318 T. SICE Vol.52 No.6 June 2016 (a) (b) (c) (a) (c) ) 11) (1) (2) 1 5) 6) 7), 8) 5) 20 11) (1
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1 Vol.52, No.6, 317/ Mathematical Model and Analysis on Human s Handwork and Their Application to Brush Motion and Junkatsu of Japanese Calligraphy Tadamichi Mawatari and Kensuke Tsuchiya There is an increasing need to automate handwork by skilled workers in industry. However, that handwork is usually difficult to express clearly in words. Furthermore, human s motion contains time and spatial perturbations, and this makes the automation even more difficult. In this report, spline functions are applied first to motion-capture data of brush strokes of Japanese calligraphy, and mathematical model of human s technique is established. Then, the isomorphism mapping method is introduced to deal with the time perturbations, and Min-Max norm was applied to deal with the spatial perturbations. Furthermore, a new analytical point of view called controllability is introduced to the mathematical model, and the target curve is derived as the goal of the automation. The motion control theory based on controllability is hereby established to automate human s handwork. Key Words: perturbation, isomorphism, min-max norm, controllability, mathematical model ) 2) 3) Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo Received June 20, 2015 Revised March 13, TR 0006/16/ c 2015 SICE
2 318 T. SICE Vol.52 No.6 June 2016 (a) (b) (c) (a) (c) ) 11) (1) (2) 1 5) 6) 7), 8) 5) 20 11) (1) X Y Z Yaw Pitch Roll P d
3 Fig. 1 Motion capture system Fig. 3 Coordinate system for motion data Table 1 Experimental data Fig. 2 Motion capture markers attached to Mouhitsu Fig. 1 Fig. 2 i 1st Phase ii (2nd Phase) iii (3rd Phase) 3 TotalData(Id, Time,X,Y,Z, Yaw, Pitch, Roll,P d) n n Time sec X Y Z mm Yaw Pitch Roll deg P d gf/mm Fig i iii Fig. 3 X Y Z 7 1 Phase Table 1 Start P 1 P 2 P 3 ID End 12) Y
4 320 T. SICE Vol.52 No.6 June 2016 P d(t) =ω[min{y } +max{y } Y(t)]. ω 3. Table 1 (1) (1) Fourier Wavelet 13) 14) (1) 9 2 Id Time 7 DenoiseData(X, Y, Z, Yaw, Pitch, Roll,P d) Cubic Spline Fig. 4 Idea of isomorphism knot Cubic Spline i iii Cubic Spline 273 Spline (2) [t a,t b ] f [τ a,τ b ] g (isomorphism) Fig. 4 t α = t a + α(t b t a), f(t α)=g(τ α) where τ α = τ a + α(τ b τ a), 3 α:0 α 1. 2 { t α = t a + α(t b t a), τ α = τ a + α(τ b τ a). 4 f g g f f g : f C[t a,t b ], g C[τ a,τ b ]
5 f g : f C[t a,t b ], g C[τ a,τ b ] (4) d dt f(tα) = τ b τ a d t b t a dτ g(τα). 6 [ ] (6) (τ b τ a)/(t b t a) f g [t a,t b ] [τ a,τ b ] t a t b τ a τ b 4 (6) X Y Z Yaw Pitch Roll (P d) 7 1 g C[τ a,τ d ] 2. 3 i iii 3 f C[t a,t d ] (X Y Z) (Yaw Pitch Roll) (P d) 7 7 Ω(t) (7) 1 (7) 1 Ω(t) [t a,t d ] 7 C 7 [t a,t d ] [ ] X(t),Y(t),Z(t), Yaw (t), Ω(t), Pitch(t), Roll(t),P d(t) C[t a,t d ] C[t a,t d ] C[t a,t d ], C 7 [t a,t d ] (Product Space). 7 Ω(t) C 7 [t a,t d ] [ ] (7) 7 [t a,t d ] (almost everywhere) (7) 7 Ω(t) k 1 k n Ω k (t) Ω k (t) k k Ω k (t) C 7 [t a,t d ] C 7 [t a,t d ] 4. 4 Sobolev (7) Ω(t) C 7 [t a,t d ] Ω(t) (Ω 1(t),, Ω 7(t)) C 7 [t a,t d ]. ( ) d d dt Ω(t) dt Ω1(t),, d dt Ω7(t). 8 7 (X, Y, Z) (Yaw, Pitch, Roll) (P d) S 1 C[t a,t d ] 2 Sobolev Ω S Ω S 7 ω i Ω i + i=1 7 d ω i+7 i=1 dt Ωi. 9 ω i 0 i =1, 2,, 14 (9) C 7 [t a,t d ] (9) C 7 [t a,t d ] (9) Sobolev
6 322 T. SICE Vol.52 No.6 June 2016 Table 2 Experimental data after isomorphism transformation 7 Ω(t) C 7 [t a,t b ] (X Y Z) Ω[1, 3](t) =(Ω 1(t), Ω 2(t), Ω 3(t)). Fig. 5 y = Z(t) and its mean (Yaw Pitch Roll) Ω[4, 6](t) =(Ω 4(t), Ω 5(t), Ω 6(t)). Table 3 Important points in brush motion n 7 R(n) = { Ω k (t) C 7 [t a,t d ]:1 k n } n-7 n C 7 [t a,t d ] dynamical system 4. 5 Table 1 Table 2 Table 1 S GP 1 GP 2 GP 3 E Table 2 GP 1 GP 2 GP 3 GP 1: [0, 2.7], GP 2: [2.7, 10.1], GP 3: [10.1, 12.3] 4. 6 Fig. 5 10) Table [P S, P 1], [P 1,P 2],, [P 5,P E ] Fig. 5 Fig. 13 Fig. 5 Table n χ 2
7 Fig. 9 Wavelet decomposition of y = d dt Z(t) Fig. 6 y = d Z(t) and its mean dt Fig. 10 y = dx(t) dt and its target curve Fig. 7 y = Z(t) and its Min-Max curve Fig. 11 y = Yaw (t) total variation Fig. 8 y = d Z(t) and its Min-Max curve dt 10) ( ) n Step [ ] ( ) () 10) [ ] [Step 1] 2 1
8 324 T. SICE Vol.52 No.6 June 2016 Fig. 12 Function y = λ T (t) Fig. 13 Function y =Φ T (t) 3 2 n-7 3 [Step 2] 1 P 1 P 2, P k P S P E Table 3 P S P 1 P 2 P k P E [P S, P 1] [P 1,P 2],, [P k, P E ] 2 [Step 3] 2 n-7 1 Q S Q 1 Q 2, Q k Q E Table 3 2 Q S Q 1 Q 2 Q k Q E [Q S, Q 1] [Q 1,Q 2],, [Q k, Q E ] [Step 4] 1 n n-7 Table 3 Table 2 13 (7) t [seconds] y = f(t) f(t) Fig. 5 Fig C 7 [t a,t d ] 6. 4 Ω(t) C 7 [t a,t d ] Min-Max 6. 1 n
9 n-7 n-7 n Min-Max n-7 n-7 Min-Max Ω k (t) k =1, 2, n (10) Ω 0 (t) max 1 k n Ωk Ω 0 S = min max 1 j n, 1 k n Ωk Ω j S. 10 5) 5) 11) n-7 (10) Min-Max Ω 0 (t) Ω 0 (t) Sobolev center Min-Max n (10) 0 n-7 0 (11) (11) 3 2 (11) n-7 (i) (ii) I(n ratio) (iii) I(n index) I(n pfy) (Skill level) I(n R) = min 1 j n, max 1 k n Ωk Ω j S, I(n R) I(n ratio) = Ω 0 (t) if S Ω0 (t) S 0, I(n index) = log(i(n ratio)) if Ω 0 (t) S 0, I(n pfy) =round (I(n index), 2). 11 I(n R) = , I(n ratio) = , I(n index) = , I(n pfy) = ( 0.01) 0.56 Fig. 7 Fig. 8 Ω k (t) k =1, 2, 13 Min-Max (10) Min-Max 14 Min-Max Min-Max
10 326 T. SICE Vol.52 No.6 June (11) 4 8 6) n-7 (Lattice) 15) 7. 1 (c) 7) 7 9 n n-7 Z Fig. 7 Fig. 8 1 Fig. 7 (a-1) (b-1) Fig. 8 (c-1) (d-1) 2 (a-1) (c-1) (a-2) (b-2) (a-2) (a-1) (c-2) (a-k) (a-(k+1)) k 17) n-7 R(n) = { Ω k (t) C 7 [t a,t d ]:1 k n } where Ω k (t) =(Ω k 1(t),, Ω k 7(t)) C 7 [t a,t d ]. Ω j (t) = ( Ω j 1 (t),, Ωj 7 (t)) C 7 [t a,t d ] Ω j i (t) Ω 17) [ΩA ΩD] =wavedec(ω,n, name ). 15 Ω name N 2 [ΩA ΩD] ΩA ΩD Ω(t) (Core Orbit) Ω C (t) = ( Ω C 1 (t),, Ω C 7 (t) ). 16 (14) Ω j (t) Ω C (t) Fig. 9 Fig Fig. 9
11 ) (7) (controllable) (1) (5) Ω(t) =(Ω 1(t), Ω 7(t)), t [t a,t d ] Φ(t) t Φ(t) [Ω 7(t) ϕ(t)]dt t a where { 6 }1 ( ) 2 2 d ϕ(t) =λ(t) δ k dt Ω k(t). 17 k=1 λ(t) Φ(t) (18) (19) δ k k =1, 2,, 6 λ(t) 0 for all t [t a,t d ], { 0 for all t [t a,t d ], Φ(t) =0 at t = t d (18) (19) (Condition of controllability) λ(t) Φ(t) (Adjustment function) (Verification function) 8 (3) ( ) (1) (3) (1) (2) 18) [t a,t d ] 18) n-7 (3) (18) (19) (1) (3) [I] [III] (Target curve)
12 328 T. SICE Vol.52 No.6 June 2016 Ω T (t) = ( Ω T 1 (t), Ω T 2 (t),, Ω T 7 (t) ) [I] Ω T k (t) =(1+ρ(t)) Ω c k(t), 1 k 7, t [t a,t d ]. 20 ρ(t) ρ(t) I(n ratio) [II] td d td dt ΩT k (t) dt d dt Ωj k (t) dt, 21 t a min 1 j n t a 4 k 6. Ω j (t) = ( Ω j 1 (t), Ωj 2 (t),, Ωj 7 (t)) j(1 j n) [III] t Φ T (t) [Ω T 7 (t) ϕ T (t)]dt t a where { 6 ( ϕ T (t) =λ T d (t) δ k dt ΩT k (t) k=1 ) 2 }1 2 Φ T (t) 0 t [t a,t d ], Φ T (t d )=0. λ T (t) =μ T (t) 1 n λ j (t), t [t a,t d ]. n j= λ j (t) j (1 j n) μ T (t) n-7 Ω k (t) (17) λ j (t) λ T (t) (19) Ω T (t) Ω T (t) =μ T (t)ω C (t) Ω T (t) Fig. 10 Fig. 11 [I] [III] (a) (d) (a) (20) ρ(t) ρ(t) 0.059, t [t a,t d ] I(n ratio) (cf (12)) 26 Ω T k (t) Fig. 10 I (b) (21) Fig. 11 II (c) (24) μ T (t) (24) λ T (t) Fig. 12 λ T (t) (18) (d) (c) λ T (t) (22) Φ T (t) Fig. 13 Φ T (t) (19) (23) (24) III (a) (d) I III (1) C 7 [t a,t d ] (2) 1 (3) 7 9 i)
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