Bisser Raytchev {tamaki,bisser,kin}@hiroshima-u.ac.jp, amano@is.naist.jp. R n R

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1 (MIRU9) 9 7 R R R 1 R, R,..., R R Bisser Raytchev {tamaki,bisser,ki}@hiroshima-u.ac.jp, amao@is.aist.jp 3 3 R R, R 3,..., R R R, R 3, R, R,..., R R R R,,, Abstract Ca R estimate a rotatio matrix R more accurately tha R? A method for estimatig a rotatio matrix R by usig R, R, R 3,... obtaied by a oe-shot measuremet Toru TAMAKI, Bisser RAYTCHEV, Toshiyuki AMANO, ad Kazufumi KANEDA Departmet of Iformatio Egieerig, Graduate School of Egieerig, Hiroshima Uiversity Kagamiyama, Higashi-hiroshima, Hiroshima, Japa Graduate School of Iformatio Sciece, Nara Istitute of Sciece ad Techology Takayama, Ikoma, Nara, Japa {tamaki,bisser,ki}@hiroshima-u.ac.jp, amao@is.aist.jp I this paper, we show that a more accurate estimatio of a 3 3 rotatio matrix R ca be achieved by appropriately decomposig higher-order rotatio matrices: R, R 3, ad so o. First we discuss a agle estimatio of a rotatio matrix ispired by the Electroic Distace Measuremet. The we reformulate the problem for a 3 3 rotatio matrix: if oise-cotamiated measuremet matrices R, R,..., R are give, fid a appropriate rotatio matrix R. I the proposed method, the give measuremet matrices are first trasformed to rotatio matrices by usig the polar decompositio. The the rotatio agles are obtaied by usig a eige decompositio of the rotatio matrices. Fially, the ambiguity of the obtaied rotatio agle is removed. Numerical simulatios ad pose estimatio experimets show that the use of R results i more accurate estimates tha whe R itself is used. Key words pose estimatio Rotatio matrix, Measuremet with higher frequecy, Electroic distace measuremet, View-based 1. R t 3

2 [1], [] F E R t [3] [1], [] ICP [4] [5] [6] [8] [9], [1] R R SO(3) [11] [14] R R R, R 3,... (Electroic Distace Measuremet, EDM) [15], [16] 3 3 R, R 3,... R R, R,..., R R 8 [] 8 1 R, R,..., R [17] 1 R R [6] [8] R R, R 3,... R, R, R 3, R R. 1 R R R 1 R θ R R θ R R R. 1 [16], [18] 1 Dist λ 1 1 θ 1 λ 1 > Dist Dist = θ 1 π λ 1, < = θ 1 < π (1)

3 1 Dist λ 1 λ k 1 k θ θ π θ = θ + π θ θ + π θ θ + πk θ 1 ( ) ( ) cos θ 1 cos( θ mi + πk ) k =,1 si θ 1 si( θ. (5) + πk ) θ 1 π/1 Dist λ 1 /1 λ (< λ 1 ) θ ( ) θ Dist = π + k λ, < = θ < π () k λ λ λ λ /1 k Total Statio Geodimeter Tellurometer [19] (beatig). R R R R θ R 1 cos θ 1, si θ 1 R cos θ, si θ 1 θ θ 1 θ θ θ R R θ 1 θ θ θ 1 1 < = θ < π λ 1 = π θ θ θ 1 λ = π θ = θ 1 π λ 1 = θ 1, (3) ( ) θ θ = π + k λ = θ + πk, (4) θ 1 θ + πk R R θ R θ θ θ θ λ = π/ ( ) θ θ = π + k λ = θ + π k, (6) k =, 1,..., 1 k ( ) ( ) cos θ 1 cos( ˆk θ = argmi + π k ) k si θ 1 si( θ + π k ) θ + π ˆk (7) R 3 3 R R ( = 1,,...) R 3 3 R R R R R R [], [1] mi R R F + µ R T R I F (8) µ

4 polar decompositio [] R R = U Σ V T R polar decompositio R = (U V T )(V Σ V T ). (9) polar part U V T (8) U V T 1 U V T R { U V R T if U V = +1, = (1) U (HV ) T if U V = 1, H = diag(1, 1, 1) polar decompositio R U V T V T V I G V T G 1 GV polar part 1 H V T H 1 HV H T H = I H = 1 U V = 1 polar decompositio R = (U (HV ) T )(HV Σ V T ) (11) 3. R R expoetial map [] cos θ = trr 1 expoetial map π θ π cos si ω = (b, c, d) T 3 expoetial map [] ω = (r 3 r 3, r 13 r 31, r 1 r 1 ) T, (1) r ij R ij R R 1 [1], [] R P D [1], [], [3] R = P D P H, (13) i = 1 b P = 1 c ω d c +d bc+id ω (c +d ) bd ic ω (c +d ) c +d bc id ω (c +d ) bd+ic ω (c +d ) (14) D = diag(1, e iθ, e iθ ) (15) P 1 D e ±iθ R P = D θ P H P R R = P DP H R P H P = I R = P DP H P DP H = P D P H P R P ω 1,..., ω media 1 1 ω med P med ω med = med ω i (16) i=1,..., media media 3. 3 [4], [5] Matlab S S = 1 1 i 1 i, S H S = SS H = I. (17) R = P D P H = P SS H D SS H P H, (18) = P D P T, (19) P = P S D = S H D S P 1 media

5 P P T = P T P = I D D = 1 cos θ si θ si θ cos θ. () θ D = P T R P 3. 4 R 1 R θ 1 θ θ ( ) ( ) cos θ 1 cos( ˆk θ = argmi + π k ) k=,1, si θ 1 si( θ..., 1 + π k ) ˆθ = θ + π ˆk (1) R 1 R θ 1 θ 1 θ 1 π k θ 1 θ 1 π 1 k 1 R = P med cos ˆθ si ˆθ P T si ˆθ cos ˆθ med () 4. R ( = 1,,...) R 1 (R 1 ) R R R R R R R [6] 1 4 agle error [deg] F orm 3 F orm ±.1 ± ±.1 ± ±.1 ±.5 P med P 1 1 R 1 R ±.1 ±.5

6 θ, ω ˆθ, ˆω 3 θ, ω (π θ), ω ω ˆω 9 π θ ˆθ, ω T ˆω < π agle error = θ (π ˆθ), (3) otherwise agle error [deg] [deg] R = 1 R 1 > 1 R 1 R,..., R 1 (±.1,.,.3,.4,.5) R, R 3,... R 1 R R R R 1 R 3 R media P med media R P med P 1 4 R R 1 R 3 P P 4. R R R 1 R 1 (R 1 ) R ±.1 ±.5 F orm ±.1 ±.5 R R R 1 R R 1 R R 1, R,... R [6] [8] [5] [9], [1] 3 R R, R,..., R 8

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