*Article* **sEMG-Based Gain-Tuned Compliance Control for the Lower Limb Rehabilitation Robot during Passive Training**

**Junjie Tian 1, Hongbo Wang 1,2, Siyuan Zheng 1, Yuansheng Ning 1, Xingchao Zhang 1, Jianye Niu 1,\* and Luige Vladareanu 3,\***


**Abstract:** The lower limb rehabilitation robot is a typical man-machine coupling system. Aiming at the problems of insufficient physiological information and unsatisfactory safety performance in the compliance control strategy for the lower limb rehabilitation robot during passive training, this study developed a surface electromyography-based gain-tuned compliance control (EGCC) strategy for the lower limb rehabilitation robot. First, the mapping function relationship between the normalized surface electromyography (sEMG) signal and the gain parameter was established and an overall EGCC strategy proposed. Next, the EGCC strategy without sEMG information was simulated and analyzed. The effects of the impedance control parameters on the position correction amount were studied, and the change rules of the robot end trajectory, man-machine contact force, and position correction amount analyzed in different training modes. Then, the sEMG signal acquisition and feature analysis of target muscle groups under different training modes were carried out. Finally, based on the lower limb rehabilitation robot control system, the influence of normalized sEMG threshold on the robot end trajectory and gain parameters under different training modes was experimentally studied. The simulation and experimental results show that the adoption of the EGCC strategy can significantly enhance the compliance of the robot end-effector by detecting the sEMG signal and improve the safety of the robot in different training modes, indicating the EGCC strategy has good application prospects in the rehabilitation robot field.

**Keywords:** sEMG; lower limb rehabilitation robot; compliance control; training mode; MOTOmed; continuous passive motion; straight leg raise; feature analysis

#### **1. Introduction**

Lower limb motor dysfunction is a common sequela of stroke patients. The elderly is a high-risk group for stroke, and as the population ages, the incidence of stroke increases dramatically [1,2]. The plasticity of the human brain and central nervous system is the basis of rehabilitation medicine. Through the training exercise of specific tasks and the use of the motor relearning program of the nervous system, the motor function of the patient's lower limbs can be effectively restored [3–5]. Rehabilitation robotics, as an emerging technology developed in the rehabilitation field, has advantages in clinical and biomechanical measurements compared with conventional therapy [6]. In addition, the rehabilitation robot is relatively easy to manage and control, which can help patients perform predetermined training actions accurately and repeatedly and improve the effectiveness of rehabilitation treatment [7]. In recent years, the design and control strategies of rehabilitation robots have become research hotspots in the fields of rehabilitation engineering and robotics.

With the development of robotics and rehabilitation theory, various lower limb rehabilitation robots have been designed. Lower limb rehabilitation robots are mainly divided

**Citation:** Tian, J.; Wang, H.; Zheng, S.; Ning, Y.; Zhang, X.; Niu, J.; Vladareanu, L. sEMG-Based Gain-Tuned Compliance Control for the Lower Limb Rehabilitation Robot during Passive Training. *Sensors* **2022**, *22*, 7890. https://doi.org/ 10.3390/s22207890

Academic Editor: M. Osman Tokhi

Received: 14 September 2022 Accepted: 14 October 2022 Published: 17 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

into exoskeleton type and end-effector type [8]. In the exoskeleton robot system, there is a one-to-one correspondence between the robot and human joints. The exoskeleton robot system can be worn on the human body and usually has a compact structure [9]. The lower limb exoskeleton robot MotionMaker adopts the integrated design of the seat and lower limb motion mechanism, which can carry out passive, semi-active, and active training modes [10]. Li et al. designed a lower limb exoskeleton rehabilitation robot which can assist the patient in carrying out gait training [11]. Feng et al. designed a lower limb rehabilitation robot for passive training of stroke patients, and the moving seat can be adjusted or separated from the robot to meet the rehabilitation demands of patients at different stages [12]. Akdo ˘gan et al. produced a therapeutic exercise robot Physiotherabot, which can perform active and passive movements and learn specific exercise movements [13]. In the end-effector robot system, pedals or platforms are used to generate limb motion from the distal end of the lower limb without requiring alignment between the robot and human joints. Wang et al. designed a rigid-flexible end-effector lower limb rehabilitation robot, which consists of a rigid mobile device and a flexible drive system, which can realize the adduction/abduction and internal/external rotation movement of the lower limb [14]. Bouri et al. developed a parallel robot Lambda that can be used to guide the movement of the lower limb and carry out rehabilitation training of the hip, knee, and ankle joints [15]. Saglia et al. developed a 3-UPS/U parallel mechanism, which can perform rehabilitation training of the human ankle joint [16].

According to the active participation degree of patients, rehabilitation training can be divided into three categories: passive training, semi-active training, and active training [17]. In the passive training process, the rehabilitation robot guides the affected limb to move along a predetermined trajectory for rehabilitation training [18]. For the passive training modes of lower limb rehabilitation robots, the typical ones include MOTOmed training mode, continuous passive motion (CPM) training mode, and straight leg raise (SLR) training mode [19–22]. In the MOTOmed training mode, the end trajectory of the robot is a circular trajectory; In the CPM training mode, the end trajectory of the robot is a linear trajectory; In the SLR training mode, the end trajectory of the robot is an arc trajectory. In order to improve the safety and comfort of patients during passive rehabilitation training, numerous studies have been conducted on the compliance control strategy of the lower limb rehabilitation robot. Wang et al. [23] proposed a fuzzy sliding mode variable admittance controller based on safety evaluation and supervision for the cable-driven lower limb rehabilitation robot, which can switch between active training mode and passive training mode and adjust the parameters of the admittance controller. Li et al. [24] designed a multi-modal control scheme for exoskeleton rehabilitation robots, including robot-assisted mode, robot-dominant mode, and safety-stop mode, and verified the effectiveness of the scheme in upper-limb and lower-limb exoskeleton robot systems. Zhou et al. [25] proposed a trajectory deformation algorithm, which can realize the desired trajectory planning of participants based on the interaction force in the process of human-robot interaction and improve robot compliance and motion smoothness. Chen et al. proposed a reference trajectory adaptive compliance control algorithm, which combines impedance control and motion trajectory planning [26]. Huo et al. developed a lower limb exoskeleton impedance modulation strategy, which can provide proper power and balance assistance during sitto-stand movements [27]. Compared with the position control strategy, the compliance control strategy is beneficial in avoiding excessive force between the human and the robot and has a wider application in the field of rehabilitation robots [28].

The sEMG-based control strategies of the lower limb rehabilitation robot mainly include the sEMG-based continuous control strategy and the sEMG-triggered control strategy [29]. Many studies have been carried out on the sEMG-based continuous control strategy, in which the lower limb motion intention recognition is performed using the sEMG signal and torque assistance proportional to the sEMG signals is provided to generate the desired motions. Khoshdel et al. proposed an sEMG-based robust impedance control strategy for the lower limb rehabilitation robots and the sEMG signals were used

to estimate the exerted force [30]. Yao et al. developed an adaptive admittance control scheme consisting of an admittance filter, an inner position controller, and an sEMG-driven musculoskeletal model [31]. Xie et al. proposed an adaptive trajectory planning method based on sEMG signals and interactive forces for lower limb rehabilitation robots and planned three periodic trajectories using sEMG signals [32]. Different from the sEMG-based continuous control, the robot assistance is triggered when the sEMG signals reach a certain threshold in the sEMG-triggered control strategy. Meng et al. proposed an active interactive controller based on motion recognition and adaptive impedance control. Using the root mean square (RMS) feature of the sEMG signal integrated with the support vector machine (SVM) classifier, it can predict the motion intention of the lower limbs and trigger robot assistance [33]. Lin et al. designed an sEMG-triggered controller for the artificial muscle-driven lower limb rehabilitation robot, and the methods of discrete wavelet transformation and the support vector machine are used to predict the lower limb movement intention [34]. Compared with force and position signals, the sEMG signals can reflect the activity level of specific muscle groups, which can monitor and control the movement of limbs in more detail [35].

However, the above-mentioned compliance control strategies for lower limb rehabilitation robots using sEMG signals are mainly aimed at active training scenarios. Existing passive training control strategies mainly rely on force and position information and lack the intelligent sEMG-based compliance adjustment function, resulting in an unsatisfactory safety performance of lower limb rehabilitation robots [36]. Moreover, in the passive training process of lower limbs, the essential purpose of adopting different training modes is to perform specific training effects on different muscle groups. The fusion of the force, position and sEMG signals in the compliance control strategy, monitoring the muscle activation degree in real time, and controlling the motion of the robot, encompass a significant problem to be solved in the control strategy development of the lower limb rehabilitation robot [36].

Aiming at the problems above, based on the hybrid end-effector lower limb rehabilitation robot (HE-LRR) developed in our research group [37], this paper proposes an sEMG-based gain-tuned compliance control (EGCC) strategy. In the passive training process, the lower limbs follow the robot end effector to move in three-dimensional space. The human body keeps the lower limbs relaxed and does not actively contract muscles. The sEMG signal collected under this condition is intended to monitor the muscle condition and protect the patient by enhancing robot compliance. The rationality of the control strategy is verified through simulation and experimental research under three training modes: MOTOmed, CPM, and SLR. The rest of this paper is organized as follows. Section 2 contains the introduction of the configuration design of the HE-LRR. The EGCC strategy is proposed in Section 3. The simulation research of the EGCC strategy without sEMG information is performed in Section 4. In Section 5, the sEMG acquisition and feature analysis are carried out, and the EGCC strategy comprehensive experiment is conducted. Section 6 presents the conclusions and prospects for the EGCC strategy.

#### **2. Robot Configuration**

There are mainly three types of lower limb movement for the human body, namely moving in the sagittal plane, stepping in the coronal plane, and turning around the longitudinal axis of the human body [38]. HE-LRR is designed in accordance with ergonomic considerations, which includes a base frame, a hybrid (2UPS+U)&(R+RPR) mechanism, and a pedal unit. Here U, P, R, and S represent a universal pair, a prismatic pair, a revolute pair, and a spherical pair, respectively. Figure 1 shows the virtual prototype of the HE-LRR and the pedal unit. The HE-LRR allows people to sit or lie on the opposite side of the machine while their feet are connected to the robot end effector, and they receive rehabilitation training.

**Figure 1.** (**a**) The virtual prototype of the HE-LRR; (**b**) structure of pedal unit.

According to the simplified rotation characteristics of the hip joint where two rotation axes are orthogonal, the parallel part of the lower limb rehabilitation robot is designed as a (2-UPS+U) mechanism, including two UPS branches and one U branch chain. Using linear actuators, the parallel part is driven to rotate around the cross axis, thereby assisting the lower limbs in achieving rehabilitation training in the sagittal and coronal planes. In order to realize the rotary motion of the knee joint, the RPR branch chain is introduced into the parallel part, and the linear actuator is used as the driving unit. Rehabilitation training requirements for patients with multiple degrees of freedom can be met by the coordinated movements of (2-UPS+U)&(R+RPR) mechanisms. The (R+RPR) mechanism is superior to rotary motor driving, and it can reduce the mass and inertia of the kinematic joint of the robot and increase its bearing capacity.

The pedal unit is composed of a foot pedal, a pedal shaft, connecting plates, a tension compression sensor, and an angle sensor. The foot pedal is utilized to guide the distal end of the lower limb to move while the pedal shaft is used to connect the pedal unit with the hybrid mechanism. The tension compression sensor is embedded in the pedal unit to record the man-machine contact force, and the angle force is installed on the connecting plate to acquire the angle information of the pedal unit.

#### **3. EGCC Strategy**

There are two typical impedance control strategies applied in rehabilitation robots: the force-based impedance control strategy and the position-based impedance control strategy. Although the force-based impedance control strategy can realize force tracking, the controller relies on the dynamic characteristics between the robot and the environment, making it difficult to implement control in practice. Compared with the force-based impedance control, the position-based impedance control has more stable performance [39,40]. In this section, the passive training of the lower limb rehabilitation robot adopts a position-based impedance control strategy. The impedance control model is as follows:

$$M\_{\rm d} \Delta \ddot{X} + B\_{\rm d} \Delta \dot{X} + K\_{\rm d} \Delta X = F \tag{1}$$

where, *M*d, *B*d, *K*<sup>d</sup> are the target inertia matrix, damping matrix, and stiffness matrix of the impedance model; *F* is the man-machine contact force acting on the robot end effector; Δ*X* is the position correction amount of the robot end effector.

Using Laplace transformation, the position correction amount in the Laplace domain can be derived as follows:

$$
\Delta X(s) = \frac{F(s)}{M\_{\rm d}s^2 + B\_{\rm d}s + K\_{\rm d}} \tag{2}
$$

where, *s* is the complex number frequency parameter.

The block diagram of the position-based impedance control is shown in Figure 2. The man-machine contact force *F* passes through the impedance control model to generate the position correction amount Δ*X*, which is superimposed on the reference position *X*<sup>r</sup> to generate the desired position *X*d, which is sent to the position controller after the inverse kinematics solution, so that the actual position tracks the desired position.

**Figure 2.** Position-based impedance control strategy diagram.

The above position-based impedance control strategy is suitable for not only controlling the robot to move along a preset trajectory, but also maintaining a certain flexibility during the movement. The method is to convert the end contact force into the position correction amount through the impedance control model. In order to improve the compliance and safety of the control strategy, the sEMG information needs to be integrated into the above position-based impedance control strategy. The modified EGCC strategy diagram is shown in Figure 3.

**Figure 3.** EGCC strategy diagram for the HE-LRR.

From the original sEMG signal of the patient's target muscle group to the gain parameter, it needs to go through two processes: data preprocessing and function mapping. In the process of data preprocessing, the high-frequency and low-frequency signals are filtered out of the sEMG signal through the band-pass filter, and then the time-domain features with the intuitive physical significance are obtained through feature extraction. The root mean square (RMS) can reflect the average power of the sEMG signal, so the RMS feature value is used to evaluate the characteristics of the sEMG signal, and the calculation formula is as follows:

$$RMS\_j = \sqrt{\frac{1}{W} \sum\_{i=1}^{W} \mathbf{x}\_i^2} \tag{3}$$

where *j* represents the *j*-th segment in the original sEMG data sequence, *xi* is the *i*-th original data in the segment data, and *W* is the sliding window width.

In order to improve the generalization ability of the model, the sEMG signals after feature extraction need to be normalized. The normalization calculation formula is as follows:

$$RMS\_{\rm R} = \frac{RMS - RMS\_{\rm min}}{RMS\_{\rm max} - RMS\_{\rm min}} \tag{4}$$

where *RMS* represents the sEMG signal after feature extraction; *RMS*min and *RMS*max are the minimum and maximum values of *RMS*, respectively; *RMS*n is the normalized sEMG signal. *RMS*min and *RMS*max are constants in different training modes and can be obtained through sEMG signal acquisition and feature analysis (see Section 5.2). Here the gain parameter *G* is set to be 1, that is, the sEMG signal is not included in the control strategy during the sEMG acquisition experiment.

After the normalization processing, the normalized sEMG signals *RMS*<sup>n</sup> of different muscle groups can be obtained according to Equation (4) respectively. In the "Function mapping" block, the maximum value of the muscle groups' normalized sEMG signals is compared with the threshold value of the normalized sEMG signal *RMS*t, and the gain parameter *G* can be calculated according to the following equation:

$$G = \begin{cases} 1 & \text{RMS}\_{\text{n}} \le \text{RMS}\_{\text{t}}\\ a \left( \text{RMS}\_{\text{n}} - \text{RMS}\_{\text{t}} \right)^2 + 1 & \text{RMS}\_{\text{n}} > \text{RMS}\_{\text{t}} \end{cases} \tag{5}$$

When the normalized sEMG signal does not exceed the threshold value, the gain parameter is equal to 1. Otherwise, there is a quadratic functional relationship between the gain parameter and the normalized sEMG signal. Thus, in the passive training process of the lower limb rehabilitation robot, the position correction amount is jointly affected by the inertia parameter, damping parameter, stiffness parameter, and gain parameter. When the normalized sEMG threshold is constant, the maximum value of the gain parameter *G* is determined by the parameter *a*. If the parameter *a* is too large, the position correction amount will be too large, it will become more difficult for the robot end effector to move near the set trajectory, and the patient will not be able to receive standardized rehabilitation training. If the parameter *a* is too small, the position correction amount is too small, and the robot end effector will have no apparent sEMG-based compliance enhancement effect in the EGCC strategy. Therefore, the parameter *a* should be kept within a moderate range.

#### **4. Simulation and Results**

#### *4.1. Impedance Control Parameter Influence Analysis*

In the passive training process, it is important to select appropriate inertia parameters, damping parameters, and stiffness parameters when applying the impedance control model. Therefore, it is necessary to analyze the influence of impedance control parameters on the control performance. The transfer function of the impedance control model is:

$$G(s) = \frac{\Delta X(s)}{F(s)} = \frac{1}{M\_{\rm d}s^2 + B\_{\rm d}s + K\_{\rm d}}\tag{6}$$

For the convenience of analysis, considering the impedance control model in a single direction, Equation (6) can be simplified to Equation (7):

$$G(s) = \frac{1}{ms^2 + bs + k} \tag{7}$$

where, *m*, *b*, and *k* are the inertia parameter, damping parameter, and stiffness parameter, respectively. Equation (7) is transformed into the standard form:

$$G(s) = \frac{1}{k} \frac{\omega\_n^2}{s^2 + 2\xi\omega\_n s + \omega\_n^2} \tag{8}$$

where, *ω<sup>n</sup>* is the undamped natural frequency; *ξ* is the damping ratio.

The response curves of position correction amount under different inertia parameters are shown in Figure 4. The simulation parameters are set to {*F* = 1 N, *b* = 0.10 N·s/mm, *<sup>k</sup>* = 0.25 N/mm}. When *<sup>m</sup>* = 0.001 N·s2/mm, *<sup>ξ</sup>* > 1, the system is in the overdamped state; when *<sup>m</sup>* = 0.01 N·s2/mm, *<sup>ξ</sup>* = 1, the system is in the critically damped state; when *<sup>m</sup>* = 0.02, 0.03 N·s2/mm, *<sup>ξ</sup>* < 1, the system is in the underdamped state. The response curves of the position correction amount under different damping parameters are shown in Figure 5. The simulation parameters are set to {*<sup>F</sup>* = 1 N, *<sup>m</sup>* = 0.01 N·s2/mm, *<sup>k</sup>* = 0.25 N/mm}. When *b* = 0.20 N·s/mm, *ξ* > 1, the system is in the overdamped state; when *b* = 0.10 N·s/mm, *ξ* = 1, the system is in the critically damped state; when *b* = 0.03, 0.05 N·s/mm, *ξ* < 1, the system is in the underdamped state. When the system is in the overdamped or critically damped state, the response curve has no overshoot and oscillation, and the rise time and settling time of the critically damped system are shorter than those of the overdamped system. When the system is in the underdamped state, as the damping ratio decreases, the overshoot increases and the settling time becomes longer.

**Figure 4.** Response curves of position correction amount under different inertia parameters (*m*: N·s2/mm; *<sup>b</sup>*: N·s/mm; *<sup>k</sup>*: N/mm).

**Figure 5.** Response curves of position correction amount under different damping parameters (*m*: N·s2/mm; *<sup>b</sup>*: N·s/mm; *<sup>k</sup>*: N/mm).

The response curves of the position correction amount under different stiffness parameters are shown in Figure 6. The simulation parameters are set to {*<sup>F</sup>* = 1 N, *<sup>m</sup>* = 0.005 N·s2/mm, *b* = 0.06 N·s/mm}. When *k* = 0.12 N/mm, *ξ* > 1, the system is in the overdamped state; when *k* = 0.18 N/mm, *ξ* = 1, the system is in the critically damped state; when *k* = 0.24, 0.30 N/mm, *ξ* < 1, the system is in the underdamped state. With the change of the stiffness

parameter, it is found that the steady-state value of the response curve changes significantly. As the stiffness parameter increases, the steady-state value decreases, that is, the position correction amount becomes smaller, thus the robot's compliance worsens.

**Figure 6.** Response curves of position correction amount under different stiffness parameters (*m*: N·s2/mm; *<sup>b</sup>*: N·s/mm; *<sup>k</sup>*: N/mm).

Through the above analysis, applying the impedance control model to the passive training of the rehabilitation robot is to improve the compliance of the rehabilitation robot and achieve the purpose of protecting the patient. The response curve needs to show no overshoot and no oscillation. In addition, the settling time should be shortened as much as possible. Therefore, the impedance model parameters should be set to the critically damped state. Since the steady-state value of the position correction amount is only affected by the stiffness parameter, the stiffness parameter can be reduced to increase the robot's compliance.

#### *4.2. Impedance Control Strategy Simulation*

When simulating the passive training impedance control strategy, it is necessary to add the impedance control model on the basis of the previous position control simulation. In the simulation environment, the man-machine contact force is set to be:

$$\begin{cases} F\_y = \sin t + \sin 2t + \sin 4t\\ F\_z = \cos t + \cos 2t + \cos 4t \end{cases} \tag{9}$$

where, *Fy* and *Fz* are the components of the man-machine contact force in the *Y*-direction and *Z*-direction, respectively.

In the MOTOmed training mode, the reference trajectory of the robot end effector is a circular trajectory. The reference trajectory parameters are set to {the center coordinates (*x*0, *y*0, *z*0) = (0, −670, 470) and the radius *r* = 90.00 mm}. The impedance model parameters are selected from a set of parameters in the critically damped state: {*<sup>m</sup>* = 0.01 N·s2/mm, *b* = 0.10 N·s/mm, *k* = 0.25 N/mm}. The comparison between the reference trajectory and the simulated trajectory of MOTOmed training is shown in Figure 7a. It can be seen that under the action of the man-machine contact forces *Fy* and *F*z, the simulated trajectory has a certain degree of offset compared with the reference trajectory, the coordinate where the maximum position offset occurs is (0, −689.77, 567.59) and the maximum offset is 9.73 mm (*Y*-direction: −0.32 mm, *Z*-direction: 9.72 mm). In the CPM training mode, the reference trajectory of the robot end effector is a beeline trajectory, and the coordinates of the starting point and the end point are set to be (0, −575, 300) and (0, −775, 300), respectively. The impedance control parameters and the contact force function are the same as those of the circular trajectory. The comparison between the CPM training reference trajectory and the simulated trajectory is shown in Figure 7b. Compared with the reference trajectory, the coordinate of the maximum position offset on the simulated trajectory is (0, −602.70, 309.72) and the maximum offset is 9.73 mm (*Y*-direction: −0.33 mm, *Z*-direction: 9.72 mm). In the

SLR training mode, the reference trajectory of the robot is an arc trajectory, the coordinate of the starting point of the reference trajectory is (*x*0, *y*0, *z*0) = (0, −822.5, 613.5), and the coordinate of the end point is (*x*0, *y*0, *z*0) = (0, −639.8, 326.3), the radius *r* = 892.00 mm. The comparison between the SLR training reference trajectory and the simulated trajectory is shown in Figure 7c. Compared with the reference trajectory, the coordinate of the maximum position offset of the simulated trajectory is (0, −774.39, 567.70), and the maximum offset is 9.73 mm (*Y*-direction: −0.32 mm, *Z*-direction: 9.72 mm). From the above analysis, it is found that in the three training modes, the maximum offset values of the simulated trajectories are the same, which is related to the same settings of man-machine contact force and impedance control parameters in the simulation.

**Figure 7.** Reference trajectories and simulated trajectories under different training modes: (**a**) MO-TOmed training; (**b**) CPM training; (**c**) SLR training; (*m*: N·s2/mm; *<sup>b</sup>*: N·s/mm; *<sup>k</sup>*: N/mm).

The contact force and position correction amount in *Y*-direction are shown in Figure 8a. It can be seen that within the simulation time of 0–10 s, the *Y*-direction contact force fluctuates within a certain range, and at the time of 6.80 s, the contact force reaches the maximum value of 2.23 N. The fluctuation trend of the position correction amount in the *Y*-direction is consistent with that of the contact force, but there is a certain delay between the position correction amount and the contact force. At the moment of 7.20 s, the position correction amount reaches the maximum value of 6.74 mm. The contact force and position correction amount in the *Z*-direction are shown in Figure 8b. Within the simulation time of 0–10 s, the position correction amount lags behind the contact force. At the time of 6.28 s, the contact force in the *Z*-direction reaches the maximum value of 3.00 N. At the moment of 6.63 s, the *Z*-direction position correction amount achieves the maximum value of 9.72 mm. By comprehensive analysis of the above results, the time at which the maximum position offset occurs is 6.63 s in the three training modes. The maximum offsets in the three training modes are the same, indicating that the position offset is determined by the man-machine contact force and not affected by the training mode. Through the above simulations of MOTOmed training, CPM training, and SLR training, it can be shown that under the action of man-machine contact force, the rehabilitation robot shows a certain compliance by generating the position correction amount to adapt to changes of the man-machine contact force.

**Figure 8.** The comparison of the contact force and the position correction amount in different directions (**a**) *Y*-direction; (**b**) *Z*-direction.

#### **5. Experimental Verification**

#### *5.1. Robot Prototype and Control System*

The control system of the lower limb rehabilitation robot consists of the controlling unit, the driving unit, the actuating unit, the sensing unit, the sEMG acquisition unit, and the power unit, as shown in Figure 9.

**Figure 9.** Frame diagram of lower limb rehabilitation robot control system.

The biosignal acquisition tool (PLUX wireless biosignals S.A., Biosignals Researcher, Lisbon, Portugal) collects sEMG signals in real-time through electromyography electrodes pasted on the target muscle groups of the lower limbs and transmits the signals to the upper computer (DELL Technologies Co., Ltd., Vostro 5370, Round Rock, TX, USA) through Bluetooth. Filter processing and feature value calculation are carried out within the set time period, and the feature value is transmitted to the controller through the Ethernet. The industrial controller (Advantech Technology Co., Ltd., IPC610, Suzhou, China) is used as the controller. In addition to receiving instructions from the upper computer in real-time, it can also receive signals from the tension compression sensor (HY chuangan Technologies Co., Ltd., HYLY-019, Bengbu, China) and the angle sensor (BEWIS Sensing Technologies Co., Ltd., BWK220, Wuxi, China). At the same time, the controller sends instructions to the DC motor driver (Magicon Intelligent Technologies Co., Ltd., MC-FBLD-6600, Shenzhen, China), and drives the linear actuators (Suzhou Yuancheng mingchuang Electromechanical

Equipment Co., Ltd, LEC606, Suzhou, China) to perform telescopic movement. The linear actuator has a built-in incremental encoder, which can record the motion position of the DC motor to facilitate the position-based closed-loop control of the linear actuator. Angle sensors, tension compression sensors, and DC motor drivers require 12 V or 24 V DC voltage, which is provided by the power unit.

The prototype of HE-LRR was manufactured and integrated with the control system, which is shown in Figure 10. Universal casters with brakes are installed at the bottom of the base frame to facilitate the movement of the robot and improve the stability during rehabilitation training. The patient's feet are placed on the foot pedal to carry out the rehabilitation training. During the implementation of this study, five healthy participants (age: 24–31 years old; height: 1670–1870 mm; thigh length: 405–455 mm; calf length: 385–420 mm) were recruited to take part in the experiment following the procedures for healthy participants as approved by the China Rehabilitation Research Center (CRRC-IEC-RF-SC-005-01), and the basic information of the participants is listed in Table 1. There were no known muscular or neurological disorders among the healthy participants. All participants completed the experimental protocol safely and reported no physical discomfort.

**Figure 10.** Prototype of the HE-LRR and the control system.


**Table 1.** Basic information of the participants in the experiments.

The experimental procedure is shown in Figure 11. In the subsection of Signal Acquisition and Feature Analysis, the experimental processes include signal acquisition preparation, signal acquisition, signal preprocessing. and signal characteristic analysis. In the subsection of EGCC Strategy Comprehensive Experiment, the research is carried out in the order of the determination of model parameters, experimental verification, comparative analysis of experimental results. and experimental conclusion.

**Figure 11.** Flowchart of the experimental procedure.

#### *5.2. Signal Acquisition and Feature Analysis*

Before the sEMG signal acquisition experiment, the biceps femoris (BF), rectus femoris (RF), tibialis anterior (TA), and peroneus longus (PL) were selected as the target muscle groups of the lower limbs, and the surface electrodes were pasted on the corresponding skin positions of the muscle groups. The positions of the four target muscle groups of the lower limbs and the sensor sticking positions are shown in Figure 12. In the sEMG signal acquisition process, the subjects were given instructions to keep their lower limbs relaxed and not to contract their muscles actively. Their feet followed the robot end effector to move in space. Each subject participated in 12 groups of experiments for each training mode (MOTOmed, CPM or SLR). Impedance parameter settings in the 12 groups of experiments are shown in Table 2. In each group of experiments, the subjects performed 10 cycles of training.

The sampling frequency of the sEMG acquisition unit is 1000 Hz, and the sampling period is 1 ms. The collected original sEMG signals are in the range of 0–10 μV. After passing through the band-pass filter with a passband of 10–500 Hz, the feature value is extracted from the filtered sEMG signal and the RMS feature value is used for the timedomain quantitative analysis of the sEMG signal. Figure 13 shows the sEMG signals before and after RMS feature extraction. It can be seen that the signal characteristic of violent fluctuations is eliminated after RMS feature extraction. At the same time, the sEMG signal after the RMS feature extraction can well reflect the change trend of the original signal (before RMS feature extraction) and shows good regularity and stability. The maximum RMS values of the sEMG signal of the subjects in different training modes are extracted and statistical analysis is carried out to obtain the average value and standard deviation.

**Figure 12.** (**a**) Target muscle groups and sensor sticking positions; (**b**) MOTOmed training mode; (**c**) CPM training mode; (**d**) SLR training mode.


**Table 2.** Impedance parameter setting in the 12 groups of experiments.

**Figure 13.** sEMG signals before and after RMS feature extraction.

Figure 14 displays the maximum RMS values of the sEMG signal in different training modes. In the MOTOmed training mode, under the condition of different impedance control parameters, the maximum RMS values of the four muscle groups are shown in Figure 14a. It can be seen that when the damping parameter and stiffness parameter are fixed values (*b* = 0.10 N·s/mm, *k* = 0.25 N/mm), the RMS values of the four muscles are at a higher level under the underdamped state (*<sup>m</sup>* = 0.02, 0.03 N·s2/mm) and the RMS values of the four muscles are at a lower level when under the overdamped or critically damped state (*<sup>m</sup>* = 0.001, 0.01 N·s2/mm). Similarly, when the inertia parameter and stiffness parameter are fixed values (*<sup>m</sup>* = 0.01 N·s2/mm, *<sup>k</sup>* = 0.25 N/mm), the four muscles obtain relatively high RMS values of the sEMG signals in the underdamped state. When the inertia parameter and damping parameter are fixed values (*<sup>m</sup>* = 0.005 N·s2/mm, *<sup>b</sup>* = 0.06 N·s/mm), the maximum RMS values of the muscle groups except for the PL muscle increase with the increase of the stiffness parameter. This is because when the stiffness parameter increases, the offset degree of the robot in response to the action of the man-machine contact force decreases, and the compliance of the HE-LRR robot is reduced, resulting in the situation where the muscle activation level cannot be released and maintained at a high level.

In the CPM training mode, under different impedance control parameters, the maximum RMS values of the four muscle groups are shown in Figure 14b. It can be seen that different muscles can obtain higher RMS values in the underdamped state, which is similar to the MOTOmed training mode. The difference is that the maximum RMS value is 8.26 ± 0.25 μV (TA muscle) in the CPM training mode, while the maximum RMS value is

9.24 ± 0.23 μV (BF muscle) in the MOTOmed training mode. In the SLR training mode, under different impedance control parameters, the maximum RMS values of the four muscles are shown in Figure 14c. It can be seen that, when *<sup>m</sup>* = 0.005 N·s2/mm, *<sup>b</sup>* = 0.06 N·s/mm, *k* = 0.30 N/mm, the maximum RMS value of the sEMG signal is 8.30 ± 0.24 μV (BF muscle), and when *<sup>m</sup>* = 0.005 N·s2/mm, *<sup>b</sup>* = 0.06 N·s/mm, *<sup>k</sup>* = 0.24 N/mm, the maximum RMS value is 7.89 ± 0.28 μV (RF muscle).

**Figure 14.** Maximum RMS value of sEMG signal in different training modes (**a**) MOTOmed mode; (**b**) CPM mode; (**c**) SLR mode (*m*: N·s2/mm; *<sup>b</sup>*: N·s/mm; *<sup>k</sup>*: N/mm).

Comprehensive analysis, when the participants participate in the three training modes in a relaxed state, the RMS range of sEMG for target muscle groups is 0–9.24 μV in MO-TOmed training, the RMS range of sEMG is 0–8.26 μV in CPM training, and the RMS range of sEMG is 0–8.30 μV in the SLR training (since the minimum values of RMS of different subjects were close to zero, here the lower bound value of the RMS range is determined to be zero). In the MOTOmed training mode, *RMS*min and *RMS*max are determined as 0 μV and 9.24 μV; in the CPM training mode, *RMS*min and *RMS*max are determined as 0 μV and 8.26 μV; in the SLR training mode, *RMS*min and *RMS*max are determined as 0 μV and 8.30 μV. The feature analysis results show that there exists a difference in the RMS range under different training modes, which proves that adopting different training modes can carry out targeted rehabilitation training for different muscle groups, so as to achieve a better effect of lower limb rehabilitation training. In particular, after RMS feature extraction, the regularity and stability of the sEMG signals are further improved, which can meet the needs of the EGCC strategy. Moreover, taking the maximum RMS values in this subsection as the reference values for normalization processing can improve the generalization ability of the EGCC strategy.

#### *5.3. EGCC Strategy Comprehensive Experiment*

In order to verify the control effect of EGCC strategy, validation experiments were carried out under different training modes. The participants kept their lower limbs in a relaxed state during the training process. After normalization processing, the normalized sEMG threshold was set at 0.50, 0.75, and 1.00, respectively. The inertia parameter *m*, damping parameter *<sup>b</sup>*, and stiffness parameter *<sup>k</sup>* in the EGCC strategy were set at 0.01 N·s2/mm, 0.10 N·s/mm, and 0.25 N/mm, respectively. For the convenience of comparison, the coefficient *a* was set to be 5 in the following EGCC strategy comprehensive experiment.

The experimental results of the actual end trajectory and gain parameter of the lower limb rehabilitation robot under different training modes are shown in Figure 15. It can be seen that in the MOTOmed training mode, the actual trajectories in the three groups of experiments deviate to a certain extent compared with the reference trajectory (Figure 15a). When the normalized sEMG threshold is 0.50, 0.75, and 1.00, the maximum values of the position correction amount are 17.22 mm, 12.03 mm, and 8.33 mm, respectively. As can be seen from Figure 15b, when the normalized sEMG thresholds are 0.50 and 0.75, the gain parameter fluctuates locally. When the normalized sEMG threshold is 0.50, the maximum value of the gain parameter is 2.09. When the normalized sEMG threshold is 0.75, the maximum value of the gain parameter is 1.24, which shows that the decrease of the normalized sEMG threshold is beneficial in improving the compliance of HE-LRR.

In the CPM training mode, when the normalized sEMG thresholds are 0.50, 0.75, and 1.00, the maximum values of the position correction amount are 21.75 mm, 13.71 mm, and 7.69 mm, respectively, while the maximum values of the gain parameter are 2.09, 1.24, and 1.00, respectively. In the SLR training mode, when the normalized sEMG thresholds are 0.50, 0.75, and 1.00, the maximum values of the position correction amount are 16.98 mm, 11.74 mm, and 5.92 mm, and the maximum values of the gain parameter are 2.08, 1.27, and 1.00, respectively. Comparing the results in the three training modes, although the maximum values of the position correction amount are different, the gain parameters are relatively close to each other. This is because when the normalized sEMG thresholds are 0.50, 0.75, and 1.00, the gain parameters have a maximum value of 2.25, 1.3125, and 1.00, respectively, which enables the position correction amount of the lower limb rehabilitation robot to be maintained within a certain range to prevent secondary damage caused by excessive offset.

In addition, it can be seen from Figure 15b,d,f that the gain parameter is larger than 1.00 in a relatively short time. Since there is a clear functional relationship between the gain parameter and normalized sEMG threshold, it shows that the normalized sEMG can recover below the threshold in a short time. This is due to the fact that as the gain parameter increases, the position offset occurring in the direction of the man-machine contact force increases and the compliance of the lower limb rehabilitation robot is enhanced, which is conducive to the recovery of muscle activation. When the normalized sEMG threshold is set as 1.00, the EGCC strategy can be used to identify abnormal sEMG signals and increase the compliance of the lower limb rehabilitation robot to protect the participant. In conclusion, the EGCC strategy can play a significant role in regulating the compliance of the lower limb rehabilitation robot and increasing the safety of the participant.

**Figure 15.** *Cont*.

**Figure 15.** EGCC experimental results of HE-LRR. (**a**) Actual trajectories in the MOTOmed training mode; (**b**) gain parameters in the MOTOmed training mode; (**c**) actual trajectories in the CPM training mode; (**d**) gain parameters in the CPM training mode; (**e**) actual trajectories in the SLR training mode; (**f**) gain parameters in the SLR training mode.

#### **6. Conclusions and Future Work**

Aiming at the problems of insufficient physiological information and unsatisfactory safety performance in the existing compliance control strategies for the lower limb rehabilitation robot during passive training, this paper developed an sEMG-based gain-tuned compliance control strategy and carried out simulation and experimental research based on this control strategy. The main conclusions are as follows:


compliance and safety of the lower limb rehabilitation robot, which validates the rationality of the EGCC strategy.

Although the control strategy in this paper was verified in the end-effector robot system, the basic methodology can also be applied in the exoskeleton lower limb robot system. There are still some shortcomings in the current research work, for example, the simulation and experimental research of the EGCC strategy were mainly carried out in three training modes: MOTOmed, CPM, and SLR, and the normalized sEMG threshold was required to be set manually. Future research work will be committed to solving the problems of the EGCC strategy validation in various training modes as well as the autonomous learning and optimization of the EGCC strategy model.

**Author Contributions:** Methodology, J.T. and J.N.; software, S.Z. and Y.N.; validation, X.Z.; writing—original draft preparation, J.T. and S.Z.; writing—review and editing, J.T. and J.N.; project administration, J.N. and L.V.; funding acquisition, J.N. and H.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially funded by the National Key Research and Development Program under Grant 2019YFB1312500, National Natural Science Foundation of China under Grant U1913216, Science and Technology (S&T) Program of Hebei under Grant 216Z1803G and E2020103001, and is also partially funded by Shanghai Clinical Research Center for Aging and Medicine (19MC1910500).

**Institutional Review Board Statement:** The study was conducted in accordance with the Declaration of Helsinki and approved by the Ethics Committee of China Rehabilitation Research Center (protocol code: 2020-006-1 and date of approval: March 2020).

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** The original data contributions presented in the study are included in the article; further inquiries can be directed to the corresponding authors.

**Acknowledgments:** The authors would like to take this opportunity to express thanks to Doctor Ying Liu from Qinhuangdao Haigang hospital for his assistance in the design and implementation of experimental schemes.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**


#### **References**


**Musong Lin 1, Hongbo Wang 1,2,3, Congliang Yang 1, Wenjie Liu 1, Jianye Niu 1,4,\* and Luige Vladareanu 5,\***


**Abstract:** A stroke is a common disease that can easily lead to lower limb motor dysfunction in the elderly. Stroke survivors can effectively train muscle strength through leg flexion and extension training. However, available lower limb rehabilitation robots ignore the knee soft tissue protection of the elderly in training. This paper proposes a human–robot cooperative lower limb active strength training based on a robust admittance control strategy. The stiffness change law of the admittance model is designed based on the biomechanics of knee joints, and it can guide the user to make force correctly and reduce the stress on the joint soft tissue. The controller will adjust the model stiffness in real-time according to the knee joint angle and then indirectly control the exertion force of users. This control strategy not only can avoid excessive compressive force on the joint soft tissue but also can enhance the stimulation of quadriceps femoris muscles. Moreover, a dual input robust control is proposed to improve the tracking performance under the disturbance caused by model uncertainty, interaction force and external noise. Experiments about the controller performance and the training feasibility were conducted with eight stroke survivors. Results show that the designed controller can effectively influence the interaction force; it can reduce the possibility of joint soft tissue injury. The robot also has a good tracking performance under disturbances. This control strategy also can enhance the stimulation of quadriceps femoris muscles, which is proved by measuring the muscle electrical signal and interaction force. Human–robot cooperative strength training is a feasible method for training lower limb muscles with the knee soft tissue protection mechanism.

**Keywords:** rehabilitation robot; human–robot interaction; admittance control; robust control; active strength training

#### **1. Introduction**

The independent walking ability of the elderly is the basic premise to ensure the quality of life [1]. However, limb weakness increases with age and the impact of cardiovascular disease often leads to physical disability in the elderly [2]. According to statistics, there are more than millions new incident stroke cases worldly in every year, and there is a high probability of losing walk ability among the survivors [3,4]. Facing such a large number of disabled people, more rehabilitation physicians and rehabilitation training equipment are needed to help them regain lower limb strength, stand up again and return to society [5,6]. As a new type of intelligent medical robot, rehabilitation robot can effectively improve limb disabilities caused by aging or sequela and their therapeutic effect has been proved by many clinical experiments [7–9].

**Citation:** Lin, M.; Wang, H.; Yang, C.; Liu, W.; Niu, J.; Vladareanu, L. Human–Robot Cooperative Strength Training Based on Robust Admittance Control Strategy. *Sensors* **2022**, *22*, 7746. https://doi.org/ 10.3390/s22207746

Academic Editor: Carlo Alberto Avizzano

Received: 25 August 2022 Accepted: 8 October 2022 Published: 12 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Muscle weakness is well established as the primary impairment that affects walking after stroke, and strength training can effectively promote the recovery of muscle strength [10,11]. The effectiveness of strength training has also been proven by some resistance training [12,13]. The quadriceps femoris is the biggest human skeletal muscle at the front of the thigh, and it plays a vital role in extending the knee, flexing the hip and maintaining an upright position. Leg flexion and extension is a strength training exercise that can effectively enhance the quadriceps, so the research of related equipment has also attracted much attention. The American Harley Company proposed a rehabilitation device X-10, which uses variable pressure technology to reduce the pain in the patient's treatment and improve the patient's joint mobility [14]; Another similar device is a sitting rehabilitation device developed by King Wangut University of Technology, which is suitable for home training but only has one free rotation and a small range of motion [15]. They all belong to the same type of rehabilitation equipment using a moving platform. They send the terminal force and joint torque into the control feedback loop to ensure the safety of training respectively, but the training effect on the hip joint is not obvious. The University of Tsukuba and a partner company developed an exoskeleton robot called HAL, which can directly provide active or passive lower limb flexion and extension training to bedridden patients. The range of motion and the walking ability of the patients is improved after training, but the strength of the quadriceps femoris does not change significantly [16,17]. Italy and Poland developed a new 3-DOF bionic exoskeleton, which can be used for rehabilitation after joint surgery, ligament, and cartilage injuries [18].

In robot-assist active training fields, the robot needs to be able to extract the patient's motion intention according to the interaction information and assist the patient to complete the training action. The effectiveness of impedance control and force-position hybrid control have been verified on the rehabilitation machine LOKOMAT, and these methods have improved the interactivity of human–machine cooperative training [19]. Wu et al. developed an admittance control strategy induces the active participation of patients [20]; an optimization method based on admittance control was proposed to compensate the weight and friction of the exoskeleton [21]. A lower rehabilitation robot called LOPES II allows different active training intensities through admittance control [22]. Impedance controllers are also applied in robot-assist active training for joint or lower limb rehabilitation [23,24]. Some researchers use sEMG (surface electromyography) or EEG (electroencephalography) signals for guiding rehabilitation robots to complete active training [25–27]. Courtney et al developed an algorithm for adjusting functional electrical stimulation to help patients taking active training [28].

Including the research mentioned above and other we can find, none have mentioned the protection of the knee soft tissue. However, the physiological functions of the elderly gradually degenerate, and soft tissues such as the meniscus, cartilage and ligaments are relatively fragile [29,30]. For the main user groups of rehabilitation therapy, it is necessary to avoid damage to their joint soft tissues during rehabilitation strength training. The National Strength and Conditioning Association has studied knee joint biomechanics during the human squat and pointed out the conclusion. That is, the tibiofemoral compressive force will peak at 130 degrees of knee flexion, and the menisci and articular cartilage bear significant amounts of stress [31]. Soft tissue such as ligaments are at great risk of injury at this moment [32]. Patellofemoral compressive force, tibiofemoral compressive force and tibiofemoral shear force will gradually decrease with knee extension, while quadriceps muscle activity will peak at approximately 80 to 90 degrees of knee flexion and remain relatively stable thereafter [33,34]. In the human–machine cooperative leg flexion and extension training, it is necessary to timely control the interaction force depending on the knee joint angle in order to reduce the possibility of joint soft tissue injury.

In this paper, a human–machine cooperative leg flexion and extension training based on a robust admittance control strategy is proposed, which fully considers the protection of knee soft tissue based on biomechanics. The performance device is the sitting and lying lower limb rehabilitation robot (LLR-II) developed by our team. In this training, LLR-II responds according to the interaction force and assists the patient performs a full lower limb flexion and extension similar to a leg press. Compared with single knee flexion and extension training, this training can maintain and improve the mobility of each joint of the lower limbs, and can effectively exercise the muscles of the hips, knees and ankles. Firstly, according to the biomechanics of the knee joint, the change law of the stiffness of the main admittance model is designed, and the flexibility of the training is increased by the subsidiary admittance control. The controller will adjust the model stiffness according to the joint angle during the training, and it could avoid excessive compressive force on the soft tissue and increase the stimulation of the quadriceps. Then, the joint tracking performance is improved by two-input robust motion control by compensating the motion control disturbances caused by model uncertainty, interactive forces, and external noise. Finally, the testing experiment of this human–machine cooperative leg flexion and extension training is conducted.

#### **2. LLR-II Rehabilitation Robot**

The LLR-II is an intelligent robotic system that can intervene early and provide a variety of rehabilitation training and more details can be found in our published papers [35,36]. LLR-II can be divided into four modules which include two symmetrical training modules, a seat module and an electric control module, as shown in Figure 1. The LLR-II is assembled by connecting the underframe of each module and each module can be moved independently for installation and transportation. The right training module is equipped with a touch display and an emergency stop button and the width between the two training modules can be adjusted according to the user's body shape. The height of the seat module is adjustable and it can help medical staff transfer patients. In addition, in order to adapt to different people, the length of the upper and lower mechanical legs can be adjusted through the internal electric linear actuator.

**Figure 1.** Structure of LLR-II.

#### *2.1. Structural Design of LLR-II*

The mechanical leg of LLR-II is a three joint series mechanical mechanism, and the three joints correspond to the hip, knee and ankle joints of the human body, respectively. Its joint drive train is composed of flange structures, as shown in Figure 2. The high torque motors of the hip and knee joints adjust the fixed positions through timing belts, which are located at the bottom of the training module and the rear end of the mechanical leg respectively. Hip and knee joint transmission structures are similar, and both of them are consist of a synchronous pulley, a reducer and a torque sensor (Figure 2a). The ankle joint equips with a frameless motor, and the integration of the ankle joint is effectively improved by directly connecting the motor and the reducer (Figure 2b).

**Figure 2.** Section view of joint drivetrain: (**a**) knee joint and (**b**) ankle joint.

The electric control system of LLR-II can be divided into four parts as follows: central control section, drive control section, sensor feedback section and human–robot interaction section (Figure 3). The central control section mainly includes the host computer and related data acquisition equipment, which is responsible for the advanced operations and coordinates other parts. The drive control section is mainly composed of the joint motor, the electric linear actuator and the related communication control equipment. The sensor feedback section mainly includes the torque sensor, the angle sensor of the joint, the six-dimensional force sensor and the potentiometer. The interaction operation is mainly realized through a touch display screen. In addition, the LLR-II also has multimedia functions such as virtual reality and voice control.

**Figure 3.** Electric control system. The arrow represents the direction of information transmission.

#### *2.2. Mechanical Leg Model Analysis*

The mechanical leg of LLR-II is a series manipulator working in the sagittal plane, and its physical model can be simplified as a 3R structure, as shown in Figure 4.

**Figure 4.** Mechanism model of the mechanical leg.

Establish a global coordinate system {*O*-*X*0*Y*0*Z*0} at the hip joint rotation center point *A*. *B* and *C* represent the rotation centers of the knee and ankle joints respectively. *q*1, *q*<sup>2</sup> and *q*<sup>3</sup> are the joint variables of the three rotating joints, *l*1, *l*<sup>2</sup> and *l*<sup>3</sup> respectively represent the distance between the rotating joints, *l*<sup>0</sup> represents the distance between the counterweight mess center and the hip rotating joint; *R*1, *R*<sup>2</sup> and *R*<sup>3</sup> represent the distance between the link mass center and the rotation center, respectively. The kinematic model of LLR-II is the same as the standard 3R mechanism, and its kinematics forward and inverse solutions can be calculated by the D-H method and geometric method. The results are shown in Equations (1) and (2) below:

$$T = \begin{bmatrix} n\_x & o\_x & a\_x & p\_x \\ n\_y & o\_y & a\_y & p\_y \\ n\_z & o\_z & a\_z & p\_z \\ 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} \cos q\_{123} & -\sin q\_{123} & 0 & l\_3 \cos q\_{123} + l\_2 \cos q\_{12} + l\_1 \cos q\_1 \\ \sin q\_{123} & \cos q\_{123} & 0 & l\_3 \sin q\_{123} + l\_2 \sin q\_{12} + l\_1 \sin q\_1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix},\tag{1}$$

$$\begin{aligned} q\_1 &= \text{Atom2}(K, \pm\sqrt{1-K^2}) - A \tan 2(A, \text{ }B) \\ q\_2 &= \text{Atom2}(B - l\_1 \sin q\_1, \text{ }A - l\_1 \cos q\_1) - q\_1 \\ q\_3 &= \text{Atom2}(n\_{y\_1}, n\_x) - q\_1 - q\_2 \end{aligned} \tag{2}$$

where

$$\begin{array}{c} A = p\_x - l\_3 n\_x \\ B = p\_y - l\_3 n\_y \\ K = \frac{A^2 + B^2 + l\_1^2 - l\_2^2}{2l\_1\sqrt{A^2 + B^2}} \\ q\_{1\dots i} = q\_1 + \dots + q\_i \end{array}$$

The dynamic equation of the mechanism can be obtained through the Lagrangian equation, and the controlled system model of the robot can be obtained as follows:

$$\mathcal{M}(q)\ddot{q} + \mathcal{C}(q,\dot{q})\dot{q} + \mathcal{g}(q) + \mathcal{J}^T(q)\mathcal{F} = \mathfrak{u},\tag{3}$$

where *M*(*q*) is a diagonal matrix consisting of the inertia matrix and drive train inertia. *C*(*q*, . *q*) represents the matrix of Coriolis and centrifugal forces and *g*(*q*) represents the gravitational vector. .. *q*, . *q* and *q* are the joint acceleration, velocity and position vectors. *J*(*q*) is the Jacobian matrix of the mechanism and *F* represents the human–robot interaction force. *u* is the control input vector.

Unlike the standard 3R structure, this mechanism has a counterweight used for lightening the motor load, as the yellow line shown in Figure 4. The leg length adjustment function is adjusting the position of the rotation center point *A* and it means that the relative position of the link mass center point *R*<sup>1</sup> in the global coordinate system will change under the influence of the counterweight and the leg length change. The Lagrangian quantity change caused by the mass center position change will exacerbate the system uncertainty

in the dynamic control. Therefore, it is necessary to calculate the mass center position of the link *l*1, as shown in Equation (4) below:

$$R\_1 = \frac{m(l\_1 - l\_0) - 2m\_0 l\_0}{2m + 2m\_0},\tag{4}$$

where *m*<sup>0</sup> is the mass of the counterweight and *m* is the mass of the first link (without counterweight). In Lagrangian dynamics, the change of the mass center position will directly change the translational kinetic energy and gravitational potential energy of the first link, and the moment of inertia in the angular kinetic energy term also needs to be recalculated according to Equation (4) after length adjustment.

#### **3. Robust Admittance Control Strategy**

The control strategy of the rehabilitation robot is different from the general industrial robot; it needs to fully consider human–robot interactions to ensure the safety of patients. Rehabilitation robots should be able to respond to different levels of interaction and maximize the movement potential of patients. Biomechanical research has shown that pushing force should be avoided when the knee is flexed at a wide angle, and there is also an efficient training range for the quadriceps. In addition, due to the large interactive force of the active training, it has high requirements for the robustness of the control algorithm. Therefore, a robust admittance control strategy for lower limb strength training is proposed, which combines robust control and admittance control. The strategy block diagram is shown in Figure 5. This strategy indirectly controls the user's force through variable stiffness admittance control, and it can avoid excessive compressive force on the joint soft tissue and increase muscle group stimulation. Dual input robust control adds an error compensation term that can be used for compensating force interference, and it improves the tracking performance of the machine joints.

**Figure 5.** Admittance robust control block diagram.

#### *3.1. Variable Stiffness Admittance Control*

Admittance control is a control strategy that describes the relationship between force and motion through a spring damping model, and both admittance control and impedance control use the same model. The input and output of admittance control are force and position, respectively. The end force of the series robot can be easily obtained by force sensors, so this method is often used in human–robot interactions. The admittance control strategy proposed in this paper includes two control laws, and its function is shown in Figure 6. The effect of the main control law is changing the model stiffness according to the knee joint angle; it can protect the knee joint and increase the stimulation of the quadriceps muscle (the stiffness change is plotted on a trajectory with color mapping in Figure 6, the bright part indicates high stiffness). The subsidiary control law allows a small deviation of the training trajectory, which increases the flexibility of the training action. When the user's vertical force will lead to a large deviation of the trajectory, the subsidiary control law will ensure the trajectory by resisting the user's force (as arrows shown in Figure 6).

**Figure 6.** Working schematic diagram of admittance controller.

The input of the subsidiary admittance control is the interaction force *Fy*(s) in the vertical direction, and the output is the end position *Py*(s) in the vertical direction. Its transfer function is shown in Equation (5):

$$G(s) = \frac{P\_{\mathcal{Y}}(s)}{F\_{\mathcal{Y}}(s)} = \frac{1}{M\_{\text{dy}}s^2 + B\_{\text{dy}}s + K\_{\text{dy}}},\tag{5}$$

where *M*dy, *B*dy and *K*dy represent inertia, damping and stiffness. The system will compare the expected position with the set offset threshold. If output exceeds the set allowable offset, the excess part will be limited. Two outputs of the subsidiary control and main control will be sent to the trajectory generator, and the generator will obtain the actual end position based on vector calculation before inverse kinematics. With smaller model parameters, the compliant and constrained training trajectory can be achieved.

The model of main admittance control is a second-order model with varying stiffness along the motion trajectory and the output .. *P* is the desired end acceleration. When receiving an interaction force exceeding the threshold, the end of the machine will accelerate. When the interaction force is deficient, the end of the machine will decelerate to stop according to the admittance parameters. Its control law is designed as follows:

$$M\_{\rm dx}\ddot{P} + B\_{\rm dx} \| \dot{X} \|\_{2} + \mathbf{K}\_{\rm var}(q\_{2})D = F\_{\rm x} \tag{6}$$

where *F*x is the extracted effective interaction force; *D* is a constant with the same dimension as the end position; . *X* represents the end velocity vector of the robot. *M*dx and *B*dx are the inertia and damping of the model and the model stiffness *K*var is a piecewise function of the knee joint angle, designed as follows:

$$\mathbf{K}\_{\text{var}} = \begin{cases} \begin{aligned} &k\_1 + \frac{\left(k\_2 - k\_1\right)\left[L\left(q\_2\right) - L\_0\right]^2}{\left[L\_1 - L\_0\right]^2} & -120^\circ \le q\_2 < -100^\circ\\ &\frac{k\_2 + k\_3}{2} - \frac{k\_3 - k\_2}{2}\cos\left(\pi \frac{L\left(q\_2\right) - L\_1}{L\_2 - L\_1}\right) & -100^\circ \le q\_2 < -90^\circ\\ &k\_3 - k\_4 \exp\left[k\_5 \frac{L\left(q\_2\right) - L}{L - L\_2}\right] & -90^\circ \le q\_2 \le 0 \end{aligned} \end{cases}$$

*L*(*q*2) is a function of the knee joint angle and leg length; it represents the end position of the robot. *L*0, *L*<sup>1</sup> and *L*<sup>2</sup> represent the end position scalars of the robot when the knee joint is at −120◦, −100◦ and −90◦; *L* represents the total length of the training trajectory. The constant coefficients *k*<sup>i</sup> (i = 1, 2 ... ) are all parameters of this function and the amplitude of stiffness can be adjusted by changing these parameters.

In this training process, the motion range of the knee joint is −120◦ to 0◦, which covers 80% normal motion range of the human body. The purpose of this design is to stretch the muscles of the knee joint and maintain joint mobility. In addition, the controller will adjust the model stiffness in real-time according to the knee joint, and this can protect the knee soft tissue and increase the stimulation effect on the quadriceps. At the beginning of the training (120◦ of knee flexion), the model stiffness is set at a low level. This is because, in this angle range, it will put greater stress on the knee soft tissue when the leg extends with resistance forces. Training in this situation for a long time might cause damage to the knee joint. When the knee joint is flexed to 100◦, the model stiffness begins to rise rapidly. When the knee joint is flexed to 90◦, the stiffness reaches the highest level, which marks the training entering the strong stimulation phase. At this stage, the force of the lower limbs mainly depends on the contraction of the quadriceps femoris, and the training effect can be improved by correctly exerting force in this stage. In the final stage, the stiffness decreases slowly with a negative exponential trend. Considering the lower limb is not easy to exert force when it is close to full extension, this design can extend the stimulation movement and ensure that the user can complete the leg extension.

#### *3.2. Dual Input Robust Control*

Robot dynamics control needs to solve the tracking error problem caused by external disturbance or model inaccuracy. For most of the series robots, the model inaccuracy mainly comes from the uncertainty of the dynamic parameters (the deviation between the theoretical reference model and the actual model). This uncertainty is generally changeless and can be reduced by optimizing parameters through classical algorithms. The model structure of the LLR-II is rather special as the first link mass center position becomes a variable under the influence of the counterweight structure and the length adjustment. Therefore, the parameters of the dynamic model will change greatly after the mechanical leg length adjustment because the mass center change will lead to the change of Lagrangian variables. That is to say, the parameter uncertainty of the LLR-II model is also a variable. Although the reference model will be updated according to Equation (4), there is still a deviation from the actual model. Adding to the influence of the large fluctuation interaction force, common classical algorithms cannot adapt to such variable parameter systems.

This paper proposes a dual input robust control considering the s interactive force effect, and it is used for reducing the influence of model uncertainty, noise interference and the impact of interactive forces on machine tracking performance. The design control law is as follows:

$$\mathbf{u} = \mathbf{\hat{M}}(\boldsymbol{q})\mathbf{a} + \mathbf{\hat{C}}(\boldsymbol{q}, \dot{\boldsymbol{q}})\mathbf{v} + \mathbf{\hat{g}}(\boldsymbol{q}) + \mathbf{J}^{T}(\boldsymbol{q})\mathbf{\hat{F}} - \mathbf{K}\mathbf{r},\tag{7}$$

.

where *M*ˆ , *C*ˆ , *g*ˆ and ˆ *J* are estimated values defined by the corresponding symbols (theoretical reference value); *K* and **Λ** are two constant positive gain matrices; *v*, *a* and *r* are defined as follows:

$$\begin{cases} \boldsymbol{v} = \dot{\boldsymbol{q}}\_{\mathrm{d}} - \Lambda \boldsymbol{e} \\\ a = \dot{\boldsymbol{v}} = \ddot{\boldsymbol{q}}\_{\mathrm{d}} - \Lambda \dot{\boldsymbol{e}} \\\ r = \dot{\boldsymbol{q}} - \boldsymbol{v} = \dot{\boldsymbol{e}} + \Lambda \boldsymbol{e} \end{cases}$$

Another simplified form of the control input can be obtained by linearizing the parameters of Equation (4):

$$
\mu = \mathcal{Y}(q, \dot{q}, a, \upsilon)\dot{\Theta} + \mathcal{Z}(q)\,\text{tr} - \text{Kr},\tag{8}
$$

where the functions *Y* and *Z* are the regressors of the first three terms and the fourth terms on the left side of Equation (4). θˆ and *π*ˆ are the parameter vectors of the corresponding estimated model (two control inputs). Substituting Equation (8) into Equation (4) and linearizing the parameters, the designed closed-loop system equation can be obtained:

$$\mathcal{M}(q)\dot{r} + \mathcal{C}(q,\dot{q})r + \mathcal{K}r = \mathcal{Y}(\dot{\Theta} - \Theta) + \mathcal{Z}(\dot{\pi} - \pi). \tag{9}$$

As mentioned above, considering the uncertainty of model parameters, the following design is made:

$$
\hat{\Theta} = \Theta\_0 + \delta\Theta; \ \hat{\pi} = \pi\_0 + \delta\pi,\tag{10}
$$

where *θ*<sup>0</sup> and *π*<sup>0</sup> are the constant vectors of the corresponding parameter vectors (the theoretical calculation values); *δθ* and *δπ* are two design control terms used for compensating

the disturbance caused by uncertainty. For the above uncertainty (the difference between the actual value and the calculated value), it can be expressed as:

$$\|\|\boldsymbol{\Theta}\|\| = \|\boldsymbol{\Theta} - \boldsymbol{\Theta}\_{0}\|\| \le \rho; \; \|\|\boldsymbol{\tilde{\pi}}\|\| = \|\boldsymbol{\pi} - \boldsymbol{\pi}\_{0}\|\leq \sigma,\tag{11}$$

where <sup>θ</sup>) represents the parameter uncertainty of the dynamic model and *<sup>π</sup>*) represents the uncertainty of the link length and the interaction force. Selecting the upper bound constants *σ* and *ρ*. The designs of *δθ* and *δπ* are as follows:

$$\delta\Theta = \begin{cases} -\rho \frac{\mathbf{y}^T r}{\|\mathbf{y}^T r\|} & \|\mathbf{y}^T r\| > \varepsilon \\ -\rho \frac{\mathbf{y}^T r}{\varepsilon} & \|\mathbf{y}^T r\| \le \varepsilon \end{cases} \tag{12}$$

$$\delta \pi = \begin{cases} -\sigma \frac{\mathbf{Z}^T \mathbf{r}}{\|\mathbf{Z}^T \mathbf{r}\|} & \|\mathbf{Z}^T \mathbf{r}\| > \eta \\ -\sigma \frac{\mathbf{Z}^T \mathbf{r}}{\eta} & \|\mathbf{Z}^T \mathbf{r}\| \le \eta \end{cases} \tag{13}$$

where *ε* and *η* are two positive constants used to ensure the continuity of the design term.

In order to analyze the stability of the designed closed-loop system by the Lyapunov second method, the following Lyapunov function is selected:

$$V = \frac{1}{2}r^T \mathcal{M}(q)r + \mathfrak{e}^T \Lambda \mathcal{K} \mathfrak{e}.\tag{14}$$

Taking the derivative of Equation (14) along the system (9):

$$\dot{V} = -\dot{e}^T \mathbf{K} \dot{e} - e^T \boldsymbol{\Lambda}^T \mathbf{K} \mathbf{A} e + r^T \mathbf{Y} (\overline{\theta} + \delta \theta) + r^T \mathbf{Z} (\overline{\pi} + \delta \pi). \tag{15}$$

According to the Lyapunov second method, if a Lyapunov function derivative along the system direction is strictly negative definite, it can be determined that the system is asymptotically stable. No matter what state the system starts from, the error will eventually converge to zero. However, in order to ensure Equation (15) is negative definite, additional constraints need to be found. First, rewrite Equation (15) into the following form:

$$\dot{V} = -A^T Q A + r^T \mathcal{Y} (\tilde{\Theta} + \delta \Theta) + r^T Z (\tilde{\pi} + \delta \pi), \tag{16}$$

where *<sup>A</sup><sup>T</sup>* = [*eT*, . *e T* ], *Q* = diag[**Λ***TK***Λ**, *K*]. Although the first term of Equation (16) can be determined to be semi-negative definite, there are four possible combinations of the last two terms. Since the structures of these two items are similar, the last item is used as an example for analysis. First, when *<sup>Z</sup>T<sup>r</sup>* <sup>&</sup>gt; *<sup>η</sup>*, according to the Cauchy-Schwartz inequality we can obtain:

$$\left(\left(\mathbf{Z}^{\mathrm{T}}\mathbf{r}\right)^{\mathrm{T}}\left(\tilde{\boldsymbol{\pi}} + \delta\boldsymbol{\pi}\right) = \left(\mathbf{Z}^{\mathrm{T}}\mathbf{r}\right)^{\mathrm{T}}\left(\tilde{\boldsymbol{\pi}} - \boldsymbol{\sigma}\frac{\mathbf{Z}^{\mathrm{T}}\mathbf{r}}{\|\mathbf{Z}^{\mathrm{T}}\mathbf{r}\|}\right) \leq \|\mathbf{Z}^{\mathrm{T}}\mathbf{r}\| \left(\|\boldsymbol{\tilde{\pi}}\| - \boldsymbol{\sigma}\right) < 0. \tag{17}$$

When *<sup>Z</sup>Tr*<sup>≤</sup> *<sup>η</sup>*, we can be obtained:

$$\sigma \left( \mathbf{Z}^{\mathrm{T}} \mathbf{r} \right)^{\mathrm{T}} (\tilde{\boldsymbol{\pi}} + \delta \boldsymbol{\pi}) \le \left( \mathbf{Z}^{\mathrm{T}} \mathbf{r} \right)^{\mathrm{T}} (\sigma \frac{\mathbf{Z}^{\mathrm{T}} \mathbf{r}}{\left\| \mathbf{Z}^{\mathrm{T}} \mathbf{r} \right\|} - \sigma \frac{\mathbf{Z}^{\mathrm{T}} \mathbf{r}}{\eta}) = -\frac{\sigma}{\eta} \left\| \mathbf{Z}^{\mathrm{T}} \mathbf{r} \right\|^{2} + \sigma \left\| \mathbf{Z}^{\mathrm{T}} \mathbf{r} \right\|. \tag{18}$$

When the designed item is in the state of Equation (17), the judgment condition is satisfied. When the design item is in the state of Equation (18), Equation (18) can be regarded as a quadratic function about *<sup>Z</sup>T<sup>r</sup>* . Its maximum value *ση*/2 at *<sup>Z</sup>T<sup>r</sup>* <sup>=</sup> *<sup>η</sup>*/2 can be obtained, and then the conditions for guaranteeing the Equation (15) is negative definite can be obtained.

According to the designed terms *δθ* and *δπ*, two maximum values *ση*/2 and *ρε*/2 can be obtained respectively. It is not difficult to find that if *ATQA* is always greater than the sum of these two maximum values, Equation (16) is less than zero forever in all cases (four combinations). In other words, when Equation (19) is satisfied:

$$\mathbf{A}^T \mathbf{Q} \mathbf{A} \succeq (\sigma \eta + \rho \varepsilon)/2. \tag{19}$$

Using the matrix eigenvalue relation *<sup>A</sup>TQA* <sup>≥</sup> *<sup>λ</sup>*min *<sup>A</sup>* <sup>2</sup> (where *<sup>λ</sup>*min is the minimum eigenvalue of the matrix *Q*), the constraints that guarantee Equation (15) is negative definite could be obtained:

$$\|\mathbf{A}\|\,>\left[\left(\sigma\eta+\rho\varepsilon\right)/2\lambda\_{\text{min}}\right]^{1/2}.\tag{20}$$

When Equation (20) is satisfied, . *V* can be guaranteed to be less than zero. Therefore, according to Lyapunov second method, the tracking error of system (9) under the designed control law is uniformly ultimately bounded. That is to say, selecting appropriate coefficients in Equations (12) and (13) can ensure that the error continuously approaches a sufficiently small upper error bound, and a good tracking performance could be obtained.

#### **4. Experiment**

In order to verify the function, feasibility and effectiveness of the proposed lower limb flexion and extension strength training, eight stroke survivors were selected to participate in the test experiment using LLR-II. Every subject confirmed the protocol of the experiment, and research was carried out following the principles of the Declaration of Helsinki. All experiments were conducted under the premise of ensuring the subject's safety, and sufficient time was given to familiarize the subjects with LLR-II before the formal experiment. The training trajectory is a straight line passing through the hip joint and parallel to the ground, and its starting point and length are determined according to the user's leg length and knee joint rotation range. The knee joint angle range of all training trajectories in experiments was consistent. Due to the height difference of subjects, the horizontal position coordinates of the training trajectory are also different. For normalized analysis, the horizontal position in this part is represented by percentage of total track length.

To test the controller performance on guiding users to generate the force, the training interaction force was recorded through the six-dimensional force sensor. In the experiment, each subject was required to maintain higher training speed in three groups of training. Figure 7 shows the changes in knee joint angle *q*2, model stiffness *K*var and effective interaction force *F*x during training. In the experiment, the adjustment constant coefficients *k*<sup>i</sup> (i = 1, 2 ... ) of *K*var are set to 0.3, 0.8, 2.5, 1.5, 6. The average of the end interaction force was calculated, and error bars were plotted based on its standard deviation, as the red line and the orange area shown in the figure. With the stiffness change based on the knee joint angle, the interaction force also displayed a similar trend. Although the strength levels of different subjects were inconsistent, the data results show that the controller has achieved the function of guiding the user to make forces.

To analyze the tracking performance of the dual input robust controller, joint angle data in training were recorded as shown in Figure 8. Observing average error curves, it can be found that the absolute values of each joint steady-state error are close to about 0.5◦. The result shows that the controller has good tracking ability, and it is in line with the final boundedness proved before. Moreover, it can be found that the two joint errors (orange and purple lines) and error bars (yellow and green areas) appear to be fluctuations in the half of the trajectory. The maximum standard deviation of the hip joint is 0.29◦, while the knee joint is 0.16◦. This is due to the rapid force increase when the subject tries to adapt to the model stiffness change. The interaction force influence is different to two joints, but the controller can make adjustments to adapt to different sudden interference. It shows that the designed robust controller has strong robustness.

**Figure 7.** Performance testing of admittance control.

The EMG signal is a physiological indicator that can directly reflect neuromuscular activity [37–39]. This experiment verifies the effectiveness of this strength training by collecting the quadriceps EMG signal during training. The quadriceps femoris is divided into rectus femoris, vastus medialis, vastus lateralis and vastus intermedius. Since the vastus intermedius is located in the deep part of the muscle group, only the EMG signals of the other three muscles were collected in this experiment. EMG device information and electrode patch positions are shown in Figure 9. The positions of the electrode patches are selected under the doctor's guidance.

**Figure 9.** Parameters of EMG device and electrode patch positions.

Figure 10 shows the changes in the EMG signals and terminal interaction force of each muscle during a 10-min training. In order to extract the features of EMG signals, the original data was processed by high and low-pass filtering, absolute value taking and smoothing, respectively. The interaction force collected by the force sensor was also plotted in the figure. It can be found that all the data in the training action area are significantly higher.

**Figure 10.** Changes of EMG and interaction force during the training (Dashed line for value reference).

#### **5. Discussion**

The model stiffness of the admittance controller can be adjusted in real-time according to the knee joint angle during training, thereby avoiding excessive compressive force on the knee joint soft tissue and enhancing the muscle stimulation. Through the result of the controller performance experiment, it can be found that the model stiffness *K*var changes strictly depending on the knee joint angle according to the designed function during training, as the blue line shown in Figure 7. Under the effect of stiffness adjustment, the subject only needs about 90 N to maintain the target training speed when the knee joint flexes more than 90◦. This shows that this controller successfully guides users to avoid making forces in the posture that soft tissue is the main bearing object. Meanwhile, the quadriceps femoris enter the most active area when the knee extends to about 90◦, and all subjects reported that the training speed at this stage was significantly slowed down. This is due to the change in stiffness, which led to the reduction of the model output acceleration. In order to maintain the training speed, the average interaction force of subjects can reach around 200 N. The results above show that designing a variable stiffness admittance model can indirectly control the terminal interaction force, and it can reduce the possibility of joint soft tissue injury and enhance exertion force in the effective training range. Moreover, the designed robust admittance control can ensure joint tracking performance even under the strong influence of interaction force, and it makes the robot meet the task requirements of this active strength training. In experiments, the peak value of the terminal interaction force was basically above 200 N. The EMG peak value of the vastus lateralis muscle was around 150 uV; the peak value of the vastus medialis muscle was around 75 uV; the peak value of the rectus femoris muscle is around 45 uV. The signal performance of these muscles is consistent with the results of related lower body training studies [33], and obvious signal increase means that the quadriceps femoris is in an active state. These prove that the target muscle group has received effective stimulations under this active strength training method.

There are some limitations in this study and they need to be further studied. Firstly, we design a robust admittance controller strategy to guide users to generate the force correctly during training, and it can be regarded as a method of avoiding soft tissue bearing too much stress under the biomechanics theory support. However, we are unable to provide accurate data on the reduction of soft tissue stress or the actual contribution of this method so far. We planned to conduct a controlled experiment, but it may put control group subjects at risk of injury. We believe that we still need to find a non-invasive method for measuring joint stress to provide strong proof for our work. On the other hand, we selected eight stroke survivors for testing under the recommendation of doctors, and we obtained the result that this training provides effective stimulation to the target muscle group through the sEMG information. Obviously, the sample data are not enough to conclude stronger results, and we ignored to study of the intervention of this training to different types of stroke survivors. Although we believe that the training efficiency can be increased (compared to other same type training) by guiding users to generate forces intensively in an efficient range, there is still a lack of clinical trial data that can quantify the rehabilitation effect of this strength training. We have to recognize that the work shown in the paper is still preliminary research, and more testing experiments need to be carried out later.

We think the rehabilitation robot research should not only consider the training effect but ignore the potential hidden dangers, especially for the elderly group of stroke survivors. In the robot-assist rehabilitation field, few researchers have focused on knee joint protection. This research presents a solution as an attempt to this research gap, but its clinical effect needs a long-term follow-up observation. However, this research still proposes a new point to robot-assisted training: potential negative factors should be considered in order to provide better rehabilitation medical devices for the elderly.

#### **6. Conclusions**

In order to avoid an excessive compressive force on the joint soft tissues and increase the stimulation to the target muscle during the leg flexion and extension training, this paper proposes a human–robot cooperative lower limb active strength training based on a robust admittance control strategy. The robust admittance control strategy mainly includes variable stiffness admittance control and dual input robust control. The variation law of admittance model stiffness is designed according to the knee joint biomechanics. The main controller can adjust the stiffness of the model in real-time according to the angle of the knee joint and indirectly control the exertion force of users; the subsidiary admittance control can increase the training flexibility and compliance. Dual input robust control can improve joint tracking performance under the influence of the disturbance caused by the model uncertainty, interactive forces, and external noise. The experiment results show that the designed controller can effectively reduce the possibility of joint soft tissue injury and enhance the stimulation of the quadriceps, and this active training method is effective for exercising the quadriceps. In order to evaluate the efficacy of this strategy, it will be applied to more clinical experiments in future works.

**Author Contributions:** Conceptualization, H.W., J.N. and L.V.; methodology, M.L.; software, C.Y.; formal analysis, W.L.; writing—original draft preparation, M.L.; writing—review and editing, H.W. and J.N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the National Key Research and Development Program of China (2019YFB1312500), the National Natural Science Foundation of China (U1913216), and the Science and Technology (S&T) Program of Hebei under Grant, China (E2020103001, 216Z1803G).

**Institutional Review Board Statement:** The study was conducted according to the guidelines of the Declaration of Helsinki, and approved by Ethics Committee of China Rehabilitation Research Center (protocol code 2020-006-1; date of approval 25 February 2020).

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This research is partially supported by Shanghai Clinical Research Center for Aging and Medicine (19MC1910500).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

