[rad] residual error. number of principal componetns. singular value. number of principal components PCA NLPCA 0.

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1 ( ) ( CREST) Reductive Mapping for Sequential Patterns of Humanoid Body Motion *Koji TATANI, Yoshihiko NAKAMURA. Univ. of Tokyo , Hongo, Bunkyo-ku, Tokyo Abstract Since a humanoid robot takes the morphology of human, users will intuitively expect that they can freely manipulate the humanoid extremities when try to control as pilots. However, it is not realized with simple devices because it is dicult to issue multivariate control inputs to the whole body at a time. On the other hand, a small number of control inputs, such as a cam in wind-up mechanical doll, can generate various motion patterns of extremities. It is useful for motion pattern generation to get mapping functions bidirectionally between multivariate control inputs to a humanoid robot and a few control inputs that a user can intentionally operate. From a standpoint of multivariate analysis, by executing principal component analysis (PCA) in joint angle space, motion patterns are converted into low dimensional variables. The problem is to nd convenient variables not only for specic motion like walk but also multiple whole body motion patterns. This paper presents the results that 1 dimensional inputs can generate walking pattern and 3 dimensional inputs does 9 types of motion patterns with hierarchical nonlinear PCA (NLPCA). Keywords: Dimensionality Reduction, Nonlinear Principal Component Analysis, Internal Representation, Motion Pattern, Humanoid Robot 1. 1) ZMP 2) 3) 4) NLPCA (NonLinear Principal Componet Analysis) 5, 6) NLPCA NLPCA NLPCA J R N x 2J

J OJ = fx 1 x 2 g OJ RR M J OJ R OR Fig.

The rst half layers take role of injection and the latter half ones do projection. Fig.

^x i =^g(x i ) (1) x i OJ i, ^g e i = k^x i ; x i k 2 w kk x i 2 R N N M x 0 i 2 RM ^x

(x) f(w i2 f(w i1 x)) (2) ^x = h p (x 0 ) f(w p2 f(w p1 x 0 )) (3) w i1 w i2 w p1 w p2

R (0 1) M = k (2) x i x 0k i = (x i 0 x 0 1 i x 0 2 ik ) x0k i (3) ^x k i E k E k = 1

20 number of principal components [rad] residual error 1 0.8 0.6 0.4 0.

2 J OJ = fx 1 x 2 g OJ RR M J OJ R OR Fig. 1 The left/right side layer denotes input/output layer of NLPCA neural network. The rst half layers take role of injection and the latter half ones do projection. Fig. 1 OJ g. ^x i =^g(x i ) (1) x i OJ i, ^g e i = k^x i ; x i k 2 w kk x i 2 R N N M x 0 i 2 RM ^x i 2 R N ^g ^g = h i h p e i OJ S (injection) h i, S OR OJ (projection) h p x 0 = h i (x) f(w i2 f(w i1 x)) (2) ^x = h p (x 0 ) f(w p2 f(w p1 x 0 )) (3) w i1 w i2 w p1 w p2 a 0 = f(a) a 2 R K a 0 2 R K (2)(3) x i x 0 i ( OJ R singular value OR f (0 1) (0 1) (0 1) 22 NLPCA PCA NLPCA M =1 PCA R (0 1) M = k (2) x i x 0k i = (x i 0 x 0 1 i x 0 2 ik ) x0k i (3) ^x k i E k E k = 1 n nx i k^x k i ; x ik 2 (4) n E 1 x i 0 2 k number of principal components [rad] residual error PCA NLPCA number of principal componetns Fig. 2 The left is the semilog plot of singular value of motion pattern walk. The right shows the norm of residual error to compare reproduction power between PCA and NLPCA. The unit is The vertical error bar on NLPCA shows the dispersion of ten trials and the asterisks are their average.

sensory pathway descending pathway O J Fig. 3 The design of a hierarchical NLPCA. Two pathways are separately illustrated based on the dierence of functional aspect.

3 sensory pathway descending pathway O J Fig. 3 The design of a hierarchical NLPCA. Two pathways are separately illustrated based on the dierence of functional aspect. PCA NLPCA 20 HOAP- 1 7) OJ 5ms Fig. 8 NLPCA PCA Fig. 2 OR O R Fig. 2 PCA NLPCA E k k k^x k i ;x ik PCA NLPCA NLPCA =0:03 10 PCA NLPCA NLPCA k =3 NLPCA NLPCA Fig. 3 NLPCA NLPCA ^g x i (2) x 0 x i NLPCA x 0 i (3) ^x i ^x i x 0 i J ^x i

24 8) 9) NLPCA 1. 2. 3. ill-posed 3. 31 NLPCA Fig. 4 OJ R2R 1 OR i x 0 i Fig. 3 OJ x 0 =0:41 x0 =0:62 x0 =0:22 x0 = 0 235 483 945 0:35 J R R NLPCA R activation value original walk reduced walk 0.

4 24 8) 9) NLPCA ill-posed NLPCA Fig. 4 OJ R2R 1 OR i x 0 i Fig. 3 OJ x 0 =0:41 x0 =0:62 x0 =0:22 x0 = :35 J R R NLPCA R activation value original walk reduced walk step Fig. 4 Internal representation of walk (top) and its external appearance (middle) compared to its original attitude (bottom). 32 OR Fig. 4 OR wk2 OR wk3 Owk4 R Owk5 R ( Fig. 5) 1) OJ p (p = wk2 ::: wk5) OJ wk2 ( Fig. 5 ) OJ J R Owk2 OJ OJ wk2 R Fig. 5 OJ wk3 OJ wk4 OJ wk2 OJ wk5 OJ wk3 R NLPCA OJ OJ (0:2 0:6) (0:0 0:2) Fig. 5

2nd walk 3rd walk swing 0.9 throw squat 0.7 captured walks kick 0.

5 The state of varying internal representation with increment of captured

8 OJ p (p = wk2 ::: wk5) O J yaw OJ OJ p R OR Op R O R Fig.

attitude of humanoid robot. O kc R Op R 34 Fig.

walk5 walk3 Value of 2nd Unit Value of 1st Unit walk2 walk4 walk3 walk5

5 2nd walk 3rd walk swing 0.9 throw squat 0.7 captured walks kick 0.5 4th walk 5th walk Fig. 5 The state of varying internal representation with increment of captured walk patterns. 33 Fig. 5 OJ sw Oth J OJ kc Osq J Fig. 6 x 0 i (0:4 0:6) Fig. 6 (0:4 0:6) x 0 i =0:4 0:5 0:6 (0:4 0:6) sin OR OR Fig. 8 OJ p (p = wk2 ::: wk5) O J yaw OJ OJ p R OR Op R O R Fig. 6 Internal representation of each motion pattern and corresponding attitude of humanoid robot. O kc R Op R 34 Fig Value of 1st Unit swing squat walk1 kick throw walk4 walk2 walk5 walk3 Value of 2nd Unit Value of 1st Unit walk2 walk4 walk3 walk5 throw squat kick swing walk1 Value of 2nd Unit Fig. 7 Comparison of internal representation between unit-increment learning(left) and learning with constant number of units from the beginning(right). Fig. 7 R 1 ( ) 2 ( ) Fig

6 Fig. 8 Comparison of external appearances original walk OJ (top) and the regenarated motion from the 1 dimensional internal representation OR (middle) and 3 dimensional one (bottom). 2 Fig. 8 OR NLPCA NLPCA ) 20 NLPCA 11) ( ) 1) K. Kurihara, S. Hoshino, K. Yamane, and Y. Nakamura. Optical motion capture system with pan-tilt camera tracking and realtime data processing. In Proc. of IEEE International Conference onrobotics and Automation, Vol. 2, pp. 1241{1248, Washington D.C., U.S.A., May ) T. Sugihara, Y. Nakamura, and H. Inoue. Realtime humanoid motion generation through zmp manipulation based on inverted pendulum control. In Proc. of IEEE International Conference onrobotics and Automation, Vol. 2, pp. 1404{1409, ),.., Vol. 20, No. 3, pp. 335{343, ),.. 20, p. 1C37, ) M. Kramer. Nonlinear principal component analysis using autoassociative neural networks. AIChE Journal, Vol. 37, pp. 233{243, ) D. DeMers and G. Cottrell. Non-linear dimensionality reduction. In Proceedings of Neural Information Processing Systems, Vol. 5, pp. 580{587, ),,,.. 19, pp. 789{790,, September ) E.E. Fetz, S.I. Perlmutter, et al. Primate spinal interneurons: muscle elds and response properties during voluntary movement. Progress in brain research, Vol. 123, pp. 323{30, ) E. Bizzi, M.C. Tresch, et al. New perspectives on spinal motor systems. Nature Reviews Neuroscience, Vol. 1, pp. 101{108, ) G. Taga. Freezing and freeing degrees of freedom in a model neuro-musculo-skeletal system for development of locomotion. In Proceedings of XVIth International Society of Biomechanics Congress, p.47, Tokyo, Japan, August ) N. Bernstein. The co-ordination and regulation of movoments. Pergamon Press, 1967.

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ξ, ξ ξ Data number ξ 1 Polynomial Design of Dynamics-based Information Processing System Masafumi OKADA and Yoshihiko NAKAMURA Univ. of Tokyo, 7-- Hongo Bunkyo-ku, JAPAN Abstract. For the development of the intelligent robot