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# Identification Procedure for McKibben Pneumatic Artificial Muscle Systems

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Kogiso, Sawano, Itto, and Sugimoto, "Identification Procedure for McKibben Pneumatic Artificial Muscle Systems." presented in IROS 2013

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### Identification Procedure for McKibben Pneumatic Artificial Muscle Systems

1. 1. IEEE/RSJ International Conference on Intelligent Robots and Systems October 7-12, 2012 Vilamoura, Algarve, Portugal Identiﬁcation Procedure for McKibben Pneumatic Artiﬁcial Muscle Systems K. Kogiso, K. Sawano, T. Itto, and K. Sugimoto Nara Institute of Science and Technology (NAIST), Japan Oct. 10, 2012 @ WedBT5, 9:30 to 9:45 am, Regular Session, Gemini 2, Tivoli Marina Vilamoura13年1月30日水曜日
2. 2. Outline Introduction Modeling of PAM system Identiﬁcation Procedure Experimental Validation Extension Conclusion Active Link, Co. 213年1月30日水曜日
3. 3. Introduction McKibben Pneumatic Artiﬁcial Muscle (PAM) rubber tube mesh Advantage for application Disadvantage for modeling & control high power/weight ratio complex & nonlinear system (hydrodynamics) ﬂexibility empirical approximation or linearization 313年1月30日水曜日
4. 4. Introduction Motivation Mathematical modeling of PAM is a challenging issue. L0 L l nonlinearities such as hysteresis, hydrodynamics, friction,... solenoid PDC valve valve diﬃculty to explain validity of approximation or linearization. dependence on what kind of a valve you use. air compressure M M PAM system = PAM + proportional directional control (PDC) valve 0.3 Mathematical modeling of PAM system w/ constant weight. 0.2 [Itto, Kogiso: Hybrid modeling of McKibben pneumatic artiﬁcial muscle systems, IEEE ICIT&SSST, 2011] ε formulates model structure based on dynamics, but 0.1 M = 3 [kg] requires complete try and errors for identifying parameters. M = 6 [kg] M = 9 [kg] 0 hysteresis loop 100 200 300 400 500 600 700 pressure [kPa] 413年1月30日水曜日
5. 5. Introduction Motivation Mathematical modeling of PAM is a challenging issue. L0 L l nonlinearities such as hysteresis, hydrodynamics, friction,... solenoid PDC valve valve diﬃculty to explain validity of approximation or linearization. dependence on what kind of a valve you use. air compressure M M PAM system = PAM + proportional directional control (PDC) valve 0.3 Mathematical modeling of PAM system w/ constant weight. 0.2 [Itto, Kogiso: Hybrid modeling of McKibben pneumatic artiﬁcial muscle systems, IEEE ICIT&SSST, 2011] ε formulates model structure based on dynamics, but 0.1 M = 3 [kg] requires complete try and errors for identifying parameters. M = 6 [kg] M = 9 [kg] 0 hysteresis loop Outcomes 100 200 300 pressure 400 [kPa] 500 600 700 a parameter identiﬁcation procedure supported by analysis of the mathematical model, which contributes to reduce the cost for try and errors. an identiﬁed model validated by comparison with several experimental data, which well simulates behaviors of a practical PAM system. an extension to a model expressing the PAM system over a speciﬁed weight range, which is realized by interpolation of some dominant parameters in terms of a weight. 413年1月30日水曜日
6. 6. Modeling Dynamics of PAM system [Itto, Kogiso, IEEE ICIT&SSST 11] switched system with 64 nonlinear subsystems x(t) = f (x(t), u(t)) if x(t) 2 S ˙ y(t) = h(x(t)) T x := [✏ ✏ P ]T y := [✏ F ] ˙ S := {x 2 <3 | (x)  0} 2 {1, 2, · · · , 64} 513年1月30日水曜日
7. 7. Modeling Dynamics of PAM system [Itto, Kogiso, IEEE ICIT&SSST 11] switched system with 64 nonlinear subsystems x(t) = f (x(t), u(t)) if x(t) 2 S ˙ y(t) = h(x(t)) T x := [✏ ✏ P ]T y := [✏ F ] ˙ S := {x 2 <3 | (x)  0} 2 {1, 2, · · · , 64} dynamic equation (w/ friction [Kikuue, IEEE TRO 06] ) 8 > F (P, ✏, t) M g Ff (t) > < M L¨(t) = ✏ K(L0 L(1 ✏(t)))3 , if ✏(t)  L LL0 , > > : F (P, ✏, t) M g F (t), f otherwise, 8 > cv L✏(t) + cc sgn(✏(t)), ˙ ˙ if ✏(t) 6= 0, ˙ > > < c , if ✏(t) = 0 and Fo (t) > cc , ˙ c Ff (t) = > Fo (t), if ✏(t) = 0 and Fo (t) 2 [ cc , cc ], > ˙ > : cc , if ✏(t) = 0 and Fo (t) < cc , ˙ 513年1月30日水曜日
8. 8. Modeling Dynamics of PAM system PAM volume [Kagawa, CEP 97], [Minh, Mechatronics 10] [Itto, Kogiso, IEEE ICIT&SSST 11] V (t) = D1 ✏(t)2 + D2 ✏(t) + D3 switched system with 64 nonlinear subsystems x(t) = f (x(t), u(t)) if x(t) 2 S ˙ y(t) = h(x(t)) T x := [✏ ✏ P ]T y := [✏ F ] ˙ S := {x 2 <3 | (x)  0} 2 {1, 2, · · · , 64} dynamic equation (w/ friction [Kikuue, IEEE TRO 06] ) 8 > F (P, ✏, t) M g Ff (t) > < M L¨(t) = ✏ K(L0 L(1 ✏(t)))3 , if ✏(t)  L LL0 , > > : F (P, ✏, t) M g F (t), f otherwise, 8 > cv L✏(t) + cc sgn(✏(t)), ˙ ˙ if ✏(t) 6= 0, ˙ > > < c , if ✏(t) = 0 and Fo (t) > cc , ˙ c Ff (t) = > Fo (t), if ✏(t) = 0 and Fo (t) 2 [ cc , cc ], > ˙ > : cc , if ✏(t) = 0 and Fo (t) < cc , ˙ 513年1月30日水曜日
9. 9. Modeling Dynamics of PAM system PAM volume [Kagawa, CEP 97], [Minh, Mechatronics 10] [Itto, Kogiso, IEEE ICIT&SSST 11] V (t) = D1 ✏(t)2 + D2 ✏(t) + D3 switched system with 64 nonlinear subsystems contraction force [Tondu, IEEE CSM 00], [Kang, ICRA 09] x(t) = f (x(t), u(t)) if x(t) 2 S ˙  n ⇣ ⌘ o2 Cq2 Pg (t) F (P, ✏, t) = APg (t) at a C q1 1 + e ✏(t) as y(t) = h(x(t)) T x := [✏ ✏ P ]T y := [✏ F ] ˙ S := {x 2 <3 | (x)  0} 2 {1, 2, · · · , 64} dynamic equation (w/ friction [Kikuue, IEEE TRO 06] ) 8 > F (P, ✏, t) M g Ff (t) > < M L¨(t) = ✏ K(L0 L(1 ✏(t)))3 , if ✏(t)  L LL0 , > > : F (P, ✏, t) M g F (t), f otherwise, 8 > cv L✏(t) + cc sgn(✏(t)), ˙ ˙ if ✏(t) 6= 0, ˙ > > < c , if ✏(t) = 0 and Fo (t) > cc , ˙ c Ff (t) = > Fo (t), if ✏(t) = 0 and Fo (t) 2 [ cc , cc ], > ˙ > : cc , if ✏(t) = 0 and Fo (t) < cc , ˙ 513年1月30日水曜日
10. 10. Modeling Dynamics of PAM system PAM volume [Kagawa, CEP 97], [Minh, Mechatronics 10] [Itto, Kogiso, IEEE ICIT&SSST 11] V (t) = D1 ✏(t)2 + D2 ✏(t) + D3 switched system with 64 nonlinear subsystems contraction force [Tondu, IEEE CSM 00], [Kang, ICRA 09] x(t) = f (x(t), u(t)) if x(t) 2 S ˙  n ⇣ ⌘ o2 Cq2 Pg (t) F (P, ✏, t) = APg (t) at a C q1 1 + e ✏(t) as y(t) = h(x(t)) T pressure change in a PAM [Richer, JDSMC 00] x := [✏ ✏ P ]T y := [✏ F ] ˙ ˙ ˙ RT V (t) S := {x 2 <3 | (x)  0} 2 {1, 2, · · · , 64} P (t) = k1 m(t) ˙ k2 P (t) V (t) V (t) dynamic equation (w/ friction [Kikuue, IEEE TRO 06] ) 8 > F (P, ✏, t) M g Ff (t) > < M L¨(t) = ✏ K(L0 L(1 ✏(t)))3 , if ✏(t)  L LL0 , > > : F (P, ✏, t) M g F (t), f otherwise, 8 > cv L✏(t) + cc sgn(✏(t)), ˙ ˙ if ✏(t) 6= 0, ˙ > > < c , if ✏(t) = 0 and Fo (t) > cc , ˙ c Ff (t) = > Fo (t), if ✏(t) = 0 and Fo (t) 2 [ cc , cc ], > ˙ > : cc , if ✏(t) = 0 and Fo (t) < cc , ˙ 513年1月30日水曜日
11. 11. Modeling Dynamics of PAM system PAM volume [Kagawa, CEP 97], [Minh, Mechatronics 10] [Itto, Kogiso, IEEE ICIT&SSST 11] V (t) = D1 ✏(t)2 + D2 ✏(t) + D3 switched system with 64 nonlinear subsystems contraction force [Tondu, IEEE CSM 00], [Kang, ICRA 09] x(t) = f (x(t), u(t)) if x(t) 2 S ˙  n ⇣ ⌘ o2 Cq2 Pg (t) F (P, ✏, t) = APg (t) at a C q1 1 + e ✏(t) as y(t) = h(x(t)) T pressure change in a PAM [Richer, JDSMC 00] x := [✏ ✏ P ]T y := [✏ F ] ˙ ˙ ˙ RT V (t) S := {x 2 <3 | (x)  0} 2 {1, 2, · · · , 64} P (t) = k1 m(t) ˙ k2 P (t) V (t) V (t) net mass ﬂow rate of PDC valve dynamic equation (w/ friction [Kikuue, IEEE TRO 06] ) m(t) = ↵(t)mi (t) (1 ↵(t))mo (t) ˙ ˙ ˙ 8 8 r ⇣ ⌘k 1 > F (P, ✏, t) M g Ff (t) > > > A Pp k > 0 tank R k+1 2 k+1 < > > > > T ⇣ ⌘ kk 1 > M L¨(t) = ✏ K(L0 L(1 ✏(t)))3 , if ✏(t)  L LL0 , > > < if P (t)  2 k+1 Ptank , > > mi (t) = ˙ r : > > > q ⇣ ⌘k 1 ⇣ ⌘ kk 1 F (P, ✏, t) M g F (t), f otherwise, > A0 Pp > > > tank T 2k P (t) R(k 1) Ptank 1 P (t) Ptank > > ⇣ ⌘ kk 1 > 8 : if P (t) > 2 Ptank , > cv L✏(t) + cc sgn(✏(t)), ˙ ˙ if ✏(t) 6= 0, ˙ 8 r k+1 > > > ⇣ ⌘k 1 k+1 < c , if ✏(t) = 0 and Fo (t) > cc , ˙ > A P (t) k > 0p > > 2 R k+1 c > > T ⇣ ⌘ kk 1 Ff (t) = > > > 2 > Fo (t), if ✏(t) = 0 and Fo (t) 2 [ cc , cc ], > ˙ < if P (t) k+1 Pout , > : mo (t) = ˙ > q ⇣ ⌘1 r ⇣ ⌘ kk 1 > > cc , if ✏(t) = 0 and Fo (t) < cc , ˙ > A0 P (t) > > p 2k Pout k R(k 1) P (t) 1 Pout P (t) > > T ⇣ ⌘ kk 1 > > : if P (t) 2 < Pout . k+1 513年1月30日水曜日
12. 12. Analysis     Dominant parameters switched system with 64 nonlinear subsystems x(t) = f (x(t), u(t)) if x(t) 2 S ˙ y(t) = h(x(t)) x := [✏ ✏ P ]T y := [✏ F ]T ˙ 613年1月30日水曜日
13. 13. Analysis     Dominant parameters M : mass of weight [kg] D0 : natural diameter of PAM [m] switched system with 64 nonlinear subsystems L0 : natural length of PAM [m] x(t) = f (x(t), u(t)) if x(t) 2 S ˙ D1 D2 D3 : coeﬃcients for PAM volume [m^3] y(t) = h(x(t)) Ptank : source absolute pressure [Pa] x := [✏ ✏ P ]T y := [✏ F ]T ˙ Pout : atmospheric pressure [Pa] k : speciﬁc heat ratio for air [-] R : ideal gas constant [J/kg K] T : absolute temperature [K] K : coeﬃcient of elasticity [N/m^3] ✓ : initial angle btw braided thread & cylinder long axis [deg] Cq1 : correction coeﬃcient [-] Cq2 : correction coeﬃcient [1/Pa] cc : Coulomb friction [N] A0 : oriﬁce area of PDC valve [m^2] k1 k2 : polytropic indexes [-] cv : viscous friction coeﬃcient [Ns/m] 613年1月30日水曜日
14. 14. Analysis     Dominant parameters M : mass of weight [kg] D0 : natural diameter of PAM [m] switched system with 64 nonlinear subsystems L0 : natural length of PAM [m] x(t) = f (x(t), u(t)) if x(t) 2 S ˙ D1 D2 D3 : coeﬃcients for PAM volume [m^3] y(t) = h(x(t)) Ptank : source absolute pressure [Pa] x := [✏ ✏ P ]T y := [✏ F ]T ˙ Pout : atmospheric pressure [Pa] k : speciﬁc heat ratio for air [-] R : ideal gas constant [J/kg K] Analysis result: T : absolute temperature [K] For the PAM system model, : coeﬃcient of elasticity [N/m^3] K its steady-state behavior is characterized by : initial angle btw braided thread ✓ & cylinder long axis [deg] parameters: K ✓ Cq1 Cq2 cc : correction coeﬃcient [-] Cq1 and its transient behavior is characterized by Cq2 : correction coeﬃcient [1/Pa] cc : Coulomb friction [N] parameters: A0 k1 k2 cv A0 : oriﬁce area of PDC valve [m^2] k1 k2 : polytropic indexes [-] Hint: as t ! 1 , then params left or not. cv : viscous friction coeﬃcient [Ns/m] 613年1月30日水曜日
15. 15. Analysis     Dominant parameters M : mass of weight [kg] D0 : natural diameter of PAM [m] switched system with 64 nonlinear subsystems L0 : natural length of PAM [m] x(t) = f (x(t), u(t)) if x(t) 2 S ˙ D1 D2 D3 : coeﬃcients for PAM volume [m^3] y(t) = h(x(t)) Ptank : source absolute pressure [Pa] x := [✏ ✏ P ]T y := [✏ F ]T ˙ Pout : atmospheric pressure [Pa] k : speciﬁc heat ratio for air [-] R : ideal gas constant [J/kg K] Analysis result: T : absolute temperature [K] For the PAM system model, : coeﬃcient of elasticity [N/m^3] K its steady-state behavior is characterized by : initial angle btw braided thread ✓ & cylinder long axis [deg] parameters: K ✓ Cq1 Cq2 cc : correction coeﬃcient [-] Cq1 and its transient behavior is characterized by Cq2 : correction coeﬃcient [1/Pa] cc : Coulomb friction [N] parameters: A0 k1 k2 cv A0 : oriﬁce area of PDC valve [m^2] k1 k2 : polytropic indexes [-] Hint: as t ! 1 , then params left or not. cv : viscous friction coeﬃcient [Ns/m] 613年1月30日水曜日
16. 16. Analysis     Dominant parameters M : mass of weight [kg] D0 : natural diameter of PAM [m] switched system with 64 nonlinear subsystems L0 : natural length of PAM [m] x(t) = f (x(t), u(t)) if x(t) 2 S ˙ D1 D2 D3 : coeﬃcients for PAM volume [m^3] y(t) = h(x(t)) Ptank : source absolute pressure [Pa] x := [✏ ✏ P ]T y := [✏ F ]T ˙ Pout : atmospheric pressure [Pa] k : speciﬁc heat ratio for air [-] R : ideal gas constant [J/kg K] Analysis result: T : absolute temperature [K] For the PAM system model, : coeﬃcient of elasticity [N/m^3] K its steady-state behavior is characterized by : initial angle btw braided thread ✓ & cylinder long axis [deg] parameters: K ✓ Cq1 Cq2 cc : correction coeﬃcient [-] Cq1 and its transient behavior is characterized by Cq2 : correction coeﬃcient [1/Pa] cc : Coulomb friction [N] parameters: A0 k1 k2 cv A0 : oriﬁce area of PDC valve [m^2] k1 k2 : polytropic indexes [-] Hint: as t ! 1 , then params left or not. cv : viscous friction coeﬃcient [Ns/m] Note, no couplings btwn the two param groups. 613年1月30日水曜日
17. 17. Achievement Identiﬁcation procedure -5 x 10 7 PAM volume: PAM volume 6 measurable or known in advance V (t) = V [m ] 3 5 D1 ✏(t)2 + 4 M D0 L0 D1 Ptank Pout k T R D2 ✏(t) + D3 3 2 contraction ratio 0 0.1 0.2 0.3 ε steady-state behavior transient behavior Determines the parameters value: Determines the parameters value: K ✓ Cq1 Cq2 cc A0 k1 k2 cv until model maker satisﬁes. until model maker satisﬁes. 713年1月30日水曜日
18. 18. Achievement Identiﬁcation procedure -5 x 10 7 PAM volume: PAM volume 6 measurable or known in advance V (t) = V [m ] 3 5 D1 ✏(t)2 + 4 M D0 L0 D1 Ptank Pout k T R D2 ✏(t) + D3 3 2 contraction ratio 0 0.1 0.2 0.3 ε steady-state behavior transient behavior Determines the parameters value: Determines the parameters value: K ✓ Cq1 Cq2 cc A0 k1 k2 cv until model maker satisﬁes. until model maker satisﬁes. When satisﬁed? Since determination of values is subjective, observe a trend by parameter variation. 713年1月30日水曜日
19. 19. Info.  for  observing Trend by parameter variation 0.3 0.3 0.3 0.2 0.2 0.2 cq  increases cc increases ε ε ε cq  increases 0.1 0.1 0.1 cc = 0 cq  = 0.8 cq  = - 1/0.01 x10-6 cc = 4.875/P (oo) x105 cq  = 0.99 cq  = - 1/0.083 x10-6 0 cc = 9.75 / P (oo) x105 0 0 cq  = 1.2 cq  = - 1/0.2 x10-6 100 200 300 400 500 600 700 100 200 300 400 500 600 700 100 200 300 400 500 600 700 pressure [kPa] pressure [kPa] pressure [kPa] param in contraction force param in contraction force Coulomb friction coeﬃcient 0.26 0.26 0.26 ε εε 0.24 0.24 0.24 k  increases k  increases c v increases A increases 0.22 0.22 0.22 0 10 20 0 10 20 0 10 20 pressure [kPa] pressure [kPa] 550pressure [kPa] 550 550 -6 k = 1.0, k = 1.0 c v = 10 A = 0.058 x10 -6 k = 1.0, k = 1.4 A increases A = 0.099 x10 500 k = 1.4, k = 1.0 500 c v = 500 500 -6 c v = 1000 A = 0.176 x10 k = 1.4, k = 1.4 450 k  increases k  increases 450 450 0 10 20 0 10 20 0 10 20 time [s] time [s] time [s] polytropic indexes viscous friction coeﬃcient max oriﬁce area 813年1月30日水曜日
20. 20. Experimental  Validation PAM system setup for model validation proportional directional control valve How to validate: Input a step signal to the PDC valve, and check steady-state and transient responses of simulation and experiment. 913年1月30日水曜日
21. 21. Experimental  Validation Comparison: steady state behavior in e vs P 0.30 0.30 0.25 M = 4.0 [kg] 0.25 M = 5.0 [kg] 0.20 0.20 0.15 0.15 ε ε 0.10 0.10 0.05 0.05 experiment experiment model fixed at M=4 model fixed at M=5 0 0 model parametried by M model parametried by M 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 P [kPa] P [kPa] experiment 0.30 experiment 0.30 model fixed at M=7 model fixed at M=8 0.25 model parametried by M 0.25 model parametried by M 0.20 0.20 0.15 0.15 ε ε 0.10 0.10 0.05 0.05 0 M = 7.0 [kg] 0 M = 8.0 [kg] 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 P [kPa] P [kPa] 1013年1月30日水曜日
22. 22. Experimental  Validation Comparison: transient behavior in e vs t & P vs t M = 4.0 [kg] 1113年1月30日水曜日
23. 23. Extension  to  M-‐‑‒parameterized  Model Interpolation over [1, 9] in weight 1.15 18 1.1 16 1.05 14 1 12 Cq1 [-] 10 cc [N] 0.95 8 0.9 6 0.85 4 0.8 Cq1 (M ) = 0.1573 log M + 0.7974 2 cc (M ) = 1.7353M + 0.1422 0.75 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 M [kg] M [kg] x 10 7 8 40 7 38 0.7915M 0.2296 K(M ) = 109600 exp 36 ✓(M ) = 39.984M 6 34 5 K [N/m] 32 θ [rad] 4 30 3 28 2 26 1 24 0 22 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 M [kg] 12 M [kg]13年1月30日水曜日
24. 24. Extension  to  M-‐‑‒parameterized  Model Interpolation over [1, 9] in weight 1.15 18 1.1 16 1.05 14 1 12 Cq1 [-] 10 cc [N] 0.95 8 0.9 6 0.85 4 0.8 Cq1 (M ) = 0.1573 log M + 0.7974 2 cc (M ) = 1.7353M + 0.1422 0.75 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 M [kg] M [kg] x 10 7 8 40 7 38 0.7915M 0.2296 K(M ) = 109600 exp 36 ✓(M ) = 39.984M 6 34 5 K [N/m] 32 θ [rad] 4 30 3 28 2 26 1 24 0 22 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 M [kg] 12 M [kg]13年1月30日水曜日
25. 25. Extension  to  M-‐‑‒parameterized  Model Comparison: steady state behavior in e vs P 0.30 0.30 0.30 0.25 M = 4.0 [kg] 0.25 M = 4.5 [kg] 0.25 M = 5.0 [kg] 0.20 0.20 0.20 0.15 0.15 0.15 εε ε 0.10 0.10 0.10 0.05 0.05 0.05 experiment experiment model fixed at M=4 experiment model fixed at M=5 0 0 model parametried by M 0 model parametried by M model parametried by M 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 P [kPa] P [kPa] P [kPa] experiment 0.30 experiment 0.30 experiment 0.30 model fixed at M=7 model parametried by M model fixed at M=8 0.25 model parametried by M 0.25 0.25 model parametried by M 0.20 0.20 0.20 0.15 0.15 0.15 ε εε 0.10 0.10 0.10 0.05 0.05 0.05 0 M = 7.0 [kg] 0 M = 7.5 [kg] 0 M = 8.0 [kg] 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 P [kPa] P [kPa] P [kPa] 1313年1月30日水曜日
26. 26. Extension  to  M-‐‑‒parameterized  Model Comparison: transient behavior in e vs t & P vs t M = 4.5 [kg] M = 7.5 [kg] 1413年1月30日水曜日
27. 27. Conclusion Summary a mathematical model of a PAM system (PAM + PDC valve) with a constant weight, which involves 11 measurable parameters and 9 need-to-be-identiﬁed parameters. a parameter identiﬁcation procedure supported by analysis of the mathematical model, which contributes to reduce the cost for try and errors in ﬁnding the 9 parameter values. an identiﬁed model validated by comparison with several experimental data, which well simulates behaviors of a practical PAM system. a mathematical model expressing the PAM system over a speciﬁed weight range, which is also identiﬁable by using the proposed procedure plus interpolation. Future works an automatic identiﬁcation procedure that appropriately determines parameter values based on experimental sample data. an antagonistic layout of PAMs as an actuator to realize position/force controls appropriate for applications of power-assist systems or rehabilitation/training exoskeleton systems. a model reduction technique in case of practical use of many PAMs. 1513年1月30日水曜日
28. 28. Thank you for your kind attention. 1613年1月30日水曜日