2.1. Ergonomic design

From the perspective of ergonomics, knee joint movements include rotation, extension/flexion, and adduction/abduction, as shown in Fig. 2a. In the process of human walking, the main action is sagittal extension/flexion of the knee joint. Rotation and adduction/abduction degrees of freedom are mainly used to ensure the coordination, balance, and efficiency of human walking. Therefore, SR-KR will actively provide torque for extension/flexion degree of freedom. The other degrees of freedom will passively cooperate with the knee joint through the SR-KR soft actuating part. The purpose of this design is to reduce the complexity and weight of SR-KR, while ensuring its safety and comfort.

In order to further determine the functional requirements of SR-KR, literature survey was conducted on the gait of knee joint. The relationship between knee joint bending angle (extension/flexion) and required torque under human gait range is shown in Fig. 2b [Reference Sridar, Nguyen, Zhu, Lam and Polygerinos25, Reference Kadaba, Ramakrishnan, Wootten, Gainey, Gorton and Cochran31]. It can be noted that in a human gait cycle, the maximum bending angle of the knee joint is approximately 56.7°, and the maximum torque required is approximately 22 Nm. This result is also consistent with the researches of Takenagura et al. [Reference Nagura, Dyrby, Alexander and Andriacchi32] and Nordez et al. [Reference Nordez, Casari and Cornu33]. Therefore, we get two functional requirements of SR-KR:

1. 1. SR-KR can continuously carry out more than 56.7° load bending motion.

2. 2. The output torque of SR-KR should be greater than 22 Nm.

2.2. Structure design

The overall structure of SR-KR is shown in Fig. 3, which is mainly composed of a novel soft-rigid bidirectional curl actuator, a thigh clamping structure, and a crus clamping structure (Fig. 3e). They are fixed by screws and nuts.

Figure 3. Structure design of the SR-KR. (a) The wearing concept. (b) Appearance of SR-KR. (c) Bending deformation of SR-KR. (d) Components of air chamber. (e) Components of SR-KR. (f) Structure of the novel soft-rigid bidirectional curl actuator.

The design concept of the soft-rigid bidirectional curl actuator is pressure driving, airbag expansion, fabric limiting, and bending deformation. The main body of the actuator is two symmetrically distributed soft air chambers. Each air chamber is composed of a soft airbag, a central limiting layer, and an external limiting layer. These three parts are tightly pressed and fixed by screws and nuts at both ends of the chamber through matching conical fasteners (Fig. 3f).

The soft airbag is made of silicone material (ELASTOSILM 4601 A/B), and its main task is expanding under pressure. The section shape of soft airbag is semi-elliptical, which is convenient for axial bending deformation. The central limiting layer is made of non-tensile material, polyvinylchloride-coated polyester-woven fabric to limit the deformation of the soft airbag’s plane side. The external limiting layer is made of rubber-blended polyester-knitted fabric, which can be only one-way deformed. The external limiting layer limits the radial deformation of the soft airbag’s curved side. These two limiting layers are formed by laser cutting. They are stitched at edges and combined to tightly wrap the soft airbag, so that the soft airbag can only undergo axial bending deformation. The conical fastener is made of aluminium alloy materials.

The design of conical fasteners removes the boundary lines that are easily damaged in traditional air chambers. Therefore, SR-KR has the advantage of high working pressure. Experiment shows that the actuator can withstand exceeding 400 kpa pressure. The conical fasteners are equipped with air pipe connector and pressure sensor interface to facilitate subsequent control work.

The thigh and crus clamping structure consist of carbon fiber guide rod, leg fixed mechanism, leg strap, and attitude sensor (Fig. 3e). The leg fixed mechanism is made of PLA materials by 3D printing and is fixed on the carbon fiber guide rod by screws and nuts. The leg fixed mechanism can move the installation position along the guide rod. Therefore, SR-KR can match people with different statures. The leg strap is connected to the leg fixed mechanism and can be adjusted by velcro. The attitude sensor is fixed on the leg fixed mechanism to detect the bending angle of SR-KR. The thigh and crus clamping structure is the wearing part of SR-KR, which is responsible for fixing SR-KR on human body.

When users need to use SR-KR to complete passive training, they can inflate the SR-KR air chamber near the front of their body (chamber 1 in Fig. 3a), while the other air chamber does not inflate or bleed. At this time, the SR-KR will generate the torque to help the users complete the knee joint bending motion. When users need to use SR-KR for impedance training, they can inflate the SR-KR air chamber near the back of their body (chamber 2 in Fig. 3a), while the other air chamber does not inflate. So, SR-KR will generate torque in the opposite direction of knee joint movement to help users complete knee joint impedance training.

The design idea of the SR-KR’s structure is soft-rigid integration. SR-KR’s bending deformation part contacted to knee is completely made of soft materials, which aims to ensure the good axis adaptivity and wearing safety. The force transmission part of SR-KR are rigid structures, which aims to ensure the efficient transmission of output torque.

2.3. Parameter design

The simplified lower limb connecting rod diagram is shown in Fig. 4a. The four connecting rods from top to bottom represent the waist, thigh, crus, and foot, respectively. The hollow circle represents the joint of each part. $m_{w}$ , $m_{t}$ , and $m_{c}$ represent the centroids of each part. $d_{w}$ , $d_{t}$ , and $d_{c}$ represent the length from the centroid of the relative part to the previous joint.

The size of SR-KR should be determined according to each user’s actual lower limb size. Considering the wearing space of SR-KR, the whole length of SR-KR $L_{\text{whole}}$ should be longer than the distance between the centroids of thigh and crus and shorter than the total length of thigh and calf as (1). The length of SR-KR’s soft part $l_{\text{soft}}$ should cover the whole knee $l_{\text{knee}}$ but less than the length from the thigh centroid to the crus centroid as (2). The adaptable length of the thigh clamping structure $L_{\mathrm{ad}\_ \text{thigh}}$ should be longer than the length from the centroid of the thigh to the hip joint, and less than the thigh length as (3). Similarly, the adaptable length of the crus clamping structure $L_{\mathrm{ad}\_ \text{crus}}$ should be longer than the length from the centroid of the crus to the ankle joint and less than the crus length as (4). Considering of wearing comfort, the maximum thickness of SR-KR $w_{d}$ should be less than the thickness of the user’s knee joint as (5).

(1) \begin{equation} L_{t}-d_{t}+d_{c}\leq L_{\text{whole}}\leq L_{t}+L_{c} \end{equation}

(2) \begin{equation} l_{\text{knee}}\lt l_{\text{soft}}\leq L_{t}-d_{t}+d_{c} \end{equation}

(3) \begin{equation} d_{t}\leq L_{\mathrm{a}{\mathrm{d}_{\text{thigh}}}}\leq L_{t} \end{equation}

(4) \begin{equation} {L_{c}}-d_{c}\leq L_{\mathrm{a}{\mathrm{d}_{\text{crus}}}}\leq L_{c} \end{equation}

(5) \begin{equation} w_{d}\leq w_{\text{knee}} \end{equation}

In this paper, we choose a 24-year-old male as the prototype user of SR-KR. This man is 178 cm tall and weighs 71 kg. Through the above analysis, we get the prototype parameters, as shown in Table II. The wearing diagram of SR-KR prototype is shown in Fig. 4c.

Table II. Prototype parameters.

Figure 4. Parameter design of SR-KR. (a) The simplified lower limb connecting rod diagram. (b) Parameters of the SR-KR. (c) Wearing diagram of SR-KR.

3.1. Bending model of the actuator

When the air chamber at one side is inflated, the actuator will bend, as shown in Fig. 5a. The cross-section of the air chamber at this time is shown in Fig. 5b, which includes the central limiting layer, external limiting layer, soft airbag, and air chamber cavity.

Figure 5. Inflation bending diagram of SR-KR. (a) SR-KR’s bending front view. (b) Section A-A of the SR-KR in (a).

Because of the material properties, the central limiting layer is not subject to tensile deformation, so it can be regarded as the bending neutral layer. The external limiting layer can only stretch and deform in radial direction, which can restrain the expansion of the soft airbag in axial direction. Therefore, it can be approximately considered that the sectional shape of the soft air chamber remains unchanged during the bending process. So, we can establish the YOZ coordinate system as shown in Fig. 5b. The origin O of the coordinate system is the midpoint of the neutral layer, and the Z axis direction is parallel to the neutral layer.

When the actuator is inflated and bent, the moment on its section includes the external limiting layer moment $M_{e}$ , the soft airbag moment $M_{s}$ , the pressure moment $M_{p}$ , and the possible output torque $T$ . We can list the mechanical equation of the relative neutral layer when the bending is balanced, as shown in (6).

(6) \begin{equation} M_{e}+M_{s}-M_{p}+T=0 \end{equation}

Setting the sectional area of the external limiting layer, soft airbag, and air chamber cavity as $A_{e}$ , $A_{s}$ , and $A_{a}$ , we can derive each item in (6) separately, as shown in (7)–(9).

(7) \begin{equation} M_{e}=\iint _{{A_{e}}}\sigma _{e}\left(\varepsilon \right)ydA \end{equation}

(8) \begin{equation} M_{s}=\iint _{{A_{s}}}\sigma _{s}\left(\varepsilon \right)ydA \end{equation}

(9) \begin{equation} M_{p}=\iint _{{A_{a}}}PydA \end{equation}

where $\sigma _{e}(\varepsilon )$ , $\sigma _{s}(\varepsilon )$ , and $P$ are the axial tensile stress of the external limiting layer, the axial tensile stress of the soft airbag, and the working pressure of the actuator respectively.

Since the pressure in the air chamber cavity is consistent everywhere, we can simplify the pressure term as (10).

(10) \begin{equation} \iint _{{A_{a}}}PydA=\iint _{{A_{a}}}ydA\cdot P=S_{c}P \end{equation}

where, $S_{c}$ is the static moment of the air chamber cavity.

The axial tensile stress of the external limiting layer and the soft airbag can be expressed as functions of their axial strain, $\varepsilon$ . For soft materials, the stress-strain relationship is usually nonlinear. In this paper, we select cubic function to fit the relationship, as shown in (11) and (12).

(11) \begin{equation} \sigma _{e}\left(\varepsilon \right)=E_{e0}\varepsilon +E_{e1}\varepsilon ^{2}+E_{e2}\varepsilon ^{3} \end{equation}

(12) \begin{equation} \sigma _{s}\left(\varepsilon \right)=E_{s0}\varepsilon +E_{s1}\varepsilon ^{2}+E_{s2}\varepsilon ^{3} \end{equation}

$E_{e}^{(i)}$ and $E_{s}^{(i)}$ ( $i=0,1,2$ ) are obtained through experiments and listed in Table III. $\varepsilon$ is the axial strain of the soft airbag and the external limiting layer at each point of their cross-section. $\varepsilon$ can be expressed by the bending curvature at the cross-section $\kappa$ and the distance between the point and the neutral layer $y$ , as shown in (13). Furthermore, since the cross-section of the soft chamber is same everywhere, it can be considered that the curvature $\kappa$ of each cross-section is const.

(13) \begin{equation} \varepsilon =\kappa y \end{equation}

By substituting (7)–(13) into (6), we can get the relationship between the bending curvature, working pressure, and output torque of SR-KR, as shown in (14) and (15), where $C_{i}$ are consts determined by material properties and section shape.

(14) \begin{equation} \left(C_{0}\kappa +C_{1}\kappa ^{2}+C_{2}\kappa ^{3}\right)-S_{c}P+T=0 \end{equation}

(15) \begin{equation} C_{i}=E_{ei}\iint _{{A_{e}}}y^{i+2}dA+E_{si}\iint _{{A_{s}}}y^{i+2}dA \end{equation}

Therefore, we can get the conclusions as below:

1. 1. When the bending curvature of SR-KR is constant, the relationship between the output torque and the working pressure is linear. The proportional coefficient is the static moment of the air chamber cavity, which is only related to the section parameters of the air chamber.

2. 2. When the working pressure is constant, the output torque of SR-KR is polynomial with its bending curvature. Polynomial coefficients are determined by section parameters and material properties.

3. 3. When the output torque is constant, the working pressure is polynomial with its bending curvature. The polynomial coefficient is $C_{i}/S_{c}$ , ( $i=0,1,2$ ).

Table III. Material parameters of the SR-KR.

3.2. Stiffness analysis of the actuator

The stiffness of a structure represents its ability to maintain shape under external loads [Reference Yang, Chen, Li, Wang and Li34]. In this paper, the actuator stiffness $K$ is defined as the partial derivative of the actuator’s bearing moment to its bending curvature, as shown in (16). The actuator’s bearing moment means the moment acting on it, which is the opposite number of the actuator’s output torque.

(16) \begin{equation} K=\frac{-\partial T}{\partial \kappa }=C_{0}+C_{1}\kappa +C_{2}\kappa ^{2} \end{equation}

From the (16), we can know:

1. 1. The stiffness of the actuator is a quadratic function of its bending curvature. The stiffness of the actuator decreases first and then increases with the increase of curvature. This helps us to better use SR-KR.

2. 2. When the bending curvature is constant, the stiffness of the actuator is only related to the material properties and the cross-section shape. So, the stiffness of actuator can be changed by changing these two terms.

4.1. Double closed loop servo control system

In order to control and apply SR-KR more conveniently and safely, a double closed loop servo control system is built as shown in Fig. 6. This system consists of attitude servo and pressure servo. The two servos are independent of each other and work independently through their own feedback signals to achieve accurate control of SR-KR. The double closed loop structure of the system ensures the stability and safety of its operation, because every output will be double tested.

Figure 6. Control system of SR-KR. (a) The double closed loop servo control system. (b) Pressure servo. (c) Composition of the system. (d) Integration of control system and SR-KR.

The pressure information of SR-KR is collected by the pressure sensor (XGZP6847500KPG; CFSensor) which can be equipped with the pressure sensor interface of SR-KR. The attitude information is obtained through the cooperation of the attitude sensors (WT901C TTL 9 Axis IMU Sensor; WitMotion) on the thigh and crus clamping structures.

Apart from these two sensors, the system also includes control board (Arduino Mega 2560), voltage proportional valve (SMC ITV0010 ITV0030 ITV0050-0/1/2/3BCLNS), PWM to voltage module, small air pump (ARP12DC24; 8bar; 17L/min; TIDE SMART TECHNOLOGY (SHANGHAI) CO., LTD, China), and lithium battery (YSN-37044800, YISENNENG). All the above components are integrated into a 42 cm × 32 cm × 12 cm backpack. The signal transmission relationship between the components is shown in Fig. 6c.

4.2. Attitude servo

For this paper, the attitude also means the bending angle $\alpha$ of SR-KR (Fig. 7). The attitude sensor chip selected is MPU9250, which can output quaternion, $q_{0}$ , $q_{1}$ , $q_{2}$ , and $q_{3}$ after being calibrated by ellipsoid fitting method [Reference Tedaldi, Pretto and Menegatti35], as shown in (17). Through the operation between quaternions, we can obtain the spatial attitude of MPU9250. In this paper, by mirroring the placement of two MPU9250s at both ends (Fig. 8), the bending angle of SR-KR is the sum of the two sensors’ rolling angle ( $\varphi _{1}$ , $\varphi _{2}$ ). Thus, the bending angle of SR-KR $\alpha$ can be obtained, as shown in (18) and (19).

(17) \begin{equation} q=q_{0}+q_{1}i+q_{2}j+q_{3}k \end{equation}

(18) \begin{equation} \varphi =\arctan \frac{2\left(q_{0}q_{1}+q_{2}q_{3}\right)}{1-2\left({q_{1}}^{2}+{q_{2}}^{2}\right)} \end{equation}

(19) \begin{equation} \alpha =\varphi _{1}+\varphi _{2} \end{equation}

The attitude error is obtained from the difference between the input attitude preset signal $\alpha _{d}$ and the attitude feedback signal $\alpha _{F}$ .The error is processed by proportional gain $K_{{P\alpha _{i}}}$ and integral gain $K_{{I\alpha _{i}}}$ to obtain the preset pressure value. The proportional gain is added to speed up the response of the system, while the integral gain coefficient is added to eliminate the steady-state error and make the system more stable and accurate.

Figure 7. Attitude change of MPU9250s during SR-KR bending.

Figure 8. Mechanical property test of SR-KR. (a) Free bending test. (b) Motion attitude of SR-KR at the corresponding points. (c) Constant bending angle test. (d) Torque test platform. (e) Constant working pressure test. (f) Relational surface for bending angle, working pressure, and output torque for SR-KR.

4.3. Pressure servo

In the pressure servo (Fig. 6b), the control of SR-KR’s left and right air chambers is independent. First, when the preset pressure signal $P_{\alpha }$ is input, the pressure servo will filter it as shown in (20) to ensure that the pressure is distributed to the correct chamber. Then, the error is obtained by calculating the difference between the preset pressure signal ( $P_{\alpha Li}$ , $P_{\alpha Ri}$ ) and the pressure feedback signal ( $P_{\alpha FL}$ , $P_{\alpha FR}$ ). Next, the error will be amplified by the proportional gain ( $K_{P{P_{i}}}$ ) to obtain the left and right voltage proportional valve signal (0-10V). Finally, the voltage proportional valve releases the corresponding pressure to actuate SR-KR.

(20) \begin{equation} \begin{cases} P_{\alpha Li}=P_{\alpha },\text{ if }P_{\alpha }\geq 0\\[5pt] P_{\alpha Ri}=P_{\alpha },\text{ if }P_{\alpha }\lt 0 \end{cases} \end{equation}

Since the control board cannot directly output 0–10 V voltage signal, PWM wave signal is selected as the control signal of pressure intensity. The PWM signal is transmitted to the voltage proportional valve through the PWM to voltage module. At the fixed frequency, the output pressure of the proportional valve has a linear relationship with the duty cycle (21).

(21) \begin{equation} P=\beta P_{0} \end{equation}

where $P$ is the output pressure of the proportional valve, $\beta$ is the duty cycle of PWM wave, and $P_{0}$ is the input pressure of the proportional valve.

For the safe use of SR-KR, the pressure limiting value is set in the pressure servo. When the pressure feedback signal ( $P_{\alpha FL}$ , $P_{\alpha FR}$ ) is greater than 400 kPa, the voltage proportional valve signal will be multiplied by 0, that is, it will stop working.

5.1. Free bending test

In order to evaluate the bending performance of SR-KR, free bending test was carried out first. Fixing one end of SR-KR and inflating the air chamber at one side, we can get the results as shown in Fig. 8a and b. The maximum bending angle of SR-KR is 92.5°. The minimum working pressure reaching the maximum bending angle is 140 kPa. The first derivative of bending angle increases first and then decreases with the increase of working pressure. This phenomenon is in accordance with the derivation in (14).

From the free bending test, it can be seen that SR-KR meets the bending angle requirements mentioned in Section 2.

5.2. Mechanical property test

The mechanical property test is based on the torque test platform shown in Fig. 8d. This test is divided into two groups: constant bending angle and constant working pressure. According to the pre-inflation test, the safe working pressure range of SR-KR is 0–400 kPa. This test is performed within this pressure range.

Constant bending angle: The results are shown in Fig. 8c. With constant bending angle, the output torque of SR-KR is approximately linear with the working pressure, which is consistent with the derivation of (14). The proportion coefficient will decrease slightly with the increase of the bending angle. This phenomenon may be caused by slight changes in the cross-section of SR-KR at different bending angles.

Constant working pressure: The results are shown in Fig. 8e. The output torque and bending angle of the SR-KR are approximately in a negative linear relationship, and the proportion coefficient almost does not change with the working pressure.

Combining these two sets of experimental data, the relational surface for bending angle, working pressure, and output torque of the SR-KR can be obtained, as shown in Fig. 8f. It can be noted that SR-KR can provide larger output torque at smaller bending angle and higher working pressure.

From the relational surface, we can see that within the bending angle range in human gait range (≤56.7°), SR-KR can provide more than 26.3 Nm torque for user’s knee joint with the working pressure in 0 ∼ 400 kPa. In addition, SR-KR can provide at most 34.1 Nm torque with 400 kPa working pressure.

Therefore, SR-KR meets the two functional requirements in Section 2 and has ability to assist human walking movements.

It should be noted that the parameters of the prototype are not the upper limit of SR-KR performance. Based on the modeling analysis and parameter design, we know that the mechanical properties of SR-KR can be further improved by increasing the sectional area of soft chamber and changing fabrication materials. For burly users, the size of the SR-KR can be enlarged to obtain greater output torque.

5.3. Trajectory-driven test

In order to verify the controllability and stability of the control system, trajectory-driven test was conducted. Considering the impact of knee joint injury on walking speed, this paper selects the 0.2 m/s gait trajectory as the test trajectory. Of course, users can adjust the speed or change to another motion trajectories according to their own needs, such as simple bending-stretching motion.

The test result is shown in Fig. 9. During the test, the maximum error of the whole process is 3.7°, which occurs when the bending angle tends to the regional extreme value. The average error is 1.18°. The maximum proportional error is 20.59%, and the average proportional error is 6.19%. Macroscopically, there is a delay of about 0.1 s between the actual and ideal motion. The response speed of the system is the main reason for the error.

Figure 9. Track driven test. (a) Comparison of actual and ideal trajectories of SR-KR. (b) Tracking error in test.

On the whole, the overall motion trajectory of SR-KR is basically consistent with the ideal trajectory, though there are some errors. This result also shows that SR-KR has good controllability and stability under the control system above.

5.4. Lower limb wearing test

In order to verify the wearing effect of SR-KR, lower limb wearing test was conducted. Two surface electromyography (sEMG) sensors (SEN0240, DFROBOT, china) are placed at the vastus lateralis (VL) and vastus medial (VM). The subject walked with or without wearing SR-KR. The results of the middle four gait cycles in the test are shown in Fig. 10.

Figure 10. Lower limb wearing test. (a) VL test result. (b) VM test result.

It can be noted that after wearing SR-KR, the sEMG signal intensity at the VL and VM of the subject significantly decreased. The average sEMG signal intensity at VL decreased by 53.2%, and the peak value decreased by 41.9%. The average sEMG signal intensity at VM decreased by 49.1%, and the peak value decreased by 57.4%. Subject feels that the SR-KR suits well with the human lower limb and can provide objective torque support for the knee joint.

This indicates that wearing SR-KR can reduce the muscle burden during walking, demonstrating the effectiveness of SR-KR wearing. Furthermore, SR-KR has potential application value in the field of rehabilitation and human enhancement.