Disturbance observer-based terminal sliding mode control for the training safety improvement in robot-assisted rehabilitation

2.1. Notations, definitions, and lemmas

Notations: $$ \mathbb{R} $$, $$ {\mathbb{R}^{n}} $$, $$ {\mathbb{R}^{m \times n}} $$ represent the set of real numbers, $$ n $$ dimensional vectors space, and $$ m \times n $$ real matrices space, respectively. The maximum and minimum eigenvalues of the matrix $$ \Phi \in {\mathbb{R}^{m \times n}} $$ are defined by $$ {\lambda _{\max }}\left( \Phi \right) $$ and $$ {\lambda _{\min }}\left( \Phi \right) $$, respectively. For $$ {\vartheta}\in {\mathbb{R}^{n}} $$, denote $$ si{g^\iota }\left( \vartheta \right) = col\left( {si{g^\iota }\left( {{\vartheta _1}} \right), \ldots , si{g^\iota }\left( {{\vartheta _n}} \right)} \right) $$, $$ si{g^\iota }\left( {{\vartheta _i}} \right) = {\mathop{\rm sgn}} \left( {{\vartheta _i}} \right){\left| {{\vartheta _i}} \right|^\iota }\left( {i = 1, \ldots , n} \right) $$, and the signum function is given by

Lemma 1 ^[41]Assume that there exists a continuous positive definite function $$ V\left( t \right) $$ which satisfies the following inequality:

Subsequently, the function $$ V\left( t \right) $$ approaches the equilibrium position within a finite time $$ {t_f} $$:

where $$ \eta > 0 $$, $$ \mu > 0 $$ and $$ 0 < \nu < 1 $$.

2.2. System modeling

Consider describing an exoskeleton system with systematic uncertainty and bounded disturbances in the following form:

where $$ q = {[{q_1}, \ldots , {q_n}]^T} $$ denotes the angles of the five joints, $$ {\dot q} $$ and $$ {\ddot q} $$ represent joint velocity and acceleration, respectively, $$ \tau = {[{\tau _1}, \ldots {\tau _n}]^T} \in {\mathbb{R}^{n}} $$ denotes the input torque vector, and $$ {\tau _d} = {[{\tau _d}_1, \ldots {\tau _d}_n]^T}\in {\mathbb{R}^{n}} $$ represents the disturbance torque vector; the generalized inertia matrix $$ M(q)\in {\mathbb{R}^{n \times n}} $$, carioles/centripetal matrix $$ C(q, \dot q)\in {\mathbb{R}^{n \times n}} $$ and gravity vector $$ G(q)\in {\mathbb{R}^{n \times 1}} $$.

Due to the unavoidable uncertainty in the system, Equation (4) can be written as:

where

with $$ {M_q} = M\left( q \right) - {M_0}\left( q \right) $$, $$ {C_q} = C\left( {q, \dot q} \right) - {C_0}\left( {q, \dot q} \right) $$, $$ {G_q} = G\left( q \right) - {G_0}\left( q \right) $$. $$ {M_q}, {C_q}, {G_q} $$ represent the values obtained from parameter identification ^[42] and $$ {M_0}\left( q \right), {C_0}\left( {q, \dot q} \right), {G_0}\left( q \right) $$ represent the system uncertainty, respectively.

Equation (5) can be further given as:

3.1. Trajectory planning

Figure 1 illustrates the overall process of trajectory planning. In this paper, trajectory planning optimizes a constrained-time fifth-order polynomial using the minimum-jerk principle. The form of the fifth-order polynomial is:

Figure 1. Trajectory planning structure diagram.

where $$ {p_{oi}}, \ldots , {p_{5i}} $$ are constants to be designed, $$ i = 1, 2, \ldots , n $$.

Sudden and abrupt movements can exacerbate mechanical wear and precipitate resonance in exoskeletons; hence, we aspire to ensure that the exoskeleton's trajectory is smooth. However, the specifics of certain motions are contingent upon their respective constraints. The spatial and temporal properties of the path are constrained by the position, velocity, and acceleration at the beginning (time = 0) and at the end (time = d) of the motion, as well as their maximum values. The selected constraints for point-to-point motion are as follows:

where $$ {t_o} $$ is the starting moment and $$ {t_s} $$ is the ending moment, $$ {q_o} $$, $$ {q_s} $$ are the joint angles at the initial and end positions respectively, $$ {q_m} $$, $$ {{\dot q}_m} $$, and $$ {{\ddot q}_m} $$ are the maximum values of position, velocity and acceleration, respectively.

The objective is to achieve the maximum degree of smoothness, which is accomplished by minimizing the mean-square jerk. Jerk, by its mathematical definition, is the derivative of acceleration. Consequently, the minimum-jerk model can be expressed as

where $$ {j_i} = \frac{{{d^3}{q_i}\left( t \right)}}{{d{t^3}}} $$ denotes jerk.

Combining Equation (8), solving for the constant $$ {p_0}, \ldots, {p_5} $$ yields ^[43]:

where $$ {t_z} = {t_o} - {t_s} $$ represents the duration, and $$ {q_z} = {q_o} - {q_s} $$ represents the amplitude of motion.

Modifications to the $$ {t_z} $$ or $$ {q_z} $$ are employed solely for the purpose of altering the scale of the position and time axes, respectively. The image of the trajectory with $$ {t_z} = 10 $$ and $$ {q_z} = 8 $$ is shown in Figure 2, where $$ {y_2} $$ is the first order derivative of $$ {y_1} $$ and $$ {y_3} $$ is the second order derivative of $$ {y_1} $$, and the following properties can be defined:

Figure 2. Reference trajectories and their first and second order derivative images in terms of $$ {t_z} = 10 $$ and $$ {q_z} = 8 $$.

In order to enhance the safety of the entire robotic system, this paper establishes a safety stop mode. When the end-effector is detected within the safety stop area, the robot will gradually cease its training movements and return to the designated safe position to prevent any potential harm. To achieve a smooth transition between normal training and safety stopping, the following motion-related transition functions ^[30] are employed to provide a buffer zone.

where $$ {{q_{i, j}}} $$ denotes the value of $$ {q_i} $$ at the jth sampling, $$ {a_2} = {a_1} + 2, {b_1} = \frac{{{a_2}}}{2}, {b_2} = - {b_1} + 1 $$ denote the constants that satisfy the conditions and in order to guarantee the desired trajectory is at least twice differentiable, it is necessary that $$ {a_1} > 4 $$ is an even number. $$ {c_1} $$ and $$ {c_2} $$ denote the boundaries of the buffer region, respectively.

Equation (12) provides a buffer smoothing transition between $$ {c_1} $$ and $$ {c_2} $$ by a combination of two power functions. The smoothness and shape of the transition can be controlled by adjusting the values of the parameters. As shown in Figure 3, where $$ {\rm{Param{s_i} = }}\left[ {\begin{array}{*{20}{c}} {{c_1}}&{{c_2}}&{{b_1}}&{{b_2}}&{{a_1}}&{{a_2}} \end{array}} \right] $$, $$ {\rm{Param{s_1} = }}\left[ {\begin{array}{*{20}{c}} { - 1}&1&4&{ - 3}&6&8 \end{array}} \right] $$ and $$ {\rm{Param{s_2} = }}\left[ {\begin{array}{*{20}{c}} { - 2}&2&3&{ - 2}&4&6 \end{array}} \right] $$, a larger value of the exponent will make the transition steeper, while a smaller value of the exponent will make the transition smoother. This function can be utilized to regulate the robot's behavior, for instance, enabling a gradual transition from safe stop mode to normal operation mode.

Figure 3. H-function images with different parameters.

Therefore, the desired trajectory is designed as:

where $$ {q_A} = {[{q_{A1}}, \ldots , {q_{An}}]^T} $$ is stop position in safety stop mode.

Using the physical properties of the switching function, we can divide the workspace into three regions: the tracking training region, the transition region and the safe stopping region, as shown in Figure 4. When the desired trajectory $$ {q_d} $$ is smaller than the boundary of the transition region $$ {c_1} $$, it is consistent with the reference trajectory $$ {q_r} $$; when the reference trajectory is in the transition region, the switching function $$ H $$ is responsible for realizing the smooth transition from the tracking training region to the safe stopping region; and when the reference trajectory is larger than the boundary of the transition region, the desired trajectory will stay at a safe position to ensure the safety of the subject.

Figure 4. Schematic of the desired trajectory under the motion switching function.

3.2. NDO design

The disturbance signals considered in this paper are the input torque of the human and system uncertainty body and unknown external vibrations. Assume that $$ {\hat D} $$ is the estimate of the disturbance $$ D $$. The nonlinear perturbation observer is designed as follows

where $$ w $$ is a defined auxiliary variable for the convenience of NDO design and $$ {K_1} $$, $$ {K_2} $$ are the gain matrixes to be designed, $$ {K_1} = diag\left\{ {{k_{11}}, \ldots {k_{1n}}} \right\} $$, $$ {K_2} = diag\left\{ {{k_{21}}, \ldots {k_{2n}}} \right\} $$, $$ {K_3} = diag\left\{ {{k_{31}}, \ldots {k_{3n}}} \right\} $$ and $$ 0 < \iota < 1 $$.

Theorem 1: In consideration of the exoskeleton system Equation (5), the DOC is designed in accordance with Equation (14). Subsequently, the disturbance approximation error of the proposed DOC is shown to be convergent in a finite time.

Proof: Let us define $$ \tilde D = \hat D - D $$. Then, from Equations (5) and (14) it follows that:

Therefore, it can be concluded that the convergence of the disturbance error is consistent with the convergence of the auxiliary variables $$ {s_1} $$. Choose the Lyapunov function candidate as:

The time derivative of $$ V_1 $$ is given by

where $$ {k_1} = {\lambda _{\min }}\left( {{K_1}} \right) $$ and $$ {k_2} = {\lambda _{\min }}\left( {{K_2}} \right) $$, $$ {k_3} = {\lambda _{\min }}\left( {{K_3}} \right) $$ and $$ {k_3} \ge \left\| D \right\| $$.

According to Lemma 1, the auxiliary variable $$ {s_1} $$ converges to the equilibrium point in the finite time $$ {t_1} $$ defined by

In light of the finite time convergence property of the auxiliary variable $$ {s_1} $$, it can be concluded that the disturbance approximation error of the proposed disturbance observer is convergent in a finite time.

3.3. NFTSMC design

Furthermore, a NFTSM control method based on NDO is proposed as a means of enhancing the performance of the control system. In the NFTSMC design, the singular value problem is circumvented and rapid convergence is attained through a modified terminal sliding mode surface. The overview of the NFTSMC framework is illustrated in Figure 5, which provides a detailed view of the structure and key components of the control strategy.

Figure 5. Overview of the NFTSMC framework. NFTSMC: Non-singular fast terminal sliding mode controller.

The tracking error is defined as follows:

To circumvent the issue of singular values, an improved terminal sliding surface ^[44] is designed as:

where

with $$ {s^*} = \dot e + {L_1}si{g^\gamma }\left( e \right) $$, $$ {L_1} = diag\left( {{l_{11}}, \ldots , {l_{1n}}} \right) $$, $$ {L_2} = diag\left( {{l_{21}}, \ldots , {l_{2n}}} \right) $$, $$ \gamma \in \left( {0.5, 1} \right) $$, $$ \mu > 0 $$, $$ {x_1} = \left( {2 - \gamma } \right){\mu ^{\gamma - 1}} $$, $$ {x_2} = \left( {\gamma - 1} \right){\mu ^{\gamma - 2}} $$.

The nonlinear term $$ \Psi \left( e \right) $$ plays a crucial role in the design of the sliding mode surface, providing the system with additional control authority and robustness. Specifically, the desired value of the sliding mode surface $$ s $$ is denoted by $$ {s^*} $$, where $$ {s^*} = 0 $$ represents the system being in equilibrium, while $$ {s^*} \ne 0 $$ indicates that the system is in a non-equilibrium state. When the system is in equilibrium, it means that the system has reached the desired equilibrium point but still needs to remain at that point. At this stage, the objective of SMC is to ensure stability near the equilibrium point and to resist external disturbances or parameter variations. The control signal in the form of $$ si{g^\gamma }\left( e \right) $$ can quickly correct minor deviations from the equilibrium point. When the system is in a high error state, the absolute value of the error \(e \) exceeds a predefined threshold $$ \mu $$. In this case, the system needs to rapidly reduce the error to approach the desired value as soon as possible. The control signal in the form of $$ si{g^\gamma }\left( e \right) $$ provides sufficient control authority to ensure rapid convergence when the error is large.

When the error $$ e $$ is small, the system requires more precise control to avoid excessive adjustments that may lead to chattering or instability. At this stage, the nonlinear term $$ \Psi \left( e \right) $$ takes the form of a combination of linear and nonlinear components. The linear term $$ {x_1}e $$ provides stable feedback control, which helps maintain system stability, while the nonlinear term $$ {x_2}si{g^2} $$ offers additional control authority when the error approaches zero, ensuring that the system can accurately reach the desired value. The design of the nonlinear term $$ \Psi \left( e \right) $$ takes into account the system's rapid convergence, stability, and robustness. Using the same nonlinear control form $$ si{g^\gamma }\left( e \right) $$ in both equilibrium and large error phases simplifies the control strategy design while ensuring that the system can provide effective control signals in different stages. This design effectively reduces the system's convergence time and prevents oscillations or instability near the equilibrium point.

Differentiating Equation (20), and considering Equation (6) yield:

where

The NDO-based NFSMC controller is then given by:

where $$ {L_3} = diag\left( {{l_{31}}, \ldots , {l_{3n}}} \right) $$, $$ {L_4} = diag\left( {{l_{41}}, \ldots , {l_{4n}}} \right) $$, $$ {L_5} = diag\left( {{l_{51}}, \ldots , {l_{5n}}} \right) $$, $$ \alpha \in \left( {0, 1} \right) $$.

Considering the uncertain nonlinear system represented by Equation (5), which is subject to an external disturbance, a terminal sliding mode disturbance observer is designed in Equation (14). Under the NFTSMC is constructed as Equation (22), all signals of the closed-loop system are shown to converge in a finite time.

Choose the Lyapunov function V as:

Substituting Equation (22) into the derivative of the Lyapunov function, we obtain:

where $$ {l_3} = {\lambda _{\min }}\left( {{L_3}} \right) $$ and $$ {l_4} = {\lambda _{\min }}\left( {{L_4}} \right) $$, $$ {l_5} = {\lambda _{\min }}\left( {{L_5}} \right) $$ and $$ {l_5} \ge \left\| {\tilde D} \right\| $$.

Invoking Equations (23) and (24) are written as:

Based on Equation (25), we can conclude that the system is stable and the state of the system will converge to the stable point as the $$ {V_2} $$ decreases. The value of $$ {l_3} $$ and $$ {l_4} $$ has a significant effect on the rate of convergence, while the value of $$ {\alpha} $$ affects the contribution of the nonlinear term to the rate of convergence. Furthermore, from Lemma 1, it can be concluded that the system's tracking error can be driven to s = 0 within finite time.

Next, Equation (22) is substituted into Equation (6) to derive the sliding motion equations. Subsequently, a stability analysis of the sliding motion is conducted. Given the considerations of $$ \dot e = 0 $$ and $$ \tilde D = 0 $$, the following can be derived:

In finite-time control, the objective is to ensure that the error converges to zero within a finite time interval, rather than merely driving the derivative of the error to zero. Therefore, $$ \dot e = 0 $$ itself is not an attractor. Instead, the control law is designed such that both the error $$ e $$ and its derivative $$ \dot e $$ converge to zero in finite time ^[46].

4.1. System modeling

In this paper, the simulation uses a 5 degrees of freedom (5-DOF) upper-limb exoskeleton model ^[45]. The schematic diagram of the exoskeleton robot is shown in Figure 6, which illustrates the structure and key components of the device. The robot dynamics can be referred to Equation (4). The details are as follows:

Figure 6. Schematic diagram of a 5-DOF upper-limb exoskeleton robot. 5-DOF: 5 Degrees of freedom.

4.2. Trajectory planning

Consider a fifth-order polynomial for trajectory planning based on the optimization of the minimum-jerk principle, where the initial position $$ {q_1} $$ and the farthest position $$ {q_2} $$. The motion trajectory path is a reciprocal motion first from $$ {q_1} $$ to $$ {q_2} $$ and then back to $$ {q_1} $$ as shown in Figure 7. It should be noted that the reference joint angle is depicted as a constant value at the end of the movement. This is because the figure illustrates the trajectory planning between two specific target points for simplicity. In practice, the overall trajectory is decomposed into multiple smaller segments, each planned using fifth-order polynomial interpolation. By connecting these segments sequentially, a continuous and smooth trajectory is formed, allowing the exoskeleton to perform complex and dynamic tasks rather than maintaining a single position.

Figure 7. Trajectories based on minimum-jerk optimization.

If the safe position is artificially set to $$ {q_A} = {\left[ {\begin{array}{*{20}{c}} {0.9}&4&{5.5}&{5.5}&5 \end{array}} \right]^T} $$, the region beyond this position is set as the safe stopping region. According to Equation (13), the desired trajectory $$ {q_d} $$ adjusted by the switching function is shown in Figure 8. In which, the parameters in Equation (12) are set to $$ {a_1} = 4, {a_2} = 6, {b_1} = 3, {b_2} = - 2 $$.

Figure 8. Desired trajectory with safe switching.

Figure 8 shows that when the reference trajectory is close to the safe position, the desired trajectory enters the transition region and switches to the safe position under the action of the switching function, and when the reference trajectory has entered the safe stopping region, the desired trajectory stays in the safe stopping position until the reference trajectory leaves the safe stopping region, and then once again smoothly switches to the tracking training mode.

Then, the sliding variable can be easily calculated based on the tracking error between the desired trajectory and the actual trajectory. Thus, the robot can achieve perfect tracking by reducing the sliding film error.

4.3. NFSMC controller simulation

In order to underscore the enhanced efficacy of the proposed NFTSMC scheme in realizing faster and higher precision tracking, this study engages in a comparative analysis with two additional control methodologies. The control strategies selected for this comparative investigation are TSMC^[46] and non-singular TSMC (NTSMC)^[47]. The comparative experiments within this paper will assess the performance of the NFTSMC scheme against the TSMC and NTSMC approaches across a spectrum of criteria, encompassing convergence velocity, tracking accuracy, robustness to parameter variations, and the stability of output torque. It is anticipated that the outcomes will accentuate the superior performance of the NFTSMC scheme in delivering expedited and more precise tracking capabilities in contrast to the conventional TSMC and NTSMC methods. It is worth noting that the human force exerted by the patient on the exoskeleton under the passive training effect is regarded as the interference force. In consideration of the periodic interference effect that occurs in the actual training, the interference force is set as in this paper: $$ {\tau _{d1}} = 1 + 0.5\sin \left( {t + \frac{\pi }{6}} \right) $$, $$ {\tau _{d2}} = 2 + 0.5\sin \left( {t - \frac{\pi }{6}} \right) $$, $$ {\tau _{d3}} = 2 + 0.5\cos \left( {t - \frac{\pi }{6}} \right) $$, $$ {\tau _{d4}} = 1 + 0.8\sin \left( {t - \frac{\pi }{6}} \right) $$, $$ {\tau _{d5}} = 1 + 0.5\sin \left( {t - \frac{\pi }{6}} \right) $$. The disturbance observer parameters and controller parameters are given as $$ {K_1} = diag\left[ {\begin{array}{*{20}{c}} 5&{10}&3&4&6 \end{array}} \right] $$, $$ {K_2} = diag\left[ {\begin{array}{*{20}{c}} {10}&{20}&{20}&{20}&{20} \end{array}} \right] $$, $$ {L_1} = diag\left[ {\begin{array}{*{20}{c}} 5&6&3&5&6 \end{array}} \right] $$, $$ {L_2} = diag\left[ {\begin{array}{*{20}{c}} {50}&{50}&{50}&{50}&{50} \end{array}} \right] $$, $$ {L_3} = diag\left[ {\begin{array}{*{20}{c}} 3&4&5&4&6 \end{array}} \right] $$, $$ {L_4} = diag\left[ {\begin{array}{*{20}{c}} {40}&{40}&{40}&{40}&{40} \end{array}} \right] $$, $$ \iota = 0.5 $$, $$ \gamma = 0.55 $$, $$ \alpha = 0.6 $$.

Based on the control parameters previously determined, the numerical simulation results of the controller have been obtained and are presented in Figures 9-11, respectively. This figure comprehensively demonstrates the performance of the controller by comparing the desired trajectory with the actual output trajectory. We consider tracking the same segment (from $$ q1 $$ to $$ q2 $$) of the desired trajectory as a representative example.

Figure 9. Output tracking in the case of NFTSMC. NFTSMC: Non-singular fast terminal sliding mode controller.

Figure 10. Output tracking in the case of TSMC. TSMC: Terminal sliding mode control.

Figure 11. Output tracking in the case of NTSMC. NTSMC: Non-singular terminal sliding mode control.

In Figure 9, the black curve represents the desired trajectory, which is the path that the system is expected to follow. On the other hand, the blue dashed line represents the actual output trajectory, which is the path that the system follows under the control of the proposed NFTSMC method. The simulation results shown in Figure 9 indicate that the NFTSMC method effectively tracks the desired trajectory. The close alignment between the black curve and the blue dashed line demonstrates that the system's output closely matches the desired path. This alignment confirms the effectiveness of the NFTSMC method in guiding the system toward the desired trajectory with high precision.

Figure 10 presents the tracking results for the TSMC method. This figure allows for an assessment of the tracking capabilities of TSMC, including its response to disturbances and its ability to maintain stability. The plot may reveal some deviations from the desired trajectory, indicating areas where TSMC may not be as effective as NFTSMC, particularly in terms of tracking accuracy and robustness against perturbations.

Figure 11 displays the tracking outcomes for the NTSMC method. This figure is crucial for evaluating NTSMC's performance in tracking the desired trajectory, especially in terms of its terminal phase behavior and its ability to achieve fast convergence. The plot may exhibit characteristics such as chattering, which is a common issue in traditional SMCs, and may affect the smoothness of the tracking trajectory. To quantify and compare the performance of these controllers, we calculated the root mean square (RMS) and mean absolute error (MAE) and summarized the results in Table 1. NFSMC exhibited the best performance in terms of both RMS and MAE, indicating higher precision and stability in joint angle control.

The tracking performance comparison between the NFTSMC and the TSMC, as shown in Figure 12, is presented with the error extended over a longer time span to facilitate a clearer observation of the differences. As depicted in the figure, the tracking error plot for NFTSMC [Figure 12A] exhibits a clear trend towards convergence, ultimately stabilizing at a negligible value. This indicates that the system's output, under the control of NFTSMC, closely follows the desired reference trajectory with minimal deviation. In contrast, the TSMC tracking error plot [Figure 12B], while consistently maintaining a smaller value than NFTSMC, is characterized by a continuous fluctuating process. These fluctuations suggest that TSMC, despite having smaller error values, is unable to achieve the same level of stability and precision as NFTSMC. The sustained oscillations in the TSMC error curve prevent it from converging to a smaller value. Furthermore, the capacity of NFTSMC to maintain a low tracking error even when faced with parameter changes or external perturbations demonstrates its enhanced robustness.

Figure 12. Tracking error in the case of NFTSMC and TSMC. NFTSMC: non-singular fast terminal sliding mode controller; TSMC: terminal sliding mode control.

In summary, the comparative analysis of the tracking error plots reveals that NFTSMC not only achieves higher tracking accuracy by bringing the system output closer to the desired trajectory but also exhibits better stability and robustness compared to TSMC. These attributes make NFTSMC a more reliable choice for applications where precise and stable tracking is paramount.

As can be seen from Figure 13, in the experimental analysis comparing the NFTSMC and NTSMC control strategies, the system error is intentionally extended over a longer time span to facilitate clearer observation. It is observed that both methods are capable of steering the system error towards a vicinity of zero within a relatively brief timeframe. However, an examination of the error curves reveals that the NFTSMC exhibits a more rapid convergence, signifying its superior response time. Additionally, NTSMC encounters specific data constraints during experimentation, which may impair its performance in real-world applications. The NTSMC is prone to singular value issues when managing data, which can result in the instability or diminished efficacy of the control algorithm. NFTSMC circumvents these challenges by employing a non-singular sliding surface design, which bolsters the system's robustness and stability. This design accounts for the dynamic properties and potential perturbations of the system, ensuring the control strategy's efficacy amidst uncertainties and external disturbances. Furthermore, NFTSMC incorporates a rapidly converging sliding surface and an innovative control law, which not only enhances the system's response velocity but also mitigates the occurrence of chattering phenomena.

Figure 13. Tracking error in the case of NFTSMC and NTSMC. NFTSMC: Non-singular fast terminal sliding mode controller; NTSMC: non-singular terminal sliding mode control.

Figures 14 and 15 show the performance of NFTSMC and NTSMC in terms of torque output, respectively. The torque diagram of NFTSMC shows smoother torque variation than that of NTSMC, which indicates that NFTSMC generates less chattering during the control process, thus avoiding wear and tear of the actuator and reduction of energy efficiency caused by chattering. The torque diagram of NFTSMC is more stable than that of NTSMC. NFTSMC can still maintain stable torque output when facing parameter changes or external disturbances, which indicates that NFTSMC has better robustness. NFTSMC can maintain stable torque output in the face of parameter changes or external disturbances, which indicates that NFTSMC has better robustness.

Figure 14. Control input in the case of NFTSMC. NFTSMC: Non-singular fast terminal sliding mode controller.

Figure 15. Control input in the case of NTSMC. NTSMC: Non-singular terminal sliding mode control.

Authors' contributions

Made substantial contributions to the conception and design of the study and performed data analysis and interpretation: Zhang, Y.

Performed data acquisition and provided administrative, technical, and material support: Xie, W.; Ma, R.

Availability of data and materials

The data involved in this study are private and confidential due to privacy and confidentiality concerns. Therefore, we declare that the data will not be made publicly available. However, readers who require further information may contact the corresponding author to obtain the relevant data.

Financial support and sponsorship

None.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

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Copyright

© The Author(s) 2025.