Adaptive super-twisting sliding mode control for robust tracking of wearable PAM-driven robotic manipulator

Abstract

Pneumatic Artificial Muscle (PAM)-actuated rehabilitation robots provide a safe and compliant alternative to traditional rigid motor-driven systems, mimicking the inherent viscoelasticity of human muscle. However, precise trajectory tracking is a difficult control problem due to their highly nonlinear pressure–force–contraction relationship, multibody leg dynamics, friction effects, and external disturbances. This paper proposes an Adaptive Finite Time Super-Twisting Sliding Mode Control (ASTSMC) strategy for robust tracking of a leg rehabilitation manipulator capable of performing hip/knee flexion-extension exercises. The human leg is represented as a planar 2-DOF system that includes the hip and knee joints. The Euler–Lagrange formulation is used to derive a comprehensive nonlinear dynamic model that includes viscous–Coulomb friction and antagonistic PAM torque generation. The controller employs a second-order sliding mode law to ensure a continuous control signal, significantly suppressing high-frequency oscillations. Additionally, in order to manage unknown lumped uncertainties without knowing the upper bounds beforehand, an adaptive gain law is integrated. Finite-time convergence of the tracking errors is confirmed by stability analysis using the Lyapunov method. Simulation results in MATLAB/Simulink validate that the proposed ASTSMC achieves superior trajectory tracking accuracy and robustness against parameter variations compared to standard sliding mode control schemes. The substantial reduction in chattering improves patient comfort and safety, proving that the proposed framework could be applied for advanced robotic neurorehabilitation in clinical settings.

Introduction

The field of physical therapy and neurorehabilitation has experienced a significant change, moving from labor-intensive manual exercises to the incorporation of advanced robotic systems intended to enhance motor function recovery. This shift is driven by the clinical observation that the human brain and spinal cord possess a remarkable capacity for adaptation and neural reorganization following injury, provided that specific movements are practiced consistently and repetitively¹. The dosage and length of therapy sessions are often restricted by the physical strain that traditional rehabilitation techniques place on therapists. In order to improve motor outcomes and promote neuroplasticity in patients with stroke, spinal cord injuries, or aging mobility decline, robotic systems, especially lower-limb rehabilitation robots, provide quantifiable, repetitive, and task-specific training^2,3. Neural and intelligent control strategies have further enhanced the effectiveness of such rehabilitation systems⁴.

Central to the development of these systems is the choice of actuation. Conventional rehabilitation robots have historically relied on hydraulic systems or electric motors with rigid gearing⁵. While these actuators offer high precision and high torque density, they are inherently non-compliant, which poses significant safety risks in human-robot interaction. Rigid actuators require complex, high-bandwidth force-feedback loops to ensure that the robot does not exert excessive force on a patient’s limb, particularly during involuntary muscle spasms⁶. This limitation has led researchers toward the adoption of soft actuation technologies, among which Pneumatic Artificial Muscles (PAMs), specifically the McKibben-type muscle, have emerged as a premier candidate for wearable and assistive devices⁷.

Pneumatic Artificial Muscles offer a unique set of advantages that make them ideal for medical rehabilitation. Their key features include a high power-to-weight ratio, simple structure, and compliance. Since PAMs operate based on the compressibility of air, they exhibit a natural elasticity similar to that of human skeletal muscles⁸. This soft characteristic ensures increased safety and comfort for patients, as the robot can easily yield to unexpected movements⁹. However, transitioning from rigid to soft actuation introduces certain complexities. The relationship between compressed air pressure, contraction length, and contracting force in a PAM is highly nonlinear, hysteretic, and time-varying. Consequently, precise trajectory tracking requires the use of nonlinear control techniques¹⁰.

Recent work on pneumatic artificial muscle (PAM) actuators has produced a wide range of robust control approaches designed to address nonlinearities, hysteresis, uncertainties, and input constraints. Many designs are based on sliding mode control (SMC) combined with observers or intelligent approximators^11,12. Intelligent hybrid and learning-based SMC frameworks have also been explored to enhance tracking performance in complex robotic systems¹³. Integral and terminal sliding mode variants further enhance robustness and finite-time convergence while reducing chattering effects^14,15. In parallel, adaptive and learning-based control methods have been developed to handle parametric uncertainties and time-varying dynamics in PAM systems¹⁶. Adaptive super-twisting strategies have also shown improved robustness for uncertain robotic manipulators¹⁷.

In addition to sliding mode-based strategies, Active Disturbance Rejection Control (ADRC) has emerged as a powerful alternative for handling uncertainties and disturbances in rehabilitation robotics. ADRC employs extended state observers to estimate and compensate for internal dynamics and external disturbances in real time, enabling robust performance without requiring an accurate system model. Recent studies have demonstrated the effectiveness of ADRC in lower-limb exoskeleton systems, including differential flatness-based ADRC combined with impedance control for enhanced human–robot interaction¹⁸, as well as observer-based ADRC frameworks for PAM-driven rehabilitation devices¹⁹. Neural-enhanced ADRC approaches have also been proposed to further improve disturbance estimation and adaptability in nonlinear robotic systems²⁰.

Furthermore, enhanced ADRC strategies have been successfully applied to trajectory tracking control of robotic rehabilitation systems under time-varying and nonlinear conditions²¹. While these approaches provide strong disturbance rejection capabilities, they typically rely on observer tuning and may not explicitly guarantee finite-time convergence of tracking errors. In contrast, higher-order sliding mode techniques, such as the proposed Adaptive Super-Twisting Sliding Mode Control (ASTSMC), offer inherent robustness with finite-time convergence properties and continuous control action. This distinction highlights the relevance of ASTSMC as an alternative robust control framework for wearable PAM-driven rehabilitation systems.

Another important trend is the integration of SMC with fuzzy or neural approximators to estimate unknown nonlinearities and disturbances online. Adaptive fuzzy SMC and related intelligent control schemes have demonstrated improved robustness and reduced tracking error compared with conventional methods^22,23. Neural-network-based and fuzzy-enhanced SMC approaches further improve disturbance rejection and adaptability^24,25. These hybrid controllers have been successfully applied to PAM-driven robotic systems under input saturation, dead zones, and external disturbances^26–28. In a related development, adaptive fuzzy sliding mode control has been shown to effectively reduce chattering while preserving robustness in highly nonlinear and underactuated systems such as the inverted pendulum⁶.

Despite these advances, several limitations remain for wearable multi-DOF PAM-driven rehabilitation systems. Many existing approaches are limited to single-DOF systems or simulation-based validation and do not fully address joint coupling, safety, and comfort requirements in human–robot interaction^14,15,29. Furthermore, some intelligent SMC methods introduce high design complexity and may still suffer from residual chattering^13,22. Fixed-time and terminal sliding mode approaches show promising convergence but often lack validation in wearable rehabilitation scenarios^{17,24,25,27,28}.

Motivated by these challenges, this work proposes a finite-time Adaptive Super-Twisting Sliding Mode Control (ASTSMC) strategy for a multi-DOF wearable PAM-driven lower-limb rehabilitation system. The proposed controller combines robustness, reduced chattering, and adaptive gain tuning without prior knowledge of disturbance bounds^30–33. Lyapunov stability theory is employed to guarantee finite-time convergence of tracking errors under uncertainties and disturbances. Simulation results demonstrate improved tracking performance and reduced chattering, highlighting the suitability of the proposed approach for safe and compliant rehabilitation applications.

In summary, the main contributions of this work are:

1. An adaptive super-twisting sliding mode control (ASTSMC) strategy is developed for a 2-DOF PAM-driven rehabilitation robot. The proposed controller integrates a second-order sliding mode algorithm with adaptive gain tuning, enabling robust trajectory tracking without requiring prior knowledge of disturbance bounds.

2. A comprehensive nonlinear dynamic model of the PAM-actuated rehabilitation system is formulated using the Euler–Lagrange approach. The model captures key practical characteristics, including PAM nonlinearities, antagonistic actuation, viscous–Coulomb friction, and multibody dynamics.

3. The proposed ASTSMC achieves improved tracking accuracy, reduced chattering, and enhanced robustness against disturbances and parameter variations compared to conventional SMC and STSMC, as validated through simulation studies.

System modeling

Mechanical architecture of the PAM-actuated rehabilitation leg

Wearable pneumatic artificial muscle actuation.

The proposed rehabilitation manipulator utilizes a wearable Pneumatic Artificial Muscles (PAMs) as a compliant actuator for assisting lower-limb motion during therapeutic training. PAM-driven actuators are particularly suitable for human-robot interaction applications due to their lightweight construction, intrinsic compliance, and high force-to-weight ratio. These characteristics are particularly advantageous in medical robotics, as they facilitate safe and natural human–robot interaction, a critical requirement for effective lower-limb physical therapy^34–36.

A McKibben-type PAM consists of an inner elastomeric bladder surrounded by an inextensible braided mesh sleeve and terminated by rigid end fittings. When compressed air is supplied into the bladder, internal pressure causes radial expansion. However, because the braided fibers maintain nearly constant length, radial expansion results in axial shortening of the actuator. Consequently, pneumatic energy is transformed into a mechanical contraction force along the muscle axis^37,38.

Due to this unique architecture, a PAM produces a strictly unidirectional tensile force along its longitudinal axis. As the internal volume increases, the muscle shortens, creating a powerful pulling motion as in Fig. 1. Because the elastomeric bladder and mesh cannot provide structural rigidity under internal vacuum or atmospheric pressure, the actuator is incapable of generating a compressive or pushing force³⁸.

Fig. 1.

Unidirectional tensile actuation of a pneumatic artificial muscle (PAM).

The strictly contractile nature of PAMs dictates the mechanical design of the overall robotic system. To achieve bidirectional joint motion, such as knee or ankle flexion and extension, the actuators must be integrated into specific mechanical arrangements³⁹.

Antagonistic actuation mechanism

To achieve bidirectional rotational motion, each joint of the rehabilitation manipulator is actuated using an antagonistic pair of PAMs that mimics the functional behavior of human muscle groups such as the quadriceps and hamstrings. One actuator functions as the agonist muscle producing motion in one direction, while the opposing actuator generates motion in the reverse direction.

Both muscles are connected to the joint through a pulley transmission mechanism. When pressure inside one muscle increases, contraction occurs, and joint rotation is produced. Simultaneously, the opposing muscle elongates. Reversing the pressure distribution reverses joint motion as in Fig. 2. Furthermore, simultaneous pressurization of both muscles enables modulation of joint stiffness without producing motion, providing an inherent impedance regulation capability highly desirable in rehabilitation systems^37–39.

Fig. 2.

Antagonistic PAM actuation for bidirectional joint motion.

This setup requires four PAMs per leg (two for the hip and two for the knee), totaling eight PAMs for a bilateral rehabilitation system⁵. The use of antagonistic pairs offers several functional benefits beyond bidirectional torque:

• Adjustable Stiffness: By co-contracting both muscles (increasing pressure in both simultaneously), the joint stiffness can be increased, providing stability.

• Safety: The intrinsic compliance of the air within the muscles acts as a shock absorber against sudden external impacts.

• Biomimicry: The system closely replicates the torque-angle and stiffness-angle relationships of the human musculoskeletal system.

Simplified biomechanical model of the human leg

The human leg is simplified into a planar mechanism where Link 1 represents the thigh, and Link 2 represents the shank and ankle-foot complex, as illustrated in Fig. 3. A critical assumption in this constrained rehabilitation model is that the ankle joint remains mechanically fixed during the exercise session. This allows the ankle and foot segments to be lumped together with the shank, reducing the computational complexity of the dynamic model while remaining sufficient for hip and knee flexion-extension training⁵. This 2-DOF configuration is a standard approach in earlier and specialized rehabilitation models, providing a balance between control simplicity and clinical efficacy²² (Table 1).

Fig. 3.

Two-link planar manipulator model of the human leg.

Table 1.

Simplified 2-DOF human leg model description.

Link segment	Joint correspondence	Degree of freedom	Primary motion
Link 1 (Thigh)	Hip Joint	1-DOF	Flexion/Extension
Link 2 (Shank+Ankle)	Knee Joint	1-DOF	Flexion/Extension
Base Frame	Pelvis/Fixed Support	N/A	Stationary

The manipulator is wearable, meaning the PAMs are attached parallel to the human segments, and the joint axes of the robot are aligned as closely as possible with the anatomical axes of the hip and knee. This alignment is crucial to prevent joint misalignment torques, which could lead to discomfort or injury⁵.

Let the generalized coordinate vector be defined as

1

where and denote hip and knee joint angles, respectively. Proper alignment between robotic and anatomical joint axes is assumed to minimize undesired interaction torques.

Modeling of pneumatic artificial muscle actuation

Geometric model of the McKibben muscle.

The PAM used in this paper follows the McKibben design, which consists of an elastic inner bladder and a braided mesh sheath. The operating principle of the PAM is governed by geometric constraints imposed by the braided mesh structure. Assuming constant fiber length b, the geometric relation between muscle length L, diameter D, and braid angle α is expressed as

2

where n represents the number of braid turns. The braid angle satisfies

3

An increase in internal pressure enlarges the muscle diameter, increases the braid angle, and consequently reduces muscle length.

The contraction ratio is defined as

4

where denotes the initial muscle length (Figs. 4, 5).

Fig. 4.

McKibben muscle.

Fig. 5.

Geometric model of the McKibben muscle.

Pneumatic artificial muscle force modeling.

The PAM force is derived using an energy-based quasi-static formulation in which pneumatic work equals mechanical output work,

5

leading to

6

By expressing the muscle volume as and using the geometric constraint between D, L, and the braid angle α, the derivative can be obtained in terms of the contraction ratio \varepsilon. Substituting this result into Eq. (6) and rearranging yields the closed-form expression for the PAM tensile force. The resulting formulation is consistent with established McKibben muscle models reported in the literature^37,40.

7

The tensile force generated by a single pneumatic artificial muscle can be written in compact form as

8

where is a constant determined by the initial muscle diameter , P denotes internal pressure, and represents the nonlinear geometric contraction function of the McKibben muscle given by:

9

This expression highlights the nonlinear dependence of muscle force on both supplied pressure and contraction level.

In the simulation study, physical actuator limitations were considered to ensure realistic system behavior. In particular, the supplied pressure was constrained within the allowable range , where Pa, consistent with the actuator parameters listed in Table 2. Although these constraints were enforced in the simulation implementation, they were not explicitly stated in the original manuscript. This has now been clarified to improve transparency.

Table 2.

Simulation parameters of the PAM-driven rehabilitation manipulator (80 kg subject)⁴⁴.

Parameter / Definition	Formula / Source	Value
Manipulator Parameters (Anthropometric Model)
Mass of link 1 (thigh) –		8.00 kg
Mass of shank –		3.72 kg
Mass of ankle (foot) –		1.16 kg
Mass of link 2 (shank + ankle) –		4.88 kg
Length of thigh –	-	0.40 m
Length of shank –	-	0.40 m
Distance to COM link 1 –		0.173 m
Distance to COM link 2 –	, ,	0.255 m
Radius-of-gyration fraction (thigh) –	Winter Table 4.1	0.323
Radius-of-gyration (thigh) –		0.129 m
Radius-of-gyration fraction (shank) –	Winter Table 4.1	0.302
Radius-of-gyration (shank) –		0.121 m
Radius-of-gyration fraction (foot) –	Winter Table 4.1	0.475
Radius-of-gyration (foot) –		0.095 m
Inertia of link 1 about hip –	,	0.39 kg·m2
Inertia of link 2 about knee –	, , ,	0.48 kg·m2
Pulley radius – R	-	0.06 m
Gravitational acceleration – g	Constant	9.81 m/s2
PAM Actuator Parameters
Initial muscle length –		0.34 m
Initial diameter –		0.025 m
Initial braid angle –
Number of braid turns – n		2
Baseline pressure –		Pa
Maximum pressure –		Pa
Friction Parameters
Viscous friction (hip) –		0.05 N·m·s/rad
Viscous friction (knee) –		0.05 N·m·s/rad
Coulomb friction (hip) –		0.02 N·m
Coulomb friction (knee) –		0.02 N·m

In addition, the contraction ratio is inherently bounded by the geometric limits of the McKibben actuator, and the generated force remains within feasible limits as determined by the nonlinear pressure–contraction relationship. The pressure responses presented in Section 4 confirm that all control inputs remain within the defined physical bounds.

It should be noted that valve dynamics and pressure rate limitations are not explicitly modeled in this work and are considered as part of future experimental validation.

Muscle–Joint Transmission Relationship.

The linear contraction of the PAM is converted into rotational motion through a pulley mechanism of radius R. Assuming negligible cable elasticity, the muscle length varies proportionally with joint rotation according to

10

Consequently, the contraction ratio becomes

11

establishing a direct kinematic coupling between actuator deformation and joint displacement. At the nominal equilibrium configuration, both muscles share a common initial contraction ratio , where denotes the equilibrium joint angle.

Joint torque generation by antagonistic PAMs.

The tensile force generated by a PAM acts tangentially on the pulley attached to the joint axis. According to rotational mechanics, the torque produced by a single muscle is given by

12

Substituting the muscle force model yields

13

Since each joint is actuated by antagonistic muscles, the effective joint torque results from the difference between agonist and antagonist forces,

14

This formulation demonstrates that joint motion is governed by pressure differential rather than absolute pressure, while simultaneous activation of both muscles regulates mechanical stiffness.

Antagonistic pressure control strategy.

In antagonistically actuated pneumatic muscle systems, joint motion is generated through differential pressurization of two opposing artificial muscles. Let both muscles initially operate at an equilibrium pressure , producing identical contraction ratios and zero net joint torque. Controlled rotation is achieved by introducing a symmetric pressure variation about this equilibrium value (Fig. 6).

Fig. 6.

Antagonistic pressure control strategy.

The symmetric pressure control law is therefore defined as

15

16

where represents the baseline co-contraction pressure responsible for joint stiffness regulation, and Δ P denotes the differential pressure command used to generate motion.

When a joint rotation θ occurs, the pulley transmission converts angular displacement into linear cable motion. The resulting cable length variation is proportional to the pulley radius R, such that

17

Due to the antagonistic configuration, joint rotation shortens one muscle while elongating the opposing muscle. Thus, the contraction ratios of the agonist and antagonist muscles become

18

19

Accordingly, the forces produced by the agonist and antagonist muscles are written as

20

21

The torque applied to the joint through a pulley of radius R is obtained from the difference between antagonistic muscle forces,

22

Substituting the muscle force expressions gives

23

Expanding the above equation yields

24

Grouping similar terms results in the antagonistic PAM torque model

25

Nonlinear dynamic modeling of the PAM-actuated 2-DOF rehabilitation leg

The wearable robotic system is modeled as a simplified biomechanical representation of the human lower limb based on anthropometric data. Specifically, a planar 2-DOF model is considered, where Link 1 corresponds to the thigh segment and Link 2 represents the combined shank–ankle–foot complex. The hip and knee joints are modeled as revolute joints enabling flexion–extension motion.

The segment parameters, including masses, lengths, and inertias, are derived from standard anthropometric data for an 80 kg subject as summarized in Table 2. These parameters represent the physical properties of the human limb segments, consistent with established biomechanical models.

The PAM-based actuation system is attached in parallel to the human limb, and the robotic joints are assumed to be aligned with the anatomical joint axes. Under this alignment assumption, no relative motion is considered between the human limb and the wearable device. As a result, the combined system can be modeled as a unified dynamic structure, where the actuator torques directly drive the motion of the limb.

The dynamic equations for the 2-DOF system are derived based on the kinetic and potential energies of the limb segments⁸. The state of the system is defined by the joint angles . The resulting dynamic model is expressed as:

26

where:

• is the symmetric, positive-definite inertia matrix.

• is the Coriolis and centrifugal matrix.

• is the gravity vector.

• is the vector of friction torques.

• is the torque vector generated by the antagonistic PAMs.

The components of the inertia matrix for the thigh and shank links are:

27

28

29

The gravity vector is expressed as:

30

31

In these equations, represents the mass of the i-th link, is the length of the thigh, is the distance to the center of mass, and is the moment of inertia about the center of mass.

Nonlinear friction modeling

PAM-actuated systems exhibit significant nonlinear friction effects arising from internal interactions between the elastomeric bladder and braided mesh, as well as from the cable–pulley transmission mechanism. To enhance modeling accuracy and improve trajectory tracking performance, a combined viscous–Coulomb friction model is adopted:

32

where denote the viscous friction coefficients and represent the Coulomb friction coefficients associated with each joint.

Integrated nonlinear state–space model

Combining actuator torque generation with manipulator dynamics results in the integrated nonlinear system model suitable for controller synthesis. Defining the state variables as

33

the system dynamics can be written as

34

35

The obtained state-space representation incorporates pneumatic actuator nonlinearities, antagonistic muscle interaction, multibody dynamics, gravitational effects, and frictional disturbances.

Controller design

To achieve finite-time trajectory tracking for the 2-DOF PAM-driven leg manipulator under parameter uncertainties and actuator nonlinearities, an adaptive super-twisting sliding mode controller (ASTSMC) is designed. The joint torque generated by antagonistic PAM pairs is used as the control input, and the adaptive gains are updated online to ensure robustness and reduce chattering^41–43.

The overall control architecture of the proposed Adaptive Super-Twisting Sliding Mode Controller (ASTSMC) for the PAM-driven rehabilitation manipulator is illustrated in Fig. 7. The controller aims to ensure accurate trajectory tracking of the hip and knee joints despite the strong nonlinearities, actuator uncertainties, and external disturbances inherent in pneumatic artificial muscle systems. The desired joint trajectory is compared with the measured joint position to generate the tracking error, which is used to construct the sliding surface governing the closed-loop error dynamics.

Fig. 7.

Control Architecture of the Adaptive Super-Twisting Sliding Mode Control (ASTSMC) for a 2-DOF PAM Manipulator.

The adaptive super-twisting sliding-mode controller uses the sliding variable to generate the control torque applied to the manipulator. The controller consists of an equivalent control component that compensates for the nominal system dynamics and a super-twisting term that provides robustness against modeling uncertainties and disturbances. The resulting torque drives the PAM-actuated manipulator, while the measured joint motion is fed back.

Control objective

The objective of the controller is to ensure accurate trajectory tracking of the 2-DOF PAM-driven manipulator under model uncertainties, actuator nonlinearities, friction, and external disturbances applied for leg rehabilitation.

Let the desired trajectory be and define the tracking error as:

36

The control objective is to achieve

37

in finite time, T, despite model uncertainties and disturbances.

Sliding surface design

Sliding mode control achieves robustness by forcing system trajectories onto a predefined sliding manifold. The sliding surface is defined as

38

where

39

If the system reaches , then the error dynamics become

40

which yields exponential convergence of the tracking error as . Thus, stabilizing the sliding surface ensures asymptotic tracking.

Differentiating the sliding variable gives

41

Substituting the tracking error definitions,

42

Using the manipulator dynamics,

43

yields

44

This equation can be rewritten as

45

where represents lumped system dynamics and disturbances.

Equivalent control

The equivalent control represents the nominal control input required to maintain the system on the sliding manifold. Setting yields

46

This component compensates for the nominal manipulator dynamics. The friction term defined in Eq. (32) is explicitly incorporated into the equivalent control component to compensate for known nonlinear dissipation effects. This inclusion improves tracking accuracy by accounting for both viscous and Coulomb friction present in the PAM-driven system. Any residual modeling errors in the friction term are treated as part of the lumped uncertainties, which are then handled by the robust super-twisting control component.

Super-twisting sliding mode control law

Classical sliding mode control uses a discontinuous switching term which introduces high-frequency oscillations known as chattering. To overcome this limitation, the Super-Twisting Algorithm (STA) is employed (Fig. 8).

Fig. 8.

Smooth approximations of sliding mode control.

The sliding mode control signal is typically divided into two parts: equivalent control and switching control. The final control law is

47

where the STA component is defined as

48

49

with , ϕ > 0 the boundary layer thickness, δ > 0 a small constant to avoid singularity near , and the sliding surface. The STA has two important advantages: a continuous control signal and finite-time convergence of the sliding variable, which are particularly suitable for PAM-driven rehabilitation robots where smooth actuation is essential for patient safety.

Adaptive gain law

In practical PAM systems, the upper bound of uncertainties and disturbances is generally unknown due to actuator hysteresis, nonlinear pressure dynamics, and varying external loads. Consequently, selecting fixed control gains may lead to either insufficient robustness or unnecessarily large control effort. To overcome this limitation, adaptive gain laws are introduced to automatically adjust the controller gains based on the system behavior.

The adaptive laws are defined as

50

51

where is an adaptive scaling factor, are decay (leakage) coefficients, defines a time-varying boundary layer, δ > 0 is a small constant to avoid singularity, and ψ is an estimate of the sliding variable derivative: .

In practical implementation, direct numerical differentiation of the sliding variable is avoided due to its sensitivity to measurement noise. Instead, a first-order low-pass filter (dirty derivative) is employed to estimate the derivative. An auxiliary filtered state w is introduced such that

52

and the derivative is approximated as

53

where is a small filter time constant. This approach provides a smooth estimate of while attenuating high-frequency noise, and the adaptive law uses .

The adaptive scaling factor is defined as

54

and the boundary layer thickness is given by

55

with a lower bound . To estimate lumped uncertainties and enhance robustness, an auxiliary adaptive variable is introduced with the following dynamics:

56

These adaptive laws have the following important properties:

• When the sliding variable is large, the gains and increase automatically, enhancing robustness against disturbances and uncertainties.

• When the system approaches the sliding surface (), the gains decrease smoothly due to the leakage terms and , preventing excessive control effort and reducing chattering.

• The inclusion of the boundary layer ensures continuous adaptation, avoids gain overestimation, and improves numerical stability.

• The use of ψ (estimated sliding derivative) allows the controller to respond to dynamic changes, not only the magnitude of .

In the presence of persistent disturbances, it is important to ensure that the adaptive gains remain bounded and do not exhibit windup behavior. The proposed adaptive law incorporates multiple safeguards to prevent unbounded gain growth. First, the inclusion of leakage (decay) terms ensures that when the sliding variable enters the boundary layer (), the gains decrease exponentially, preventing unnecessary accumulation. Second, the boundary layer regulates the adaptation process by limiting gain updates to regions where the tracking error is significant, thereby attenuating adaptation near convergence. Third, the adaptive scaling factor is constructed from bounded signals, which inherently constrains the rate of gain increase.

In addition to these theoretical properties, practical implementation includes explicit gain saturation to ensure that the adaptive gains remain within physically meaningful limits. As a result, even under sustained disturbances, the gains increase only transiently and settle to bounded values, avoiding instability and excessive control effort. This behavior is consistent with the simulation results, where the adaptive gains adjust in response to disturbances but do not diverge.

Finite-time stability analysis

To establish the stability of the proposed adaptive super-twisting sliding mode controller, consider the sliding surface defined as . The control objective is to drive in finite time, ensuring exponential convergence of the tracking error.

Using the control law and system dynamics, the closed-loop sliding dynamics can be expressed as:

57

58

where represents lumped uncertainties satisfying .

Consider the Lyapunov function candidate:

59

where denotes the gain estimation error.

The time derivative of V along system trajectories is:

60

Using the inequality , it follows that:

61

The adaptive laws are designed such that the parameter update terms are non-positive, ensuring:

62

By selecting the adaptive gains sufficiently large to satisfy:

63

for some η > 0, the Lyapunov derivative becomes:

64

This inequality guarantees finite-time convergence. Integrating yields:

65

Thus, the sliding variable converges to zero in finite time. Once on the sliding surface, the error dynamics reduce to:

66

which ensures exponential convergence of the tracking error.

Therefore, the closed-loop system is finite-time stable.

Results and discussion

Simulation setup

This section presents the simulation studies conducted to evaluate the effectiveness of the proposed Adaptive Super-Twisting Sliding Mode Controller (ASTSMC) for the PAM-driven 2-DOF rehabilitation manipulator. Simulations are performed using the nonlinear dynamic model developed in the System Modeling Section, which incorporates manipulator dynamics, antagonistic pneumatic actuation of artificial muscles, gravitational effects and friction disturbances.

The performance of the proposed ASTSMC is evaluated through comprehensive MATLAB/Simulink simulations and compared with classical Sliding Mode Control (SMC) and Super-Twisting Sliding Mode Control (STSMC). Three distinct simulation scenarios are considered: sinusoidal trajectory tracking (Rehabilitation Mode), robustness against sudden step changes (Step Mode), and performance under external disturbances and parameter variations.

The mechanical, actuator, and controller parameters used in the simulation study are summarized in Table 2. These parameters correspond to typical values reported for PAM-driven rehabilitation manipulators and were selected to represent a realistic lower-limb assistive system.

In addition, the computational complexity of the proposed ASTSMC controller is low, involving only basic algebraic operations for the equivalent control, super-twisting terms, and adaptive gain updates. For the considered 2-DOF system, the total computational load is modest and compatible with typical real-time control rates (e.g., 1 kHz). This indicates that the proposed controller is suitable for real-time implementation on standard embedded platforms.

Step tracking performance (constant reference)

To evaluate the transient response and steady-state tracking performance of the controllers, constant reference trajectories are employed for both hip and knee joints. This setup is analogous to a step-response test commonly used in control system analysis, enabling a clear assessment of convergence speed, overshoot, and steady-state error.

While constant references do not fully represent practical rehabilitation movements, they provide a useful benchmark for comparing control performance under well-defined conditions. More realistic time-varying trajectories are considered in the subsequent section.

Figure 9 present the tracking performance of the hip () and knee () joints evaluated in terms of angular position (rad) over time (s), along with the corresponding control torque (Nm). The desired trajectories are set to for the hip joint and for the knee joint (i.e., and ), and the responses of SMC, STSMC, and the proposed ASTSMC are compared to assess accuracy and control effectiveness.

Fig. 9.

Tracking and control performance for hip () and knee () joints; ASTSMC achieves faster convergence and smoother, low-chattering control compared to SMC and STSMC.

The trajectory tracking in Fig. 9 shows that all controllers are capable of following the desired joint angles. However, their transient performance differs. The conventional SMC shows slower convergence and noticeable deviation during the transient phase before reaching the desired values. The STSMC improves the response by reducing tracking error and achieving faster convergence. In contrast, the proposed ASTSMC provides the best tracking performance, reaching the desired trajectories more rapidly with negligible steady-state error and minimal overshoot. Additionally, the ASTSMC response is smoother with reduced oscillations, indicating improved robustness against nonlinearities and uncertainties in the PAM-driven system.

Another noticeable advantage of the ASTSMC is its ability to achieve significant chattering reduction.. The control input results further highlight the superiority of the ASTSMC approach. As shown in Fig. 9, the classical SMC control torque is characterized by high-frequency switching, and hence, chattering is significant. Although STSMC reduces this effect, some fluctuations remain. The ASTSMC significantly suppresses chattering, resulting in smoother control signals with lower oscillation amplitude and faster stabilization. This leads to more efficient torque utilization and reduced mechanical stress on the system, demonstrating the effectiveness of the adaptive mechanism in improving both control smoothness and overall system performance.

The physical feasibility of the control inputs is further verified through the evaluation of the pneumatic artificial muscle (PAM) pressure commands. In the antagonistic actuation strategy, the controller generates a differential pressure command Δ P, while the actual agonist and antagonist muscle pressures are obtained from

67

68

where the baseline co-contraction pressure is bar, as listed in Table 2. Figure 10 presents the differential pressure responses during step trajectory tracking for both hip and knee joints. The maximum allowable actuator pressure bar is indicated by a dashed reference line. It can be observed that the differential pressure command remains below approximately 3.5 bar throughout the operation. Consequently, the resulting PAM pressures remain within the allowable actuator limits during both transient and steady-state operation. The largest pressure variations occur during the initial transient phase due to increased control effort and adaptive gain adjustment. These results confirm that the proposed controller achieves high tracking performance while satisfying the physical pressure constraints of the PAM actuators.

Fig. 10.

Differential pressure commands (Δ P) during step tracking with antagonistic PAM control. A baseline pressure is used to ensure feasible operation.

The evolution of the adaptive gains for the ASTSMC is shown in Fig. 11. Unlike fixed-gain controllers, the adaptive gains dynamically adjust their values based on the real-time tracking error and system uncertainties, enabling faster convergence and improved robustness without requiring prior knowledge of disturbance bounds.

Fig. 11.

Evolution of adaptive gains for the ASTSMC, showing online adjustment based on tracking error and system uncertainties.

Tracking error and sliding surface analysis

Figure 12 illustrates the tracking errors and sliding surface evolution for both joints. The tracking error results show that all controllers drive the error to zero; however, their convergence rates differ. The SMC exhibits slower error reduction with noticeable transient deviation, while STSMC improves the convergence speed. The proposed ASTSMC achieves the fastest error convergence with minimal oscillation, ensuring high tracking accuracy and robustness.

Fig. 12.

Tracking error and sliding surface responses; ASTSMC achieves faster error convergence to zero and smoother sliding dynamics with reduced oscillations compared to SMC and STSMC.

The ASTSMC achieves the fastest error convergence, with errors reaching zero in finite time and minimal steady-state deviation. The sliding surface converges rapidly and smoothly without oscillations, demonstrating improved stability and robustness. This confirms the effectiveness of the adaptive gain mechanism in maintaining optimal control performance under varying conditions.

Tracking Performance under human-like motion trajectory

To better represent realistic rehabilitation scenarios, smooth time-varying reference trajectories are employed for both hip and knee joints. Since human walking is a periodic movement, truncated low-order Fourier series are widely utilized to estimate and reconstruct continuous joint kinematics from empirical datasets. Unlike simple single-frequency sinusoidal profiles, the adopted references are designed as multi-harmonic, second-order Fourier trajectories derived from the clinical gait data compiled by Winter⁴⁴. These equations mathematically incorporate the fundamental and secondary harmonic components required to approximate the natural, asymmetric flexion–extension behavior observed during human walking and rehabilitation exercises.

Specifically, the desired joint trajectories are defined as

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where with f = 0.2 Hz. The corresponding velocity and acceleration profiles are obtained analytically to ensure smooth and continuous motion.

The selected amplitudes, offsets, and harmonic components are consistent with typical lower-limb gait characteristics reported in biomechanical studies, ensuring that the simulated trajectories remain physiologically realistic for rehabilitation applications.

Figure 13 presents the tracking performance under this human-like, time-varying trajectory. The classical SMC shows noticeable tracking errors and oscillatory behavior, particularly during rapid changes in the trajectory, indicating limited adaptability to dynamic conditions. The STSMC improves the tracking performance by reducing oscillations and enhancing convergence; however, small deviations are still observed during transition phases.

Fig. 13.

Tracking and control performance under human-like motion trajectory; ASTSMC achieves more accurate and smoother tracking with reduced oscillations compared to SMC and STSMC.

In contrast, the proposed ASTSMC demonstrates superior performance, accurately tracking the desired trajectories with minimal error and smooth response throughout the motion cycle. The controller effectively adapts to varying system dynamics, maintaining stability and precision even under complex, time-varying conditions. These results highlight the effectiveness of ASTSMC for practical rehabilitation applications requiring reliable and smooth joint motion tracking.

To further evaluate actuator feasibility under realistic operating conditions, the corresponding PAM differential pressure commands are analyzed during human-like trajectory tracking. Using the antagonistic pressure control law,

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with baseline pressure bar, the actual muscle pressures are reconstructed from the differential pressure command generated by the controller. It should be clarified that the pressure signals presented in Fig. 14 correspond to the differential pressure command Δ P rather than the absolute physical pressure inside the pneumatic artificial muscles. Therefore, negative values in the plotted signals do not represent negative physical air pressure. Instead, they indicate a reduction in pressure relative to the baseline co-contraction pressure. Since the magnitude of the differential pressure command remains below the baseline pressure level, the resulting PAM pressures remain strictly positive and within the allowable actuator operating range throughout the simulations.

Fig. 14.

Differential pressure (Δ P) responses of the antagonistic PAMs during human-like trajectory tracking, where the commanded pressures are referenced to a baseline pressure bar.

Figure 14 illustrates the differential pressure responses for both hip and knee joints during the periodic rehabilitation motion. The maximum allowable pressure limit bar is also indicated. The differential pressure magnitude remains bounded below approximately 3.5 bar throughout the motion cycle, ensuring that the resulting PAM pressures remain within the allowable actuator range. The pressure variations follow the periodic joint motion and exhibit moderate transient peaks during rapid trajectory transitions. These results confirm that the proposed ASTSMC provides safe and physically feasible actuator commands while maintaining accurate trajectory tracking under realistic rehabilitation conditions.

Robustness under external disturbance

The robustness of the controllers is evaluated by introducing an external disturbance at t = 5 seconds, as shown in Fig. 15. The disturbance consists of an additive step torque applied simultaneously to both joints, with magnitudes Nm at the hip joint and Nm at the knee joint. These values correspond to approximately 10% of the nominal gravitational torque at each joint and represent an unexpected external perturbation, such as sudden patient movement or environmental interaction. The disturbance is applied at t = 5 s and maintained for the remainder of the simulation.

Fig. 15.

Robustness under external disturbance (step torque applied at t = 5 s).

The classical SMC exhibits significant deviation from the desired trajectory after the disturbance is applied, along with increased oscillations. The STSMC shows improved disturbance rejection but still experiences noticeable transient errors.

In contrast, the ASTSMC effectively rejects the disturbance and quickly restores accurate tracking. The tracking error remains small, and the system rapidly returns to the desired trajectory. This demonstrates the strong robustness of the proposed controller against external perturbations.

The adaptive behavior of the ASTSMC gains under external disturbance is illustrated in Fig. 16. At t = 5 seconds, when the disturbance is applied, both and increase sharply to compensate for the sudden perturbation. The hip and knee joint gains adapt online, demonstrating the controller’s ability to enhance robustness without manual retuning.

Fig. 16.

Adaptive gains and for hip and knee joints under external disturbance applied at t = 5 s; gains increase rapidly to compensate for the perturbation.

Robustness under parameter variation

To further evaluate robustness, system parameters are varied by modifying the link masses based on a change in the total body mass from 80 kg (nominal case) to 100 kg. This simulates modeling uncertainties and real-world variations in human–robot interaction, such as differences in patient body mass.

The link masses are computed using the same anthropometric relationships as in the nominal model. As a result, link 1 mass increases from 8.00 kg to 10.00 kg, and link 2 mass increases from 4.88 kg to 6.10 kg, corresponding to a +25% variation. These parameter changes are applied simultaneously throughout the simulation, while the controller continues to use the nominal parameter values, introducing a structured model mismatch.

The classical SMC shows degraded performance, with increased oscillations and tracking errors. The STSMC performs better but still experiences performance degradation under parameter changes.

In contrast, the ASTSMC maintains stable and accurate tracking despite the parameter variations. The adaptive gain mechanism compensates for uncertainties in real time, ensuring consistent performance. This highlights the controller’s ability to handle modeling inaccuracies and varying system dynamics (Fig. 17).

Fig. 17.

Robustness under parameter variation (total mass increased from 80 kg to 100 kg); ASTSMC maintains stable and accurate tracking despite system uncertainties, outperforming SMC and STSMC.

Performance indices

To quantitatively evaluate the effectiveness of the proposed controller, performance indices are computed in terms of control effort and tracking accuracy.

The control effort is measured using the norm of the control input, defined as

Table 3 presents the control effort for SMC, STSMC, and ASTSMC under nominal conditions, external disturbances, and parameter variations.

Table 3.

Control effort ( norm) under different operating conditions.

Condition	SMC	STSMC	ASTSMC
Nominal	106.0990	89.0530	85.4672
Disturbance	113.2664	103.2977	100.2383
Parameter Variation	190.9611	111.3055	104.7144

From Table 3, it is evident that the proposed ASTSMC consistently yields the lowest control effort across all operating conditions. Under nominal conditions, the ASTSMC reduces the control effort compared to both SMC and STSMC, indicating smoother control action. In the presence of external disturbances, the control effort increases for all controllers; however, the ASTSMC maintains the lowest value, demonstrating improved robustness with reduced control activity.

A similar trend is observed under parameter variations, where the SMC exhibits the highest control effort due to aggressive switching behavior, while the STSMC provides moderate improvement. The ASTSMC achieves the most significant reduction, confirming that the adaptive mechanism effectively regulates control gains and minimizes unnecessary control energy, thereby reducing chattering and improving efficiency.

The tracking performance is evaluated using the Integral of Time-weighted Absolute Error (ITAE), defined as

Table 4 summarizes the ITAE values.

Table 4.

Tracking performance (ITAE) under different operating conditions.

Condition	SMC	STSMC	ASTSMC
Nominal	1.5069	0.5707	0.2148
Disturbance	1.6048	0.6345	0.2871
Parameter Variation	2.2424	0.6273	0.3046

As shown in Table 4, the ASTSMC achieves the lowest ITAE values in all scenarios, indicating superior tracking accuracy and faster convergence. Under nominal conditions, the proposed controller significantly outperforms both SMC and STSMC, reducing the tracking error by more than 50%. Under external disturbances, the ASTSMC maintains a substantially lower ITAE compared to the other controllers, demonstrating strong disturbance rejection capability. Similarly, under parameter variations, the ASTSMC continues to provide the best tracking performance, indicating robustness against model uncertainties.

Overall, the results clearly demonstrate that the proposed ASTSMC provides an improved trade-off between control effort and tracking performance. The reduced norm confirms smoother control action with minimized chattering, while the significantly lower ITAE values indicate enhanced tracking precision, faster transient response, and superior robustness under disturbances and parameter uncertainties. These results validate the effectiveness of the proposed controller for PAM-driven rehabilitation systems.

Conclusion

This study presented a robust control framework for trajectory tracking of a 2-DOF pneumatic artificial muscle (PAM)-driven lower-limb rehabilitation manipulator. An adaptive super-twisting sliding mode control (ASTSMC) strategy was developed to address the inherent nonlinearities, hysteresis effects, and parametric uncertainties associated with PAM actuators. By integrating a nonlinear dynamic model based on the Euler–Lagrange formulation with an adaptive high-order sliding mode controller, the proposed approach ensures finite-time convergence of tracking errors while maintaining robustness against external disturbances and modeling inaccuracies.

A key contribution of this work lies in the development of an improved low-chattering adaptive super-twisting algorithm with bounded adaptive gain laws and a smoothed switching mechanism. These enhancements effectively mitigate chattering phenomena while preserving the strong robustness properties of sliding mode control. The inclusion of a differentiator-based disturbance estimation and adaptive gain tuning further improves control performance without requiring prior knowledge of disturbance bounds. Simulation results demonstrated that the proposed ASTSMC approach provides superior trajectory tracking accuracy and reduced chattering compared to conventional sliding mode strategies. The controller maintains stable and consistent performance under varying operating conditions, highlighting its suitability for PAM-driven rehabilitation systems where compliance and safety are critical.

For future work, the model can be extended to a more realistic 3-DOF lower-limb configuration that includes hip, knee, and ankle joints to better represent human gait dynamics. Additionally, experimental validation on a physical PAM-driven rehabilitation platform is necessary to further assess real-time performance, robustness, and clinical applicability of the proposed control strategy.

Author contributions

Solomon Ferede Ezez: Conceptualization, Methodology, Software, Data Curation, Formal Analysis, Visualization, Investigation, Writing – Original Draft, Writing – Review & Editing.

Daniel Gosaye Tesfaye, & Tofik Kemal Mohammed: Conceptualization, Methodology, Software, Data Curation, Visualization, Investigation, Writing – Review & Editing.

R. Manivannan: Methodology, Software, Data Curation, Visualization, Investigation, Writing – Review & Editing.

Chala Merga Abdissa (corresponding author): Supervision, Conceptualization, Methodology, Validation, Data Curation, Visualization, Investigation, Writing – Review & Editing.

Funding

This work has not received any funding.

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.