B,E�Ac_ ��Bei�A�P�I4A�B,E�]UKaR�bcBebE�BeI!TcKaV7I3KLK�R!u�A�P�B\t�TUR���I!�`_LKL_MAcVYjpI�bUb�A�P�I4A�]^R�E�AUTc_a��[�A�BebxA�R�A�P�B�_aE�AUB,|�T~I3Kdb)_aEXÐ$á�Ñ^��I!E�i2[�b�_aE�|nA�P�B�_aE�AUB,|�T~I!Ac_aR�EYj7B,AUP�R¾i2R!u@�Be]^Ac_aR�EXß���Ì1BgI!TUB´I�]^AU[�I3KLKMV¬E�R�A�T�BeÒD[`_aT�Bei¬A�R'bct#Be]U_Lu�V���R�[�E�i�b�A�R2t�B,Tcu.R�T�j A�P�B1_aE`A�B,|�T~I!Ar_MRDE5ЦI>K�K�R!u�/ 1 _db�[�bcBei#Ñ~G l P�Bt�T�R��`KaB,j×R4u/bcB^KaBe]^Ac_aE�|n��RD[�E�i�bxu.R�TwIO[�E`_Lu.R�T�jÞt�Tc_aR�TP�IDbx��B,B,E7o`E�R!Ì\EnA�RµKaBeI�iYA�R�iD_LËÕ]^[`KaAcVn_aEpm1I3V�Beb�_dI!E7I!E�I>KMV�b�_db,�`I!E�i_db\T�B^u.B,T�T�BeiÛA�RpI�bµH#_aE�iDKMB,V�ϧbOt�I!T^I�i`R3ÎzÜa�XV�Ý�G�W�b\AUP�B�C�R!Ka[�j7BnR!C�B,TµÌ\P`_d]�PSA�P�Bn[�E`_Lu.R�T�j i`B,E�b�_aA¦V�_db�i`B ��E�BeiÕ_ME�]^TUBeI�bcBeb,�A�P�B�T~I!Ar_MRÛÐ$á�Ñ�i`Be]^T�BeI�bcBebkG����� 9 � �/�ZY��@[]\��/ ����� ^������� �������ã R�T�RD[�T�t�I!T~I!j7B,AUB,T´bct�ID]^B���Ì1BY[�bcBpIXbct#Be]U_«I>KâF`_ ã u�RDT�jY[`K«I4Ac_aR�E���Ì\P`_d]UP�_dbO_aE`A�T�R�i`[�]^Bei�_aE±Üa�3ßeÝ�G l P`_db�j7R¾i`B^K�P�IDb��B,B,E I!t�t`KL_MBei A�R?A�B^ξAU[�T�BÕbrB,|�j7B,E`A~I!Ac_aR�EÔR!uO_aE�AUB,E�b�_aA¦V(_ajpI4|�Beb2_aE Üa�3���à�à4Ý$��I!E�i P�I�bY��B,B,E T�Be]^B,E`AcKaVÔB^ξA�B,E�i`Bei A�RA�B^Î�A�[�T�B�j7R�i`B^KL_ME�|5I4E�iXbcB,|�j7B,E`A~I!Ac_aR�EXR4u{]^R!KaR�Tâ_Mj�I!|�Beb�ܧàD�eÝ�GWgE�_ajpI4|�B�B^KaB,j7B,E�A3�ba5Ü4� ��� Ý#T�B,t�T�BebcB,E`A~b\IYb�_aE�|!KaB�_aE`A�B,E�b�_aAcV��bcÔÜ/� ��� Ý$�3A�T�BeI!AUBeiXI�b1IYT~I!E�i`R�j C!I!Tc_dI!�`KaB�G í B�P�IeCDBnI!Ed Ú$iD_Mj¬B,E�b�_aR�E�I3K�t�I!T~I4j7B,A�B,T�bct�I�]^B��#Ì\P`_d]�P�T�B,t�T�BebrB,E�A~bµA�P�B2_aE�AUB,T~I�]^Ac_aR�E�R!u·I¬t`_LξB^KÌ1_aA�P-IYKaR�],I3K·bcB,AOR!u�E�B^_a|�P`��R�Tc_aE�|t`_LξB^Kdb,G l P�B *�,e&�#�,�R!u=I!EXF,_ ã _aE�iD_d],I!A�Beb1AUP�B´b�_aÓ,B�R!uxA�P�B\KaR�],I3K�E�B^_a|�P��#R�T�P�R`R¾i�AUP�I!A_db\]^R�E�b¦_«i`B,TUBei�G ã _a|�[�T�BY��bcP�R!ÌgbA�P�B�E�B^_a|�P`��R�T�bcB,A\A�P�I4A1_db1[�bcBei�u.R�T\AUP�BnF,_ ã R�T~i`B,T~b\]^RDE�b�_di`B,T�Bei�_aE�R�[�T1B^ξt#B,Tc_aj7B,E`A~b,Gí B�[�bcBgf-I!E�i�Ì1_LKLK=[�bcBh< � A�RYT�B,t�TUBebcB,E�AµA�P�B�j7BeI!EXI!E�i�C>I4Tc_dI!E�]^B�_aE�Y��T�Bebrt�Be]^Ac_aC�B^KaV�G ã R�T\I!E`Vp|DB,E�B,T~I3K�R�T~i`B,T1R!uF`_ ã _aE�AUB,T~I�]^Ac_aR�E�b,�`A�P�Bµ_ajpI!|�BOB^KaB,j7B,E`A·R4u¹AUP�Bji 698 t�I4T~I!j7B,A�B,T{_aE�A�B,T^I�]^Ac_aR�E¬_«b1i`B,E�R�AUBei7�`V'kml�Ð).�Ñ~GZµB,E�]^B��D_ME�|�B,E�B,T~I>KI!AgbcRDj7Bgt#R!_aE�AncÔÜ/� ��� Ý �o.���A�P�B�j¬R¾i`B^Kx_db

.p02f � 1� l � � �bl�Ð�k=laÐe.�Ñ�0`fxÑ~ñ Ц�`Ñí B-]^R�[`Kdi�I3KdbcR±]^R�E�b¦_«i`B,T`fÞI!E�io< � I�b�t�I!T�A�R!uYA�P�B�t�I!T~I!j7B,AUB,TÕbct�ID]^B�G l P`_dbÛÌ·RD[`K«i T�BeÒ�[`_aT�B?A�P�B-bcB^KaBe]^Ac_aR�E�R!uI!t�t�T�R�t�Tr_«I4A�B�t�Tc_aR�T�i`B,E�b�_aAc_aBeb,��é¹Ð�fxÑ·I!E�iYé�Ð)< � Ñ~�#I!E�i�T�BeÒD[`_MTUBnA�P�B,j�AUR7��B�_ME`A�B,|�T^I!A�Bei¬_aE-Ð$á Ñ~Gl P�BYR���bcB,T�C!I!Ac_aR�E�brt�I�]^B���ç��#_dbni`B ��E�Bei(I�bOI�C�Be]^A�R�T�AUP�I!A´]^RDT�T�Bebct#R�E�i�bOA�R5I3KLK�R!u1A�P�B2_ME`A�B,E�b�_aAcV-i�I!A~I��q.�Ü4� ��� Ý$�¾_MEbcR�j7B�TUB,|!_aR�E�YG�ZµB,E�]^B���A�P�BniD_aj7B,E�b¦_MRDE©R4uwç _«bâBeÒ�[�I3KxA�R7A�P�B�E`[�jY��B,TOR!uwt`_LÎ�B^Kdb1_aE YG

Page 4: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

x12

3

41

11

2

2 2

3

334

44

4

44

4

ã _a|�[�T�BY��î l P�B�F`_ ã t`_LÎ�B^K=E�B^_a|�P��#R�T�P�R`R¾i¬Ì1_aA�P%cÔÜ/� ��� Ý4KMR�],I!A�BeiÍ_aEXA�P�Bg]^B,E`A�B,TeG ã R�T·I4E�� 698 Ú.R�T~i`B,T1F`_ ã ��A�P�Bµt`_�Î�B^Kdb_aE5��R3Î�Beb1Ì1_aA�PSE�[�jY��B,T^b�KaBeb�b\AUP�I!EXR�T\BeÒD[�I3K=A�R��-]^RDj7t�Tc_dbcBnAUP�B�E�B^_M|DP��#R�T�P�R`R¾i�Gí B'IDb�bc[�j7B�A�P�I4A\A�P�BnE�R4_«brB't�TUR¾]^Beb�bµA�P�I!AµR¾],]^[�T~bâ_aE�A�P�BOKL_ME�BeI4T�t�T�BeiD_d]^Ac_aR�EÔЦ�`Ñ{_«b�Ù´I4[�b�b�_dI!E�G l P�BµØ¦R!_aE`Agi`B,E�b�_aAcVA�P�I!A7Ì1BS[�bcB�R>CDB,T7A�P�B5t#R!_aE�A^b¬_ME _«b¬E�R�AXI-t�TUR�t�B,T�t/iDuW�OP�R>Ì1B,C�B,T3��_MA�P�I�b¬��B,B,E�]^R�E�b�_di`B,T�Bei;I�bpI?T�BeI�brR�E�I!�`KaBI!t�t�T�R3Î`_Mj�I!Ac_aR�E5I!E�iÛ[�bcBei¬_aE5t�T�B,CD_aR�[�bgbrB,|�j7B,E`A~I!Ac_aR�E5b�]UP�B,j7Beb2Üa�e�`��à�����àDàeÝ�Gí B7R���A^I3_aE-A�P�BX]^R�j7t`KaB,A�B�E�R!_dbcBpj¬R¾i`B^K1��V�A~I!oD_aE�|5A�P�B¬t�T�R¾i`[�]^AYR!u1A�P�BXi`B,E�b�_aA¦V?B^ξt�T�BebUb�_aR�E�b2R>CDB,T�BeI�]�P?R!u\A�P�B_aE�iD_MCD_di`[�I3K�t`_LξB^Kdb,î

é¹Ð$êâë�ì=Ñ���Цà;: < � Ñ > ?@>A B .DéEDF G 0��à;< � �HJP R� .p02f 0 1� l � � � l�Ð�k=l�Ð).¹Ñm0`fxÑ��

� S TU ñ Цò`Ñ

ã R�T1A�P�B2A�B^Î�A�[�T�B�j7R�i`B^K=Ì1BnI3KdbcR7[�bcBnIÍ[�E`_�u.R�T�j t�Tc_aR�Tgi`B,E�b¦_MAcVXR�E5IÍ��R�[�E�i`BeiSt�I!T~I4j7B,A�B,T\bct�I�]^B�G����� �>�;:��$�����.�70n�)�� � ��)���$���\���������ã R�TOA�P`_dbµj7R¾i`B^K�BeID]�PÛ_Mj�I!|�BYB^KaB,j7B,E�A�TUB,t�T�BebcB,E`A~bgI¬t�R4_ME`AO_aE /�>��brt�Be]U_ ��Bei��`V-Ü/. � � . � � . � Ý=]^R`R�T~iD_aE�I!A�Bebk��Ì\P`_d]�P�Ì1B]^R�E�]U_dbcB^KaV©i`B,E�R�AUB���V��GxWOE�_aj7t`KL_«]U_aA2t�R!KaV`E�R�jY_dI3KBeÒ�[�I!Ar_MRDEX_db1T�B,t�T�BebrB,E�A�BeiSI�b

�)Ð� � ì=Ñ�� 1�N � � � N . ���� . �� .�� �� � � Ð^�e�`ÑÌ1_aA�P " 1 ��$ 1 ��� 1 � ��ñ Ð^���!Ñl P�B7]^R�E�bcA~I4E�A~bn" N �=$ N ��I!E�i�� N I!T�BYt�R�b¦_MAr_MCDBY_aE�A�B,|DB,T~b,��T�B,t�T�BebrB,E�Ac_aE�|XA�P�BYB^Î�t�R�E�B,E`A~bOR!u1BeI�]UP©C!I!Tc_dI!�`KaB�G l P�B� �[�bcBeiP�B,T�Bµ_ME�iD_d],I!AUBeb�AUP�I!A�Ì1B\P�I3C�BgI4EY_aj7t`KL_«]U_aA1u.[�E�]^Ac_aR�EXÌ1_aA�P�©I�b{A�P�B\C>I4Tc_dI!�`KaBeb,G í B\Ì1_LKLK/KdI!AUB,T�T�B^u.B,T�A�R��)Ð� � ìxÑ~�3Ì\P`_d]�PV�_aB^Kdi�b�I2E�R�E�Ó,B,T�R7C!I3Ka[�BY[�E`KMBebUb�S_db\RDEXA�P�Bnbc[�Tcu$I�]^B�G l P�B & #)( , # #1R!uA�P�B�t�R4KMV`E�R�jY_dI3Kxj7R¾i`B^K)_«bµA�P�B�jpI>Î�_ajY[�j�R!C�B,T� R!u " N �3$ N ��� N Gí _MAUPÔA�P�BXt�T�BebcB,E`AYu.R�T�jY[`KdI!Ac_aR�E���A�P�B,T�B©I4T�BpTUBei`[�E�i�I!E�A�T�B,t�T�BebrB,E�A~I4Ac_aR�E�bYR!u�A�P�B5bcR!Ka[�Ac_aR�E bcB,A~b¬Ð._$GhB�Ga��A�P�B,T�B5I4T�BjpI!E`V5t�I!T^I!j7B,A�B,T�C�Be]^AUR�T~bOA�P�I!Ani`BebU]^Tc_a��B�A�P�Bpb�I!j¬BÍbr[�Tcu8ID]^Bn_aE / � Ñ~GYq$A�_«b�t�T�R���A~I4�`KMB¬A�R5]�P�R`R�bcB7bcRDj7BYT�BebcA�Tc_d]^Ac_aR�ER!uA�P�Bnt�I!T^I!j7B,A�B,T�bct�I�]^B2A�P�I!A1u$I�]U_LKL_MA^I!A�BebOA�P�B�_ME`A�B,|�T^I!Ac_aR�E�b�_aE(Ð$á�Ñ~� ��[�AgjpI>_ME`A~I3_aE�bâu�[`KLKB^ξt�TUBeb�b�_aC�BYt�R!Ì1B,TeG í B2[�bcBA�P�Bp]^R�E�brA�T~I3_aE`A~b��3ì��A� ��I!E�i3� � � ����AUR©]^R�E�bcAUT~I3_aE-A�P�B�t�I!T~I!j¬B,A�B,Tnbct�I�]^B�A�R5IXP�I>K�u.Ú.P`V�t#B,T~bct�P�B,T�B���A�B,T�j7BeizA�P�B! �, #" #�)+#�,$" #%�& %'*#' &�Gl P�BOR���bcB,T�C!I!Ac_aR�EX]^R�E�b�_di`B,T�Bei�P�B,T�Bµ_db·I�u.[�E�]^Ac_aR�E�R!u�A�P�B´b¦_M|DE�BeiXiD_«brA~I!E�]^Beb�R!u=A�P�Bµt�R!_aE`A~b)(� u.T�R�j A�P�B�bc[�Tcu$I�]^Bi`B,A�B,T�jY_aE�Bei?��V(ì�xA�B,T�j7Bei�&*& $ ! ' �( #+" #+%�) $�G s B ��E�B -DÐ� � �)Ð� � ì=Ñ�Ñ�A�RX��B�A�P�BpiD_dbct`KdI�]^B,j7B,E`A7R!u1A�P�B7t#R!_aE�A�ÔA�RSA�P�Bbc[�Tcu$I�]^B7i`Beb�]^Tc_a��Bei?�`V5A�P�BYÓ,B,T�R5bcB,A�,-±î.�)Ð� � ìxÑ�Þ�0/�G l P�Bnu.[�E�]^Ac_aR�E3-DÐ� � �)Ð� � ì=Ñ�ÑxA~I!o�BebORDE©E�B,|`I!Ac_aC�BYC>I3Ka[�Beb�R�ER�E�Bnb�_di`BnR4uwA�P�BYbc[�Tcu$I�]^BnI!E�i�t#R�b�_aAc_aC�BnRDEXA�P�BnR�AUP�B,TeG

Page 5: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

í Bn]^R�E�b�_di`B,TOA�P�B�u.R!KLKaR!Ì1_aE�|7R���bcB,T�C!I!Ac_aR�E�bct�I�]^Bni`B ��E`_aAc_aR�E���I!E�i�R�AUP�B,T~b\I!T�B�j¬B,E�Ac_aR�E�BeiÕ_aE�Üa���3Ý$î� Ð � ìxÑ�� �

� P R Ü/-DÐ� � ��Ð � ìxÑ�Ñ.Ý� ñ Ð^�eà`Ñ

Ö RDA�B�A�P�I!A1Ì1B�[�bcB � _aE�bcA�BeI�iÛR!u=ê(Ì\P�B,E5AUP�B�R���bcB,T�C!I!Ac_aR�E�_dbgI¬b�],I3KdI!TeGW\KaA�P�RD[�|�P�Ì·B2P�I3C�Bni`B ��E�Bei�A�P�B2R���bcB,T�C!I!Ac_aR�E-bct�I�]^B�_ME�A�B,TUjpb\R4u�A�P�B'iD_dbct`KdI�]^B,j7B,E`A~b,�xIp]UKaR�brBei`Ú�u�R�TUj B^Î�t�T�Beb�b�_aR�Eu�RDT�A�P�BpiD_dbct`KdI�]^B,j7B,E`AYR!ugI�t�R!_aE`AnA�R5IÛt�R!KaV`E�R�jY_dI3Kâbr[�Tcu8ID]^Bpi`R�Beb�E�R�A�B^Î`_«brA�_aE�|DB,E�B,T~I3K$G í B¬[�bcBpIXiD_dbct`KdI�]^B,j7B,E`ABebcAc_ajpI!AUB�t�T�BebcB,E`A�Bei�_aE�ܺà��3Ý$î�-DÐ� � �)Ð� � ìxÑ�Ñ � �)Ð� � ìxÑ��� H �)Ð� � ì=Ñ � Ð^�eß`Ñ

l RYi`B ��E�B�A�P�BOE�R!_dbcB�j7R�i`B^K8�`Ì1BOB^ξt�T�BebUb�AUP�B´i`B,E�b�_aAcVp]^R�TUT�Bebct�RDE�iD_ME�|¬A�RnA�P�B�iD_dbct`KdI�]^B,j7B,E`A\R!uwI4EpR���brB,T�C�BeiYt#R!_aE�Au�TUR�j IX|!_aC�B,EÔbc[�Tcu$I�]^B�G í B7[�brBÕIÛt�T�R���I!�`_LKL_aA¦V?j7R¾i`B^K�u�RDTnT~I!E�|DB,Ú$b�],I!E�E`_aE�|�B,T�T�R�T�[�bcBei(I!E�iÛئ[�brAc_ ��Bei�_aE ܧ�3Ý$G l P�Bj7R�i`B^K)IDb�bcB,T�A~b{A�P�I!A1A�P�B�i`B,E�b�_aAcV���é�Ð)-�ëºìxÑ~��R4u=A�P�B�iD_«brt`K«ID]^B,j7B,E�AOR!u=I!EXR���brB,T�C�Bei¬t�R!_aE`A�u.T�R�j A�P�B�bc[�Tcu$I�]^B�� �)Ð� � ì=Ñ~�3_dbI�Ù´I4[�b�b�_dI!EXT~I!E�i`RDj�C>I!Tr_«I4�`KMBYÌ1_aA�PXÓ,B,TUR7j7BeI!E5I!E�iSbcR�j7B�o`E�R!Ì\EXC>I4Tc_dI!E�]^B��b< � G@!_aE�]^B�A~I!oD_ME�|YA�P�B�bc[�j R!uwbUÒ�[�I!TUBeb�R4uwÙ�I![�b�b�_dI!EXi`B,E�b�_aAc_aBebµV�_aB^Kdi�b1A�P�B�]�P`_aÚ$b�ÒD[�I!T�Bni`B,E�b¦_MAcV���AUP�B´i`B,E�b�_aAcV7[�b�_aE�|XÐ^�eà`Ñ_dbé¹Ð � ë�ì=Ñ ��� �� Ð � Ñ�� �

à ��� �� Ð)*�>à`Ñ � ��� � � � B ��� � � ñ Ð^�,á�ÑZ\B,TUB � _dbOA�P�B'br[�j7Ú.R!u.Ú$b�Ò�[�I4T�Beb1u.R�T�IY|!_aC�B,E�T�B,|!_aR�E�� Y��I!E�iSt�I!T~I!j7B,AUB,T\C>I3Ka[�B�ì��|!_aC�B,E��`V�Ð~�eà`Ñ^G�WµKdbcR�� Ð� MÑ_dbOA�P�BbcA~I!E�i�I4T~i�|�I!j7j�I�u.[�E�]^Ac_aR�E�I!E�i * � ë 7ë Ð$AUP�B�E�[�jY��B,TOR!u=B^KaB,j7B,E`A~bâ_aE 2Ñ~Gí BnI�b�b¦_M|DE7é�Ð�ìxÑ�A�RY��BYIY[�E`_Lu�R�TUj t�Tc_aR�TµR�EXA�P�Bnt�I4T~I!j7B,A�B,T1jpI4E`_�u.R!Kdi�G� Á�ÂOÃ�9��OÄ�C1ÃxÉ~Å� Å�� C´Â������ CµÄxÉ C1Ã�9��YÇ7ÂYÈÃxÉ~Å� Å���C�� ÇEC\Æ7Ä�CâÃ=É~Èq$ESA�P`_db\bcBe]^Ac_aR�E��#Ì1Bn]^R�E�b�_di`B,T´I4E�_ME`A�B,|�T^I3K=R!u=AUP�B�u.R�T�j

ï � Ð"!�Ð�ìxÑ�Ñ$# ì � Ð^���`Ñ_aE5Ì\P`_d]UP©å � / 1 I!E�i%!¬_«bOI7b�],I>K«I4Te��T�BeI3KaÚ.C!I3Ka[�Bei©ÒD[�I�i`T~I4Ac_d]µu�[�E�]^Ac_aR�E�î

& �'!/ÐUì=Ñ���ì)(+*Þì%�3$3ì%��� � Ð^�e�`ÑI!E�i � _db\IYt#R�b�_aAc_aC�BY]^R�E�Ar_ME`[�R�[�bâu�[�E�]^Ar_MRDE�Gí BnÌâ_�KLK�A�T^I!E�b�u.R�T�j A�P�B d Ú.C>I4Tc_dI!A�B�_aE`A�B,|�T~I>K{I!�#R!C�B�_aE`A�R5Ipb¦_ME�|4KMBÍ_ME`A�B,|�T^I3K=�`V©i`Be]^R�j7t#R�b�_aE�|5å _aE�A�RXbc[���bcB,A^bOR�EÌ\P`_d]�P � Ð"!�Ð�ìxÑ�Ñ\_db7I4t�t�T�R3Î�_ajpI!AUB^KMVÔ]^R�E�bcA^I!E�A3G l P`_db2_db7I�],]^R�j7t`KL_dbcP�Bei ��V�]^RDE�b�_di`B,Tc_aE�| ��Î�BeiÔC!I3Ka[�Beb,� & �xu.R�T,!/ÐUì=Ñ~�I!E�iXAUP�BnÒ�[�I�i�I4Tc_d]'br[�Tcu8ID]^Beb�_aEzå AUP�I!A\T�Bebc[`KaAµu�TUR�j�[�b�_aE�|�Ð~�e�¾Ñ~GZ\B,E�]^BYÌ1Bn]^R�E�b�_di`B,T�A�T~I4E�b�u�RDT�jY_aE�|7A�P�Bni`R�j�I3_aE5R!u_aE�A�B,|DT~I!Ac_aR�E�u.T�R�j ì�A�R & ��V�_aB^KdiD_aE�| ï � Ð & Ñ$#-! � � Ð & Ñ � Ð^�JV�ÑA�P`[�b\]^R!KLKdI!t�b�_aE�|XA�P�B Ö Ú$iD_Mj¬B,E�b�_aR�E�I3K_aE�AUB,|�T~I3K�_aE�AUR7R�E�B'iD_aj7B,E�b¦_MRDE�GÖ R!Ì�Ì1Bn]^R�E�b¦_«i`B,T�AUP�B'bcB,A\R4u·I3KLK=t�R!_aE`A~bâ_ME�A�P�B2t�I!T~I!j¬B,A�B,Tgbct�I�]^B2A�P�I!A\j�I!tX��B,AcÌ1B,B,E & K � �nI4E�i & K ЦbcB,B ã _M|D[�T�Bà`Ñ~î . K � ,�ì-î & K � ��/0!/ÐUì=Ñ21 & K /�ñ Ð^�e�`Ñq$E�I7bc[�j7jpI4Ac_aR�E���AUP�B'iD_43�B,T�B,E�Ar_«I>K5#6! � � Ð & Ñ=_dbOT�B,t�TUBebcB,E�AUBeiX�`VpA�P�BYH�B,��Bebr|�[�BYj7BeI�bc[�T�B¬Ð$R�TgI4T�BeI`ÑR!u . K G1Z\B,E�]^BÌ1Bn],I!EXÌ\Tc_aA�B ï � Ð & Ñ$#-! � � Ð & Ñ�� KL_ajI 798 � 7984:<;�O>=�? � K

� Ð & K Ñ�f=Ð . K Ñ � Ð^�eò`Ñ

Page 6: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

U

f(w )i

f(w )i-1

W

A i

ã _M|D[�T�B'à�î s Be]^R�j¬t�R�b�_aE�|�A�P�B�t�I!T~I4j7B,A�B,Tgbct�ID]^BO_ME`A�Rp]^RDE�]^B,E�A�Tr_«]�B^KLK�_at�bcR4_«i�bkG

_aE5Ì\P`_d]UP#fxÐ . K Ñ�TUB,t�T�BebcB,E`A~b1A�P�B2j7BeI�bc[�T�B�R4u . K G@!_aE�]^B !7_db\ÒD[�I�i`T~I!Ar_«]!� . K _db\I2�#R�[�E�i`Bei5bcB,A�_43 !/ÐUì=Ñ�� & K _dbµA�P�BOBeÒ�[�I!Ar_MRDEXR!uwI!EXB^KLKL_at�bcR!_di�G1q.u . K _db1[�E`��R�[�E�i`Bei��A�P�B,ES_aA´],I!E��#BÍbrB,B,E5_aE(Ð~�eò¾Ñ�A�P�I!A\AUP�Bn_aE`A�B,|�T~I>K=_aE(Ð~���`Ñ{_«bµ_aE���E`_aA�B;�)A�P�B,T�B^u.R�T�B��#Ì·BYI!T�B2R�E`KaV©]^R�E�]^B,T�E�BeiSÌ1_aA�P-],I�brBeb_aE5Ì\P`_d]UP !�Ð�ì=Ñ�� & K T�B,t�T�BebrB,E�A~bµI!EXB^KLK�_at�bcR4_«i�Gl P�Bnj¬BeI�bc[�T�B�R!u . K _db1u.R�[�E�iÛ��VXA~I4o�_aE�|7A�P�BYbcB,A�iD_>3�B,T�B,E�]^BYR!uAcÌ1RÍ]^RDE�]^B,E�A�Tr_«]2B^KLKL_at�bcR!_di�b2A�P�I!AgI!TUB�T�R�A~I!AUBeiXI!E�iA�T~I!E�b¦K«I4A�Bei7I3Ì\IeV�u�TUR�j×A�P�BµR�Tc_a|!_aE���I�b�i`B,t`_«]^AUBei¬_ME ã _a|�[�T�B�à�G�_1Be],I3KLK�A�P�I!A�A�P�B\C�R!Ka[�j7BOR!u=I!E�B^KLK�_at�bcB�_db�t�TUR�t�R�TUAc_aR�E�I3KA�R�_aA~bµI3Î�_db{KaB,E�|�A�P�b,G l R7]^R�j7t�[�AUB fxÐ . K Ñ~��Ì1B�]^B,E�AUB,T·AUP�BgB^KLKL_at�bcR!_di�b�I!A1A�P�BOR�Tc_a|!_aEXÌ1_aA�P�A�P�B^_aTgI3Î�Beb·I>K�_a|�E�Bei�Ìâ_MAUPXA�P�B]^R�RDT~iD_aE�I!A�B'I>ξBeb,Gm�Vn[�b�_aE�|nI!EYI3ËpE�B1A�T~I!E�b¦u�R�TUjpI!Ac_aR�E�R�E,!�Ð�ìxÑ�� & K Ñ~�!i`Beb�]^Tc_a�#BeiY_aEXÜ �!Ý$�3Ì1B�R���A^I3_aE'AUP�BâÒD[�I�i`T~I4Ac_d]u.R�T�j ì ó ( * ó ì ó ����`_aE5̵P`_«]UP *;ó�_dbOiD_dI!|�R�E�I3K$G l P�B�T�Bebc[`KaAc_aE�|XbcA~I!E�i�I4T~iD_aÓ,BeiXB^KLKL_Mt�brB7BeÒ�[�I4Ac_aR�EX_db� N Ð � óN" N Ñ � � � Ðcà��`Ñ

_aE5Ì\P`_d]UP " N � � �� N Ðcà��!Ñ

I!E�i � � � NÐ)$^óN Ñ �á�� N 0)��� & ñ Ðcà�à`Ñ

l P�B1C�Be]^A�R�T�� ó _«b�]^R�j7t�[�A�Bei2�`VnA�P�Bâ_ME�E�B,T�t�T�R�i`[�]^Ae� ����3_aE7Ì\P`_d]UP _dbA�P�B\]^R�T�TUBebct�R�E�iD_aE�|2jpI!A�Tc_LÎYR!u/B^_a|�B,E`C�Be]^A�R�T^bR!uA�P�B�jpI!A�Tr_�Î * GW\KdbcR#��� K T�B,t�TUBebcB,E�A^b1A�P�Bh�.Ú.AUPXB^_M|DB,E�C!I3Ka[�B'R4uwA�P�B�j�I!A�Tc_LÎ *×G l P�B�B^K�KL_at�bcB7CDR!Ka[�j7B�_db 1 1N � � " N Ðcà�ß`Ñ

_aE5Ì\P`_d]UP 1 �

D����F����G

à;: � 1 � ���d Ð d >à 0 �!Ñ�� _Lu d _«bâB,C�B,Eà � 1�� ��� � � � : � 1 � ��� � �d Ð d 0-à`ÑUÐ d 0�á�Ñ + - �� � _Lu d _«bâR¾i�i?G

Ðcà!á�Ñ

Page 7: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

l P�B©]^R`B^ËÕ]U_aB,E`A 1 ],I!E±�#B©i`B,A�B,T�jÍ_ME�Bei ��V(t�B,Tru�R�TUjY_aE�|­I!EÔB^KLKL_at�Ac_d],I3KÍ]^R`R�T~iD_aE�I!AUBXA�T~I!E�b�u.R�T�j�I!Ac_aR�E��Ì\P`_d]UPÔ_db�I|�B,E�B,T~I3KL_aÓeI!Ac_aR�E�R!ubcA~I!E�i�I!T^iXbct�P�B,Tc_d],I3K�]^R�R�T^iD_ME�I4A�Beb�Üa���3Ý$Gq$ESt�T~I�]^Ac_d]^B�Ì1Bn]^R�j7t�[�A�B2A�P�BO_aE�AUB,|�T~I3K�Ð^���`Ñ��VX]^R�E�b�_di`B,Tc_aE�|XI ��E`_aA�B'I4t�t�T�R3Î�_ajpI!Ar_MRDE5R!u=AUP�B'bc[�j _aE?Ð~�eò`Ñ~îï � Ð"!�Ð�ìxÑ�Ñ$# ì � ï � Ð & Ñ$#6! � � Ð & Ñ � �� KQ�9?

� Ð & K Ñ�fxÐ . K Ñ~ñ Ðcà��`Ñq$E�|�B,E�B,T~I>K=E�[�j¬B,Tc_d],I3K·Ò�[�IDi`T~I!A�[�T�BOu.R�T�jY[`KdI�bO],I!E©I>K«brRp�#BÍI4t�t`K�_aBeiq�)P�R>Ì1B,C�B,T3��Ì1BnP�I3C�BnRD��A~I3_aE�Bei5b�I4Ac_db�u8ID]^A�R�T�V�t�B,T�Úu�RDT�jpI!E�]^B��`VÕiD_aT�Be]^ArKMVX[�b¦_ME�|�A�P�BYbc[�j5Gí BnbcB^KaBe]^A�bcA~I!T�Ar_ME�|�I!E�i�B,E�iD_aE�|Xt�R!_aE`A~b,� & ? I!E�i & � _aE-Ðcà��`Ñ��V�jpI!oD_aE�|7A�P�BnI�bUbc[�j7t�Ac_aR�ESA�P�I!Aï 7��

��� � Ð & Ñ #6! � � Ð & Ñ � � Ðcà��`ÑI!E�i ï �

7��� Ð & Ñ$#-! � � Ð & Ñ � � ñ Ðcà�V�Ñ

Z\B,E�]^B2A�P�B�t�B,Tru�R�TUjpI!E�]^BnR4uwA�P`_db1j¬B,A�P�R¾iÕ_db\I 3�Be]^A�Bei��`VpA�P�B2T~I!A�B�I!A\̵P`_«]UP � I!t�t�TUR�I�]UP�Beb1A�P�B�R�Tc_a|!_aE�G� ; Å�ÂgÃ�9 C\Ä�=~Å ����C!D 91Æ É,ÂOÃ�9��OÄ�C1ÃxÉ~Å�Âq$ESA�P`_db\bcBe]^Ac_aR�ESÌ·Bni`BebU]^Tc_a��B7IYj7R�TUB�|�B,E�B,T~I3K=j¬B,A�P�R¾i¬u.R�T\P`_a|�P�iD_aj7B,E�b�_aR�E�I3Kx_aE�A�B,|DT~I!Ac_aR�E���̵P`_«]UP©i`R`Beb1E�R�AµT�BeÒ�[`_aT�BISt�I!T�Ac_d]^[`KdI!T¬u�R�TUj u.R�TYA�P�B�_aE�A�B,|DT~I!E�iq�1P�R!Ì1B,C�B,Te�xA�P�B�_aE�AUB,|�T~I!Ac_aR�E�bYI!TUB�KMBebUbYB^ËÕ]U_aB,E�A3G í B���T~brA7��Tc_aB� �V­i`BebU]^Tc_a��BA�P�BX|�B,E�B,T~I>KµFSR�E`A�BXy1I!TcKaR�_aE�AUB,|�T~I!Ac_aR�E(j7B,A�P�R�i�G-q�E @�Be]^Ac_aR�E±Ì1BXiD_db�]^[�b�b7I-brt�Be]U_dI3KL_aÓ,Bei±AUBe]�P�E`_dÒD[�B5A�P�I!AYR���A^I3_aE�bb�_a|�E`_ �/],I!E`A\_aj7t�T�R!C�B,j¬B,E�A1_aE�]^R�j7t�[�A~I!Ar_MRDE�I3K=t�B,Tru�R�TUjpI!E�]^BOu.R�T1A�P�Bnj7R�i`B^K�_aE`A�T�R�i`[�]^Bei�_aE-@�Be]^Ac_aR�E�à�G§ß�Gl P�B7FSR�E�AUB,Úcy·I!TcKaRY_aE`A�B,|�T~I!Ar_MRDE©j¬B,A�P�R¾iÛ_aA�B,T~I4Ac_aC�B^KaV©I!t�t�T�R3Î`_ajpI!A�BebOIXi`B ��E`_aA�BY_aE�AUB,|�T~I3K��VX[�E`_Lu.R�T�jYKaV-b�I!j¬t`K�_aE�|u�TUR�j AUP�Bâi`RDjpI3_aEYR!u�_aE�A�B,|DT~I!Ac_aR�E���I!E�iYI3C�B,T~I4|!_aE�|\A�P�B�u�[�E�]^Ar_MRDE7C>I3Ka[�BebI!AxA�P�B\b�I!j¬t`KMBebkG l P�B�_aE`A�B,|�T~I4E�i�_dbxA�TUBeI!A�BeinIDbIOT~I!E�i`R�j%C>I4Tc_dI!�`KaB�GW b�I4j7t`KL_ME�|��>I3C�B,T~I!|!_aE�|nb�]UP�B,j7B1VD_MB^Kdi�b�IOt�I!T~I!j7B,AUB,T=BebcAr_Mj�I!A�B1R!u�A�P�B1j7BeI!E��!RDTB^ξt#Be]^A�BeiYC>I>KM[�BR!u�A�P�BYT~I!E�i`RDj C>I!Tc_dI!�`KaB�G í BY[�bcBYA�P�BYj7R�bcAO��I�b�_d]7FSR�E`A�B¬y·I!TcKaR�j7B,A�P�R�i�GnFSR�T�BnB^KdI!�#R�T~I!AUBÍbU]�P�B,j7Beb�Ì1_aA�PSu8IDbcA�B,T]^R�E`C�B,T�|�B,E�]^B�T~I!A�Beb2I!T�BpiD_db�]^[�bUbcBei(_aE ܧà�V,Ý��P�R!Ì1B,C�B,Te�¹_Mj¬t�T�R!C�B,j7B,E`An_aE(A�P�BX]^R�E�CDB,T�|�B,E�]^B�T~I!A�BYu.R�T2A�P�BebcB�j7B,A�P�R�i�b_dbOt�R�b�b¦_M�`KaB�R�E`KaVSu�R�TµKaR!Ì\Ú$iD_aj7B,E�b�_aR�E�I3K\],I�bcBeb2Ð d 1 ß¾Ñ~G ã R�TOR�[�T�t�[�T�t#R�bcBeb,�xb�_aE�]^B7A�P�B7iD_aj7B,E�b�_aR�E?_«bOP`_a|�P(I!E�iSA�P�BE�[�jY�#B,T7R!u�E�B,Bei`Bei bUI!j7t`KaBeb7i`R`Beb7E�R�A7i`B,t#B,E�i±RDE±iD_Mj¬B,E�b�_aR�E��\A�P�BX��IDb�_d]5FSR�E`A�B5y1I!TcKaR�_aE�AUB,|�T~I!Ac_aR�EÔj¬B,A�P�R¾i?_dbI!t�t�T�R�t�Tr_«I4A�B�G ã RDTâI¬]^R�j7t`KaB,A�B�_aE`A�T�R�i`[�]^Ac_aR�E5AURpFSR�E�AUB,Úcy·I!TcKaR2_aE�A�B,|DT~I!Ac_aR�E���brB,BpÜa�eà3Ý$Gq$E2A�P�B·i`B,Tc_aC!I!Ac_aR�E7A�P�I4A�u�R4K�KaR!Ìgb,�!Ì1BA�TUBeI!A�A�P�B�T�B,|!_aR�EYR!u`_ME`A�B,|�T^I!Ac_aR�E'IDb=I1CDBe]^A�R�TxR!u¾T^I!E�i`R�j%C>I!Tc_dI!�`KaBeb,�`i`B,E�R�A�Bei���V� �!i`B ��E�Bei2R�E'I1[�E`_aA�]^[��#B�G=q8u/I\iD_43�B,T�B,E�AxT�B,|!_aR�E2R!u`_ME`A�B,|�T^I!Ac_aR�E�_db)E�B,Bei`Bei���I!E'I4t�t�T�R�t�Tc_dI!A�Bx_aE`A�B,|�T~I>K�A�T~I!E�b¦u�R�TUjpI!Ac_aR�E],I!EX��BYI!t�t`KL_aBei�G l I!oDB��5î ��� /�I!E�i�� (�� � G � l P�B�_aE�AUB,|�T~I3KÐ~�!Ñx_dbµT�B,t�T�BebcB,E`A�Bei5IDb

� Ð��¹Ñ � ï(��Ð��!Ñ$#���ñ Ðcà��`Ñ

l I4o�B'I�bcB,AOR!u ��_aE�i`B,t#B,E�i`B,E�A7b�I!j¬t`KMBeb�� � � � � � - + � � �w��i`T~I3Ì\ES[�E`_�u.R�T�jÍKMVSu.T�R�j�AUP�BÍbrt�I�]^B � G l P�B � 698 BebcAc_ajpI4A�BR!u � Ð��¹Ñx_«b� ! Ð���Ñ � �� !�KQ� � ��Ð"� #�Ñ~ñ Ðcà�ò`Ñ

m�V�A�P�B5brA�T�R�E�|SKdI3Ì�R!u\KdI!T�|DBXE�[�jY��B,T^b,� � � ! Ð��¹Ñ 0 � Ð��¹Ñ � � �-I�b�� � $ �1Ì1_aA�PÔt�T�R���I4�`_�KL_aAcVÔR�E�B�G y�R�E�b�_di`B,TA�P�B7C!I!Tc_dI!E�]^BXR!u1A�P�BpBebrAc_ajpI!A�B�� < �! �&%pÜ � ! Ð���Ñ�0 � Ð���Ñ�Ý � G ã T�RDjæЦà�ò¾Ñ~��Ì1B7R���bcB,T�CDB7��VSKL_aE�BeI!Tc_aAcV�AUP�I!A�%�Ü � ! Ð��¹Ñ$Ý2�%�Ü � Ð'��Ñ$Ý � � Ð��¹Ñ~G ã T�RDj AUP`_«bµR���bcB,TUC>I!Ac_aR�ESÌ1B�R���A~I3_aE5I!EXB^Î�t�T�Beb�b�_aR�EÛu�RDT\A�P�B�C>I4Tc_dI!E�]^BnR!uA�P�B�BebcAr_Mj�I!A�B7Üa�e�3Ý$�

< �! �(%�Ü � �! Ýb0-à�%�Ü � ! Ð��¹Ñ$Ý)%�Ü � Ð���Ñ$ÝJ�*%�Ü � Ð'��Ñ$Ý � � �� , � Ð�� � Ñ=0 Ü � Ð'��Ñ$Ý � /�ñ Ðcß��`Ñ;�+-,/.1012 A436587:9;7=<?>/@BAC<=@ED . AGFIHKJ

Page 8: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

l P`_db1_aE�iD_d],I!A�BebµA�P�I!AµA�P�B�B,T�T�R�Tâ_«bâT�Bei`[�]^Bei�I!A\IYT~I!A�B�R4u·�6 �xG����� �>�;:\���!���� ������!����:����$ �"( �����:\�#�qY���%�$ )����"����� ���l P�BpFSR�E`A�BXy·I4TcKaRX_aE�AUB,|�T~I!Ac_aR�E?j7B,A�P�R�iS_dbno`E�R!Ì\E-A�RSP�I3C�BpiD_LËÕ]^[`KaAcV-Ì\P�B,E(A�P�BY_aE�AUB,|�T~I!E�iz_db�t�BeI4o�Bei±Ü«�eà3Ý$G l P�BT�BeI�bcRDE5_dbOA�P�I4Agj7R`bcA�R!u�A�P�B7I!T�BeI�AUP�I!A´]^RDE�A�Tr_M��[�AUBebOA�RXA�P�B2_aE�A�B,|DT~I3Kx_«b2]^R�E�]^B,E`A�T~I!A�BeiÛ_aE�I�bcjpI3KLK�T�B,|4_MRDE���Ì\P`_d]UP5_dbjY_db�bcBei?��V�j7R`bcA�R!u\A�P�Bpb�I4j7t`KaBeb,G l P`_dbnP�IDbOKaBei-AURXj7B,A�P�R�i�b�A�P�I!AYI!A�A�B,j¬t�A�A�R5]^R�E�]^B,E`A�T~I!A�B¬A�P�Bpb�I!j¬t`KMBeb�_aE(A�P�Bt�BeI!oDBei5I!T�BeI¾Ð¦b^Ñ~�`I!E�i�_db1A�B,TUj7Bei)& " ! *�, )+ #% ( # $ *" ! ' & %1(DGã R�T�A�P�B\j7R�i`B^K�_aE5@�Be]^Ac_aR�EXà�G§ß ��Ì1B1_di`B,E�Ar_�u.VXI�T�Be]^A~I4E�|�[`KdI!T�T�B,|4_MRDE7_aE�A�P�B´i`RDjpI3_aEpR4u/_aE`A�B,|�T~I4Ac_aR�Ep̵P`_«]UPX]^R�E�A^I3_aE�bE�BeI!TcKaVpI3KLK�R!u�A�P�BOt�R!_aE`A~b�A�P�I!A�b�_a|�E`_ �/],I!E`AcKaV5]^R�E`A�Tc_a��[�A�BOA�RnAUP�B1_aE�A�B,|DT~I3K$G l P�BOT~I!E�i`R�j b�I!j7t`KL_aE�|2_db1R�E`KaV7t�B,Tru�R�TUj7Bei_aE�b�_di`B-A�P�B5TUBe]^A~I!E�|�[`KdI!T���R3Î��gI!E�i A�P�B5E`[�jY��B,TXR!u'b�I4j7t`KaBeb7T�BeÒD[`_aT�Bei _db�b�_a|�E`_ �/],I!E�ArKMV T�Bei`[�]^Bei Ð$��VÔIzu8I�]^AUR�T7R!uA�P�R�[�b�I4E�i�b�_aE5jpI!E`V�t�T~I�]^Ac_d],I3Kw],IDbcBeb^Ñ~Gã _aT~bcAYÌ1BÕ]^R�E�b¦_«i`B,T¬A�P�BÕ],IDbcBXR!u\B,C>I3Ka[�I!Ar_ME�|(A�P�B¬_aE�A�B,|DT~I3K�Ð~�!ѵu�R�T7bcRDj7BpTUB,|!_aR�E� Ìâ_MAUP­b�_aÓ,B �xG-q8ugÌ1Bp[�brBXA�P�BB^ξt�TUBeb�b�_aR�E�b�Ð~�eà`Ñ~�#Ð~�3ß`Ñ~��I4E�i�Ð~�,á Ñ~��I!E�iSI!E5I!t�t�T�R3Î`_ajpI!Ac_aR�E�u.T�R�j ܧà��3Ý$� A�P�BO_aE�A�B,|DT~I!E�iX],I!ES��BnB^Î�t�T�Beb�brBei5I�b

� Ð ì ( * ìì (�� ì Ñ � Ðcß��!Ñ

_aE-Ì\P`_d]�P0* I!E�i � I!TUB't#R�b�_aAc_aC�Bpi`B ��E`_aA�B�bcV�j¬j7B,A�Tc_d]YjpI4A�Tc_d]^Beb,��I!E�i � T�B,t�T�BebcB,E`A~bOA�P�B7]UP`_MÚ$bUÒ�[�I!TUBpi`B,E�b�_aAcV�G l P�BFSR�E�AUB�y·I!TcKaR7b�I!j¬t`K�_aE�|¬_db·I4t�t`K�_aBeiSA�RYInCDR!Ka[�j7BO_aE�A�B,|DT~I3K/AUP�I!A�_«bâR���A~I3_aE�Bei��`V7A�T~I!E�b¦u�R�TUjY_aE�|nA�P�BO_aE`A�B,|�T~I3K�R!C�B,T1A�P�Bt�I!T~I!j¬B,A�B,T´brt�I�]^B2_ME`A�R5I�C�R4KM[�j¬Bn_aE`A�B,|�T~I>KwR!C�B,TOA�P�BY[�E`_aAnP`V�t#B,T~]^[���B;�)A�P`_db�A�T~I4E�b�u�RDT�jpI!Ac_aR�E�_db2I7|�B,E�B,T^I3KL_MÓeI4Ac_aR�E-R!uA�P�Bnbct�P�B,Tc_d],I3K�]^R`R�T~iD_aE�I!A�B2A�T~I!E�b�u.R�T�j�I!Ac_aR�E�Üa���3Ý$Gí B�I!T�BB,C!I3Ka[�I!Ac_aE�|�A�P�B�]UP`_aÚ$b�Ò�[�I4T�B\I!A�C>I!Tc_aR�[�b)t�R4_ME`A~b)A�P�T�R�[�|DP�R�[�A�A�P�Bx_aE`A�B,|�T~I!Ar_MRDE�G í B1],I!E�A~I!o�B�brR�j7BjpI3Î`_ajY[�jC>I3Ka[�B�� � � � � �q< � ��bc[�]UPÔA�P�I!AYb�I!j7t`KaBXt#R!_aE�A~bÍA�P�I!AYV�_aB^Kdi±I�bc[�j¬Ú.R!u�Ú$bUÒ�[�I!TUBebnC!I3Ka[�BX|�T�BeI!AUB,T'AUP�I!E � ��]^R�E�AUTc_a��[�A�BT�B^KdI!Ac_aC�B^KaV�KL_aA�AcKaB�A�RYA�P�BO_aE�AUB,|�T~I!Ac_aR�E���b�_aE�]^B2B^_MAUP�B,Tgi`B,E�b�_aAcVÕI�brV�j7t�AUR�Ac_d],I3KLKaVÕI!t�t�TUR�I�]UP�Beb�Ó,B,TUR�G í BO[�bcB�A�P�B2y�R�T�E`_dbcP�Úã _dbcP�B,T7I!t�t�T�R3Î�_ajpI4Ac_aR�E±Ü§à��3Ý�A�RXA�P�BX]�P`_aÚ$b�ÒD[�I!T�BX]^[�jY[`K«I4Ac_aC�BÕiD_dbcAUTc_a��[�Ac_aR�E(u�[�E�]^Ar_MRDE�A�RXRD��A~I3_aE­I � �YI!A2A�P�Bpò�ò G§ò!A�Pt�B,T~]^B,E`Ac_LKaB�u�RDTnbcR�j7B��xG l P�BYKaB^u.A'b�_di`B�R!u\A�P�B7BeÒD[�I!Ac_aR�E-�#B^KaR>Ì T�B,t�TUBebcB,E�A^b�A�P�BpbrB,AnR!u\I3KLK·t�I!T~I4j7B,A�B,T�C>I>KM[�BebÍA�P�I!AV�_aB^Kdi�bc[�j7Ú.R!u.Ú$b�ÒD[�I!T�Beb�KaBeb�bµA�P�I!E � �3G Ö RDA�B�A�P�I!A1A�P`_dbâ_«bOIYbc[���bcB,AOR!uA�P�B�Tc_a|�P`Agb�_di`B�î� ì-î ì ( *Þìì ( � ì / � ��� � � ì-î ì ( * ì* "�.�Ð�ì ( � ìxÑ / � ���Ûñ Ðcß�à`Ñ

@!_aE�]^B �>ì �>� �D� * "1.¹Ð�ì ( � ì=Ñ ��* "1. B � !/Ð � Ñ � � � � Ðcß�ß`Ñ_aE©ÌµP`_«]UP3* "�. B � !�Ð � Ñ�TUB,t�T�BebcB,E`A~b\AUP�B'j�I3Î�_ajY[�j B^_a|�B,E`C>I3Ka[�B7R!u � G l P�B,T�B^u.R�T�B��#A�P�BYTc_a|�P`A´b¦_«i`B¬��B^KaR!Ì×i`Beb�]^Tc_a��Beb2IbcR!KL_di(B^KLK�_at�bcR4_«i��\]^B,E`A�B,T�Bei(I!A¬ì ���)�xÌ\P`_d]�PÔB,E�]UKaR�bcBeb¬I3KLKâAUP�Bpt#R!_aE�A~b�_aE(A�P�Bpt�I4T~I!j7B,A�B,TYbct�I�]^B�A�P�I!AYb�_a|�E`_ �/],I!E�ArKMV]^R�E`A�Tc_a��[�A�BnAUR'AUP�BO_ME`A�B,|�T^I!Ac_aR�E�î � ì�î ì ( *Þì

ì (� ì / � � � �� ì�î�ì ( * ì / � � � � � Ðcß!á�ÑH�B,A , � � � � � � ñeñ3ñ � � 1 /Yi`B,E�R�A�BnAUP�BnB^_a|�B,E`C>I3Ka[�Beb�R4u * ��_aESR�T~i`B,TOR!u=_aE�]^T�BeIDb�_aE�|XjpI!|�E`_aA�[�i`B�G1WµKdbcR�� KMB,A��Ôi`B,E�R�A�B2A�P�B]^R�T�T�Bebrt�R�E�iD_aE�|5B^_a|�B,E`C�Be]^A�RDTnjpI!AUTc_Lι��Ì\P`_d]UP-_db�IÛT�R�A~I!Ar_MRDE5jpI!AUTc_LΩA�P�I!AnI>K�_a|�E�b�A�P�B7B^KLKL_at�bcR!_di(Ì1_aA�P-A�P�B�]^R�R�T^iD_ME�I4A�BI3Î�BebOЦiD_dI!|�R�E�I>K�_aÓ^_aE�| * Ñ~G l I!oDBYì ����1�¾I4E�i�Ì·B�RD��A~I3_aE

ì ( *Þì ��Ð�����Ñ ( * Ð����Ñ���� ( � ( *���� ��� (�� � Ðcß��`Ñ_aE5Ì\P`_d]UP � � #1��"6! , � � � � � � ñeñ3ñ � � 1 /�G } b�_aE�|��·�`A�P�B�B^KLKL_at�bcR!_di�BeÒD[�I!Ac_aR�E5�#Be]^R�j7Beb� K � K�� �K / � � � �3ñ Ðcß��`Ñ

Page 9: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

l P�B�P�I3KLuKaB,E�|�A�P�bµR!uA�P�B�t�Tc_aE�]U_at`KMB¬B^K�KL_at�bcR!_di-I>ξBeb\I4T�B$ K � � � � � �� K ñ Ðcß�V�Ñ

l P�B�T�Be]^A~I!E�|�[`KdI!T7bc[���brB,AnR!u / 1 AUP�I!AnP�I�b2]^R�T�E�B,T~b�KaR�],I!A�Bei(I!A']^R`R�T~iD_aE�I!A�Beb�� $ K B,E�]UKaR�brBebnAUP�BpB^KLKL_at�bcR!_di��\I!E�i?A�P�BT�Be]^A~I!E�|D[`K«I4T\u$I�]^BebOI!T�B�A~I!E�|DB,E�A\AURpA�P�B2B^KLK�_at�bcR4_«i?bc[�Tcu$I�]^B�G í B'],I4E©I!t�t`KaV5AUP�B�_aE�CDB,T~bcBYR!uA�P�B'bct�P�B,Tr_«],I>K·]^R�R�T^iD_ME�I4A�BA�T~I!E�b¦u�R�TUjpI!Ac_aR�ESA�RXjpI!t�A�P�B7]^R�T�E�B,T^bOR!u·A�P�BY�#ReÎS_aE�A�R knG l P�BebrBYu�RDT�j I7TUBe]^A~I!E�|�[`KdI!TYbc[���bcB,Ae�qk ó ��R!u+k%_aE-Ì\P`_d]�PA�P�Bn]^R�T�E�B,T^b1P�I3C�Bn]^R�RDT~iD_aE�I!A�BebµÌ·Bni`B,E�RDA�B���V��6>à�� � K G �l P�B1t�R�TUAc_aR�E7R!u�t�I!T~I!j7B,AUB,T�bct�I�]^B1A�P�I!A�b�_a|�E`_ �/],I!E`AcKaVp]^R�E`A�Tc_a��[�A�BebxAUR�A�P�B\T�Bebc[`KaAc_aE�|�_ME`A�B,|�T^I3K�KL_MBeb{Ì1_aA�P`_aE kµód��P�B,E�]^BÌ1B�R�E`KaVXE�B,BeiXA�R7i`T~I3Ì b�I4j7t`KaBeb1[�E`_Lu�R�TUjYKaVXu.T�R�j k\ó.G } b�_aE�|�A�P�BebcB2T�Bebc[`KaA~b,��AUP�BO_ME`A�B,|�T^I3K�Ð~�!Ñ�],I!EX�#B']^R�j¬t�[�A�BeiX�`V���� � � ���� K � K � �K� ( � (�� ��� ñ Ðcß��`Ñ

Wg�#R!C�B����%T�B,t�T�BebcB,E`A~b1A�P�BOT~I!Ar_MRYR!uxA�P�BnI!T�BeI2R!u�k ó A�Rkn��I!E�i �Û_«bâA�P�BgE`[�jY��B,T\R4uwb�I!j7t`KaBeb,� �·�`A�P�I!A1I!T�BO[�bcBei�G l P�B�×I3KdbcR¬T�B,t�T�BebcB,E`A~b1A�P�BOu$I�]^A�RDT·�`VXÌ\P`_d]UP5A�P�B�E`[�jY��B,T\R!uT�BeÒD[`_aT�Bei�b�I!j7t`KaBeb�_dbµT�Bei`[�]^Bei�Gl R±]^RDj7t�[�A�B-I!E _aE�A�B,|DT~I3KO_ME;A�P�B-i`B,E�R�jY_aE�I!A�R�TXR4upÐ$á�Ñ^�\Ì1B5[�bcB?A�P�B-bcjpI3KLKaBebcAXT�Be]^A~I4E�|�[`KdI!TXT�B,|!_aR�E��nk ó ��ÐcI!E�i]^R�T�T�Bebrt�R�E�iD_aE�|�T�R�A~I!Ar_MRDE ��ÑR!uA�P�B�AcÌ·RYT�B,|4_MRDE�b �OI!E�i� �3Gq8u�A�P�I!A1T�B,|!_aR�EÛ_«b �µA�P�B,E5A�P�BO_aE`A�B,|�T~I>K�_dbg]^R�j¬t�[�A�Bei��V ���� � � ��� � K � K�� �K� ( � (�� � ��� ��� � ( � ( *�� ��� ( � (�� � ��� Ðcß�ò`Ñ_aE5Ì\P`_d]UP5A�P�B � K I!T�B�A�P�B�B^_a|�B,E`C>I3Ka[�BebOR!u2*���I!E�i ��_dbâ_MA^b1B^_a|�B,E�CDBe]^A�R�T\jpI4A�Tc_LιG� ���E7�91ÄxÉ�� 9\ÂOÃ�C =49��EC�� 7B= 92Dq$ESA�P`_dbgbcBe]^Ar_MRDEXÌ·B���Tr_MB� �VSt�T�BebcB,E`AgbcR�j¬BnbcB,|�j7B,E`A~I!Ac_aR�EXTUBebc[`KaA~b1A�P�I!AµÌ·B,TUB�R���A~I3_aE�BeiXÌâ_MAUP5I7]UKa[�bcA�B,Tc_aE�|5I3Ka|�R�Tr_MAUP�jA�P�I!A��#B,|!_aE�bYÌ1_aA�P�IÛt�I!T�Ac_aAc_aR�E-R4u·A�P�B¬_ajpI!|�BY_aE`A�R©brjpI3KLK�T�B,|!_aR�E�b,�I!E�i©_MAUB,T~I!Ac_aC�B^KaV-|�TUR�[�t�b�T�B,|4_MRDE�b�A�P�I!A�P�I3C�B¬A�P�BP`_a|�P�BebcAOt�T�R���I!�`_LKL_aA¦VSR!uP�R�j7R�|�B,E�B^_aAcV���I�bât�T�Beb�]^Tc_a��BeiS�`V5Ð�á�Ñ~G l P�BebrBnbcB,|�j7B,E`A~I!Ac_aR�E�bk��I!E�i�R�A�P�B,TµB^ξt#B,Tc_aj7B,E�A^bOA�P�I!AÌ1B�P�IeCDB�t�B,Tcu.R�T�j7Bei��#T�BeÒD[`_aT�BnP`_a|�P�Ú$iD_aj7B,E�b�_aR�E�I>KgÐ.u.T�R�j�ßYA�R�à��`Ñt�I!T~I!j¬B,A�B,Tgbct�I�]^B�_aE�A�B,|DT~I!Ac_aR�E�b1[�b¦_ME�|XAUP�B�j7R¾i`B^Kdbt�T�BebcB,E`A�Bei�_aE?@�Be]^Ac_aR�E©à�Gl P�B ��T~bcA2A¦Ì1R�T�R!ÌgbµR!u ã _a|�[�T�BB�XbcP�R!ÌÞA¦Ì1R�T~I!E�|�B2_ajpI!|DB'B^Î�t�B,Tr_Mj¬B,E�A~b2A�P�I4A�[�bcBYA�P�BY_aj7t`KL_d]U_MA�Ò�[�I�i`Tr_«]¬bc[�Tcu$I�]^Bj7R�i`B^KgÐ�@�Be]^Ac_aR�EÔà�G§ß¾Ñ~G ã _a|�[�T�Beb8� G§IXI!E�i ��Gºi-bcP�R!Ì×bcV`E�A�P�B,Ar_«]¬T�B,E�i`B,Tc_aE�|�b2R!u\A�P�B�ß s i�I!A~I5brB,A~b,G ã _a|�[�TUBeb8��Gh�(I!E�i��GhBXbcP�R!Ì I!E ЦI4[�A�R�jpI!AUBei#Ñ\_aE`_aAc_dI3K´t�I4T�Ac_aAc_aR�E±R4ugA�P�B�_ajpI!|�B���R�EÔÌ\P`_d]UP±A�P�B5]UKa[�bcAUB,Tc_aE�|-_db¬t�B,Tcu.R�T�j7Bei�G l P�BB��E�I>KbcB,|�j7B,E`A~I!Ar_MRDE7T�Bebc[`KaA~b,�`bcP�R!Ì\EY_aE ã _M|D[�T�Beb0��Gº]·I4E�i ��G ur��I!TUB\R���A~I3_aE�Bei7I>u�A�B,T�t�B,Tcu.R�T�jÍ_ME�|2A�P�Bg]UKa[�bcA�B,Tr_ME�|�I!E�i7I�b�_aj7t`KaB��R�[�E�i�I4T�V�KaR¾],I>K�_aÓeI!Ar_MRDEÔR�t�B,T~I4Ac_aR�E±Üa�e�3Ý$G ã _a|�[�T�Beb8� Gh|�Ú�_·bcP�R!ÌÞA�B^Î�A�[�T�BpbrB,|�j7B,E`A~I!Ac_aR�E-TUBebc[`KaA~b�R�Ez_ME`A�B,E�b�_aAcV�_ajpI!|DBebÐ$[�b�_aE�|�A�P�Bnj7R�i`B^Kx_ME(@�Be]^Ac_aR�E-à�Gºà`Ñ�A�P�I!AµÌ·B,TUBnR���A~I>_ME�BeiÛ��V5]UKa[�bcAUB,Tc_aE�|5R�E�_ajpI!|DBeb\A�P�I4AgÌ1B,T�BO_aE`_aAc_dI3KLKaV-t�I!T�Ac_aAc_aR�E�BeiÌ1_aA�P7b�ÒD[�I!T�B�|DTc_di�G ã _a|�[�TUBeb ��G ØcÚ�K¾bcP�R!Ì­brR�j7B·brB,|�j7B,E`A~I!Ac_aR�E�b)R!u�_aE�AUB,E�b�_aA¦V�_ajpI!|�Bebx�`VnI!t�t`KaV�_aE�|YIgÒD[�I�i`Tc_d]�t�I!T~I!j¬B,A�Tc_d]t�R!KaV`E�R�jY_dI3Kj7R�i`B^K=R!u1@�Be]^Ac_aR�E©à�G«�´iD_aT�Be]^ArKMVSA�RYA�P�BO_aE�AUB,E�b�_aAc_aBeb,G� �Å�Â'È2=,ÇED)É~Å�ÂEDí B�P�I3C�B�t�T�BebcB,E`A�Bei�_aE�A�B,|DT~I!Ac_aR�E2j7B,A�P�R�i�bxA�P�I!Axt�B,T�A~I>_MEYA�ROI\Ì1_di`Bg]UKdI�bUb�R!u/brA~I!Ac_dbcAc_d],I3K`_ajpI!|�B�j7R�i`B^K«bkGq$EYt�I!T�Ar_«]^[`KdI!T3�A�P�BebcB�j7B,A�P�R�i�b�P�I3C�B5�#B,B,E bc[�],]^Beb�b¦u�[`KLKaV I!t�t`KL_aBei A�R(A�P�BS_aj7t`KL_«]U_aA�t�R4KMV`E�R�jY_dI3K7bc[�Tcu$I�]^B5j¬R¾i`B^K$�OA�P�B�t�I!T~I!j¬B,A�Tc_d]Ð$B^Î�t`K�_d]U_aA,Ñ7t#R!KaV�E�R�jÍ_«I>K�bc[�Tru8I�]^BSj7R�i`B^K8�\I4E�i±I-F�I!T�oDR>C?T~I!E�i`R�j ��B^Kdi j7R�i`B^K$G l P�BebcBX_aE`A�B,|�T~I4Ac_aR�E±j7B,AUP�R¾i�b7I4T�BA���� 9 7 ����� 7��'@"<?>! #"�$ < �%� <���7�&�' >�@BA67)( < � <?9;7'@B7 � &*(C<+�'7:9 < , $,' 7 � "6@*&-'/. 7 @BA670" >�' @ A , (C7 � &*(6A 7 � 7 J%1KA67'>�@BA67)2 � &B@ <�3#'/&4'/& �5� " >!.�@BA <?@6'/&� " @*&-'/.67�3�@BA67 � 7'9 <�' >�' >! 87 8 < � 70&B7'@ @ � @BA679' � 9 <�3#' 9:" 9<; <�$/"67�3�=+>@?4J

Page 10: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

I�G ��G ]!G

i�G B�G u�G

|�G P�G _8G

Ø�G o�G Kã _a|�[�T�Bnß�î1@�RDj7BO_ajpI!|�BnbcB,|Dj7B,E�A^I!Ac_aR�EXT�Bebc[`KaA~bkG

Page 11: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

]^T�[�]U_dI3Ku.R�TgAUP�BÍm1I3V�Beb�_dI!E-]^R�j¬t�[�A~I!Ac_aR�E�T�BeÒD[`_aT�BeiÛ_ME?R�[�T�T�B^KdI!A�Bei�bcB,|�j¬B,E�A~I4Ac_aR�E5Ì1R�T�o-Ü«���3Ý$��̵P`_«]UP-[�bcB�Ð$á Ñ~G�@!_aE�]^BA�P�B7j7R�i`B^Kdb�t�T�BebrB,E�A�Bei©_ME @�Be]^Ac_aR�E­àXI4T�BYE�BebcA�BeiSu$I!jY_LKL_aBeb,�)_MA2Ì1R�[`Kdi���BY_aE`A�B,T�BebcAr_ME�|Û_aE�u�[�AU[�T�B7Ì1R�T�oXA�R5brA�[�i`V�A�P�BI!t�t`KL_d],I!Ac_aR�E�R!u1Цà`Ñxu�RDT·j¬R¾i`B^K�bcB^KaBe]^Ac_aR�E�G� ? È��ÍÂ'Å�� = 91Æ � 9�� 9µÂOà Dl P`_db\Ì1R�TUo7Ì\I�b1bct�R�E�brR�T�BeiX�`V Ö @ ã [�E�i`B,TO|�T~I!E`A��nq)_âq$Ú$ò����3��à�V>��G� 9���91Ä�9µÂ'È092DÜa�,Ý�F(G�Wµ_MAUo�_aE�G2.xR�bcA�B,Tc_aR�T\m1I3V�Beb�u$I�]^A�R�T^b,G����2*�"� #'� &)+ )�& $')�& ( #'� &*�(+& #�)+"!��m2��ß`Ð~�!Ñ^îa���������,á�à����eòDò���Gܧà!Ý s G�WYG�m�B,T�T�V��xq~G í GxvCD_MAUAe�=I!E�i _�G�.w_ME�]UP`_aE�GS@�A~I!Ac_dbcAc_d],I3K�_aE`u�B,T�B,E�]^B¬_aE­]^Tr_Mj¬B7_aE`C�BebcAc_a|�I!Ar_MRDE�b�[�b�_aE�|-i`B,R3ξV`Tc_aÚ��RDE�[�]UKaB^_d]7I�]U_diXt�TUR���KL_ME�|#G�����*�"� #'� &)+ �)�& $ )�& ( #'� &*�(-& #�)+"!��y�á���î¿á�ò�ò�� �Dß����`�eò�ò�à�Gܧß!Ý�Q�Gm�Beb�I!|�I!E�i .xGxQ�G�ÙgT�B,B,E�GS@�t�I!Ac_dI3KgbcA^I!Ac_dbcAc_d],bnI4E�i�m1I3V�Beb�_dI!EÔ]^R�j7t�[�A~I4Ac_aR�E�G����2*�"� #'� &)+ �)�& $')�& ( #'� &*�(+& #�)+"!�m2���`Ð~�4Ñ~î§à�����ß�V����eò�òDß�GÜhá�Ý .xG�Q�G�m�Beb�K·I!E�i`_�G{ygG/QDI3_aE�G7@�B,|�j¬B,E�A~I4Ac_aR�E�A�P�T�R�[�|�P�C!I!Tc_dI!�`KaB,Ú.R�T~i`B,TYbc[�Tcu$I�]^B ��A�Ac_aE�|�G���������� , #% $�� �)+)+#�,�%

! %� #'"�# �(%$ &�%�#&��%�)+#+' '"a���e�`Ðcà`Ñ~îa�e�!V'�/�eò`����F�I!T~]UP��eò�����GÜ �!Ý s G�F(G�m{KMR`R�j5G)( &�%�# �,*! ' (�#,+', � #% &.-!# *#" #�)+, "!G\y1I!jY��Tc_di`|�B } E`_aC�B,T^b�_aA¦V�.{T�Beb�b,���eò!V>��Gܧ�!Ýg_�G1F(Gwm�R!KLKaB©I4E�i s GmOG1y�R`R�t�B,TeG Ê\E R�t�Ac_ajpI3KLKaV±]^R�jY�`_aE`_ME�|(t`_aBe]^Beb�R!u\_aE`u.R�T�jpI!Ar_MRDE���Ì1_aA�P I!t�t`KL_d],I!Ac_aR�E A�RBebcAc_ajpI4Ac_aE�|5ß!Ú s ]^R�j7t`KaB^Î�Ú.R���ØcBe]^Agt#R�b�_aAc_aR�ESu�TUR�j T^I!E�|�Bni�I!A~I G��������/� , #% $�� )+)+#�, %0! % #'"�# ('$ & % #1� % )+#+' '"a��`Ð �¾Ñ~î§���eò2�¾��ß����`@�B,t�AUB,j'�#B,T'�3ò�����GÜ VeÝg_�GF(G=m�R!KLKaB5I!E�i(mgG{ygG�JB,jY[�Tc_$G?Ê\E�AUP�T�B,B,Ú$iD_aj7B,E�b�_aR�E�I3K´br[�Tcu8ID]^B7T�Be]^R�E�bcAUT�[�]^Ac_aR�E(j7B,A�P�R�i�b,G3�������4� , #% $�

)+)+#�, %.! % *'5�# ('$ & % #6��%�)+#+' '5«���3ß`Ð~�!Ñ~î«���/�eà���QDI!E�[�I4T�V©�3ò�ò���Gܧ�!Ý�mOG�.xGxy·I!TcKL_aE��=_�G�v\G�N�IDb�b,� ã G�Q�I3C�_aB,TnH�B,T~]UP���I!E�i�mOG=_�G�Zµ[�|�B,[�E�I!T~i�G8.{T�BeiD_d]^Ac_aE�|5Ì1R�T�oD_aE�|Xj7B,j7R�TUV�u8I3_LKa[�T�B�îW bc[�� ئBe]^Ac_aC�B�m·I3V�Beb¦_«I4E�I!t�t�T�R�ID]�P-AURXj7R¾i`B^K\bcB^KaBe]^Ac_aR�E�G3��7! " #�, & ( #%8 &)+ )�& $')�& ( #'9! $ $'*�(-& )�& *#%#����V�î§ß �eò��¾ß�à!V���eò��Dò�Gܧò!Ý�NYG�.xGhÚc@�Gxy�P�I!E�I!E�i?ygG)Ù'G)Ù'G#W\_aA�o�B,E�G2v�bcAc_ajpI4Ac_aR�E�R!u�A�P�BY��I3V�Beb,Ï u8I�]^AUR�T\_aE-I¬u.R�T�B,E�b�_d]Yb�]U_aB,E�]^B7t�T�R��`KaB,j5G:�

&)+ )�& $');�<0*#" !>= );� &�" = '5«��ß�ß�î§à4á�ò��¾à��4á����eò�ò��DGÜa�e�!Ý ã G)@�Gxy�R�P�B,E-I!E�i s G/mOG)y�R`R�t�B,TeGÍ@!_Mj¬t`KMB¬t�I!T~I3KLKaB^K·P`_aB,T~I!T^]�P`_d],I3K·I!E�iST�B^KdI3Î�I!Ac_aR�E-I3Ka|�RDTc_aA�P�jpbµu.R�T´brB,|�j7B,E`Ac_aE�|E�R�E�],I![�bUI3KwF�I!T�o�R!CD_dI!EXT~I!E�i`R�j ��B^K«i�bkG��������/� , #% $�� �)+)+#�,�%?! %� #'"�# �('$ & % #6��%�)+#+' '"a��ò`Цà¾Ñ~îa�eò��2�¾à��eò���F�I!T^]�P�eò��!V�GÜa���,Ý s G)ÙgB,jpI!E-I!E�i?@�G)ÙgB,jpI4E�Gn@�AUR¾]UP�I�bcAc_d]2T�B^KdI3Î�I!Ac_aR�E��xÙµ_M����b2iD_dbcA�Tc_a��[�Ac_aR�E�b,�xI!E�iSA�P�B7m1IeVDBeb�_dI!E�T�BebcA�R�T^I!Ac_aR�EXR!u_ajpI!|�BebkG*�������@� , *% $�� �)+)+#�,�%?! %� #'"�# �('$0&�%�#A� % )+#+' '"a���`Ðc�`Ñ~î V>à ���bVeá���� Ö R!C�B,jY��B,Tn�eòD�!á�GÜa�eà!Ý�F(G�Z2G�N�I3KaR�b\I!E�i�.xG�W'G í P`_aAcKaR¾]Uo�G�# *#% )+#B<0 �, ' *C# #�)D$ * &�$,G í _LKaB,V�� Ö B,ÌFER�T�o�� Ö EY���eò���� GÜa�eß!Ýg_�G�HG�N�I�bcP`V�I!t�I!E�i`_�Gxy�P�B^KLKdI!t�t�I�Gnv�bcAc_ajpI!Ar_MRDE©I!E�i�]UP�R!_d]^B'R4u�E�B^_M|DP��#R�T~bµ_aE-bct�I!Ac_dI3KaÚ�_aE�AUB,T~I�]^Ac_aR�E-j¬R¾i`B^KdbOR!u_ajpI!|�BebkG*�������@� , *% $����%'%'*�, " �)�& **%G��$ # * , "!��à�ò¾Ð~�!Ñ~î§�D���bV>à`��Q�I!E`[�I!T�V��eò��Dß�GÜa�,á�Ýg_�G\vµG�N�I�b�b�I!E�i;@�G·NYG�J�I3_di`V�I!E�I4A�P�I!E�G�WOt�t�T�R3Î`_Mj�I!A�B©m1I3V�Beb2u$I�]^A�RDT~b7I!E�i R�T�AUP�R�|�R�E�I3K\t�I!T^I!j7B,A�B,T~bk�1Ì1_aA�PI!t�t`KL_d],I!Ac_aR�E�A�R2A�BebcAc_aE�|7BeÒD[�I3KL_aA¦V�R!u=AcÌ1RY�`_aE�R�E�jY_dI3K=t�T�RDt�R�T�Ar_MRDE�b,G����*�"� #'� &)+ �)�& $ )�& ( #'H *�(+& #�)+"!��m2�!á�îa�3à�ò��/�,áDá���eò�òDà�GÜa���!Ý2@�G�F�G�H�I!J�I>K�KaB�GOW×m1I3V�Beb�_dI!EXu.T~I!j¬B,Ì·RDT�oYu.R�T´]^RDE�b�_di`B,Tc_aE�|Xt�T�R���I!�`_LKL_aA¦V-iD_dbcA�Tr_M��[�Ar_MRDE�bOR!u_ajpI!|�BYbcB,|�j7B,E`A~bOI!E�ibcB,|�j¬B,E�A~I4Ac_aR�E�b,G�F�I�bcAUB,TeϧbAUP�Beb�_db,� } E`_aC/G#R!u=q8KLK�_aE�R!_db,� } TU��I!E�I ��y�P�I4j7t�I3_a|�E�� s Be]^B,jY��B,T2�eò�ò�à�G

Page 12: KdIeC!I3KLKaB!fg],b,Gh[` a[ ]!GhBei`[ - LaValle: Motion ...msl.cs.illinois.edu/~lavalle/papers/LavMorHut93.pdf · u RDTâIÍ|!_aCB,E©iI!A^I'brB,Ae Ì\PB,EXA PB j¬R¾i`B^KdbOI!T

Üa�e�!Ý2@�G\F(G�H�I!J�I>K�KaB�I4E�i�@�G�WYG·Zµ[�A~]UP`_ME�brR�E�G%W ��I3V�Beb�_dI!E bcB,|�j7B,E`A~I!Ac_aR�E j7B,A�P�R�i`R!KaR�|�V u�R�T�t�I!T~I4j7B,A�Tc_d]�_ajpI4|�Bj7R�i`B^Kdb,G�� % ! , # ! �, �)�& *#%����3ò�ò�ß�GÜa�JVeÝ s G�JYG�H#_aE�iDKaB,V�G1W brA~I!Ac_dbcAc_d],I3Kxt�I!T~I�i`R3Î�G0��& *#" #�)+, &��� ���á�á#îa�e��V'���eò�à����eò ��V�GÜa�e�!Ý�v\G�F�I�bcT�VXI!E�iz@�G�y·I4j'��I4E`_«bkG l T~I!t#B,Ó,R!_di�I3K{FSR�E`A�B¬y·I!TcKaRY_aE`A�B,|�T~I!Ar_MRDE�G� �� ! # ���� = " #�, �! % #'"«��à�V�Ð^�!Ñ~î§à�à ���à!á�� � ã B,��T�[�I!T�V��eò�òD��GÜa�eò!Ý�F(G/Q G/FÛ_aT�ÓeI7I!E�i�NYG�H{G/m�R!V�B,TeG1WOEX_aE`u.R�T�jpI!Ar_MRDE5A�P�B,R�TUB,Ac_d]�T�R���[�brA´bcBeÒD[�B,E`Ac_dI3Kwt�TUR¾]^Bei`[�T�B2u.R�Tgbc[�Tru8I�]^B2j7R�i`B^KR�T~i`B,T7bcB^KaBe]^Ac_aR�E _aEÔE�R!_dbcV-T~I!E�|DBÕi�I!A^I�G5q�E , *�(,���������<0*#%�%% *#% <0**" ! ���& $ & *#% #%b& �)+); �# ( *�( «��t�I!|DBebß����2�¾ß�V����#�eò�ò�à Gܧà��!Ý s G�NYG0.I4E!ئÌ\I!E`_gI!E�i Ù'G�ZµBeI3KaB,V�G } E�bc[�t#B,T�C�_dbcBei bcB,|Dj7B,E�A^I!Ac_aR�EÔR!u�A�B^Î�A�[�T�BeiÔ]^R!KaR�T¬_ajpI!|�BebY[�b¦_ME�|(jpI!TUo�R!CT~I!E�i`RDj ��B^Kdi�b,G�q�E? , *�(2�������� <0*#%�%% **%3<0*#" ! ���& $ & *#% *%b&1 2 �)+);��# ( *�( «�!t�I!|�Beb V;V>��� V;V;V�� Ö B,Ì ER�T�o���Q![�E�B�eò�òDß�Gܧà��,Ý�HG�q^G&.xB,Ac_aAeGOm1IeVDBeb1u$I�]^A�R�T^b�u�RDTgR�[�ArK�_aB,T~bOj¬R¾i`B^KdbO[�b�_aE�|XA�P�BYi`B,C�_d]^B7R!u_ajpI4|!_aE�I!T�VXR���bcB,TUC>I!Ac_aR�E�bkGA��9! "E#�,�& ( #%

&)+ )�& $')�& ( #'�! $ $ *�(+& �)�& **%����!V�î �!á�� � �!á�á����3ò�ò�à�Gܧà�à!Ý�Q�G ã G1@!_LKaC�B,T�jpI!E�I!E�i s G�mgGµy�R`R�t�B,T3G m1I3V�Beb�_dI!E±]UKa[�bcA�B,Tr_ME�|(u.R�T7[�E�br[�t�B,T�CD_dbcBei BebcAc_ajpI4Ac_aR�E±R!unbc[�Tcu$I�]^B�I!E�iA�B^Î�A�[�T�B�j7R�i`B^Kdb,G)�������/� , #% $�� �)+)+#�,�%?! %� #'"�# �('$ & % #6��%�)+#+' '"a�#�e�`Ð$á�Ñ^îhá���à��`á�ò����!Q![`KaV��eòD����Gܧà�ß!ÝOWYG ã G�F(G#@�jY_aA�P�G·m1I3V�Beb�_dI!E5]^RDj7t�[�A~I!Ac_aR�E�I>K¹j¬B,A�P�R¾i�bkG* �$ & '")� , #% $���&� &*�(,�(�*#%b& �!2��ß�ß�V�îºß���ò��¾ßD���`���eò�ò���Gܧà!á�ÝOWYG ã G�F(G�@�jÍ_MAUP©I!E�izÙ'G�ÊnGq_1R��#B,T�A~b,G\m1I3V�Beb�_dI!E5]^R�j¬t�[�A~I!Ac_aR�ESC�_dI¬A�P�B7Ùµ_M����bOb�I!j¬t`KMB,T�I!E�iÛT�B^KdI!A�Bei�F�I!T�o�R!C]UP�I3_aE©FSR�E`A�BYy·I!TcKaR7j¬B,A�P�R¾i�bkG*���2*�"� #'� )+ �)�& $')�& ( #'� *�(+& #�)+"!��m2���`Ð~�4Ñ~î§ß��¾àDß����eò�ò��DGܧà��!ÝOWYG ã G/F(G)@�jY_aA�P-I!E�i s G/Q�Gx@�t#B^_a|�B^KaP�I3KaA�B,TeG7m1I3V�Beb1u$I�]^A�RDT~bgI4E�i©]UP�R!_d]^Bp]^Tc_aA�B,Tr_«I¬u.R�TµKL_ME�BeI4Tnj7R�i`B^Kdb,G1��7�2*�"� #'

&)+ )�& $')�& ( #'� *�(-& #�)+">��m�á�à�îºà��eß��¾àDà������eò��D��Gܧà��!Ý2Ù'G l I![��`_aE�I!E�i s G§mOG\y�R�R�t#B,TeG _1Be]^RD|�E`_aAc_aR�E I!E�i t�R`b�_aAc_aR�E`_aE�| R4u'ß s t`_aBe]^B,Ì1_dbcBÔI3Ka|�B,��T~I3_d]�RD��ئBe]^A^b7[�b�_aE�|v[�]UKL_di`BeI!E _aE`C>I!Tr_«I4E�A~bkG q$E 2, *�(2���*�,��$�$ * ! *#% )D$ #���% )+#e(�, �)�& *#% * %� = " #�, & ( #'! #% & &" "C+-**' & ( < *#" !>= )�& %1(

# #�)D$ * & $���@�I4T~I!A�R�|`IY@�t�Tc_aE�|�b,� Ö E'��Q4[`KMV-�3ò�ò���Gܧà�VeÝ2@�G�E�I!o�R!Ì1_aA�Ó���Q�G�vµG�NOTc_aj7j7B^K$��I!E�i ã G#@�Ó^_di�I!T�R!C�bco�V�G í B^_a|�P�AUBei5FSR�E`A�BYy·I!TcKaRY_aE`A�B,|�T~I4Ac_aR�E�G* �� !&# ��� = " #�,

! %� #'"a�����`Ðc�`Ñ~îa�eàD��ò��/�eß!��� � s Be]^B,j'�#B,Tn�eò�V!��Gܧà��!Ý�Q�G�ZnG��/I!T3G�WOt�t�T�R3Î�_ajpI4Ac_aR�E�b1u.R�TOA�P�Bnt#B,T~]^B,E`A~I!|�B2t�R!_aE`A~b\R4u�A�P�B']UP`_aÚ$b�Ò�[�I4T�Bei�iD_«brA�Tc_a��[�Ac_aR�E�G&��9! !#! '" &)+ �)�& $ );«�à�V�Ðcß`Ñ~î§à��D���¾à�ò�� ���eò!V>��G