0 3

i................................................................ 3.3................................ 4 5. SOM........................ 5........................... 4.3 SOM mnsom................ 8.4 MPFIM........ 7 3 SOAC 35 3................................... 35 3................................. 38 3.3............................... 38 3.4............................. 39 3.5............................. 4 4 SOAC 45 4........................... 45 4................................... 46 4.3................................. 47 4.4................................. 50 4.5............................. 54 4.6....................... 58

ii 4.7..................................... 6 5 SOAC 7 5................................... 7 5................................... 7 5.3................................. 73 5.4................................. 74 5.5............. 76 5.6................ 8 5.7..................................... 8 6 85 6.............................. 85 6. SOAC.............................. 86 6.3 SOAC............................ 87 6.4 SOAC.............................. 88 7 9 93 95

iii AANN Auto-Associative Neural Network 3 (5)L-AANN 3 (5) Layer-AANN ASSOM Adaptive Subspace SOM BMU Best Matching Unit BMM Best Matching Module BP Back Propagation CFC Conventional Feedback Controller EM Expectation Maximization ESOM Evolving Self-Organizing Map GTM Generative Topographic Mapping HMM Hidden Markov Model IDML Inverse Dynamics Model Learning MAE Mean Absolute Error MDS Multi Dimensional Scaling MLP Multi-Layer Perceptron mnsom modular network SOM MMRL Multiple Model Reinforcement Learning MOSAIC MOdular Selection And Identification for Control MPFIM Multiple Paired Forward-Inverse Models (G) NG (Growing) Neural Gas NNC Neural Network Controller PCA Principal Component Analysis RBF Radial Basis Function RNN Recurrent Neural Network RP Responsibility Predictor SOAC Self-Organizing Adaptive Controller SOM Self-Organizing Map VQ Vector Quantization WTA Winner-Takes-All

iv SOM x i w k X A ξ k φ φ k i ψi k n η σ(n) W X Ψ Modular network x i ŷi k ŷ i w k Ei k Ē i p k i T η ψ H C r mnsom x ij y ij D i f i L (f, g) p( ) ỹ k Ei k ki ψi k ξ k i-th data vector k-th reference vector input data set lattice of SOM coordinate vector in the map space index of the BMU neighborhood function (learning rate) k-th learning rate for i-th data normalized learning rate iteration number [step] learning coefficient neighborhood radius at step n reference matrix (set of reference vectors) input matrix (set of input data) matrix of learning rates i-th vector output of k-th module for i-th input vector total output of the network weight vector of k-th module error of k-th module for i-th input vector total error of the network probability selected k-th module for i-th input vector annealing temperature learning coefficient learning rate set of neighborhoods average distance between neighborhoods j-th input vector belonging to i-th class j-th output vector belonging to i-th class data subset belonging to i-th class function of i-th class distance between function f and function g probability density function (pdf) output of k-th module ensemble average of k-th module for i-th class BMM for i-th class learning rate of k-th module for i-th class coordinate vector of k-th module

v MPFIM x t, ẋ t, ẍ t ˆx t, ˆẋ t, ˆẍ t u u k u fb u ff σ f g( ) i g( ) η( ) f w i w δ h( ) K P, K D, K A x l π λ ε SOAC t n K ξ x ˆx x x p actual position, the velocity, and the acceleration at time t desired position, the velocity, and the acceleration at time t total control command control command of k-th inverse model feedback control command feedforward control command standard deviation of Gaussian function of a forward model function of an inverse model function of an RP parameter of a forward model parameter of an inverse model parameter of an RP function of dynamics of a controlled object feedback gains of Proportion, Differential, and Acceleration predicted state prior probability likelihood posterior probability learning coefficient time [sec] iteration number [step] number of modules coordinate vector in the map space state vector desired state predicted state input vector to predictors * index of the BMM p f( ) function of a predictor c f( ) function of a controller p w k c w k σ(t) σ 0 σ τ ψ φ p η c η ε weight vector of k-th predictor weight vector of k-th controller neighborhood radius at time t initial neighborhood radius final neighborhood radius time constant learning rate normalized learning rate (in the execution phase) learning coefficient of a predictor learning coefficient of an NNC damping coefficient

.

Jacobs Jordan [0, ] mixture of experts Narendra [39, 40] Gomi [6] Wolpert [57] Haruno [8, 9] Jacobs mixture of experts Narendra Narendra Kohonen Self-Organizing Map : SOM SOM modular network SOM : mnsom mnsom mnsom Self-Organizing Adaptive Controller : SOAC

. 3. SOAC... A B A B. mnsom Jordan Narendra Wolpert

4.3.3 SOM.4 MPFIM SOAC 3 SOAC 3. SOAC 3. SOAC 3.3 SOAC 3.4 3.5 SOAC 4 SOAC SOAC SOAC 5 SOAC 3 6 SOAC 7

5. SOM Self-Organizing Map : SOM Willshaw von der Malsburg [56] Amari[] Willshaw von der Malsburg [3] Kohonen [7] Kohonen Kohonen [45, 38] [3] [9] [43] [6] SOM Kohonen Kohonen SOM 3 SOM

6 SOM unit. SOM Multi Dimensional Scaling : MDS [48] MDS SOM. SOM k-th w k ξ k ξ SOM, 3 SOM d x i = [x i,..., x id ] T I X = {x,..., x i,..., x I } X d X R d A w k = [w, k..., wd k]t K SOM X SOM () () w k () x i Best Matching Unit : BMU Winner-Takes-All : WTA BMU

. SOM 7 k BMU k = arg max x T i w k (.) k BMU k = arg min x i w k (.) k Kohonen w k SOM Vector Quantization : VQ * () SOM BMU BMU BMU w k = ηφ k (n)(x i w k ), k A (.3) φ k φ k ξ (n) = exp ( k ) ξk σ(n) (.4) * SOM w k

8 (a) (b) (c) 0.8 0.6 0.4 0. 0 5 4 3 0 3 4 5 0.8 0.6 0.4 0. 0 5 4 3 0 3 4 5 0.8 0.6 0.4 0. 0 5 4 3 0 3 4 5 5 4 3 0 BMU 3 4 5 5 4 3 0 3 4 5 5 4 3 0 BMU 3 4 5 5 4 3 0 3 4 5 5 4 3 0 BMU 3 4 5 5 4 3 0 3 4 5 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. unit. (a) σ = 4 (b) σ = (c) σ = η σ(n) σ(n) n. σ η () () SOM.3 SOM (.3) Kohonen Luttrell[9, 30] Bishop[5] Expectation Maximization : EM Generative Topographic Mapping (GTM) (.5) (.5) η

. SOM 9 input vector reference vector coordinate vector of each unit.5 0 0.5 8 0 0.5 6 4.5 0 0 4 6 8 0 input space (d=3) map space (two dimension).3 SOM SOM w k = ψ k i = I ψi k x i (.5) i= φ k i I i = φk i (.6) W = XΨ (.7) W = {w,..., w k } (.8) X = {x,..., x I } (.9) ψ ψ K Ψ =..... (.0) ψi ψi K

0... BMU k = arg min x i w k (.) k. 3. φ k ξ (n) = exp ( k ) ξk σ(n) (.) w k = ηφ k (n)(x i w k ), k A (.3). BMUs ki = arg min x i w k (.4) k. * φ k i = exp ( ξk i ξ k σ ) (.5) ψ k i = φ k i I i = φk i (.6) 3. w k = I ψi k x i (.7) i=.. SOM SOM * (.5) ξ k i ξ k i

. SOM. d ov d uc g e t h z o h ag w l o o awk f d olf c ion c se l ox og at ow e iger orse ebra h e n e k w l is small 0 0 0 0 0 0 0 0 0 medium 0 0 0 0 0 0 0 0 0 0 0 0 big 0 0 0 0 0 0 0 0 0 0 0 nocturnal 0 0 0 0 0 0 0.5 0 0.5 0.5 0 0 0 0 herbivorous 0.5 0.5 0.5 0.5 0 0 0 0 0 0 0 0 0 has likes to legs 4 legs hair hooves mane feathers stripes hunt run fly swim 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 dog tiger lion horse zebra wolf cow cat fox hen dove eagle owl hawk duck goose.4 SOM SOM SOM. [55] *3 *3. eagle 0

input vector reference vector input vector reference vector 0.6 0.6 0. 0. x x 0. 0. 0.6 0.6 0.6 0. 0. 0.6 x (a) 0.6 0. 0. 0.6 x (b).5 SOM (a) (b) 6 6 I = 6 d = 6 6 SOM.4 SOM 00 Cr k = H w k w h (.8) h H h k-th H h Cr k owl hawk [55]

. SOM 3 SOM U-matrix dog wolf dog goose SOM d = SOM 4.5 SOM SOM SOM 4 4 SOM SOM K = 4 0..5(a) SOM SOM SOM SOM SOM SOM

4 SOM.5(b) SOM K = 5 σ =.8 4 SOM k-means SOM SOM SOM SOM SOM. SOM SOM Jacobs Jordan [0] Jacobs mixture of experts

. 5 Jordan[] mixture of experts Jacobs mixturre of experts MLP [8,, 7, 8, 9, 45, 39, 40, 44, 47, 49, 50, 57] [] SOM Gomi [7] mixture of experts Narendra [39, 40] Narendra Wolpert Kawato[57] Multiple Paired Forward-Inverse Models : MPFIM Narendra Narendra MP- FIM MPFIM.5...6 3 3 Layer Auto-Associative Neural Network : 3L-AANN 3L-AANN MLP N D N H

6 Output : Error : Error of each submodel Probability of each submodel Output of each submodel Data space Input :.6 N H < N D MLP 3L-AANN 3L-AANN 3L-AANN 3L-AANN Auto-encoder 3L-AANN Principal Component Analysis : PCA PCA 3 3 0 3 7 I {x i }(x i R N D ) K x i

. 7 yi k 3L-AANN x i Ei k Ei k = x i yi k (.9) x i k-th p k i p k i = exp[ Ek i /T ] k exp[ E k i /T ] (.0) T T K ŷ i = p k i yi k (.) k= ŷ i = ŷ k i (.) k = arg max p k i (.3) k Ē i = K p k i Ei k (.4) k= 3L-AANN BP k-th

8 w k w k = η Ēi w k (.5) { } = η p k Ei k K i w k + Ei k p k i w k (.6) = ηψ k i k = E k i w k (.7) BP ψ k i ψk i k-th ψ k i ψ k i = p k i { + } T (Ēi Ei k ) i-th (.8) T SOM.3 SOM mnsom.3. Kohonen SOM 3 [7] SOM SOM SOM

.3 SOM mnsom 9 SOM [5, 53, 54, 4] SOM modular network SOM : mnsom Kohonen SOM mnsom SOM mnsom SOM SOM SOM Multi-Layer Perceptron : MLP Radial Basis Function : RBF Recurrent Neural Network : RNN SOM Neural Gas : NG [3] MLP MLP mnsom MLPmnSOM [5, 53] mnsom SOM SOM MLP-mnSOM MLP AANN Kohonen [6] SOM Adaptive Subspace SOM : ASSOM 5 AANN 5L-AANN SOM Non-Linear ASSOM : NL-ASSOM [5] ASSOM SOM SOM ASSOM ASSOM 3L-AANN-mnSOM 3 mnsom 5 5L-AANN-mnSOM ASSOM NL-ASSOM

0 input output.7 MLP-mnSOM RNN-mnSOM MLP-mnSOM [4, 53, 4, 4] MLP-mnSOM RNN-mnSOM mnsom SOM NG SOM SOM SOM NG SOM [] [3] [5] mnsom mnsom MLP mnsom MLP-mnSOM mnsom.3. mnsom mnsom.7 MLP mnsom MLP MLP-mnSOM

.3 SOM mnsom I systems I datasets : unknown : observed K modules system sampling system i system I MLP-mnSOM.8 MLP-mnSOM I J y i = f i (x) (.9) D i = {(x ij, y ij )} (.30) x ij = [x ij,..., x ijdi ] T (.3) y ij = [y ij,..., y ijdo ] T. (.3) f i i-th D i-th x ij y ij i-th j-th d i d o f i I J mnsom 3. f i. 3.. mnsom I Best Matching Module : BMM. BMM

SOM 3. mnsom.8 SOM MLP-mnSOM L (f, g) = f(x) g(x) p(x)dx (.33) p( ) mnsom mnsom [53] (.33) f g.3.3 mnsom MLP-mnSOM mnsom 4 BMM BMM MLP Evaluative process (.33) f (.33) E k i = J J y ij ỹij k (.34) ỹij k i-th j-th k-th j=

.3 SOM mnsom 3 Competitive process BMM ki = arg min Ei k (.35) k Cooperative process ψ k i = exp [ ξi k ξ k /σ(n) ] I i = exp [ (.36) ξi k ξ k /σ(n) ] σ(n) n I σ σ(n) = σ + (σ 0 σ ) exp ( n ) τ (.37) σ 0 n = 0 σ n = τ σ Adaptive Process Back Propagation : BP w k = η I i= ψ k i E k i w k (.38) w k k-th k-th E k E k = I ψi k Ei k (.39) i= (.39) (.34) (.33) mnsom

4 class a b c 0. 0. 0. -0. -0. -0. 3 0.4 0.0 0.0 4-0.4 0.0 0.0 5 0.0 0.0 0.4 6 0.0 0.0-0.4.9 f f 3 f 5 class class 0 0 0.5 0 0.5 0.5 0 0.5 x x class 3 class 4 0 0 0.5 0 0.5 0.5 0 0.5 x x class 5 class 6 0 0 0.5 0 0.5 0.5 0 0.5 x x f f 4 f 6 3. MLP-mnSOM Parameters of MLP Number of input unit Number of hidden units 5 Number of output unit Learning coefficient η 0.05 Parameters of mnsom Map size 00 (0 0) Initial value of neighborhood radius σ 0 0.0 Final value of neighborhood radius σ.0 Time constant τ 50 Iteration number 300 k-th g k g k (x) = I ψi k f i (x) (.40) i= 4

.3 SOM mnsom 5 i BMM for i-th class 4 module number... 0 5 6 3 (a) 9... (b) 00.0 MLP-mnSOM 3 (a) (b) 8 module number... 0 Root mean square error x 0-3.5.5 0.5 9... 00 0. mnsom mnsom

6.3.4 mnsom mnsom 3 3 a, b, c 6 I = 6 3.9 y i = f i (x) (.4) f i (x) = ax 3 + bx + cx (.4) mnsom x {.0, 0.99, 0.98,..., 0.99,.0} y i (.4) 0 J = 0 MLP-mnSOM.0..0 (a) mnsom (b) (.40) 8 3 4 5 6 BMM BMM BMM mnsom ỹ k ŷ k. 6-th 0.008 mnsom J J ŷ k ỹ k (.43) j=

.4 MPFIM 7 current state forward model predicted trajectory desired trajectory inverse model control command controlled object actual trajectory..4 MPFIM [4] Kawato [3, 5] M x x u(t) x(t) x = F (u) u = F (x) ˆx x = F (u) = F (F (ˆx)) = ˆx.

8 Albus[] Kuperstein[8] Miller[33] Atkeson[4] [4] Jordan Rumelhart[] Kawato[3, 5] Conventional Feedback Controller : CFC CFC MPFIM.4. Kawato[3, 5]

.4 MPFIM 9 inverse model desired trajectory + + + - CFC controlled object actual trajectory.3.3 CFC ˆx CFC CFC u fb = K p (ˆx x) + K D (ˆẋ ẋ) + K A (ˆẍ ẍ) (.44) x, ẋ, ẍ K P, K D, K A ˆx ˆẋ ˆẍ i w u ff = i g( i w, ˆx, ˆẋ, ˆẍ) (.45) i g MLP RBF CFC u fb u ff h(x, ẋ, ẍ) = u fb + u ff (.46) h dw/dt = ε( u ff / i w) T u fb (.47)

30 Widrow-Hoff u fb û (û u ff ) T (û u ff ) i w dw/dt = ε( u ff / i w) T (û u ff ) (.48) (.47) (.48) CFC u fb (û u ff ) CFC Kawato [3] Miyamura[37] Gomi [6] [36] [7].4. MPFIM Multiple Paired Forward-Inverse Models : MPFIM MOSAIC MOdular Selection And Identification for Control Wolpert Kawato [57] Wolpert Kawato MPFIM Gomi [7] Gomi mixture of experts Wolpert Kawato MPFIM soft-max Gomi Haruno [8, 9] Gomi MPFIM

.4 MPFIM 3 contextual signal desired state efference copy of motor command K forward likelyhood model state prediction model forward prediction error likelyhood model model forward likelyhood model responsibility prior model x predictor responsibility prior x predictor responsibility prior module feedforward predictor inverse control command module feedforward model inverse control command module feedforward model inverse control command model x x x posterior x normalization + + control command controlled object next state + - CFC feedback control command.4 MPFIM MPFIM Doya Samejima MPFIM Multiple Model Reinforcement Learning : MMRL [8, 47] Wolpert Kawato MPFIM MPFIM Hidden Markov Model : HMM Haruno [8].4.3 MPFIM MPFIM.4 MPFIM K 3 Responsibility Predictor : RP x k t+ = f g( f w k t, x t, u t ) (.49)

3 w k t f g x k t x t l k t l k t = P (x t f w k t, u t, k) = πσ e x t x k t /σ (.50) σ soft-max l k t K k = lk t (.5) 0 MPFIM contextual signal RP y t RP πt k = η(δt k, y t ) (.5) δ k t η λ k t = πt k lt k K k = πk t lt k (.53) λ k t 3 3 RP MPFIM 3 ˆx k-th u k t = i g( i wt k, ˆx t ) (.54)

.4 MPFIM 33 K K u t = λ k t u k t = λ k t ig( i wt k, ˆx t ) (.55) k= k= i wt k = ελ k d i g k t d i wt k (û t u t ) ε duk t dvt k λ k t u fb (.56) f wt k = ελ k d f g k t d f wt k (x t x k t ) (.57).4.4 MPFIM MPFIM. π k t π k t = η(δ k t, y t ) (.58). l k t l k t = πσ e x t x k t /σ (.59) 3. λ k t λ k t = πt k lt k K k = πk t lt k (.60) 4. u t K u t = λ k t u k t (.6) k= 5. i wt k = ε duk t dvt k λ k t u fb (.6) f wt k = ελ k d f g k t dwt k (x t x k t ) (.63)

35 3 SOAC 3. Jacobs Jordan [0] mixture of experts Jordan Jacobs mixture of experts [] mixture of experts Narendra [39, 40]

36 3 SOAC Wolpert [57] Narendra Wolpert soft-max Narendra Gomi [7] mixture of experts Kawato [3, 4, 5] Wolpert Multiple Paired Forward-Inverse Models : MPFIM Wolpert Wolpert SOAC SOAC Tokunaga [5, 53, 54] SOM mnsom mnsom mnsom mnsom Nishida [4, 4]

3. 37 SOAC modules BMM for object B switch BMM for object A object A object B 3. SOAC BMM BMM BMM SOAC () () (3) () mnsom () SOAC mnsom (3) SOAC mnsom 3. SOAC 3.3 SOAC 3.4

38 3 SOAC 3. SOAC SOAC SOM SOAC SOM Best Matching Module : BMM BMM BMM 3. 3.3 SOAC predictor controller 3. k-th controlled object current state x(t) control signal u(t) t predicted state x k (t + t) * x k (t + t) = p f k (x(t), u(t)) (3.) k-th x(t) desired state ˆx(t) u k (t) u k (t) = c f k (x(t), ˆx(t)) (3.) SOAC * * SOAC * predictor-map controller-map

3.4 39 Predictor D x x^ current state x desired state u Controller Predictor Controller predicted state D x ~ k time delay u k Winner Takes All Controlled Object Predictor Controller D u control signal 3. 3.4 3.4. I I I {x i (t), u i (t)}(i =,..., I) mnsom [, 5] mnsom MLP Multi-Layer Perceptron p w k I p E k i = T T 0 x i (t) x k i (t) dt (3.3) x k i (t) p E k i i-th k-th T

40 3 SOAC BMM Best Matching Module : BMM BMM i = arg min p Ei k (3.4) k ψ k i ψk i ψ k i = exp[ ξ k ξ i /σ ] I i = exp[ ξk ξ i /σ ] (3.5) ξ k, ξ i k-th i-th BMM σ σ n ( σ(n) = σ + (σ 0 σ ) exp n ) τ (3.6) σ 0 σ τ ψ k i p w k = p η p η I i= ψ k i p E k i p w k (3.7) 4

3.4 4 u(t) control signal x(t) current state x(t) ^ desired state - + predictor CFC NNC D time delay ~ x k (t) predicted state u k (t) control signal 3.3 3.4. SOAC Kawato feedback-error-learning [3, 5] Conventional Feedback Controller : CFC SOAC 3.3 CFC Neural Network Controller : NNC NNC CFC 3.4.3 x(t) x k (t) p e k (t) = ( ε) p e k (t t) + ε x(t) x k (t) (3.8) p e k (t) 0 < ε ε

4 3 SOAC ε ε ε BMM BMM p e k (t) * (t) = arg min p e k (t) (3.9) k BMM φ k = exp[ ξ k ξ /σ ] K k = exp[ ξk ξ /σ ] (3.0) σ BMM BMM NNC CFC cfc u(t) *3 K u(t) = φ k u k (t) + cfc u(t) (3.) k= u k (t) = c f k (ˆx(t)) (3.) cfc u(t) = cfc W (ˆx(t) x(t)). (3.3) cfc W NNC c w k cfc u φ k c w k = c η φ k c f k c w k cfc u (3.4) 3.5 SOAC BMM *3 (3.) BMM

3.5 43 p i-th p i p k ψ k i p i p k = I ψi k p i (3.5) i= p parameter-map 3.5. BMM (3.4) BMM BMM BMM (0) = arg min p p k (3.6) k (3.9) BMM BMM 3.5. BMM BMM BMM p 5

45 4 SOAC SOAC SOAC SOAC Wolpert Multiple Paired Forward-Inverse Models : MPFIM 4. I

46 4 SOAC 4. B [kg/s] K [kg/s ] M [kg] p.0.0.0 p 6.0.0.0 p 3 0.0.0.0 p 4.0 6.0.0 p 5 6.0 6.0.0 p 6 0.0 6.0.0 p 7.0 0.0.0 p 8 6.0 0.0.0 p 9 0.0 0.0.0 p A 6.00 6.00.0 p B 6.3 6.79.0 p C 4.48 6.50.0 p D 4.9 3.86.0 p E 8.59 4..0 p F 8.83 8.83.0 B M =.0 [kg] K [kg/s ] 0 8 6 4 M used object parameter unused object parameter 4 6 8 0 B [kg/s], x(t) K u(t) 4. M i ẍ + B i ẋ + K i x = u (4.) x [m] u [N] M i, B i, K i i-th [kg] [kg/s] [kg/s ] 4. p p 9 p A p F p 5 p A 4 runge-kutta h=0.00 4.. x ẋ u 3 { 0., 0.05, 0, 0.05, 0.} 5

4.3 47 4. Number of classes 9 Map size 9 9 ( dimension) Initial neighborhood radius σ 0 9.0 Final neighborhood radius σ.5 Time constant τ 00 Iteration number N 000 Learning coefficient η 0.000 =9 5 3 =5 4.3 4.3. x p = [x, ẋ, u] T y = ẍ T k-th p w k = [ p w, k p w, k p w3] k T y k = x T p p w k (4.) 4. 4.3. 4. 4. 4. 4. BMM 4.4 n=0 00 BMM n=000

48 4 SOAC p p 3 p p p 3 p p p 3 p p p 3 p 5 p 7 p p 6 iteration number n=0 p 8 p 5 (a) p 9 p 4 n=00 p 4 p 7 p 5 p 6 p 6 p 8 p 9 n=300 p 4 p 7 p 5 p 6 p 8 p 9 n=500 p 4 p 7 p 8 p 9 K [kg/s] K [kg/s] K [kg/s] K [kg/s] 0 9 8 7 6 5 4 3 3 4 5 6 7 8 9 0 B [kg/s ] 0 9 8 7 6 5 4 3 3 4 5 6 7 8 9 0 B [kg/s ] 0 9 8 7 6 5 4 3 3 4 5 6 7 8 9 0 B [kg/s ] 0 9 8 7 6 5 4 3 3 4 5 6 7 8 9 0 B [kg/s ] (b) 4. (a) (b) 0 00 300 500 (a) p i (i =,..., 9) p i i-th BMM

4.3 49 module 3 p iteration number n=000 p 4 p 7 0 used object parameter p 7 p 8 p 9 9 8 p p 5 p 8 K [kg/s] 7 6 5 p 4 p 5 p 6 4 p 3 p 6 (a)... module 73 74 75 8 p 9 3 p p p 3 3 4 5 6 7 8 9 0 B [kg/s ] (b) 4. (a) (b) desired position [m] desired velocity [m/s] desired acceleration [m/s ] 0 0 5 0 5 0 5 30 time [sec] 0 0 5 0 5 0 5 30 time [sec] 5 0 5 0 5 0 5 0 5 30 time [sec] 4.3

50 4 SOAC 4.3 Damping coefficient ε.0 Neighborhood radius σ.5 Learning coefficient η 0.0 Learning time t 9000 [sec] 4.4 4.4. Ornstein-Uhlenbeck [msec] 30 [sec] 30000 4.3 4.4. SOAC Kawato NNC NNC ˆx ˆẋ ˆẍ nnc u k 3 nnc u k = ˆx T c w k. (4.3) ˆx = [ˆx, ˆẋ, ˆẍ] T c w k = [ c w k, c w k, c w k 3] T PDA cfc W = [k x, kẋ, kẍ] = [5, 0, 0.5] 4.4.3 p p 9 9 5 [sec] 30 300 = 9000 [sec] 4.3 [msec] 4.3 4.4 4.4

4.4 5 K [kg/s] K [kg/s] K [kg/s] 0 9 8 7 6 5 4 3 0 9 8 7 6 5 4 3 3 4 5 6 7 8 9 0 B [kg/s ] t = 00 [sec] 3 4 5 6 7 8 9 0 B [kg/s ] t = 500 [sec] 0 9 8 7 6 K [kg/s] 9 8 7 6 5 4 3 time t = 0 [sec] K [kg/s] K [kg/s] K [kg/s] 0 9 8 7 6 5 4 3 0 9 8 7 6 5 4 3 0 9 8 7 6 3 4 5 6 7 8 9 0 B [kg/s ] t = 3000 [sec] 3 4 5 6 7 8 9 0 B [kg/s ] t = 5000 [sec] 5 5 4 4 3 3 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 B [kg/s ] B [kg/s ] used object parameter t = 9000 [sec] p 7 p 8 p 9 0 p 4 p 5 p p p 3 3 4 5 6 7 8 9 0 B [kg/s ] p 6 t = 000 [sec] 4.4

5 4 SOAC position [m] trajectory error of CFC [m] 0.5 0 0.5 0 0 0 30 time [sec] 0.5 0 (a) CFC p A p B p C p D p E p F p A p B p C p D p E p F desired trajectory actual trajectory p A p B p C p D p E p F 0.5 0 0 0 30 time [sec] position [m] trajectory error of SOAC [m] 0.5 0 0.5 0 0 0 30 time [sec] 0.04 0.0 0 (b) SOAC desired trajectory actual trajectory p A p B p C p D p E p F 0.0 0 0 0 30 time [sec] 4.5 (a) CFC (b) SOAC 4.4 BMM (for i-th class) p B k p K k p M k c B k c K k c M k (p ).08.8.0005.0084.0034.000 37 (p ) 5.9978.5 0.999 6.0000.53.08 73 (p 3 ) 9.8888..0006 9.999.0076.009 5 (p 4 ).04 5.9990.000.069 5.9707 0.9895 4 (p 5 ) 6.000 6.007.0004 5.9979 6.000.00 77 (p 6 ) 9.8904 6.005.003 9.864 6.093 0.9957 9 (p 7 ).4 9.888 0.9993.006 9.985 0.993 45 (p 8 ) 5.9996 9.8898.0000 5.9896 9.806 0.974 8 (p 9 ) 9.8870 9.889 0.9999 9.9856 9.996 0.99 NNC φ k K k= φk ( c w k )

4.4 53 K [kg/s ] 0 8 6 4 object parameter 4 6 8 0 B [kg/s] 9 8 7 6 5 4 3 9 module BMM 3 4 5 6 7 8 9 8 (module number) 80. 74 73 4.6 BMM BMM 4.4.4 p A p F 6 p A p 5 5 [sec] BMM ε=0.00 4.5 (a) CFC (b) SOAC 30 [mm] SOAC BMM SOAC SOAC BMM BMM BMM 4.6 BMM

54 4 SOAC 4.5 MPFIM SOAC MPFIM SOAC Number of classes 9 9 Damping coefficient ε.0.0 Learning coefficient of predictor p η 0.005 0.000 Learning coefficient of NNC c η 0.005 0.05 p A p B BMM BMM BMM SOAC BMM 4.5 4.5. MPFIM Wolpert [57] Multiple Paired Forward- Inverse Models SOAC MPFIM soft-max MPFIM 4.5. MPFIM SOAC 4.3. 4.4. MPFIM RP SOAC

4.5 55 forward model (predictor) inverse model (NNC) used object parameter 0 module 9 module 8 module 7 0 module 3 module 7 module 8 8 8 K [kg/s ] 6 module 6 module 5 module 4 K [kg/s ] 6 module 5 module module 9 4 4 module 3 module module module module 6 module 4 4 6 B [kg/s] 8 0 4 6 B [kg/s] 8 0 0 module 9 module 8 module 7 0 module 3 module 7 module 8 8 8 K [kg/s ] 6 module 6 module 5 module 4 K [kg/s ] 6 module 5 module module 9 4 4 module 3 module module module module 6 module 4 4 6 B [kg/s] 8 0 4 6 B [kg/s] 8 0 (a) SOAC (b) MPFIM 4.7 SOAC MPFIM (a) SOAC (b) MPFIM SOAC 4.5 MPFIM SOAC σ 9000 [sec]

56 4 SOAC (a) SOAC (b) MPFIM 0.5 desired trajectory actual trajectory 0.5 desired trajectory actual trajectory position [m] 0 position [m] 0-0.5-0.5-9000 9005 900 905 900 905 9030 time [sec] - 9000 9005 900 905 900 905 9030 time [sec] 0.0 0.0 error 0.0 0.0 error trajectory error [m] 0-0.0-0.0-0.03 trajectory error [m] 0-0.0-0.0-0.03-0.04 9000 9005 900 905 900 905 9030 time [sec] -0.04 9000 9005 900 905 900 905 9030 time [sec] 4.8 (a) SOAC (b) MPFIM 4.5.3 SOAC MPFIM Mean Absolute Error : MAE MAE K = 9 SOAC MPFIM 4.7 4.8 4.7 MPFIM SOAC MPFIM

4.5 57 0 00 times average of RMSEs 0 0 3 SOAC MPFIM 0 0 40 60 80 00 Number of modules 4.9 SOAC MPFIM 9, 5, 49, 8 MAE 00 9000 [sec] MPFIM SOAC 4.8 MAE 0.0058 SOAC MPFIM MPFIM SOAC 00 MAE SOAC 0.0058 MPFIM 0.094 SOAC 4.5.4 SOAC SOM

58 4 SOAC 0 training class untrained class class 3 8 K [kg/s ] 6 4 class class A class 4 6 8 0 B [kg/s] 4.0 MPFIM 9,5,49,8 00 MAE 4.9 SOAC MAE MPFIM MAE MPFIM 4.6 SOAC SOM h(x; θ + θ ) h(x; θ ) + h(x; θ ). (4.4)

4.6 59 4.6 P 3 m i= Ei /3 m E A SOAC P 3 m i= Ei /3 + m EA First level 0.30.5766.8886 Second level 0.30.08.438 Third level 0.079 0.065 0.0444 MPFIM P 3 m i= Ei /3 m EA P 3 m i= Ei /3 + m EA First level 0.0057 3.053 3.0570 Second level 0.0057.489.546 Third level 5.5436.953 7.4689 h x θ SOAC k-th õ k o i B, K, M 3 o i = [B i, K i, M i ] Model Error m E k i := o i õ k (4.5) SOAC 4.0 3,, 3 A M M = B K (B ) + (K 0) = 8, B, K 0 SOAC MPFIM SOAC 5 MPFIM 3 SOAC MPFIM 4.5.4

60 4 SOAC MPFIM 4. SOAC MPFIM 3 SOAC (a) MPFIM (b) (c), (d) 3 (e), (f) 4.6 SOAC -st 3-rd 5-th 3 MPFIM -st -nd 3 3-rd SOAC SOM MPFIM 4.6 A SOAC 4-th MPFIM 3 3-rd SOAC MPFIM soft-max st winner nd winner e = ( ε)e + ε x x (4.6) e = ( ε)e + ε x x (4.7) o ip = e o + e o e + e (4.8) ( ), ( ) MPFIM 3 A /

4.6 6 3 3 3 3 p w = p η p w x x (4.9) (4.9) 4.(e), (f) 3 4 4.(e) SOAC 4-th A SOM -st 3-rd 5-th MPFIM 3 4 4.6 4.(f) SOAC SOAC MPFIM SOAC

6 4 SOAC 4.7 4.7. p B, p K, p M c B, c K, c M 4. SOM 8 4.3 SOAC NNC 4. 4.7. SOAC BMM mnsom

4.7 63 5 σ σ.0,.5,.0 4.4 σ σ SOM σ =.0 σ σ σ.0 mnsom CFC BMM BMM BMM 4.7.3 MPFIM SOAC MPFIM SOAC SOM MPFIM [46] SOAC MPFIM

64 4 SOAC MPFIM SOAC SOAC mnsom mnsom [54] MPFIM SOAC 4.5 MPFIM SOAC MPFIM 4.5 9 5 SOAC MPFIM SOAC MPFIM 4.5(c) MPFIM SOAC

4.7 65 4.5(d) 9000 [sec] [8, 9] MPFIM SOM SOAC MPFIM

66 4 SOAC SOAC MPFIM 0 5 class 3 0 class 3 First level : learning phase K [kg/s ] 8 6 4 class 4 model error class A 3 class K [kg/s ] 8 6 4 class class A 3 model error class 4 6 8 0 B [kg/s] (a) 4 6 8 0 B [kg/s] (b) 0 interpolated object by soft-max 5 class 3 0 interpolated object by soft-max nd winner class 3 Second level : interpolation K [kg/s ] 8 6 4 class st winner 4 model error nd winner class A 3 class K [kg/s ] 8 6 4 class st winner 3 class model error class A 4 6 8 0 B [kg/s] (c) 4 6 8 0 B [kg/s] (d) Third level : incremental learning phase K [kg/s ] 0 8 6 4 5 class 3 class A 4 class 3 class 4 6 8 0 B [kg/s] (e) K [kg/s ] 0 8 6 4 model error class 3 3 class A model errors class class 4 6 8 0 B [kg/s] (f) 4. SOAC MPFIM 3 (a) SOAC (b) MPFIM (c) SOAC (d) MPFIM (e) SOAC (f) MPFIM

4.7 67 predictor NNC 0 8 K [kg/s ] 6 4 4 6 8 0 B [kg/s] 4. predictor NNC 0 8 K [kg/s ] 6 4 4 6 8 0 B [kg/s] 4.3

68 4 SOAC predicto r NNC.5.5.5 K [kg/s ] class class class class class class K [kg/s ] K [kg/s ].5 3 4 5 6 7.5 B [kg/s].5 3 4 5 6 7.5 B [kg/s].5 3 4 5 6 7 B [kg/s].5 K [kg/s ] class class K [kg/s ] class class K [kg/s ] class class.5 3 4 5 6 7.5 B [kg/s].5 3 4 5 6 7.5 B [kg/s].5 3 4 5 6 7 B [kg/s].5 K [kg/s ] class class K [kg/s ] class class K [kg/s ] class class.5 3 4 5 6 7 B [kg/s].5 3 4 5 6 7 B [kg/s].5 3 4 5 6 7 B [kg/s] (a) predictor (b) NNC (c) predictor and NNC 4.4 (a) (b) (c) σ = {.0,.5,.0} (a) (b)nnc (c) NNC

4.7 69 forward model (SOAC) inverse model (SOAC) K [kg/s ] 0 8 6 4 0 8 6 B [kg/s] 4 0 (a) 00 K [kg/s ] 000 800 600 400 iteration number [time] 0 8 6 4 0 8 6 B [kg/s] 4 0 (b) 000 4000 time [sec] 6000 8000 forward model (MPFIM) inverse model (MPFIM) K [kg/s ] 0 8 6 4 0 8 6 B [kg/s] 4 0 (c) 000 time [sec] 000 K [kg/s ] 0 8 6 4 3000 0 8 6 B [kg/s] 4 0 (d) 000 4000 time [sec] 6000 8000 4.5 SOAC MPFIM (a) SOAC (b) MPFIM (c) SOAC (d) MPFIM

7 5 SOAC 5. SOAC SOAC SOAC 5. (M + m)ẍ + ml cos θ θ ml θ sin θ + fẋ = a u (5.) ml cos θ ẍ + (I + ml ) θ mlg sin θ + C θ = 0 (5.) x [m] ẋ [m/s] ẍ [m/s ] θ [rad] θ [rad/s] θ [rad/s ] x = [x, θ, ẋ, θ] T u M [kg] m [kg] l [m] f [kg/s] C [kgm /s] g [m/s ] a [N/V] I

7 5 SOAC l m C x u M f m [kg].8.4 0.6 0. 0.6 0.9. l [m].5.8 learning phase execution phase p = [0.6, 0.] p A = [.0,.00] p = [., 0.] p B = [.5, 0.9] p 3 = [.8, 0.] p C = [.4,.8] variable parameter p 4 = [0.6,.0] p D = [0.93, 0.94] p i p i = [l i [m], m i [kg]] p 5 = [.,.0] p E = [0.93,.3] l i : length to the mass center p 6 = [.8,.0] p F = [.6,.49] m i : pendulum mass p 7 = [0.6,.8] p G = [.7,.4] p 8 = [.,.8] p H = [.75, 0.64] p 9 = [.8,.8] p I = [.0, 0.0] M cart mass 5.0 [kg] C friction coefficient of pendulum 4.0 0 4 [kgm /s] f friction coefficient of cart 0.0 [kg/s] g gravity acceleration 9.8 [m/s ] a gain 5 [N/V] 5. I = ml /3 [kgm ] 5. 4 runge-kutta h=0.0 5.. x(t) u(t) x(t + t) 5 4 x

5.3 73 5. Number of classes 9 Map size 9 9 ( dimension) Initial neighborhood radius σ 0 0.0 Final neighborhood radius σ.8 Time constant τ 00 Iteration number N 000 { 0., 0, 0.} 3 t [sec] =9 3 5 =87 5.3 5.3. θ = 0 x(t) u(t) t [sec] x(t + t) 5 4 t = 0.0 [sec] 5. p η 5.3. 5. SOAC SOAC CFC CFC

74 5 SOAC 9 73 [rad] 0.0 0-0.0 0 5 [sec] 8 BMM for training data 5. CFC CFC cfc W = [k x, k θ, kẋ, k θ] = [ 0.5, 5.64, 0.67,.03] 8 5.4 5.4.

5.4 75 cart position [m] pendulum angle [rad] 0 0. 0 0. actual response desired response 0 0 0 30 40 50 60 70 80 90 time [sec] 5.3 5 [sec] 0 (x T Qx + u T Ru)dt (5.3) cfc W k = R B k P k (5.4) P k (4 4) Riccati (A k ) T P k + P k A k + Q P k B k R (B k ) T P k = 0 (5.5) A k B k k-th Q R 4 4 R = 5.4. 5.3 BMM 5.4 5.4

76 5 SOAC 3 4 5 6 7 8 SOAC module BMM 9 3 4 5 6 7 8 9 5.4 5.3 5.5 3.5. BMM SOAC BMM 5.5 ψi k p i p i SOM p i p k 5.5 BMM

5.5 77.8.6.4. m [kg] 0.8 0.6 0.4 0. 0.6 0.8..4.6.8 l [m] 5.5 5.6 BMM BMM BMM p = [l, m ] = [.8, 0.] p = [l, m ] = [0.6,.8] p 0 [sec] p BMM BMM BMM 5.7 BMM BMM 0.9 [sec] BMM

78 5 SOAC 3 4 5 6 7 8 SOAC module BMM 9 3 4 5 6 7 8 9 5.6 5.8 ε = 0.00 ε BMM 0.9 [sec] BMM BMM BMM BMM BMM BMM BMM { P (kff (tc < t ) = U(t c t s ) = s ) 0 (t c t s ) { 0 P (kfb) (tc < t = U(t c t s ) = s ) (t c t s ) (5.6) (5.7)

5.5 79 3 4 5 6 7 8 9 3 4 5 6 7 8 9 5 0 0.9 5 time [sec] feedback selection feedforward selection 5.7 3 4 5 6 7 8 9 0 5 0 0.9 5 0 time [sec] feedback selection feedforward selection BMM BMM BMM P (kff ) k ff P (kfb ) kfb t c t s t s [sec] t s ε t s ε 5.9 p = [l, m ] = [.8, 0.] p = [l, m ] = [0.6,.8] p 0.8[kg]

80 5 SOAC pendulum angle [rad] cart position [m] 0 0. 0. 0. 0 5 0 5 0 time [sec] 0 feedback selection feedforward selection feedback selection feedforward selection 0. 0 5 0 5 0 time [sec] 5.8 BMM BMM 3 4 5 6 7 8 9 3 4 5 6 7 8 9 visual effects error 5 0 5 time [sec] 5.9 BMM BMM (5.7) t s = 5.9 [sec] BMM BMM 0 [sec] BMM [sec]

5.6 8.8 estimated parameter.8 estimated parameter true parameter.6.6.4.4.. m [kg] m [kg] 0.8 0.8 0.6 0.6 0.4 0.4 0. 0.6 0.8..4.6 l [m] 0. 0.6 0.8..4.6 l [m] 5.0 BMM 5.6 p = [.60,.50] SOAC BMM BMM p p = [.65,.54] p p p = 0.064 50 0.05 5.4

8 5 SOAC 5.0 5.7 5.7. SOAC SOAC SOM SOM mnsom SOAC 0 BMM

5.7 83 5.7. SOAC BMM SOAC 5.7.3 4 3 NNC GOMI [6] IDML Inverse Dynamics Model Learning IDML SOAC IDML [34, 35]

85 6 6. SOAC () () Jacobs Jordan mixture of experts Narendra Wolpert MPFIM () MPFIM () mixture of experts SOAC Narendra Wolpert SOAC Narendra Wolpert MPFIM SOAC mnsom SOAC MPFIM

86 6 6. SOAC SOAC SOAC MPFIM SOAC MPFIM SOAC SOAC SOAC

6.3 SOAC 87 SOAC 6.3 SOAC mnsom mnsom SOAC SOAC SOAC SOAC mnsom SOM mnsom [5, 53] 3 5 4.7.

88 6 4.7. Jordan Rumelhart [] SOAC mnsom mnsom MPFIM SOAC mnsom [54] SOAC SOAC 6.4 SOAC SOAC mnsom mnsom Tokunaga[5] mnsom MLP RBF RNN SOM NG

6.4 SOAC 89 SOM NG [3, 45] SOAC SOM SOM MPFIM MPFIM mnsom SOM NG SOM Evolving SOM : ESOM [7] Furukawa[, 3] SOM NG SOAC SOM SOM SOM SOM SOM SOAC SOM SOM NG SOM NG GNG Growing Neural Gas [0] ESOM SOAC 6.4. mnsom SOAC [54]

90 6 SOAC SOAC A B SOAC SOAC A B SOAC MMRL Multiple Model Reinforcement Learning MMRL MPFIM MPFIM [47] Doya [8] MMRL SOAC MPFIM SOAC SOAC SOAC SOM SOAC

9 7 SOAC SOAC SOM mnsom SOAC SOAC MPFIM SOAC SOAC SOAC.. SOAC

9 7 7. SOAC MPFIM MPFIM SOAC MPFIM 7. MPFIM SOAC MPFIM SOAC SOM MPFIM MPFIM MPFIM SOAC SOAC mnsom MPFIM SOAC SOAC MPFIM

93 4 4 SOAC MPFIM COE mnsom

94 7 Sandor M. Veres SOAC Veres COE J9 No. 750093

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