Adaptive Neural Backstepping Control for Harmonic Drive System Based on Modified LuGre Friction Model

To improve the control accuracy and stability of the harmonic drive system under the influence of nonlinear friction and external disturbances, we have developed an adaptive neural backstepping control approach with friction compensation. During the design process, we employ a nonlinear observer based on a novel modified LuGre model with friction compensation. This observer effectively reduces the influence of nonlinear friction on the harmonic drive system, even under dynamic changes in the environment. Additionally, we utilize a Chebyshev neural network to approximate unknown disturbances applied to the harmonic drive system. To prevent violations of output constraints, we introduce a tangent barrier Lyapunov function. Furthermore, to address the challenges of “explosion of complexity” and poor precision associated with first-order filters in backstepping, we integrate the inverse hyperbolic sine function tracking differentiator into this controller. Finally, we employ the Lyapunov criterion to prove that all errors in the closed-loop system are uniformly bounded. Simulation results confirm the feasibility of the proposed control scheme and demonstrate better closed-loop behavior compared to that obtained using a radial basis function dynamic surface controller.


I. INTRODUCTION
The harmonic drive (HD) system is widely utilized in various fields such as aerospace and industrial robotics due to its compact structure, high transmission ratio, smooth operation, and high efficiency [1], [2]. However, the HD system faces challenges caused by the intricate sliding friction contact between the wave generator and the inner wall of the flexible wheel, as well as between the teeth of the flexible wheel and the teeth of the rigid wheel [3], [4]. These factors contribute to strong nonlinear friction throughout the reducer, which in turn makes the system sensitive to unknown external disturbances. Therefore, there is immense theoretical significance and practical value in studying the dynamic characteristics The associate editor coordinating the review of this manuscript and approving it for publication was Jinquan Xu . of the HD system and devising effective control strategies to achieve precise compensation for nonlinear friction and external disturbances. The LuGre friction model is widely used in model-based friction compensation of control systems due to its ability to accurately describe various dynamic and static characteristics of nonlinear friction, including presliding displacement, frictional lag, and varying break-away force [5]. In LuGremodel-based friction compensation, the primary control scheme involves an adaptive control method based on an observer, which estimates the unmeasurable state variables of the LuGre model [6]. Various control methods have been proposed in this context. For example, an iterative learning control method based on a dual-observer is presented to ensure that the tracking error converges to zero in the L2-norm sense [7]. An adaptive control method based on a projection observer is introduced to achieve asymptotic tracking [8]. Another adaptive control method employs an observer with filtered states to guarantee that the tracking error converges to zero under a persistent excitation condition for the dynamical function of friction [9]. Additionally, adaptive control methods based on neural networks [10] and parameter identification [11] are also commonly employed for LuGre-modelbased friction compensation. The aforementioned control methods primarily address the issues of asymptotic tracking [7], [8], [9] when the exact values of parameters in the dynamic equation of the LuGre model are known, or approximate tracking [10], [11] when all parameters of the LuGre model are unknown. However, they overlook the dynamic changes of these parameters under varying external conditions. Therefore, a new focus of research is to develop better approaches for describing the dynamic behavior of the LuGre model.
Friction compensation using modified LuGre models has been extensively studied by researchers. For instance, an adaptive robust controller based on a modified LuGre friction observer is proposed for uncertain linear motors to ensure a sufficiently small tracking error [12]. By utilizing the modified LuGre model, the vector field of the friction equation becomes zero when the speed exceeds a preset limit, leading to improved dynamic friction compensation across a wide range of velocity regulation. Building upon this, another modified LuGre model combined with a zero-velocity crossing window is proposed to address lag and model oscillation in force control systems [13]. Other modification methods, such as those based on function-based approximation theory, replace the non-smooth LuGre model with a smooth friction model [14]. Interested readers can find more information in [14], [15], and [16]. While the aforementioned modified LuGre models with friction compensation enhance dynamic friction performance to some extent, most of them do not explicitly consider the influence of external disturbances.
The backstepping control (BC) framework, an adaptive control technique, is a powerful strategy for nonlinear servo systems under nonlinear friction and external disturbance conditions. Unlike the sliding-mode control (SMC) framework [17], which suffers from internal chattering that limits its expansion, BC has been proven to overcome this limitation [18]. Additionally, incorporating specific control tools in the controller design process can significantly enhance controller performance. Dynamic surface control (DSC) technology [19] addresses the ''explosion of complexity'' in adaptive backstepping control of high-order nonlinear systems by introducing a first-order filter (FOF) into the BC framework. While the FOF can eliminate the drawbacks of BC, its accuracy is often low. To compensate for this, a tracking differentiator (TD) [20] is employed to achieve higher precision. Neural networks (NNs) [21], [22], [23], [24] are widely utilized in controller design because they can approximate unknown smooth functions with arbitrary precision. Common types include radial basis function (RBF) neural networks and fuzzy basis function (FBF) neural networks. Furthermore, when combined with DSC, radial basis function dynamic surface control (RBFDSC) and fuzzy basis function dynamic surface control (FBFDSC) exhibit notable advantages in solving practical control problems. However, these NNs often require additional parameters, prior knowledge of approximation functions, and real-time parameter updates. In contrast, the Chebyshev neural network [25] offers significant advantages as its inputs only rely on a subset of Chebyshev polynomials. In practical applications, the harmonic drive (HD) system often needs to consider output constraint issues. A barrier Lyapunov function (BLF) [26], proposed for constraint handling in nonlinear systems, is introduced due to its property of approaching infinity when its arguments approach certain limits. Previous works on BLF typically employ logarithmic BLF (BLF-Log) to design control systems [27]. However, if the constraint of the initial error approaches convergence to zero, BLF-Log will also converge to zero and replace the quadratic form. Considering practical needs and theoretical challenges, constructing the BLF-tan for the HD system is more appropriate.
Based on our current understanding, the combination of adaptive BC, NN, TD, and BLF has not been previously applied to control the friction compensation of the HD system. Consequently, in this paper, an adaptive neural backstepping control (ANBC) scheme is designed specifically for the HD system. The main contributions of this paper are given as follows: 1. The proposed ANBC scheme in this paper combines the Chebyshev neural network, Inverse Hyperbolic Sine Function Tracking Differentiator (IHSFTD), Tangent Barrier Lyapunov Function (TBLF), and minimum learning parameters. Compared to other neural networks [21], [22], [23], [24], the Chebyshev neural network can approximate unknown functions without the need to determine the centers of the basis functions. This simplifies the network structure and training process, making it more efficient. The IHSFTD is preferred over FOF [19] as it provides a higher accuracy in estimating the virtual input derivative. This improves the overall performance of the control system, especially in accurately tracking the system dynamics. In terms of BLF, the TBLF used in this study exhibits a wider applicability compared to other types [27]. The TBLF can handle a broader range of constraints, making it suitable for a variety of control scenarios. 2. The ANBC scheme proposed in this paper is built upon a novel modified LuGre model. One notable advantage of this model is the ability to anticipate the effects of nonlinear friction in advance. By introducing only two adaptive parameters, the influence of external uncertainties on friction compensation can be mitigated. In traditional approaches [12], [13], [14], [15], [16], the consideration of nonlinear friction and the reduction of uncertainties in friction compensation typically require more complex models 96094 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. and a larger number of adaptive parameters. The modified LuGre model used in the ANBC scheme simplifies the control process while still accounting for the influence of nonlinear friction and reducing the impact of external uncertainties. This paper is organized in the following manner. Section II states system formulation and preliminaries. In Section III, the design of the ANBC is given, and the stability analysis is executed. Then, in Section IV, simulation analyses are given. Finally, the conclusion is given in Section V.

II. SYSTEM FORMULATION AND PRELIMINARIES A. SYSTEM DYNAMIC MODEL
Considering the influence of nonlinear factors such as flexible deformation, nonlinear friction, and unknown external disturbances, a second-order nonlinear dynamic model can be established for HD mechanisms [28], as described in equation (1).
where F f (q l ) denotes the nonlinear friction torque, J l , c l , d l are the rotational inertia, the damping coefficient, and the unknown external disturbance of the driving mechanism, J m , c m , d m are the rotational inertia, the damping coefficient and the unknown external disturbance of the motor, K represents the flexible stiffness, N denotes the reduction ratio of the reducer component, q l ,q l ,q l denote the displacement, speed and acceleration of the drive mechanism respectively, q m ,q m ,q m represent the displacement, angular speed and angular acceleration of the motor respectively, and u is the output torque of the motor. The nonlinear friction force F f (q l ) in the system (1) can be described by the LuGre friction model as where σ 0 , σ 1 , σ 2 is the stiffness coefficient, the damping coefficient and the viscous friction coefficient of the bristles in contact surface, z denotes the average deflection of bristles, g (q l ) represents bounded function, q s is the Stribeck angular speed, F c denotes the Coulomb friction, F s denotes the stiction force.
In model (2), for the convenience of calculation, the coefficients σ 0 , σ 1 , σ 2 are usually regarded as constants. However, in actual working conditions, the coefficients σ 0 , σ 1 will vary with the contact of the micro convex body and the proportion of load the oil film bears, and the coefficient σ 2 will be affected by the changes in the relative velocity and the temperature. Therefore, in order to more accurately reflect changes in internal friction of the reducer with the external environment, parameters ω and υ were introduced to improve the classical LuGre friction model. Then, the modified LuGre model can be expressed as In the equation (3), ω is used to reflect changes the average deflection of bristles. υ is used to reflect changes in the coefficient of viscous friction. They meet 0 < ω min < ω < ω max and 0 < υ min < υ < υ max .
Substituting equation (3) into equation (2) yields Then, assuming that the actual LuGre model can be represented as However, the actual values of these parameters are difficult to obtain. If their nominal values are used to represent the actual model, equation (1) can be rewritten as Define the system state variables x 1 = q l , x 2 =q l , x 3 = q m , x 4 =q m . Then the dynamic equation (1) is rewritten as follows For any given continuous signal x 1d , the four dynamics surfaces are defined as where α i , i = 2, 3, 4 is the virtual control inputs.
Control Goal: All the signals of the closed HD system are uniform and ultimately bounded with the output position signal x 1 is restricted in the set = {x 1 ∈ R ||x 1 | < a, ∀t > 0 }. Assumption 1: The reference trajectory x 1d is bounded by In order to avoid the explosion of complexity in the controller design, the tracking differentiator based on inverse hyperbolic sine function is introduced [29], which is expressed as where α (t) is the input signal. v 1 (t) , v 2 (t) are the state variables of tracking differentiator. R, ℓ 1 , ℓ 2 , λ 1 , λ 2 are the positive constants. With properly design constants R, ℓ 1 , ℓ 2 , λ 1 , λ 2 and R is big enough, when the input signal α passes through the differentiator (9), the following theorem holds Lemma 1 [30]: For the following system The solution of system (10) satisfies where g (·) denotes a smooth bounded function. R > 0 represents a design constant. α (t) is a bounded integrable function. Lemma 1 ensures that when R is large enough, the solution v 1 (t) of equation (10) can fully approximate the input signal α (t) in arbitrary finite time, thus the following formula holdṡ Lemma 2 [31]: If the condition v 1 (t) − α (t) ≤ κ with κ > 0, then it has positive constant l v2 which satisfies the following inequality Assumption 2: For facilitating controller design and avoiding the explosion of complexity. Assuming that x 1d = α 1 , soα i , i = 1, 2, 3, 4 can be written by letting pass through the IHSFTD as: where |v i2 −α i | ≤ l vi2 , and R i , ℓ i1 , ℓ i2 , λ i1 , λ i2 , l vi2 are the positive design constants

C. CHEBYSHEV NEURAL NETWORK
The Chebyshev polynomial can be obtained from the following second order recursive equation [25] where X ∈ R, T 1 (X ) is usually expressed as X , 2X , 2X −1 or 2X + 1.
The Chebyshev neural network is capable of approximating any nonlinear continuous function F * (x) over a compact set to any degree of accuracy. A strengthened pattern about the Chebyshev multinomial for X = [x 1 , . . . , x m ] T ∈ R m is presented as where T i x j , i = 1, . . . , n, j = 1, . . . , m, n stands for the order, m denotes the Chebyshev multinomial, ζ (x) represents the Chebyshev multinomial basis function vector. According to the above descriptions, the arbitrary unknown nonlinear function can be formulated as where W * denotes the desired weight vector, δ is the approximation error, and there exists known constants δ 0 , which satisfies 0 < |δ| < δ 0 . W * can be defined as where W and D X are the compact set of reasonable bounds of W and X , respectively. Assumption 3: There is a positive constant δ M which satisfies |δ i | ≤ δ M , i = 1, 2.

D. TANGENT BARRIER FUNCTION
For the sake of ensuring that system state is bounded in a desired region, a tangent barrier function y tan (y) is employed in this paper, where tan (·) stands for the tangent function. It is obvious that the tangent barrier function satisfies the characteristics listed as below: +∞ > y tan (y) ≥ 0 for y ∈ (−π/2, π/2) According to the above descriptions, we can formalize the results for general forms of tangent barrier function in Lyapunov synthesis satisfying y tan (y) → ∞ as y → −π/2 or y → π/2.

III. ADAPTIVE CONTROL BASED ON BACKSTEPPING A. CONTROLLER DESIGN
According to the above-mentioned dynamics system, the whole design process consists of four phases. Then, the controller design process is described in detail.
Step 1: Choosing the first Lyapunov function as where the design parameter r 1 = a − d > 0 (a > d) denotes the constraint on e 1 . That is, e 1 ∈ (−r 1 , r 1 ).
With (20), the time derivative of V 1 is expressed as: where M = tan π 2r 1 e 1 + π 2r 1 e 1 sec 2 π 2r 1 e 1 96096 VOLUME 11,2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
With (24), the time derivative of V 2 is expressed as the following inequation: By using the Chebyshev neural network to approximate the unknown external disturbance d l , it can be obtained that where δ 1 is the approximation error and satisfies |δ 1 | ≤ δ M . By utilizing the Young's inequality, one can obtain where ϑ 1 = ∥W 1 ∥ 2 = W T 1 W 1 , which can reduce the numbers of weights for the Chebyshev neural network, ϑ 1 and ∥·∥ are the unknown variable and the 2-norms of W 1 , respectively, and m 1 is the design constant.
Substituting (27) into (25) yieldṡ Since the average deflection of bristles cannot be measured directly, a nonlinear state observer is used to estimate z, one hasż whereẑ is the estimated value of z, andz =ẑ − z is the estimation error of z. The estimated value of friction torque is: whereF f is the estimated value of F f . Then, the virtual control input α 3 and adaptive lawθ 1 ,ω,υ are given as follows where k 2 > 0 and c 1 > 0 are the design constants.
Step 3: Choosing the third Lyapunov function as Computing the derivative of V 3 , it giveṡ Then, defining the virtual control α 4 as where k 3 > 0 is the design constant. Substituting (38) into (37), one haṡ Step 4: Choosing the fourth Lyapunov function as where 2 > 0, and the definitionθ 2 =θ 2 − ϑ 2 is similar to that ofθ 1 .
Next, the time derivative of V 4 is calculated aṡ By using the Chebyshev neural network to approximate the unknown external disturbance d m , it can be obtained that where δ 2 is the approximation error and satisfies |δ 2 | ≤ δ M . Similarly, utilizing the Young's inequality, it obtains where m 2 > 0 is the design constant andθ 2 = Ŵ 2 2 . Substituting (43) into (41), one haṡ Then, the control input u with adaptive lawθ 2 is designed as where k 4 > 0 and c 2 > 0 are the design constants.
Proof: The differentiation of the Lyapunov function candidate in relation to t is derived aṡ VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Let β > /p, Solving equation (50) yields: Equation (51) indicates that V (t) is ultimately bounded by β > /p. Therefore, all errors e i ,θ j ,z , j = 1, 2 in the closed-loop system are ultimately uniformly bounded. In addition, by adjusting the value of k i , j , m j , c j , R i , l ij , λ ij , the value of /p can be arbitrarily small. Therefore, the position tracking error e 1 can be arbitrarily small.
Remark 1: The larger k i and j can obtain good tracking performance. But too k i and j can result in large control input, which may be far beyond physical limitations of HD systems. Meanwhile, the large R i and small l ij , λ ij can improve the accuracy of tracking differentiator. However, too large R i can make tracking position have obvious step oscillation at the beginning. Besides, the relatively large m j and small c j can improve the approximation performance of neural network.
In addition, the Chebyshev basis functions are selected by The external disturbances are applied at the load and motor ends as follow The initial value of the state variables of the system are all set as x 0 = [0, 0, 0, 0], and the reference trajectory is defined as Figs. 1(a)-(b) illustrate the comparison of tracking effects between ANBC and RBFDSC on reference trajectories under output constraints, with fixed LuGre friction parameters. As shown in Fig. 1(a), both ANBC and RBFDSC can accurately track the reference trajectory. However, upon closer   inspection in the magnified image, it is apparent that the trajectory of ANBC closely aligns with the reference trajectory. This observation is further supported by the error data presented in Table 3 and Fig. 2(b). The MAX, AVG, and SD of the tracking errors obtained using the proposed ANBC are smaller than those achieved with RBFDSC. Additionally, it is worth noting that the tracking error of ANBC remains within the predefined region bounds, unlike that of RBFDSC. Consequently, we can deduce that when output is constrained, ANBC can enhance the tracking performance of a given signal.
Figs. 2(a)-(b) illustrate the compensation of the neural network for the external unknown disturbance torques at both the load and motor ends. As observed in Figs. 2(a)-(b), T2SFNN1 and T2SFNN2 exhibit a swift response, accurately tracking the desired trajectory with minimal error, even after the application of an external disturbance torque.   To verify the dynamic performance of the proposed ANBC. We reset the reference trajectory as follows: Case 1: x 1d is reset as a mix frequency sinusoidal function x 1d = 1 − cos (2t) + 0.75 sin (0.25t).
Case 2: x 1d is reset to a square wave function with Amplitude =1 and Frequency =0.25. Fig. 3 shows that the ANBC can still achieve effective tracking when the reference trajectory is set as the mix frequency sinusoidal function or a square wave function.
To further validate the robustness of the proposed adaptive control algorithm, we analyzed the tracking performance on   the LuGre model with dynamic parameters. The dynamic changes in the LuGre friction model parameters were selected and are listed in Table 4.
Figs. 4(a)-(b) demonstrate the position tracking and tracking errors of the two controllers when the parameters of the LuGre friction model dynamically change. Observing these figures, it is evident that the RBFDSC controller deviates noticeably from the desired trajectory, while the ANBC controller effectively tracks the desired signal. Additionally, the tracking error of ANBC remains within the predefined region bounds. Moreover, as indicated in Table 5, the ANBC controller outperforms the RBFDSC controller in terms of MAX, AVG, and SD of the tracking error in the same scenario.
Figs. 5(a)-(b) showcase the friction compensation effects and friction compensation errors of the two controllers under dynamic parameter changes in the LuGre model. It is observed that when the parameters of the LuGre model change dynamically, the proposed modified adaptive method accurately compensates for the actual friction force, while the RBFDSC method, without modified parameters, exhibits more pronounced deviations in friction compensation. In summary, the simulation results confirm that the proposed ANBC controller effectively tracks the desired signal in the presence of disturbances, output constraints, and dynamic LuGre friction parameters. Furthermore, the comparative analysis between ANBC and RBFDSC demonstrates that ANBC achieves higher precision tracking performance for the harmonic drive system. These findings support the conclusion that the adaptive controller proposed in this paper enables accurate control and exhibits robustness.

V. CONCLUSION
To address the challenges of nonlinear friction and external disturbances in the HD system, an ANBC scheme with friction compensation, incorporating a modified LuGre model, has been developed. This scheme aims to reduce the effect of nonlinear friction by introducing average mane deformation and viscous friction coefficients. Additionally, a Chebyshev neural network is integrated into the control strategy to enhance robustness against external disturbances at both the load and motor ends. Simultaneously, the TBLF ensures that the tracking error approximates the desired signal within a small margin, while adhering to certain restrictions. Stability analysis of the closed-loop HD system is conducted using the Lyapunov criterion. Simulation results demonstrate that the proposed ANBC method exhibits greater robustness compared to RBFDSC against changes in friction parameters and external disturbances, leading to enhanced position-tracking accuracy in the HD system. Future research will focus on designing adaptive controllers that utilize neural network approximations for friction compensation while considering the impact of factors such as hysteresis and nonlinear stiffness on the HD system.