Modeling and compensation control of a single tendon-sheath actuated system with time-varying parameters

Abstract

The tendon-sheath actuated system (TSAS) has been widely adopted in many cases due to its merits of compliance, dexterity, and remote transmission. However, realizing precise control of the distal end poses remains a significant challenge due to inherent nonlinear phenomena caused by friction and hysteresis. In this study, a novel adaptive sliding mode compensation control (ASMCC) scheme based on the inverse transmission model was proposed to achieve accurate trajectory tracking of the TSAS. Based on the Coulomb friction model, the static and dynamic models of TSAS were analyzed and the correctness of system’s output displacement and output force characteristics was confirmed through simulation and experimental results. Based on the assumptions of the constant curvature of the sheath and the pre-tensioned tendon, the inverse model was calibrated by offline measurements with sensors installed at the proximal end. A novel control strategy was developed based on the inverse model combined with an adaptive algorithm according to Lyapunov stability theory with sliding mode controller for the online estimation of time-varying parameters. Finally, trajectory tracking experiments were conducted with/without external springs to evaluate the feasibility of the proposed control strategy, and its effectiveness and accuracy in parameter identification and accuracy compensation for the single tendon-sheath actuated system was confirmed.

Introduction

The tendon-sheath actuated system (TSAS) is composed of a hollow sheath and an internal steel tendon, where the two ends of the sheath are generally fixed and the inner tendon approximately axially slides inside the sheath. Taking advantage of its inherent compliance and remote power transmission, the TSAS has been extensively applied in various domains such surgical operation treatment^1,2, medical rehabilitation^3–5, and functional areas of human assistance^6,7. Whereas, due to the elastic deformation of the inner steel tendon and its friction with the outer sheath, nonlinear problems such as hysteresis, backlash, and dead-band arise. These nonlinear characteristics not only reduce the transmission efficiency of TSAS, but also seriously affect its precise transmission^8–10. Hence, to overcome these undesirable weaknesses in tendon-sheath actuated systems and satisfy new application demands, it is crucial to conduct the dynamic characteristic analysis and compensation control research.

The mathematical models of tendon-sheath actuated systems have been explored to reveal their transmission characteristics. Kaneko et al. originally proposed a simple transmission model for a single-tendon-sheath actuated system, but they only explored tensile force and ignored the position error caused by elastic deformation of internal steel tendon¹¹. Based on the Coulomb friction model, Chen et al. studied nonlinear transmission problems and discovered the transitive relationship between force and position of a single-tendon-sheath actuator¹². However, considering that tendon sheath transmission is a dynamic process, it is unilateral to describe the dynamic behavior of hysteresis and backlash with a static model. To deal with the viscoelastic effects of the transmission system and explore the force–deformation properties of tendons, Palli et al. employed the LuGre-like dynamic model to describe the friction effects generated by the interaction between the tendon and its path curvature^13,14. On this basis, Wu et al. developed a general mathematical suitable for tendon-sheath systems under various loading conditions and successfully described the displacement characteristics of TSAS^15,16. Nonetheless, it is worth noting that the assumed constant sheath shape and the required sensors to be placed at the distal end of the system are the drawbacks of this method. To study the distribution of frictional force in any form of the tendon sheath, Liu et al. experimentally analyzed the transmission characteristics of two-dimensional tendon sheath with analytical modeling methods and the model’s accuracy was evaluated ¹⁷. However, this technique required a large quantity of computation and the calculated values of the dynamic model differed significantly from experimental results.

It is significant to develop the efficient tracking control for TSAS^5,18–20. The technologies for nonlinear compensation and model parameter identification have been widely concerned. In order to solve the model’s uncertainty caused by the variation of the tendon-sheath bending angle during the transmission process, Shao et al. designed a static model to perform proportional-integral-derivative (PID) compensation control on tendon-sheath artificial muscles^21,22. To estimate the friction force and improve the position tracking performance of a tendon-sheath actuated surgical robot, Do et al. proposed a feedforward tracking control based on the inverse hysteresis model²³. Herbin et al. proposed a feed-forward compensation control method to minimize the influence of friction in the tendon-driven rehabilitation exoskeleton²⁴. Similarly, Yin et al. developed a closed-loop control algorithm to realize grasping tasks for a tendon-sheath transmission-based robotic hand²⁵. This compensation control based on inverse models could partly ensure the accuracy of the system, but it could not provide an ideal result in case of interference and parameter variations. Wang et al. proposed an adaptive control scheme for a tendon-sheath actuated manipulator and identified key parameters in real-time²⁶. Nevertheless, this control algorithm might result in instability when the system was severely disturbed. In order to suppress the generated interference torque of the tendon-sheath actuated manipulator, Zhang et al. proposed a sliding mode control (SMC) algorithm²⁷. SMC had the merits of robustness and was insensitive to changes in system structure, but its inherent chattering problem could not be ignored. Although various feasible techniques have been proposed for the control of nonlinear tendon-based transmission systems, the uncertainty of system models and nonlinear problems were seldom explored.

On the basis of previous studies, it is feasible to integrate the adaptive control scheme with SMC techniques for the purpose of investigating the nonlinear tendon-sheath actuated system with parametric uncertainties. The study aims to unveil a developed adaptive sliding mode compensation control algorithm with an inverse model and realize the efficient tracking control of the TSAS. The adaptive sliding mode control (ASMC) algorithm is an ideal strategy with a simple control structure and strong robustness and can adaptively adjust the parameter changes of tendon-sheath-based transmission systems. The main contributions of the paper are presented as follows. Firstly, compared with an existing model⁷, the improved dynamic model of the single tendon-sheath transmission system could more accurately describe its behaviors. Secondly, the compensation control method was based on an inverse model with an excellent tracking performance and could ensure the stability the closed loop system. Thus its control could be easily implemented without any feedback at the distal end. Thirdly, to examine the effectiveness of the proposed control strategy based on an inverse transmission model, a single-tendon-sheath driven experimental platform was constructed.

Model descriptions

In the connection of structural components in the tendon-sheath transmission system, the distal force and position are significant. Therefore, it is necessary to explore dynamic and static models of TSAS based on its structural and transmission characteristics. Advanced control algorithms can ensure the basic requirements of accuracy. The model of the tendon-sheath transmission system under general conditions is conducted as follows.

Static model of the tendon-sheath actuated system

As illustrated in Fig. 1, the single-tendon-sheath system can be simplified into any shape with fixed ends and a smooth middle, where the proximal end serves as the power input and the distal end is directly connected to the load through a hollow outer sheath. The direction from the power output to the input along the tendon-sheath is defined as the positive direction, and L, F(0, t), x(0, t), F(L, t), and x(L, t) are the total length, the input force, input displacement, output force, and output displacement of the tendon-sheath transmission system, respectively. It is worth noting that due to the equal tension between adjacent tendon-sheath microelements, the system structure can be equivalent to any shape of tendon-sheath transmission in a plane.

Fig. 1.

Force balance analysis of the tendon-sheath microelement.

Taking a tendon-sheath element at any length s for force analysis, when the element is small enough, its force relationship can be expressed as:

1

where N(s, t), f (s, t) and μ respectively denote the positive pressure, friction force and friction coefficient between the inner tendon and outer sheath; F(s,t) is the axial tension of the inner tendon; dθ(s, t) and ds represents the deflection angle and arc-length of the tendon-sheath element, respectively; λ = sgn(v) and v is the system transmission speed. Note that the tendon-sheath works in the contraction phase when the output speed v is positive, and it works in the relaxation phase when the output speed v is negative. Otherwise, the tendon-sheath is working in the dead-band phase. Subsequently, by eliminating f and N(s,t) in Eq. (1), we can get that:

2

where κ(s, t) represents the curvature of the tendon-sheath.

Assuming that the tension and deformation of the inner tendon comply with Hooke’s law, the deformation of the tendon-sheath element under tension can be described as:

3

where E and A denote the elastic modulus and cross-sectional area of the internal steel tendon, respectively.

Considering that the input displacement of the tendon-sheath changes direction at t₀, its output displacement unable respond simultaneously due to the presence of friction resulting in motion hysteresis or transmission dead-band. In addition, the output displacement begins to respond until the sum of the frictional force and the input tension is less than the output tension at t₁. Thus, the transmission characteristics of the tendon-sheath can be summarized as:

4

5

where t₀ < t < t₁ and x(0, t₀) represents the initial input displacement of the tendon-sheath at the dead-band phase; Δs(L, t₀) and Δs(L, t) denote the elongation of the entire internal steel tendon at time t₀ and t, respectively. It should be noted that the values of F(0, t₀) and Δs(L, t₀) will be updated when the TSAS enters into a new dead-band phase.

The curvature κ(s, t) is related to position and time, requires a large quantity of computation, and increases the difficulty of measurements due to its time-varying property. For the purpose of simplification, the total curvature θ of outer sheath is expressed as:

6

Consequently, Eqs. (4) and (5) yield:

7

8

where Δx represents the input displacement variation at the dead-band phase and Δx(0, t) = x(0, t) − x(0, t₀).

Therefore, the relationship between the input displacement, output displacement, and elongation of the tendon-sheath can be described as:

9

Considering the above analysis, the transmission characteristics of the single tendon-sheath are related to the full curvature of the outer sheath, the elastic modulus of the inner tendon, the friction coefficient and the load stiffness.

Dynamic model of the tendon-sheath actuated system

The static model mentioned above only describes the relationship between displacement and force at the distal end of the tendon-sheath transmission system, but it cannot accurately describe dynamic characteristics such as hysteresis effect and the influence of velocity on friction force. Therefore, it is necessary to further investigate and establish an accurate dynamic model.

As shown in Fig. 2, the tendon-sheath transmission system is regarded as several sequentially connected spring-mass-damper systems, where m_i, c_i, k_i, f_i, and x_i respectively denote the mass, damping coefficient, stiffness coefficient, dynamic friction force, and displacement of the i-th tendon-sheath element. Besides, F_in, x_in, F_out, and k_L represent the input force, input displacement, output force, and stiffness coefficient of the load spring of the system, respectively. It should be noted that the direction of force and displacement along the tendon sheath path from the distal end to the proximal end is considered to be positive, and the displacement of each internal tendon element is zero as well as the tensile is equal to F₀.

Fig. 2.

Analysis of transmission characteristics of the dynamic model of the TSAS.

According to the previous study²⁸, an analysis of the i-th element in Fig. 2 was conducted. This micro-element was influenced by the springs and dampers on both left and right sides, indicating the combined effect of the i-th and (i + 1)-th segments. Hence, the axial force of the i-th tendon-sheath element can be expressed as follows:

10

where 1 ≤ i ≤ n, x₀ = x_in, k_n+1 = k_L, and c_n+1 = 0. represents the inertial force of the differential element ( is the acceleration of the element); and respectively indicate the restoring force of the left-side spring and the damping force of the left-side damper; and respectively represent the restoring force of the right-side spring and the damping force of the right-side damper; the negative sign indicates that the direction of these forces is opposite to the direction of motion; k_i and k_i+1 are respectively the stiffness coefficients of the springs on the left and right sides; c_i and c_i+1 are respectively the damping coefficients of the dampers on the left and right sides; x_i−1, x_i, and x_i+1 denote the positions of the adjacent differential elements; , , and denote the velocities of the adjacent differential elements.

Since the nonlinear variation patterns of displacement and force transmission characteristics are consistent, only the force transmission characteristics are analyzed. For the first segment of the rope microelement, c₀ = 0, k₀ = 0, with x₀ and F₀ corresponding to the input displacement x_in and the input tension F_in, respectively. For the n + 1 segment of the microelement, m_n+1 = 0, c_n+1 = 0, k_n+1 = k_L, and f_n+1 = 0, with x_n and F_n+2 corresponding to the output displacement x_out and the output tension F_out, respectively. By applying Hooke’s Law, the dynamic equilibrium equation for the n + 1 segment of the rope microelement can be established, from which the force output relationship of the lasso transmission system can be derived:

11

12

where ki and ki+1 are respectively denote the stiffness coefficients of the springs on the left and right sides, xi−1 and xi represent the positions of the adjacent differential elements.

The structural parameters of each spring-mass-damper system are the same, so Eq. (10) can be rewritten as:

13

where M is the n-order diagonal matrix with M = diag(m₁, m₂,…, m_n); X = [x₁, x₂,…, x_n]^T; P is the n-dimensional column vector with ; and F = [f₁, f₂,…, f_n]^T; K and C are n-order Square Matrixes and can be expressed as:

14

where K(i, j) = -2k and C(i, j) = -2c when i = j < n; when |i-j|= 1, K(i, j) = k and C(i, j) = c; when i = j = n, K(i, j) = -2k and C(n, n) = -c; otherwise, K(i, j) = C(i, j) = 0.

Assuming that the stiffness of the bristles on one side of the surface is much greater than that on the other side, the equivalent analysis of frictional force can be conducted by choosing the deformation on the side with lower stiffness. Lugre friction model can effectively describe the hysteresis effect, Stribeck effect, and pre-sliding phenomena of the relative moving surface, the friction force F in Eq. (13) can be described as follows:

15

where z denotes the average deformation of bristles on the contact surface of each tendon-sheath element; v represents the velocity of the tendon-sheath element relative to the hollow sheath; F(t) is the frictional force applied on the tendon-sheath element from the hollow sheath; V_s indicates Stribeck velocity; σ₀, σ₁ and σ₂ denote the stiffness, micro-damping coefficient, and viscous friction coefficient of the bristles, respectively.

Subsequently, based on the input and output tensions of the system with the tension between tendon-sheath microelements, a column vector is expressed as:

16

where 1 ≤ i ≤ n and ; when i = n + 1, there will be t_i = k_Lx_n.

The i-th tendon-sheath microelement needs to bear the positive pressure N_i from the inner wall of the hollow sheath due to the presence of axial tension and bending angle, which can be described as:

17

In Eq. (15), the Coulomb friction vector F_c and the static viscous friction force F_s can be written as:

18

where F_ci and F_si respectively denote the Coulomb friction and the static viscous friction force applied on the i-th tendon-sheath element and can be expressed as:

19

where μ_s and μ_d respectively represent the static friction coefficient and the dynamic friction coefficient of the tendon-sheath.

Hence, based on Eqs. (13), (15), and (18), the dynamic state of TSAS can be described as:

20

Meanwhile, the input tension, output tension, and output displacement of the system can be described as follows:

21

Model validation

The experimental principle and the physical platform for verifying the tendon-sheath transmission system are shown in Fig. 3. The platform mainly includes a dSPACE-RCP system, several tension/displacement sensors, a set of motor and deceleration device, a tendon-sheath transmission mechanism, and a load spring. In particular, the reduction ratio of the harmonic reducer is i_m = 120:1, the radius of the motor turntable is r₁ = 27.1 mm, and the stiffness of the load spring k_L = 1.579 N/mm. Besides, the equivalent rotational inertia of the system at the turntable is J = 0.1296 g·m², and the equivalent damping of the system along the tendon-sheath transmission direction is B = 0.007 N s/mm.

Fig. 3.

(a) Experimental principle and (b) physical platform for the model verification of the single TSAS.

The motor was connected to the harmonic reducer to provide the power input for the TSAS. The proximal end of the tendon-sheath was fixed to the harmonic reducer turntable, and the other end was connected to the loading spring through the hollow sheath. Tension sensors and laser displacement sensors were installed at the proximal and distal ends of the system. Two laser displacement sensors were respectively used to measure the input and output displacements of the tendon-sheath and two S-type tension sensors were respectively used measure the input and output tension of the tendon-sheath. In the meantime, the dSPACE system platform collected the sensor data in real-time through the DS2003 analog acquisition board and controlled the motor to actuate the tendon-sheath transmission mechanism to move strictly according to the predetermined trajectory. Finally, the host computer software was connected to the DS1007 board of the dSPACE via Ethernet cable to collect and record the data, which were compared and verified with the model simulation results.

In the experiment, the motor was operating in position mode with a given input displacement of x_in = 10sin(ωt − π/2) + 10 mm. Relevant parameters of the tendon-sheath transmission mechanism are illustrated in Table 1. Table 2 shows the experimental variables of angular frequency, curvature radius of the sheath, and total curvature of the sheath for four different input displacement signals. Figure 4 shows the comparison of simulation calculations and experimental results of the static and dynamic models of the tendon-sheath actuated transmission system.

Table 1.

Parameters of the TSAS.

Parameters	Values	Units
Diameter of inner tendon d	1.2	mm
Elastic modulus of inner tendon E	60	Gpa
Outer sheath inner diameter d1	0.8	mm
Outer sheath outer diameter d2	5	mm
Frictional coefficient in static model μ	0.125	–
Number of simulation microelements n	50	–
Bristle stiffness σ0	100	N/mm
Micro damping coefficient σ1	0.31622	N s/mm
viscous friction coefficient σ2	4.0 × 10–4	N·s/mm
Stribeck velocity νs	0.01	mm/s
Dynamic friction coefficient μd	0.125	–
Static friction coefficient μs	0.13	–
Damping coefficient c	0.1	N s/mm
Tendon sheath linear density ρ	0.25	kg/(100m)

Table 2.

Conditions of different input displacement signals.

Groups	Angular frequency ω (rad/s)	Radius of curvature R (mm)	Total curvature of the sheath θ (rad)
a	1	250	π
b	2	250	π
c	1	200	π
d	1	200	π/2

Fig. 4.

Model validation results of experimental groups (a–d) under four different conditions.

The static and dynamic models of the tendon-sheath transmission system can well describe its actual motion characteristics (Fig. 4). The actual output displacement of the system is slightly smaller than theoretical calculation values. The difference might be interpreted as follows. The elongation of the steel tendon outside the sheath caused by tension was not considered in the experiment. Additionally, the dynamic model of the improved transmission system yielded the similar data to the calculation results of the static model. The similarity might be interpreted as follows. Although the dynamic model considered some microscopic characteristics, the transmission speed of the system was relatively small and the mass of the rope was relatively light. Compared with the existing tendon sheath transmission dynamics model based on the Lugre model⁷, the improved dynamics model in this study could more accurately describe the tension transmission characteristics measured in the experiment, especially at Point P illustrated in Fig. 4. In general, the comparison between experimental data and calculation results confirmed that it was feasible to use the displacement at the proximal end of the tendon-sheath as the input in the dynamic model.

Compensation control based on an inverse model

The obvious nonlinear problems of the tendon-sheath seriously affect its transmission accuracy and various novel closed-loop control methods based on end sensors are limited by its limited installation space and extreme working environments¹⁶. Therefore, the high-precision compensation control without end sensors is particularly important for the tendon-sheath transmission system.

Inverse model and parameter identification

The sensorless compensation control method proposed in this study was required for realizing the transmission characteristics of TSAS. Based on the expected output signal of the system, the friction force between the sheath and the steel tendon as well as the elongation of the steel tendon were calculated to adjust the system’s input in real-time. Subsequently, the obtained static model was chosen to compensate and control the tendon-sheath transmission system. Given that the load spring is linearly correlated with a stiffness of k_L and the expected output force of the tendon-sheath is F_outd, according to Eqs. (8) and (9), the output displacement of system inverse model can be described as:

22

It should be noted that the input of the TSAS static model is the displacement x_in at the proximal end, and its output is the tension F_out at the distal end. On the contrary, the expected output force of the inverse model in Eq. (22) is regarded as the input of the system and the required input displacement is considered as the output of the inverse model.

The proximal end of tendon-sheath was fixed to the motor reducer and its distal end passed through the sheath and was connected to a linear load spring (Fig. 5). The motor was installed on the harmonious gear reducer and drove the inner tendon to pull the load spring through the turntable. When the influence of internal tendon elongation on the viscous friction of the tendon-sheath is neglected, the single TSAS can be expressed as:

23

where τ_in denotes the system input torque, J and r₁ respectively indicate the equivalent rotational inertia and the radius of the tendon turntable, B represents the equivalent damping, and F₀ is the pre-tightening force of the tendon-sheath.

Fig. 5.

Structural diagram of the single TSAS.

The subsequent compensation control is based on the output displacement of the inverse model and the stiffness of the harmonic reducer driver is relatively high. Therefore, with the input displacement of the inverse model, the system can eliminate the influence of frictional force. Then, Eq. (21) can be simplified as:

24

When the motor is operating in the speed mode, the system input torque τ_in can be formulated as:

25

where the equivalent stiffness k_r = 2πr₁k_m/(60i_m) that i_m represents the reduction ratio of the harmonic reducer and k_m denotes the equivalent stiffness of the tendon-sheath at the proximal.

The schematic diagram of TSAS at different phases is illustrated in Fig. 6. Figure 6a shows the tension transmission characteristics of the tendon-sheath transmission system and Fig. 6b shows the system stiffness at the proximal end. As shown in Fig. 6a, the system state is in a counter-clockwise direction (a-b-c-d-e-f-a); the key points (Points a and d) denote the midpoints of the dead-band phase; Points b, c, e, and f represent the transition critical points between different phases of the system. Besides, the tension of the steel tendon at Point a is F₀ and the displacement of the steel tendon is zero. The slope of each line segment in Fig. 6b represents the equivalent stiffness value k_m of the system’s input. The corresponding values in each phase are treated as follows.

Fig. 6.

Schematic diagram of the TSAS in different phases: (a) F_in-F_out mapping and (b) x_in-F_in mapping.

According to Eq. (7), the input tension of the tendon-sheath at Points b and f can be expressed as:

26

where F_inf represents the input tension at the proximal end of point f.

Similarly, the input forces of tendon-sheath at Points c, d, and e are respectively described as:

27

where x_outc denotes the output displacement of the tendon-sheath at Point c, and noting that the output displacement of the system at Points c, d, and e is the same since the existence of displacement dead-band.

Moreover, substituting Eqs. (26) and (27) into Eq. (8) yields the elongation of the internal tendon when the system is respectively in Points b, c, e, and f. Then the corresponding values of Δs_b., Δs_c, Δs_e, and Δs_f can be expressed as:

28

According to Eqs. (26) and (28), the equivalent stiffness k_m of the input end in the f-b phase can be rewritten as:

29

Similarly, it can be concluded that the value of k_m in the c-e phase is the same as that in the f-b phase. Furthermore, k_m in the b-c phase can be depicted as:

30

where δ_bc = k_L[exp(μθ) − 1]L.

From Eq. (30), it can be seen that the equivalent stiffness of the b-c phase is independent of the output displacement x_outc, and the elongation of the inner tendon δ_bc is neglectable compared to its stiffness. Therefore, Eq. (30) can be simplified as:

31

Similarly, k_m in the e–f phase can be described as:

32

In general, the equivalent stiffness k_m at the proximal end can be expressed as:

33

According to the relationship between k_m and k_r in Eq. (25), it can be concluded from Eq. (33) that:

34

By substituting the given parameter into Eqs. (33) and (34), the unknown parameter values of the system at each phase were obtained (Table 3). Subsequently, tension and displacement data at the proximal end of the tendon-sheath were collected. With the linear fitting method, the offline identification of the parameter k_m was performed. The identification results are depicted in Fig. 7.

Table 3.

Values of k_m and k_r in each tendon-sheath transmission phase.

Parameters	Contraction phase	Dead-band phase	Loosening phase
km (N/mm)	2.338	175.1	1.066
kr	0.553	4.141	0.0252

Fig. 7.

Identification of equivalent stiffness parameters of the target system.

The slopes of the curves in Fig. 7 denote the equivalent stiffness value k_m at the input end of the system. In particular, the identification results of k_m in the e-f phase, f-b phase, b-c phase, and c-e phase were 0.96 N/mm, 101.96 N/mm, 2.35 N/mm, and 220.63 N/mm, respectively. The identification results were close to the corresponding values in Table 3, indicating that the actual identification results of the equivalent stiffness were almost consistent with the theoretical values. Unfortunately, the identification results of the dead-band phase were slightly different from the results of Table 3. The slight difference might be interpreted as follows. The short duration of the dead-band phase and the significant input force variation of the tendon-sheath resulted in significant measurement errors of the sensor. The value of parameter k_r should be selected according to the characteristics of the system. According to Eq. (29), a small equivalent stiffness k_m may reduce the sensitivity of the system and cause system chattering. However, the identification results showed that excessive equivalent stiffness k_m caused a significant deviation from the true value, so it was difficult for the system to cope with external disturbances. Hence, the two parameters were chosen as k_m = 52 N/mm and k_r = 1.23 based on the consideration of the stability and accuracy of the target system.

Considering the errors caused by external interferences and the inherent modeling errors, it can be concluded from Eqs. (24) and (25) that:

35

where d(t) represents the external disturbance, and it is assumed that there is a constant D that satisfies |d(t)|< D.

The system state variables are given as , and the Eq. (35) can be rewritten as:

36

where B = [0,0, k_rr₁²/J] ^T; E = [0,0, -d(t)] ^T; .

Controller design

From Eqs. (35) and (36), it can be obtained that:

37

where ϕ_i (i = 1, 2, 3) represents the generalized unknown parameters with ϕ₁ = J/k_rr₁², ϕ₂ = k_L/k_r, and ϕ₃ = B/k_r; d₁ = Jd(t)/k_rr₁² denotes the generalized external disturbance and there exists a positive constant D₁ that satisfies |d₁|< D₁.

Assuming the expected output force of the TSAS is F_outd, the difference between the output displacement x_inv of the system inverse model and the actual input displacement x_in is the error e. Thus, the error vector of the system is given as:

38

According to Eqs. (37) and (38), the control object can be described as

39

Given the estimated value of ϕ_i is , then its corresponding estimation error can be expressed as:

40

According to the system error vector of Eq. (38), a sliding surface is constructed as²⁹:

41

where c₁ and c₂ represent the parameters of the sliding surface with c₁ > 0 and c₂ > 0, which determine the dynamic performance of the system on the sliding surface.

After both sides of Eq. (41) are multiplied by parameter ϕ₁, taking its derivative yields:

42

Employing the exponent approaching law, can be described as:

43

where ε and k_s denote gain coefficients and are the constants satisfying ε > D₁ and k_s > 0.

After ϕ_i is replaced with , the unknown parameters are updated according to the following adaptive laws:

44

where ψ_i > 0 (i = 1,2,3) represents the adjustment parameters.

Based on Eqs. (42–44), the adaptive sliding mode controller of the system can be obtained as follows:

45

In order to improve the system’s response performance, a feedforward controller based on the transfer function G(s) of the harmonic drive was designed for compensation. The transfer function of the harmonic reducer could be identified with the collected data x_inv (mm) and u (rpm) in MATLAB software, G(s) is expressed as follows:

46

The constant term in the denominator is small because the high reduction ratio of the harmonic reducer results in substantial stiffness. G⁻¹(s) is represented as:

47

where the integral term 42.25s constitutes the main feedforward component.

With the reduction ratio of the harmonic reducer (i_m = 120:1) and the radius of the coupling disk (r₁ = 27.1 mm), the relationship between and the motor speed u is established as:

48

Therefore, the feedforward controller was designed as:

49

where

50

where k_V and k_A respectively represent the damping gain and inertia gain of the feedforward system, and their specific values are determined with offline identification methods; s_L denotes the Laplace operator.

Based on Eqs. (47), (49), and (50), the feedforward controller of the tendon-sheath transmission system can be described as:

51

Therefore, with the adaptive sliding mode compensation controller Eq. (45) and feedforward controller Eq. (51), the controller of the system can be re-expressed as:

52

Stability analysis

Take the candidate Lyapunov function of the system as:

53

Hence, the time derivative of Eq. (48) will be written as:

54

Each phase of the tendon-sheath transmission process is strictly carried out in a certain order. Therefore, it can be considered that the unknown parameter ϕ_i is bounded and slowly varies with . Besides, based on Eq. (40) and Eqs. (42)- (45), the Eq. (54) can be re-expressed as:

55

where ϕ₁ > 0, k_s > 0 and |d₁|< ε, indicating that V is positive and is constant. Therefore, the closed-loop control system is stable.

Experimental setup and results

Based on the above analysis, the flowchart diagram of adaptive sliding mode control based on feedforward compensation is shown in Fig. 8. In general, for the sake of overcoming system’s nonlinearity and resisting external disturbances, the inverse model of the system is first utilized to derive the desired input displacement signal. Then, the sliding mode compensation control is implemented based on the displacement feedback signal x_in from the displacement sensor at the system input (drive end), and a feedforward compensation controller based on the transfer function of the reducer has been added to reduce the pressure on the sliding mode controller. Finally, precise compensation for the tendon-sheath transmission system is realized through the controller designed in this study.

Fig. 8.

Control flowchart of the proposed ASMCC.

Verification the of proposed controller

To demonstrate the stability and accuracy of the proposed control scheme, a variable frequency signal was introduced as the output force for tracking performance experiments with the testing platform shown in Fig. 3. The reference input signal was F_outd = 18sin(ωt − π/2) + 43 N, and its angular frequency ω within one cycle was 2 rad/s and 1 rad/s for 0 ~ 2π seconds and 2π ~ 4π seconds, respectively. The total curvature, length, and preload of the tendon were θ = π, L = 785 mm, and F₀ = 25N, respectively. The parameters of the ASMCC scheme were chosen as c₁ = 900, c₂ = 1, ε = 1, and k_s = 1115. The initial and bounded values of system parameters are selected as shown in Table 4, and their corresponding adjustment coefficients were chosen as ψ₁ = 1 × 10^–8, ψ₂ = 2 × 10^–4 and ψ₃ = 1 × 10^–6, respectively.

Table 4.

Initial and bounded values of parameters .

Parameters	Initial values	Lower limit values	Upper limit values
	0.005	0.0005	0.05
	18	1.8	180
	0.02	0.002	0.2

The actual output force asymptotically converged to the reference signal, while the uncertain parameters of the system were dynamically changed and eventually stabilized (Fig. 9). The system’s output force tracking error was significant in the first 4 π s. This change might be ascribed to the significant difference between estimated parameters and actual values in the early stage of adjustment, indicating the obvious parameter estimation adjustments under the updating effect of adaptive laws. In the subsequent 4 π s, it provided a small-scale asymptotic tracking estimate as the estimated parameters were close to the actual system. Finally, the parameter estimation of the controller was consistent with the target system and the tracking performance and corresponding errors of the system tended to periodically and steadily change within a small range. Consequently, the control algorithm proposed in this study was efficient.

Fig. 9.

Experimental results of the variable frequency signal response with the ASMCC controller: (a) tracking trajectory of the output force under the proposed controller and (b) self-adjustment of the system parameters.

Comparative verification

To further evaluate the superiority of the designed controller, the performances of conventional PID compensation control (PIDCC) and conventional sliding mode compensation control (SMCC) schemes were tested for comparison. During the experiment, the expected output force signal was F_outd = 18sin(t − π/2) + 43 N. Based on trial results, the parameters of the PIDCC were set as K_P = 25, K_I = 65, and K_D = 1, whereas the structural parameters of conventional SMCC was chosen as c₁ = 800, c₂ = 1, ε = 1, and k_s = 1200. For comparison purposes, the structural parameters of the proposed ASMCC were consistent with those of the SMCC controller, and its system parameter values φ_i and adjustment coefficients ψ_i were almost the same as the previous experimental values. Based on the experimental platform shown in Fig. 3, comparative experiments were conducted on the control performance of different controllers with/without external disturbances at the proximal end of the tendon sheath.

Case 1: Compensation experiments without disturbance.

In this case, different control algorithms were tested for the end output force tracking control of the tendon-sheath transmission system on the experimental platform shown in Fig. 3, and the corresponding results are shown in Fig. 10.

Fig. 10.

Comparison results of different controllers under undisturbed conditions: (a) tracking trajectories of output force under different controllers and (b) estimation of parameters under the action of ASMCC controller.

The undisturbed tendon-sheath actuated system performed well under the action of different controllers, and all controllers could ensure that the output force of the system changed with the command signal (Fig. 10). In particular, it could be inferred from Fig. 10a that the tracking error of the SMCC algorithm (− 1.75 N to 1.45 N) was lower than that of the PIDCC method (− 1.96 N to 2.34 N). The tracking error of ASMCC decreased with the increase in the stability of time-varying parameter estimation and its stabilized error (− 0.99 N to 0.68 N) was smaller than that of the other two controllers. The system parameters were all bounded under the action of ASMCC controller (Fig. 10b) and it is their update during the control process that improves the performance of the tendon-sheath transmission system.

Case 2: Compensation experiments with disturbances.

To verify the system performance under external disturbances, the system output under different controllers after the introduction of a tension spring with a stiffness of 12.5 N/mm at the proximal end of the tendon sheath is shown in Fig. 11.

Fig. 11.

Comparison results of different controllers under disturbed conditions: (a) tracking trajectories of output force under different controllers and (b) estimation of the parameters under action of ASMCC controller.

Three controllers could still ensure the good tracking performance of the system’s output force and their error curves exhibited the periodic stability (Fig. 11a). The tracking error of the SMCC controller (− 1.99 N to 1.48 N) was smaller than that of the PIDCC controller (− 3.2 N to 1.54 N), whereas the tracking error of the ASMCC controller (− 1.52 N to 1.08 N) was smaller than that of the other two controllers. The maximum force tracking error of PIDCC control increased from 2.34 to 3.2 N, whereas the changes of the tracking error of the other two controllers were slight (Figs. 10a and 11a). In other words, the PIDCC method could ensure the accurate output force of the tendon sheath transmission system under specific conditions, but the influences of parameter changes or external disturbance could not be suppressed. Due to the robustness of the SMC scheme and its insensitivity to the changes in system parameters, both SMCC and ASMCC methods could greatly improve the dynamic performance of the system and eliminate external disturbances. External disturbances increased the nonlinearity of the system and made the changes in time-varying parameters more significant (Figs. 10b and 11b). By adjusting the adaptive parameters ϕ_i appropriately, the ASMCC controller could achieve the better control performance than the SMCC.

Quantitative evaluation of the controller performance

To provide the more intuitive illustration of the performance of different controllers, the root mean square error (RMSE) values were used to quantitatively evaluate the trajectory tracking performance of each controller. Table 5 presents the results for comparative analysis.

Table 5.

Root mean square values of different controllers.

Comparative experiments	Controller types	RMSE values
Disturbance-free conditions	PIDCC	1.4654
	SMCC	0.8271
	ASMCC	0.6328
Disturbed conditions	PIDCC	0.8910
	SMCC	0.6745
	ASMCC	0.5735

The root mean square values for trajectory tracking decreased sequentially across the three controllers (Table 5). According to the definition of root mean square, a smaller value indicated a smaller deviation between the actual output and the desired input and suggested the better performance. Therefore, under two different operating conditions, the performance of the controllers could be ranked as follows: ASMCC > SMCC > PIDCC.

Conclusions and future work

In this paper, the static and improved dynamic models of the single TSAS were studied for the purpose of overcoming the shortcomings of parametric uncertainty and low transmission accuracy of the system. The system input was converted from force into displacement based on the stiffness relationship so as to improve the accuracy of the model and the correctness of the model was verified by relevant experiments. Subsequently, based on the sliding mode controller and an adaptive control approach, ASMCC method of the TSAS was presented. The generalization of the control method was further enhanced with the time-varying parameters obtained from the adaptive adjustment mechanism. Finally, controller performance testing experiments were conducted under the conditions with/without interference. It was confirmed that the proposed ASMCC had strong robustness and high accuracy. This study provided a theoretical basis for the application of single TSAS in the field of high-precision transmission. In the future, we will introduce this algorithm and the tendon-sheath actuated system into exoskeleton robots to assist with movement retraining.

Author contributions

M.Y. and Q.W. completed the programming of algorithm, and were the major contributors in writing the manuscript. H.W. did the simulation analysis and prepared figures. T.Z. and X.Z. conceptualized the study. All authors conceived the experiments and discussed the results.

Data availability

The datasets used during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.