2.1. Anatomy of the knee joint

The knee joint is a very important joint in the body. Not only does it provide support for the body to resist gravity but it also has to generate flexion and extension for many forms of human locomotion.

According to anatomy of knee, the movement in the sagittal plane mainly depends on its agonistic and antagonistic muscle groups. The extension mainly originates from the agonistic muscle group that includes quadriceps femoris muscle group. The flexion arises from the antagonistic muscle group that consists of the hamstrings. The agonist muscle group is approximately three times stronger than the antagonist muscle group due to the continuous contraction to resist gravity and provide negative acceleration for stopping flexion [Reference Hamill and Knutzen23].

However, patients with stroke or spinal injuries often fail to obtain a normal gait due to abnormal knee motion. The knee cannot flex and extend properly, resulting in the affected side not clearing the road well in the swing phase. This could greatly increase the risk of falling to the ground. Therefore, it is important to better assist the knee joint to complete knee flexion and extension movements to help patients walk normally. It is important to reduce the chance of secondary injury in a fall.

In this paper, we proposed a knee exoskeleton with Bowden cable to assist the knee joint based on the form of agonist–antagonist muscle groups acting on human joints. The contraction of the Bowden inner cable relative to the outer sheath simulates the mechanism of muscle generation of motion, and the knee joint is assisted by a lightweight wearable structure without introducing a new additional mechanical joint, as shown in Fig. 1. This design without additional rigid joints will reduce joint constraints and reduce the risk of injury.

Figure 1.

The knee joint exoskeleton robot diving the shank.

2.2. Dynamic model of the control system

The dynamic model always represents the relationship between movement parameters like position, velocity, acceleration, and force parameters, including all kinds of force and torque. Specifically, taking the human body as an example, the movement parameters include not only joint angle but also the displacement of the center of mass (CoM) of shank and foot and their corresponding speed and acceleration. Of course, joint force and torque that come from muscle contractions produce the corresponding movement of the body, like walking, jogging, jumping, and so on. Hence, it is very imperative to generate highly precise motion and path via this kind of dynamic model.

There are several methods to build the dynamic model, such as the Newton –Euler method [Reference Musić, Kamnik, Zanchi and Munih24], Kane method [Reference Yang, Huang, Ye and Han25], Lagrange method [Reference Vundavilli and Pratihar26], and so forth. As for complex multirigid modeling, the Lagrange method has a uniform form and appeals to plenty of researchers.

In this paper, the shank-foot and the worn exoskeleton are assumed to be rigid linkages. Then they all rotated around the knee joints. Because of mimicking and assisting the knee flexion motion, the thigh has been fixed in the vertical direction, as shown in Fig. 1. Then dynamic model has been derived according to the Lagrange method as follows.

According to the Lagrange method and Fig. 1, the CoM position of the shank is described as $(x,y)$

(1) \begin{align}\left\{ \begin{matrix} {x = - l\sin \theta } \\ {y = \;\;l\cos \theta } \end{matrix} \right.\end{align}

where $l$ is the length from origin to CoM of the shank, and $\theta $ means the knee joint angle. Next, taking the derivative of the position regarding time, we got the velocity of CoM:

(2) \begin{align}\left\{ \begin{matrix} \dot x = - l\cos \theta \dot \theta \\ {\dot y = \; - l\sin \theta \dot \theta } \end{matrix} \right.\end{align}

where $\dot \theta $ represents the angular speed. Then kinetic energy of the system $V$ can be described as

(3) \begin{align} V & = {V_m} + {V_r} \nonumber \\[4pt] & = {1 \over 2}M({{\dot x}^2} + {{\dot y}^2}) + {1 \over 2}I{{\dot \theta }^2} \\[4pt] & = {2 \over 3}M{l^2}{{\dot \theta }^2} \nonumber \end{align}

where ${V_m}$ and ${V_r}$ are translational energy and rotational energy, respectively. $M$ is the total mass of shank, $I$ is the moment of inertia of shank, and $I = {1 \over 3}M{l^2}$ .

Considering $\theta = 0$ as the reference posture (the thigh is vertical), the potential energy of the system can be defined as follows:

(4) \begin{align}E = Mgl(1 - \cos (\theta ))\end{align}

Hence, the Lagrange function can be expressed as follows:

(5) \begin{align}L = V - E\end{align}

(6) \begin{align}{d \over {dt}}\left( {{{\partial L} \over {\partial \dot \theta }}} \right) - {{\partial L} \over {\partial \theta }} = \tau \end{align}

(7) \begin{align}{{4M{l^2}} \over 3}\ddot \theta - Mgl\sin (\theta ) = \tau \end{align}

It also can be written as follows:

(8) \begin{align}\ddot \theta - {{3g} \over {4l}}\sin (\theta ) = {3 \over {4M{l^2}}}\tau \end{align}

where $\tau $ represents the whole driving torque, including muscles producing torque ${\tau _{hum}}$ and motors generating torque ${\tau _{ex}}$ .

(9) \begin{align}\tau = {\tau _{hum}} + {\tau _{ex}}\end{align}

2.3. The design of controller

The ideal model of the system was expressed as Eq. (7); however, the actual situation is always more complex. There are other factors that may exert a certain influence on the performance of the system. One of the most important factors among these is the joint viscous damping coefficient $B$ . So, the ideal model can be changed to

(9) \begin{align}{{4M{l^2}} \over 3}\ddot \theta + B\dot \theta - Mgl\sin (\theta ) = \tau \end{align}

The next step was to design the controller to track the reference trajectory. We redefined the variances:

(10) \begin{align}\left\{ \begin{matrix} {{z_1} = \theta } \\ {{z_2} = \dot \theta } \\ {u = \tau } \end{matrix} \right. = {\dot z_1}\end{align}

The goal of the controller was to trace the reference input, so the object of the controller can be described as

(11) \begin{align}\theta \to {\theta _d}\;{\rm{or}}\;{z_1} \to {z_{1d}}\end{align}

where ${\theta _d}$ and ${z_{1d}}$ are the desired variance. Then the traced error and derivative of error can bedefined as

(12) \begin{align}e = {z_{1d}} - {z_1}\end{align}

(13) \begin{align}\dot e = {\dot z_{1d}} - {\dot z_1}\end{align}

Of course, we hope the traced error was closed to zero, which means the reference trajectory is traced precisely. Next, according to the second Lyapunov method, we defined positive definite energy function ${V_1}(e)$ :

(14) \begin{align}{V_1}(e) = {1 \over 2}{e^2}\end{align}

Derivative of ${V_1}(e)$ with respect to time:

(15) \begin{align}{{\dot V}_1}(e) & = e\dot e \nonumber \\ & = e({{\dot z}_{1d}} - {{\dot z}_1}) \nonumber\\ & = e({{\dot z}_{1d}} - {z_2}) \end{align}

In order to get a stable system, it must subject to ${\dot V_1}(e)$ < 0, merely negative definite. So, let:

(16) \begin{align}{\dot z_{1d}} - {z_2} = - {k_1}e,\;\;\forall {k_1} \gt 0\end{align}

where ${k_1}$ is the control parameter, to get desired ${z_{2d}}$ , let

(17) \begin{align}{z_{2d}} = {\dot z_{1d}} + {k_1}e\end{align}

The system can maintain stable and the error is enough small. Similarly, we seek a new goal:

(18) \begin{align}{z_2} \to {z_{2d}}\end{align}

And the new error was defined as

(19) \begin{align}\delta = {z_{2d}} - {z_2}\end{align}

(20) \begin{align}\dot \delta = {\dot z_{2d}} - {\dot z_2}\end{align}

Eq. (9) (17)(10) substituted into Eq. (20):

(21) \begin{align}\dot \delta & = {{\dot z}_{1d}} + {k_1}e + {{3B} \over {4M{l^2}}}{z_2} - {{3g} \over {4l}}\sin ({z_1}) - {3 \over {4M{l^2}}}u \nonumber\\ &= {{\ddot z}_{1d}} + {k_1}({{\dot z}_{1d}} - {z_2}) + {{3B} \over {4M{l^2}}}{z_2} - {{3g} \over {4l}}\sin ({z_1}) - {3 \over {4M{l^2}}}u \end{align}

To let error close to zero, we redefined the positive definite energy function ${V_2}(e,\delta )$ :

(22) \begin{align}{V_2}(e,\delta ) = {V_1}(e) + {1 \over 2}{\delta ^2}\end{align}

Derivative of ${V_2}(e,\delta )$ with respect to time:

(23) \begin{align} {{\dot V}_2}(e,\delta ) & = {{\dot V}_1}(e) + \delta \dot \delta \nonumber\\ & = e({{\dot z}_{1d}} - ({z_{2d}} - \delta )) + \delta \dot \delta \nonumber\\ & = e({{\dot z}_{1d}} - {z_{2d}}) + \delta (e + \dot \delta ) \nonumber \\ & = - {k_1}{e^2} + \delta (e + \dot \delta ) \end{align}

When ${\dot V_2}(e,\delta )$ < 0 at balance, the system shows asymptotic stability. And this can make sure that the error approaches zero. Only need Eq. (23) subject to:

(24) \begin{align}e + \dot \delta = - {k_2}\delta ,\;\;\forall {k_2} \gt 0\end{align}

Let Eq. (24) substitute into Eq. (21):

(25) \begin{align}e & + {{\ddot z}_{1d}} + {k_1}({{\dot z}_{1d}} - {z_2}) + {{3B} \over {4M{l^2}}}{z_2} \nonumber \\ &- {{3g} \over {4l}}\sin ({z_1}) - {3 \over {4M{l^2}}}u = - {k_2}\delta \end{align}

Eq. (12)(19)(20) substituted Eq. (25):

(26) \begin{align} u & = {{4M{l^2}} \over 3}(e + {{\ddot z}_{1d}} + {k_1}({{\dot z}_{1d}} - {z_2}) + {{3B} \over {4M{l^2}}}{z_2}\nonumber \\ & \quad - {{3g} \over {4l}}\sin ({z_1}) + {k_2}\delta ) \nonumber\\ & = {{4M{l^2}} \over 3}({z_{1d}} - {z_1} + {{\ddot z}_{1d}} + {k_1}({{\dot z}_{1d}} - {z_2}) + {{3B} \over {4M{l^2}}}{z_2} \nonumber \\ & \quad - {{3g} \over {4l}}\sin ({z_1}) + {k_2}(({{\dot z}_{1d}} + {k_1}({z_{1d}} - {z_1})) - {z_2})) \end{align}

Finally, we replace the variance with angular parameters:

(27) \begin{align}\tau & = {{4M{l^2}} \over 3}({\theta _d} - \theta + {{\ddot \theta }_d} + {k_1}({{\dot \theta }_d} - \dot \theta ) + {{3B} \over {4M{l^2}}}\dot \nonumber \\ & \quad - {{3g} \over {4l}}\sin (\theta ) + {k_2}(({{\dot \theta }_d} + {k_1}({\theta _d} - \theta )) - \dot \theta )) \end{align}

where ${\theta _d}$ , ${\dot \theta _d}$ , and ${\ddot \theta _d}$ are the reference angular position, angular speed, and angular acceleration, respectively. The control torque can be represented as Fig. 2.

Figure 2.

The control scheme of the knee exoskeleton.

2.4. Output torque of the actuator

The extern torque comes from the motor acting on the knee joints by Bowden cable. The sum torque was the combined result of bilateral agonist–antagonist tension. So, the cooperation control of the two motors is the key factor to generator desired torque acting on the joints.

As shown in Fig. 1. We recorded the initial angle between the thigh L ₁ and shank L ₂ as ${\varphi _0}$ when the $\theta $ is zero, according to the definition of torque:

(28) \begin{align}{\tau _{ex}} = {F_2} \cdot H - {F_1} \cdot R\end{align}

where ${F_1}$ and ${F_2}$ are the tensions generated by two motors driving Bowden cables contraction, respectively. $H$ and $R$ are their responding force arm. $R$ is the knee joint’s equivalent radius and can be measured in anthropometry. $H$ can be calculated as

(29) \begin{align}H = {{{L_1}{L_2}\sin ({\varphi _0} - \theta )} \over {\sqrt {L_1^2 + L_{\rm{2}}^2 - 2{L_1}{L_2}\cos ({\varphi _0} - \theta )} }}\end{align}

where ${L_1}$ and ${L_2}$ are structural parameters shown in Fig. 1.

The shank-foot movements are always a combination of work from joint muscles and exoskeleton robots. While there is some special condition that the subject in the passive condition. Their muscles cannot offer any force to drive their body to finish the desired movement, just like early stroke patients in complete myasthenia ( ${\tau _{hum}} = 0$ ) or in the passive rehabilitation mode. Considering these situations, the external force and torque can help them achieve their goal. And the relationship of ${F_1}$ , ${F_2}$ and their force arms determines what movement can be achieved:

1. 1) ${\tau _{ex}} \gt 0$ , knee flexion;

2. 2) ${\tau _{ex}} \lt 0$ , knee extension;

3. 3) ${\tau _{ex}} = 0$ , maintaining the posture.

When exerting force to the Bowden cable, the two motors connected to the Bowden cable are designed in consistent with knee joint movement. The active cable tensor can be calculated according to Eq. (28) and wanted joints torque ${\tau _{ex}}$ .

3.1. Parameter tuned

Before simulation and experiments, some parameters concerned with the human body need to be determined. According to related researches about biomechanics [Reference Winter27–Reference Robertson, Caldwell, Hamill, Kamen and Whittlesey29], almost all body parameters, say a mass of body segments, length, and so forth, are subjected to certain functions with their whole body mass and height.

According to anthropometric data, the mass of the leg (not including the tight) and foot is 6.1% of the whole-body mass. And the mass is at 60.6% of the leg and foot’s length (Femoral condyles are the proximal end). The shank and foot total weight can be supposed as 8 kg, and the length as 0.5 m. The damper of the joint is set to a series value from 0.1 to 0.8.

As for the real experiments, it is essential to measure the real values before the experiments start. For the sake of uniformity, we set the simulation parameters and the real experiment parameters to be the same. In all, the setting of all the parameters is shown in Table I.

Table I.

Parameter setting.

3.2. Simulation

Simulation is always a highly effective approach to verify the validity of some algorithm, or control system. Here, we simulate the control system to trace two different desired trajectories in the MATLAB 2019b. One test trajectory is a continuous sinusoidal signal with a frequency of 0.5 Hz. And the other is a step-wise signal with 4 s. The general PID method was also be tested as a comparison.

There is an obvious improvement when the control system follows the continuously changing joint angle (shown in Fig. 3 a). The trace error quickly approaches zero within a second. However, the traditional PID method is slower, and there is some error finally within test time, as shown in Fig. 3 (b). As for the step-wise reference path, though two kinds of methods exist some error because of the sharply changing (shown in Fig. 4 (a). They all can follow the path roughly, as shown in Fig. 4 (a).

Figure 3.

(a) Is the results of the control system tracing the sinusoid curve, there are two method results that follow the desired path: one is our method, and the other one is the traditional PID method. They all have a nice performance; (b) is the tracing error, with the viewpoint of error, our method has the better following capability.

Figure 4.

(a) Is the results of the control system tracing the step-wise curve, there are two method results that follow the desired path: one is our method, and the other one is the traditional PID method. They all have a nice performance; (b) is the tracing error, with the viewpoint of error, both of them have some error when following the desired step-wise curve.

3.3. Experimental test bench

To validate the feasibility of the proposed control scheme, a test bench that includes a wearable knee joint exoskeleton (as shown in Fig. 5), controller PC, and sensor subsystem have been built. The exoskeleton has been worn on a customized mannequin.

Figure 5.

The prototype of knee exoskeleton in the side view and the front view.

3.3.1. Driving source and sensor

Consider the knee joint’s normal rotation movement (including flexion and extension). In our application, high response speed and precision of motor were selected to meet the requirements of knee joint rotation. Two 180W DC servo motors (EC-i 52, Maxon, Switzerland) for each joint were used to drag the shank movement by Bowden cable. Corresponding parameters in detail about the motor are shown in Table III.

To output the proper torque that exerts on the knee joints, it is indispensable to connect a load cell at the end of an inner steel cable directly. Considering the control need, different force sensors have a crucial influence on system performance. Here the LCM300 (FUTEK, 0∼1500N, USA) load cell was selected, with IAA200 (FUTEK, USA) current transmitter. The total weight of the sensor is 300 g, which is very suitable for a lightweight exoskeleton.

The knee joint angle needs to be monitored when the exoskeleton extern force and torque to the joints. While there is no fixed axis for the exoskeleton to install encoders. The indirect measure approach is more workable for our application. Hence, two inertial measurement units (IMU, LPMS-CURS2, LP-RESEARCH, Japan) have been adopted to sense the posture of the thigh and shank, then by calculating related angle to get the knee joint angle, all the IMUs communicated via the Controller Area Network (CAN) with 1M baud rate.

The control unit is an industrial personal computer (IPC, Beckhoff German) with the EtherCAT communication protocol. The maximum control instruction cycle is 1 ms, which could meet system control demand.

Table II.

Key parameters of the Bowden cable.

Table III.

Parameters of the servo motor.