Motor function impairments frequently stem from neurological conditions such as spinal cord injuries and strokes [1–3]. In the rehabilitation context, designing appropriate exercises plays a crucial role in helping patients regain motor functions necessary for daily activities [4,5]. However, traditional rehabilitation methods heavily rely on one-on-one manual treatment provided by physiotherapists, who subjectively determine exercise difficulty levels and adaptations based on their expertise [6]. This labor-intensive approach highlights the challenges posed by the subjective nature of therapist perception [7]. Consequently, there is a growing interest in robotic-assisted devices as they have the potential to alleviate muscle atrophy in patients and reduce the physical demands placed on physiotherapists [8]. In addition, those devices can ensure a certain level of rehabilitation automation highly required for special types of disabilities.

Robot-assisted systems which involve a high level of interaction between the device and the patient, have become a significant area of interest in robotics, particularly in the context of rehabilitating patients with neurological disabilities [9–11]. Exoskeleton robots have been utilized in the implementation of rehabilitation programs to support physiotherapists and individuals recovering from post-stroke impairments.

The control strategies employed in rehabilitation robots are essential for achieving optimal performance in various rehabilitation treatments [12–14]. Zhao et al. [15] investigated a back stepping controller based on extended state observer for a hydraulic driven lower limb exoskeleton robot to enhance the functionality of load-bearing capacity and to reduce fatigue. Zhang et al. [16] proposed an estimated time-delay approach using Neural Network (NN) of a model-free proportional-derivative controller for a lower-limb exoskeleton. They validated the efficiency of their control system using a simulated model of an lower-limb exoskeleton. Yang et al. [17] proposed an NN controller for a lower-limb exoskeleton robot to conduct trajectory tracking tasks to compensate for external disturbances and unknown parameters. They validated their control system through simulation in MATLAB. Al-dujaili et al. [18] presented an active disturbance rejection control scheme for knee-joint motion control in an exoskeleton medical robot, utilizing both linear and nonlinear extended state observers for disturbance estimation. Asl et al. [19] also developed an NN feedback trajectory tracking controller for a robotic exoskeleton to compensate the nonlinear dynamics of the exoskeleton. Wu et al. [20] proposed a novel control strategy combining Sliding Mode Controller (SMC) and dynamic movement primitives for a reconfigurable upper-limb rehabilitation exoskeleton. Through experimental validation, they demonstrated that sliding mode control under a combinational reaching law outperforms traditional methods such as Proportional-Integral-Derivative (PID) and power reaching law-based sliding mode control in trajectory tracking. Alwand et al. [21] introduced a hybrid control strategy combining active disturbance rejection control and SMC to enhance the tracking performance of a lower limb exoskeleton for hip and knee rehabilitation. Simulation results demonstrated that their controller outperformed conventional approaches by achieving faster tracking, improved disturbance rejection, reduced chattering, and lower control effort. Motivated by the reliable estimation ability of the NN, this work proposed a NN estimation method to determine the unknown parameters.

Table 1 provides a comparative analysis of several existing studies that have developed controller methods for various applications. The focus of the studies, shown in Table 1, was on the control methods, the application as well as the study highlights including advantages and challenges of those studies. By examining Table 1, it can be observed that the surveyed works were relatively new, ranging from 2018 to 2024. In addition, those studies were concerned with the upper-limb and lower-limb as they were the most common motor disabilities. Challenges such as stability, robustness, uncertainties and disturbances were tackled with different levels of accuracy. In Table 1, controller methods are given by: Finite-time Fractional-order Nonsingular Fast Terminal Sliding Mode Control (FONFTSMC), Neural-Fuzzy Adaptive Controller (NFAC), Model-Free based Neural Network Control with Time-Delay Estimation (TDE-MFNNC), Repetitive Learning Control (RLC), Non-singular Fast Terminal Sliding Mode Control method based on Nonlinear Disturbance Observer (NFTSMC-NDO), Iterative Learning Control (ILC), Fractional-Order Ultra-local Model-based NN Sliding Mode Controller (FO-NNSMC), Adaptive Fuzzy Control (AFC) and Robust Adaptive Radial Basis Function Neural Network (RABFNN) have been used.

From the literature presented in Table 1, the researchers have investigated several controller strategies to increase the tracking performance under external disturbances. Although prior controllers in Table 1 have contributed significantly to the field of rehabilitation robotics, several critical limitations persist. In terms of disturbance handling, some approaches lack explicit disturbance observers, making them less robust against unpredictable human-robot interaction forces. Others rely on the assumption of constant disturbances, which reduces their effectiveness under dynamic patient movements. Regarding stability guarantees, certain methods require prior knowledge of disturbance bounds while others adopt overly conservative designs that compromise responsiveness. Clinical adaptability is also a concern, as some controllers require manual tuning for different rehabilitation modes, and others fail to adjust to a patient’s recovery progression over time. These limitations point to the need for a more adaptive, robust, and clinically scalable control strategy.

Motivated by the weakness of the existing works in Table 1, the novelty of this paper lies in the combination of an adaptive controller, Radial Basis Function Neural Network (RBFNN), and disturbance observer to enhance control performance as well as address unknown disturbances and uncertainties in a wearable assistive upper-limb exoskeleton. More explicitly, our paper presented the following contributions:

An adaptive neural network controller was established based on a high-dimensional integral-type Lyapunov function and associated stability strategies. This design allowed the tracking error to remain within a prescribed boundary while ensuring efficient trajectory tracking.

These contributions collectively strengthened the capabilities of the adaptive neural network control system, enabling more effective and reliable performance for upper-limb exoskeleton application.

The remainder of this article has been structured as follows: In Section 2, the structure of the upper-limb exoskeleton and the dynamic model have been given. Section 3 addresses the adaptive neural network controller development. Section 4 represents validation of the adaptive neural network controller on an upper-limb exoskeleton prototype. Section 5 concludes this paper.

The main purpose of upper-limb exoskeleton is to aid patients with muscle injuries in improving their motor capabilities for daily activities. In this section, we have presented a comprehensive overview of the mechanical design and dynamics of the prototype exoskeleton.

This study presents an adaptive neural network controller using a high-dimensional integral-type Lyapunov function for a wearable assistive upper-limb exoskeleton. The objective of the adaptive controller is the integration between Lyapunov function, RBFNN estimator, and disturbance observer for increasing robustness and consider unknown disturbances and uncertainties. This section presents the development of the adaptive neural network controller along with its stability analysis. Fig. 2 shows the logic diagram of the proposed adaptive neural network controller.

The tracking error is defined as: (14)e=q−qdwhere qd represents the desired trajectory and q is the actual trajectory. The tracking error z of the control system is given by: (15)z=Λe+e˙

Here, Λ=diag[λ1,λ2,…,λ3] is a positive definite symmetric matrix. According to Eq. (15), several linear differential functions can be defined to ensure that e converges when z=0. Consequently, e˙→0 as t→∞. By differentiating Eq. (15), we obtain: (16)z˙=Λe˙+e¨=Λe˙+q¨−q¨d

By substituting Eq. (9) into Eq. (16), the following Eq. (17) is obtained: (17)z˙=Mk−1(ℋ(x)+𝒬+τc)+vwhere v=[v1,v2,…,vn]T and z=[z1,z2,…,zn]T. The elements of v are defined as: (18)vi=λie˙i−q¨dwhere ei is the elements of tracking error, ei(i=1,2,…,n). An efficient controller can ensure actual trajectory tracks the desired trajectory and guarantees system stability in the presence of unknown disturbances. To achieve this, the controller’s output, τc, is designed to converge to zero, i.e., z⟶0 [27,30].

In upper-limb exoskeleton control for rehabilitation, desired trajectories are very important in guiding patient movement and ensuring therapeutic effectiveness. In this context, desired trajectories may vary depending on the patient or the stage of rehabilitation of the same subject. In the case of the same subject, four case scenarios may illustrate the range of control needs. First, passive rehabilitation involves the exoskeleton following a pre-defined trajectory without patient effort, ideal for early-stage recovery or patients with limited motor function. This case-scenario is usually faced during the beginning of the rehabilitation exercises or when the patient power is low. Second, active-assisted control adapts the trajectory based on partial user input, where the exoskeleton supports and guides the limb along a desired path when the patient initiates motion but lacks full strength. Third, in resistive training, the exoskeleton imposes resistance along a target trajectory to strengthen muscles, commonly used in advanced rehabilitation stages. Lastly, task-specific trajectories involve complex, real-world movements (e.g., reaching or grasping), requiring the exoskeleton to follow trajectories that mimic daily activities, promoting neuroplasticity and functional recovery. Each scenario demands precise trajectory planning and real-time adaptability to patient condition, making control design a key challenge in upper-limb rehabilitation robotics.

To test the effectiveness of our proposed controller in trajectory tracking, four case scenarios were created artificially to imitate the above-cited cases. Without loss of generality, the generated trajectories were expected to replicate the upper-limb joint movements of the same subject but at different stages of the rehabilitation or four different subjects. The scenarios were created only for simulation purposes.

The gains of our adaptive neural network controller, namely Λ, K, Kl, α, σ, and Ξ, were proportional gains that significantly affected the system. For example, selecting larger values for the gains increased the system’s convergence rate but also led to larger oscillations. Conversely, choosing smaller values for the gains resulted in slower convergence but reduced oscillation. In this study, the parameters were selected using either a trial-and-error approach or based on prior experience to achieve a balance between convergence rate, oscillation, and maintaining a limited range of tracking error. Considering system response, robustness, and tracking error, the controller parameters were determined as Ξ=diag[1.51.51.51.5], Λ=diag[200200200200], α=I4×4, and σ=0.55. For disturbance compensation, the disturbance observer parameters were selected as K=diag[4444] and Kl=[3.23.23.23.2]. The choice of controller parameters significantly influenced system behavior. For example, higher gain values led to faster convergence but result in greater oscillations, whereas lower gains reduced oscillations at the cost of slower convergence. In this simulation, the parameters were chosen through a trial-and-error process.

The dynamic model of the upper-limb exoskeleton can be described as in Eq. (1). Because it is challenging to obtain the exact mathematical model of the system to simulate precise simulation model of the robot, we assumed that the following external disturbances and the interaction forces between the human and the exoskeleton are present: (73)τh=[0.4q˙1+sin⁡(2q1)+0.3sin⁡(q˙1)1.2q˙2−1.5sin⁡(q2+q1)+0.9sin⁡(q˙2)−1.6q˙3−2.2sin⁡(q3)+0.2sin⁡(q˙3)0.7q˙4−1.3sin⁡(q4−q2)+0.4sin⁡(q˙4)] (74)ℱext(t)=[1.5cos⁡(t)+sin⁡(2t)2cos2⁡(t)+1.8cos⁡(t)2.5sin⁡(t)−cos⁡(t)sin⁡(2t)1.2sin⁡(2t)+0.5cos⁡(3t)]

The artificially generated trajectories qd have been shown in Fig. 4 for SR, SP, EP, and WY. It should be recalled here that the abbreviations SP, SR, EP, and WY have been used to respectively represent shoulder pitch, shoulder roll, elbow pitch, and wrist yaw. The actual trajectories, represented as q, have been shown in Fig. 4. In what follows, the four cases were considered as scenarios in the context of the conducted simulations.

The desired trajectory were generated based on our previous study in [37]. An analysis of Fig. 4 reveals that the tracking error e stayed within a defined range, confirming the effectiveness of the adaptive neural network controller. The output of the controller has been represented in Fig. 5.

Fig. 5 presented the raw output of the controller, where the observed chattering reflects the controller’s effort to suppress disturbances affecting the system. Also, the controller output remained within a bounded and stable range. The controller output increased as the generated desired trajectory changed more drastically. For example, as represented in Fig. 4, the trajectory corresponded to a movement of the SR joint more frequently than SP, resulting in the controller applying a higher output to the actuators. This efficient adaptation of the control system can be observed. Furthermore, Fig. 6 illustrates the estimated disturbance of SR, SP, EP, and WY.

The estimated disturbance was calculated based on the measured data from actual trajectory and the measured torque data for each joint. Frequent moves in trajectory introduce heightened disturbances to the joints, elevating angular velocity and external forces on the arm. This creates additional disturbances in the control system. analysis of Figs. 4 and 6 revealed the adaptive neural network controller adeptly observed and responded to these disturbances.

In another experiment, a similar procedure was conducted with predefined periodically desired trajectories. The results were then compared with a conventional PID controller mentioned in [38,39]. The conventional PID controller had the following parameter values: Kp=diag[3333], Ki=diag[2.52.52.52.5], and Kd=diag[0.10.10.10.1]. Fig. 7 illustrates the comparison of the tracking trajectory between the adaptive neural network controller and the conventional PID controller.

The Root Mean Square (RMS) of the tracking error had been determined as shown in Table 2. This table compares the propsed daptive neural network controller, PID controller from [38], NFAC from [22], as well as AFC from [27].

The adaptive neural network controller exhibited a smaller RMS value compared to the conventional PID controller. The adaptive neural network control scheme, which incorporated a disturbance observer, established a more instinctive interaction between the human and robot than the method mentioned in [22,27]. For instance, the performance of the Adaptive Neural Network Controller for Joint 4 (WY) was more efficient by 15.79%, 389.47%, and 268.42% than the NFAC controller by [22], the AFC controller by [27], and the Conventional PID controller, respectively. Therefore, our adaptive neural network controller method enabled efficient trajectory tracking even in the presence of human interaction forces and unknown disturbances.

Our adaptive neural network controller efficiently tracked defined desired trajectory that was adopted from our previous study in [37]. The disturbance observer effectively mitigated unknown disturbances, ensuring system stability using a high-dimensional integral Lyapunov function. The neural network estimator accurately estimated uncertain parameters of the nonlinear dynamic system. In summary, our adaptive neural network controller constrained the tracking error within a specified range, demonstrating its overall effectiveness. In conclusion, our adaptive control scheme has exhibited satisfactory performance and holds significant potential in the field of upper-limb exoskeletons.

This paper presentd a novel adaptive neural network controller designed for a four DoFs wearable assistive upper-limb exoskeleton. The controller ensured system stability using a high-dimensional integral-based Lyapunov function. The RBFNN estimated uncertain parameters, while a disturbance observer handled unknown disturbances. The adaptive neural network controller aimed to accurately track the exoskeleton’s desired trajectory for target joints.

Four human-interaction-based scenarios were conducted to assess the controller’s effectiveness. Results revealed that the adaptive neural network controller bounded tracking errors within a specific range, surpassing the conventional PID controller in terms of efficiency and performance. The proposed controller achieved a significant improvement in RMS tracking error over several benchmark methods, reducing the error by 73% compared to the conventional PID controller, by 15.8% compared to the NFAC, and by an impressive 389% compared to the AFC method.

Although our adaptive neural network controller exhibited impressive performance, recognizing its limitations is crucial for future enhancements. Future research could focus on refining control parameters to attain a higher level of precision and adaptability. Intelligent algorithms based on human interaction movements could be employed to train disturbance observer parameters. Also, future studies could explore hybridizing our control framework with reinforcement learning for adaptive parameter tuning or Gaussian processes for uncertainty quantification. Incorporating the wearer’s estimation strategy into training may further enhance the disturbance observer’s efficiency and overall performance.