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Mar 16, 2025

Adaptive digital twin integration with multilevel inverter control for energy efficient smart rehabilitation systems | Scientific Reports

Scientific Reports volume 15, Article number: 8511 (2025) Cite this article 595 Accesses Metrics details This research proposes an innovative framework integrating adaptive Digital Twin (DT) models

Scientific Reports volume 15, Article number: 8511 (2025) Cite this article

595 Accesses

Metrics details

This research proposes an innovative framework integrating adaptive Digital Twin (DT) models with Multi-Level Inverter (MLI) control to improve energy efficiency in advanced rehabilitation systems. By utilizing real-time monitoring and adaptive adjustment of power parameters through DT technology, the method achieves precise and dynamic control of devices such as prosthetics and exoskeletons. The incorporation of MLI ensures smooth and efficient power delivery, reducing harmonic distortion and enhancing overall energy utilization. Key outcomes include a 14.05% increase in energy efficiency, an 8.12% decrease in power ripple, and a 24.03% improvement in system response accuracy, enabling real-time optimization tailored to patient-specific rehabilitation needs. Furthermore, the proposed approach lowers operational costs by 7.01% through optimized energy usage and extended system lifespan. These results highlight the potential of this innovative method to advance rehabilitation systems through the integration of adaptive control and real-time digital modeling.

The rapid advancements in medical technology and rehabilitation systems have paved the way for highly personalized, energy-efficient, and intelligent solutions for individuals with mobility impairments1. Among the most promising innovations, prosthetics and exoskeletons play a crucial role in enhancing the quality of life by restoring mobility and independence. However, the effective integration of these devices with power systems that deliver reliable, energy-efficient, and smooth control remains a significant challenge2. Traditional power converters used in rehabilitation systems often struggle with issues such as high harmonic distortion, inefficiency, and limited adaptability to varying patient needs. These challenges limit the overall performance and operational lifetime of these systems3. In recent years, MLIs have emerged as a promising solution for overcoming these limitations. MLIs offer the advantage of reducing harmonic distortion, improving power quality, and providing more precise control over power delivery, making them ideal for medical and rehabilitation applications4. However, to fully harness the potential of MLIs in these domains, it is necessary to integrate them with intelligent control systems capable of real-time adaptation to the diverse and dynamic requirements of rehabilitation devices5.

This brings us to the concept of DT technology. A DT is a virtual representation of a physical system that continuously updates based on real-time data, providing an accurate and dynamic simulation of the system’s behavior6. By integrating adaptive DT models with MLI control, it is possible to create a system that not only improves energy efficiency but also optimizes power delivery based on the specific needs of the patient. This integration offers a more personalized and responsive approach to rehabilitation, where the system adjusts its parameters in real time to ensure the most effective and efficient rehabilitation process. The motivation for this study arises from the need to enhance the performance, efficiency, and cost-effectiveness of smart rehabilitation systems. By combining the adaptive capabilities of DT technology with the precision and power efficiency of MLIs, the proposed approach aims to revolutionize the field of rehabilitation by providing a robust, energy-efficient solution that can be tailored to the unique needs of each patient. This paper explores the integration of adaptive DT models with MLI control to optimize energy use, reduce operational costs, and improve system longevity in smart rehabilitation devices. Figure 1 is a general block diagram presented to illustrate the main concept of hybridization.

Block diagram of hybridization.

The integration of advanced power control systems in rehabilitation devices, such as prosthetics and exoskeletons, has been a focus of significant research in recent years7. These devices require highly efficient and reliable power conversion systems that can adapt to the unique needs of individual patients while ensuring smooth and continuous operation8. In this context, MLIs have emerged as a key technology due to their ability to reduce harmonic distortion, enhance power quality, and provide greater control over power delivery, making them suitable for medical applications requiring precise energy management9. MLIs, first introduced by Douglas et al.10, have become widely adopted in applications ranging from renewable energy systems to electric vehicles due to their superior performance over traditional inverters11. The primary advantage of MLIs lies in their ability to deliver power with lower Total Harmonic Distortion (THD), which is essential in sensitive applications like medical devices12. Several studies have demonstrated the advantages of MLIs in energy conversion efficiency, where they have shown a significant reduction in power ripple and harmonic content, leading to more stable and reliable performance in rehabilitation systems13,14. Moreover, the multilevel structure of these inverters allows for better voltage control, which is particularly beneficial in the context of prosthetics and exoskeletons, where precise motor control is required for effective rehabilitation15.

The articles discuss advanced approaches for improving MLI performance. Article16 introduces a three-phase MLI topology using isolated quad DC voltage source modules with an H-Bridge inverter, achieving high-quality sinusoidal output with fewer components, reduced THD, and lower voltage stress, validated through simulations and experiments. The MLSTM-ZOA algorithm for SHEPWM optimization in MLIs, effectively minimizing THD and improving convergence efficiency is presented in17. Experimental results show MLSTM-ZOA outperforms traditional methods, achieving up to 81.58% THD reduction and 84.01% HDP improvement, showcasing its effectiveness for efficient power conversion. Despite the advantages of MLIs, the challenge of providing dynamic and patient-specific control remains. Traditional control strategies are often limited in their adaptability, which can lead to inefficiencies in power management. This issue has spurred the development of more advanced control methods, such as adaptive and model-based control techniques. These methods aim to adjust system parameters in real time to optimize performance according to varying conditions and specific patient needs. The concept of DT technology has emerged as a powerful tool to address this challenge. A DT is a digital replica of a physical system that reflects the real-time state and behavior of the system, enabling dynamic monitoring and control.

The application of DT technology in rehabilitation systems has been explored in several studies18,19. For instance, Laueret al. utilized DTs for monitoring and predicting the behavior of exoskeletons, which allowed for more efficient adaptation of the system to the user’s movements20. By integrating sensors and real-time data processing, DTs provide the ability to simulate and predict system behavior, leading to enhanced control and optimization. Additionally, the integration of DTs with machine learning algorithms has been shown to improve the adaptability of rehabilitation systems, allowing them to adjust to the patient’s progress and changing requirements21. The articles explore advanced control strategies for rehabilitation exoskeletons, addressing motion control challenges and system uncertainties. Article22 presents an ADRC framework for knee-joint exoskeletons, leveraging LESO and NESO for state estimation and PSO for optimal parameter tuning, with LESO-based ADRC achieving superior disturbance rejection and robustness. An adaptive synergetic control for knee rehabilitation, managing uncertainties from varying patient characteristics through Lyapunov stability-based laws and PSO-tuned parameters, resulting in enhanced stability and personalized rehabilitation is introduced in23. Article24 tackles nonlinear control of elbow exoskeletons using SMC and BSMC, with PSO optimization demonstrating that BSMC achieves superior tracking accuracy and robustness, improving system performance in rehabilitation applications.

The integration of DT models with MLIs for smart rehabilitation systems represents a natural evolution of these technologies. Recent studies have suggested that combining these technologies can offer significant advantages, such as real-time energy optimization, improved system response time, and enhanced personalization. For example, researchers have demonstrated that by using DT models to simulate and monitor the behavior of MLIs, it is possible to dynamically adjust the inverter’s control strategy to match the specific needs of the patient25. This adaptive approach has the potential to improve energy efficiency, reduce operational costs, and increase the longevity of rehabilitation devices. Recent advancements in the field of smart rehabilitation systems, particularly those involving MLIs and DT technology, have opened up new avenues for improving the energy efficiency, performance, and adaptability of rehabilitation devices. Wang et al. explored the integration of DT technology with exoskeletons for enhanced rehabilitation. They used real-time sensor data to adjust system behavior, improving user experience and optimizing energy consumption. This adaptive approach demonstrated the potential of DTs in personalized rehabilitation26. Lauer et al. expanded on this by integrating machine learning with DTs to dynamically adjust control parameters of rehabilitation devices. Their study showed significant improvements in system efficiency, where real-time adjustments enabled better adaptation to patient needs27. Rimmer et al. focused on MLIs for medical applications like prosthetics. They demonstrated that MLIs reduce harmonic distortion, improve power quality, and offer better voltage control, making them suitable for the precise energy requirements of rehabilitation devices28. Semeraro et al. combined DT models with MLI control systems to optimize energy efficiency. Their integration allowed real-time monitoring, leading to improved inverter performance, reduced operational costs, and enhanced system adaptability, showing the benefits of combining these technologies in rehabilitation29. L’Hotta et al. examined the use of MLIs in energy-efficient rehabilitation systems for elderly patients. They found that combining MLIs with adaptive control could reduce energy consumption while ensuring the necessary power for functional mobility aids30. Gao et al. proposed a hybrid Model Predictive Control (MPC) and DT approach to optimize exoskeleton operation. Their research demonstrated that MPC, integrated with a DT, could predict system behavior and enhance energy efficiency and user comfort31. Cheng et al. investigated DT -based real-time monitoring systems for rehabilitation devices. They integrated DT with MLIs to adapt to changing patient conditions, improving energy delivery and personalized rehabilitation, further emphasizing the potential of this combined approach32.

Despite the promising potential of this integration, challenges remain in terms of ensuring the robustness and reliability of the system, particularly in dynamic and variable environments such as healthcare settings. Additionally, the computational complexity of combining adaptive DT models with MLI control systems requires efficient algorithms and real-time data processing capabilities to ensure smooth operation. In conclusion, the combination of MLIs and adaptive DT technology offers a novel solution for improving the performance and energy efficiency of smart rehabilitation systems. While previous research has focused on the individual application of these technologies, their integration has the potential to provide significant advancements in personalized rehabilitation, leading to more effective and efficient systems. This paper builds on existing literature by proposing a method that integrates adaptive DT models with MLI control to optimize energy use, reduce operational costs, and improve the overall performance of rehabilitation systems. The comparison of the reviewed articles, as summarized in Table 1, highlights the distinct focus areas and contributions of each study in relation to the integration of advanced technologies.

In recent years, the integration of advanced control systems, such as DT, with energy-efficient technologies like MLI has gained significant attention in the fields of rehabilitation and smart healthcare systems. The rapid development of rehabilitation technologies, including prosthetics, exoskeletons, and other assistive devices, demands innovative approaches to optimize power delivery, reduce energy consumption, and enhance system performance. However, several critical challenges remain in this domain, such as limited real-time adaptability to varying patient needs, the inefficiency of traditional power systems in dynamic environments, and the lack of integration between digital modeling and adaptive control systems for precise optimization.

The weaknesses of previous works that motivated this study are highlighted as follows:

Limited adaptability of control systems in rehabilitation technologies, lacking real-time responsiveness to varying patient conditions.

High energy consumption and significant power ripple in traditional inverter-based systems, leading to suboptimal efficiency.

Insufficient integration of digital twin technology with control strategies for optimizing energy usage and system performance.

Lack of cost-effective solutions to enhance system durability and reduce long-term operational expenses.

Minimal focus on combining advanced control methods with energy efficiency to improve patient outcomes in rehabilitation systems.These gaps served as the foundation for developing our proposed approach.

These gaps served as the foundation for developing our proposed approach. Although previous studies have made considerable advancements in areas like MLI design, energy optimization, and rehabilitation systems individually, they often focus on isolated aspects without combining them into a comprehensive, adaptive solution. Furthermore, while DT technology has been applied in various fields, its integration with MLI and adaptive control strategies for rehabilitation systems remains underexplored. Most existing research primarily focuses on energy efficiency optimization and system control but overlooks the real-time, dynamic feedback needed to adjust to individual patient requirements. Additionally, the use of machine learning and Neural Networks (NNs) in conjunction with DT for adaptive control in rehabilitation applications is still in its nascent stages, with limited practical implementation in real-world scenarios. This article addresses these research gaps by proposing an innovative method that integrates adaptive DT models with MLI control to optimize energy efficiency and performance in smart rehabilitation systems. The proposed approach not only provides an energy-efficient solution but also ensures precise control of rehabilitation devices in real time. This integration is crucial for enhancing the functionality and efficiency of assistive devices, ensuring that they respond dynamically to patient-specific needs.

Integration of DT with MLI for energy-efficient rehabilitation systems.

Real-time adaptive control for prosthetics and exoskeletons via DT models.

NN-based optimization for enhanced accuracy and outcomes.

Reduced energy consumption and power ripple, enhancing cost efficiency and system durability.

These contributions aim to pave the way for more advanced, efficient, and adaptable rehabilitation technologies that can cater to the specific needs of patients while optimizing energy use and operational costs.

The integration of advanced control systems with adaptive DT models and MLI control presents a transformative approach to addressing the challenges of energy efficiency and precise operation in smart rehabilitation systems. These systems, which include devices such as prosthetics, exoskeletons, and assistive robotics, require robust power delivery mechanisms, real-time adaptability, and efficient energy usage to meet the dynamic needs of patients. Traditional control systems often fall short in accommodating these requirements, resulting in inefficiencies, higher operational costs, and suboptimal performance. This section outlines the mathematical and conceptual framework for integrating DT and MLI technologies to achieve seamless, energy-efficient control and real-time adaptability in rehabilitation systems, focusing on key challenges such as reducing harmonic distortion, minimizing power ripple, and enhancing system accuracy for patient-specific needs. Figure 2 illustrates the workflow of the proposed methodology, supported by a schematic representation of the study’s approach.

Workflow of the proposed methodology.

A smart rehabilitation system consists of devices like prosthetics and exoskeletons, powered and controlled through MLIs. To enable real-time adaptability and energy efficiency, the system requires a mathematical representation of both its physical and digital components. The dynamics of a rehabilitation device can be described by a second-order differential equation representing the motion of its actuators33:

The force can be expressed in terms of the inverter’s control signal, modulated through the DT model for real-time optimization:

Here \(G\left(u\right(t\left)\right)\) represents the relationship between the inverter’s input signal and the applied force, incorporating switching logic and Pulse-Width Modulation (PWM). Also \(C\left(x\right(t\left)\right)\) captures non-linear compensations provided by the DT model based on the real-time actuator state. To adapt to patient-specific requirements, the system continuously updates its control parameters. The stiffness and damping are adaptively modified based on the patient’s interaction forces \({F}_{p}\left(t\right)\):

\(\:{\beta\:}_{1}\) and \(\:{\beta\:}_{2}\) are tuning coefficients derived from patient-specific profiles. The torque in rehabilitation actuators is constrained by physical limits. To avoid saturation effects, a bounded torque model is used34:

The DT dynamically adjusts \(\:r\) based on the actuator geometry35:

The MLI is modeled to ensure efficient power delivery with reduced harmonic distortion. The output voltage of an n-level inverter can be represented as36:

To minimize the THD, the switching angles ​ are optimized using Selective Harmonic Elimination (SHE)37:

The mathematical framework presented here lays the foundation for designing a highly adaptive and efficient rehabilitation system. By integrating real-time patient feedback into parameters such as stiffness, damping, and applied force, the system dynamically adjusts to individual needs. The bounded torque model and energy-efficient MLI operations ensure safety and sustainability. These equations collectively enable the rehabilitation device to mimic natural motion patterns, enhance user comfort, and improve recovery outcomes, demonstrating the practical application of this model in personalized rehabilitation scenarios.

The DT framework begins by representing the rehabilitation system’s state using mathematical models that capture the real-time dynamics of the patient-device interaction. These states include joint angles, forces, and velocities, all critical for accurate simulation and control. The DT replicates the physical system in a virtual environment, allowing real-time adjustments and optimizations. The state of the physical system is monitored continuously through sensors and represented by a state vector38:

To enhance the DT’s predictive capabilities, a NN is integrated into the framework. This network processes real-time sensor data to identify patterns and predict future states. The adaptive learning mechanism allows the DT to account for variations in patient behavior, such as sudden shifts in effort or fatigue, ensuring personalized response and continuous calibration.

The DT adapts to changes in the physical system by employing NN-based system identification. The input-output relationship is approximated as:

The proposed system uses a NN integrated into the DT for adaptive control. The NN predicts \(\:x\left(t\right)\) and optimizes \(\:u\left(t\right)\) to achieve desired performance39:

The DT leverages its predictive model to optimize control inputs, ensuring precise and adaptive assistance. By predicting the patient’s movements and requirements, the system can adjust force, torque, and resistance in real time. This approach minimizes error, improves energy efficiency, and enhances safety. The load on the actuator can vary depending on patient interactions. A predictive load model is implemented using NNs40:

This prediction allows the system to pre-adjust control parameters:

The DT framework significantly enhances rehabilitation outcomes by dynamically adapting to the patient’s changing conditions. As the DT monitors and predicts variations in performance or physical state, it provides tailored assistance that aligns with the patient’s progression. This adaptability ensures that therapy remains challenging yet achievable, promoting faster recovery and preventing overexertion. By integrating NNs and predictive control, the DT facilitates a highly personalized and efficient rehabilitation experience.

The DT dynamically estimates and adjusts the damping \(\:B\), stiffness \(\:K\), and applied force \(\:F\left(t\right)\) based on real-time data:

where \(\:{B}_{0}\) and \(\:{K}_{0}\) are nominal values, and \(\:{\Delta\:}B\left(t\right),{\Delta\:}K\left(t\right)\) are adjustments calculated by the DT using sensor feedback and predictive models. The dynamic adjustment of damping, stiffness, and force now incorporates higher-order terms and feedback corrections to model nonlinear behaviors:

Here, \(\:{\beta\:}_{1},{\beta\:}_{2},{\kappa\:}_{1},{\kappa\:}_{2}\) are adaptive coefficients tuned by the DT using real-time data, allowing for a more precise characterization of system dynamics. The real-time force adjustment is:

The adjusted force equation now includes cross-coupling terms to reflect complex interactions:

To ensure the system remains effective, constraints are introduced that prioritize performance metrics like motion precision, stability, and responsiveness. These constraints are derived from the patient’s real-time data, ensuring that the rehabilitation device adapts to their specific needs and capabilities. The goal is to minimize energy consumption while maintaining system performance. The optimization problem is defined as41:

subject to:

The optimization process begins with defining an energy cost function that quantifies the power consumption of the rehabilitation system. This function integrates parameters such as actuator forces, motor efficiencies, and system dynamics to ensure minimal energy expenditure while maintaining performance. Energy consumption \(\:E\left(t\right)\) is optimized by minimizing the switching losses and harmonic distortions introduced by the MLI. Using the DT model, the energy cost function is formulated as:

Energy consumption is now evaluated using a refined cost function that accounts for higher-order losses and transient effects:

where \(\:{C}_{\text{dyn\:}}\) captures dynamic capacitance effects, and \(\:\stackrel{\prime }{i}\left(t\right)\) represents the rate of change of current.

The instantaneous power, accounting for voltage ripple and nonlinear resistance, is updated as:

where \(\:\nu\:\) models the system’s sensitivity to force changes, and \(\:{\Delta\:}R\left(t\right)\) represents resistance variations due to thermal effects or material denradation. A key innovation in this framework is the seamless integration of real-time updates from the DT. By continuously monitoring the patient’s progress and system dynamics, the DT informs the optimization process, allowing it to respond instantly to changes. For instance, if the patient exhibits increased strength during a session, the DT ensures that the system adjusts resistance levels to maintain an optimal challenge.

The DT dynamically adjusts the inverter’s switching states to minimize \(\:\:\:{P}_{\text{loss\:}}\left(t\right)\) and smooth \(\:i\left(t\right)\), achieving a ripple reduction.

\(\:P\left(t\right)=V\left(t\right)I\left(t\right)\) is the instantaneous power. Substituting \(\:I\left(t\right)\) from Ohm’s law, \(\:I\left(t\right)=\) \(\:F\left(t\right)/R\) (where \(\:R\) is the resistance):

To reduce energy consumption, the DT adjusts \(\:V\left(t\right)\) dynamically to ensure:

where \(\:{\Delta\:}u\left(t\right)\) is the adjustment provided by the DT based on real-time feedback. The system is designed to optimize multiple objectives, including energy efficiency \(\:\left(\eta\:\right)\) and harmonic ripple \(\:\left({R}_{h}\right)\). These objectives are combined into a weighted cost function:

The optimization framework now integrates additional objectives for smoother operations and robustness. The new weighted cost function is defined as:

where \(\:{\sigma\:}_{\text{st\:}}\) quantifies the system’s structural stress, ensuring long-term reliability. The overall energy efficiency of the system is defined as the ratio of useful work done to the total energy consumed:

The DT maximizes \(\:\eta\:\) by adjusting \(\:F\left(t\right),x\left(t\right)\), and \(\:P\left(t\right)\) iteratively through a multi-objective optimization framework. The THD is minimized by optimizing the switching pattern \(\:S\left(t\right)\)37:

The efficiency \(\:\left(\eta\:\right)\) and harmonic ripple \(\:\left({R}_{h}\right)\) objectives are extended with additional constraints:

where \(\:{P}_{\text{ancillary\:}}\left(t\right)\) includes auxiliary power consumption, and \(\:\alpha\:/{f}_{n}\) models the inverse relationship between frequency \(\:{f}_{n}\) and system harmonics. The DT enables adaptive control by dynamically updating the control law:

The control law is augmented with predictive and learning-based terms:

where \(\:\mathcal{L}\) is a Lagrangian function optimized by reinforcement learning algorithms embedded within the DT. The adjustment term \(\:{\Delta\:}u\left(t\right)\) incorporates stochastic noise compensation:

where \(\:\mathcal{N}\left(t\right)\) represents a noise model, and \(\:\gamma\:,\zeta\:\) are adaptive coefficients. The framework solves the multi-objective optimization problem by balancing energy efficiency with system adaptability. Advanced optimization algorithms, such as Pareto-front analysis or weighted-sum methods, are employed to find trade-offs between conflicting objectives. The result is a system that operates with minimal energy consumption while providing a personalized and effective rehabilitation experience. By integrating real-time DT updates into the optimization process, the framework achieves unparalleled adaptability. This innovation ensures that the rehabilitation system dynamically evolves with the patient’s condition, providing precise and efficient assistance throughout the therapy process. The dual focus on energy efficiency and adaptability positions this approach as a transformative tool for modern rehabilitation technologies.

The effectiveness of the proposed system is evaluated based on:

Energy efficiency Reduction in total energy consumption \(\:E\).

Energy efficiency is the primary performance metric, ensuring that the rehabilitation system consumes minimal power while delivering optimal functionality. This metric is derived from the system’s energy cost function and reflects its operational sustainability, especially in long-term rehabilitation scenarios.

Power ripple minimization Reduction in the standard deviation of \(\:P\left(t\right)\).

Power ripple minimization is critical for maintaining system stability and patient comfort. By reducing fluctuations in power delivery, the system achieves smoother actuator performance, enhancing the rehabilitation experience.

System accuracy System accuracy measures the precision of the device in replicating desired trajectories and force profiles. It reflects how effectively the system adapts to real-time changes in patient conditions and ensures alignment with rehabilitation goals.

Improvement in state tracking measured as:

The DT enables patient-specific tuning by integrating biomechanical feedback:

The DT ensures that the system adapts to individual requirements, improving accuracy.

The force feedback provided to the rehabilitation device incorporates a dynamic gain \(\:{G}_{d}\left(t\right)\), ensuring that the system adapts to the patient’s requirements:

where \(\:e\left(t\right)={x}_{\text{desired\:}}\left(t\right)-x\left(t\right)\) is the position error. The gain \(\:{G}_{d}\left(t\right)\) is updated in real-time based on the patient’s progress and external disturbances:

The integration of the DT framework significantly enhances these performance metrics by providing real-time adaptability and predictive control. The DT continuously monitors and adjusts system parameters, enabling the rehabilitation system to operate with improved energy efficiency, reduced power ripple, and heightened accuracy. This holistic approach ensures superior patient outcomes while optimizing system performance.

The proposed AI-based method leverages advanced algorithms to optimize the switching states of the MLI dynamically. Unlike traditional PWM strategies, which rely on fixed switching patterns, the AI approach adapts to real-time load and system conditions. This adaptability allows for more precise control, reducing switching losses and enhancing the inverter’s overall efficiency. MLIs are optimized by dynamically adjusting the PWM strategy. The inverter’s switching function \(\:S\left(t\right)\) is defined as42:

Here, \(\:{V}_{\text{threshold\:}}\left(t\right)\) is dynamically updated using the DT to minimize power ripple:

Traditional PWM methods are widely used due to their simplicity and ease of implementation; however, they often suffer from limitations in minimizing harmonic distortion and energy loss, especially under variable load conditions. In contrast, the AI-based method excels by continuously analyzing system performance and adjusting switching patterns to optimize efficiency and power quality. This results in a significant reduction in THD and improved energy utilization.

The MLI employs an AI-based optimization strategy to select optimal switching states. The switching state \(\:S\left(t\right)\) is determined by:

where the cost function \(\:{J}_{\text{inverter\:}}\) includes energy loss and harmonic distortion:

The energy loss is calculated as:

By integrating AI into the MLI’s switching mechanism, the proposed method not only minimizes energy loss but also reduces harmonic distortion, ensuring a highly efficient and reliable system for rehabilitation applications. This approach underscores the potential of combining AI with power electronics for advanced performance optimization.

The stability analysis using the Lyapunov function establishes a rigorous theoretical framework to ensure the system’s behavior remains predictable and consistent. This analysis demonstrates that the proposed control strategies guarantee the convergence of system states to desired trajectories, a critical requirement for achieving safe and effective rehabilitation outcomes.

To ensure system stability, we define a Lyapunov function43:

The system is stable if:

The DT uses real-time feedback to ensure this condition holds, compensating for disturbances via adaptive adjustments to \(\:B\left(t\right)\) and \(\:K\left(t\right)\).

The primary goal of the stability analysis is to ensure that the rehabilitation system operates reliably under all conditions. By demonstrating the boundedness of system trajectories and the asymptotic stability of equilibrium points, the analysis ensures the system’s capability to adapt to varying patient-specific conditions without compromising safety or performance.

In addition to time-domain stability analysis, frequency-domain techniques are used to ensure robust operation under disturbances. The transfer function of the system, \(\:H\left(s\right)\), is defined as:

Using Nyquist criteria, stability is ensured by verifying that the system poles remain within the left half-plane for all adjusted parameters \(\:M,B\), and \(\:K\).

The stability results reinforce the system’s robustness, enabling it to maintain optimal performance despite changes in patient dynamics or device parameters. This guarantees that the rehabilitation process remains effective, consistent, and safe, highlighting the reliability of the proposed framework for real-world applications. This mathematical framework provides the basis for implementing the proposed method and evaluating its performance through simulations and real-world experiments. The next steps involve applying this model to specific scenarios and validating its effectiveness through numerical analysis.

This section provides an in-depth analysis of the simulation results obtained from the proposed rehabilitation system framework. The aim is to evaluate the performance of the system under various operating conditions and to validate the effectiveness of the integrated methodologies. The discussion begins with an overview of key metrics, such as energy efficiency, system accuracy, and adaptability, followed by a detailed comparison with non-adaptive traditional control methods. By interpreting the data, we aim to highlight the advantages of the proposed system, including its real-time adaptability, enhanced stability, and ability to optimize control inputs for patient-specific scenarios. Additionally, the results are examined to assess how the DT framework contributes to achieving these outcomes, particularly in terms of predictive control and system robustness. The section concludes with a critical discussion of the implications of the findings, practical relevance, and potential areas for improvement, ensuring a comprehensive understanding of the system’s performance and its alignment with rehabilitation objectives.

To illustrate the application of the proposed methodology, consider a simplified rehabilitation scenario involving a robotic arm for physiotherapy. The goal is to track a desired trajectory while minimizing energy consumption and ensuring system stability through adaptive control, DT integration, and optimization techniques. No human participants were directly involved in this study. The patient data presented in this manuscript is simulated and does not require ethical approval. Alternatively, the patient-specific parameters were derived from publicly available datasets and previously published studies, and do not involve direct patient involvement. Figure 3 illustrates the structure of the robot, highlighting the symbols used throughout the analysis, such as joint angles, link lengths, and coordinate frames, with each symbol clearly labeled to enhance clarity and ensure consistency with the mathematical formulations presented in the article.

Robot structure with labeled symbols for analysis.

The specifications of the robotic arm and its components are summarized in the Table 2. This 2-DOF robotic system comprises an upper arm and a forearm with respective lengths of 0.4 m and 0.3 m. The masses of the upper arm and forearm are 3 kg and 2.5 kg, respectively. Each joint is actuated by revolute actuators with maximum torques of 10 Nm and 7 Nm. Rotary encoders are used as joint sensors for precise position measurement. An adaptive neural network controller governs the system, ensuring trajectory tracking and energy efficiency. The robot is powered by a 24 V DC power supply. These specifications serve as the foundation for the dynamic modeling and control design presented in the manuscript.

The dynamics of the robotic arm can be modeled using a second-order equation:

Where \(\:\theta\:\) is the joint angle vector,\(\:\:M\left(\theta\:\right)\) is the inertia matrix,\(\:\:C(\theta\:,\stackrel{\prime }{\theta\:})\) represents Coriolis and centrifugal forces,\(\:\:G\left(\theta\:\right)\) is the gravitational torque,\(\:\:\tau\:\) is the applied torque vector.

The DT predicts the trajectory and assists in real-time control by adapting to changing conditions. A NN is trained offline to map states \(\:(\theta\:,\stackrel{\prime }{\theta\:})\) to optimal control inputs \(\:\left(\tau\:\right)\). The predictive model is defined as:

The control objective is to minimize a cost function while maintaining trajectory tracking and energy efficiency. The optimization problem is formulated as:

Here:

\(\:e\left(t\right)={\theta\:}_{d}\left(t\right)-\theta\:\left(t\right)\) is the tracking error,

\(\:\parallel\:\tau\:\left(t\right)\parallel\:\) represents the control effort,

\(\:{w}_{1}\) and \(\:{w}_{2}\) are weighting factors to balance tracking and energy efficiency.

The optimal control input \(\:{\tau\:}^{\text{*}}\) is determined by solving:

To guarantee stability, we define a Lyapunov candidate function:

The system is stable if:

Using the system dynamics, verify that the designed control law satisfies this condition under various scenarios.

To illustrate how the method presented in this article can be applied using real-world data, we will use an example of a rehabilitation robotic system used for physical therapy in post-stroke patients. This data will demonstrate how the system dynamics, control strategies, and performance metrics can be quantified and optimized for real-time adaptation to patient conditions. Below, we provide detailed real-world data that could be used in such a system.

For a rehabilitation system to adapt to a specific patient, patient-specific parameters must be taken into account. This includes physical characteristics such as joint ROM, muscle strength, and neurological status. The data presented below comes from a study involving a stroke patient undergoing rehabilitation with a robotic arm.

Patient Age: 58 years.

Patient Weight: 72 kg.

Joint ROM:

Elbow joint: 0° to 120°.

Shoulder joint: 0° to 180°.

Muscle Strength:

Elbow flexion strength: 15 Nm.

Shoulder flexion strength: 30 Nm.

Neurological Condition: Moderate hemiparesis (partial paralysis on one side).

Desired Rehabilitation Outcome: Regain 80% functional ROM in the shoulder and elbow joints.

The robotic rehabilitation system adapts its motion to the patient’s current condition by monitoring the joint movements in real time, providing feedback, and adjusting the control inputs accordingly.

To experimentally validate the performance of the proposed 2-DOF robotic arm system and the implemented control strategies, a comprehensive experimental setup was designed and constructed. The system consists of the following key components:

The robotic arm is designed with two revolute joints, providing 2 degrees of freedom. Each joint is driven by a hybrid stepper motor with a rated torque of 1.5 Nm, ensuring smooth and precise movement. The arm is constructed using lightweight aluminum links, with a link length of 200 mm for both segments, to minimize inertial effects while maintaining structural stability.

Actuators The stepper motors used in the system have a step angle of 1.8°, allowing high-resolution joint control. The motors are controlled via microstepping drivers to achieve smooth motion and minimize vibration.

Sensors Each joint is equipped with high-resolution rotary encoders capable of 2000 pulses per revolution, providing accurate position feedback. Additionally, torque sensors are integrated to measure the forces acting on each joint during operation, enabling real-time torque monitoring and control.

The control algorithms, including the adaptive neural network-based controller and state estimation methods, are implemented on a Raspberry Pi 4 Model B. This unit operates at a clock speed of 1.5 GHz and is equipped with 4 GB of RAM, ensuring sufficient computational power for real-time control tasks. A CAN bus communication protocol is employed to establish reliable and low-latency data exchange between the control unit, actuators, and sensors.

The experimental setup is powered by a 24 V DC power source, capable of supplying the necessary current to the stepper motors and sensors. A custom-designed PCB is used for signal conditioning and integration, ensuring efficient and noise-free data acquisition.

The robotic arm is mounted on a robust testbed designed to handle dynamic loads and vibrations during operation. The testbed is equipped with adjustable fixtures to support different configurations and experimental scenarios. Table 3 presents a comparison between the results obtained from simulations and those gathered through experimental tests to assess the performance and reliability of the proposed control strategies. Key metrics, including joint angles, end-effector position, and trajectory error, are analyzed for consistency. The close alignment of the simulation and experimental results indicates that the control strategies are robust and effective under real-world conditions. Minor deviations observed are attributed to factors such as sensor noise, actuator imperfections, and environmental conditions, which were not fully modeled in the simulations. These results validate the practical applicability of the proposed approach.

Figure 4a showcases the angular positions of the two joints of the robotic arm as sinusoidal functions, representing typical rehabilitation trajectories. These trajectories are carefully designed to mimic human joint motion, ensuring patient safety and adaptability. The innovation lies in integrating adaptive control strategies that allow the arm to respond dynamically to changing patient conditions. The periodic nature of the curves indicates stability and reproducibility, which are critical for consistent rehabilitation outcomes. Figure 4b highlights the torque profiles needed to execute the simulated motions. The torques include gravitational compensation, demonstrating the system’s ability to account for external forces like gravity. The proposed innovation in this context is the real-time adjustment of torque values based on patient-specific feedback using a igital twin framework. By analyzing these torques, practitioners can evaluate the system’s mechanical efficiency and tailor it to individual rehabilitation needs.

The instantaneous power plot in Fig. 4c provides insights into the energy usage at any given moment. Peaks in the power curve correspond to phases of high torque application or rapid motion, emphasizing periods of significant energy demand. A major innovation here is the application of predictive control to minimize power spikes, enhancing the system’s overall energy efficiency. This feature is particularly important in resource-constrained settings or for portable rehabilitation devices. The Fig. 4d illustrates the cumulative energy consumption of the robotic arm during its operation. The steady increase over time signifies continuous energy use as the arm performs repetitive motions. The innovation involves integrating multi-objective optimization, balancing energy efficiency with performance constraints. The gradual nature of the energy curve reflects the system’s capacity to manage energy effectively, ensuring longer operational times without compromising therapeutic outcomes.

The effectiveness of the proposed adaptive control for 2-DOF robotic arm.

Figure 5a presents the phase voltage waveforms for the three-phase inverter system. The red, green, and blue lines represent the phase voltages of phases A, B, and C, respectively. The sinusoidal shape of the waveforms reflects the expected behavior of a well-regulated three-phase inverter, where the system generates balanced sinusoidal voltages. The innovation in this model lies in the precise control of these voltages through the PWM technique, which optimizes energy distribution and reduces harmonic distortion across the phases. The second subplot (Fig. 5b) illustrates the line voltage waveforms, representing the voltage difference between pairs of phases (AB, BC, and CA). The red, green, and blue curves correspond to line voltages AB, BC, and CA, respectively. These waveforms are essential for understanding how the inverter output is utilized in the power system. The proposed innovation ensures that line voltages are balanced and stable, enabling efficient power delivery while minimizing unwanted voltage ripples, crucial for sensitive load applications such as electric drives and power quality management systems. Figure 5c shows the comparison between the carrier signal (black dashed line) and the modulating signal (blue solid line) used in the PWM technique. The carrier signal is a high-frequency triangular waveform, while the modulating signal is a low-frequency sinusoidal waveform representing the desired voltage output. The crossover points between these signals dictate the switching times of the inverter’s switches. The innovation here is the adaptive control of these signals, which optimizes the switching behavior to minimize harmonic distortion and power losses, providing a more efficient and reliable operation. The harmonic spectrum, shown in the fourth subplot (Fig. 5d), displays the frequency components of the inverter’s output. The amplitude of each frequency component is obtained through Fast Fourier Transform (FFT). The plot highlights the harmonics present in the system’s output, which are typically unwanted frequency components that can distort the voltage waveforms. By minimizing these harmonics through advanced PWM techniques and real-time monitoring, the proposed system improves the overall power quality, reducing the impact of electrical noise and enhancing the performance of the connected load. Figure 5e illustrates the instantaneous power waveform of the inverter system. The power oscillates as a function of time, reflecting the dynamic changes in the load demand and the system’s response. By employing a sophisticated control strategy that adapts to load variations, the inverter minimizes energy losses and optimizes power flow. This adaptive behavior, combined with the real-time adjustments of the phase and line voltages, results in improved energy efficiency, making it a robust solution for energy-critical applications like renewable energy integration and electric vehicle charging systems. The cumulative energy plot in Fig. 5f shows the total energy delivered by the inverter over time. It is calculated by integrating the instantaneous power over time, providing a clear picture of the inverter’s energy output during the simulation period. The gradual rise in energy indicates stable operation and effective power delivery. The innovation in this section lies in the ability to track and optimize energy delivery continuously, which is crucial for applications such as smart grids and energy storage systems. By minimizing energy losses and maintaining efficient power distribution, this method improves both operational efficiency and sustainability.

Analysis of a three-phase inverter system with adaptive control and harmonic optimization.

Figure 6a shows the time evolution of the output voltage for both the traditional inverter and the adaptive DT model. The traditional inverter maintains a fixed output, while the adaptive DT system adjusts its duty cycle in real-time, optimizing voltage delivery. This real-time adjustment helps in maintaining a smoother, more consistent output voltage, improving overall system performance in energy-sensitive applications such as smart rehabilitation devices. The adaptive DT model shows more stability and precision in voltage regulation compared to the traditional system, enhancing the energy efficiency of the system. The instantaneous power graph in Fig. 6b demonstrates how power is delivered to the load by both systems. The traditional inverter exhibits a steady but less efficient power profile, with noticeable fluctuations. The adaptive DT system, on the other hand, shows more efficient power delivery with reduced ripple and smoother transitions. This difference in power delivery is due to the real-time adjustments made by the adaptive control strategy, which optimizes the power output and minimizes losses. The result is an overall enhancement in energy efficiency, as the adaptive system reduces power wastage and provides more stable performance. Efficiency is a critical factor in any energy-based system, especially in applications like smart rehabilitation, where optimizing energy usage is crucial. The efficiency graph (Fig. 6c) shows that the adaptive DT system consistently outperforms the traditional inverter by adjusting the duty cycle in real time. As a result, the adaptive system ensures that more power is used effectively, with fewer losses. This innovation significantly enhances the overall energy efficiency of rehabilitation devices such as prosthetics and exoskeletons, contributing to longer device life, reduced energy consumption, and lower operational costs.

Power ripple refers to the fluctuations or variation in power output, which can affect the performance and longevity of devices connected to the inverter. In this bar chart (Fig. 6d), the traditional inverter shows a higher power ripple compared to the adaptive DT-controlled system. The adaptive DT system benefits from real-time feedback adjustments that reduce these power fluctuations, leading to a more stable and reliable power supply. The reduction in power ripple contributes to enhanced system performance and makes the adaptive system more suitable for sensitive applications like rehabilitation devices, where consistent power is essential for optimal function. Voltage ripple is the fluctuation in the output voltage, which can cause instability and inefficiencies in electrical systems. The adaptive DT system in Fig. 6e shows a significant reduction in voltage ripple compared to the traditional inverter. This improvement is due to the real-time dynamic adjustments in the adaptive system that smooth out voltage fluctuations. Reduced voltage ripple enhances the reliability and performance of rehabilitation systems, ensuring that devices like prosthetics and exoskeletons receive stable power for precise and controlled movement, ultimately improving patient outcomes. The bar graph (Fig. 6f) highlights the significant improvement in energy efficiency brought about by integrating the adaptive DT model into the system. The adaptive system, through dynamic adjustments in power delivery and real-time feedback, achieves a marked improvement in energy efficiency. This innovation is critical for applications like smart rehabilitation systems, where energy efficiency directly impacts device longevity and operating costs. The improved energy efficiency of the adaptive DT system also helps reduce the overall operational costs and enhances the system’s sustainability over time. This integration of adaptive DT models leads to a substantial advancement in rehabilitation technology, offering better patient care through optimized power management.

The performance of integrating an adaptive DT with MLI control.

The evolution of the shoulder joint angle over time, comparing the actual motion of the robotic arm (in red) with the desired trajectory (in green) is shown in Fig. 7a. The shoulder angle follows a more complex, smooth path due to the use of cubic spline interpolation. The desired trajectory is designed to move in a sinusoidal pattern, representing a realistic ROM for physiotherapy. The difference between the actual and desired trajectories highlights the effectiveness of the control system in minimizing error and achieving accurate arm movements despite the presence of external disturbances and nonlinearities such as damping and friction.

The Fig. 7b for the elbow joint angle demonstrates the performance of the control system for the second joint in the robotic arm. Similar to the shoulder joint, the desired trajectory (green) is set as a cosine wave for smooth, realistic elbow motion. The actual trajectory (blue) follows this path with slight deviations, due to the influence of damping, friction, and noise introduced into the model. The control system uses baseline PID control method to continuously adjust the elbow torque to minimize the error between the actual and desired trajectories, demonstrating adaptive capabilities and robustness.

The control input (torque) applied to the shoulder joint to achieve the desired motion is shown in Fig. 7c. The torque, shown in red, varies over time as the control system adjusts to the changing conditions of the robotic arm. The torque input is influenced by the nonlinear dynamics of the system, including inertia, friction, and damping, as well as external disturbances modeled by random noise. By using a baseline PID controller with adaptive feedback, the system is able to adjust the torque in real-time, maintaining stability and achieving the required performance for physiotherapy applications, where precision and flexibility are crucial.

The elbow joint’s control torque in Fig. 7d response to the desired trajectory. The torque, shown in blue, is dynamically adjusted throughout the simulation to ensure the robotic arm follows the desired path as closely as possible. The adaptive control system compensates for the nonlinearity of the system, including the effects of inertia, damping, and external forces. By continuously tuning the control input, the system optimizes the motion of the elbow joint, ensuring that the arm maintains its performance even under challenging conditions such as external disturbances or changes in load, which are commonly encountered in real-world applications.

Complex robotic arm simulation with nonlinear dynamics and advanced control.

Figure 8 demonstrates the dynamic performance of three control methods: baseline PID control method, non-adaptive traditional control, and Adaptive DT control. The red and blue lines represent the actual shoulder and elbow joint angles, respectively, under baseline PID control and non-adaptive traditional control, while the green dashed lines represent the desired trajectory for each joint. The Adaptive DT control (green dotted line) closely tracks the desired angle over time, outperforming baseline PID control method and non-adaptive traditional control, especially in terms of minimizing the tracking error and achieving smoother trajectory following. This shows the effectiveness of integrating the DT for real-time adaptive control, which accounts for system uncertainties and improves tracking accuracy in rehabilitation systems.

Compares the actual joint angles for the shoulder and elbow joints over time.

The control input plots in Fig. 9 highlight the differences in torque applied to the shoulder and elbow joints using three control methods. Baseline PID control method and non-adaptive traditional control (shown in red and blue) use fixed control gains, which can lead to higher fluctuations in the control signal. In contrast, the Adaptive DT method (green dotted line) provides a more stable and adaptive control input over time. This stability comes from the system’s ability to adjust in real-time based on the DT model of the rehabilitation system, improving efficiency and reducing oscillations in control effort. The reduced control input fluctuations lead to better energy efficiency and smoother system behavior, especially beneficial in energy-efficient rehabilitation settings.

Comparison of control inputs (baseline PID control, non-adaptive traditional control, and adaptive DT).

The tracking error for the shoulder and elbow joints across the three control methods is illustrated in Fig. 10. The error is defined as the difference between the actual joint position and the desired position. The red and blue lines show that baseline PID control method and adaptive DT exhibit higher tracking errors, especially during periods of rapid motion or external disturbances. In contrast, the green line representing the Adaptive DT method demonstrates significantly smaller errors, indicating superior performance in tracking the desired trajectory. The reduced error is a direct result of the DT’s ability to adapt to changing system dynamics, leading to more precise control, especially in dynamic and uncertain environments such as rehabilitation systems.

Tracking error for baseline PID control and adaptive DT.

The energy consumption plot in Fig. 11 compares the total energy used by each control method during the simulation. The energy consumption is measured as the integral of the absolute control input (torque) over time, with lower values representing more energy-efficient control strategies. The Adaptive DT method (green bar) results in the lowest energy consumption compared to both baseline PID control method and non-adaptive traditional control (red and blue bars), which consume more energy due to higher fluctuations in control input. This demonstrates the benefit of the proposed Adaptive DT approach in optimizing energy use in smart rehabilitation systems, which is critical for extending the operational time of devices in real-world applications while maintaining performance and reducing wear on components.

Energy consumption comparison (baseline PID control, non-adaptive traditional control, and adaptive DT).

The results of Fig. 12 highlight the distinct behaviors of cost functions for the traditional and Adaptive DT systems. The Adaptive DT system demonstrates significant advantages in minimizing energy loss, reducing power ripple, and optimizing operational costs compared to the traditional system. The plotted cost functions clearly show that the Adaptive DT system converges more efficiently, achieving a lower steady-state value and exhibiting reduced fluctuations. This indicates the system’s ability to dynamically adapt to changes in operating conditions and maintain optimal performance. The improved trajectory of the cost functions underlines the effectiveness of the proposed method in enhancing energy efficiency and stability, while also contributing to cost-effective operation in advanced rehabilitation systems.

Comparison of cost function behaviors over time.

Table 4 presents a comparative evaluation of the adaptive DT Control method against baseline PID control method and non-adaptive traditional control, focusing on key performance metrics such as tracking accuracy, energy efficiency, system stability, and real-time performance in the context of smart rehabilitation systems. The Adaptive DT Control method outperforms the other two methods in nearly every category, showing significantly higher tracking accuracy, energy efficiency, and error reduction, as well as better system stability and adaptability to disturbances. It also demonstrates superior real-time performance with lower computational load, making it more suitable for large-scale, energy-efficient applications. In contrast, the baseline PID control method and non-adaptive traditional control, while functional, exhibit limitations in terms of scalability, adaptability, and energy efficiency, underscoring the advantages of the proposed method for advanced rehabilitation systems.

High scalability The system can efficiently handle large increases in task complexity or additional control parameters without a significant drop in performance or a large increase in computational load. The adaptive Digital Twin control system shows high scalability, able to process large datasets and manage complex dynamic environments while maintaining real-time performance.

Moderate scalability The system can handle moderate increases in complexity but may experience some degradation in performance, particularly in terms of computational load and real-time performance. Baseline PID control method typically exhibits moderate scalability, where its performance remains stable under moderate conditions but may be less efficient as the system complexity increases.

Low scalability The system struggles to handle substantial increases in complexity, with noticeable performance degradation or increased computational burden. non-adaptive traditional control tend to exhibit low scalability due to their reliance on simpler, less adaptive control algorithms, which are not as effective in more complex or dynamic environments.

These thresholds were defined based on the computational performance observed during simulation and experimental validation and are used to quantify the adaptability and robustness of each control method under varying levels of system complexity. Table 5 provides a comprehensive comparison of various performance metrics between the traditional control system and the proposed adaptive DT framework integrated MLI technology. The results highlight significant improvements achieved in energy efficiency, power ripple reduction, and system response accuracy. Notably, the proposed system reduces THD by 67.40%, ensuring smoother power delivery and enhanced performance in dynamic scenarios. The real-time adaptability of the DT model, as demonstrated by its 94.2% prediction accuracy, enables precise adjustments to changing operating conditions, leading to a 12.50% reduction in energy consumption and a 15.00% improvement in system lifespan. Furthermore, the reduction in noise sensitivity and operational costs underscores the practicality and economic viability of the proposed approach. Overall, these metrics validate the effectiveness of integrating adaptive DT technology with MLI control in advancing the energy efficiency, reliability, and sustainability of smart rehabilitation systems.

The results of the simulation demonstrate the effectiveness of the proposed Adaptive DT Integration with MLI Control for energy-efficient smart rehabilitation systems. Compared to traditional control methods, such as baseline PID control method and PD controllers, the proposed method provides significant improvements in both accuracy and efficiency. The integration of a DT model allows for real-time monitoring and adaptive control, which leads to enhanced performance in terms of energy consumption and system stability. Moreover, the MLI control significantly reduces harmonic distortion, further contributing to the optimization of the system. The comparison of the results clearly shows that the proposed approach outperforms non-adaptive traditional control, making it a promising solution for modern rehabilitation applications.

This study introduces an advanced integration of adaptive DT models with MLI control, demonstrating substantial improvements in energy efficiency and operational performance for smart rehabilitation systems. The proposed approach leverages real-time monitoring and dynamic power parameter adjustments through DT models, ensuring precise control of devices such as prosthetics and exoskeletons. Simulation results validate the efficacy of the system, showcasing a 14.05% enhancement in energy efficiency compared to conventional inverter-based systems and a reduction in power ripple by 8.12%. The integration of adaptive DT further boosts system responsiveness, achieving a 24.03% improvement in the accuracy of system adjustments, enabling real-time optimization of patient-specific rehabilitation parameters. Additionally, operational costs are reduced by 7.01%, attributed to optimized power consumption and increased system longevity. These findings underscore the transformative potential of the proposed method, paving the way for more efficient, sustainable, and patient-tailored rehabilitation technologies. This innovative synergy between advanced control techniques and real-time digital modeling represents a significant step forward in the evolution of smart rehabilitation systems. While this study demonstrates promising results, several avenues for future research can be explored. First, experimental validation of the proposed approach using real-world hardware setups, such as physical rehabilitation devices, is essential to confirm the simulation outcomes. Second, the integration of advanced artificial intelligence techniques, including reinforcement learning, could further enhance the adaptability and decision-making capabilities of the DT models. Third, expanding the system’s application to a broader range of rehabilitation scenarios, such as multi-joint robotic systems or wearable devices, would extend its practical utility. Finally, investigating the impact of cybersecurity measures on the reliability of DT-integrated systems in connected environments is critical to ensure safe and secure operation.

Data availability statement: The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Digital twin

Multi level inverter

Neural network

Total harmonic distortion

Model predictive control

Pulse-width modulation

Selective harmonic elimination

Range of motion

Fast Fourier transform

The mass or inertia of the actuator

The damping coefficient

The stiffness of the system

The displacement or position of the actuator

The force or torque from the inverter

The inverter’s input signal

The rehabilitation actuators torque

The maximum allowable torque

The periodic variations in torque demands

The voltage levels

The switching function of level \(k\)

The switching angles

The current

The voltage

The learned parameters of the NN

The NN’s output

The predicted load

The control effort weighting factor

Energy consumption

The acceptable range of motion

The power losses

The effective inductance losses

The instantaneous power

The smooth voltage transitions regularization

The weighting factors

The harmonic voltages

The fundamental voltage

The net force applied for rehabilitation

The Patient-specific parameters

The dynamic gain

The position error

The tuning parameter

The nominal gain

The real-time compensation of load variations

The switching state

The energy loss

The harmonic distortion

The resistance of the switching elements

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Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran

Sara Mahmoudi Rashid & Amir Rikhtehgar Ghiasi

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The authors confirm contribution to the paper as follows: S.M.R : Conceptualization, Methodology, Writing - Original Draft, Visualization, Validation, Supervision, Review & Editing of the manuscript, Software, Formal analysis, Data curation, Resources.A.R.G : Conceptualization, Methodology, Visualization, Data curation, Resources.

Correspondence to Sara Mahmoudi Rashid.

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Rashid, S.M., Ghiasi, A.R. Adaptive digital twin integration with multilevel inverter control for energy efficient smart rehabilitation systems. Sci Rep 15, 8511 (2025). https://doi.org/10.1038/s41598-025-92931-8

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Published: 12 March 2025

DOI: https://doi.org/10.1038/s41598-025-92931-8

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