Journal of Robotics and Control (JRC) Volume 6.
Issue 3, 2025 ISSN: 2715-5072.
DOI: 10.
18196/jrc.
Robust Velocity Control of Rehabilitation Robots Using Adaptive Sliding Mode and Admittance Strategies Merkuryev 1.
Duishenaliev 2.
Guijun Wu 3*.
Zh.
Zh.
Dotalieva 4.
Orozbaev 5 Department of Robotics.
Mechatronics.
Dynamics and Strength of Machines.
National Research University Moscow Power Engineering Institute.
Moscow,111250.
Russia
3, 4, 5
Department of Mechanics and Industrial Engineering.
Kyrgyz-German Technical Institute.
Kyrgyz State Technical University named after I.
Razzakov.
Bishkek, 720044.
Kyrgyzstan Department of Mechanical Engineering.
Anyang Institute of Technology.
Anyang.
Henan, 455000.
China Emailo1 merkuryeviv@mpei.
ru, 2 duyshenaliyevt@mpei.
ru, 3 wuguijun957@gmail.
com, 4 zh.
dotalieva@kstu.
*Corresponding Author
1, 2
AbstractAiThis paper investigates a velocity tracking control strategy for a planar rehabilitation training robot equipped with two independent linear actuators along the X and Y axes.
dual-loop control framework is proposed by combining admittance control and adaptive sliding mode robust control to facilitate compliant and accurate humanAerobot interaction during active rehabilitation.
In the outer loop, admittance control converts the interaction force applied by the patient into a reference velocity, enabling compliant force-to-motion In the inner loop, an adaptive sliding mode controller augmented with a disturbance observer is designed to ensure robust tracking performance under model uncertainties and external disturbances.
Lyapunov theory is employed to prove the closed-loop stability, ensuring that tracking errors asymptotically converge to zero.
Compared to conventional PID control, the proposed method reduces the root mean square tracking error (RMSE) from 0.
2113 m/s to 0.
0747 m/s.
, decreases the maximum velocity error from 4553 m/s to 0.
2057 m/s.
8% reductio.
, and shortens the recovery time after disturbances from 1.
26 s to 0.
81 s, as validated through MATLAB simulations.
Preliminary experimental results on a planar upper-limb rehabilitation robot demonstrate the controllerAos real-time applicability and confirm its effectiveness in improving interaction responsiveness and motion stability.
Nevertheless, the implementation introduces increased computational complexity and may require real-time optimization for deployment on embedded systems.
Furthermore, while this study focuses on planar motion, the control framework can be extended to multiDOF systems and integrated with physiological signal-based intention recognition to enable more personalized These results indicate that the proposed strategy offers a promising solution for enhancing the performance, robustness, and adaptability of rehabilitation robots in clinical and home-care applications, though clinical trials have not yet been conducted.
KeywordsAiRehabilitation Robot.
Admittance Control.
Adaptive Sliding Mode Control.
Disturbance Observer.
Velocity Tracking.
HumanAeRobot Interaction.
Lyapunov Stability.
MultiDOF Systems.
INTRODUCTION
Neurological disorders such as stroke and spinal cord injury have resulted in a growing global population suffering from motor impairments and reduced mobility .
Ae.
Clinical studies have demonstrated that high-intensity, repetitive, and task-specific rehabilitation training can effectively promote neuroplasticity and enhance motor recovery .
, .
To overcome the limitations of traditional manual therapyAisuch as low training intensity, inconsistent therapist input, and limited scalabilityAirehabilitation robots have emerged as intelligent solutions that integrate mechanical systems .
Ae.
, control theory, and artificial intelligence .
, .
These systems provide consistent, highfrequency interventions, offering great promise in improving patient outcomes .
Ae.
Despite their advantages, most existing rehabilitation robots face two critical challenges.
First, it is difficult to achieve compliant and adaptive humanAerobot interaction (HRI) in the presence of variable patient input .
, .
, fatigue, or intent.
Second, these systems often exhibit low robustness to model uncertainties and external disturbances, which can compromise tracking accuracy and safety .
Ae .
Therefore, among compliance-oriented strategies, admittance control has gained widespread application due to its ability to convert human-applied interaction forces into motion commands.
Several studies have enhanced admittance control by incorporating adaptive and personalized mechanisms.
For example.
Wu et al.
a minimal-intervention strategy to support patient autonomy .
Han et al.
proposed a muscle-strength-based adaptive scheme to respond to fatigue .
Zhang et al.
EMG-based motion intention recognition to personalize training .
and Lee et al.
employed convolutional neural networks to optimize admittance parameters from patient motion data .
However, these methods remain vulnerable to external disturbances and unmodeled dynamics, and often lack mechanisms to ensure robust tracking under real-world In contrast, sliding mode control (SMC) is well known for its robustness against system uncertainties and perturbations .
Ae.
By constraining system trajectories onto a sliding surface.
SMC guarantees convergence even in the presence of bounded disturbances.
Research in recent years has improved SMC performance through dynamic sliding surfaces .
, adaptive gain tuning .
, and the Journal Web site: http://journal.
id/index.
php/jrc Journal Email: jrc@umy.
Journal of Robotics and Control (JRC) ISSN: 2715-5072 integration of disturbance observers, such as extended state observers (ESO) and nonlinear disturbance observers (NDO) .
, .
These techniques enhance robustness and reduce chattering effects.
Nonetheless, classical SMC still suffers from residual chattering and assumes ideal disturbance estimation, which may not hold under dynamic, nonlinear humanAerobot interactions .
Ae.
A Simulation and experimental results validate the system's compliance and adaptability under spastic or varying patient force input.
Traditional PID controllers, despite their simplicity and popularity, struggle with model mismatches and fail to maintain stable tracking performance during episodes of spasticity or parameter variation .
The rest of the paper is structured as follows.
Section II presents the system modeling and the dual-loop control Section i reports simulation and experimental validation results.
Section IV discusses performance, limitations, and future work.
Section V concludes the paper.
While previous efforts have attempted to combine admittance control and SMC, a systematic framework that unifies compliant interaction with adaptive robustness and disturbance rejection remains underdeveloped .
Ae.
Many existing hybrid approaches lack adaptive tuning, depend on heuristic parameter settings, or omit the explicit role of disturbance estimation .
Ae.
Moreover, few existing studies evaluate the trade-offs between compliance, tracking accuracy, and computational overhead, which are critical considerations for real-time implementation in embedded systems used in rehabilitation scenarios .
, .
Addressing this balance is essential to ensure both control effectiveness and practical feasibility.
To address these limitations, this paper proposes a robust velocity tracking control strategy for a two-degree-offreedom planar rehabilitation robot based on a dual-loop A An outer-loop admittance controller transforms interaction force into a reference velocity to achieve compliant HRI .
, .
A An inner-loop adaptive sliding mode controller, augmented by a disturbance observer, ensures robust and accurate tracking performance under model uncertainties and external disturbances.
A A Lyapunov-based stability analysis is provided to verify the asymptotic convergence of velocity tracking errors.
Compared with conventional PID and fixed-gain SMC controllers, the proposed method demonstrates superior tracking accuracy, improved robustness, and better adaptability under varying interaction forces, as validated by simulation and preliminary experimental results.
This control framework offers a practical solution for enhancing the responsiveness and reliability of rehabilitation robots in active, patient-centered training.
The major contributions of this paper are summarized as A A dual-loop control strategy combining admittance control and adaptive SMC is proposed for planar rehabilitation robots.
A A disturbance observer is integrated to enhance robustness and reduce chattering under uncertain This hybrid strategy improves safety and tracking precision, enabling safer, higher-intensity training for patients with motor impairments or muscle spasms.
II.
METHOD
System Modeling and Controller Design Before designing the controller, we need a mathematical model that describes how the robot moves in response to applied forces.
This model serves as the foundation for developing precise and responsive control strategies .
, .
In this section, the dynamic model of the planar rehabilitation robot is first established.
Then, the admittancebased outer-loop control mechanism, which converts humanAe robot interaction force into reference velocity, is explained.
Finally, the design of the adaptive sliding mode robust controller is presented, including the definition of the sliding surface, derivation of the control law, and construction of the disturbance observer.
Dynamic Modeling To capture the robot's motion, we analyze how forces along the X and Y axes cause acceleration, based on Newton's second law.
The system is simplified using decoupling assumptions to reduce computational complexity .
, .
The rehabilitation robot in this study is designed as a 2-DOF planar system, consisting of two orthogonally arranged and independently actuated translational modules.
To analyze the dynamic interactions between the two axes, a general coupled dynamic model is formulated as:
co ycycu ycoycuyc ycuO ycoyc ] .
cO ] .
ccycycu yccycuyc ycuN ] [ ] = .
a ] yccyc ycN The terms ycoycuyc ,ycoycycu Uyccycuyc ,and yccycycu represent the crossaxis inertial and damping coupling terms.
This matrix-based formulation is widely used in multivariable mechanical systems, including robotic manipulators and mechatronic platforms .
, .
Preliminary experimental tests and structural symmetry analysis indicate that these coupling terms are relatively small.
Thus, the coupled model can be approximated as dynamically decoupled:
ycoycu UI ycuO = yaycu ycoyc UI ycO = yayc This simplification allows for independent controller design on each axis and significantly reduces computational complexity during real-time implementation.
The structure of the two-dimensional upper limb rehabilitation robot discussed in this study is shown in Fig.
A Lyapunov-based stability analysis is provided for the proposed controller.
Merkuryev.
Robust Velocity Control of Rehabilitation Robots Using Adaptive Sliding Mode and Admittance Strategies Journal of Robotics and Control (JRC) ISSN: 2715-5072 compute the robotAos motion response based on the force applied by the human .
Ae.
In this study, a second-order linear admittance model is adopted to convert the interaction force into the desired velocity command.
For each axis, virtual inertia ycAycc and damping coefficient yaAycc are defined to represent the expected dynamic compliance relationship.
When a force yayceycu is applied in a given direction, the expected velocity response of the robot satisfies the following admittance model equation:
ycAycc ycN ycc yaAycc ycycc = yayceycu Here, ycycc is the desired velocity of the robot along the X-axis as calculated by the admittance controller, and ycN ycc is its rate of change .
, acceleratio.
Equation .
describes a firstorder inertiaAedamping system: when a patient applies a constant interaction force to the handle, the handle of the rehabilitation robot will move at a steady velocity, the magnitude of which is determined by yaAycc .
At steady state, ycN ycc = 0:
Fig.
Two-dimensional upper limb rehabilitation robot discussed yaAycc ycycc = yayceycu The upper-limb rehabilitation robot supports four training modes: active movement, passive movement, resistive mode, and assistive mode.
Among these, the active movement mode is the most frequently used.
In this mode, the patient is required to actively move the handle within the planar workspace to perform tasks such as drawing circles or moving along straight lines, which demands the highest level of velocity tracking accuracy from the control system.
The two linear drive axes are treated as dynamically decoupled based on the platformAos orthogonal linear guide structure and minimal mechanical coupling .
, .
ycoycu ycO ycu = yayco yayceycu Oe yccycu where ycO ycu denotes the acceleration of the handle along the Xaxis, and ycoycu represents the equivalent inertial mass of the handle in the X direction, i.
, the mass of the handle.
yayco is the driving force applied by the linear actuator along the Xaxis.
yayceycu is the interaction force exerted by the patient along the X-axis.
and yccycu represents the total disturbance and uncertainties along the X-axis, which may include unmodeled friction, parameter variations, and external environmental disturbances, etc.
Similarly, the dynamic model in the Y-axis direction is:
ycoyc ycO yc = yayco yayceyc Oe yccyc The physical meanings of each variable are similar to those in the X-axis, differing only in the direction of action.
Since the two axes are independently controlled, the subsequent controller design process is identical for both the X and Y axes.
The following sections will focus on the derivation of the control method for the X-axis.
Admittance Control Outer-Loop Design The outer-loop admittance controller translates the human-applied interaction force into a target velocity for the This mimics how a physical system with mass and damping responds to force, making interaction more natural and compliant.
Admittance control introduces an equivalent mechanical impedance model into the robot control system to When the applied force increases suddenly, the virtual inertia ycAycc limits the instantaneous rate of velocity change, making the acceleration process smoother and preventing unsafe rapid motion.
By adjusting the values of ycAycc and yaAycc , different equivalent mechanical admittance characteristics can be set:
A larger yaAycc provides greater damping .
, lower compliance, requiring the patient to apply more force to move the robot quickl.
A smaller yaAycc results in higher compliance .
, allowing the robot to respond to smaller forces more The inertia ycAycc affects the inertial response of the system Ai if too large, the system may respond too if too small, the system may become overly sensitive to small force variations.
In this study, appropriate admittance parameters are selected based on the needs of rehabilitation training .
, .
, ensuring that the robot's response to the patientAos force output is neither too stiff nor too sensitive, thus maintaining smooth and safe humanAerobot interaction.
In implementation, the admittance controller serves as the outer loop, taking the real-time measured humanAerobot interaction force yayceycu as input and solving the differential equation .
online to obtain the desired velocity ycycc .
discrete-time systems, numerical integration methods can be used to iteratively calculate ycycc .
for example, using the Euler method .
Ae.
, the following is executed at each control period yuuyc:
c yuuy.
= yc.
a Oe yaAycc yc.
) ycAycc yceycu To compute how the robot should move in response to force in real time, we use numerical integration to update the desired velocity at each control step .
Ae.
Although the Euler method is used for real-time numerical integration due to its simplicity, its accuracy is controlled by adopting a sufficiently small sampling interval .
, 1 m.
Given the relatively slow dynamics of upper-limb motion, this highfrequency regime ensures numerical stability and acceptable Merkuryev.
Robust Velocity Control of Rehabilitation Robots Using Adaptive Sliding Mode and Admittance Strategies Journal of Robotics and Control (JRC) ISSN: 2715-5072 accuracy for real-time admittance control, as validated in simulation and hardware-in-the-loop tests.
Through this outer-loop admittance control, the robot is able to "yield" to the force applied by the patient, moving with the specified inertia and damping characteristics, force-to-velocity transformation .
, .
Inner-Loop Design of Adaptive SMC The outer-loop admittance control generates the desired velocity signal ycycc , but the robotAos actual motion velocity ycycn = ycN ycn still needs to be tracked by the inner-loop controller.
Considering the presence of model uncertainties and external disturbances, this study adopts SMC as the inner-loop velocity tracking strategy, and introduces adaptive tuning and a disturbance observer to enhance robustness and reduce chattering .
Ae.
The design process of the sliding mode controller is detailed below.
Sliding Surface and Error Definition:
Let the difference between the actual velocity ycycn of the Xaxis and the desired velocity ycycc be defined as the tracking yceycn = ycycc Oe ycycn Since the control objective is to ensure that the actual velocity closely follows the desired velocity, it is expected that yceycn Ie 0.
To ensure accurate tracking and fast response, we define a sliding surface that combines the current tracking error, its rate of change, and the accumulated error over time .
This allows the controller to react to both sudden changes and long-term trends in the error, enhancing stability and eliminating steady-state error.
To improve the dynamic response and robustness against disturbances, a first-order proportionalAeintegral (PI) sliding surface is introduced, defined as:
yc yc = yceycn yuIyceNycn yuU O yceycn .
where yuI > 0 is the proportional gain, and yuU > 0 is the integral gain, used to enhance system stability and the ability to eliminate steady-state errors.
yayco = yayceyc yaycyc Oe yayceycu The controller consists of three components:
HumanAerobot interaction feedforward term yaex The interaction force yaex directly applied by the human is introduced as a feedforward term in the control system, which helps enhance the systemAos sensitivity and responsiveness in humanAerobot collaboration .
Ae.
This whole-body dynamic control structure takes into account the characteristics of humanAerobot coupling and is more suited to the control needs in active rehabilitation training scenarios.
Equivalent control term yayceyc This term is used to describe the ideal system dynamics .
xcluding disturbances and uncertaintie.
, and is designed based on the differentiation of the sliding surface under the condition ycN = 0, yielding:
yayceyc = yco.
cN ycc yuIyceN yuUyce ) .
This term ensures that the system can accurately track the reference trajectory under disturbance-free conditions, serving as the main component of the controller.
Sliding mode robust compensation term yaycyc This term is used to suppress unknown disturbances yccycu , modeling errors, and other uncertainties, and is constructed as follows:
yaycyc = ycycu yayc UI yuo y.
Where ycycu is the estimated value of the unknown disturbance yccycu by the disturbance observer.
yayc is the adaptive sliding mode gain, which dynamically adjusts according to the error yuo > 0 is the boundary layer parameter, yu > 0 is the frequency tuning parameter.
The sliding mode term adopts a continuous fractional function form:
yc yuo y.
This sliding surface not only includes the velocity tracking error yceycn , but also incorporates its rate of change yceNycn and an integral term, enabling it to better reflect the dynamic variation of the tracking error in the system.
The control objective is to ensure yc Ie 0, thereby achieving yceycn Ie 0.
which can effectively alleviate the chattering problem caused by traditional sign functions, while maintaining sufficient .
Control Law Design:
To improve accuracy and smoothness, a disturbance observer is introduced to estimate external disturbances in real time, allowing the controller to adjust accordingly .
Ae .
To reduce the chattering caused by sliding mode switching and improve steady-state performance, a linear disturbance observer is designed to estimate the total disturbance term yccycu .
Let the estimated disturbance be denoted by ycycu , and based on inverse dynamics, the disturbance observer is constructed as:
The total control force is designed to track the reference velocity generated by the admittance controller, while compensating for external disturbances and ensuring stability .
, .
When constructing the controller, this paper considers the combined effects of external interaction force yaex and unknown disturbance yccycu acting on the system.
The control objective is to stably track the reference velocity trajectory ycycc generated by the admittance controller.
The following composite controller is constructed:
Disturbance Observer Design ycycu = yayco yayceycu Oe ycoycN ycn Merkuryev.
Robust Velocity Control of Rehabilitation Robots Using Adaptive Sliding Mode and Admittance Strategies Journal of Robotics and Control (JRC) ISSN: 2715-5072 .
Adaptive Sliding Mode Gain Adjustment To balance system responsiveness and robust control performance, this paper adopts a concise and effective linear adaptive gain design strategy:
yayc = ya0 yco1 UI .
Where ya0 > 0 is the base sliding gain, providing minimum control capability.
yco1 > 0 is the responsiveness factor for sliding mode gain adjustment.
Indicates the current deviation magnitude of the sliding surface.
As shown in Fig.
2, the proposed control system for the rehabilitation robot adopts a dual-loop architecture.
The outer loop employs an admittance controller to convert the interaction force yayceycu exerted by the patient into a desired velocity ycycc , thereby enabling compliant and natural humanAe robot interaction.
The inner loop consists of an adaptive sliding mode controller (ASMC), which ensures accurate tracking of the desired velocity.
A disturbance observer (DO) is integrated to estimate model uncertainties and external disturbances in real time, enhancing the robustness of the control system.
The control process is as follows:
The interaction force yayceycu is applied to the handle of the The admittance controller computes the corresponding desired velocity ycycc .
The actual velocity yc is compared with ycycc to obtain the tracking error yce = ycycc Oe yc, which is then fed into the adaptive sliding mode controller.
The controller dynamically adjusts the sliding mode gain based on the error The control law combines the equivalent control term, sliding mode compensation, and disturbance estimation to generate the control signal.
The actuator executes the control input, driving the system while being subject to external disturbances.
To mitigate the influence of such disturbances, the disturbance observer estimates the total disturbance in real time based on the system's velocity and applied control force.
This estimation is fed back into the control law for The entire structure forms a closed-loop control system that ensures high-precision and robust velocity tracking performance even in the presence of external disturbances and model uncertainties.
Lyapunov Stability Analysis To mathematically guarantee that the system remains stable despite disturbances, we use Lyapunov theory, a standard method for proving convergence and robustness in control systems .
Ae.
To verify the stability of the proposed adaptive sliding mode controller under conditions of model uncertainties and external disturbances, the Lyapunov method is employed for analysis.
The following Lyapunov function is selected as a candidate function:
ycO = yc2 ayc Oe yayc O )2 yccEycu where yccEycu = yccycu Oe ycycu is the disturbance estimation error, and yayc O denotes the minimum switching gain required to completely reject the actual disturbance.
Fig.
Dual-loop control architecture with admittance and adaptive SMC Taking the time derivative of ycO, we obtain:
ycON = yc UI ycN .
ayc Oe yayc O )yaNyc yccEycu yccENycu Each term in the Lyapunov function derivative ycON is analyzed for its sign to ensure that the Lyapunov derivative is negative definite.
First term: yc UI ycN Due to the design of SMC, the tracking error yceycn gradually decreases over time, causing yc to approach zero.
The design of the sliding mode gain ensures that ycN decreases continuously and eventually tends to zero.
According to SMC theory, the evolution of the sliding surface yc and its ycN can be expressed using the following .
O ya1 .
cN | O ya2 .
Therefore, yc UI ycN can be expressed as:
yc UI ycN O Oeyu1 yc 2 .
Second term: .
ayc Oe yayc O )yaNyc yu The gain yayc is dynamically adjusted by the control law, so its rate of change yaNyc is related to the error between gains yayc Oe yayc O .
The difference yayc Oe yayc O gradually decreases and can be expressed using the following inequality:
ayc Oe yayc O | O ya3 .
aNyc | O ya4 .
ayc Oe ya O yc | Merkuryev.
Robust Velocity Control of Rehabilitation Robots Using Adaptive Sliding Mode and Admittance Strategies Journal of Robotics and Control (JRC) ISSN: 2715-5072 Therefore, the gain term .
ayc Oe yayc O )yaNyc can be expressed yu .
a Oe yaycO )yaNyc O Oeyu2 .
ayc Oe yaycO )2 yu yc .
Third term: yccEycu yccENycu The disturbance estimation error yccEycu will gradually approach the actual disturbance based on the system dynamics, and its derivative will also gradually tend to zero.
Due to the design of the disturbance estimation, yccEycu yccENycu will tend to zero.
Therefore, it can be concluded that:
yccEycu yccENycu O Oeyu3 yccEycu2 .
Comprehensive Analysis By combining the three terms, we obtain:
ycON O Oeyu1 yc 2 Oe yu2 .
ayc Oe yaycO )2 Oe yu3 yccEycu2 Since each term is negative, the derivative of the Lyapunov function is always less than zero.
Therefore:
ycON O Oeyuyc 2 In summary, the proposed adaptive sliding mode controller ensures closed-loop system stability under the presence of external disturbances and system uncertainties.
The tracking error asymptotically converges to zero, and the disturbance estimation error also converges to zero, thereby achieving global asymptotic stability of the system.
Despite incorporating adaptive gain tuning and a disturbance observer, the proposed dual-loop control architecture remains computationally efficient.
All control laws are algebraic and rely on first-order derivatives or integrals, making them suitable for real-time implementation.
Although the current study focuses on simulation and earlystage experiments, a preliminary deployment analysis suggests that the control algorithm can be implemented on common embedded platforms such as ARM Cortex-M Future work will include precise timing evaluation and code optimization for hardware deployment.
EXPERIMENTS AND DISCUSSION
To verify the effectiveness of the proposed admittanceAe adaptive SMC strategy, simulation experiments were conducted on the rehabilitation robot system.
In order to evaluate the compliance of the admittance control strategy and the robustness of the proposed adaptive sliding mode controller (ASMC) in suppressing unknown frictional disturbances.
MATLAB-based simulation experiments were first carried out .
Admittance and Super Sliding Mode Combined Control The control parameters are listed in Table 1.
To evaluate the performance of the proposed control method under both smooth and abrupt disturbance conditions, a comprehensive simulation scenario was constructed.
Specifically, a periodic interaction force Fm y10cosO2A*0.
5tO and a sinusoidal disturbance dx=3sinO2A*0.
5tO were applied to the system.
Additionally, a step disturbance of 3 N was introduced at t=4s to simulate sudden external interference.
The combined effects of these inputs aim to verify the compliance and robustness of the proposed controller.
The corresponding simulation results are shown in Fig.
TABLE I.
SIMULATION CONTROL SYSTEM PARAMETERS
Parameter Name Physical mass Proportional gain Integral gain Boundary layer parameter Frequency parameter Admittance mass Admittance mass Admittance mass Gain ratio Symbol yco yuI yuU yuo yu ycoycc ycaycc ya0 yco1 Value Unit ycoyci ycoyci ycA UI yc/yco AiAi A Fig.
: Shows the consistency between the handleAos motion velocity and the desired velocity, reflecting excellent tracking performance under compliant control.
The actual speed closely follows the sinusoidal reference trajectory across the entire duration.
A Fig.
: Illustrates the comparison between the actual disturbance and the estimated disturbance.
The green dashed curve almost completely overlaps the true disturbance, confirming that the disturbance observer achieves high accuracy in real-time estimation.
A Fig.
: Displays the time evolution of the sliding surface yc.
A red rectangle highlights the transient peak caused by the step disturbance at 4s.
The inset plot provides a magnified view of this region, showing how s spikes sharply before rapidly converging back toward This behavior demonstrates the strong disturbance rejection capability and fast error attenuation of the proposed adaptive sliding mode controller.
A Fig.
: Depicts the variation of the adaptive sliding gain yayc over time.
A blue rectangle marks the disturbance response region, and the zoomed inset shows a sharp gain increase in response to the error spike.
Subsequently, yayc decreases smoothly and stabilizes, confirming that the adaptive gain mechanism dynamically adjusts control strength based on the error magnitude.
This ensures accurate and robust tracking even under sudden To quantitatively evaluate the superiority of the proposed control strategy compared to conventional PID control, a baseline PID controller was implemented with identical system parameters.
The proportional, integral, and derivative gains were set to yaycy =40, yaycn =5, and yaycc =1.
5, respectively.
Fig.
4 presents a comparison of the velocity tracking errors between the proposed controller and the PID approach under the same external force and disturbance conditions.
Furthermore, three key performance metrics were computed based on simulation data to assess the control effectiveness: .
the root mean square error (RMSE) of the tracking velocity, .
the maximum tracking error under disturbance, and .
the recovery time, defined as the duration required for the tracking error to return within a A0.
05 m/s threshold after the application of a step disturbance.
The Merkuryev.
Robust Velocity Control of Rehabilitation Robots Using Adaptive Sliding Mode and Admittance Strategies Journal of Robotics and Control (JRC) ISSN: 2715-5072 quantitative results of both control schemes are summarized in Table II.
significantly lower than that of conventional control methods, reflecting reliable trajectory tracking performance.
The disturbance observer enhances compensation accuracy, while the sliding surface rapidly converges during disturbances, ensuring precise dynamic response.
In addition, the adaptive adjustment of the sliding gain ycoyc enables the controller to dynamically regulate control strengthAiintensifying control when large errors occur and reducing it as the system stabilizesAithus achieving both accuracy and smoothness.
Experimental Validation To further verify the effectiveness of the proposed control method, experimental validation was carried out on an upperlimb rehabilitation robot.
Fig.
5 shows the compliant control of the handle during active training mode.
A force sensor installed at the bottom of the handle was used to detect the participantAos training The motor drives the handle to follow the interaction force in a compliant manner.
Fig.
Simulation results of admittance and adaptive SMC Fig.
Prototype experimental verification In Fig.
5, a rubber band is fixed to the handle, and only a small interaction force is needed to realize compliant motion As shown clearly in Fig.
6, the handle velocity dynamically follows the changes in interaction force, demonstrating the compliant forceAevelocity mapping of the Fig.
Tracking error comparison: PID vs.
SMC
TABLE II.
PERFORMANCE COMPARISON BETWEEN PID AND ADAPTIVE
SMC CONTROLLERS
Method RMSE.
Max Error.
PID
ASMC DO
Recover Time.
Compared to PID control, the proposed controller achieved: A 64.
6% reduction in RMSE .
2113 m/s to 0747 m/.
A 54.
8% reduction in maximum error .
rom 4553 m/s to 0.
2057 m/.
, and A shorter recovery time .
rom 26 s to 0.
following a step disturbance.
These findings confirm that the proposed controller achieves not only excellent tracking accuracy and improved system stability, but also stronger robustness.
The RMSE is Fig.
Variation curves of robot interaction force and velocity Merkuryev.
Robust Velocity Control of Rehabilitation Robots Using Adaptive Sliding Mode and Admittance Strategies Journal of Robotics and Control (JRC) ISSN: 2715-5072 The experimental results confirm that the handle can respond rapidly to changes in interaction force while maintaining stable motion, thereby validating the effectiveness of the combined admittance and adaptive SMC The experiment was conducted using a healthy adult male .
ge: .
in active tracking mode.
A rubber band was used to simulate compliant resistance.
The systemAos sampling frequency was 1 kHz, and the force sensor had an accuracy of A0.
1 N.
VI.
CONCLUSION
This paper proposes a robust velocity control method with adaptive disturbance compensation for rehabilitation robots, aiming to achieve compliant velocity tracking during patientinitiated movement.
The control strategy adopts a dual-loop architecture: an outer-loop admittance controller enables the robot to respond compliantly to external forces applied by the patient, imparting a desired massAedamping behavior to the an inner-loop adaptive sliding mode controller ensures high-precision and robust velocity tracking under model uncertainties and external disturbances.
The combination of adaptive sliding mode control and a disturbance observer significantly mitigates the chattering problem commonly associated with traditional sliding mode The proposed control strategy offers a key advantage by integrating compliant humanAerobot interaction with robust and precise control performance:
A The outer-loop admittance controller allows the robot handle to actively follow the patientAos motion intention.
A The inner-loop adaptive sliding mode controller ensures accurate velocity tracking even under varying patient forces and disturbances.
This dual-loop structure is particularly suitable for rehabilitation training scenarios, as it enhances the intensity of active movement while maintaining the accuracy and safety of execution.
Simulation and preliminary experimental results validate the effectiveness of the proposed method in improving response speed, tracking precision, and disturbance rejection capability.
Nonetheless, several limitations remain in the current A The system is limited to a planar two-degree-of-freedom .
-DOF) configuration, consisting of two perpendicular translational axes, which restricts its functionality.
A The modeling of human input signals, such as motion intention, is simplified and may not fully reflect the complexity of real-world humanAerobot interaction.
A Clinical validation has not yet been conducted, and the method lacks experimental data from actual patients.
Future research will focus on:
A Extending the control framework to multi-degree-offreedom .
ulti-DOF) systems, enabling 3D rehabilitation motion control.
A Integrating patient-specific physiological signals, such as electromyography (EMG) or intention recognition, to make the control strategy more intelligent and In conclusion, the proposed adaptive disturbancecompensated robust velocity control method provides an effective solution for active rehabilitation training.
achieves a balance between compliant humanAerobot interaction and robust control performance, and holds strong potential for application in other types of rehabilitation robots, contributing to improved patient engagement and rehabilitation outcomes.
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