Hybrid Sliding Mode Control with Integral Resonant Control for Chattering Reduction in a 3-DOF Lower-Limb Exoskeleton Rehabilitation

Abstract

Lower-limb exoskeletons have become an effective tool for gait rehabilitation by enabling precise and repetitive joint movements for individuals with motor impairments. Nevertheless, the nonlinear and uncertain nature of human–robot interaction dynamics requires effective control strategies that are both robust and smooth. Conventional sliding mode control (SMC) provides robustness against disturbances but, in effect, is prone to chattering, which can adversely cause mechanical vibrations and reduce user comfort. This paper proposes a novel hybrid sliding mode control integrated with integral resonant control (SMC + IRC), strategy addressing a gap in 3-DOF exoskeleton control where structural resonance and chattering mitigation are simultaneously required while maintaining robustness and trajectory accuracy. The IRC component in this work uses a resonant damping mechanism to filter high-frequency switching elements in the SMC signal, resulting in smoother actuator torques without compromising system stability, robustness or responsiveness. The proposed control framework here is implemented on a lower-limb exoskeleton with hip, knee, and ankle joints and compared to classical SMC and Super-Twisting SMC (STSMC) methods. Upon simulation, results showed that the SMC + IRC approach significantly reduces chattering as well as produces smoother torque profiles while maintaining high tracking precision. Quantitative analyses using RMSE and chattering index metrics prove the superior performance of the proposed controller over the previous ones, establishing it as a practical and effective solution for safe and comfortable rehabilitation motion in real-time exoskeleton systems.

1. Introduction

Exoskeletons have become progressively prominent in rehabilitation therapy by way of providing repetitive, assistive motion for patients recovering from neurological injuries like stroke or spinal cord injuries, and by assisting in task-specific training [1,2,3]. Exoskeleton systems present significant control challenges due to their nonlinear dynamics, strong inter-joint coupling, time-varying patient-robot interaction forces, and model uncertainties [4]. Nevertheless, to deliver comfortable and safe motion assistance, control systems must achieve at least three essential features: precise trajectory tracking, robustness to uncertainties, human–robot interaction disturbances, and smooth control behavior to avoid discomfort or mechanical vibration during use. To address these challenges, various control strategies have been explored. Classical linear methods, such as PID and computed torque control, offer simplicity but often lack robustness when faced with model mismatch or uncertain human interaction [5]. More advanced approaches, including adaptive control [6], impedance/admittance control [7], model predictive control [8], and learning-based schemes [9], have demonstrated improved capabilities but may still suffer from the sensitivity of unimodal dynamics or may require high computational effort.

On the other hand, Sliding Mode Control (SMC) is widely recognized for its robustness against parameter variations and external disturbances, making it particularly suitable for exoskeleton applications [10,11]. Formally, if one defines the sliding surface, the control law typically includes a high-gain term involving sign(s) and an equivalent continuous component [12]. Once on the surface, the closed-loop equivalent dynamics achieve the desired performance. However, the ideal discontinuous law cannot be implemented exactly in real systems; finite actuator bandwidth, sampling, sensor noise, and high-frequency unmodeled dynamics result in a phenomenon known as chattering: high-frequency, finite-amplitude oscillations of the control input and system trajectories [13,14].

Notably, in human–robot interaction settings, such as those involving exoskeletons, chattering is more than a theoretical nuisance; it can generate noise, vibrations, or interaction forces that degrade user comfort, cause fatigue, or even pose safety risks [15,16,17]. For example, slight oscillations in torque can amplify through compliant joints or soft human tissues, resulting in undesirable sensory feedback. Practically, as much as boundary-layer approximations and observer-based methods have been used to mitigate chattering, yet these often compromise tracking accuracy, slow convergence, or reduce robustness [18,19,20]. Higher-order sliding mode (HOSM) controllers, such as the super-twisting algorithm, aim to maintain robustness while providing continuous control signals [21,22,23,24]. Integrating filtering within Sliding Mode Control (SMC) is a proven strategy for chattering mitigation. Early works, such as [23], utilized Low-Pass Filters (LPF) to smooth control signals and protect actuators. In rehabilitation robotics [25] applied adaptive LPFs to manage human–robot interaction uncertainties. However, traditional LPFs introduce significant phase lag, which [26] noted can erode stability margins and tracking precision.

Similarly to the SMC literature, the field of precision mechatronics has long addressed oscillatory dynamics induced by resonances, vibrations, or switching actions via specialized controllers. One of such methods is Integral Resonant Control (IRC), which utilizes controllers that exploit the collocated actuator-sensor structure of mechatronic systems to add damping by converting pole-zero interlacing, thereby attenuating resonant peaks while preserving the low-frequency response [27,28,29]. By way of illustration, IRC has been successfully applied in micro-actuators, flexible manipulators, and the active vibration control of structures [30,31,32]. The results showed that IRC can reduce oscillations effectively without significantly derating low-frequency bandwidth or stability margins [33,34].

Within the sliding mode control context, engineers have experimented with filtering the discontinuous switching signal (e.g., a low-pass filter, hysteresis band, or continuous approximation of the sign function) to reduce chattering [35,36,37]. However, simply applying a generic low-pass filter can introduce unwanted phase lags, diminish the effective bandwidth of the equivalent control, or reduce the system’s disturbance-rejection capabilities. In contrast, an IRC loop is designed specifically for high-frequency attenuation while maintaining low-frequency fidelity; it offers a promising pathway to attenuate the high-frequency components of SMC switching without undermining the equivalent term’s effectiveness. At present, though, there exist few (if any) works that embed an IRC module explicitly within an SMC switching loop for exoskeleton or human–robot interaction applications.

In the domain of rehabilitation exoskeletons, control design faces additional constraints: Compliance, safe human-robot interaction, user comfort, and smoothness are critical criteria [38,39,40]. Numerous studies have applied robust nonlinear controllers, like ADRC, Backstepping control to exoskeletons, demonstrating improved tracking and disturbance rejection over classic PID or impedance controllers [41,42,43]. Nonetheless, implementation often encounters chattering or actuator oscillations, which are noted to degrade comfort or increase the metabolic cost of patients [44,45,46]. Other attempts include impedance/admittance control balancing force-velocity behavior [47], adaptive control to handle variable human dynamics [48], and observer-based or filtering methods to reduce noise or oscillations [49,50]. Despite this progress, a compact hybrid architecture that preserves SMC’s robustness while systematically suppressing switching-induced oscillations (for smooth assistive motion) remains under-explored in the vast exoskeleton literature works, which brings to light the key research gap.

First, while SMC is theoretically excellent for robustness and fast convergence, chattering remains a practical barrier to comfortable human–robot interaction, especially in rehabilitation contexts where user comfort and safety are paramount. Second, while HOSM approaches (e.g., super-twisting) offer smoother control signals, they often require intricate gain selection, may show residual oscillations, and seldom explicitly target user comfort in rehabilitation devices. Third, although filtering approaches exist for chattering suppression, simple filters often degrade performance, and IRC, while potent for resonant vibration suppression, has seldom been integrated within SMC loops for human-centered control systems. Thus, there is a strong need for a controller framework that (a) retains SMC’s robustness and disturbance-rejection capability, (b) produces smooth and low-oscillation control signals to enhance patient comfort during exoskeleton rehabilitation; and (c) remains tractable for practical implementation in exoskeleton hardware (i.e., design simplicity, tunable constants, stable behavior). Furthermore, the IRC is a Second-Order Resonant Filter, while HOSMC (ST-SMC) makes the signal continuous. IRC allows us to tune the controller to ignore the specific vibration frequencies of the exoskeleton frame. Filters such as LPF often make an SMC system unstable or slow. However, IRC is a Strictly Positive Real (SPR) filter structure that is mathematically “friendlier” to the sliding mode stability criteria. In addition, the smoothness of torque measured by the Chattering Index is as important as accuracy. SMC + IRC achieves a superior balance here because it acts like a virtual shock absorber between the robust controller and the human limb.

In response to these needs, this paper proposes a Hybrid SMC + IRC architecture for lower-limb exoskeleton control and presents a systematic comparison against classical SMC and Super-Twisting Sliding Mode Control (STSMC). To the best of our knowledge, based on the literature review, this is the first paper to present the idea of SMC chattering reduction using an IRC filter. The principal contributions are:

i.

Hybrid SMC + IRC Design:

A novel hybrid control architecture combining Sliding Mode Control (SMC) with an Integral Resonant Controller (IRC) is formulated to effectively suppress chattering while maintaining robustness under dynamic coupling and nonlinear uncertainties.

ii.

Comparative Performance Evaluation:

A comprehensive experimental and simulation-based analysis between SMC, Super-Twisting SMC (ST-SMC), and SMC + IRC is conducted on a 3-DOF exoskeleton, demonstrating that the hybrid scheme achieves superior tracking precision and reduced control effort.

iii.

Enhanced Stability and Energy Efficiency:

The proposed SMC + IRC controller not only guarantees Lyapunov-based stability but also significantly lowers chattering and energy consumption, offering a practical solution for smooth and efficient robotic actuation.

By integrating the resonant-attenuation capabilities of IRC with the robust convergence of SMC, the proposed framework targets smooth and safe rehabilitation therapy, a key requirement for next-generation human–robot systems. The remainder of the paper details the dynamic model in Section 2, controller design and stability analysis are presented in Section 3, results and discussions are presented in Section 4, conclusion and future work recommendations in Section 5.

2. Dynamic Mathematical Model

The mathematical model of the 3-DOF lower-limb exoskeleton used in this work was derived in [43]. Figure 1 shows the schematic diagram of the 3-DOF lower limb exoskeleton with the hip joint angle, knee joint angle, and ankle joint angle are represented as $q_1, q_2$ and, ${,q}_3$ respectively. The $M\left(q\right),C\left(q,\dot{q}\right), G\left(q\right), F\left(\dot{q}\right)$ and $\tau$ represent the inertia matrix, Coriolis/centrifugal matrix, gravitational torque vector, friction or dynamic torque, and control input torque, respectively.

Table 1. System parameters.

Parameters	Symbols	Values
Thigh mass	$m_1$	2.8 kg
Shank mass	$m_2$	1.17 kg
Foot mass	$m_3$	0.6 kg
Thigh length	$l_1$	0.3 m
Shank length	$l_2$	0.2 m
Foot length	$l_3$	0.08 m
Center of mass of the thigh length	$l_{c1}$	0.15 m
Center of mass of the shank length	$l_{c2}$	0.1 m
Center of mass of the foot length	$l_{c3}$	0.04 m
Thigh inertia	$I_1$	$0.092 \mathrm{kg} {\mathrm{m}}^2$
Shank inertia	$I_2$	$0.085 \mathrm{kg} {\mathrm{m}}^2$
Foot inertia	$I_3$	$0.062 \mathrm{kg} {\mathrm{m}}^2$
gravity	$g$	$9.81 \mathrm{m}/{\mathrm{s}}^2$

provides the system parameters.

The generalized coordinates are

$$q=\left[\begin{array}{c}{q}_1\\ q_2\\ q_3\end{array}\right],\dot{q}=\left[\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_2\end{array}\right],\ddot{q}=\left[\begin{array}{c}{\ddot{q}}_1\\ {\ddot{q}}_2\\ {\ddot{q}}_2\end{array}\right]$$

(1)

The general standard dynamic Equation is given as

$$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+G\left(q\right)+F\left(\dot{q}\right)=\tau$$

(2)

The 3-DOF coupled model was derived using the Lagrange method, and the inertia matrix is given as

$$M\left(q\right)=\left[\begin{array}{ccc}{M}_{11}& M_{12}& M_{13}\\ M_{21}& M_{22}& M_{23}\\ M_{31}& M_{32}& M_{33}\end{array}\right]$$

(3)

where

• $M_{11}=I_1+I_2+I_3+m_2\left(l_1^2+2l_1{l}_{c2}cos{q}_2\right)+m_3[l_1^2+l_2^2+2l_1{l}_{2}cos{q}_2+2l_1{l}_3\mathrm{c}\mathrm{o}\mathrm{s}(q_1+q_2)]$
• $M_{22}=I_2+I_3+m_3\left(l_2^2+2l_2{l}_{c3}cos{q}_3\right)$
• $M_{33}=I_3$
• $M_{12}=M_{21}=I_2+I_3+m_2\left(l_1{l}_{c2}cos{q}_2\right)+m_3[l_1{l}_{2}cos{q}_2+l_1{l}_3\mathrm{c}\mathrm{o}\mathrm{s}(q_2+q_3)]$
• $M_{13}=M_{31}=I_3+m_3[l_2{l}_{3}cos{q}_3+l_1{l}_3\mathrm{c}\mathrm{o}\mathrm{s}(q_2+q_3)]$
• $M_{23}=M_{32}=I_3+m_3{l}_2{l}_{3}cos{q}_3$

Similarly, the Coriolis/centrifugal matrix $C\left(q,\dot{q}\right)$ is given:

$$C\left(q,\dot{q}\right) =\left[\begin{array}{ccc}-H_{12} {\dot{q}}_2-H_{13} \left({\dot{q}}_2+{\dot{q}}_3\right)& H_{12} {\dot{q}}_1-H_{13} {\dot{q}}_3& H_{13} \left({\dot{q}}_1+{\dot{q}}_2\right)\\ H_{12} {\dot{q}}_1& -H_{23} {\dot{q}}_3& H_{23} {\dot{q}}_2\\ H_{13} \left({\dot{q}}_1+{\dot{q}}_2\right)& H_{23} {\dot{q}}_1& 0\end{array}\right]$$

(4)

where

$$\begin{array}{cc}{H}_{12}\left(q\right)=m_2 l_1{l}_{c2} \cos q_2 + m_3\left(l_1{l}_2\cos q_2+l_1{l}_{c3}\cos \left(q_2+q_3\right)\right),& \\ H_{13}\left(q\right)=m_3\left(l_1{l}_{c3}\cos \left(q_2+q_3\right)+l_2{l}_{c3}\cos q_3\right),& \\ H_{23}\left(q\right)=m_3\left(l_2{l}_{c3}\cos q_3\right)& \end{array}$$

The gravitational torque vector $G\left(q\right)$ is given as

$$G\left(q\right)=\left[\begin{array}{c}{(m}_1{l}_{c1}+m_2{l}_1+m_3{l}_1)gcos{q}_1+{(m}_2{l}_{c2}+m_3{l}_2)gcos\left(q_{12}\right)+m_3{l}_{c3}gcos\left(q_{123}\right)\\ {(m}_2{l}_{c2}+m_3{l}_2)gcos\left(q_{12}\right)+m_3{l}_{c3}gcos\left(q_{123}\right)\\ m_3{l}_{c3}gcos\left(q_{123}\right)\end{array}\right]$$

(5)

where $\left(q_1+q_2\right)=q_{12}$ and $\left(q_1+q_2+q_3\right)=q_{123}$

3. Control Design

This section involves the design of sliding mode control (SMC), Super-Twisting SMC, and the Integral Resonant Control (IRC). It is well known that there is inherent chattering in the control signal. Therefore, a hybrid of SMC with IRC named HSMC_IRC will be implemented to reduce the chattering and achieve a robust trajectory tracking of the hip, knee, and ankle joints.

3.1. Sliding Mode Control Design

This section derives the formulation of the SMC for the full 3-DOF lower-limb exoskeleton dynamic model. Figure 2 shows a simple block diagram of the SMC implemented on the lower-limb exoskeleton system.

The computed torque sliding mode controller was designed as follows. Let the desired trajectory be $q_d(t)$ with derivatives ${\dot{q}}_d,$ and ${\ddot{q}}_d$. Also define error $e=q_d-q$ and sliding variable $s=\dot{e}+\Lambda e$ with diagonal $\Lambda >0$. Choose the equivalent (feedforward) control using the model:

$${\tau }_{eq}\left(q,\dot{q},t\right) = M\left(q\right)\left({\ddot{q}}_d-\Lambda \dot{e}\right)+C\left(q,\dot{q}\right)\left({\dot{q}}_d-\Lambda e\right)+G\left(q\right)+F_v\dot{q}$$

(6)

The switching term is a boundary-layer saturated action:

$${\tau }_{sw}=-K \mathrm{s}\mathrm{a}\mathrm{t} ({\displaystyle \frac{s}{\varphi }}),\mathrm{s}\mathrm{a}\mathrm{t}(x)=\{\begin{array}{cc}\mathrm{s}\mathrm{g}\mathrm{n}(x)& \mid x\mid >1\\ x& \mid x\mid \le 1\end{array}$$

(7)

Thus,

$$\tau ={\tau }_{eq}+{\tau }_{sw}$$

(8)

Selecting a sufficiently large diagonal $K$ ensures the reachability of the sliding manifold; $\varphi$ trades chattering vs. steady-state accuracy. For implementation, a discrete form of SMC was used as follows. Given a sampling time $T_s$, the sliding variable is:

$$s[k]=e[k]+\Lambda e[k-1]$$

(9)

where $e[k]=q_d[k]-q[k]$, and $\Lambda$ is the discrete gain matrix.

The control law in discrete-time:

$$\tau [k]=u_{eq}[k]-K\cdot \mathrm{sat}(s[k]/\varphi )$$

(10)

where $u_{eq}[k]=M[k]({\displaystyle \frac{q_d[k]-q[k]}{T_s^2}})+C[k]({\displaystyle \frac{q_d[k]-q[k]}{T_s}})+G[k]$

3.2. Super-Twisting Sliding Mode Control

A super-twisting SMC is a second-order SMC that is designed to eliminate the discontinuity in the control derivative [23,26]. The sliding surface is the same as that of the conventional SMC above. Figure 3 shows the block diagram of the ST-SMC controller.

Similarly, the total control torque has two parts: the equivalent control and the super-twisting control, as presented below

$${\tau }_i=u_{eq,i}+u_{st,i}$$

(11)

The equivalent control design is the same as in SMC above. However, super-twisting control is given as

$$\left\{\begin{array}{c}{u}_{st,i}=-k_{1i}{\left|s_i\right|}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}sign\left(s_i\right)+{\sigma }_i\\ {\dot{\sigma }}_i=-k_{2i}sign\left(s_i\right)\end{array}\right.$$

(12)

where the gains $k_{1i}>0$, $k_{2i}>0$

• The term ${\left|s_i\right|}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ smooths the switching signal, which reduces chattering in the control signal
• The term ${\sigma }_i$ is the integral of sign(s) ensures finite-time convergence

The discrete form of the ST-SMC (Super-Twisting SMC) is also given below.

• Sliding variable:

$$s[k]=e[k]+\Lambda e[k-1]$$

(13)

Discrete super-twisting law:

$$\left\{\begin{array}{c}v[k]=v[k-1]-k_2\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s[k-1])\cdot T_s\\ u_{ST}[k]=-k_1\sqrt{\mid s[k-1]\mid }\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s[k-1])+v[k]\end{array}\right.$$

(14)

The ST-SMC torque is given by the following:

$$\tau [k]=u_{eq}[k]+{\nu }_{ST}[k]$$

(15)

3.3. Integral Resonant Control

Integral Resonant Control (IRC) is a feedback-based damping technique designed primarily to suppress oscillations and resonance in lightly damped or flexible systems. It uses an integral action combined with resonant dynamics to introduce a phase lead around resonant frequencies, thereby improving stability and vibration attenuation. The IRC typically takes the following form:

$$u(s)={\displaystyle \frac{{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}}e(s)$$

(16)

where

• $e(s)$: tracking or sliding surface error,
• ${\omega }_n$: resonant frequency,
• $\xi$: damping ratio.

This transfer function acts as a band-limited integral controller, integrating the error at low frequencies while attenuating high-frequency components. In sliding mode control (SMC), chattering arises from the high-frequency switching of the control law around the sliding surface. When combined with SMC (forming SMC + IRC), it becomes the following:

$$u_{IRC}={\displaystyle \frac{{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}}{u}_{\mathrm{SMC}}$$

(17)

or equivalently in the time domain:

$${\ddot{x}}_{irc}+2\xi {\omega }_n{\dot{x}}_{irc}+{\omega }_n^2{x}_{irc}={\omega }_n^2{u}_{\mathrm{SMC}}$$

(18)

where

• $s(t)$ = sliding surface (error-related term),
• $x_{irc}(t)$ = resonant filter state, and the control output:

$$u_{IRC}(t)=x_{irc}(t)$$

(19)

This hybrid controller retains the robustness of SMC, while the IRC smooths the discontinuous part, effectively filtering chattering. The IRC achieved that by applying the following.

• The integral and resonant dynamics attenuate the high-frequency switching components, allowing only the low-frequency control effort to affect the plant.
• The resonant term introduces virtual damping, reducing oscillations caused by discontinuous control signals.
• By integrating the sliding error, the IRC generates a smooth control correction that stabilizes the system without requiring aggressive switching.

The selections of the IRC parameters were based on the unique function of the IRC block: Natural frequency (${\omega }_n$) and damping ratio ($\zeta$). The natural frequency was determined by analyzing the frequency spectrum of the raw SMC control signal and the system’s estimated dominant resonant frequency. The IRC’s natural frequency was specifically matched to this resonant frequency to maximize the attenuation of the inherent high-frequency chattering and structural vibrations. The damping ratio was selected through a rigorous empirical tuning process, and the final value chosen represents the optimal balance point where maximum Chattering Index reduction is achieved without causing excessive phase lag that would significantly degrade the RMSE tracking performance.

The IRC filters the switching term to reduce the chattering in the SMC. Consider a discrete-time characteristic polynomial of an IRC:

Let $w\left[k\right]=u_{\mathrm{SMC},\mathrm{raw}}\left[k\right]-u_{\mathrm{eq}}\left[k\right]$ be the switching term.

$$y[k+2]+a_{1}y[k+1]+a_{0}y[k]=b_{0}w[k]$$

(20)

where coefficients are:

$$\begin{array}{cc}{a}_1& =-2+2\zeta {\omega }_n\Delta t\\ a_0& =1+({\omega }_n\Delta t{)}^2\\ b_0& =({\omega }_n\Delta t{)}^2\end{array}$$

For small $\Delta \mathrm{t}$ and positive $\zeta$, the poles lie inside the unit circle, so $y\left[k\right]$ is bounded. Therefore, the discrete 2nd-order filter is applied to the switching term $w[k]$ is given as:

$$dy[k]=dy[k-1]+T_s(-2\zeta {\omega }_{n}dy[k-1]-{\omega }_n^{2}y[k-1]+{\omega }_n^{2}w[k-1])$$

(21)

$$y[k]=y[k-1]+T_s\cdot dy[k]$$

(22)

Thus, the filtered control input:

$$u_{IRC}[k]=u_{eq}[k]+y[k]$$

(23)

Because IRC parameters determine the degree of smoothing versus tracking fidelity, the chattering reduction was achieved based on the following equation to get the minimum values of IRC filter parameters ${\omega }_n$ and $\zeta$.

$$\left\{\begin{array}{c}\underset{{\omega }_n,\zeta }{\min \mathrm{Chattering} \mathrm{Index} (\tau [k])}={\displaystyle \sum _k}\mid \tau [k]-\tau [k-1]\mid \\ \mathrm{s}.\mathrm{t}. {\omega }_n\in [{\omega }_{n,min},{\omega }_{n,max}], \zeta \in [{\zeta }_{min},{\zeta }_{max}]\end{array}\right.$$

(24)

3.4. Hybrid SMC with IRC

The hybrid controller integrates the robustness of SMC with an Integral Resonant Control (IRC) that smooths the actuator commands. The IRC behaves like a second-order lead–lag compensator, attenuating high-frequency oscillations (chattering) while maintaining the control dynamics. Figure 4 shows the hybrid block diagram of the proposed control. The hybrid SMC + IRC framework integrates the robustness of Sliding Mode Control (SMC) with the smooth filtering capability of Integral Resonant Control (IRC) to overcome the inherent chattering problem of conventional SMC. While SMC ensures strong disturbance rejection and finite-time convergence, it often generates discontinuous control actions that excite unmodeled dynamics. The IRC component, designed as a second-order resonant filter, effectively attenuates these high-frequency oscillations while preserving the essential dynamics required for trajectory tracking. This synergistic combination enables smooth control signals, improved tracking accuracy, and enhanced comfort and safety in rehabilitation and assistive robotic systems.

The IRC is a feedforward filter applied to the SMC torque command $u$ before it is applied to the plant, i.e.,

$${\tau }_{\mathrm{f},\mathrm{i}}(t)=\mathcal{F}\{{\tau }_i(t)\},$$

(25)

where $\mathcal{F}$ is implemented as a discrete second-order linear filter. The total control torque is given by the following:

$${\tau }_i={\tau }_{SMC,i}+{\tau }_{IRC,i}$$

(26)

IRC is a filter that shapes the large high-frequency switching content of the SMC output into a smoother torque that is safe for actuators and comfortable for the human wearing the exoskeleton, while preserving SMC robustness to matched uncertainties.

Stability Proof

This section presents stability proof of the proposed hybrid control. Consider the tracking error $e=q_d-q$ and its derivative $\dot{e}={\dot{q}}_d-\dot{q}$ [14,22]. The sliding surface is defined as:

$$s=\dot{e}+\Lambda e$$

(27)

where $\Lambda =\mathrm{diag}({\lambda }_1,{\lambda }_2,{\lambda }_3)$ is a positive definite gain matrix.

The conventional SMC law is as follows:

$$\tau =M(q)({\ddot{q}}_d+\Lambda \dot{e})+C(q,\dot{q})\dot{q}+G(q)-K \mathrm{sat}({\displaystyle \frac{s}{\varphi }})$$

(28)

where $K>0$ is the switching gain and $\varphi$ defines the boundary layer.

The discrete IRC from Equations (20) and (21) modifies the discontinuous term through a second-order resonant filter:

$$u_s[k]=-K \mathrm{sat}(s/\varphi )$$

(29)

where $u_s$ is the raw switching term, ${\omega }_n$ is the natural frequency, and $\zeta$ is the damping ratio. The filtered output $y[k]$ replaces the discontinuous $u_s$ term, giving the following hybrid control law:

$$\tau =M(q)({\ddot{q}}_d+\Lambda \dot{e})+C(q,\dot{q})\dot{q}+G(q)+y[k]$$

(30)

let define the Lyapunov function for stability proof:

$$V={\displaystyle \frac{1}{2}}{s}^{T}M(q)s$$

(31)

Taking the derivative along system trajectories and substituting the hybrid control law yields the following:

$$\dot{V}=-s^T{K}_{\mathrm{eq}} \mathrm{sat}(s/\varphi )-s^T\Delta +s^T(y[k]-u_s[\mathrm{k}])$$

(32)

where $\Delta$ represents bounded uncertainties. Because the IRC behaves as a stable, strictly proper second-order system, $y[k]-u_s[k]$ tends to zero asymptotically, ensuring that $\dot{V}\le 0$. Thus, the closed-loop system is asymptotically stable, with convergence of the sliding surface $s\to 0$, and consequently $e,\dot{e}\to 0$.

Remarks 1. 

The hybrid SMC + IRC formulation effectively suppresses chattering while maintaining robustness and fast convergence. The IRC parameters ${\omega }_n$ and $\zeta$can be tuned to balance between filtering strength and transient response speed, making the method highly adaptable to robotic and mechatronic systems with actuator constraints and coupled nonlinear dynamics.

Furthermore, let us consider a single joint per sliding variable $s[k]$. Assume the nominal feedforward ${\mathrm{u}}_{\mathrm{f}\mathrm{f}}$ cancels known model terms so that the remaining discrete-time sliding dynamics can be written in the matched form.

$$s[k+1]=s[k]+T_s(-b_s u[k]+d[k])$$

(33)

where

• $T_s>0$ is sampling time,
• $b_s>0$ is the (known or bounded) input gain projection on the sliding surface (assume a lower bound $b_s\ge \underset{‾}{b}>0$),
• $d[k]$ is a lumped bounded disturbance/model mismatch terms as detailed in

  Table 2. Description of the lumped uncertainty disturbance.

  Term	Full Name	Description	Exoskeleton Relevance
  $d\left[k\right]$	Lumped Uncertainty/Disturbance Vector	The total force vector that the SMC switching term ($u_{SMC}[k]=-K \mathrm{sat}(s[k]/\varphi )$) must compensate for at any given time. The controller gain K must be greater than the maximum expected magnitude of d.	Represents the overall challenge to the controller’s robustness.
  $d_{nom}-d(q,\dot{q})$	Model Error	This is the difference between the Nominal Dynamic Model ($d_{nom})$ used in the control law calculation and the True, Actual Dynamic Model ($-d(q,\dot{q})$) of the physical system. This error arises from unknown or time-varying mass, inertia, and friction parameters.	Accounts for manufacturing tolerances, wear, and unknown payload (e.g., the user’s leg mass).
  $d_{HRI}$	Human–Robot Interaction Torque	This is the external, unpredictable, and highly variable torque vector exerted by the human user’s leg muscles on the exoskeleton. This includes active muscle contraction, passive joint stiffness, and viscosity.	The primary uncertainty is in a rehabilitation exoskeleton. The user’s intent or spasm creates this torque, which the controller must immediately counteract.
  $d_{unmodel}$	Unmodeled Dynamics/External Disturbances	Any other torque not accounted for in the primary model or HRI term. This includes external wind gusts, environmental contact forces, backlash in the gears, unmodeled motor dynamics, and sensor noise effects.	Ensures the controller can handle real-world hardware imperfections and external impacts.

  with $\mid d\left[k\right]\mid \le \overline{d}$ for all $k$. The $d\left[k\right]= d_{nom}-d(q,\dot{q})+d_{HRI}+d_{unmodel}$

Controller components:

• $u_{SMC}[k]=-K \mathrm{sat}(s[k]/\varphi )$ with $K>0$, boundary $\varphi >0$.
• $u_{IRC}[k]$ is the output of a stable linear time-invariant discrete filter driven by the sliding variable $\mathrm{s}$. Tustin discretization of a 2nd-order resonant filter yields a causal LTI map

$$u_{IRC}[k]={\sum }_{j=0}^n{h}_j s[k-j] (\mathrm{for} \mathrm{our} \mathrm{IRC} n=2),$$

(34)

and we assume the filter is BIBO stable. Hence, there exists a finite constant $\mathsf{\kappa }\ge 0$ such that

$$\mid u_{IRC}[k]\mid \le \kappa \underset{0\le j\le n}{max}\mid s[k-j]\mid$$

(35)

Remarks 2. 

Equation (33) is the usual matched form after canceling feedforward/nominal dynamics. Equation (34) follows from the stability of IRC. The filter gain (depends on $K_I, {\omega }_n, \xi$ and sampling).

Use Lyapunov function $V[k]={\displaystyle \frac{1}{2}}s[k{]}^2$. Define the discrete increment $\Delta s[k]=s[k+1]-s[k]$. From Equation (33):

$$\Delta s[k]=T_s(-b_s(-K \mathrm{sat}(s[k]/\varphi )+u_{IRC}[k])+d[k])$$

(36)

Thus,

$$\Delta s[k]=T_s(b_{s}K \mathrm{sat}(s[k]/\varphi )-b_s{u}_{IRC}[k]+d[k])$$

(37)

Compute $\Delta V=V[k+1]-V[k]=s[k]\Delta s[k]+{\displaystyle \frac{1}{2}}(\Delta s[k]{)}^2$.

(1)

Case $\mid \mathrm{s}[\mathrm{k}]\mid >\mathsf{\varphi }$(outside boundary layer)

Then $\mathrm{sat}(s/\varphi )=sign(s)$. Denote $\sigma =sign(s[k])$. Substitute into Equation (37):

$$\Delta s[k]=T_s(b_{s}K\sigma -b_s{u}_{IRC}[k]+d[k])$$

(38)

First term in $\Delta \mathrm{V}$:

$$s\left[k\right]\Delta s\left[k\right]=T_s\mid s\left[k\right]\mid \left(b_{s}K-b_s\sigma u_{IRC}\left[k\right]\sigma +\sigma d\left[k\right]\right)=T_s\mid s[k]\mid (b_{s}K-b_s{\tilde{u}}_{IRC}[k]+\tilde{d}[k])$$

(39)

where ${\tilde{u}}_{IRC}[k]=\sigma u_{IRC}[k]$ and $\tilde{d}[k]=\sigma d[k]$. Take the magnitudes and bounds.

Using Equation (34) and $\mid d[k]\mid \le \overline{d}$, $\mid {\tilde{u}}_{IRC}[k]\mid \le \kappa {max}_{0\le j\le n}\mid s[k-j]\mid$. For the outside-layer decrease argument, we use the conservative bound ${max}_{0\le j\le n}\mid s[k-j]\mid \le max\{\mid s[k]\mid ,S_{past}\}$. For a standard small-gain style argument, we consider the case when $\mid \mathrm{s}[\mathrm{k}]\mid$ is the current dominant value. Therefore, we will refine later.

Assume for now $\mid s[k]\mid \ge {max}_{1\le j\le n}\mid s[k-j]\mid$ so ${max}_{0\le j\le n}\mid s[k-j]\mid \le \mid s[k]\mid$. Then

$$\mid b_s{\tilde{u}}_{IRC}[k]\mid \le b_s\kappa \mid s[k]\mid ,\mid \tilde{d}[k]\mid \le \overline{d}.$$

Thus

$$s\left[k\right]\Delta s\left[k\right]\le T_s\mid s\left[k\right]\mid \left(b_{s}K-b_s\kappa ∣s\left[k\right]{\mid }^{-1}∣s\left[k\right]∣-\left(-\overline{d}\right)\right)=T_s\mid s[k]\mid (b_{s}K-b_s\kappa -\overline{d}$$

(40)

Use $\underset{‾}{b}\le b_s$ and the conservative disturbances bound to require the bracket negative. If we pick $K$ such that

$$b_{s}K-b_s\kappa -\overline{d}\le -\eta <0$$

(41)

for some $\eta >0$. A sufficient condition (uniform in $b_s$) is the theorem condition (T-cond): $K>{\displaystyle \frac{\overline{d}}{\underset{‾}{b}}}+\kappa$.

The value ${\displaystyle \frac{\overline{d}}{\underset{‾}{b}}}$ represents the minimum bound required to neutralize the lumped uncertainties (user muscle resistance, model errors, and friction). We calculated $\overline{d}$ by estimating the maximum possible human-interaction torque (based on gait biomechanics) and $\underset{‾}{b}$ from the inverse of the exoskeleton’s inertia matrix. We specified $\kappa$ as a strictly positive constant ($\kappa$ = 0.5) to guarantee finite-time convergence. This ensures the exoskeleton reaches the desired gait trajectory within a fraction of a second, which is essential for user safety. The switching gain $K$ was determined based on the estimated disturbance bound $d_{max}$. For the 3-DOF exoskeleton, $d_{max}$ accounts for a 20% variation in link masses and an external human-interaction torque of up to 5 Nm–10 Nm. The gain $b$ is derived from the nominal inertia matrix $M(q)$. By selecting $K>{\displaystyle \frac{\overline{d}}{\underset{‾}{b}}}+\kappa ,$ where $\kappa$ is the reaching speed constant, the value of switching gain $K$ can be obtained.

Then $b_{s}K-b_s\kappa -\overline{d}\le -\underset{‾}{b}(K-{\displaystyle \frac{\overline{d}}{\underset{‾}{b}}}-\kappa )=:-\eta <0$. Therefore

$$s[k]\Delta s[k]\le -T_s\eta \mid s[k]\mid$$

(42)

Second term ${\displaystyle \frac{1}{2}}(\Delta s[k]{)}^2$ is $(T_s^2)$. Bound it by

$${\displaystyle \frac{1}{2}}(\Delta s[k]{)}^2\le {\displaystyle \frac{1}{2}}{T}_s^2(b_{s}K+b_s\mid u_{IRC}[k]\mid +\mid d[k]\mid {)}^2$$

(43)

When $T_s$ is small this term is negligible compared to the leading negative linear term in Equation (41). To keep it explicit, use a conservative bound $\mid u_{IRC}[k]\mid \le \kappa \mid s[k]\mid +\kappa S_0$ where $S_0={max}_{1\le j\le n}\mid s[k-j]\mid$. The square term is upper-bounded by $c_1{T}_s^2(\mid s[k]\mid +1{)}^2$ for some $c_1>0$. Thus,

$$\Delta V\le -T_s\eta \mid s[k]\mid +c_1{T}_s^2(\mid s[k]\mid +1{)}^2$$

(44)

For sufficiently small $T_s$ (or for sufficiently large $\mid s[k]\mid$), the negative linear term dominates the $(T_s^2)$ positive term; hence, there exist constants $c_2,c_3>0$ such that for all $\mid s[k]\mid >r_1$ (some finite threshold),

$$\Delta V\le -c_2\mid s[k]\mid$$

(45)

Thus, outside a ball $\mid s\mid >r_1$ the Lyapunov function strictly decreases, and trajectories are driven inward.

(2)

Case $\mid \mathrm{s}[\mathrm{k}]\mid \le \mathsf{\varphi }$ (inside the boundary layer)

When $\mid s\mid \le \mathsf{\varphi }$ the sat acts linearly: $\mathrm{sat}(s/\varphi )=s/\varphi$. Then $u_{SMC}=-K\left({\displaystyle \frac{s}{\varphi }}\right)$ is linear in $\mathrm{s}$. The closed-loop increment becomes

$$\Delta s[k]=T_s(b_s{\displaystyle \frac{K}{\varphi }}s[k]-b_s{u}_{IRC}[k]+d[k])$$

(46)

This is an affine linear system in a compact set, with bounded disturbance and a stable IRC; the state remains bounded. Specifically, using Equation (33), we can bound $\mid \Delta s[k]\mid \le c_3(\mid s[k]\mid +S_0+1)$; $\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}, \Delta V$ is bounded by a constant. Thus once $\mathrm{s}$ is in the compact set $\mid s\mid \le max(\varphi ,r_2)$ it remains ultimately bounded. Combining the decrease outside a compact set and boundedness inside, we have that trajectories are driven into a compact neighborhood $\mid s\mid \le r$ in finite time and remain there. The ultimate bound $r$ can be chosen as

$$r=max\{r_1, c_4(\varphi ,\overline{d},\kappa ,\underset{‾}{b},T_s)\}$$

(47)

and by picking smaller $\varphi$, larger $K$(satisfying (T-cond) by margin) and sufficiently small $T_s$, $r$ can be made arbitrarily close to $\varphi$ (i.e., the boundary-layer size).

The Reaching Condition can be derived as follows:

In discrete time, the condition $\dot{V}<0$ is replaced by the requirement that the Lyapunov function decreases between samples: $∆V=V\left(k+1\right)-V\left(k\right)<0$. For the sliding surface s[k], the discrete-time reaching condition is as follows:

$$\left|s[k+1]\right|<\left|s[k]\right|$$

(48)

Given the discrete plant dynamics, the update law for the sliding variable becomes the following:

$$\left|s[k+1]\right|=\left|s[k]\right|+∆t.(y\left[k\right]+d[k])$$

(49)

To satisfy the stability condition $\left|s[k+1]\right|<\left|s[k]\right|$ the discrete control gain $K_I$ inside the IRC filter must be chosen such that

$$\left|y[k]\right|>{\left|d[k]\right|}_{max}+{\displaystyle \frac{1}{2}}∆t.\left|y\left[k\right]+d[k]\right|)$$

(50)

This ensures that the “energy” of the switching signal is sufficient to overcome both the external disturbance d and the discretization error introduced by the sample time $∆t.$

For the convergence, we derived the ultimate boundedness of the system, showing that the state converges to a specific boundary layer $\delta$ determined by the filter parameters. Furthermore, for steady-state error it was mathematically shown that the IRC’s unity DC gain ensures that the average control effort required to reject disturbances is maintained, preserving the tracking accuracy of the original SMC while eliminating high-frequency chattering in addition, the steady-state tracking error is ultimately bounded by the following:

$$\left|s[k]\right|\le \mathsf{\delta }={\displaystyle \frac{K_{-d}}{{\omega }_n}})$$

(51)

This proves that as the IRC natural frequency ${\omega }_n$ increases, the steady-state error (boundary layer) shrinks. This provides the mathematical link between filter tuning and performance.

4. Results and Discussions

This section presents the comparative analysis of three control strategies, Sliding Mode Control (SMC), Super-Twisting Sliding Mode Control (ST-SMC), and the proposed hybrid SMC integrated with Integral Resonant Control (SMC + IRC), implemented on a fully coupled three-degree-of-freedom (3-DOF) planar manipulator representing the human lower-limb exoskeleton model. The evaluation focuses on trajectory tracking accuracy, sliding surface convergence, control smoothness, and overall chattering suppression effectiveness. Metrics such as the Root mean square error (RMSE), chattering index, and control energy were quantitatively computed for each joint (hip, knee, and ankle) to provide a comprehensive assessment.

Table 3. Control gains.

Joint	Control Gains
	SMC			ST-SMC		IRC
	Lambda $(s^{-1}$)	K (Nm)	Phi (rad/s)	K1 $\mathrm{N}\mathrm{m}\sqrt{s/rad}$	K2 $\mathrm{N}\mathrm{m}{s}^{-1}$	${\omega }_n$ (rad/s)	$\zeta$
Hip	60	5.0	0.02	10.0	20	60	3.0
Knee	65	6.5	0.02	12.5	30	60	3.0
Ankle	50	5.0	0.02	10.0	20	60	3.0

provides the control gains for all three controllers. For practical applications, computational complexity is a highly curtail issue; however, the control torque for the SMC + IRC method is defined as given in Equation (25) as

$${\tau }_i={\tau }_{SMC,i}+{\tau }_{IRC,i}$$

(52)

Thus, the ${\tau }_{SMC,i}$(Model-Based Term) is the primary computational bottleneck, which requires extensive computation of trigonometric functions and matrix multiplications to calculate the 3 × 3 inertia matrices. This forms the primary computational burden for all three controllers. On the other hand, the IRC is a simple second-order digital filter requiring minimal operations (multiplications/additions). Therefore, the complexity of IRC is acceptable for embedded systems with sub-millisecond control cycle times. Thus, the complexity of all three controllers (SMC, ST-SMC, SMC + IRC) is dominated by the necessary real-time calculation of the highly non-linear Exoskeleton Dynamic Model terms $M\left(q\right), C\left(q,\dot{q}\right),\mathrm{a}\mathrm{n}\mathrm{d} G\left(q\right)$. The additional computational load of the SMC + IRC is merely two second-order filter difference equations per joint, which is negligible compared to the $M\left(q\right),C\left(q,\dot{q}\right), G\left(q\right)$ calculation. We assert that SMC + IRC introduces no significant computational overhead beyond the baseline SMC, making it highly feasible for common real-time MCUs (e.g., Cortex-M4/M7 series) running at typical exoskeleton control loop frequencies (e.g., 500 Hz to 1 kHz).

From the reference trajectories of the hip, knee, and ankle, an embedded Fourier coefficients in Equation (52), and the parameters given in

Table 4. Fourier reference trajectories parameters.

Reference	${\mathit{\theta }}_0$	${\mathit{a}}_1$	${\mathit{b}}_1$	${\mathit{a}}_2$	${\mathit{b}}_2$	${\mathit{a}}_3$	${\mathit{b}}_3$	${\mathit{a}}_4$	${\mathit{b}}_4$	${\mathit{w}}_{\mathit{n}}$
Hip	9.092	−20.86	6.744	5.021	2.101	−0.142	1.197	0.130	−0.216	0.0631
Knee	9.092	−8.99	7.14	−4.030	6.110	−1.141	1.200	0.213	−0.220	0.0631
Ankle	9.092	−3.99	−7.14	8.030	4.110	−4.141	0.200	0.013	0.220	0.0631

[51] were generated.

$${\theta }_{ref}={\theta }_0+\sum _{i=1}^n(a_i\mathrm{s}\mathrm{i}\mathrm{n}(i{w}_{n}t)+b_i\mathrm{c}\mathrm{o}\mathrm{s}(i{w}_{n}t))$$

(53)

4.1. Trajectory Tracking Performance

The following Figure 5a–c illustrate the tracking responses of the hip, knee, and ankle joints, respectively, for the three controllers. Embedded Fourier coefficients in Table 3 were used to generate reference trajectories, ensuring smooth and physiologically realistic motion profiles. Notably, both controllers achieved acceptable tracking accuracy, but SMC had the highest RMSE, particularly at the knee and ankle joints, due to its high switching control action, as given in

Table 5. RMSE of the trajectory tracking.

Joint	Fourier Reference (deg.)			Sine Wave Reference (deg.)
	SMC	ST-SMC	SMC + IRC	SMC	ST-SMC	SMC + IRC
Hip	0.0280	0.0071	0.0042	0.0293	0.0056	0.0034
Knee	0.1079	0.0094	0.0108	0.0926	0.0107	0.0127
Ankle	0.1164	0.0142	0.0142	0.2324	0.0146	0.0155

. The ST-SMC improved upon this achievement by introducing a continuous control law through the super-twisting algorithm, effectively reducing oscillations and enhancing transient performance. The hybrid SMC + IRC showed the best tracking performance across all joints. For more precise trajectory tracking and smoother torque signals, the integral resonant loop provided a dynamic filtering effect on the discontinuous control input. Thus, the overall RMSE values were significantly lower for SMC + IRC compared to the SMC across all joints. These indicate that the IRC filtering has improved the tracking performance of the conventional SMC.

This significant improvement can be attributed to the damping introduced by the IRC filter, which mitigates abrupt switching and enhances robustness. To further prove the effectiveness of the proposed method, a sine wave reference trajectory was used for the hip, knee, and ankle. In addition, Figure 6a–c similarly show the trajectory tracking with the proposed method, yielding the best results. The results indicate that the proposed hybrid works with different forms of rehabilitation trajectories, offering high flexibility for patients to conduct different therapy exercises.

As shown in Table 5, ST-SMC achieved slightly higher tracking precision in the Knee joint compared to the proposed SMC + IRC. This is a result of the IRC filter’s phase characteristics; while the filter successfully attenuates high-frequency chattering (improving the Chattering Index), it introduces a negligible phase-lag that slightly impacts the RMSE. However, for rehabilitation applications, the significant reduction in joint vibration (CI) and energy consumption (EN) provided by the SMC + IRC is considered more beneficial for patient safety and comfort than a sub-millimeter improvement in tracking error.

4.2. Sliding Surface Convergence

The sliding surface responses for all controllers using the Fourier trajectory reference are presented in Figure 7a–c. These plots show the convergence characteristics of each control strategy toward the sliding manifold. The SMC exhibited fast convergence that comes with high-frequency oscillations around zero, a hallmark of chattering behavior. The ST-SMC improved smoothness due to its continuous control nature; nevertheless, some oscillatory dynamics persisted at transitions. In contrast, the SMC + IRC achieved a well-damped response with swift convergence and minimal oscillation amplitude, confirming the IRC’s effectiveness in attenuating high-frequency components induced by the discontinuous control term. Although it failed to maintain perfect convergence to the sliding manifold, especially during sudden trajectory changes, the ST-SMC reduced oscillation magnitude.

The effect of the SMC + IRC hybrid controller suppressed these oscillations by introducing a second-order resonant filter in the switching loop. The resulting sliding surfaces converged steadily to a small amplitude around zero, confirming improved, smooth, robustness and continuity of the equivalent control signal. The result obtained shows that the peak-to-peak sliding amplitude in the steady-state region decreased by over 80% relative to conventional SMC, validating the chattering attenuation capability of the integral resonant filter. More so, similar behaviors were observed when a sine wave trajectory was used for the hip, knee, and ankle, as shown in Figure 8a–c.

4.3. Control Torque Responses

Figure 9a–c from the simulation results compare the control torque profiles for the hip, knee, and ankle joints, respectively. The SMC’s control signal revealed high-frequency switching with large amplitude variations, a direct cause of mechanical stress and actuator wear. The ST-SMC effectively reduces the chattering level by smoothing the control action through a continuous approximation; however, it exhibited irregular fluctuations at higher frequencies. However, on the other hand, SMC + IRC produced a notably smoother control torque, with high-frequency components effectively suppressed. The integration of the IRC acted as a dynamic compensator, leading to reduced actuator effort and improved energy efficiency by absorbing rapid torque changes. The results demonstrated visual clarity of the torque plots, including their zoomed sections, and emphasized the SMC + IRC’s capability to maintain control smoothness without sacrificing tracking precision. From Figure 10a–c, similar responses were observed with the sine wave reference trajectory shown. Furthermore, the zoomed-in views of the control torque and sliding surfaces for the last 1 s are given Figure 11a–c and Figure 12a–c, these figures further highlighting that SMC + IRC’s steady-state error region was almost negligible compared to the others. This behavior shows a more robust and stable closed-loop performance under coupled system dynamics. The chattering index, measured as the total absolute derivative of the control torque, revealed a significant reduction with the proposed hybrid scheme, as given in Table 5. In addition to chattering suppression, the hybrid approach significantly reduces the control of energy consumption. This energy efficiency is particularly advantageous for rehabilitation robots, where excessive torque oscillations can cause patient discomfort or actuator overheating.

4.4. Quantitative Performance Evaluation

Table 6. Chattering and Control Energy Indices.

Joint	Controller	Chattering Index (Nm)	Reduction vs. SMC (%)	Control Energy (Nms)	Reduction vs. SMC (%)
Hip	SMC	47,897.457	-	1510.4285	-
	ST-SMC	16,152.901	66.3%	1306.9233	13.5%
	SMC + IRC	229.68025	99.5%	1280.9737	15.2%
Knee	SMC	83,834.476	-	769.17701	-
	ST-SMC	53,041.697	36.7%	347.33722	54.8%
	SMC + IRC	550.78939	99.3%	66.594739	91.3%
Ankle	SMC	50,597.934	-	259.34885	-
	ST-SMC	33,996.612	32.8%	118.78576	54.2%
	SMC + IRC	416.12706	99.2%	3.5165859	98.6%

gives computed performance indices for the Fourier and sine wave references, including RMSE, chattering index, and control energy for each joint. Across all joints, the SMC + IRC outperformed the SMC and ST-SMC in all three metrics. The formulas used for the calculations of the chattering index and energy consumption are as follows:

i.

Control Energy Consumption (EN)

This metric evaluates the total effort exerted by the actuator(s) over the entire simulation time. It is the integral of the absolute value of the control torque.

$$EN={\int }_0^{T_f}\left|\tau (t)\right|dt\approx \sum _{k=1}^N\left|{\tau }_k\right|∆t$$

where ${\tau }_k$, $∆t$, and $N$ are the magnitude of the control torque applied at time step k, the simulation time step (i.e., 0.002), and the total number of simulation steps, respectively.

ii.

Chattering Index (CI)

This metric quantifies the discontinuity and high-frequency content of the control signal, directly measuring actuator wear and tear. It is the sum of the absolute changes in the control torque signal between consecutive time steps.

$$CI=\sum _{k=1}^N\left|{\tau }_k-{\tau }_{k-1}\right|$$

where ${\tau }_k \mathrm{a}\mathrm{n}\mathrm{d} {\tau }_{k-1}$ are the control torque at time step and control torque at time step, respectively.

• RMSE: The proposed SMC + IRC achieved the lowest RMSE across all joints, indicating superior tracking accuracy compared to SMC. The ST-SMC similarly shows a better tracking performance compared to SMC.
• Chattering Index: Results showed that the hybrid controller drastically reduced the chattering index, confirming that the resonant feedback loop effectively filtered high-frequency oscillations.
• Control Energy: Results revealed that the total control energy for SMC + IRC was the smallest, demonstrating better efficiency due to smoother torque profiles and less actuator effort.
• Generally, the quantitative and visual analyses jointly confirm that the SMC + IRC provides a robust, high-precision, and energy-efficient control solution for coupled nonlinear systems, namely, lower-limb exoskeletons.

Summary of the results in Table 6 indicates that SMC + IRC achieves the lowest chattering index and energy consumption across all joints, confirming smoother and more efficient control.

Comparatively, ST-SMC offers a notable improvement in smoothness over conventional SMC; it remains sensitive to measurement noise and dynamic coupling. The SMC + IRC hybrid control, nonetheless, achieves a desirable compromise, maintaining the robustness and finite-time convergence properties of SMC while introducing resonant damping through the IRC to minimize chattering. More so, the hybrid controller maintained consistent, smooth stability throughout the entire trajectory range without introducing phase lag or delay that could degrade performance. These results confirm that integrating an IRC within the SMC framework enhances control smoothness and safety, two essential features for human-interactive robotic systems.

5. Conclusions and Future Work Recommendation

5.1. Conclusions

The outcome of this study reveals a comparative analysis of conventional SMC, ST-SMC, and a hybrid SMC + IRC control framework applied to a fully coupled 3-DOF exoskeleton model. The proposed hybrid structure in the work combines the robustness of sliding mode control with the chattering reduction capability of the integral resonant controller. Simulation results demonstrated that while both SMC and ST-SMC ensured satisfactory tracking, they suffered from chattering and energy inefficiency. In contrast, result analyses proved that the hybrid SMC + IRC achieved superior tracking precision, faster sliding surface convergence, and significantly smoother control signals. Quantitative metrics were used to further validate the SMC + IRC reduced RMSE, chattering index, and control energy simultaneously across all joints. The research findings result, highlight the hybrid controller’s effectiveness, robustness, and smoothness, making it a promising approach for real-time implementation in wearable robotic systems and other coupled nonlinear electromechanical platforms. Particularly, the proposed controller achieved the lowest RMSE across all joints, reduced chattering indices by more than 60%, and maintained torque smoothness suitable for safe physical human–robot interaction. These developments directly address one of the major challenges in applying sliding mode-based controllers to rehabilitation robots, balancing robustness with user comfort.

5.2. Future Recommendations

Even though the proposed hybrid controller yielded excellent simulation results, the remarkable achievements were not without room for future improvement. Several areas remain open for exploration to enhance real-world applicability:

• Experimental or Hardware-in-the-Loop (HIL) Validation: In our next phase of future work, the proposed control SMC + IRC will be implemented on a physical exoskeleton prototype to evaluate real-time performance, robustness to sensor noise, and hardware-induced nonlinearities.
• Adaptive and Learning-Based Tuning: Incorporating adaptive gain scheduling will go a long way in improving its reinforcement learning that could be used to optimize the IRC and SMC parameters online to handle time-varying dynamics. Therefore, while the current fixed gains provide excellent performance, a comprehensive parameter sensitivity and adaptivity analysis would be considered to significantly enhance the robustness of the tuning process.
• Human–Robot Interaction Modeling: Extending the control strategy to include muscle dynamics and interaction torques would go a long way in creating room for more natural and responsive motion assistance works.
• Energy-Aware Optimization: Further minimizing actuator energy consumption and exploring the field of battery quality improvement through optimization-based IRC design could improve battery life in wearable robotic systems.
• Hardware-in-the-Loop Simulation: Integrating hard ware in the loop testing and synthesis would bridge the gap between simulation and experimental deployment, ensuring stability under real-time computational constraints.