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1 ! *1 19/08/ :! $%&' (&) 19/10/08 :! 01 (&) 0 (1 * 0 #$ %& '$ () *+, - #./ (NAMO) ( - (8 - $ NP-Complete NAMO. ( ( *+, #$ )+, ( #$ > - *.+) =+ );< :( 9 #$ *. *F '- $ % ( #F, F % F ( $ BC+ ) BD 'EA)?+@ NAMO?+@ N (+:- A+1 5 LH KG/ $ I : 'EA) (G)H - H+0 1C F GE P) Q # N8 ( N >O A+1 : *EA). N (1CF '-.0 % ( #F, F $ %.(R,R G LH KG/ I : #$ ( : #$ %& '$ () *+, - #./ ( *+, + #$ (X (> N. ( %) #G/ (+ #X %_ R NR % ( $ *+, : #$. O 0 I) #$ )( *+ O ) - (R (_ $ #G/.0 )' ' NAMO F, ':Q0a #_ ` & *E. :( %_ 11- ',D G) (F.D )'G+0 c 'EA) (b T9$ BG+ (S ' : ' _H $ (G0 )-: + $ B1+ G) ' #$ - H+0 d1 $ + S 51-1 SC - #$ %0 D - / *+, B. 'O SC T$ N8-0 (1 *.0 % ( #$ ( - (F9$ %.+) X (0> (0 - * 1 :( 18 sh.k.moghaddam@gmail.com Z[O %>E ' ) ' ) ' X (0> :(R$ FX ' ' >.111:(+O P@ +0 - masehian@modares.ac.ir :(R$ +R1O - Navigation Among Movable Obstacles - Visibility Graph,VG 5- Penetration Depth, PD 19 1

2 A) - (, : NR LP 1 N8 B, dorh ` - N R+ N -a o, ( [5] 8.0 a 0 $ N. >a N 9 CH (8.[] +, : HRP- G' #$ C, A *E ;. # 'a - fo %_ #CC< N8 N) [8,] 9 ` '-A) #X N A+1 8 _ > (LP.[9] : NAMO N8 N (sg+ + ;< - NAMO N8 *1$ [10] (S<. AGF + );< 0-0O 'a 1C * 8 A+1 ;C, #$ $ NR NS+ % N ;C, R $. ' N) % ( 0 > RG/ ` - 8 A+1. (G *Go > ( #/[D *+ (0 + );< NAMO [11] * 8 A+1 #$H * 0 A (G/ $ % NR (+$< 1C (G [11]. $ ( %_ *G) $ $ 'a _ (_ 'R $< #/[D + ( Q.0. O 0 > j I D ;< - +0 (>N9 < 9= 5 >?5= - b ( % $ *E* ' 0 (1 *.( O $a $ e0 )'G+0 (F.D [ $ N9 * - PO () I D ;< )#$ NAMO N.( % % :( AG& $ 9 (0 + $ O $ (+.( - ' - ( 0 #.b [1] fo.0 NP-Complete % * 0 T ' [] 'a - $ g BD () NR % 'a.$( G NP-Ccomplete +) $ A *E *1$ [] T, :( ` '-A) #X I) - - /G_ +.0 I$ F CH - ) '0 'a - fo $ 1 I) (G * B *.( c[ I) n) (- (+. + #$ RO $ * #$ (RG #$ $ o, - v 'E A E BC+ ( $ $ $a $ I) G0. I) G0 () 1$ 0 % ( Q >a (8. N A> (> j '$a 9 8 G' #$ [] SC k@ *-la SC -. % ( * $ (O $NG/?m$ 0 - (> N _ H+0 $NG/ # $ 1 ( # ;< (> ' > '/ n) *. NR ( ( *EA). (G *Go 9 ` 9$ #$ AC+ l. +, - ()$ : - N 9 [5] 8 LP 1 N8. LP 1 N8 % NAMO N8 ( ;C, ( -a o, A) l ` R$. N&+ R A o, F. ( Hm$ (:po *+, :. P) NAMO N8 _+ o, ;< - Resolution Complete - Probabilistically Complete 1- Navigation Motion Planner - Manipulation Motion Planner 15 19

3 N8 - NAMO ) %_ - * - ) N ( $ AC +0 $* -.0 +, : 9 (0 N8 0 ( D (+:- A+1 * N - (X N $ AC R$ - H ) N - fo $ _ E N8 '0 _ )N N8.( Q R j ) 0 ' (1) NR :'G) (0-$ (+:- A+1 * (,. (> 'R *F ;C, #$ (>+ $ 1$ F9 Q *.( A #X fy0 $ ( I) G0 + ( c[ ` A j@+ N (+:- A a (+ ' - ( N ( D :( R w. - )%:.#$ (>+ $ (+ RO *F.1 RO G0 *-la RO - *+,..I) $< #$ (G (0..0 A (> RO *F..0 $< #$.1$ *+, : F9 *F.5 *F (> RO A BC+..#$ ;0 'a $ *+,) $ )%: z, )%: ' - G) - ( A (> RO *F *+, : ` +0O.+) + '/ I : - 1C * #$ o, fy0.0 H+0 C LH KG/ %>H I) RO G0 j@+ % (> F9 ( G. : *F ( $ I) G0 D@ >EFGH 5 < 9=-1- I 1 $ + ( 'R '/ n) :(Bi) $J.( j< + ;< - (8 (/ $ 9 :(Mj) M ( (X $ ( #X* l. $ +, : >a #$.( *0 (1) S K.D C -a o, :(Cfree) #$ >a () RO /G_ - 0 #./. N - n) C free = C \ ( UCBi ) (1) i 0 () RO /G_ :cl(cfree) OP) A% % N LH n) '$ #$ >a.( O G A E CC o, * :(C-obstacle) Q A% C- RO :.0 C free cl(c free) AG+ DC A E #$ 9$ obstacle ( S).( LH % 'a CC $ + CB = { q A( q) B } () $ b % RO o,) o, 1C * (>+ $ (+ ) RO Fg$ $ ( +0O $ F $ #X $ k@ N.9 - #$ (1C+ # w, X % 0 9 #$.0 (G. % P E 9 $ > $ w_ (. A ( ' ;C, `<1 ) #$ NAMO RS RE 'TQ <U&V! - CC NAMO - #[R - (R *+, ( j< 1C * (X I) - (R )'R F P) Q0 R+ c$ x o, ' - % #CC< ( - %>1 $ PO 1C.0 _+ - Roadmap 1- Fixed Obstacles 19 1

4 Q Z [U% TU Q U & >5& $8E 8 P OQ $O '8 $'5 M < T Q [ C) $O '8 '5 & >5& $8E 8!5 [U%W $ C\ Z [ C) T Q M < XU N [ C) $ 1$ *+, : F9 '0 'R. I) 0 1$ *+, : F9 d1 *F (> F9 A (_ 'R (_ A $ $ 'a (G SC d1 0 ` A. I) 0 0 (R $ A 0 1$ *+, : F9. $ 'a (>+ RO A BC+ UW <U&V! 8 X :(1) R () S K.D.0 )C-obstacle 8 %G $ #$ $ +) (GC+ );O ( N&+ A) v j $ v i e ij I : * )B R :.[1] eij 0 sv i + (1 s) v j cl ( Cfree ), s [0,1] () R : $ * (SO #./ #$ RO -a o, N #X ( N&+ &V' YW-1- )&@.0 C I : (X 'a ) : 0 * I : ( N&+ A) B R +0 #$ RO -a o, {C %G *EA) $.+) I : ) : - : N9 F $ RO o, +0 I : I : ) :.(? F F+ )(FgE N (> RO #$ (+ RO N v i 1 19

5 - F $ N.9 I : - (G () NR.)( ' ;< A *+, ` '(G 0 k@ NR (?1) G9 D'G)., (>+ RO (+ RO - n) )(FgE T[g (+.0 9$.( +, : ` I : )B '/ ^= _P` -- ' *F X, F (PD) LH KG/ ' *+ +.0 DC+ % * DC 'R * LH KG/ #$ DC N+ RO ) X, d:( + (0 9$ CB N #$ - RO) C free #$ - RO) -a RO *..0< (0 9$ Q $ P j< (FgE $ *+ + #X * )' σ ( P, Q ) a d > $ d d +0 (+ Q 'a 0 (+, - LH R N Q $ P - (CD (FgE $ * LH KG/ σ d ( P, Q ) P. R LH KG/ N9.( d +0 Q $ σ ( P, Q ) N9 CC d *RG KG/ % σ( P, Q).0 Q $ P * σ ( P d, Q ) d #> %G *+, : `.[1] ( + 1 (1C+ LH LH KG/ PD - +, :%_ )1F, + B $ A DC+ A $. G (PD t ) (1C+ t :(? F - NR () S PD (A) % - (R BG/ BC+ * +G.[1](B) A - 'a -0 t PD ( A, B) = min{ d int erior( A+ d) I B = } () ( )A+1 t PD :- +0 * +.0 j< l $ j< % * O Minkowski Sum B, O )A+1 t *+, 0 PD.0< CC. SC Minkowski Sum ~S0 SC * B $ A ) - /G_ :.[1] 1- Translational penetration depth %_.[1]&V ' YW Z P :() R O n ( log( n)) ( I : 0 * '( I : 0 - fo. #$ (>+ $ (+ ) RO ' +R $ A* 'EA) (_+ )c$ - H+0 PG I : () NR.,.0 % % ' - ( $ % N #$ o, % : 9 < *EA).+) b % 0 (R + o, * $ I) F9 '0 -a n) % %G C-obstacle 8 : (F * I : 0 ( $ b - A/) d1 $ )@ A) +0 I : : 9 `. O #$ N - / '(G I : ) : 1C * +, : c$ (> RO #$ (+ RO N % - A/ ) C-obstacle %G 8 $ #$ ) : B& +0.0 $ b o, % (, R % *+, `. $ #$ -./ )B ( }/ ;< 9 ` I : % N. : 19 18

6 o, 9 $ L1 ; O Minkowski Sum (F ' M1 SC. M1 B #$ RO Minkowski () G9.0 Minkowski Sum - X, * +G NX Sum G9.0.0< (d) LH KG/ N9 'G) Minkowski Sum 'G) d M 1 )' ().0 d > $ M1 A )ao bu - ;0 # - +, : N8 G 1C * ()+, ;C,. -0. MATLAB,% # C >a C 'R % F $ NAMO N '- $ % ( #F, N F c$ - H+0 ( ) D'G).+) d )+, - (R N9 1C * 8 N.0 +,.> # m+ +,. /G_ S K.D >a Minkowski Sum A B = { a+ b a A, b B} : 0 $ 0 ' () NR D'G) (5) Minkowski Sum $ RO o, % >H. (R ($ CC RE UO& &' ^= _P` 5 &V' YW '=UO-- NAMO RS {. N.9 )P@ 8 #<g 0 k@ Minkowski Sum $ LH KG/ G0 j@+ - (+G9 DC #X '( _ A E (> RO. I D ( * +G DC * A $ o, (?1) G9 (5) NR - L1 ; O $ 0 PG A * a M1 A j@+ L1 ( CB M (j) G9. DC 0 (@+ 1) (j) (?1).Q<9 &V' YW Z P (c).q<9 &V' YW Z P (B!) :() R ' e% A < >5& 8 Q A% B 5 A < 5' d) P :() R.[1B] XE A < Q\ Minkowski Sum

7 L1 M1 (j) (?1) d () M1 5 L1 O g [UP8.(h). d >C%' R8 'C) M c b a 1 d ( M1) L1 ( CB M.(c).'&' d) 1) ().(').$O '' ip d >j &' ' P RS RE bu :(1) X5 c 'C) M b : Not Given a: Our method (Intel Core i5.50ghz with GB of memory) b: [5] Pentium, GHz) c:[] (Intel Pentium M, 1.Ghz) d: [9] (1. GHz Intel T5500 processor with 1GByte of memory) e:[8] (Pentium.0GHz with 1GB of memory) a 5 1 e 9 d 0 / 01 0 / 0 M1 < L1 f & :(5) R ( J) c 0 / / 08 0 / 05 0 / 10 b / 5 9 a 0 / 09 1 / 1 / 9 / 9 0 / 00 0 / 5 0 / 809 / 8 0 / / 0 / 98 5 / 'C) M

50 P)$ O * [5] [] [9] [8] 0 RE 0 0 10 0 11 1 1 M 'C) 5.! [ &' RS RE 5 > ' &' ' RS RE :() R P)$ O * [5] [] M 'C) 5 1 0 1 M 'C).

8 50 P)$ O * [5] [] [9] [8] 0 RE M 'C) 5.! [ &' RS RE 5 > ' &' ' RS RE :() R P)$ O * [5] [] M 'C) M 'C).! [ 5 > ' &' ' RS &' 'C) :() R 9 M >C%' R8 'C) P)$ O * [5] [] [9] M 'C) 9.! [ 5 > ' &' ' RS &' >C%' R8 'C) :(8) R 1 19

9 *+, : 9 9 -a o, - $ B& B ) (0 $ w[.9 B0 A $ #$.( (0 ` +0O #X #$ o, 1C * $ I : %>H $ - H+0 $ 0 +, : % (> RO c[ LH KG/ *G) +) #$ H+0. : k@ +G '- I X $ +0O o, A (> F9 *+, ( LH KG/ %>H - - H+0 E 0 Q (0 $ A %: %: ( - %>H *. $ v { Q - [ 1] Wilfong, G., Motion Planning In The Presence Of Movable Obstacles. Annals of Mathematics and Artificial Intelligence,, : p [ ] Demaine, E.D., M.L. Demaine, and J. O Rourke, pushpush and push1 are hard in d. in In Proceedings of the 1th Canadian Conference on Computational Geometry,, 000: p [ ] Chen, P.C. and Y.K. Hwang, Practical Path Planning among movable obstacles. Proceedings of the IEEE International Conference on Robotics and Automation, 1991: p. -9. [ ] Okada, K,.et al., Environment manipulation planner for humanoid robots using task graph that generates action sequence Proceedings of 00 1EEElRS.J International Conference on Intelligent Robots and Systems, 00. [ 5] Stilman, M. and J.J. Kuffner, Navigation Among Movable Obstacles: Real-Time Reasoning In Complex Environments. International Journal of Humanoid Robotics, 005.(): p [ ] Stilman, M., et al., Planning and Executing Navigation Among Movable Obstacles. in IEEE/RSJ Int. Conf. On IntelligentRobots and Systems (IROS 0), 00: p N j+ '- # E ) F (G #CC< 0. 1C * (1$ 8 )A+1 9 (.0 ()'- 0 9 ( D (+:- A+1 N '- ( (+ $ N N8 BCF $ () NR D'G).( >a R$ - H+0 1C * 0 ' (8) ( #F, F $ % F (@+ N8. $ R = %.+, P) QP 5 WM U-5 % ' - ( (0 1C * % - :'G). + O (NAMO) $ ( Hm$ $ NAMO 0 k@ CC. %_ ' (+ % ( 9 0$ ) $ #$ - * H+0 $( `+ 0 % ( - % $NG/ $ ( ', - '-A) -. (D k@ I) *-la SC k@ +@A) $ ƒ );< * - fo '( (E* );< - G.( ',D $ 11- N0 'EA) (F.D 9$ ( T9$. ) 'R ' %F - ( _+ - o, (X #[R - (R 0 % (> N1 *G).( j< NAMO - $< % o, ' )$ O $< _ #$ o, ' $<.(. ) j o, ' 1C * $ )P)$ O - n) (> 'R 0 +, : #X >a $ + G R (G k@ A %: %: #X A _+ c$ - H+0 { ' 9 fy0 $ v > - Solution Space 19

10 [ ] Stilman, M. and J.J. Kuffner, Planning Among Movable Obstacles with Artificial Constraints. In Proc. th Int. Workshop on the Algorithmic Foundations of Robotics, 00: p [ 8] Nieuwenhuisen, D., A.F.van der Stappen, and M. H. Overmars, An Effective Framework for Path Planning amidst Movable Obstacles. Algorithmic Foundation of Robotics 008.VII: p [ 9] Berg, J.v.d., et al., Path Planning among Movable Obstacles: a Probabilistically Complete Approach. Workshop on Algorithmic Foundation of Robotics (WAFR VIII), 008: p [ 10] Wu, H.N., M. Levihn, and M. Stilman, Navigation Among Movable Obstacles in Unknown Environments. in IEEE/RSJ Int. Conf. On Intelligent Robots and Systems (IROS 010). [ 11] Levihn, M., Navigation among Movable Obstacles in Unknown Envrionments, 011, Georgia Institute of Technology. p. 8 [ 1] Choset, H., et al., Principles of Robot Motion-Theory, Algorithms, and Implementation, 005, Cambridge, Massachusetts: The MIT Press. [ 1] Dobkin, D., et al., Computing the intersection-depth of polyhedra. Algorithmica, 199.9: p [ 1] Zhang, L., et al., Generalized penetration depth computation. Computer-Aided Design, 00.9(8): p

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