We have proposed a precise hybrid nonlinear model of the PAM by replacing a Coulomb friction coefficient with a pressure-dependent one. It was confirmed that the proposed model can express the nonlinear behaviors of several commercial PAMs. This slide was used at my presentation at IEEE Multi-conference on Systems and Control, Sydney, Australia, 2015.

1. September 23, 2015 @WeB02.4, 15:20 to 15:40 pm, Regular CCA Session, Modeling, Room Cutler
Hybrid Nonlinear Model of McKibben
Pneumatic Artificial Muscle Systems
Incorporating a Pressure-Dependent
Coulomb Friction Coefficient
Kentaro Urabe Nara Institute of Science and Technology, Japan
Kiminao Kogiso University of Electro-Communications, Japan
2015 IEEE Multi-Conference on Systems and Control
21-23 September 2015, Novotel Sydney Manly Pacific, Sidney, Australia
Supported by
JSPS Grant-in-Aid for Young Scientists (A)

2. Outline
2
Introduction
Model of PAM System
Analysis
Validations
Conclusion
PAMs (ActiveLink, Co.)

3. Introduction
3
McKibben Pneumatic Artificial Muscle
high power/weight ratio, flexibility
appropriate use for power assist and
rehabilitation system, as a soft actuator.
nonlinearities such as hysteresis,
hydrodynamics, friction,…
Modeling and control of the PAM are known
to be challenging issues. [Tondu, CSM 00]
solenoid
air
L
M
M
L l0
compressure
valve
100 200 300 400 500 600 700
0
0.1
0.2
0.3
pressure [kPa]
ε
M [kg]= 3
M [kg]= 6
M [kg]= 9
rubber tube mesh
hysteresis
expansion
contraction

4. Nonlinear model under inner pressure range of 0.2 to 0.7 [kPa] (flexible & stiff)
Hybrid modeling with a proportional directional control valve
Heuristic parameter estimation based on model analysis
Gray-box modeling with consideration of load-dependent parameters
Gray-box modeling in game-theoretic learning
Numerical consideration of PID vs MPC using our model
Our interests: We can get the model more precise? Our proposed model is general?
Basically, linear model under inner pressure range of 0.5 to 0.7 [kPa] (stiff)
Tracking control using inverse model and state feedback
Positioning control for polynomial model using PI-type hysteresis compensator
Maxwell-slip model to capture the force/length hysteresis
Piecewise-affine modeling and constrained control
Introduction
4
Existing works
Our previous works
[T.V. Minh et al., 10]
[L. Udawatta et al., 07]
[T. Itto, et al., ICIT11]
[Kogiso, et al., IROS12]
[Kogiso, et al., IROS13]
[Kogiso, et al., AIM13]
[T.V. Minh et al., 11]
[Urabe, et al., ISIC15]
[G. Andrikopoulos et al., 13]

5. Introduction
5
Objective of this study
To obtain a precise model of the PAM system,
propose to replace a Coulomb coefficient with a pressure-dependent one,
check if the proposed model enables to have behaviors of different commercial PAMs.
100 200 300 400 500 600 700
0
0.05
0.10
0.15
0.20
0.25
pressure [KPa]
contractionratio
Satisfy the previous result?
ActiveLink TAA10（Φ10 0.25 m）
What if the PAM was discontinued?
Simulation
Experiment
&

6. Analysis
6
Dominant parameters
˙x(t) = f (x(t), u(t))
y(t) = h(x(t))
x(t) 2 S
x := [✏ ˙✏ P]T
y := [✏ F]T
Switched system with 12 subsystems
if L0
D0
D1 D2 D3
M
K
R
T
✓
cc
A0
Cq1
Cq2
k1 k2
cv
Ptank
Pout
k
: natural diameter of PAM [m]
: natural length of PAM [m]
: source absolute pressure [Pa]
: coefficients for PAM volume [m^3]
: atmospheric pressure [Pa]
: specific heat ratio for air [-]
: ideal gas constant [J/kg K]
: absolute temperature [K]
: coefficient of elasticity [N/m^3]
: initial angle btw braided thread
& cylinder long axis [deg]
: correction coefficient [-]
: correction coefficient [1/Pa]
: Coulomb friction [N]
: viscous friction coefficient [Ns/m]
: orifice area of PDC valve [m^2]
: polytropic indexes [-]
: mass of weight [kg]
Analysis result:
For the PAM system model,
its steady-state behavior is characterized by
and its transient behavior is characterized by
A0 k1 k2 cv
load-dependent parameters:
parameters:
K(M) (M) Cq1(M) Cq2(M) cc(M)
h cc
P (t)h
updated!
[Kogiso, et al., IROS12]

7. Validation
7
Equipment
proportional
directional
control valve

8. Validation: Model Precision
8
0.18
0.20
0.22
0.24
0.26
0.28
0 5 10 15 20 25 30 35 40
0 5 10 15 20 25 30 35 40
time [s]
300
350
400
450
500
550
600
650
700
contractionratiopressure[kPa]
experimental result
simulation result
Comparison
transient responsesteady-state response
proposed
previous
c0
c = h
cc
P(t)
100 200 300 400 500 600 700
0
0.05
0.10
0.15
0.20
0.25
pressure [KPa]
contractionratio
100 200 300 400 500 600 700
0
0.05
0.10
0.15
0.20
0.25
pressure [KPa]
contractionratio
ActiveLink TAA10（Φ10 0.25 m）
c0
c = cc

9. Validation: Model Precision
9
2
4
6
8
10
1 2 3 4 5 6 7 8 9
M [Kg]
cc (pressure-dependent)
cc (consntant)
x 103
errorbetweenresponses
0.5
1.0
1.5
2.0
2.5
3.0
3.5
x 105
1 2 3 4 5 6 7 8 9
M [Kg]
cc (pressure-dependent)
cc (consntant)
errorbetweenresponses
Errors over up to 9 kg weights
steady-state response transient response
proposed
previous
previous
proposed
ActiveLink TAA10（Φ10 0.25 m）

10. Model Validation
10
ActiveLink TAA10（Φ10 0.25 m） FESTO DMSP-10-250N（Φ10 0.25 m）
FESTO DMSP-20-200N（Φ20 0.20 m） Kanda AirMuscle（Φ1.25 in 0.20 m）
Four types of commercial PAMs
(discontinued)

11. Model Validation: Parameters
11
ActiveLink TAA10 DMSP-10-250N DMSP-20-200N Kanda AirMuscle
[Kogiso, et al., IROS13]
cc

12. Model Validation
12
DMSP-10-250NActiveLink TAA10
0
0.05
0.10
0.15
0.20
0.25
contractionratio
100 200 300 400 500 600 700
pressure [kPa]
0
0.05
0.10
0.15
0.20
0.25
contractionratio
100 200 300 400 500 600 700
pressure [kPa]
100 200 300 400 500 600 700
pressure [kPa]
0
0.05
0.10
0.15
0.20
0.25
contractionratio
100 200 300 400 500 600 700
pressure [kPa]
0
0.05
0.10
0.15
0.20
0.25
contractionratio
Kanda AirMuscleDMSP-20-200N
Steady-state response

13. Model Validation
13
DMSP-10-250NActiveLink TAA10
Kanda AirMuscleDMSP-20-200N
0
0.05
0.10
0.15
0.20
0.25
contractionratio
100 200 300 400 500 600 700
pressure [kPa]
0
0.05
0.10
0.15
0.20
0.25
contractionratio
100 200 300 400 500 600 700
pressure [kPa]
100 200 300 400 500 600 700
pressure [kPa]
0
0.05
0.10
0.15
0.20
0.25
contractionratio
100 200 300 400 500 600 700
pressure [kPa]
0
0.05
0.10
0.15
0.20
0.25
contractionratio
Steady-state response

14. Model Validation
14
Kanda AirMuscleDMSP-20-200N
250
300
350
400
450
500
0.18
0.20
0.22
0.24
0.26
5 10 15 20 25 30 35 400
time [s]
5 10 15 20 25 30 35 400
pressure[kPa]contractionratio
0.04
0.06
0.08
0.10
0.12
0.14
5 10 15 20 25 30 35 400
time [s]
5 10 15 20 25 30 35 400
250
300
350
400
450
500
pressure[kPa]contractionratio
5 10 15 20 25 30 35 400
time [s]
5 10 15 20 25 30 35 400
250
300
350
400
450
500
pressure[kPa]
0.08
0.10
0.12
0.14
0.16
contractionratio
5 10 15 20 25 30 35 400
time [s]
5 10 15 20 25 30 35 400
250
300
350
400
450
500
pressure[kPa]
0.12
0.14
0.16
0.18
0.20
contractionratio
DMSP-10-250NActiveLink TAA10
Transient response

15. Model Validation
15
Kanda AirMuscleDMSP-20-200N
DMSP-10-250NActiveLink TAA10
250
300
350
400
450
500
0.18
0.20
0.22
0.24
0.26
5 10 15 20 25 30 35 400
time [s]
5 10 15 20 25 30 35 400
pressure[kPa]contractionratio
0.04
0.06
0.08
0.10
0.12
0.14
5 10 15 20 25 30 35 400
time [s]
5 10 15 20 25 30 35 400
250
300
350
400
450
500
pressure[kPa]contractionratio
5 10 15 20 25 30 35 400
time [s]
5 10 15 20 25 30 35 400
250
300
350
400
450
500
pressure[kPa]
0.08
0.10
0.12
0.14
0.16
contractionratio
5 10 15 20 25 30 35 400
time [s]
5 10 15 20 25 30 35 400
250
300
350
400
450
500
pressure[kPa]
0.12
0.14
0.16
0.18
0.20
contractionratio
Transient response

16. Conclusion
16
Summary
Introduction
Model of PAM System
replaced Coulomb coefficient w/ the pressure-dependent one.
Analysis
introduced dominant parameters of the proposed model.
Validation
enable to get model’s precision better.
enable to catch behaviors of commercial McKibben PAMs.
Future works
to consider an effective model reduction, in order to obtain
an appropriate model for controller design.
to develop a flexible actuator of antagonistic pairs of PMAs,
in order to realize a position/force control system.

17. 17
Thank you for your attention.