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JRM Vol.38 No.3 pp. 928-937
(2026)
doi: 10.20965/jrm.2026.p0928

Paper:

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Friction Parameter Identification and Feedforward Compensation for Joint Motors in Lower Limb Rehabilitation Robots

Aihui Wang* ORCID Icon, Xiang Zhang*, Hengyi Li* ORCID Icon, Shengda Gao** ORCID Icon, and Jinkang Dong*

*School of Automation and Electrical Engineering, Zhongyuan University of Technology
No.41 Zhongyuan Road, Zhengzhou 450007, China

**Ritsumeikan University
1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan

Received:
May 30, 2025
Accepted:
January 30, 2026
Published:
June 20, 2026
Creative Commons License
Keywords:
motor drive, Stribeck friction model, feedforward compensation control, Kalman filter, firefly algorithm
Abstract

The inherent nonlinear characteristics of friction adversely affects the control accuracy of joint motor drive systems in lower limb rehabilitation robots. Recognizing this challenge, this study proposes an improved friction model and further designs a feedforward compensation control scheme to mitigate motor friction on the basis of the friction model. Compensating for motor control utilizing the friction model, typically the Stribeck friction model, is a promising solution. To overcome the inherent limitation discontinuities of the Stribeck friction model, this study introduces an improved friction model by incorporating the sigmoid function into it. The friction parameter of the model is identified based on the data collected during the experiment, specifically the motor velocity and current. And to enhance the precision of the parameter identification, Kalman filtering algorithm is applied to mitigate the Gaussian noise generated during the experiment. Subsequently, the firefly algorithm is employed for offline identification and curve fitting of the friction parameters in the improved model. Based on the improved friction model, a feedforward compensation controller is further designed by integrating the traditional three-closed-loop PID motor control method with real-time friction compensation. The system employs humanoid gait patterns as input signals to achieve precise position tracking of the robot’s joint motors. Compared with the conventional PID control, the proposed feedforward compensation control reduces both position and current tracking errors. These results confirm that the feedforward compensation strategy, based on the refined Stribeck friction model, effectively mitigates the adverse effects of nonlinear friction, thereby improving the control performance of joint motor drive systems.

Feedforward compensation controller

Feedforward compensation controller

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Cite this article as:
A. Wang, X. Zhang, H. Li, S. Gao, and J. Dong, “Friction Parameter Identification and Feedforward Compensation for Joint Motors in Lower Limb Rehabilitation Robots,” J. Robot. Mechatron., Vol.38 No.3, pp. 928-937, 2026.
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1. Introduction

Lower limb exoskeleton rehabilitation robots play a crucial role in assisting patients with lower limb dysfunction in regaining mobility 1,2,3. However, nonlinear friction in joint actuators significantly affects their performance, especially during low-velocity operation and directional switching 4,5,6. This friction, primarily arising between the motor and the reduction mechanisms, reduces system control accuracy and consequently impairs the rehabilitation outcomes for patients. Although certain mechanical methods, such as adjusting air gaps or bearings, can effectively reduce the impact of friction in joint motors, they often introduce challenges such as increased production costs, stringent precision requirements, and operational complexity 7. Therefore, friction compensation control algorithms that maintain the original mechanical structure and characteristics have become a primary solution for mitigating the friction-related issues 8,9,10.

In practical control systems, constructing a friction model and decoupling the system from friction forces is an intuitive and effective control strategy 11. The effectiveness of this control strategy depends on the accuracy of the friction model and its parameters. Consequently, many researchers have conducted extensive studies on developing accurate friction models. Currently, widely recognized friction models include static friction models 12, such as Coulomb friction, viscous friction, and Stribeck friction, as well as classical dynamic friction models 13,14,15, including LuGre friction, Karnopp friction, and LIP friction. Although these dynamic friction models provide more accurate descriptions of friction characteristics, their practical applications are hindered by the inclusion of unmeasurable variables. For instance, the LuGre friction model employs a system of differential equations to describe friction behavior in both stationary and sliding phases (e.g., stick-slip motion, pre-displacement). This model effectively combines the dynamic and static characteristics of friction; however, the inclusion of the unmeasurable variable “bristle” necessitates a nonlinear observer in theoretical derivation 16, complicating its practical application. Therefore, in engineering practice, the Stribeck friction model remains the most widely adopted due to a fitting accuracy of up to 90% in representing real friction forces 17,18,19.

This study focuses on the joint motor drive system as the control object, aiming to identify the friction model and its parameters through experiments and design a feedforward friction compensation controller to mitigate the effects of nonlinear friction. By doing so, the study enhances control accuracy and improves the overall performance of the system.

The main contributions of this paper are as follows. First, an improved friction model based on the Stribeck model is developed. This model accurately characterizes the friction behavior of the system across different velocity regions and provides precise friction predictions for the design of effective control strategy. Second, a feedforward compensated controller based on the improved friction model is implemented. The controller is applied to a lower limb rehabilitation robot experimental platform. And experimental results demonstrate a significant improvement in motion accuracy and reduction in friction effects.

The structure of the paper is as follows. Section 2 reviews the related work on friction compensation and provides the background necessary for this study. Section 3 explains the construction of the friction model, the identification method of the friction model parameters, and the design of a feedforward controller to compensate for the robot joint motors. Section 4 provides a detailed analysis of the experimental results. Section 5 concludes by summarizing the study.

3. Methodology

The research methodology encompasses friction model development, parameter identification, and feedforward control design. Initially, an improved friction model is developed by refining the classical Stribeck model, providing a more precise theoretical foundation. Following this, parameter identification is conducted to establish the relationship between velocity and friction force. Ultimately, the identified model is integrated into a feedforward control scheme to improve system stability and enhance control precision.

3.1. Fundamentals of the Improved Stribeck Friction Model

The Stribeck friction model, first proposed by German engineer Richard Stribeck in 1902, provides an accurate description of the negative resistance characteristics of friction 36. The model employs an exponential velocity function to capture the nonlinear relationship between friction and velocity. In the classical Stribeck friction model, the relationship is mathematically expressed as follows:

\begin{equation} F_f\left(\dot{\theta}\right) = \left[ F_c + \left(F_s - F_c\right) e^{-\left(\frac{\dot{\theta}}{\dot{\theta}_s}\right)^2} \right] \operatorname{sgn}\left(\dot{\theta}\right) + \sigma \dot{\theta}, \label{eq:friction_model} \end{equation}
where \(\dot{\theta}\) is the relative velocity of the contact surface; \(F_f(\dot{\theta})\) is a nonlinear function, constant greater than zero with asymmetry, indicating the friction moment; \(\sigma\) is the coefficient of viscous friction; \(F_c\) and \(F_s\) are respectively Coulomb friction and the maximum static friction; \(\dot{\theta}_s\) is the critical velocity of Stribeck; \(\operatorname{sgn}(\dot{\theta})\) is a sign function which varies with velocity. The relationship of the Stribeck friction model is illustrated in Fig. 1.
figure

Fig. 1. Stribeck model.

The Stribeck friction model has inherent limitations, as illustrated in Fig. 1. The model generates a discontinuous friction curve with significant variations around zero velocity.

In order to address the discontinuous issue, the Sigmoid–Stribeck friction model (SSFM) is designed by incorporating a sigmoid to enhance the smoothness of the model, as is illustrated in Eq. \(\eqref{eq:friction_model2}\).

\begin{align} \label{eq:friction_model2} F_f ^{\mathit{ssf}}\left(\dot{\theta}\right) &= \left[ F_c + \left(F_s - F_c\right) e^{-\left(\frac{\dot{\theta}}{\dot{\theta}_s}\right)^2} \right] f_{\dot{\theta}} + \sigma \dot{\theta},\\ \end{align}
\begin{align} \label{eq:theta_dot} f_{\dot{\theta}} &= \frac{1 - e^{-\beta \dot{\theta}}}{1 + e^{-\beta \dot{\theta}}}, \end{align}
where \(f_{\dot{\theta}}\) defines the sigmoid function; \(\beta\) is the smoothing factor, and a smaller \(\beta\) results in a smoother transition. Considering the necessity of smoothness and precision, \(\beta\) is set to 70. As the velocity approaches the zero point, the value of sigmoid function smoothly transitions from \(-1\) to 1.

This improvement effectively avoids motor quivering caused by significant variations in friction force as velocity approaches zero, resulting in a smoother Stribeck friction model curve. As illustrated in Fig. 2, the SSFM provides a gentler transition around the zero-velocity region, ensuring a more continuous and realistic representation of the friction behavior.

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Fig. 2. SSFM.

3.2. Parameter Identification of the Friction Model

Identifying friction parameters is a complex task, consisting of three main steps: constant velocity tracking, Kalman filter design, and parameter identification. As shown in Eq. (3), four friction parameters \(F_c\), \(F_s\), \(\dot{\theta}_s\), and \(\sigma\) need to be identified, and the equation contains only one independent variable \(\dot{\theta}\), and one dependent variable \(F_f^{\mathit{ssf}}(\dot{\theta})\). Therefore, measuring the friction force at different joint motor velocities provides essential data for subsequent identification of friction parameters. During the constant velocity tracking experiment, the friction force on the joint motor is measured under no-load conditions while the motor rotates at different velocities. The Kalman filter is applied to mitigate Gaussian noise introduced during the experiment, improving data accuracy. With the accurate experimental data obtained from the first two steps, the friction parameters of the joint motor can be precisely identified.

3.2.1. Constant Velocity Tracking

In the robot joint motor drive unit, the motor’s motion state is jointly determined by the driving torque and friction torque, as described by Newton’s second law:

\begin{equation} M \ddot{\theta} = F - F_f^{\mathit{ssf}}, \label{eq:motion_equation} \end{equation}
where \(M\) is the load mass; \(\ddot{\theta}\) and \(F\) are respectively the motor acceleration and the driving torque. When \(\ddot{\theta} = 0\), i.e., the motor rotates at a constant velocity, it follows from the nature of the motor driving torque that:
\begin{equation} F = F_f^{\mathit{ssf}} = KI, \label{eq:force_equation} \end{equation}
where \(K\) and \(I\) are the motor’s torque constant and driving current, respectively.

When the robot joint rotates at a constant velocity, the total friction torque can be determined by measuring the current value of motor during stable operation. By controlling the robot joint motors to rotate at different velocities under no-load conditions, and recording their corresponding total friction torque values, the Stribeck friction curve applicable to the joint motors of the lower limb rehabilitation robot can be fitted using the experimental data.

3.2.2. Design of the Kalman Filter

In the robot joint motor control system, high-frequency noise is commonly introduced by amplifier circuitry and analogue-to-digital converter of the micro controller unit when capturing the drive current 37. Therefore, in order to improve the quality of the acquired signal, it is necessary to filter the current signal. This study introduces the Kalman filtering algorithm to accurately obtain the current of the robot joint motor driver.

The algorithm is an optimal estimation algorithm that estimates the state of a dynamic system from noisy measurement data through recursive calculations 38. It is particularly effective to deal with Gaussian white noise generated during the sampling process, filter high-frequency noise, and ensure that the filtered data remain undistorted. The core idea of the algorithm is to combine the state equation of the system with the measurement model, using the previous state estimate and current measurement value to calculate the current optimal state estimate. In the robot joint motor drive system, the state equation is defined as:

\begin{equation} x_k = A_k x_{k-1} + B_k u_k + w_k, \label{eq:state_space} \end{equation}
where the subscripts \(k\) and \({k-1}\) represent the moments corresponding to the states; \(x_k\) is the state value of the motor drive; \(A_k\) and \(B_k\) are the system coefficients; \(u_k\) is the input of the control to the system; \(w_k\) is the process noise, which follows a Gaussian distribution, i.e., \(w_k \sim N(0,Q_k)\). Here, \(Q_k\) represents the process noise covariance, which was selected to reflect the modeling uncertainty of motor dynamics and to balance estimation responsiveness against noise suppression. The measured values for the real state of the drive satisfy the following:
\begin{equation} z_k = H_k x_k + v_k, \label{eq:linear_observation} \end{equation}
where \(z_k\) is the observed motor drive force; \(H_k\) is the system observation coefficient; \(v_k\) is the measurement noise, which follows a Gaussian distribution, i.e., \(v_k \sim N(0,R_k )\). Here, \(R_k\) represents the system measurement noise covariance, which was determined from the variance of sensor measurements to account for measurement disturbances realistically. The design of Kalman filter consists of two primary stages: prediction and update.

In the prediction phase, the Kalman filter employs the state estimate from the previous moment to predict the state at the current moment.

\begin{align} \label{eq:state_estimation} \hat{x}_ {\overline{k}} &= A_k \hat{x}_{k-1} + B_k u_k,\\ \end{align}
\begin{align} \label{eq:kalman_prediction} {P}_{\overline{k}} &= A_k P_{k-1} A_k^T + Q_k, \end{align}
where \(\hat{x}_ {\overline{k}}\) is the priori state estimate at time \(k\); \(\hat{x}_{k-1}\) is the posteriori state estimate; \({P}_{\overline{k}}\) is the priori state estimate covariance, corresponding to the covariance of \(\hat{x}_ {\overline{k}}\); \(P_{k-1}\) is the posteriori estimate covariance, representing the covariance of \(\hat{x}_{k-1}\), which represents the uncertainly of the state as a filtering outcome, with the initial value set as \(P_0 = 1\).

During the phase of update, the Kalman filter integrates new measurement data with the a priori knowledge from the prediction phase. It employs the new measurements to amend the a priori estimates, consequently reducing estimation errors and increasing the reliability of the estimates.

\begin{align} \label{eq:kalman_gain} K_k &= \frac{{P}_{\overline{k}} H_k^T}{H_k {P}_{\overline{k}} H_k^T + R_k},\\ \end{align}
\begin{align} \label{eq:kalman_filter} \hat{x}_k &= \hat{x}_ {\overline{k}} + K_k \left(z_k - H_k \hat{x}_ {\overline{k}}\right),\\ \end{align}
\begin{align} \label{eq:placeholder} P_k &= \left(1 - K_k H_k\right) {P}_{\overline{k}}. \end{align}

When the robot joint motors rotate at a uniform velocity, the Kalman filter effectively mitigates the Gaussian noise generated during motion, enabling accurate extraction of the motor drive current. The performance of the Kalman filter can be optimized by adjusting the values of \(Q_k\) and \(R_k\). The value of \(Q_k\) represents the process noise covariance, influencing both the system’s convergence and the weight of trust assigned to the predicted values. The value of \(R_k\) represents the measurement noise covariance, affecting both the filter response speed and the reliability of the measured values. In this experiment, the parameters are set as follows: \(A_k=1\), \(B_k=0\), \(u_k=0\), \(H_k=1\), \(Q_k=1\times10^{-6}\), and \(R_k=0.1\). \(Q_k\) and \(R_k\) were selected based on the statistical characteristics of the system and sensor noise. Specifically, \(Q_k=1\times10^{-6}\) was chosen to reflect the relatively small uncertainty in the motor dynamics, while \(R_k=0.1\) corresponds to the variance estimated from the sensor measurement noise in the experimental setup. These values balance estimation accuracy and noise suppression. The waveform of the motor drive current after Kalman filtering is illustrated in Fig. 3. It demonstrates that the noise component is significantly suppressed compared to the original current signal, leading to improved stability and accuracy of the current measurements.

Since the friction of the joint motors exhibits stronger nonlinearity at low velocities and approximately linear behavior at higher velocities, a denser sampling strategy is adopted in the low-velocity range. In this experiment, the low-velocity range (0.05 rad/s to 0.5 rad/s) is sampled at intervals of 0.05 rad/s, with 10 observation points taken; the high-velocity range (0.5 rad/s to 2 rad/s) is sampled at intervals of 0.25 rad/s, with 6 observation points taken. To ensure data reliability, each velocity point is measured multiple times. By recording the drive currents of the robot joint motors during uniform rotation at various velocities and filtering the collected signals, as illustrated in Fig. 4, this data acquisition provides the basis for analyzing the relationship between velocity and driving current.

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Fig. 3. Kalman filter effect on joint motor current.

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Fig. 4. Current acquisition.

To further analyze the relationship between velocity and driving current, the current variations at different velocities (0.1 rad/s, 0.3 rad/s, 0.5 rad/s, and 1 rad/s) are depicted in Fig. 4. As velocity increases, the average current exhibits an upward trend. For instance, at 0.1 rad/s, the current is primarily around 0.25 A, whereas at 1 rad/s, it rises to approximately 0.3 A. Additionally, the experimental results indicate that the motor’s current curve fluctuates at different velocities, meaning the current value is not constant but oscillates around a certain mean. These fluctuations stem from the dynamic response of the control system. In particular, during the PID adjustment process, the system continuously corrects errors, leading to slight current variations. This indicates that the motor requires higher driving current to sustain motion at higher velocities, aligning with the expected behavior of the SSFM.

According to the measured data, the total friction force at each velocity is accurately calculated, establishing the corresponding relationship between friction force and velocity. These experimental data provide valuable support for the subsequent parameter identification of the friction model.

3.2.3. Identification of Friction Parameters Based on the Firefly Algorithm

Parameter identification for the friction model is a multiple objective problem that involves complex calculations. From the SSFM, it is evident that in the robot joint motor structure, the total friction force is determined by four uncertain parameters \(F_c\), \(F_s\), \(\dot{\theta}_s\), and \(\sigma\) in addition to the motor’s rotational velocity. The exact values of these parameters cannot be directly determined through simple calculations. With the accurate experimental data collected above, advanced identification methods can be employed to determine the friction parameters. Traditional friction parameter identification methods, such as least squares and genetic algorithms, often suffer from slow convergence and a tendency to fall into local optima when dealing with complex nonlinear models.

To address the above issues, this paper applies the firefly algorithm to identify the friction model parameters for the joint motors of the lower limb rehabilitation robot in an offline setting 39. In the algorithm, each firefly represents a potential solution, with its position corresponding to the solution vector of the problem to be optimized, while the luminous intensity of the firefly is related to the fitness value of the objective function. The firefly algorithm is well-suitable for friction model parameter identification due to its advantages of few parameters, a simple evolutionary process and strong global search capability. Accordingly, the objective function is defined as follows:

\begin{equation} J = \sum_{i=1}^{N} \left( \tilde{F}_f^{\mathit{ssf}} - F_f^{\mathit{ssf}} \right)^2, \label{eq:sum_squared_differences} \end{equation}
where \(\smash{\tilde{F}_f^{\mathit{ssf}}}\) is the friction value calculated from the recognized parameters and motor velocity; \(J\) is the objective function value, which is the sum of the accumulated differences between the motor friction \(\smash{{F}_f^{\mathit{ssf}}}\) and the recognized friction \(\smash{\tilde{F}_f^{\mathit{ssf}}}\) at various experimentally collected velocities. A smaller value of \(J\) indicates better recognition performance.

The firefly algorithm is essentially a randomized search method characterized by its simplicity. The algorithm initializes its parameters and firefly positions, calculates mutual attraction based on brightness, and iteratively updates their positions and attraction until the termination conditions are satisfied, ultimately identifying the friction parameters. In this study, the parameters of the firefly algorithm are configured as follows: population size \(= 20\), step factor \(= 0.5\), light absorption coefficient \(= 0.8\), initial attraction \(= 1\), and maximum iterations \(= 5000\). In this experiment, these parameter settings enable the algorithm to efficiently identify friction parameters in a complex search space. The results of the firefly algorithm’s parameter identification are as follows, which are summarized in Table 1.

Table 1. Firefly algorithm parameter identification results.

figure

The curve fitting based on this friction model achieves a coefficient of determination of 0.998, indicating an excellent fit to the experimental data, as illustrated in Fig. 5. The results demonstrate that the fitted curve accurately captures the variation of robot joint motor friction with motor velocity. Notably, in the low velocity interval, the model effectively fits nonlinear friction and accurately reflects the relationship between motor friction and velocity.

figure

Fig. 5. Joint motor friction and velocity fitting results.

3.2.4. Design and Implementation of the Feedforward Compensation Controller

To address friction factor during the rotation of robot joints, a feedforward compensation controller is designed based on the friction model and the results of the recognized parameters are described above. By introducing a friction compensation value, the unfavorable effect of nonlinear friction on motor operation is mitigated.

On the basis of the traditional PID control method with three closed-loop negative feedbacks of position, velocity, and current, the feedforward compensation mechanism based on the Stribeck friction model further enhances control accuracy and system performance. Specifically, the current loop, as the inner loop, precisely controls the motor current to ensure stability and response velocity; the velocity loop limits the maximum current value of the system by restricting the maximum rotational velocity and velocity slope to ensure smooth operation; and the position loop, as the outer loop, focuses on the precise adjustment of the motion angle of the robot joints. During motor operation, sensors capture the motor’s current state (e.g., current, velocity, and position), which is integrated with the feedforward compensation controller to optimize the motor control system. Sensor feedback enables the controller to adjust the friction compensation value in real time, effectively counteracting the impact of nonlinear friction on motor operation. This feedforward compensation control mechanism effectively mitigates the impact of nonlinear friction, enhancing the overall dynamic performance and robustness of the joint motor control system.

Feedforward control can improve the dynamic performance of the system, reduce the steady state error, and does not constitute a closed loop, thereby not affecting the stability of the system. The introduction of the feedforward compensation controller on the basis of the three closed-loop control can effectively reduce the motor tracking error, thereby enhancing the kinematic performance and operational accuracy of the whole robot system, as illustrated in Fig. 6. In the figure, \(\theta_t\) is the desired displacement; \(L\) and \(R\) are respectively the motor inductance and resistance; \(K\) is the motor torque constant; \(\theta\) is the actual displacement.

figure

Fig. 6. Feedforward friction compensation controller.

With the introduction of feedforward friction compensation, the actual drive current of the motor is the sum of the PID operation result and the compensation current:

\begin{equation} I = I_q + I_c, \label{eq:linear_observation2} \end{equation}
where \(I\) is the actual drive current; \(I_q\) is the current value calculated by the velocity loop PID; \(I_c\) is the compensation current value through using the SSFM.

4. Experimental Validation

To assess the proposed friction model and the designed feedforward controller, experimental validation was conducted on a lower-limb rehabilitation robot platform. The robotic system comprises a self-developed four-joint motor drive module, a data processing center, and a frame connection structure, as illustrated in Fig. 7. Additionally, four DC motors regulate the lengths of the robot’s thigh and calf segments.

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Fig. 7. Lower limb rehabilitation robot experiment platform.

The host computer transmits motion commands and collects feedback data via the CAN bus, enabling the execution of humanoid gait operations and subsequent result analysis 40, and the design framework of the robot is illustrated in Fig. 8. In this study, experimental tuning was performed to determine the initial PID controller parameters. The parameters are as follows: current loop PID controller coefficients \(K_{\text{IP}} = 0.6\); velocity loop PID controller coefficients \(K_{\text{VP}} = 0.6\), \(K_{\text{VI}} = 0.2\); position loop PID controller coefficients \(K_{\text{PP}} = 30\), \(K_{\text{PI}} = 5\).

figure

Fig. 8. Robot design framework.

In order to verify the effectiveness of this compensation mechanism, a control experiment was conducted on the lower limb rehabilitation robot platform to compare the control performance with and without the feedforward compensation method. The humanoid motion gait signal closely resembles the actual application scenario of the system, enabling observation of the friction compensation effect on the robot joints. The results of the position tracking experiment are presented in Fig. 9.

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Fig. 9. Position tracking experiment.

The friction compensation method mitigates the impact of friction on the joint motors of the lower limb rehabilitation robot during commutation. Without friction compensation, low-velocity commutation of the joint motors results in unsmooth peak transitions, which leads to jerks during the actual operation of the robot. With compensation, the tracking curve remains smooth and stable even during low-velocity commutation. To further evaluate the impact of friction compensation on motion accuracy, the position tracking error of the joint motor during commutation is analyzed. By comparing the compensated and uncompensated cases, it can be observed that friction compensation significantly reduces fluctuations and ensures smoother transitions. Introducing compensation reduces the peak position error during commutation from 0.27° to 0.25°, as illustrated in Fig. 10, alleviating the jitter during robot commutation.

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Fig. 10. Position tracking error.

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Fig. 11. Current tracking error.

The drive current of the motor directly influences its torque output. Under three closed-loop PID control, the maximum drive current error of the robot joint motor is 19.35 mA. With the introduction of the feedforward compensation mechanism, this error is reduced to 8.59 mA, as illustrated in Fig. 11. This compensation mechanism enhances the system’s torque output capability. In addition, the reduction of the maximum error enables the motor to output a more stable and accurate torque, enhancing system control accuracy while minimizing irregular vibration and noise during operation.

5. Conclusion

The nonlinear friction in mechanical systems adversely affects the motion accuracy and smoothness of lower limb rehabilitation robot joints. To address this challenge, this paper proposes a feedforward compensation mechanism based on the SSFM utilizing the sigmoid function, Kalman filtering algorithm, and firefly algorithm. The effectiveness of the proposal is evaluated through experimental analysis of the robot joint drive system’s position tracking performance and velocity response. Experimental results demonstrate that the proposed feedforward compensation mechanism effectively reduces friction-induced disturbances at low velocities, enhancing motion smoothness. Compared with conventional three-loop PID control, the proposed approach achieves superior position tracking accuracy and faster velocity response. These findings validate the effectiveness of the proposed compensation strategy and provide a foundation for further improvements in the control precision of rehabilitation robots. Future work aims to extend the method to more complex motion scenarios and integrate adaptive friction compensation techniques to enhance robustness and adaptability.

Acknowledgments

This research was funded by the Henan Province Key Research and Development Project (Grants No.241111312000), Henan Province Key International Science and Technology Cooperation Project (Grants No.251111520400), the Key R&D Project of Henan Province (251111220900), and the Henan Province Key Technologies Research Development project (Grants Nos.252102211106, 252102320281).

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