Map Generation of Mobile Robot by Probabilistic Observation Model Considering Occlusion

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1 Map Generation of Mobile Robot by Probabilistic Observation Model Considering Occlusion *, **, **, * Kazuma HARAGUCHI Nobutaka SHIMADA Yoshiaki SHIRAI Jun MIURA *,{haraguti,jun}@cv.mech.eng.osaka-u.ac.jp **,{shimada,shirai}@ci.ritsumei.ac.jp :. observation distribution of obstacle 2: Result of update when the robot gets the observation of the obstacle of substantial margin of error [, 2, 3, 4] Occupancy Grid Map Occuluded Area 2 2 A Obstacle : Occuluded Area

2 3.. 4 ( Marcov Chain ) 4: Lines Of View 4 j j + L l 2. j e l j = { Ej l, j} Ēl O l j ( 3) j + P (e l j, el j+ O) O o P (e l j, el j+ o, O) 3: Occupancy Grid Map m mathbfo P (e i O) l o l l n e n = {E n, Ēn} m l ( Ω l ) l E n Ē n O.,2 P (e l j, el j+ o, O) O P(e l j, e l j+ o, O) = α l P(o l e l j, e l j+, O)P(e l j, e l j+ O) (2) o P (e n o, O) α l = /P (o l O) (e i o, O) = α i P(o i e i, O)P(e i O) () P(e l j, el j+ O) P(ol e l j, el j+, O) P(o [2] l e l j, el j+, O) o i i α i = 3.2. /P (o i O). P(o l e l j, el j+, O) l 3. P(o l e l j, ej+,o) l = P(o l m l, e l j, ej+,o)p(m l l e l j,ej+,o) l (3) m l Ω l l j j + N l 2 N l

3 3.2.. j P (o l e l j, el j+, O) j j + P (o l Ej, l Ēl j+, O) = P(o l Fk)P(F l k E l j, l O) (9) P (o l E j l, El j+, O) P (o l Ej l, Ēl j+, O) P (ol Ēl j, El j+, O) P (ol Ēl j, Ēl j+, O) j P (o l Ej l, El j+, O) P (o l Ēl j, Ej+, l O) = P(o l Fk)P(F l k Ēl l j, O) + P(o l Fj+)P(F l j+ O) l (0) j P (o l Ēl j, Ēl j+, O) = P(o l Fk)P(F l k Ēl l j, O) 5: Occluded area behind the j-th grid (8) () P (o l Fk l j j + ) 5 j + P(Fk l j Ēl j, O) P(F k l El j, O) P(F k l O) O(2 N l ) j + P (o l e l j, el j+, O) P(o l Ej, l Ej+, l O) = P(o l Ej, l O) (4) O(N l ) j k { F k : E0 (k = 0) P (Fk l F k = El j, O) P (F k l Ēl j, O) P (F k l O) Ē 0 Ē... Ēk E k (k > 0) P(o l Ej l, O) F k F 0, F,..., F k,... P (Fk l El j, O) P (F k l Ēl j, O) P (F k l O) P(F k E l j, l O) =P(Ēl 0 Ēl, O)P(Ēl Ēl 2, O)P(Ēl 2 Ēl 3, O) j P (o l Ej, l O) = P(o l Fk, l Ej, l O)P(Fk E l j, l... O) (5) P(Ēl k Ek, l O)P(Ek E l j, l O) (2) P(F k Ēl l j, O) =P(Ēl 0 Ēl, O)P(Ēl Ēl 2, O)P(Ēl 2 Ēl 3, O) P(o l Fk l, El j, O) F k l (... P(Ēl k Ek, l O)P(E k Ēl l j, O) (3) k P(F k O)P(Ēl l 0 Ēl, O)P(Ēl Ēl 2, O)P(Ēl 2 Ēl 3, O) 6 k... P(Ēl k Ek, l O)P(Ek O) l (4) P(o l Fk, l Ej, l O) = P(o l Fk, l O) (6) P(Ek l O) l k + P(e l k, el k+ O) P(Ek O) l = P(Ek, l Ek+ O) l + P(Ek, l Ēl k+ O) (5) 6: Occluded area behind the k-th grid P(Ēl k El k, O) P(Ēl q Ēl q+, O) (0 q < k ) P(o l Fk, l O) = P(o l Fk) l (7) P(Ēl k E k,o) l = P(Ēl k, El k O) P(Ek l, El k O) + P(Ēl k, El k O) (6) (4) (7) j P (o l Ej, l Ej+, l O) = P(o l Fk)P(F l k E l j, l O) (8) P(Ēl q Ēl q+,o) = P(Ek l Ēl j, O) k+ j P (o l Ej l, Ēl j+,o) P (ol Ēl j, El j+,o) e P (o l Ēl j, Ēl j+,o) P (ol Fk l k+,j l = { ek+ l ek+2 l... ej 2 l ej } l ( Φ l k+,j ) ) P(Ek l El j, O) + k=j+ P(o l F l k)p(f l k O) () P(Ēl q, Ēl q+ O) P(Ēl q, Ēl q+ O) + P (El q, Ēl q+ O) (7) P(Ek l El j, O)

4 P(Ek E l j,o) l P(Ek l =, El j O) P (Ej l, El j+ O) + P (El j, Ēl j+ O) = P(Ek l, el k+,j, El j O) P (E ek+,j l j l, El j+ O) + P (El j, Ēl j+ O) pixel True disparity 7: Probability of the Φl 0 k+,j observation of the disparity 8: observation model k + j Q O(2 Q ) P(Ek l Ēl j, O) () P(Ek l El j, O) P(El k Ēl j, O) [2, 3, 4] l e l q el q+ cl q,q+ k c l q,q+ = P (El q, Eq+ O) l P (Eq O)P l (Eq+ O) l o l k = o E P (Eq O)P l (Ēl q O)P (Eq+ l O)P (Ēl q+ O) o l k = oē P (o l k el k ) j P(Ek, l Ej O) l = P(Ek O)P(E l j O) l + α k,j cq,q+ l (8) P (o l k = o E Ek) l = P (o l Fk) l q=k P (o l k = o E Ēl k) = P ( T ) j P (o l P(Ek, l Ēl j O) = P(E k O)P(Ēl l j O) α k,j cq,q+ l k = o Ē Ek) l = P ( T ) (9) P (o l q=k k = o Ē Ēl k) = P (T ) { P (o l Fk, l T )} α k,j = P(E k l O)P(Ēl k O)P(El j O)P(Ēl j O) 9 2 Q 0 Q 9(a) O(Q). P (o (2) (9) (8) l Ek l )/P (ol Ēl k ) 9(b) () (2) ( ) (+) cm cm P (o l Fk l) 0 T T P (o l Fk) l = P (T )P (o l Fk, l T ) + P ( T )P (o l Fk, l T ) (20) P (T ) = 0.9,P ( T ) = 0. P (o l Fk l, T ) P (o l Fk l, T ) (a) Likelihood (b) Result of map update 60) 9: Result of map update of previous(+) and our( ) method when the robot get the observation of the P (o l Fk l) ol =, 3, 0, 25 8 obstacle in the uniformly-distributed probablity area. (cell) The gray curve in (b) represents prior probablity. 5cm (8) () P (o l e l j, el j+ ) (20) P (o l Fk l) P (el j, el j+ O) (80 20 ) 3 (+) likelihood ratio probability

5 m i P (e i ) = P (e i, E m ) + (e i, Ēm) ( ) 0(b) m (a) Likelihood (b) Result of map update : Result of map update of previous(+) and our( ) method when the robot get the observation whose error is bigger than the distribution of obstacles The gray curve in (b) represents prior probablity ( ) 90% (20) 2 (+) ( ) A B C D E (a) D (376 ) ( ) (b) E (59 ) 5% 70% 0 X D 3(a) ( ) ( ) D E 3(b) X (a) Likelihood (b) Result of map update : Result of map update of previous(+) and 0 our( ) method when the robot get the observation in th occuluded area. The gray curve in (b) represents 4(a) prior probablity. 4(b) 3.4. Y D D D E 3(b) likelihood ratio likelihood ratio probability probability

C E C E B B A D A D (b) At E 2: Truth Of Obstacle Map (b) At E 3: Result of map

(b) At B (b) At E 7: Result of map update without considering 4: Result of map

8: Result of map update considering occlusion A 5 B C D B 5.

Thrun, Robotic Mapping: A Survey, in Exploring Artificial Intelligence in the New

(a) View from A in 5 (b) View from D in 5 6: Robot s view [2] J. Miura, Y.

Shirai, Mobile Robot Map Generation by Integrationg Omnidirectional Stereo and

6 C E C E B B A D A D (b) At E 2: Truth Of Obstacle Map (b) At E 3: Result of map update without considering occlusion D (b) B D 7(a) A D B 7(b) A D 8(a) A B 8(b) A (b) At B (b) At E 7: Result of map update without considering 4: Result of map update considering occlusion occlusion Y 4(a) Y 4(b) (b) At B : Result of map update considering occlusion A 5 B C D B 5. A C D B 5: Rough map of obstacles [] S. Thrun, Robotic Mapping: A Survey, in Exploring Artificial Intelligence in the New Millenium (G. Lakemeyer and B. Nebel, eds.), Morgan Kaufmann, (a) View from A in 5 (b) View from D in 5 6: Robot s view [2] J. Miura, Y. Negishi, and Y. Shirai, Mobile Robot Map Generation by Integrationg Omnidirectional Stereo and Laser Range Finder, in IROS, pp , [3] S. Thrun, Learning occupancy grids with forward 6(a) A sensor models, in Autonomous Robots, (b) B [4] H. P. Moravec, Sensor fusion in certainty grids for 7 mobile robots, in AI Magazine, Summer, pp. 6 8 (a) 74, 988.

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