Coordinated energy-efficient walking assistance for paraplegic patients by using the exoskeleton-walker system

2.1. The optimization of the supporting force

In this subsection, the real-time optimization of the supporting force is presented, which aims to find the optimal supporting force provided by the support joint of the robotic walker. The structure of the HEW system and the energy cost calculation during walking are outlined in Section 2.1.1, and the implementation of the human-in-the-loop optimization for the supporting force is detailed in Section 2.1.2.

2.1.1. Energy calculation of the HEW system with the supporting force

In this subsection, the energy cost of the human-exoskeleton system is presented, where the energy is determined by the power of the active joints of the exoskeleton. The structure of the HEW system is shown in Figure 3, where the exoskeleton robot includes hip, knee and ankle joints to drive the human-exoskeleton system walking forward. The thighs, calves, and feet of the exoskeleton are interconnected with the corresponding segments of the pilot's lower limbs, ensuring full synchronization between the pilot's lower limb movements and those of the exoskeleton. The supporting force $$ F_S $$ can be supplied by the support joint of the robotic walker, which is used to move the support joint vertically and support the weight of the human-exoskeleton system. The support joint of the robotic walker is actuated with a spring-based mechanism and a stepping motor, which provides a variable supporting force for the human-exoskeleton system with different motor positions. In addition, $$ f $$ represents the horizontal resistance of the robotic walker; $$ F_x $$ and $$ F_z $$ represent the horizontal and vertical components of the supporting force offered by the support leg, respectively. $$ l_T $$ and $$ l_S $$ represent the lengths of the thigh and shank, respectively. $$ \theta_H $$ and $$ \theta_K $$ represent the hip and knee joint angles of the exoskeleton's support leg, respectively. Note that in this paper, the hip and knee joint angles are sampled from the healthy subjects as the control reference, and the hip and knee joints of the exoskeleton robot are driven by the DC motors with PID position controllers. The ankle of the exoskeleton robot is a passive joint with a spring mechanism, which can be used to avoid the foot drop of the patients.

Figure 3. The schematic diagram of HEW system. HEW: Human-exoskeleton-walker.

In this paper, we focus on the movement of the HEW system in the Sagittal plane, and the Center of Mass (COM) of the human-exoskeleton system has a good correspondence in the sagittal plane with the hip joints ^[19]; therefore, the COM of the human-exoskeleton system is set to the center of two hip joints. The horizontal movement of the COM is solely driven by the exoskeleton support leg, while the vertical movement is influenced by both the support joint and the exoskeleton support leg. Assuming that the ankle joint of the support leg is set as the origin of the Cartesian frame, the COM's position of the human-exoskeleton system can be described when the predefined gait trajectories $$ \theta_H (t) $$ and $$ \theta_K (t) $$ are employed to drive the exoskeleton:

(1) $$ \begin{bmatrix} x_\mathrm{com}(\theta_H, \theta_K)\\ z_\mathrm{com}(\theta_H, \theta_K) \end{bmatrix} = \begin{bmatrix} -\sin(\theta_H)&\sin(\theta_K-\theta_H)\\ \cos(\theta_H)&\cos(\theta_K-\theta_H) \end{bmatrix} \begin{bmatrix} l_T\\ l_S \end{bmatrix}, $$

where the $$ x_\mathrm{com} $$ and $$ z_\mathrm{com} $$ represent the horizontal and vertical positions of the COM, respectively. Then, the horizontal and vertical supporting forces from the support leg $$ F_x $$ and $$ F_z $$ are expressed as follows:

(2) $$ \left\{ \begin{array}{lr} F_x=\ddot x_\mathrm{com}(m_P+m_W+m_E) + f, \\ F_z=(\ddot z_\mathrm{com}+g)(m_P+m_E)-F_s, \end{array} \right. $$

where $$ m_P $$, $$ m_W $$ and $$ m_E $$ represent the masses of the human subject, the robotic walker, and the exoskeleton robot, respectively. $$ \ddot x_\mathrm{com} $$ and $$ \ddot z_\mathrm{com} $$ are the second derivatives of the COM's position $$ x_\mathrm{com} $$ and $$ z_\mathrm{com} $$. $$ g $$ is the constant gravitational acceleration. The torques of the hip and knee joints of the exoskeleton support leg can be calculated as follows:

(3) $$\begin{bmatrix} \tau_H\\ \tau_K \end{bmatrix} = \begin{bmatrix} l_T&l_S\\ 0&l_S \end{bmatrix} \begin{bmatrix} \cos(\theta_H)&\sin(\theta_H)\\ \cos(\theta_H-\theta_K)&\sin(\theta_H-\theta_K) \end{bmatrix} \begin{bmatrix} F_x\\ F_z \end{bmatrix}, $$

where $$ \tau_H $$ and $$ \tau_K $$ are the torques of the hip and knee joints. As a result, the power of the exoskeleton robot is determined by the joint torques $$ \tau_H $$ and $$ \tau_K $$ and the angular velocity of the joint $$ \dot \theta_H $$ and $$ \dot \theta_K $$. Let us take the hip joint as an example; the power of the hip joint's motor $$ P_m $$ is calculated as:

(4) $$ \left\{ \begin{array}{lr} P_m=I^2 \cdot R_m + \tau_m \cdot \dot \theta_m, \\ \tau_m = \tau_H/r = K_m \cdot I, \\ \dot \theta_m=\dot \theta_H/r, \end{array} \right. $$

where $$ \tau_m $$ and $$ \dot \theta_m $$ represent the torque and angular velocity of the motor in the hip joint, respectively. $$ I $$ represents the current of the motor, and $$ r $$ is the reduction ratio. Besides, $$ R_m $$ and $$ K_m $$ represent the resistance and torque constant of the motor, respectively. The first item of $$ P_m $$ in Equation (4) represents the thermal power, while the second item indicates the mechanical power. With the power of the motors in the joints of the support leg, the energy consumption of the exoskeleton support leg during the stance phase can be calculated as follows:

(5) $$ E=\int_{0}^{T}(P^H_m+P^K_m) dt, $$

where $$ P^H_m $$ and $$ P^K_m $$ denote motor power of the hip and knee joints, respectively. $$ T $$ signifies the duration of the stance phase in one gait cycle.

Based on the Equations (2)-(4), we can find that the torques of the hip and knee joints are decreased as the supporting force $$ F_s $$ increases, i.e., the energy cost of the exoskeleton robot is decreased with increasing $$ F_s $$. However, if the supporting force is too large, the human-exoskeleton system will be lifted off the ground, and the friction between the ground and the exoskeleton's foot will be reduced, resulting in an abnormal walking posture of the human-exoskeleton system and even with slipping over the ground. Therefore, finding the appropriate supporting force to minimize the energy consumption and prevent slipping is critical. Now, let us construct an objective function to denote the energy efficiency:

(6) $$ J(\cdot)=E/S, $$

where $$ E $$ denotes the energy consumption of both hip and knee joints of the support leg during the stance phase in one gait cycle; $$ S $$ represents the stepping length for one step. Consider the value of the objective function as Total Cost of Transport (TCoT). Now, let us find a way to solve the objective function and find the optimal supporting force.

2.1.2. Real-time optimization of the supporting force

In this subsection, the real-time optimization of the supporting force, which employs the discrete-time ESC approach, is presented. ESC is a model-free adaptive control method that finds an optimum set-point in order to minimize/maximize an objective function, whose analytical expression might be unknown^[20–23]. Kumar et al. proposed a modified structure of the discrete-time ESC by introducing a stepper motor with an integrator^[24,25]. In this modified structure, the ESC integration is performed by the motor dynamics itself. Moreover, the stepper motor has the same characteristics as the zero-order holder; it does not have any closed-loop encoder feedback for position control. Instead, it accepts a variation in the motor location as an input command rather than the final motor location. In this paper, we used the variation in the motor location to tune the supporting force of the support joint.

The block diagram of the modified ESC used in this paper is shown in Figure 4, where the workflow of the ESC is as follows. Firstly, a periodic disturbance signal of small amplitude $$ d_1(k)=-a\omega\sin(\omega t_k) $$ called the dither signal is added to the commanded change in the motor location $$ \Delta \theta(k) $$ in the modulation step.

Figure 4. The block diagram of ESC. ESC: Extremum Seeking Control.

Assuming that the stepper motor dynamics is modeled as a cascade connection of a zero-order holder and a continuous-time integrator. The zero-order holder holds the sample $$ \Delta \theta (k)+d_1(k) $$ constant for one sampling interval $$ \Delta T $$. Denoting that $$ t_k $$ is the sampling time, the expression for $$ \dot{\theta} (t_k) \approx \Delta \theta(k)/\Delta T-a\omega\sin(\omega t_k) $$; then, the integrator dynamics of the stepper motor outputs $$ \theta(t_k)+a\cos(\omega t_k) $$, where $$ \theta (t_k) $$ is the stepper motor's location at $$ t_k $$. The output of the stepper motor is used to tune the supporting force by changing the position of the stepper motor in the support joint. Then, the torques of the hip and knee joints are sampled in one gait cycle, and the power of the hip and knee joints is calculated according to the Equation (4). Next, the power is multiplied by the sampling interval $$ \Delta T $$, and we summed them up to get the total energy consumption of the hip and knee joints as $$ E $$. Therefore, the objective function can be rewritten as:

(7) $$ J(\cdot)=E/S=J((\theta(k)+a\cos(\omega k))), $$

and the Taylor series approximation of $$ J(\cdot) $$ is expressed as follows:

(8) $$ \begin{align} J(\cdot) & \approx J(\theta(k)) + J'(\theta(k))a\cos(\omega k)+ \frac{J''(\theta(k))}{2}a^2 \cos^2 (\omega k) \\ &=J(\theta(k))+J'(\theta(k))a\cos(wk)+\frac{a^2 J''(\theta(k))}{4}(1+\cos2(\omega k)), \end{align} $$

where $$ J' $$ and $$ J'' $$ are the first and the second derivatives of $$ J(\cdot) $$ with respect to $$ \theta $$. Then, the objective function measurements in Equation (8) are passed through a high-pass filter HPF to remove the DC components $$ J(\theta(k)) $$ and $$ a^2 J''(\theta(k))/4 $$ to give

(9) $$\xi(k)=J'(\theta(k))a\cos(\omega k) + \frac{a^2 J''(\theta(k))}{4}\cos(2\omega k). $$

In the demodulation step, $$ \xi(k) $$ is multiplied by another dither signal $$ b\cos(\omega k) $$ and scaled by a gain $$ -\lambda $$ to generate

(10) $$ \begin{align} \Delta\theta(k)&=-\lambda[J'(\cdot)a\cos(\omega k)+\frac{a^2J''(\cdot)}{4}\cos(2\omega k)]b\cos(\omega k)] \\ &=-\lambda[\frac{abJ'(\cdot)}{2}[1+\cos(2\omega k)]+\frac{a^2bJ''(\cdot)}{4}\cos(2\omega k)\cos(\omega k)], \end{align} $$

where $$ \Delta\theta(k) $$ indicates the amount that the stepper motor should be moved to minimize the TCoT, which realizes the supporting force adaptive tuning. From Equation (10), it can be seen that $$ \Delta\theta(k) $$ consists of a DC component, which is proportional to $$ J'(\cdot) $$, and contains other higher frequency terms.

Following standard manipulations^[26], the equations of ESC can be updated as follows:

(11) $$ \xi(k)=-h\xi(k-1)+J(\theta(k))-J(\theta(k-1)), $$

(12) $$\Delta\theta(k)=-\lambda[\xi(k)b\cos(\omega k)], $$

where $$ \lambda $$ is the adaptation gain, and $$ h\in(0, 1) $$ is the HPF cut-off frequency.

Overall, with the proposed real-time optimization approach, the optimal supporting force can be found in real time by walking several steps overground with the HEW system. As a result, the energy cost of the HEW system will be reduced to the minimum value without slipping overground.

2.2. The coordinated motion planning of the robotic walker

The coordinated motion planning is to make the human-exoskeleton system and the robotic walker move coordinately and avoid the "pull" or "push" between them. As shown in Figure 2, the coordinated motion planning is based on the hip and knee joint angles of the exoskeleton robot. As shown in Figures 5 and 6, there are four wheels of the robotic walker, and all wheels rotate around the $$ Y $$ axis.

Figure 5. The simulation model of the HEW system. HEW: Human-exoskeleton-walker.

Figure 6. The wheels of the robotic walker.

As we mentioned in Figure 1, the exoskeleton robot is connected to the robotic walker with a solid cantilever; therefore, the horizontal movement of the robotic walker is the same as the COM of the human-exoskeleton system. Note that if the wheels are passive without any power, there is no active movement of the robotic walker, and the human-exoskeleton system has to pull or push the robotic walker while walking overground. If the wheels are actuated with the DC motors, they can drive the robotic walker to follow the movement of the human-exoskeleton system and avoid the movement conflict between the human-exoskeleton system and the robotic walker. Therefore, the question is how to control these wheels to drive the robotic walker to coordinately follow the movement of the human-exoskeleton system.

Based on Equation (1), the movement of the COM during walking can be calculated with the joint angles of the exoskeleton robot. Additionally, the horizontal movement of COM can be discretized with a constant unit time $$ \Delta t $$, taking $$ i $$ as an index for discretization, then

where $$ \Delta t $$ can be set to some small positive value such as $$ \Delta t =0.005{s} $$. The COM horizontal displacement from $$ t_{i-1} $$ to $$ t_i $$ can be calculated as follows:

where $$ x_i $$ represents the horizontal position of the COM at $$ t_i $$. Based on the COM horizontal displacement $$ \Delta x_i $$, the increment of the joint angles for the wheels from $$ t_{i-1} $$ to $$ t_i $$ is calculated as follows:

where $$ R_w $$ is the radius of the wheels on the robotic walker. Now, with the horizontal movement of the human-exoskeleton system, the reference joint angles of the wheels can be obtained, and the human-exoskeleton system and the robotic walker can be moved coordinately.

3.1. Experimental setup

To evaluate the performance of the proposed CEEC approach, several experiments were designed in the simulation experiments. Firstly, the hip and knee joint angles used in the experiment are sampled from the healthy subject, as shown in Figure 7, where the joint angles in swing and stance phases are for the swing leg and the support leg, respectively. The mass distribution of the human-exoskeleton system (torso, thigh, shank, and foot) follows average human anthropometry ^[27], as shown in Table 1, and the length of the thigh and shank is set to 0.45 m, which is similar to the subject with the body height of 1.75 m.

Figure 7. The joint angles of the exoskeleton's hip and knee joints.

Overall, the total mass of the human subject, the exoskeleton, the robotic walker, and other parameters for the simulation experiments are shown in Table 2. Note that the parameters for the motors in hip and knee joints refer to the manual of the DC motors used in our exoskeleton robots shown in Figure 1.

To evaluate the proposed CEEC approach and compare it with others, four experiments were designed, each assigned a distinct name.

● The first one is the "baseline"; there was no active assistance of the robotic walker, i.e., the human-exoskeleton system had to pull the robotic walker forward during walking.

● The second one is the Coordinated Motion Planning ("CMP"); there was only the active assistance from the wheels with the generated coordinated motion planning, i.e., the wheels were controlled with the reference joint angles generated in Section 2.2. In addition, there was no supporting force from the support joint.

● The third one is the "ESC"; there was only the supporting force from the support joint under the ESC strategy, with no active assistance from the wheels.

● The last one is the "CEEC"; there was active assistance from both the support joint and wheels; the supporting force was optimized with the ESC strategy; the wheels were controlled with the reference joint angles in Section 2.2.

To evaluate the adaption of the proposed CEEC approach for different subjects, three subject simulation models with various masses are employed in the experiment [Table 3]. Note that the ESC strategy is an online iterative algorithm, and the initial value of the supporting force should be set at the beginning of the experiment. Therefore, two distinct initial supporting forces were given [Table 3].

For each trial of the experiment, seventeen steps (the first step and eight gait cycles) were conducted to test the efficiency of the proposed approach. Note that the first is a special step from the standing upright posture to walking; therefore, the control strategy only works in the last sixteen steps. The sampling rate is 20 Hz, and the gait cycle is two seconds with two steps. The TCoT was computed with the sampled torque of the support leg's hip and knee joints after each gait cycle. The parameters of ESC were selected as follows: a = 1.6, b = 0.8, $$ \omega $$ = 0.8 Hz, h = 0.4 Hz, $$ \lambda $$ = -6.

3.2. Simulation experiments recording

The snapshots of the four experiments (baseline, CMP, ESC, CEEC) are shown in Figure 8, where each row corresponds to one of the four different experiments, i.e., baseline, CMP, ESC and CEEC, respectively. It is significant that with diverse control strategies, the COM's tracking performances vary.

Figure 8. The comparison of walking experiments with different control strategies. CMP: Coordinated Motion Planning; ESC: Extremum Seeking Control; CEEC: Coordinated Energy-Efficient Control.

The snapshots of the experiments with three different subjects and the CEEC control approach are shown in Figure 9. Since the CEEC is adaptive to various scenarios, the walking performance is similar for subjects with varying masses.

Figure 9. The comparison of walking experiments for different subjects.

For a more detailed presentation of the whole walking experiment, please refer to the Supplementary Video.

4.1. Experimental results for the comparison with the COM's trajectory tracking and the energy cost

First of all, the comparison of COM's trajectories tracking performance with baseline, CMP, ESC and CEEC during the whole walking experiments are shown in Figure 10. Note that the HEW system is moving in the Sagittal plane; therefore, only the $$ x $$ and $$ z $$ positions of the COM are presented. The black solid curve is the reference trajectory of the COM calculated from the reference joint angles based on the Equation (1), the dashed purple curve is the COM's trajectory with the baseline (with no supporting force and no assistance from wheels), the solid red curve is the COM's trajectory with the CMP (with only active assistance from wheels and no supporting force), the dashed green curve is the COM's trajectory with the ESC (with only supporting force and no assistance from wheels), and the solid blue curve is the COM's trajectory with the CEEC (with both supporting force and assistance from wheels).

Figure 10. The comparison of COM trajectories tracking performance with different control strategies. (1) Reference: the reference COM trajectories; (2) Baseline: the human-exoskeleton walks without any assistance of the robotic walker; (3) CMP: the HEW system walks with only the active wheel movements; (4) ESC: the HEW system walks with only the supporting force under the ESC strategy; (5) CEEC: the HEW system walks with the proposed coordinated energy-efficient control approach. CMP: Coordinated Motion Planning; ESC: Extremum Seeking Control; CEEC: Coordinated Energy-Efficient Control; COM: Center of Mass; HEW: human-exoskeleton-walker.

To compare the trajectory tracking of four cases, the mean squared error (MSE) of four cases relative to the reference COM trajectories was calculated as:

The calculated MSE of four cases was shown in Table 4.

From the trajectory tracking comparison, we can find that in the baseline case, due to no active assistance from the robotic walker, the human-exoskeleton system has to pull the robotic walker forward during walking, the COM's trajectory tracking is bad, and the final position of the COM is far away from the desired position. In the CMP case, since there is active assistance of the wheels with the coordinated motion generated with the reference COM trajectory, the tracking performance of the COM is good. For the ESC and CEEC cases, since a supporting force exists, the COM's trajectory tracking is better than the baseline, and especially for the CEEC case, the COM's trajectory tracking is better than the ESC case. Note that in the first several steps, the supporting force is not optimal, and the ESC algorithm is tuning to find the optimal supporting force, which results in a bad performance. However, after several steps of optimization, the optimal supporting force is found, and the COM's trajectory tracking is better. This is the reason why the MSE of CEEC is a little bigger than the MSE of CMP.

Above all, for the COM's trajectory tracking, CEEC is better than the ESC and baseline but worse than the CMP. Now, let us see the energy cost during walking with these different strategies [Figure 11]. From the bars presented in Figure 11, we can see that in the baseline case, the energy cost is much higher than in any other method. In the CMP case, the energy cost is always similar during the walking. In ESC and CEEC cases, the energy cost is very high at the beginning of the walking and decreases after several steps; this is because, at the first several steps, the ESC algorithm needs to iteratively update the supporting force and find the optimal one, which leads to a bad walking performance and high energy cost. After several steps, the energy cost is reduced, and the CEEC is better than the ESC; this is because the CEEC not only optimizes the supporting force but also provides horizontal coordinated walking assistance by wheels.

Figure 11. The comparison of energy cost with different control strategies. CMP: Coordinated Motion Planning; ESC: Extremum Seeking Control; CEEC: Coordinated Energy-Efficient Control.

Overall, considering the COM's trajectory tracking performance and the energy cost, the CEEC is the best approach for the HEW system to finish coordinated energy-efficient overground walking.

4.2. Experimental results for the comparison of subjects with different masses

To validate the adaptive capacity of the CEEC algorithm in various scenarios, subjects with varying masses are employed and different initial support forces for CEEC are given [Table 3]. As mentioned before, there are eight gait cycles of each experiment; the hip and knee joint torques of the support leg were recorded with the sampling frequency of 20 Hz for the CEEC algorithm.

The variation of the supporting force for different subjects and initial supporting force settings are shown in Figure 12, and all support forces will be iteratively updated and converged to the optimal one after several steps of walking. Note that for distinct subjects, the final optimal support forces vary and mainly depend on the masses of the subject; this is because a bigger supporting force is needed for a heavier subject. So for the heaviest subject (A3 and B3) with the mass of 80 kg, the required supporting force is much bigger than the lightest subject (A1 and B1) with the mass of 40 kg.

Figure 12. The variation of the supporting force during walking.

The variation of the TCoT for different subjects and initial supporting force settings are shown in Figure 13. The TCoT can be calculated with the Equations (4)-(6), where the joint torques for during the walking can be found in Figure 14. Then, the support leg's energy consumption in one gait cycle can be obtained by integrating the power of hip and knee joints over the gait cycle; the stepping length is also obtained. Similar to the variation of the supporting force, the TCoT for diverse subjects could converge to an almost constant value after three gait cycles. Note that for the same subject, the TCoT converged to a closed value after several steps; this is because the CEEC is an iterative updated algorithm and will tune the supporting force online; even with different initial supporting force settings, the final optimal supporting force only depends on the subject's masses in these experiments.

Figure 13. The value of the TCoT during walking. TCoT: Total cost of transport.

Figure 14. The variation of the hip and knee joint torques during walking for different subjects.

The final converged supporting force and the improvement of the energy efficiency are shown in Table 5, where the TCoT in the baseline case was chosen to be compared with the CEEC case. To confirm that these values are optimal, the three control groups in Table 5 were selected to be compared with the optimal supporting force. In the experiments, the different supporting forces were set for the support joint, whose values are selected in the interval with the converged supporting force as the midpoint. The selected values of constant supporting force and the comparison of the TCoT with the converged one are shown in Table 6. From Table 6, when the converged supporting force was increased or decreased, the TCoT always increased. Looking at Equations (2)-(4), it is intuitive that the elevation of the supporting force leads to the reduction of the energy cost. However, if the supporting force is bigger than the optimal one, it may cause severe slippage in the human-exoskeleton system. Despite the reduced energy cost, the TCoT also rises. When the supporting force is smaller than the optimal one, the energy cost increases without slippage, similarly causing the TCoT to increase. As a result, the converged supporting forces of the three control groups are optimal.

The optimal supporting force in the table is the average final converged value in Figure 12, and the optimal TCoT is the average final converged value in Figure 13. Compared with the baseline, the CEEC reduced the TCoT by 58%, 60.9% and 61.6%, respectively. In other words, the average improvement of the energy efficiency can be calculated with these three values, i.e., 60.16%.

The variations of the hip and knee joint torques in the experiments are shown in Figure 14. With the iterative update of CEEC, the peak joint torques of both hip and knee joints in all scenarios are reduced significantly after several steps of walking with increasing supporting force, which means the CEEC is an online real-time optimization approach for the HEW system. Note that for different subjects and initial settings of the supporting force, the joint torques can be online optimized, and for subjects with varying masses, the required joint torques are distinct. For example, in Figure 14, the red solid curve with a lighter subject mass is smaller than the black dashed curve with a heavier subject mass.

Overall, the experimental results indicate the CEEC approach proposed in this paper can realize the coordinated energy-efficient walking assistance for the HEW system, resulting in a significant improvement. However, If you are only concerned with the COM tracking effect, CMP is the better approach. Methods can be chosen according to the actual situation.

Authors' contributions

Conception and design of the study and performed data analysis: Yang C, Zou C, Peng Z

Manuscript writing: Yang C, Zou C

Materials support and experiments set up: Zhang X, Cheng H

Performed data acquisition and analysis of the study: Zhang X, Zhang L

Manuscript review and correction: Huang R, Cheng H

Availability of data and materials

Not applicable.

Financial support and sponsorship

This work was supported by the National Key Research and Development Program of China under Grant 2018AAA0102504, the National Natural Science Foundation of China (No. 62203089, No. 62003073, No. 62103084, No. 62303092), the Project funded by China Postdoctoral Science Foundation under Grant 2021M700695, the Sichuan Science and Technology Program (2022NSFSC0890, 2021YFS0383, 2023YFG0024, 2022YFS0570, 2022NSFSC0865), and the Fundamental Research Funds for the Central Universities (ZYGX2022YGRH003, ZYGX2021YGLH003).

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

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Copyright

© The Author(s) 2024.