2.1. Exoskeleton robot structure

The design of a knee unloading exoskeleton mechanism must consider not only the physical interaction forces between the exoskeleton and the human body, specifically, the supportive forces exerted on the body, but also the kinematic characteristics of the knee joint and the H-E kinematic compatibility. The primary motion of the knee joint is flexion–extension in the sagittal plane, with a time-varying ICR. To address this, an adaptive knee mechanism for the knee unloading exoskeleton has been proposed, as illustrated in Figure 1.

The exoskeleton mechanism consists of a thigh attachment, a shank attachment, linkage rods, and a linear actuator. The thigh and shank attachments are securely attached to the thigh and shank, respectively. On both the left and right sides of the attachments, four linkage rods are connected to form a rhombus configuration, with all components joined by revolute joints (denote it with the symbol R). A linear actuator reduces the distance between the two central revolute joints (R $_2$ and R $_4$ ), causing the vertical distance between the upper and lower revolute joints (R $_3$ and R $_1$ ) to increase, thereby achieving active support motion. When the linear actuator is inactive, the entire mechanism passively follows the knee joint’s motion, enabling adaptive knee motion.

2.2. DOF of the exoskeleton mechanism

The knee exoskeleton mechanism, composed of linkage rods and a linear actuator, forms a multi-loop coupled parallel structure. The linear actuator is modeled as an equivalent prismatic joint (denote it with the symbol P). As shown in Figure 2, the shank attachment is designated as the fixed platform, while the thigh attachment serves as the moving platform. Fixed and moving coordinate systems are established accordingly. The origin of the fixed coordinate system, denoted as $O$ , is located at the center of the revolute joint R $_1$ . The $X$ -axis aligns with the axis of R $_1$ , the $Z$ -axis is oriented vertically upward, and the $Y$ -axis follows the right-hand rule, forming the fixed coordinate system $O-XYZ$ . Similarly, the origin of the moving coordinate system, denoted as $P$ , is positioned at the center of the revolute joint R $_3$ . The axes of the moving coordinate system $P-xyz$ are aligned with those of $O-XYZ$ .

Figure 2.

Diagram of the mechanism coordinate system and screw system. (a) Mechanism coordinate system. (b) Twists and wrenches of the branch R $_1$ R $_2$ R $_3$ .

H-E motion matching requires that the exoskeleton mechanism possess sufficient DOF to accommodate or adapt to knee joint movements. To ascertain this, a DOF analysis is imperative. Additionally, for multi-loop mechanisms, the presence of over-constraints can complicate the DOF analysis. The screw theory, however, stands out for its clear physical interpretations in mechanism analysis. It enables the convenient identification of constraints acting on the mechanism and allows for the accurate determination of its DOF. Therefore, the screw theory is employed to analyze the DOF of the proposed mechanism. To ensure consistent active support forces on both the medial and lateral sides of the knee joint, the exoskeleton mechanism is configured symmetrically. Therefore, only one side of the mechanism needs to be analyzed. The twist system $\mathbb{S}_{1}$ for the branch R $_1$ R $_2$ R $_3$ can be expressed as

(1) \begin{equation} \mathbb{S}_{1}= \begin{cases} \boldsymbol{\$}_{11} =\begin{bmatrix} 1 & \, 0 & \, 0 & \, 0 & \, 0 & \, 0 \end{bmatrix}^T \\ \boldsymbol{\$}_{12} =\begin{bmatrix} 1 & \, 0 & \, 0 & \, 0 & \, f_{12} & \, -e_{12} \end{bmatrix}^T \\ \boldsymbol{\$}_{13} =\begin{bmatrix} 1 & \, 0 & \, 0 & \, 0 & \, f_{13} & \, -e_{13} \end{bmatrix}^T \end{cases} \end{equation}

where the variables $f_{12}$ , $f_{13}$ , and those in other vectors of this subsection all represent position variables related to the axes of the mechanism joints.

For the branch R $_1$ R $_4$ R $_3$ (see Figure 3(a)), the twist system $\mathbb{S}_{2}$ can be expressed as

(2) \begin{equation} \mathbb{S}_{2}= \begin{cases} \boldsymbol{\$}_{21} =\begin{bmatrix} 1 & \, 0 & \, 0 & \, 0 & \, 0 & \, 0 \end{bmatrix}^T \\ \boldsymbol{\$}_{22} =\begin{bmatrix} 1 & \, 0 & \, 0 & \, 0 & \, f_{22} & \, -e_{22} \end{bmatrix}^T \\ \boldsymbol{\$}_{23} =\begin{bmatrix} 1 & \, 0 & \, 0 & \, 0 & \, f_{23} & \, -e_{23} \end{bmatrix}^T \end{cases} \end{equation}

The twist system $\mathbb{S}_{3}$ of branch R $_2$ PR $_4$ (see Figure 3(b)) can be expressed as

(3) \begin{equation} \mathbb{S}_{3}= \begin{cases} \boldsymbol{\$}_{31} =\begin{bmatrix} 1 & \, 0 & \, 0 & \, 0 & \, f_{31} & \, -e_{31} \end{bmatrix}^T \\ \boldsymbol{\$}_{32} =\begin{bmatrix} \, 0 & \, 0 & \, 0 & \, 0 & \, e_{32} & \, f_{32} \end{bmatrix}^T \\ \boldsymbol{\$}_{33} =\begin{bmatrix} 1 & \, 0 & \, 0 & \, 0 & \, f_{33} & \, -e_{33} \end{bmatrix}^T \end{cases} \end{equation}

According to the reciprocal relationship between the motion screw and the constraint screw, the wrench system of the branches R $_1$ R $_2$ R $_3$ , R $_1$ R $_4$ R $_3$ , and R $_2$ PR $_4$ can be written as

(4) \begin{equation} { \mathbb{S}_{123}^{r}= \begin{cases} \boldsymbol{\$}_{11}^{r} =\boldsymbol{\$}_{21}^{r} = \boldsymbol{\$}_{31}^{r}=\begin{bmatrix} 1 & \, 0 & \, 0 & \, 0 & \, 0 & \, 0 \end{bmatrix}^T \\ \boldsymbol{\$}_{12}^{r} = \boldsymbol{\$}_{22}^{r}= \boldsymbol{\$}_{32}^{r}=\begin{bmatrix} 0 & \, 0 & \, 0 & \, 0 & \, 1 & \, 0 \end{bmatrix}^T \\ \boldsymbol{\$}_{13}^{r} = \boldsymbol{\$}_{23}^{r}= \boldsymbol{\$}_{33}^{r}=\begin{bmatrix} 0 & \, 0 & \, 0 & \, 0 & \, 0 & \, 1 \end{bmatrix}^T \end{cases} } \end{equation}

Figure 3.

Schematic diagram of a branch screw system. (a) The twist system and wrench system of the branch R $_1$ R $_4$ R $_3$ . (b) The twist system and wrench system of the branch R $_2$ PR $_4$ .

Obviously, the wrench system of Eqs. (1), (2), and (3) are equal, indicating that the constraints on the three branches are the same. The constraint screws are the constraint force in the $X$ -axis, the constraint couple in the $Y$ -axis, and the constraint couple in the $Z$ -axis. The constraints on the opposite side are the same as those on this side. The mechanism as a whole has 15 redundant constraints. The remaining motions of the mechanism are rotation about the $X$ -axis, translation along the $Y$ -axis, and translation along the $Z$ -axis. So, the mechanism has three DOFs.

The above analysis has obtained the property of the DOF of the mechanism. The number of DOF of the mechanism can be obtained by using the modified G–K formula [Reference Huang, Li and Ding45]:

(5) \begin{equation} M=6(n-g-1)+\sum _{i=1}^{g} f_i+v =6 \times (14-18-1)+18+15 =3 \end{equation}

where $n$ is the number of components, $g$ is the number of joints, $\sum _{i=1}^{g} f_i$ is sum of the number of DOF of all joints of the mechanism, $v$ is the overconstraint. Overall, this exoskeleton mechanism has two translational and one rotational DOF within the plane. It is capable of adapting to the changes in the ICR of the knee joint in the sagittal plane.

2.3. DOF of the H-E closed-loop mechanism

Since the exoskeleton mechanism is strapped to the thigh and calf, its motion is constrained by the human leg. Therefore, it is necessary to consider the human leg and the exoskeleton as a closed-loop mechanism to analyze its DOF. The rhombus linkage mechanism has three DOFs. To simplify the analysis of the DOF of the H-E closed-loop mechanism, the exoskeleton mechanism can be equivalent to a planar serial 3-DOF mechanism, such as a planar serial 3R mechanism (see Figure 4). Project the closed-loop mechanism composed of the serial 3R mechanism and the human lower limb onto the sagittal plane. Obviously, the H-E closed-loop mechanism is a planar four-bar mechanism. Thus, the DOF of the H-E closed-loop mechanism is one.

It is worth noting that although the rhombus linkage mechanism and the serial 3R mechanism have the same DOF and similar kinematic properties, they have different advantages and disadvantages in the application field of knee exoskeletons. The advantage of the serial 3R mechanism is its simple and compact structure, which is easy to design. However, to achieve support for the knee joint, it is necessary to drive the rotation of its rotary joints. In contrast, although the closed-loop rhombus linkage mechanism has a more complex structure, it can support the knee joint without driving the rotary joints. Specifically, the support can be realized by driving the prismatic joint between joints R $_2$ and R $_4$ . In addition, when the angle $\alpha$ between the connecting rods is greater than $45^{\circ }$ , the driving force of the linear actuator can amplify the support force. A detailed discussion can be found in Section 4.2. This characteristic of force amplification means that the driving force of the linear actuator does not need to be substantial. Therefore, to a certain extent, it can reduce the performance requirements for the actuator.

Figure 4.

Schematic diagram of the H-E closed-loop mechanism. (a) The equivalent mechanism of the rhombus linkage mechanism, that is, the 3R mechanism on the right. (b) Knee exoskeleton closed-loop mechanism.

3.1. Exoskeleton mechanism kinematics

The inverse kinematics solution of the exoskeleton mechanism is to solve the length and rotation angle parameters of the driving device based on the known position and orientation parameters of the moving platform (thigh attachment). The geometric schematic diagram of the mechanism is shown in Figure 5. The establishment of the coordinate system is the same as in the previous section. The pose of the platform can be represented by the angle $\gamma$ between the $Y$ -axis of the coordinate system $O-XYZ$ and the $y$ -axis of the coordinate system $P-xyz$ , along with the coordinates of point $P$ , that is,

(6) \begin{equation} ^{0}\mathbf{P}= \left[\begin{array}{l@{\quad}l@{\quad}l} 0 & P_Y & P_Z \end{array}\right]^T \end{equation}

The lengths of the connecting rods $A_1B_1$ , $A_1B_2$ , $B_1C_1$ , and $B_2C_1$ are all $l$ . Given the pose of the platform, the input parameters $\theta _1$ , $l_2$ , and $\theta _3$ can be solved through geometric relationships. According to the geometric relationships in Figure 5, we have

(7) \begin{align} \theta _1=\arccos \frac {P_Y}{\| ^{0}\mathbf{P} \| } -\arccos \frac {\| ^{0}\mathbf{P} \|}{2l} \\[-28pt] \nonumber \end{align}

(8) \begin{align} l_2=\sqrt {4l^2-\| ^{0}\mathbf{P} \|^{2}} \\[6pt] \nonumber \end{align}

where $\theta _1$ is the angle between the $A_1B_2$ and the base, $l_2$ is the length of the $B_1B_2$ , and $\theta _3$ is the angle between the $B_2C_1$ and the platform.

The coordinates of the point $B_2$ in the moving coordinate system are

(9) \begin{equation} ^{P}\mathbf{B}_2= \left[\begin{array}{l@{\quad}l@{\quad}l} 0 & l\cos \theta _3 & l\sin \theta _3 \end{array}\right]^T \end{equation}

Similarly, the coordinates of the point $B_2$ in the base coordinate system are

(10) \begin{equation} ^{0}\mathbf{B}_2= \left[\begin{array}{l@{\quad}l@{\quad}l} 0 & l\cos \theta _1 & l\sin \theta _1 \end{array}\right]^T \end{equation}

The rotation matrix of the moving coordinate system $P-xyz$ with respect to the base coordinate system $O-XYZ$ is written as

(11) \begin{equation} ^{0}_{P}\mathbf{R}= \left[\begin{array}{l@{\quad}l@{\quad}l} 1 & 0 & 0 \\ 0 & \cos \gamma & -\sin \gamma \\ 0 & \sin \gamma & \cos \gamma \end{array}\right] \end{equation}

By expressing the coordinates of point $B_2$ in the moving coordinate system $P-xyz$ as coordinates in the base coordinate system $O-XYZ$ , there is

(12) \begin{equation} ^{0}\mathbf{B}_2= ^{0}_{P}\mathbf{R} \cdot ^{P}\mathbf{B}_2+^{0}\mathbf{P} \end{equation}

Thus, $\theta _3$ can be solved as

(13) \begin{equation} \theta _3=\arccos \left [ \frac {l\cos (\theta _1-\gamma )-P_Y\cos \gamma -P_Z\sin \gamma }{l} \right ] \end{equation}

The forward kinematics model of the exoskeleton mechanism is determined by three parameters $\theta _1$ , $l_2$ , and $\theta _3$ , which are used in solving the platform pose ( $\mathbf{P}$ and $\gamma$ ):

(14) \begin{align} \| ^{0}\mathbf{P} \| =\sqrt {4l^2-l_{2}^{2}} \\[-24pt] \nonumber \end{align}

(15) \begin{align} P_Y=\| ^{0}\mathbf{P} \| \cos \left [ \arcsin \Big ( \frac {l_2}{2l} \Big )+\theta _1 \right ] \\[-24pt] \nonumber \end{align}

(16) \begin{align} P_Z=\| ^{0}\mathbf{P} \| \sin \left [ \arcsin \Big ( \frac {l_2}{2l} \Big )+\theta _1 \right ] \\[-24pt] \nonumber \end{align}

(17) \begin{align} \gamma =- \left\{\pi - \left [ \arcsin \Big ( \frac {l_2}{2l} \Big )+\theta _1 \right ] +\theta _3 -\arcsin \Big (\frac {l_2}{2l} \Big ) \right\} \\[6pt] \nonumber \end{align}

Figure 5.

Geometric diagram of exoskeleton kinematics.

3.2. H-E closed-loop kinematics

The distance of the active support motion required by the exoskeleton mechanism is related to the motion distance of the driving joints. Moreover, during the stance phase of the gait cycle, the mechanism needs to support the thigh and the calf. During the swing phase of the gait cycle, the mechanism has to swing along with the lower limb, with the driving joints in a free state to adapt to the movement of the knee joint. Thus, the motion pattern of the driving joints is associated with the knee angle in the gait. As depicted in Figure 6, on the left is the simplified model of the thigh, calf, and knee joint, while on the right is the simplified model of the exoskeleton mechanism. Point A represents the revolute joint on the calf attachment, point B is the simplified human biological knee joint, and point C is the revolute joint on the thigh attachment. The lengths of the four connecting rods of the exoskeleton are all $l$ . The length $l_2$ of the connection point between the thigh and calf attachments can be expressed as

(18) \begin{equation} l_2=\sqrt {l_{s}^{2}+l_{t}^{2}-2l_sl_t \cos (\pi -\theta _k)} \end{equation}

where $l_t$ is the distance from the revolute joint on the thigh attachment to the center of the knee joint, $l_s$ is the distance from the revolute joint on the calf attachment to the center of the knee joint, and $\theta _k$ is the knee angle.

In the quadrangle $AB_1CB_2$ , one has

(19) \begin{equation} l_{a}^{2}+l_2^{2}=4l^{2} \end{equation}

Combining Eqs. (18) and (19), the connecting rod length of the actuator can be expressed as

(20) \begin{equation} l_{a}=\sqrt {4l^{2}-l_{s}^{2}-l_{t}^{2}-2l_{s}l_t \cos \theta _k} \end{equation}

where $l$ is the length of the linkage of the rhombus linkage mechanism. The right side of Eq. (20) contains only one variable, the knee angle, so the relationship between the length of the actuator and the knee angle is obtained.

Figure 6.

Schematic illustration of simplified mechanisms for lower limbs (left) and exoskeleton (right).

3.3. Numerical example

To further illustrate the kinematic characteristics of the mechanism, numerical simulations are carried out using MATLAB software and RecurDyn dynamics software. The simulation model in RecurDyn software is shown in Figure 7(a). Here, the length parameter of the rod is given as $l=200$ mm, the angle of the platform $\gamma =0^{\circ }$ , and the motion applied to the thigh attachment is defined as

(21) \begin{equation} P_z=10\sin \left(\frac {2\pi t}{10} \right)+280 \end{equation}

where $t$ represents the motion time. As shown in Figure 8, the calculation results of MATLAB software are consistent with the simulation results of RecurDyn, verifying the correctness of the mechanism kinematics. The changes in the three joint angles of the mechanism exhibit a sinusoidal curve, which is consistent with the given motion trend of the platform.

Figure 7.

Simulation model. (a) Exoskeleton mechanism. (b) H-E closed-loop mechanism, where the yellow cylinder is the lower limb.

Figure 8.

Comparison diagram between theoretical calculation and simulation curves of the joint angles.

According to Eq. (20), the length of the actuator connecting rod is related to the dimensions of the mechanism and the strapping positions on the thigh and calf. Given the strapping distances relative to the knee joint $l_s=140$ mm, $l_t=140$ mm, as $\theta _k$ varies from zero to $\pi /2$ , the calculated and simulated curves of the actuator length variation are shown in Figure 9. As the knee flexion angle gradually increases, the length of the actuator connecting rod also gradually increases. This variation relationship is quite intuitive because the flexion of the knee joint will inevitably reduce the distance between point A and point C in Figure 6, thus increasing the distance between points B $_1$ and B $_2$ .

Figure 9.

The relationship between the length of the actuator rod and the knee flexion angle.

Typically, within a walking gait cycle, the knee angle first increases, then decreases, then increases again, and finally decreases [Reference Rose and Gamble46]. Based on this, the curve relationship between the actuator rod length and the knee angle in the gait cycle can be obtained, as shown in Figure 10. The variation law of the actuator rod length is very similar to that of the knee angle in the gait because the relationship between the actuator rod length and the knee angle is close to a linear one (see Figure 9). During the stance phase, the change in the actuator rod length is relatively small, while during the swing phase, the change in the actuator rod length is relatively large, and the driving joint needs to follow the swinging motion of the lower limb.

Figure 10.

The relationship between the length of the actuator rod and the knee angle during the gait cycle.

Figure 11.

Exoskeleton system wearing schematic diagram.

4.1. Mechanical modeling of cable-driven system

The output force of the cable-driven system will be affected by the friction between the rope and the Bowden sheath, as well as the bending path of the Bowden sheath. Therefore, it is necessary to establish a mechanical model of the cable-driven system to analyze the relationship between the input force and output force of the cable-driven system. The schematic diagram of the Bowden cable-driven unit is shown in Figure 12. The input side includes a motor and a winch, and the output side is connected to the exoskeleton anchor point. The input force and output force are defined as $F_{in}$ and $F_{out}$ , respectively, and the length of the Bowden sheath is $L$ .

Figure 12.

Schematic diagram of the Bowden cable-driven unit.

To establish the mechanical model of the Bowden cable, a microelement d $s$ of the Bowden cable is taken, and its force analysis is carried out, as shown in Figure 13. It is assumed that the mass of the internal cable of the Bowden cable, as well as the impacts of temperature changes and material fatigue issues on the system, can be neglected. There is only relative motion along the axial direction of the sheath between the outer sleeve and the internal rope of the Bowden cable, and the friction force between the rope and the inner wall of the sheath conforms to the sliding friction formula. The force balance equation of this microelement can be written as

(22) \begin{align} f(s)+F(s+\mathrm{d}s) \cos \frac {\mathrm{d}\theta }{2} - F(s) \cos \frac {\mathrm{d}\theta }{2} = 0 \\[-24pt] \nonumber \end{align}

(23) \begin{align} N(s)-F(s+\mathrm{d}s) \sin \frac {\mathrm{d}\theta }{2} - F(s) \sin \frac {\mathrm{d}\theta }{2} = 0 \end{align}

where $F(s)$ is the force on the input side of the microelement, $f(s)$ is the friction force between the rope and the sheath, $N(s)$ is the normal positive pressure of the rope on the sheath, $F(s+\mathrm{d}s)$ is the force on the output side of the microelement, and $\mathrm{d}\theta$ is the central angle corresponding to the arc length of the microelement.

Figure 13.

Mechanical model of the Bowden cable.

Defining the tension difference between the output side and the input side of the microelement as $\mathrm{d}F(s)=F(s+\mathrm{d}s)-F(s)$ , the force balance equation of the microelement can be rewritten as

(24) \begin{align} f(s)=-\mathrm{d}F(s)\cos \frac {\mathrm{d}\theta }{2} \\[-24pt] \nonumber \end{align}

(25) \begin{align} N(s)=\mathrm{d}F(s) \sin \frac {\mathrm{d}\theta }{2} + 2F(s) \sin \frac {\mathrm{d}\theta }{2} \\[6pt] \nonumber \end{align}

When the microelement is small enough, one has $\mathrm{d}F(s) \sin \frac {\mathrm{d}\theta }{2} \approx 0$ , $\sin \frac {\mathrm{d}\theta }{2} \approx \frac {\mathrm{d}\theta }{2}$ , $\cos \frac {\mathrm{d}\theta }{2} \approx 1$ , thus,

(26) \begin{align} f(s)=-\mathrm{d}F(s) \\[-24pt] \nonumber \end{align}

(27) \begin{align} N(s)=F(s)\mathrm{d}\theta \\[6pt] \nonumber \end{align}

According to the friction formula, one has

(28) \begin{equation} f(s)=\mathrm{sign}(a) \mu N(s) \end{equation}

where $a$ represents the motion direction of the rope relative to the Bowden sheath. It is considered positive when its direction is consistent with that of the tension at the input side, and negative otherwise. Also, $\mu$ is the friction coefficient.

According to the relationship between the arc length of the element and the center angle, one has

(29) \begin{equation} \mathrm{d}\theta =\frac {\mathrm{d}s}{R(s)}=\kappa (s)\mathrm{d}s \end{equation}

where $R(s)$ is the radius of the element and $\kappa (s)$ is the curvature of the element.

Combining Eqs. (26), (27), (28), and (29), one has

(30) \begin{equation} \frac {\mathrm{d}F(s)}{\mathrm{d}s}=-\mathrm{sign}(a) \mu F(s) \kappa (s) \end{equation}

Define $F^{'}(s)$ as the derivative of the tension difference at both ends of the element with respect to the arc length, that is,

(31) \begin{equation} F^{'}(s)=\frac {\mathrm{d} F(s)}{\mathrm{d} (s)} \end{equation}

Furthermore, we can obtain

(32) \begin{equation} \frac {F^{'}(s)}{F(s)}=-\mathrm{sign}(a) \mu \kappa (s) \end{equation}

Integrating both sides of it over the interval $[s,\ s+\mathrm{d}s]$ , one has

(33) \begin{equation} \int _{s}^{s+\mathrm{d}s} \frac {F^{'}(s)}{F(s)} \mathrm{d} s = -\mathrm{sign}(a) \mu \int _{s}^{s+\mathrm{d}s} \kappa (s) \mathrm{d} s \end{equation}

If the element curvature $\kappa$ can be regarded as a constant, then the tension relationship between the input side and output side of the element can be obtained

(34) \begin{equation} {F(s+\mathrm{d}s)}=F(s) e^{-\mathrm{sign}(a) \mu \kappa (s) \mathrm{d}s} \end{equation}

Figure 14.

Diagram of the relationship between the output force of the Bowden system and the active support force.

Extending to the general case, assuming the same curvature, that is, $\theta =\int _{0}^{L} \kappa (s) \mathrm{d}s$ , then the relation between the input force and the output force is

(35) \begin{equation} F_{out}=F_{in} e^{-\mathrm{sign}(a) \mu \theta } \end{equation}

Considering the movement direction $a$ of the rope relative sheath and rewriting (35), the relationship between the input force and the output force of the Bowden cable can be finally obtained as

(36) \begin{equation} \begin{cases} F_{out}=F_{in} e^{-\mu \theta } & a\gt 0 \\ F_{out}=F_{in} e^{\mu \theta } & a\lt 0 \end{cases} \end{equation}

The internal rope of the Bowden cable is affected by the friction force from the external Bowden sheath. Moreover, the direction of the friction force is related to the direction of the motion tendency, which results in the characteristic of output force hysteresis in the cable-driven system. The relationship between the input force and the output force of the cable-driven system will be described in detail in Section 5.

4.2. Active supporting force of the exoskeleton

The tension of the Bowden cable is transformed into the active supporting force for the human body through the exoskeleton mechanism. The magnitude of this active supporting force is related to the output force of the cable-driven system. As shown in Figure 14, the output force of the Bowden system is converted into a vertical supporting force along the leg. Through the thigh and calf attachment modules of the exoskeleton, the supporting force is transmitted to the thigh and calf, allowing part of the human body’s gravitational load to bypass the knee joint. In other words, the exoskeleton and the knee joint jointly bear the human body’s gravity, thereby reducing the load on the knee joint. Conducting a force analysis on the exoskeleton mechanism, we have

(37) \begin{equation} \begin{gathered} F_a=2F_c \sin \left(\frac {\pi }{2}-\alpha \right) \\ F_s=2F_c \cos \left(\frac {\pi }{2}-\alpha \right)-F_t \end{gathered} \end{equation}

where $F_a$ and $F_s$ are the output force of the Bowden cable and the active support force of the exoskeleton, respectively.

$F_t$ is the elastic force generated by the tension spring between joints R $_1$ and R $_3$ , which satisfies $F_t=k \Delta x_s$ . Here, $k$ and $\Delta x_s$ are the stiffness coefficient and the deformation of the spring, respectively. By eliminating the component force $F_c$ along the connecting rod direction, the relationship between the active supporting force and the output force of the cable-driven system is obtained as

(38) \begin{equation} F_s=F_a \tan \alpha -k \Delta x_s \end{equation}

The above analysis has dealt with the relationship between the output force and the input force of the cable-driven system, as well as the relationship between the actuator force of the exoskeleton and the active supporting force. Next, the input–output relationship of the entire exoskeleton system from the motor torque to the exoskeleton supporting force will be established. Considering that the output shaft of the motor is connected to a winch to convert the motor torque into the input force of the cable-driven system, we have

(39) \begin{equation} F_{in}=\frac {M}{R} \end{equation}

where $M$ is the motor torque and $R$ is the radius of the winch to spool the rope. To simplify the model, it is assumed that the curvature of the Bowden cable path is the same, combining Eqs. (36), (38), and (39), the relationship between the motor torque and the active support force of the exoskeleton can be written as

(40) \begin{equation} F_{s}=\frac {M}{R} e^{- \mu \theta } \tan \alpha - k \Delta x_s \end{equation}

In addition, considering the rotation angle $\alpha _m$ of the motor and referring to Figure 6, under the action of the cable-drive system, the distance between joints R $_2$ and R $_4$ can be expressed as

(41) \begin{equation} l_{a}^{\, '}=\frac {1}{2}(l_{a}-\alpha _mR) \end{equation}

Combined with Eq. (41), the relationship between the exoskeleton support displacement $X$ and the motor rotation angle can be expressed as

(42) \begin{equation} X=\sqrt {4l^2-(l_a-\alpha _mR)^2} - l_2 \end{equation}

6.1. Motion compatibility test

This study first selected a healthy adult male with a height of 180 cm and a body mass of 73 kg for exoskeleton wearability and compatibility testing. As shown in Figure 23, a phased testing approach was adopted: First, the subject wore only the main structure of the exoskeleton (without the cable-driven actuation system) and sequentially performed typical movements, including high knee lifts, deep knee flexion, lunges, normal walking, and sit-to-stand transitions. The experimental results demonstrated that the exoskeleton did not restrict or interfere with the subject’s natural motion in any of the tested movements, and the subject reported no discomfort, confirming the exoskeleton system’s excellent biomechanical compatibility. Notably, the exoskeleton mechanism incorporates an innovative misalignment-free design, effectively avoiding the technical challenge of knee joint rotational axis alignment in traditional exoskeletons, thereby enhancing the adaptability and wearing comfort of the human-exoskeleton system.

6.2. Exoskeleton electromechanical system

The overall structure of the rhombic exoskeleton prototype is shown in Figure 24, where the rod length of the rhombus mechanism is 160 mm. The exoskeleton mechanism connects to the thigh and shank through customized braces at both ends. To achieve optimal fit between the braces and lower limbs, we first acquired 3D geometric data of the lower limbs using laser scanning. Based on this scan data, we designed anatomically conforming braces that precisely match the limb contours, which were then fabricated using additive manufacturing technology. The inner surface of each brace incorporates a flexible liner with silicone particle padding. This design serves dual purposes: enhancing wearer comfort while preventing slippage during motion through improved frictional properties.

Figure 24.

Exoskeleton electromechanical system and its overall wearing illustration. (a) Rhombic mechanism part of the exoskeleton. (b) Overall wearing illustration of the exoskeleton. (c) Actuating and control unit. (d) Input end of the Bowden cable-drive system (including the motor, winch, and PE wire). (e) Force-sensitive resistor unit for detecting gait events.

The actuation and control unit employs a modular integrated design, with all components mounted on a backpack. The system core consists of an STM32 F407 embedded controller. Beneath the control board lies a lithium battery that powers both the controller and motors. The actuation system utilizes a Bowden cable mechanism. At the input end, a motor (Xiaomi CyberGear, China) coupled with a 20 mm radius winch forms the driving component (see Figure 24(d)). Each Bowden cable comprises an outer sheath and an inner core made of ultra-high molecular weight polyethylene (PE) fishing line (Aosite, 1.2 mm diameter, China). The output ends of the Bowden cables connect to joints R $_2$ and R $_4$ of the rhombic mechanism. During operation, motor rotation tightens the PE lines via the winch, converting the motor torque into tensile force on the PE lines. This force reduces the distance between joints R $_2$ and R $_4$ , thereby generating a vertical support force to alleviate knee joint loading. Concurrently, extension springs (with a stiffness coefficient of 0.09 N/mm) between joints R $_1$ and R $_3$ store energy. Upon motor release, the springs’ elastic restoring force returns the system to its initial state, completing a full support cycle.

To enable the exoskeleton to provide supporting force at the appropriate timing, a force-sensitive resistor (FSR) unit (Ruiboyi, 30 $\times$ 60 mm, China) was installed at the heel area as the triggering sensor for the actuation system. As shown in Figs. 24(e) and 25(c), this sensor precisely detects the initial contact moment of the stance phase in the gait cycle by monitoring pressure-induced resistance changes (manifested as resistance drop) during heel strike. To ensure measurement accuracy, a specialized protective structure was implemented: both surfaces of the thin-film FSR were reinforced with rigid circular plates, and the entire assembly was encapsulated using waterproof adhesive tape. This design effectively eliminates measurement errors caused by sensor bending deformation while simultaneously ensuring durability and reliability.

6.3. Human-in-the-loop optimization

The FSR-triggered control strategy for the exoskeleton system is illustrated in Figure 25(a). When heel strike occurs and the FSR detects plantar pressure exceeding the predefined threshold, the control motor activates, causing the Bowden cable system to contract and generate tension in the PE lines. When the pressure falls below this threshold, the motor releases and the Bowden cable system resets.

Figure 25.

Human-in-the-loop optimization experiment. (a) Control strategy of exoskeleton system triggered by FSR. (b) Exoskeleton system and Bowden cable output force measurement sensor (1-D force sensor). (c) FSR pasted on the heel

Since direct measurement of the exoskeleton’s supportive force on human limbs is challenging without installing sensors on the thigh or shank, an indirect approach was adopted by monitoring the tension in the PE lines (Bowden cable output force) to evaluate the support force (the conversion relationship between Bowden cable driving force and support force is detailed in Section 4.2). At the Bowden cable output end, between joints R $_2$ and R $_4$ , 1-D force sensors (ZNLBS-VI-50 KG, Znsensor, China, 50 Hz) were installed on both medial and lateral sides of the knee to collect line tension data. As shown in Figure 25(b) and (c), the system employs an FSR-triggered control strategy. The FSR sensor attached to the heel monitors the initial contact phase of gait in real-time, while the motor operates in position control mode. Through human-in-the-loop optimization, control parameters were iteratively adjusted. During optimization, subjects walked on a treadmill at 3 km/h for approximately 1-min per test. After multiple iterations, the optimal control parameters were determined (motor rotation: 13 rad, speed: 25 rad/s). This configuration provides substantial support while maintaining natural gait patterns. Figure 26 displays the averaged and normalized Bowden cable output force curve, derived from a complete gait cycle analysis during one minute of walking tests.

During the initial stance phase of gait, the Bowden cable output force demonstrates an approximately sinusoidal increasing trend, reaching its peak during mid-stance. Experimental data show that the peak tension values recorded by the medial and lateral knee sensors were 38.2 $\pm$ 5.6 N (mean $\pm$ SD) and 39.2 $\pm$ 5.2 N (mean $\pm$ SD), respectively. As the gait transitions into the swing phase (accounting for 60–100% of the gait cycle), the system actively releases the motor, allowing cable tension to rapidly decay to zero. This dynamic force control strategy ensures unrestricted knee flexion–extension during the swing phase, thereby inherently avoiding gait perturbation.

Figure 26.

Human-in-the-loop optimized Bowden cable output force profile.

6.4. Gait experiment

To validate the adaptability of the optimized Bowden cable output force profile across different individuals, this study conducted a multi-subject gait adaptation experiment involving five healthy male participants (coded S1-S5) with a mean height of 178 $\pm$ 2.5 cm (mean $\pm$ SD) and body mass of 69.8 $\pm$ 3.2 kg (mean $\pm$ SD). The experiment adopted a controlled design where baseline natural gait data were first collected during 1-min walking trials at 3 km/h without exoskeleton assistance, followed by trials under identical conditions with exoskeleton engagement (experimental setup shown in Figure 27). A high-precision optical motion capture system (Mars, Nokov, China) recorded lower-limb kinematic data in both conditions. To ensure data reliability, dozens of gait cycles collected within 1 min were temporally normalized to percentage-based gait phases and ensemble averaged, ultimately generating the sagittal-plane joint angle profiles presented in Figure 28.

Figure 27.

Experimental scene of the experimenter walking while wearing the exoskeleton.

Figure 28.

Joint kinematic data of participants (S1–S5), where solid lines represent the averaged joint angle curves from 1-min walking trials, and shaded areas indicate $\pm$ 1 standard deviation.