2.1. Mapping of knee joint and exoskeleton

The knee joint is not only a very special way of movement, but also its joint force is variable stiffness. During the movement, the knee joint will change the joint stiffness of the knee joint through the interaction of muscles, connective tissue, and bones, so as to meet the needs of the knee joint stiffness of the human lower limbs in different motion states [Reference Li, Yuan, Tang, Mao, Zhang, Huang and Tan22, Reference Walker, Kurosawa, Rovick and Zimmerman23].

In the process of walking, according to the function of the lower limb contact with the ground, a gait cycle is divided into eight stages, which are initial landing (IC), load-bearing reaction period (LR), middle support period (MSt), end support period (TSt), early swing period (PSw), early swing period (ISw), middle swing period (MSw), and end swing period (TSw). During the load-bearing reaction period and the middle of the support phase in a gait cycle, in order to alleviate the impact force of the lower limb landing, the knee joint will have a process of first flexion and then extension. In this process, the knee joint produces a large joint torque to maintain the stability of the knee joint at this time, and the internal pressure of the knee joint also reaches a peak during this stage. Joint angle change, joint torque change, and intra-articular pressure change of the knee joint in a gait cycle are shown in Fig. 1 (a), (b), and (c), respectively.

Figure 1. Knee joint movement and force. (a) Changes of knee joint angle during gait cycle. (b) Changes of knee joint torque during gait cycle. (c) Intra-articular pressure changes of knee joint during gait cycle.

In the realm of wearable robotics, the exoskeleton is conceptualized as a unified system shared between the human body and exoskeleton itself. Within this system, the human serves as a cognitive decision-maker and motion planner, while the exoskeleton serves to assist, compensate, or replace human motion. In relation to the knee joint’s particular traits, the exoskeleton must first possess the ability to respond passively to transition while adapting to the knee joint’s movements, thus mitigating and obstructions present during lower limb movement. Secondly, the exoskeleton must exhibit the same variable stiffness characteristics as the knee joint; in doing so, the stiffness changes of the knee joint can be more efficiently matched, ultimately creating optimal assistance. Lastly, the exoskeleton must possess the ability to switch the assist torque, catering to distinct requirements of varied gait moments.

2.2. The configuration of knee joint assisted exoskeleton

To fulfill the design [Reference Chen, Xu, Yang, Yang, Hu, Chai and Wang24] demands originating from the physiological attributes of the knee joint, a novel passive assisted exoskeleton is proposed to assist the human–machine interaction of the knee joint. The exoskeleton comprises three primary components, including an eight-bar mechanism, a cam mechanism, and a switching device. The exoskeleton diagram and wearing diagram are shown in Fig. 2, while the structure diagram of the exoskeleton is depicted in Fig. 3.

Figure 2. Exoskeleton structure diagram and wearing diagram.

Figure 3. Brief diagram of exoskeleton mechanism.

The eight-bar mechanism is composed of slider components A, B, C, D and rod components AB, CD, EF, which is mainly responsible for the drive of the passive instantaneous center of the exoskeleton. In this mechanism, the rod EF is bound to the human calf and is connected to the rod CD and the rod AB through the revolute pairs E and F, respectively. The instantaneous center points of rod AB and rod CD are located at point M and point N, respectively. The instantaneous center P of rod EF is the intersection of instantaneous center lines NE and MF. The exoskeleton has a single degree of freedom, which meets the motion requirements of the human knee joint.

The cam mechanism is mainly composed of a cam and a roller, which mainly realizes the passive variable stiffness of the exoskeleton. The switching device includes a catapult rod, a push rod, a limit rod, a ratchet, a pawl, a Bowden line, and a steering wheel, which realizes the switching of the assist moment and meets the movement needs of the human body at different gait moments.

Figure 4. Changes of exoskeleton joint stiffness during gait cycle.

The switching device employs the number of knee extensions as the switching condition in order to regulate the exoskeleton’s assistance phase. This ensures that the exoskeleton provides support during the support phase without interfering with the movement of the knee joint in the swing phase. The workflow of the exoskeleton across the gait cycle (GC) is illustrated in Fig. 4, which is primarily divided into four stages:

1. 1. Knee extension, leg ready to touch the ground (GC: 0–5%). The exoskeleton middle rod EF collides with the push rod, and the ejector rod moves up to make the ratchet pawl in the exoskeleton contact, preparing for the cam pull-down spring when the knee joint is bent and the exoskeleton assists.

2. 2. Knee flexion, human leg to resist the impact of the ground (GC: 5–31%). The cam begins to pull down the spring, and the exoskeleton provides a nonlinear torque to assist. At this time, the exoskeleton joint presents a certain nonlinear stiffness change.

3. 3. Knee extension, human legs support the upper body (GC: 31–50%). The rod EF collides with the push rod again, and the push rod moves down to separate the ratchet and pawl in the exoskeleton. When the knee joint bends again, the exoskeleton will no longer assist, thus not affecting the normal swing of the human leg.

4. 4. Knee flexion, foot off the ground (GC: 50–100%). The cam movement will no longer deform the spring, and the exoskeleton joint stiffness is 0. When the swing phase ends, the knee joint is straightened again and enters the next cycle.

2.3. The working principle of exoskeleton

The rod EF of the eight-bar mechanism is bound with the human calf. As the shank drives the movement of the rod EF, it, in turn, propels the movement of the rods AB and CD. By manipulating the position of the instantaneous center points M and N, the location of the instantaneous center lines NE and MF shifts accordingly, subsequently altering the instantaneous rotation center point P of the exoskeleton.

The cam mechanism realizes the nonlinear change of exoskeleton joint stiffness during the support phase. As the leg rotates, the cam connected with the rod EF follows its downward movement. At this time, the cam will drive the spring to stretch through the pull-down roller. By designing the contour of the cam, the deformation ΔX of the spring varies nonlinearly. The joint torque of the exoskeleton is calculated as follows:

(1) \begin{equation} M=F_{T}{\Delta} l=\left(k{\Delta} x\right){\Delta} l \end{equation}

where M is the exoskeleton joint torque, F _T is spring elasticity, Δl is the distance between the elastic force and the center of rotation, k is spring stiffness, and Δx is a spring shape variable.

In accordance with Eq. (1), as the deformation Δx of the spring varies nonlinearly, the joint torque M of the exoskeleton will also change nonlinearly. As a result, the stiffness of the exoskeleton joint undergoes nonlinear changes during the support phase.

The working principle of the exoskeleton is mainly divided into two stages, namely, the energy storage stage and the energy discharge stage. In the energy storage stage, the exoskeleton moves with the knee joint when the knee joint bends. At this time, the cam mechanism in the exoskeleton stretches the spring to generate nonlinear torque to assist the knee joint movement. While assisting the knee joint to resist the buckling effect caused by the ground reaction force, the exoskeleton converts the kinetic energy in this stage into elastic potential energy and stores it. In the energy discharge stage, when the knee joint ends flexion and begins to return to the straight state, the exoskeleton releases the energy just stored to help the knee joint extend, thus helping the knee joint in this stage. The energy storage and discharge stages of the exoskeleton are shown in Fig. 5.

Figure 5. Changes of exoskeleton joint stiffness during gait cycle.

Figure 6. Kinematic analysis of mechanism.

3.1. Kinematic model of knee exoskeleton

To investigate the consistency of human–machine motion, kinematic modeling of the eight-bar mechanism is carried out to analyze its instantaneous center trajectory, as shown in Fig. 6. The exoskeleton’s thigh rod serves as a fixed frame. The coordinate system O-XY is a plane coordinate system established with the intersection of the trajectory extension lines of sliders A and B as the origin O. The trajectory extension lines of sliders C and D intersect at points (x ₀, y ₀), whereas the instantaneous center lines of sliders A and B intersect at the point M, and those of sliders C and D intersect at the point N. The instantaneous rotation center of the exoskeleton is the instantaneous center of the rod EF when the exoskeleton is worn. The instantaneous center of the rod EF is the intersection point P of the instantaneous center lines MF and NE, and the coordinate of the instantaneous center point P is (x _p , y _p ). The position of the sliders A, B, C, and D is described by the position of the rotation center of the revolute pair. The positions of the sliders A, B, C, and D are (x ₁, y ₁), (x ₂, y ₂), (x ₃, y ₃), (x ₄, y ₄), respectively. The distance between the sliders A and B and the coordinate origin is l ₁, l ₂, respectively. The distance between the sliders C and D and the point (x ₀, y ₀) is l ₃, l ₄, respectively. Let the angle between the sliders A, B, C, and D and the positive direction of the X axis be θ ₁, θ ₂, θ ₃, θ ₄, respectively, and the length of the rod AB, CD, and EF be l _AB , l _CD , l _EF , respectively. The relationship between points E and F on the rods AB and CD is expressed as l _AF = e ₁ l _FB , l _CE = e ₂ l _ED , with coordinates for points E and F being (x ₅, y ₅) and (x ₆, y ₆), respectively. The instantaneous center points M and N of the rods AB and CD are described by coordinates (x ₇, y ₇) and (x ₈, y ₈), respectively. When the knee joint is straightened, the angle between the rod EF and the vertical direction is θ ₀. The angle of rotation of the rod EF relative to the vertical direction is θ.

In this coordinate system, the coordinates of sliders A and B are expressed as follows, respectively:

(2) \begin{equation} \left\{\begin{array}{l} y_{i}=l_{i}\cos \theta _{i}\\[5pt] x_{i}=l_{i}\sin \theta _{i} \end{array}\right. \end{equation}

where y _i and x _i represent the coordinates of sliders A and B, l _i represents the distance between sliders A and B and the origin of coordinates, and θ _i represents the positive angle between sliders A and B and X axis, where i = 1, 2.

In this coordinate system, the coordinates of the sliders C and D are shown as follows, respectively:

(3) \begin{equation} \left\{\begin{array}{l} y_{j}=y_{0}+l_{j}\cos \theta _{j}\\[5pt] x_{j}=x_{0}+l_{j}\sin \theta _{j} \end{array}\right. \end{equation}

where y _j and x _j represent the coordinates of C and D, l _j denotes the distance between the C and D distance points (x ₀, y ₀) of the slider, and θ _j denotes the angle between the sliders C and D and the positive axis, where j = 3, 4.

According to the positional relationship l _AF = e ₁ l _FB and l _CE = e ₂ l _ED of points E and F on the rod CD and rod AB, the coordinates of points E and F are obtained as

(4) \begin{equation} \left\{\begin{array}{l} y_{{i_{1}}}=\dfrac{y_{{n_{1}}}+e_{i}y_{{n_{2}}}}{1+e_{i}}\\[12pt] x_{{i_{1}}}=\dfrac{x_{{n_{1}}}+e_{i}x_{{n_{2}}}}{1+e_{i}} \end{array}\right. \end{equation}

where y _i1 and x _i1 represent the coordinates of points E and F, y _n1 and x _n1 represent the table slider A and C coordinates, and y _n2 and x _n2 represent the coordinates of sliders B and D, where i ₁ = 5, 6, n ₁ = 1, 3, n ₂ = 2, 4.

According to the angle θ + θ ₀ between the pole EF and the vertical direction, the following relationship between points E and F can be deduced:

(5) \begin{equation} \left\{\begin{array}{l} x_{6}-x_{5}=l_{EF}\sin\!\left(\theta +\theta _{0}\right)\\[5pt] y_{6}-y_{5}=l_{EF}\cos\!\left(\theta +\theta _{0}\right) \end{array}\right. \end{equation}

where l _EF is the length of the rod EF, θ ₀ is the angle between the EF and the vertical direction when the leg is straight, and θ is the angle of the rod EF relative to the vertical direction.

According to the relations in Eqs. (3) and (4), the relations between l ₃, l ₄ and l ₁, l ₂ are obtained, respectively:

(6) \begin{equation} \left\{\begin{array}{l} l_{3}=\dfrac{\left(e_{2}+1\right)\left[\left(e_{1}+1\right)l_{EF}\cos\!\left(\theta +\theta _{2}\right)+l_{1}\sin\!\left(\theta -\theta _{2}\right)\right]}{\left(e_{1}+1\right)\sin\!\left(\theta _{1}-\theta _{2}\right)}\\[12pt] l_{4}=\dfrac{-\left(e_{2}+1\right)\left[\left(e_{1}+1\right)l_{EF}\cos\!\left(\theta +\theta _{1}\right)+e_{1}l_{2}\sin\!\left(\theta _{1}-\theta _{2}\right)\right]}{e_{2}\!\left(e_{1}+1\right)\sin\!\left(\theta _{2}-\theta _{1}\right)} \end{array}\right. \end{equation}

Through the instantaneous center line perpendicular to the slider trajectory, the coordinates of the instantaneous center points M and N of the rods AB and CD satisfy the following relationship:

(7) \begin{equation} \left\{\begin{array}{l} \dfrac{y_{i2}-y_{\mathrm{n}1}}{x_{i2}-x_{n1}}=-\dfrac{1}{\tan \theta _{{n_{1}}}}\\[12pt] \dfrac{y_{i2}-y_{n3}}{x_{i2}-x_{n3}}=-\dfrac{1}{\tan \theta _{{n_{2}}}} \end{array}\right. \end{equation}

where y _i2 and x _i2 represent the coordinates of the instantaneous center points M and N and y _n3 and x _n3 represent the coordinates of the recalculated slider C and slider D, where i ₂ = 7, 8; n ₃ = 3, 4.

According to Eq. (8), the coordinates of the instantaneous center points M and N are expressed as

(8) \begin{equation} \left\{\begin{array}{l} y_{i2}=\dfrac{x_{n1}-x_{n3}+y_{n1}\cdot \tan \theta _{n1}-y_{n3}\cdot \tan \theta _{n2}}{\tan \theta _{n1}-\tan \theta _{n2}}\\[12pt] x_{i2}=\dfrac{x_{n3}\cdot \tan \theta _{n1}-x_{n1}\cdot \tan \theta _{n2}+\left(y_{n3}-y_{n1}\right)\cdot \tan \theta _{n1}\cdot \tan \theta _{n2}}{\tan \theta _{n1}-\tan \theta _{n2}} \end{array}\right. \end{equation}

Assuming that the instantaneous center lines FM and NE pass through the instantaneous rotation center of the knee joint (taking the average x = 15, y = 117),

(9) \begin{equation} \frac{y_{i2}-117}{x_{i2}-15}=\frac{y_{i1}-117}{x_{i1}-15} \end{equation}

According to Eq. (10), the relationship between l ₁ and l ₂ can be obtained (see in Appendix A):

(10) \begin{equation} l_{1}=\frac{\sqrt{N_{11}l_{2}^{2}+N_{12}l_{2}+N_{13}}+N_{21}l_{2}+N_{22}}{N_{31}} \end{equation}

(11) \begin{equation} l_{2}=\frac{M_{41}-\sqrt{M_{11}l_{2}^{2}+M_{12}l_{2}-M_{13}}+M_{12}l_{2}+M_{22}}{M_{31}} \end{equation}

Based on the relationship between the instantaneous rotation center lines FM and NE and the instantaneous rotation center point P, the relationship between the instantaneous center point P and the instantaneous center points M and N can be obtained:

(12) \begin{equation} \left\{\begin{array}{l} \dfrac{y_{7}-y_{P}}{x_{7}-x_{P}}=\dfrac{y_{6}-y_{P}}{x_{6}-x_{P}}\\[12pt] \dfrac{y_{8}-y_{P}}{x_{8}-x_{P}}=\dfrac{y_{5}-y_{P}}{x_{5}-x_{P}} \end{array}\right. \end{equation}

where (x _p , y _p ) is the coordinate of the instantaneous rotation center P of the exoskeleton.

The coordinates of points M and N are substituted into Eq. (13) to obtain the instantaneous rotation center P of the exoskeleton:

(13) \begin{equation} \left\{\begin{array}{l} x_{P}=\dfrac{y_{5}y_{6}\left(x_{8}-x_{5}-x_{7}\right)+y_{7}y_{8}\left(x_{5}-y_{5}+x_{7}-x_{6}\right)}{\left(y_{6}-y_{7}\right)^{2}\left(y_{5}-y_{8}-x_{5}+x_{8}\right)}+\dfrac{\left(x_{7}y_{6}^{2}-y_{7}y_{6}x_{6}\right)\left(y_{5}-y_{8}-x_{5}+x_{8}\right)}{\left(y_{6}-y_{7}\right)^{2}\left(y_{5}-y_{8}-x_{5}+x_{8}\right)}\\[12pt] \qquad +\,\dfrac{+\left(x_{7}y_{7}y_{6}-y_{7}^{2}x_{6}\right)\left(y_{8}-y_{5}+x_{5}+x_{8}\right)}{\left(y_{6}-y_{7}\right)^{2}\left(y_{5}-y_{8}-x_{5}+x_{8}\right)}\\[11pt] y_{P}=\dfrac{y_{5}y_{6}\left(x_{8}-x_{5}-x_{7}\right)+y_{7}y_{8}\left(x_{5}-y_{5}+x_{7}-x_{6}\right)}{\left(x_{6}-x_{7}\right)\left(y_{5}-y_{8}\right)-\left(x_{5}-x_{8}\right)\left(y_{6}-y_{7}\right)} \end{array}\right. \end{equation}

3.2. Human–machine coupling kinematic model of knee exoskeleton

To facilitate the analysis of the intricate movement of the lower limbs when wearing the exoskeleton, a simplification is performed whereby the exoskeleton and lower limbs are treated as two separate entities connected by a thigh rod and a calf rod through a specialized motion joint. The thigh portion of the human–machine is fixed in place, while the connection of the calf part is visualized as a PS branch chain. The simplified human–machine lower limb model and the human–machine calf space single closed chain with one degree of freedom are shown in Fig. 7.

Figure 7. Simplified lower limb model after wearing exoskeleton and single closed chain of human–machine calf space.

The kinematics of the spatial single closed chain comprised of human–machine calf rods is described as follows: The initial position coordinate system of the exoskeleton is denoted as O _p -x _p y _p z _p . The initial position coordinate system of human body is represented by O _o -x _o y _o z _o . The rotation center coordinate system after the rotation of the calf is labeled by O ₁-x ₁ y ₁ z ₁. The slider position coordinate system is designated as O ₂-x ₂ y ₂ z ₂, while the coordinate system at the connection between the ball pair and the exoskeleton bar is O ₃-x ₃ y ₃ z ₃. Finally, the coordinate system of the rotation center position after the exoskeleton rotates is registered as O ₄-x ₄ y ₄ z ₄. The kinematic description of spatial single closed connection is shown in Fig. 8.

Figure 8. Kinematic description of single closed chain in man–machine calf space.

After the exoskeleton rotates $\beta$ degree, the transformation matrix for the coordinate O ₄-x ₄ y ₄ z ₄, coupled with the rotation center of the exoskeleton, compared to the coordinate O _p -x _p y _p z _p aligned with the initial point of the exoskeleton, is given as follows:

(14) \begin{equation} \begin{array}{l} {}^{p}_{4}{\boldsymbol{T}}{}={}^{p}_{0}{\boldsymbol{T}}{}^{0}_{1}{\boldsymbol{T}}{}^{1}_{2}{\boldsymbol{T}}{}^{2}_{3}{\boldsymbol{T}}{}^{3}_{4}{\boldsymbol{T}}{}\\[5pt] =\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \mathrm{c}\alpha \cdot \mathrm{c}\theta -\mathrm{s}\alpha \cdot \mathrm{s}\theta & -\mathrm{c}\alpha \cdot \mathrm{s}\theta -\mathrm{s}\alpha \cdot \mathrm{c}\theta & 0 & -\mathrm{c}\alpha \cdot \mathrm{s}\theta \cdot l_{e}-\mathrm{s}\alpha \cdot \mathrm{c}\theta \cdot l_{e}+l_{m}\cdot \mathrm{s}\alpha +x_{\alpha }\\[5pt] \mathrm{s}\alpha \cdot \mathrm{c}\theta +\mathrm{c}\alpha \cdot \mathrm{s}\theta & -\mathrm{s}\alpha \cdot \mathrm{s}\theta +\mathrm{c}\alpha \cdot \mathrm{c}\theta & 0 & -\mathrm{s}\alpha \cdot \mathrm{s}\theta \cdot l_{e}+\mathrm{c}\alpha \cdot \mathrm{c}\theta \cdot l_{e}-l_{m}\cdot \mathrm{c}\alpha +y_{\alpha }\\[5pt] 0 & 0 & 1 & 0\\[5pt] 0 & 0 & 0 & 1 \end{array}\right) \end{array} \end{equation}

With the exoskeleton’s rotation by $\beta$ degree, the rotation angle of the exoskeleton leg with respect to its initial point can be determined by combining it with Eq. (16), resulting in

(15) \begin{equation} \left\{\begin{array}{l} \cos \beta =\cos \alpha \cos \theta -\sin \alpha \sin \theta \\[5pt] \sin \beta =\sin \alpha \cos \theta +\cos \alpha \sin \theta \end{array}\right. \end{equation}

where $\beta$ is the rotation angle of the exoskeleton calf rod.

According to Eq. (17), the relationship between the rotation angle $\beta$ of the exoskeleton calf and the rotation angle $\alpha$ of the human calf is

(16) \begin{equation} \beta =\alpha +\theta \end{equation}

Table I. Exoskeleton size parameters.

When the exoskeleton rotates $\beta$ , according to the position of the rotation center of the exoskeleton at this time, combined with Eq. (16), it can be seen that

(17) \begin{equation} \left\{\begin{array}{l} x_{\beta }=-l_{e}\sin \beta +l_{m}\sin \alpha +x_{\alpha }\\[5pt] y_{\beta }=l_{e}\cos \beta -l_{m}\cos \alpha +y_{\alpha } \end{array}\right. \end{equation}

where ( $x_{\beta}$ , $y_{\beta}$ ) is the coordinate of the exoskeleton rotation center when the exoskeleton rotates $\beta$ .

According to Eq. (17), the distance l _m of the rotation center of the knee joint after the slider rotates $\alpha$ relative to the human knee joint is shown as

(18) \begin{equation} \left\{\begin{array}{l} l_{m}=\dfrac{x_{\beta }+l_{e}\sin \beta -x_{\alpha }}{\sin \alpha }\\[12pt] l_{m}=\dfrac{-y_{\beta }+l_{e}\cos \beta +y_{\alpha }}{\cos \alpha } \end{array}\right. \end{equation}

The relationship between the exoskeleton rotation angle $\beta$ and the human knee joint rotation angle $\alpha$ is obtained by eliminating l _m in Eq. (19):

(19) \begin{equation} x_{\beta }+l_{e}\cdot \sin \beta =\left(-y_{\beta }+l_{e}\cdot \cos \beta \right)\cdot \tan \alpha +y_{\alpha }\cdot \tan \alpha +x_{\alpha } \end{equation}

Assuming the human knee joint is not rotated, the distance between the exoskeleton and the center of rotation of the knee joint is considered to be the length l _e from the exoskeleton to the center of rotation. Wearing the exoskeleton, the deviation at the leg binding is displaced as

(20) \begin{equation} z=l_{e}-l_{m} \end{equation}

where z is the dislocation at the leg binding.

Based on the correlation between the rotation angle $\beta$ of the exoskeleton and the rotation angle $\alpha$ of the human knee joint, as outlined in Eq. (18), if one of these angles is known, the other can be derived. Utilizing Eqs. (18) and (20), the position deviation and rotation angle between the exoskeleton shank and the human shank can be obtained.

In order to verify the correctness of the man–machine kinematic model, the simulation analysis is carried out. The exoskeleton size parameters are shown in Table I.

According to the size parameters in Table I, the instantaneous center trajectory of the exoskeleton under this size is calculated as shown in Fig. 9, and the degree of deviation dislocation at the human–machine binding is obtained by simulation as shown in Fig. 10. The simulation and theoretical results of the degree of rotation dislocation are shown in Fig. 11.

Figure 9. Human–machine instantaneous center trajectory.

Figure 10. Theoretical results and simulation results of sliding degree.

Figure 11. Rotation degree theoretical results and simulation results.

According to Figs. 9 and 10, it can be seen that the simulation of the human–machine kinematic model is basically the same as the theoretical trend. The human–machine kinematic model proposed in this paper can reflect the degree of deviation and rotation at the binding between human and machine to a certain extent.

4.1. Dynamic model of knee exoskeleton

During the LR and MSt phases of human gait, the thighs in the lower limbs remain relatively immobile, while the calves revolve around the knee joint in response to the ground reaction force. As a result, the lower limbs, after wearing the exoskeleton, are simplified, where the human–machine thigh components become consolidated, while the human–machine calf components act in concert to counteract the impact force imposed on the ground. The interplay between angle relationship and force relationship of the human–machine lower limb is depicted in Fig. 12.

Figure 12. The angle of each part during knee flexion.

The dynamic equation [Reference Chen, Peng, Wang, Tu, Hu, Wang, Cheng and Zhu25] of the exoskeleton calf rod is written by Newton–Euler method:

(21) \begin{equation} \left\{\begin{array}{l} m_{e}a_{n}=-F_{T}\cos \beta +m_{e}g\cos\!\left(\beta -\theta _{t}\right)+f_{d}\cos \theta -f'_{\!\!ex}\\[5pt] m_{e}a_{t}=-F_{T}\sin \beta +m_{e}g\sin\!\left(\beta -\theta _{t}\right)+f_{d}\sin \theta -f'_{\!\!ey}\\[5pt] J_{e}\ddot{\theta }_{e}=F_{T}l_{T}\sin \beta +m_{e}gl_{me}\sin\!\left(\theta _{t}-\beta \right)+f_{d}l_{e}\sin \theta -f'_{\!\!ey}l_{e} \end{array}\right. \end{equation}

where m _e is the mass of the exoskeleton shank. F _T is the elastic force generated by the spring in the exoskeleton. θ _t is the angle between the thigh and the vertical direction. $f'_{\!\!ex}$ is the projection of the human–computer interaction force in the direction perpendicular to the calf rod during the human–machine interaction process. $f'_{\!\!ey}$ is the projection of human–machine interaction force along the direction of the calf rod in the process of human–machine interaction. J _e is the rotational inertia of the exoskeleton calf rod. $\ddot{\theta }_{e}$ is the angular acceleration of the exoskeleton calf rod. l _T is the projection of the distance from the roller center to the exoskeleton rotation center along the direction of the calf rod. l _me is the projection of the distance from the center of mass of the exoskeleton calf rod to the center of rotation of the exoskeleton along the direction of the calf rod.

The motion trajectory of the roller is selected as

(22) \begin{equation} s=a\sin\!\left(b\beta \right)+c\cos\!\left(d\beta \right)+e \end{equation}

where s is the trajectory of the roller and a, b, c, d, and e are parameter variables, respectively.

The spring elastic force generated by the exoskeleton pulling spring is given as

(23) \begin{equation} F_{T}=k_{0}\cdot \bigtriangleup l \end{equation}

where k ₀ is the stiffness coefficient of the spring and Δl is the compression of the spring, where Δl = s.

The force produced by the deformation of human leg can be calculated according to the Voith element simulating viscoelasticity:

(24) \begin{equation} f_{d}=kd+\dot{d}b \end{equation}

where k is the stiffness of human skin, d is the displacement of the binding, $\dot{d}$ for displacement velocity, and b is the friction force at the man–machine binding.

An auxiliary line is established linking the center of the roller in the exoskeleton, the midpoint of the rod EF, and the rotation center of the exoskeleton, as illustrated in Fig. 13.

Figure 13. Auxiliary line in exoskeleton.

Based on the geometric relationship illustrated in Fig. 13, the following conditions can be obtained:

(25) \begin{equation} \varepsilon =\arctan\!\left(\frac{y_{lt}}{x_{lt}}\right) \end{equation}

(26) \begin{equation} \gamma =\varepsilon -\frac{\pi }{2}+\beta \end{equation}

(27) \begin{equation} \eta =\arctan\!\left(\frac{x_{lt}}{y_{lt}}\right) \end{equation}

(28) \begin{equation} \delta =\pi -\gamma -\eta \end{equation}

According to the sine theorem, the distance from the roller center to the exoskeleton rotation center in the direction of the exoskeleton calf rod is expressed as

(29) \begin{equation} l_{T}=\frac{\sin \eta }{\sin \delta }\sqrt{x_{lt}^{2}+y_{lt}^{2}} \end{equation}

Assuming that the exoskeleton shank is a homogeneous rod, the length of the rod is shown as follows:

(30) \begin{equation} l_{a}=l_{e}+\cos \gamma \cdot \sqrt{x_{lt}^{2}+y_{lt}^{2}} \end{equation}

where l _a is the length of the exoskeleton calf rod.

The distance from the center of mass of the exoskeleton shank to the center of rotation is given as follows:

(31) \begin{equation} l_{me}=\frac{l_{a}}{2}-l_{e} \end{equation}

According to Eqs. (25), (26), (31), and (33), the other unknowns in the exoskeleton dynamic Eq. (23) have been obtained. So far, the human–machine interaction force at the binding can be obtained through Eq. (23).

The dynamic equation of human calf is written by Newton–Euler method:

(32) \begin{equation} \left\{\begin{array}{l} m_{h}a_{N}=m_{h}g\sin\!\left(\alpha -\theta _{t}\right)+f_{ey}+f_{ky}\\[5pt] m_{h}a_{T}=m_{h}g\cos\!\left(\alpha -\theta _{t}\right)+f_{ex}+f_{kx}+f_{d}\\[5pt] J_{h}\ddot{\theta }_{h}=M_{G}-M_{h}-m_{h}gl_{h}\sin\!\left(\theta _{t}-\alpha \right)/2-f_{ey}l_{m} \end{array}\right. \end{equation}

where m _h is the mass of human calf. f _ex is the projection of human–machine interaction force perpendicular to the calf direction in the process of human–machine interaction. f _ey is the projection of human–machine interaction force along the calf direction in the process of human–machine interaction. J _h is the moment of inertia of the human calf. $\ddot{\theta }_{h}$ is the angular acceleration of the knee joint. f _kx is the force along the x direction at the knee joint. f _ky is the force along the y direction at the knee joint. l _h is the leg length. M _G is the ground reaction force to produce knee flexion moment. M _h is the torque of the knee joint to resist the flexion effect generated by the ground reaction force.

4.2. Static model of human body

When the knee joint muscle force resists the flexion effect caused by the ground reaction force, the muscle force drives the femur and tibia closer together, thereby increasing the internal pressure within the joints. To mitigate this effect, the exoskeleton assists the knee joint to resist the ground reaction force by applying human–machine interaction force to the binding site, reducing the muscle force and intra-articular pressure of the knee joint. To analyze the changes in pressure within the knee joint during this process, a static analysis is performed on the knee joint while under a certain instantaneous state, which establishes a relationship between the pressure within the knee joint and the muscle force. The static model of the knee joint is shown in Fig. 14.

Figure 14. Static model of knee joint.

According to Eq. (33), the joint torque M _h of the knee joint is calculated. By consulting the relevant medical literature, the force of the quadriceps femoris of the knee joint applied to the tibia to the force arm l _PT of the instantaneous rotation center of the knee joint is obtained. According to the torque formula, the force applied to the tibia by the quadriceps of the knee joint is shown as follows:

(33) \begin{equation} f_{PT}=\frac{M_{h}}{l_{PT}} \end{equation}

where f _PT is the force exerted on the tibia by the quadriceps and l _PT is the torque applied to the tibia by the quadriceps to the instantaneous rotation center of the knee joint.

During a certain moment in the weight-bearing reaction period, the knee joint maintains the balance for the lower limbs of the human body through the combined influence of ground reaction force, knee joint force, and knee joint pressure. Applying the principles of polygon force distribution, the internal pressure within the knee joint can be obtained as

(34) \begin{equation} f_{TF}=\sqrt{f_{PT}^{2}+f_{D}^{2}-2f_{PT}f_{D}\cos\!\left(\pi -\alpha -\tau \right)} \end{equation}

where f _TF is the force exerted by the femur on the tibia, f _PT is the force applied to the tibia by the quadriceps femoris, f _D is the ground reaction force, and $\tau$ is the angle between the ground force and the vertical direction.

The angle $\tau$ representing the deviation between the ground force and the vertical direction remains relatively stable throughout the entirety of the weight-bearing reaction period, thereby displaying a static discrepancy. When the knee joint angle α undergoes alterations, the internal pressure of the knee joint f _TF can be determined based on the internal pressure relationship described in Eq. (34), especially when the knee joint rotates at different angles α.

The exoskeleton size parameters and spring stiffness under the condition of high fitting degree of human joint torque are selected as simulation parameters. The joint torque and intra-articular pressure of knee joint under the action of exoskeleton are simulated and analyzed. The exoskeleton size parameters are shown in Table II.

Table II. Exoskeleton size parameters and spring stiffness.

Under this size parameter and spring stiffness, the fitting degree of the knee joint torque is shown in Fig. 15. The human–machine interaction force is shown in Fig. 16. The theoretical results and simulation results of the joint torque are shown in Fig. 17. The theoretical results and simulation results of the intra-articular pressure are shown in Fig. 18.

Figure 15. Torque of human knee joint.

Figure 16. Human–machine interaction force.

Figure 17. Theoretical results and simulation results of joint torque.

Figure 18. Theoretical and simulation results of knee joint internal pressure.

Through the simulation analysis of joint torque and intra-articular pressure, it can be seen that the human body mechanic model is basically correct. The human body dynamic model and static model in the human body mechanic model can predict the changes of knee joint torque and intra-articular pressure.

5.1. Human–machine interaction performance optimization

To achieve greater consistency in human and machine, it is essential to enhance the assist effect of exoskeleton. Drawing upon the human–machine kinematic model and human mechanic model previously developed, a genetic algorithm (GA) is employed to meticulously optimize the size parameters and spring stiffness of the exoskeleton. This optimization endeavor aims to achieve optimal movement and assist performance of exoskeleton when worn by the user.

(1) Design variable

The design variables are independent variables in the optimization process. As described in Eqs. (21), (22), and (33), there are 18 parameters that significantly impact the human–machine kinematic model and dynamic model. Thus, these parameters are selected as the design variables in the exoskeleton optimization model to ensure the optimal performance of the exoskeleton.

(35) \begin{align} \boldsymbol{x} & =\left(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8},x_{9},x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{16},x_{17},x_{18}\right)^{\mathrm{T}}\nonumber\\[5pt] & =\left(l_{AB},l_{CD},l_{EF},\theta _{A},\theta _{B},\theta _{C},\theta _{D},\theta _{0},x_{01},y_{01},e_{1},e_{2},a,b,c,d,e,k_{0}\right)^{\mathrm{T}} \end{align}

(2) Objective function

The higher the degree of human–machine interaction after wearing the exoskeleton, the smaller the obstacle to the movement of the calf, thus ensure the comfort and safety of wearing the exoskeleton. Additionally, a smaller joint torque of the knee joint helps to improve the assist performance provided by the exoskeleton to the knee joint. Hence, it is crucial to construct an objective function that reflects the optimization degree of both objectives. This objective function ensures the optimal human–machine interaction motion and exoskeleton assist performance after wearing the exoskeleton.

To minimize the sliding distance and relative rotation angle of the exoskeleton along the lower leg, an objective function that reflects the degree of human–machine interaction motion is established. This objection function utilizes the ideal coordinates of the instantaneous center of the human knee joint every 10° based on the knee flexion angle.

(36) \begin{equation} S_{1}=\sum _{i=1}^{13}\frac{\sqrt{Z^{2}+\beta ^{2}}}{2} \end{equation}

where S ₁ is the objective function of instantaneous center trajectory optimization, Z is the sliding distance of the exoskeleton binding along the leg, and $\beta$ is the relative rotation angle between human and machine.

To minimize the joint torque and intra-articular pressure of the knee joint simultaneously, an objective function S2 is established, reflecting the assist performance of the exoskeleton. The torque of the human knee joint is taken every 1% based on the gait cycle

(37) \begin{equation} S_{2}=\sum _{i=1}^{11}\frac{\sqrt{M_{h}^{2}+f_{PT}^{2}}}{2} \end{equation}

where S ₂ is the objective function of joint stiffness optimization, M _h is the knee joint torque, and f _PT is the pressure in the knee joint.

Using the objective functions S ₁ and S ₂, an overall objective function S is constructed that can reflect the fitting degree of the two objective functions:

(38) \begin{equation} S=\sqrt{\left(\frac{S_{1}}{13}\right)^{2}+\left(\frac{S_{2}}{11}\right)^{2}} \end{equation}

where S is the objective function that reflects the degree of human–machine coupling motion and exoskeleton assist performance.

(3) Constraint condition

The movement of the exoskeleton is mainly realized by an eight-bar mechanism. The composition and motion conditions of the eight-bar mechanism are expressed as

(39) \begin{equation} \left\{\begin{array}{l} 10\leq \theta _{1}-\theta _{2}\leq 80\\[5pt] 10\leq \theta _{3}-\theta _{4}\leq 80 \end{array}\right. \end{equation}

The exoskeleton size parameters are constrained according to the Chinese adult human body size parameters, and the exoskeleton spring stiffness is selected according to the knee joint torque [26, Reference Jiang27]. The exoskeleton parameter constraints are shown in Table III.

Table III. Constraint conditions of exoskeleton parameters.

The GA toolbox in the MATLAB software optimization toolbox is used to optimize the size parameters and spring stiffness of the exoskeleton. The optimized exoskeleton size parameters and spring stiffness are shown in Table IV.

Table IV. The optimized parameters of exoskeleton.

According to the optimization results, the instantaneous center trajectory and joint torque of the exoskeleton are calculated and compared with the knee joint. Human–machine instantaneous center trajectory and joint torque are shown in Figs. 19 and 20, respectively.

Figure 19. Comparison of human–machine instantaneous center trajectory.

Figure 20. Comparison of exoskeleton joint torque and knee joint torque.

From Figs. 19 and 20, it is evident that the optimized exoskeleton is designed to precisely follow the movement of the human knee joint during use. The joint torque of the optimized exoskeleton also matches the trend of human knee joint torque. However, the actual power torque of the exoskeleton should be greater than the knee joint torque.

Based on the results of the optimization, the deviation and rotation angle of the human–machine binding can be evaluated and visualized, as shown in Figs. 21 and 22, respectively.

Figure 21. The deviation degree of human–machine binding.

Figure 22. Rotation angle of human–machine binding.

From Figs. 19, 21, and 22, it is evident that the optimized exoskeleton exhibits reduced strain on the knee joint and effectively aligns with the natural movement of the knee joint. Consequently, the optimized exoskeleton demonstrates superior characteristics in terms of human–machine interaction.

According to the exoskeleton joint torque shown in Fig. 20, the human–machine interaction force and knee joint dynamics at the binding are obtained, as shown in Figs. 23 and 24, respectively.

Figure 23. Human–machine interaction force at the binding.

Figure 24. Knee joint torque after assist.

By comparing Figs. 24 and 20, it is evident that despite the optimized exoskeleton joint torque surpassing the knee joint torque, the knee joint torque experiences a substantial reduction due to the influence of human–machine interaction force. This observation implies that the exoskeleton effectively provides assistance to the human knee joint.

5.2. Kinematic and dynamic simulation analysis of exoskeleton

The three-dimensional model of the knee joint exoskeleton is established in SolidWorks software, and the structural parameters are set. Then the model is imported into ADAMS to construct a dynamic simulation model for simulation analysis, as shown in Fig. 25.

Figure 25. The exoskeleton model imported into ADAMS.

In order to estimate the instantaneous motion trajectory of the exoskeleton since it cannot be directly obtained, the motion of the slider in the virtual prototype is used as a substituted in Eqs. (3) and (14). This allows us to calculate the instantaneous position of the exoskeleton and compare it with the theoretical trajectory. The results are shown in Fig. 26. Moreover, the exoskeleton joint torque is analyzed by comparing the changes in spring elasticity, and the spring elasticity comparison results are shown in Fig. 27.

Figure 26. Comparison of instantaneous center trajectory.

Figure 27. Comparison of spring elasticity.

Figure 28. The simulation model is established according to the wearing condition of the exoskeleton.

Figure 29. Knee joint torque before and after wearing exoskeleton.

Figure 30. Knee joint pressure before and after wearing exoskeleton.

Figure 31. The strength of each muscle. (a) Rectus femoris muscle strength before and after wearing exoskeleton. (b) Lateral femoral muscle strength before and after wearing exoskeleton. (c) Medial femoral muscle strength before and after wearing exoskeleton. (d) Middle femoral muscle strength before and after wearing exoskeleton.

The comparison between Figs. 26 and 27 reveals that the instantaneous center trajectory and the changes in spring elastic of the exoskeleton align closely with the theoretical results. This consistency indicates that the optimized exoskeleton size parameters and spring stiffness are accurate and reliable.

5.3. Biomechanical simulation analysis based on OpenSim

The human model provided in OpenSim database is selected, and combined with the situation after exoskeleton is worn, the human–machine interaction model after exoskeleton is built, as shown in Fig. 28.

Using OpenSim simulation, the knee joint torque is evaluated before and after the exoskeleton is worn, as shown in Fig. 29. According to the research [Reference Guo, Li and Jiang28], the main force stage of the knee joint during human walking is when the muscle is performing positive work, resulting in positive joint torque. Subsequent to wearing the exoskeleton, the maximum forward joint torque experienced by the human body is markedly decreased, while the joint torque experienced during other stages does not exhibit significant increases. These finding indicate that the exoskeleton has a valuable assistive effect. The exoskeleton is found to have reduced the human knee joint torque by a total of 48.42%. The pressure changes occurring within the knee joint during exercise are shown in Fig. 30.

During walking, the positive torque generated by the knee joint is mainly generated by the extensor muscles of the knee joint. The extensor muscles of the knee joint are mainly composed of the rectus, medialis, lateralis, and medialis muscles. To further verify whether the exoskeleton has an auxiliary effect on the extensor muscle group, we analyzed the torque variation of each muscle in the extensor muscle group. The strength of rectus femoris, lateral femoris, medial femoris, and middle femoral before and after wearing the exoskeleton are shown in Fig. 31 (a), (b), (c), and (d) respectively.

After calculation, the muscle strength of rectus femoris, vastus lateralis, vastus medialis, and vastus intermedius decreased by 26.89%, 11.55%, 7.51% and 0.31%, respectively, after wearing exoskeleton. The exoskeleton mainly assists the knee joint during the load-bearing reaction period and the middle stage of the support phase, reducing the positive torque that causes the knee joint to extend during this stage. According to the relevant medical literature, the rectus femoris, vastus lateralis, and vastus medialis jointly provide the power to produce positive torque, so the strength of the three decreases significantly, while the vastus medialis plays a role in fastening the three muscles, so the decline is very small. By analyzing the force of each muscle that produces the knee joint extension torque, it is further proved that the exoskeleton has the performance of assisting the knee joint.