Research on Bowden Cable–Fabric Force Transfer System Based on Force/Displacement Compensation and Impedance Control

Abstract

Bowden cable–fabric is a key force transfer device for flexible exoskeletons, and its precise control of force/displacement is a significant factor in the human–machine interaction of flexible exoskeletons. In this paper, a force/displacement control method based on friction compensation and impedance control was proposed based on a flexible Bowden cable–fabric force transfer testbed system. First, a set of in vitro experimental platforms simulating Bowden cable–fabric force transfer was built according to a typical flexible exoskeleton force transfer system, and following the walking gait of lower limbs, the expected force and knee joint motion were set. Secondly, the Bowden cable–fabric force transfer friction model was constructed as the basis of the system’s force transfer compensation. In addition, the stiffness model of Bowden cable–fabric and the lower leg movement model were established and combined with impedance control to realize the precise control of system displacement. Finally, the damping and stiffness parameters suitable for the system were obtained through the impedance control simulation. In terms of the experiment, an in vitro Bowden cable–fabric force transfer experimental platform was built, and the expected force with the input peak value of 40 N, 50 N, and 60 N was set. Through the friction and position compensation model of Bowden cable–fabric force transfer and impedance control, the relative root-mean-square errors of the output force and expected force were obtained as 2.53%, 2.16%, and 2.07%, respectively. Therefore, the effectiveness of the proposed method is verified, which provides a foundation for the engineering application of flexible exoskeletons.

1. Introduction

The flexible lower limb exoskeleton is a mechanical device that can provide strength assistance to the wearer in walking, running, bearing, carrying, and other sports by sharing the load of the hip joint and knee joint. The use of an exoskeleton can reduce the metabolic consumption generated by exercise and improve the human body’s long-term exercise ability. In recent years, the application of flexible wearable exoskeletons has made a series of breakthroughs in the military, health care service, transportation, and other aspects of human life [1,2,3].

The power-assisted types of flexible exoskeletons can be divided into pneumatic muscle assist and flexible material assist. Pneumatic muscle assist has the characteristics of high efficiency, density, light weight, and flexibility, but it is difficult to have high control accuracy due to its high nonlinearity. In flexible material assist systems, Bowden cable is commonly used to function as the transmission mechanism. The Bowden cable transmission mode operates by transferring the tension and position through the motor contract line distance. Bowden cable transmission has the advantages of small size, remote transmission, and better position control accuracy of the motor, which has become increasingly used in flexible exoskeleton design [4]. The following are some applications of Bowden cable drive in flexible exoskeletons: Hofmann [5] proposed a novel Remote Execution System (RAS) based on Bowden cables, featuring a high power-to-mass ratio, power-to-volume ratio, simple components, and a compact mechanical design, with waterproof, dustproof, and hygienic characteristics. John [6] designed equipment consisting of four motor-driven cable-driven actuators to assist four joint movements. The unique feature of this equipment is the crossover of cables in the front and rear, along with the introduction of the concept of crossover cable assistance. After testing, this device is capable of generating motion in six directions around user joints, including internal and external rotation. Mooney [7] developed an autonomous powered leg exoskeleton, which, compared to not wearing the device, reduces walking metabolic costs by 6% to 11%. Liu [8] employed cable-driven and series elastic actuators to design a flexible lower limb assistive exoskeleton, providing support for the elderly and individuals with lower limb disabilities when navigating stairs. Ma [9] devised a cable-driven hip-joint-powered assistive exoskeleton system that significantly reduces muscle consumption during walking, offering effective assistance in human locomotion. Chen [10] developed an exoskeleton utilizing cable-driven knee braces that synergistically assist both the knee and hip joints. The assistive force directly acts on hip flexion and knee extension, while also indirectly impacting hip extension. Additionally, a Parameter Optimization Iterative Learning Control (POILC) method was introduced to minimize errors caused by differences in wearing positions and wearers’ biological characteristics. Metabolic reductions of 9.86%, 12.48%, and 22.08% were observed when compared to situations without wearing the soft exoskeleton.

In summary, Bowden cable drive has been widely used in flexible exoskeletons, providing effective force assistance and reducing human consumption by combining sensors and control algorithms. However, the key challenge of Bowden cable transmission is how to achieve accurate force control and flexibility. In the Bowden cable system, there is friction between the external hollow wire sheath and the internal wire rope, which makes the Bowden cable system often have tension attenuation at the beginning, and it will cause the attenuation of the output tension too. Therefore, it is necessary to analyze and model the Bowden cable force transmission, so as to accurately control the assisting process. In terms of Bowden cable force transmission, Phee [11] investigated the recoil, hysteresis, and nonlinearity caused by cable friction and proposed a method that solely relies on force and position sensors at the drive end to estimate actuator parameters at the output end. Validation results demonstrated an approximate maximum full-scale error of only 7% for this method. Wang [12,13] addressed issues related to force transmission modeling in flexible cable systems, precise position control, and elongation studies. Furthermore, they conducted work on modeling tactile feedback properties and introduced a bidirectional actuation system using two independent cables. They applied a model-based feedforward compensation method for open-loop control, eliminating the need for any feedback information from the remote end, allowing the system to adapt to variations in transmission path configurations for precise position control. Palli [14] proposed a tendon sheath-driven system for robotic hands, analyzed its static force transmission characteristics, established a Dahl dynamic friction model, and introduced a force control algorithm incorporating feedforward friction compensation based on the static friction model. Ohnishi [15] examined the tactile feedback of a flexible actuator, assessed the force and displacement transmission of a Bowden cable system at various bending angles, and applied error compensation using disturbance observer technology. Wang [16,17,18,19] conducted extensive research on the nonlinear aspects of friction in Bowden cable transmission. They introduced a Coulomb friction model based on static force balance for studying Bowden cable transmission characteristics. The LuGre dynamic friction model was employed to describe the motion of a single Bowden cable transmission system, and a control algorithm combining sliding mode and friction compensation was proposed based on the Coulomb friction model. Wu [20] developed a Bowden cable friction model using micro-element analysis and performed experiments to investigate how velocity, radius of curvature, friction factor, and total curvature affect the force transmission characteristics of Bowden cables. Additionally, they suggested new friction compensation and displacement compensation algorithms. Although the above research has set up the force transfer of the Bowden cable, established the friction model, and analyzed the friction between the wire rope and the wire sheath, the main research is purely Bowden cable drive or applied to robot control. Flexible exoskeletons usually have a piece of fabric behind the Bowden cable, and the fabric will eventually deliver force to the human limb. The addition of fabric not only brings a certain force loss but also causes the position to be unable to be accurately controlled due to its elastic deformation. Therefore, the precise force/displacement control of the Bowden cable–fabric is a key technology for the practical application of flexible exoskeletons.

This paper focuses on the precise control of the force/displacement of the flexible exoskeleton Bowden cable–fabric. The research contents are as follows: (1) By analyzing the force transmission process of a human wearing an exoskeleton, we built an in vitro experimental platform which utilized the knee joint motion data from walking and set the motion angle curve as the expected force curve; (2) to solve the force transfer loss in the Bowden cable–fabric force transfer system, the deformation and force of the Bowden cable and fabric in the process of movement were analyzed, and the force transfer model of the Bowden cable and fabric was established; (3) to address the displacement lag of the Bowden cable–fabric transmission, a stiffness model of the Bowden cable–fabric transmission system was established and the impedance control was conducted by compensating the movement displacement of the lower leg; (4) the impedance parameters of the system were selected through the force transmission simulation after force/displacement compensation, and the flexibility and accuracy of the system output were verified. Using the off-body Bowden cable–fabric force transmission experimental platform, a Bowden cable–fabric force transfer experiment was performed to verify the accuracy of force tracking and displacement control by our proposed force/displacement compensation and impedance control methods.

2. Simulation of Bowden Cable–Fabric Force Transfer and Analysis of Lower Limb Motion

Considering the progressive development in Bowden cable–fabric force transmission control and the safety of man–machine interactions, this paper adopts the method of an in vitro experimental platform to research the force transmission route of the flexible lower limb exoskeleton. By modeling the force transmission and compensation control, the precise force transmission system can be identified. To obtain a force curve matching the human body movement, the biomechanical information of the lower limb of the human body was studied first, and our experiment was conducted using a three-dimensional motion capture system and a force measurement table.

2.1. In Vitro Simulation of Force Transfer with Bowden Cable–Fabric

The model of the flexible lower limb knee joint assist device was designed as shown in Figure 1. The motor inside the electric control box drives the Bowden cable to pull the fabric at the knee joint, which connects the lower leg movement. In this process, there is friction inside the Bowden cable and friction between the fabric and the lower leg strap. At the same time, the movement of the lower leg and the deformation of the fabric will lead to the displacement lag of the force transmission.

In order to investigate the friction loss and displacement hysteresis from the swing of the lower leg during walking, an in vitro experimental platform was built according to the transfer path of the Bowden cable and the fabric worn on the human body, as shown in Figure 2. In the experiment, the knee joint is simulated by a rotating cylinder mounted on the shaft of a DC brushless motor. The knee joint model is fixed together with the model of the lower leg. The knee joint model can rotate with the tension of the driving end. A DC brushless motor is used to drive the knee joint model and drive the lower leg model to simulate the position change in human lower limbs during walking. A servo motor that sets the desired force curve drives the Bowden cable–fabric to provide power-assisted control for the lower leg. The start force sensor is used as the feedback of the control, and the end force sensor is mainly used to verify the accuracy of the output force value.

2.2. Angle and Torque Fitting of Knee Joint

To simulate the knee joint and lower leg movement of human walking on the experimental platform, the dynamic capture of an adult walking was carried out. The scene in this article is set as a human body walking on flat land, using a three-dimensional motion capture system (PhaseSpace, San Leandro, CA, USA), which is equipped with a high-resolution digital camera. The data on human joint motion and foot pressure were collected by the force measuring table system (Bertec FIT, Columbus, OH, USA). Figure 3 shows the motion capture of walking joints and the collection of plantar force measurements.

The beginning of the gait cycle is usually defined by the maximum extension angle of the hip joint. The percentages of gait in the following graphs of the joint angles and joint torque are divided by the maximum hip joint angle, and the beginning of the gait cycle is assisted when the hip joint angle reaches the maximum. In the process of testing the walking human body, the data collected by the 3D motion capture system and the force measuring table system were input into the OpenSim V4.0 software, the musculoskeletal model was constructed, and the joint torque was calculated using the inverse dynamic method. In order to more clearly see the movement angle and moment of human lower limb joints in a single gait cycle, the movement of lower limb joints in a gait cycle was analyzed, and the data curve was smoothened. Figure 4 shows the curve of knee joint angle and torque change during a gait cycle. This curve is very similar to the sine function, so the next choice was to use the sine function to fit the force curve.

The Cftool in MATLAB 2018b software was used to fit the knee joint angle curve, and different function types, such as the exponential function approximation, Gaussian function approximation, polynomial function approximation, and sine function approximation, were used to fit, etc. Finally, it was found that the general model Sin3 had a higher degree of fitting. Therefore, the function between the knee joint angle and time is as follows:

$${\theta }_k\left(t\right)=a_1\sin \left(b_{1}t+c_1\right)+a_2\sin \left(b_{2}t+c_2\right)+a_3\sin \left(b_{3}t+c_3\right)$$

(1)

The specific values of parameters in Formula (1) are shown in

Table 1. Parameters of the Sinusoidal fitting curve.

Amplitude	Value	Period	Value	Phase	Value
$a_1$	48.16	$b_1$	0.01951	$c_1$	−3.112
$a_2$	525.2	$b_2$	0.1239	$c_2$	−0.02836
$a_3$	540.2	$b_3$	0.1232	$c_3$	−3.159

. The fitting curve of this function is shown in Figure 5. To simulate the angle of the human body during walking, a fixed knee joint motion curve and gait time was generated in the form of a leg model driven by a motor.

Figure 4b shows that the maximum torque was at the 20% gait period, and the torque is positive between 10% and 30%. Therefore, the expected force should be maximum at the 20% gait period. The goal is to provide force only between 10% and 30%, preventing interference with the wearer’s movement. A unimodal force curve starting at 10% and ending at 30% was fitted according to the torque of the knee joint. The simulation of knee joint torque converted the lower limb knee joint torque during walking to the experimental platform. Since this experiment was an in vitro experiment, the test torque on the experimental platform was set as 10% of the actual knee joint torque after consideration of factors such as the lower leg mass of the experimental platform and the actual knee joint torque. And the auxiliary efficiencies of 40%, 50%, and 60% were used to set the expected force curve.

We used Formula (2) to decompose the peak force of knee torque in Figure 4b into two independent parametric sinusoids, and they represent a portion of the simplified biological knee force. By using this method, the system can better fit the torque curve of the knee extension required by the human body in the walking process through the combination of two sine curves.

$$F_d\left(t\right)=\left\{\begin{array}{cc}{F_d}^p\mathit{sin}\left(\frac{\pi \left(t-{T_d}^0\right)}{2\left({T_d}^p-{T_d}^0\right)}\right)& {T_d}^0\le t<{T_d}^p\\ {F_d}^p\mathit{sin}\left(\frac{\pi \left({T_d}^e-t\right)}{2\left({T_d}^e-{T_d}^p\right)}\right)& {T_d}^p\le t<{T_d}^e\\ 0& otherwise\end{array}\right.$$

(2)

where $F_d$ and ${F_d}^p$ are the magnitude of expected force and peak force, respectively, $t$ is the current gait cycle, and ${T_d}^0$, ${T_d}^p$, and ${T_d}^e$ are the start time, peak time, and end time of the force curve.

According to Figure 4b, the actual peak torque of the knee joint is 40 Nm. According to the parameters of the lower leg model, the expected peak was set to 40 N, 50 N, and 60 N when the exoskeleton auxiliary efficiencies were 40%, 50%, and 60%. By substituting the peak value of the expected force into Formula (2), the expected force curve of the knee joint under three different peak values was obtained and is shown in Figure 6.

3. Modeling Analysis of Bowden Cable–Fabric Force Transfer

After completing the kinematics and dynamics simulation of the exoskeleton knee joint, we evaluated the force loss in the process of force transmission between the Bowden cable and the fabric, and force transmission modeling between the Bowden cable and the fabric was set up to obtain the compensation corresponding to the friction loss.

3.1. Modeling of Force Transfer and Friction Compensation for Bowden Cable

A Bowden cable consists of a wire rope and an outer wire sheath. Because the wire rope is located in the wire sheath, there is friction in different directions during reciprocating movement. In the application of real-time control, we usually choose to use the static force transfer model based on the Coulomb friction model. Therefore, the force transmission of the Bowden cable was analyzed using the element method, and the following assumptions were made in the modeling:

• The mass and inertia of the Bowden cable are ignored;
• The wire rope moves axially along the inner wall of the wire sheath, and there is no extra-axial movement;
• The friction between the wire rope and the wire sheath is Coulomb friction;
• The elastic deformation of the wire rope is in line with Hooke’s law and within a certain range.

The static force analysis of Bowden cable is shown in Figure 7. According to the basic relationship expression of the force on the element segment of the Bowden cable [20], the transmission model of the output tension was derived as follows:

$$F_{out}=\left\{\begin{array}{cc}{F}_{in}{e}^{{-\mu }_1\theta \left(t\right)}& \dot{v}>0\\ F_{in}& \dot{v}=0\\ F_{in}{e}^{{\mu }_1\theta \left(t\right)}& \dot{v}<0\end{array}\right.$$

(3)

where $F_{in}$ and $F_{out}$ are the input tension and output tension at both ends of the Bowden cable with length L, ${\mu }_1$ is the friction coefficient between the rope and the sheath in the Bowden cable, $\theta \left(t\right)$ is the corresponding central angle at time t, and $\dot{v}$ is the acceleration.

Based on the static tension transfer model derived from Formula (3), it is easy to obtain the tension relationship between the input end and the output end of the Bowden cable. Therefore, as long as the static tension transfer model is established, the corresponding friction compensation value is designed and added to the input signal, so as to ensure that the output tension can statically track the expected tension curve. So, the given input tension signal should be equal to the expected output force plus the offset of friction.

According to the difference between $F_{in}$ and $F_{out}$, the friction force when the output force is $F_{out}$ can be obtained; then, the friction force when the output force is $F_d$ can be expressed as follows:

$$f_L=\frac{F_{in}-F_{out}}{F_{out}}{F}_d=\left\{\begin{array}{cc}{F}_d\left[e^{{\mu }_1\theta }-1\right]& \dot{v}>0\\ 0& \dot{v}=0\\ F_d\left[e^{{-\mu }_1\theta }-1\right]& \dot{v}<0\end{array}\right.$$

(4)

where $f_L$ is the friction force of the Bowden cable under the expected tension, $F_d$ is the expected tension, and $\theta$ is the bending angle of the Bowden cable.

Therefore, the actual tension signal given by the Bowden cable should be

$$F_A=F_d+f_L$$

(5)

where $F_A$ is the actual given tension of the Bowden cable.

3.2. Force Transfer Modeling and Friction Compensation of Fabric

The fabric, as the end of the force-transmitting device, is connected with the Bowden cable and the human lower limb. When the knee joint of the human lower limb is bent, the contact part of the fabric will have friction and extrusion with the knee joint, which is called the contact section of the fabric bending stage. To this end, it is necessary to conduct a micro-element force analysis on the fabric contact section. In order to better analyze the fabric contact section, the following assumptions were made:

• The weight, compressibility, and Poisson’s ratio of the fabric are ignored;
• The movement direction of the fabric is along the tangential direction of the column;
• It is assumed that the fabric sample is a homogeneous flexible material with flexural stiffness and linear elasticity.

The element force analysis of the fabric contact section is shown in Figure 8. According to the classical friction law and the Capstan equation [21,22], the tension relationship between the two ends of the fabric is as follows:

$$T_2=T_1{e}^{{\mu }_2\phi }$$

(6)

where $T_1$ and $T_2$ are the tension at both ends of the fabric, $\phi$ is the angle of the AB segment of the fabric, and ${\mu }_2$ is the friction coefficient between the fabric and the leg binding.

Through Formula (6) combined with the tension $T_2=T_1$ when the fabric is at rest, the tension control model of the input and output ends can be obtained as follows:

$$T_{out}=\left\{\begin{array}{cc}\begin{array}{c}{T}_{in}{e}^{{\mu }_2\phi \left(t\right)}\\ T_{in}\\ T_{in}{e}^{-{\mu }_2\phi \left(t\right)}\end{array}& \begin{array}{c}\dot{\epsilon }>0\\ \dot{\epsilon }=0\\ \dot{\epsilon }<0\end{array}\end{array}\right.$$

(7)

where $T_{in}$ and $T_{out}$ are the tension at the input and output ends of the fabric, $\phi \left(t\right)$ is the function of the fabric angle with time, and $\dot{\epsilon }$ is the strain rate of the fabric.

According to the difference between $T_{in}$ and $T_{out}$, the friction force when the output force is $T_{out}$ can be obtained; then, the friction force under the expected pulling force of the fabric when the output force is $F_d$ can be expressed as follows:

$$f_T=\frac{T_{in}-T_{out}}{T_{out}}{F}_d=\left\{\begin{array}{c}{F}_d\left[e^{{-\mu }_2\phi \left(t\right)}-1\right]\\ 0 \\ F_d\left[e^{{\mu }_2\phi \left(t\right)}-1\right]\end{array}\right.\begin{array}{c}\dot{\epsilon }>0\\ \dot{\epsilon }=0\\ \dot{\epsilon }<0\end{array}$$

(8)

where $f_T$ is the friction force of the fabric under the expected pulling force, and $F_d$ is the expected pulling force.

In order to compensate the nonlinear friction loss of the fabric in the process of passing around the column, the corresponding friction compensation is added to the control model as follows:

$$T_A=F_d+f_T$$

(9)

where $T_A$ is the actual given tension of the fabric.

4. Displacement Analysis of Bowden Cable–Fabric Transmission

Drawing upon the theoretical setup of force transfer loss through the friction compensation model, we move on to the accurate control of displacement. The displacement error is caused by the nonlinear deformation in the Bowden cable–fabric force transmission system and the swing of the lower leg. In the following, the impact of these factors will be modeled and compensated to achieve the flexibility and accuracy of force and displacement control through the impedance control method.

4.1. Stiffness Model of Bowden Cable–Fabric System

When the auxiliary force is generated by the Bowden cable–fabric force transmission system, it is nonlinear because of the friction between the wire rope and the wire sheath, the friction between the fabric and the knee joint, the inelastic deformation of the fabric, and other uncertain factors which exist in the force transmission system. Therefore, it is necessary to establish the stiffness model of the Bowden cable–fabric force transfer system by testing the internal flexible link, so that the controller can compensate the nonlinear deformation in the Bowden cable–fabric force transfer system.

Using the desired reference force as input, the required displacement of the system is tested, and then, the stiffness model of the force-transmitting system is obtained. In order to obtain the stiffness model, the method used was to capture the relationship between the tension and the encoder displacement during the power-assisted experiment. Because we used an in vitro experiment, the lower leg model was moved to the position at 13% of the gait cycle for the test [23]. In each test, the control motor generated a force of 20 N to 60 N at a constant speed of about 0.06 m/s and released the wire rope at the same speed. The displacement change from the motor encoder during five wire rope pull-and-release cycles under this experimental condition was recorded.

The output force of the motor was taken as the horizontal coordinate, and the displacement of the motor encoder was taken as the vertical coordinate. By drawing the scatter plot and analyzing and fitting the data, it was found that the quadratic function had the highest degree of fitting. Therefore, the quadratic function was used to fit the force–displacement curve, and the fitting results are shown in Figure 9, where the dotted line is the original data from the five experiments, and the solid line is the fitting model. The function expression is shown in Formula (10).

$$x_1=a_k{F}_k^2+b_k{F}_k+c_k$$

(10)

where $F_k$ is the output force of the motor, $x_1$ is the displacement of the motor encoder, $a_k$ is the quadratic coefficient, $b_k$ is the primary coefficient, and $c_k$ is a constant.

After data fitting, it was determined that the value of $a_k$ is −0.0017 mm/N², the value of $b_k$ is 0.3 mm/N, and the value of $c_k$ is 9.7 mm. After calculation, we obtained the regression judgment coefficient ${\mathrm{R}}^2=0.9868$, which indicates that the regression model has a good fitting ability.

4.2. The Swing Model of the Lower Leg

In the flexible exoskeleton, the force is provided by the motor driving the contraction of the Bowden cable to drive the end fabric. One end of the Bowden cable is fixed to the motor, and the other end is fixed to the fabric. In the walking process of the human body, the lower limb will continue to repeat the swing process. When the knee joint flexes, the wire rope is moved by the pull from the lower leg and then increases the length of the transmission mechanism. So, it is necessary to establish the swing model of the lower leg.

The swing model of the lower leg is mainly used to compensate for the interference caused by lower leg movement when providing assistance during walking. In order to minimize this interference, it is necessary to measure the Bowden cable–fabric length at different leg swing angles to avoid the impact on the lasso-fabric movement during human movement. In order to establish the motion model, the servo motor outputs a constant 20 N preload force, and the DC brushless motor at the knee joint drives the leg lever to swing slowly for 10 cycles. The encoder position values of the two motors were recorded. The fitting results are shown in Figure 10. The dotted line is the raw data from ten sets of experimental tests, and the solid lines are the fitting models with linear functions.

$$x_2=k_2{\theta }_2+p_2$$

(11)

where $x_2$ is the desired displacement of the wire rope, ${\theta }_2$ is the angle of the lower leg swing, $k_2$ is the primary term coefficient, and $p_2$ is a constant.

The 0 degree is in the upright position, the negative angle indicates knee flexion, and the positive angle indicates knee extension. After data fitting, the parameters $k_2$ and $p_2$ are determined to be 0.706 mm/deg and 22.17 mm. Through calculation, the regression decision coefficient ${\mathrm{R}}^2=0.998$ was obtained, indicating that the regression model has a good fitting effect.

4.3. Knee Motion Impedance Control

In the power-assisted control of exoskeletons, the application of a Bowden cable–fabric force transmission system should also be combined with impedance control. On the one hand, it can achieve accurate control of force and displacement and reduce the energy consumption of human walking; on the other hand, it can be more compatible with the law of human movement, improve user comfort, and achieve the flexibility of human–computer interaction.

In impedance control, the most common mathematical expression in the form of a dynamic equation is as follows:

$$M\left({\ddot{x}}_r-\ddot{x}\right)+B\left({\dot{x}}_r-\dot{x}\right)+K\left(x_r-x\right)=F_e$$

(12)

where $M$, $B,$ and $K$ are inertial, damping, and stiffness parameters of the impedance model; $\ddot{x}$, $\dot{x},$ and $x$ are the actual acceleration, velocity, and position of the end; ${\ddot{x}}_r$, ${\dot{x}}_r,$ and $x_r$ are the expected acceleration, velocity, and position of the end; and $F_e$ represents the difference between the expected force and the actual force of the end.

In this paper, the impedance control strategy based on the inner loop position control was adopted. In this impedance control, the end force of the Bowden cable–fabric system is controlled by the change in position. If $x_r-x=\Delta x$ is substituted into Formula (12), it can be expressed as follows:

$$M\Delta \ddot{x}+B\Delta \dot{x}+K\Delta x=F_e$$

(13)

For position-based impedance control, there are two main components: one part is the position control of the inner loop, which is generally realized by proportional integral differential (PID) control; the other part is the impedance control of the outer loop. When the Bowden cable–fabric force transfer system is applied to the body part, the motor outputs the expected force. Through the end of the force transfer device, the force is output to the lower limb, while also creating a contact force between the body and the device. An impedance controller based on force feedback is used to input the actual force into the impedance model and convert it into the displacement correction of the Bowden cable–fabric. Then, the expected position, actual position, and displacement correction are input into the inner loop position controller, whose main function is to accurately track the input reference position.

This can be obtained through the transformation of Formula (13):

$$F_e\left(s\right)=\left(M{s}^2+Bs+K\right)x\left(s\right)$$

(14)

The relationship between force and displacement, that is, the transfer function, can be expressed as follows:

$$H\left(s\right)=\frac{\Delta x}{F_e}=\frac{1}{\left(M{s}^2+Bs+K\right)}$$

(15)

$F_e$ is the force input into the impedance model, that is, the error between the expected force and the actual output force. And $\Delta x$ is the amount of displacement that is converted by the impedance control loop to the desired amount, and then, it compensates to the position control of the inner loop. Because the system is slow in actual motion and the inertia parameters are relatively small, the effect in the system is negligible.

5. Simulation and Experimental Research of Bowden Cable–Fabric Force Transfer Control Based on Friction/Displacement Compensation

In Section 3 and Section 4, we described a model which compensates the force transfer loss and displacement loss of an exoskeleton in the process of human walking and introduced the impedance control model to achieve flexible and accurate control. In this section, the effect of the impedance control system on force and displacement control will be evaluated with a simulation and experiment.

5.1. Simulation of Bowden Cable–Fabric Force Transfer Control Based on Friction/Displacement Compensation

The assistance of knee joint motion is controlled based on the position inner loop and impedance outer loop control algorithm, and the position inner loop is PID control [24]. In actual discrete control systems, such as a servo motor, the differential can easily amplify the noise; compared with PID control, proportional integration (PI) control is simpler in the modeling of the correction link and can more easily achieve zero pole cancellation. Therefore, the parameter setting of PI control is also more convenient. In order to test the position-based feedforward impedance control feedback system, PI position control was used in the inner loop to track the transmission force and Bowden cable–fabric displacement. The impedance control block diagram based on inner loop PI, as shown in Figure 11, was adopted as a whole, in which the lower leg model is a model of force related to angle and displacement.

According to the impedance control block diagram based on the inner loop PI established in Figure 11, the control module of the Bowden cable–fabric force transmission system was built using the Simulink module in MATLAB. It consists of a Bowden cable friction compensation module, fabric friction compensation module, stiffness model module, lower leg swing angle compensation module, servo motor module, impedance module, and lower leg model. The angle of the knee joint was input into the lower leg swing angle compensation module and the fabric friction compensation module. The expected force curve was taken as the input for the other modules, and the output voltage of the motor was limited and protected, as shown in Figure 12.

The desired force curve is the tension value when the peak value is 40 N, and the joint angle adopts the angle value in Formula (1). Since there is a three-phase motor module in the control system, it needs the Powergui module to discretize it, so the maximum sampling step length was set to 0.00001 s, and the automatic solver was used to solve it, and the influence of different impedance parameters on the control performance of the system was analyzed.

When other parameters are unchanged, when the damping parameter is set to a larger value, although the system can quickly produce a response, it will produce a large overshoot and produce a certain jitter when the motor is stable and cannot achieve stability control. The value of the stiffness parameter reflects the degree of rigidity or flexibility of the system in contact with the external environment. When the value of the stiffness parameter is large, the output value will overshoot for a long time near the zero point, and the desired control effect cannot be achieved.

It is assumed that the joint angle is a fixed rotation angle, which is the ideal state. However, the actual joint angle in practical application often deviates from the expected angle. After the experimental test and adjustment, the damping parameter $B$ was set to 0.1 and the stiffness parameter $K$ was set to 0.3. In order to verify the control effect of the impedance control model on the system output, the simulation comparison of the impedance-free control was conducted based on the simulation model in Figure 12, as shown in Figure 13. We can see that compared with impedance-free control, the output force curve with impedance control is more in line with the expected force curve and is more flexible when the force changes. And there is no large fluctuation or excessive adjustment phenomenon, so the system output is more stable and flexible.

5.2. Experimental Verification and Result Analysis

In order to verify the effectiveness and accuracy of the Bowden cable–fabric force transfer control based on friction/displacement compensation, experiments were conducted according to the experimental platform shown in Figure 2. In the experiment, a 48 V DC power supply (Mingwei, LRS-400-48) was used to power the servo motor (Pulse Tower Intelligence, RD-X6). A 24 V DC power supply (Mingwei, LRS-800-24) supplied power to the DC brushless motor (Times Superstar 57 BL, 75S10-230) at the knee joint. The servo motor is directly controlled by the Single Chip Micyoco (Wildfire, Sunny STM32F407IGT6); the DC brushless motor is controlled by the SCM and a brushless motor drive board (ZM-6505A). In addition, the DC brushless motor is connected to the reducer (Times Superstar, PX6-57-1:20) to more accurately simulate knee joint movement. The servo motor outputs the auxiliary force through the Bowden cable and fabric to the lower leg. In the process of force transmission, there are internal friction losses of the Bowden cable and friction losses between the fabric and the clothing at the knee joint. The tension sensor (constant distance sensor, HYLY-019) collects the force signal and inputs it into the SCM through the analog-to-digital conversion module (Wildfire, AD7192PC). The PC is connected to the SCM through the serial port as the upper computer, displaying the data collected by the tension sensor and processed by the SCM. The compliance and accuracy of the control are judged by the data of the start and end force sensors. The end force sensor data are only used as reference information to verify the following effect of the control system and do not directly participate in the system as a feedback quantity. The tension information of the end of the control system is calculated by injecting the tension value of the start force sensor and the joint angle value into the force transfer model. The actual motion displacement of the Bowden cable–fabric is returned by the motor encoder output value. The in vitro Bowden cable–fabric force transfer experimental platform is shown in Figure 14, in which Figure 14a is the overall picture of the experimental platform, Figure 14b is the range of lower leg model movement, and Figure 14c is the schematic diagram of friction at the knee joint model. Figure 14b shows that the lower leg can be rotated 90°. In this experiment, there is only one lower leg model, and it is at the position of 0°; another leg model is added to indicate the limit position. Figure 14c simulates the friction of clothing and fabric during movement at the knee joint.

The tension curve with a peak value of 40 N was selected to conduct the experiment. The information of the end force sensor was recorded for verification and compared with the simulation. The results are shown in Figure 15.

By comparing the simulation and experiment when the peak value was 40 N, it was found that the internal friction causes a small error when the motor starts loading; Because the force transfer model is related to acceleration and strain rate, the force will produce a small sudden change at the peak. Because the end force sensor is cantilever-mounted, taking into account the weight of itself and the wire rope, the output tension will remain in the range of less than 1 N jitter when the expected tension is zero.

In order to test the force tracking effect of the force transmission system under different force peak values, experiments with peaks of 50 N and 60 N were selected at the same time. For each peak value, the force signal measured by the end sensor was recorded, and the force errors were calculated, as shown in Figure 16.

Under the three different peak values of 40 N, 50 N, and 60 N, the relative root-mean square errors of the Bowden cable–fabric force transmission control based on friction/displacement compensation are 2.53%, 2.16%, and 2.07%, respectively. The results demonstrate that the control method we introduced has good reliability in precision and flexibility.

6. Conclusions

This paper focuses on the investigation of the accurate control of force/displacement caused by friction loss and elastic deformation in the process of flexible exoskeleton Bowden cable–fabric force transfer. We conducted a kinematic analysis of human lower limbs, force transfer analysis, and transmission displacement analysis of the Bowden cable–fabric and presented a friction model of the Bowden cable–fabric system. Based on the friction/displacement compensation method, a simulation and experiment based on the Bowden cable–fabric force transfer control were conducted to evaluate the performance under different settings. Our conclusions are as follows:

• To account for the friction loss in the force transmission of Bowden cable–fabric, a tension model under quasi-static conditions was derived, which can be used as feedforward force compensation in the control system;
• The stiffness model of the transmission system and the inherent displacement model of the leg swing were introduced to express the displacement hysteresis in the process of Bowden cable–fabric transmission, and impedance control was added to improve the flexibility of human–machine interaction;
• The force/displacement compensation was introduced into the system control model, and the output force with impedance control was compared to the force without impedance control. In addition, an in vitro Bowden cable–fabric force transfer experimental platform was built to evaluate the effectiveness of the force/displacement compensation and impedance control presented in this paper. The experimental results show that the relative root-mean-square errors of friction/displacement compensation for Bowden cable–fabric force transfer control are 2.53%, 2.16%, and 2.07% when the peak values are 40 N, 50 N, and 60 N, respectively. These results suggest good effectiveness of the force/displacement compensation and impedance control methods we proposed in this paper.