Adaptive fuzzy dynamic surface control for pneumatic muscle systems with full-state constraints and disturbances

Abstract

In the era of intelligent revolution, pneumatic artificial muscle (PAM) actuators have gained significance in robotics, particularly for tasks demanding high safety and flexibility. Despite their inherent flexibility, PAMs encounter challenges in practical applications because of their complex material properties, including hysteresis, nonlinearity, and low response frequencies, which hinder precise modeling and motion control, limiting their widespread adoption. This study focuses on fuzzy logic dynamic surface control (DSC) for PAMs, addressing full-state constraints and unknown disturbances. We propose an improved neural DSC method, combining enhanced DSC techniques with fuzzy logic system approximation and parameter minimization for PAM systems. The introduction of a novel barrier Lyapunov function during system design effectively resolves full-state constraint issues. A key feature of this control approach is its single online estimation parameter update while maintaining stability characteristics akin to the conventional backstepping method. Importantly, it ensures constraint adherence even in the presence of disturbances. Lyapunov stability analysis confirms signal boundedness within the closed-loop system. Experimental results validate the algorithm’s effectiveness in enhancing control precision and response speed.

Graphical abstract

Keywords

adaptive fuzzy control / tracking control / PAM system / state constraints / input saturation

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Nomenclature

Abbreviations
BLF	Barrier Lyapunov function
DSC	Dynamic surface control
FLS	Fuzzy logic system
IAE	Integrated absolute error
ITAE	Integrated time absolute error
NI	National Instruments
NN	Neural network
PAM	Pneumatic artificial muscle
PID	Proportional–integral–derivative
Variables
b1	Coefficients of damping polynomial components (order 1)
b2	Coefficients of damping polynomial components (order 2)
bik	kth-order polynomial coefficient for ith-order damping
bi(p)	ith-order damping coefficient
d0	Calibration coefficient (pressure to voltage)
d2	Disturbance Term
e1	Primary tracking error variable
e2	Secondary error variable
f2(·)	Nonlinear function
fce	Nominal value of f1
fk	kth-order polynomial coefficient for contractile force
f(p)	Force component
g	Gravitational force
g1	Primary control gain
g10	Nominal control gain of the first subsystem
g2	Secondary control gain
g2(·)	Nonlinear function
g20	Nominal control gain of the second subsystem
gj0	Generalized nominal control gain for the jth subsystem
kik	kth-order polynomial coefficient for ith-order stiffness
ki(p)	Coefficients for the ith-order spring components
kλi	Respective positive upper bound for i = 1,2
k_{x_1}	State feedback gain
k_{x_{i+1}}	Recursive state gain
kη	Compensator gain
k_{{\omega }_{i+1}}	Filter gain
l_{\text{1}}	Length of link AB in linkage mechanism
l_{\text{2}}	Length of link BC in linkage mechanism
l_{\text{3}}	Length of link CD in linkage mechanism
l_{\text{4}}	Geometric parameter in kinematic mode
m	Nominal Mass
n	Orders of the spring polynomials
N	Degree of the polynomial approximation used
p0	Nominal pressure
p	Actual control input affecting the system
p(t)	Pressure of the injected air
s(ν)	Smooth hyperbolic tangent function
u(ν)	System saturation input
u(t)	Saturated control input
V1	Primary Lyapunov function
V2	Lyapunov function
vc	Thigh velocity at point C
x1(t)	Angle state
x2(t)	Velocity state
y	Pneumatic muscle’s initial length
yt	Reference signal
θ	Function representing unmodeled dynamics or disturbances
θ2	Angle between the pneumatic muscle and the fixed connector
\hat{\theta }	Estimation for θ
\tilde{\theta }	Estimation error for θ
φ3	Angular displacement the thigh
φ30	Initial angular displacement of the thigh
φ(s)	Combines state and filter dynamics
ω1	Output of the primary designed filter
ω2	Output of the second low-pass filter
ωDC	Angular velocity of link DC (link 3)
ωBC	Angular velocity of link BC (link 2)
ωAB	Angular velocity of link BC (link 1)
α1	Input of the primary designed filter
α2	Virtual control input
ϕ1	Basis function subset
ψ3	Angular displacement of thigh rotation induced by PAM contraction
ψ30	Initial angular position of thigh when PAM is slack
ψ4	Auxiliary angle in kinematic analysis
ψ40	Initial value of ψ4 when PAM is unpressurized
ψ(s)	Basis function vector
ε1m	Upper bound of approximation error
{\epsilon }_2^{\ast }(\cdot )	State-dependent approximation error
εm	Global maximum approximation error across all design steps
σ1	Adaptive law tuning parameter
σ2	Adaptive law leakage term
ν	Actual control input voltage to the proportional valve
\varsigma	Composite variable in Lyapunov stability analysis
μ	Linearization parameter in mean-value theorem for input saturation handling.
λi	Compensating variable for tracking errors
η1, η2	Compensator signals
τ2	Filter time constant
Δfc1	Mismatch of f1
Δm	Uncertain part of m

Acknowledgements

This study was supported by the National Key Research and Development Program of China (Grant No. 2022YFF0708903), in part by the Ningbo Key Technology Research and Development Program of China (Grant No. 2023Z018), in part by the National Natural Science Foundation of China (Grant No. 52275043), and in part by the Zhejiang Lab Open Research Project of China (Grant No. 121001-AB2212).

Conflict of Interest

The authors declare no conflict of interest.

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