Adaptive Neural Backstepping Control of Variable Stiffness Actuator

In this paper, we provide a novel adaptive neural network backstepping control scheme for a special variable stiffness actuator (VSA) based on lever mechanisms with saturation inputs, output constraints and disturbances is presented here. In the controller designing, the prescribed performance-tangent barrier Lyapunov function (PP-TBLF) is introduced to ensure that both the prescribed performance bound of tracking error and the output constraints are not violated. In specific steps of backstepping control scheme, the Chebyshev neural network and the Nussbaum-type function are used to solve the unknown nonlinearities and unknown gain sign. Meanwhile, the inverse hyperbolic sine function tracking differentiator is exploited to solve the “explosion of complexity” caused by the differentiation of virtual inputs and also approximate the complex partial derivative caused by the auxiliary control signals. Finally, the stability of the whole scheme is proved by Lyapunov criterion and the simulation results are presented to illustrate the feasibility of the raised control strategy.


Introduction
Variable stiffness actuators (VSAs) have received substantial attention in numerous robotic applications because they can transmit torque and change stiffness simultaneously. VSAs also have many advantages over traditional fixed stiffness actuators, including increases in the safety of human-robot interactions, the capacity for positioning precision and disturbance resistance, and the energy efficiency for locomotion [1][2][3][4][5][6]. According to the design principle, current VSAs can be divided into two categories such as antagonistic-type VSAs and serial-type VSAs [7].
Antagonistic-type VSAs (VSA-I, VSA-II, etc.) mimic the skeletal muscle arrangement of human joints, drive at least two elastic units together, and are placed in an antagonistic location, to adjust the position and stiffness [8,9]. However, the limitations of this location are rooted in complex synchronous control method, which leads to control complexity and high energy consumption [10]. In series-type VSAs (MACCEPA, VS-Joint, FSJ, etc.), the position and the stiffness controls are separately achieved by two different control driving units: one position control driving unit changes the equilibrium position, and the other stiffness control driving unit modifies the output stiffness independently. The main feature of this scheme is that the controllable equilibrium position and adjustable compliance elements are mechanically connected in series [11][12][13][14]. However, the stiffness control driving unit is limited by the inherent spring stiffness, which decreases the range of stiffness adjustment and increases the complexity of the stiffness calculation. Recently, the variable lever arm mechanism (VLAM) has been proposed to solve this problem of serial-type VSAs to improve the operation range of motion. These hybrid variable stiffness actuators (HVSAs: CompAct-VSA, SVSA, SVSA-II, vsaUT, vsaUT-II, AwAS, AwAS-II, etc.) with VLAM can adjust stiffness by changing the lever arm length ratio, which leads to a wide range of stiffness adjustment. On this basis, multiple researches are dedicated to deduce the dynamic equations by kinetic, stiffness and resistance modeling, and analyze the performance of the proposed model with control methods combined [15][16][17][18][19][20][21][22][23][24][25].
Undeniably, these studies have greatly enriched the dynamic theories and structure design of VSAs, while little was done in the study of control strategies after the analysis.
At present, the control schemes associated with VSAs are confined to PID controller, however, it may not be able to present strong robustness with disturbance for highly nonlinear dynamic systems. To address this problem, a neural network adaptive control strategy [26,27] and a command-filtered backstepping controller [28] based on feedback linearization (FL) are proposed.
On the basis of the FL, the disturbance observer (DO) is introduced to improve the tracking accuracy and anti-disturbance characteristics [29]. Furthermore, prescribed performance control (PPC) is introduced to guarantee that the output error can converge to a predefined arbitrarily small residual set [30].
However, good quality of the control effect based on FL is limited by a precise dynamic model. If there are many imprecise nonlinearities in the model, it will be difficult to guarantee the robustness of the controller designed according to FL. Thus, it is necessary to address these issues by devising a better control scheme to directly control the nonlinear model.
Backstepping control (BC) and sliding-mode control (SMC) are the major control methods for nonlinear models so far. In contrast to SMC, which has internal chattering that limits its expansion, BC has been proved to overcome this obstacle [31][32][33]. However, the "explosion of complexity" rising from the repetitive differentiation of virtual signals and the highcomplexity caused by the unknown nonlinearities are often encountered in the design of backstepping controller [34]. For solving these issues, the inverse hyperbolic sine function tracking differentiator (IHSFTD) and the Chebyshev neural network (CNN) are proposed [35,36]. For the first issue, IHSFTD is exploited to acquire the derivatives of virtual inputs and obtain a higher precision than traditional first-order filters [37]. As for the second one, The CNN can approximate unknown nonlinear terms with arbitrary precision and shows significant advantages because its inputs only rely on a subset of Chebyshev polynomial [36,[38][39][40]. Additionally, in practical applications, nonlinear systems often need to consider the problems of saturation input and the output constraints. It has been reported in the other nonlinear dynamic systems that the hyperbolic tangent function (HTF) can realize the bounded of control input [41] and the Nussbaum-type function (NF) can overcome the problem of unknown control gain and direction [42]. Furthermore, the prescribed performance-tangent barrier Lyapunov function (PP-TBLF) can ensure that both the bounds of tracking error and output are not violated [30,42,43]. Thus, the HTF with NF and PP-TBLF are introduced here.
In summary, in order to realize direct control for the VSAs, we design an adaptive neural network backstepping control (ANNBC) scheme for a special VSA here. The contributions of this article are given as follows: 1. Compared with the traditional FL control strategies [26][27][28][29], the ANNBC proposed can achieve the aim to directly control the nonlinear model without using the FL. The controller for VSAs designed in this way will have higher accuracy and robustness.
2. This paper designs the ANNBC by the merging of the CNN, IHSFTD, HTF, NF, PP-TBLF and minimal learning parameterization. Compared with other neural networks [34,38], the CNN can approximate unknown nonlinear terms without determining the center of the basis function. In comparison with the first-order filters [37], the IHSFTD can obtain the higher precision derivative of virtual input. These methods can reduce the complexity of the raised controllers.
3. To prespecify the bounds of tracking error, combining prescribed performance control (PPC) with TBLF and creating the PP-TBLF ensures that the prescribed performance bound of tracking error and output constraints are not violated. In contrast with the PPC [30,[43][44][45], the PP-TBLF proposed is more efficient than there without computing the transformed error. This paper is organized in the following manner. Section 2 states system formulation and preliminaries. In Section 3, the design of ANNBC is given and the stability analysis is executed. Then, in Section 4, Simulation analysis are given.
Finally, the conclusions are given in Section 5.  Figure 1 presents the schematic diagram of the VSA with gravity and load, which can be illustrated as follows [18]:

System model
where q is the angular position of the output link, p  denotes the where the equivalent disturbances in the equation (7) Then, by calculating with equation (8), the state space model corresponding to the system dynamics model (7) is the equivalent disturbance vector, m represents the mass of the output link, g is the acceleration of gravity, d denotes the distance from the rotating axis of the output components assembly to its center of mass. Additionally, the elastic torque, the resistant torque and the stiffness of the elastic mechanism are derived as   Remark 1 When s k , l , n take the fixed value, K is the function of the variable 5 x , therefore, when K is restricted, 5 x is restricted simultaneously. In order to simplify the derivation of the controller, instead of directly considering the output error restriction between the output K and the reference stiffness tracking signal d K , we will first consider the output error restriction between the output 5 x and the reference angular position tracking signal 5d x . Since the complexity of coupling angular-torque relationship in dynamic model, the current control methods are all deduced on the basis of feedback linearization, which can make controller design and parameter adjustment easier. However, feedback linearization is limited to the precise dynamics model. The robustness of the controller designed by feedback linearization is difficult to be guaranteed if existing lots of imprecise nonlinearities in dynamic system. Thus, the control objects of this paper can be summarized as: In addition, the position output error 1 e and stiffness output error 5 e can keep in prescribed performance region during the whole dynamic process, i.e.

Prescribed performance-tangent barrier Lyapunov function
In order to ensure that the output and tracking error converge strictly within the performance boundary, we use a novel PP-TBLF [42], which is expressed as with the performance boundary function () i rt is designed as where 0 0

Assumption 1 The desired position trajectory 1d
x and its derivative 1d x , and the stiffness trajectory 5d x and it derivative 5d x , are continuous and differentiable, and constrained by 15 , bb satisfying 1 1 0 ab  and 5 5 0 ab are constants.

Inverse hyperbolic sine function tracking differentiator
In order to avoid the explosion of complexity in the controller design, the tracking differentiator based on inverse hyperbolic sine function is introduced [35], which is expressed as  and R is big enough, when the input signal  passes through the differentiator (13), the following theorem holds Lemma 1 [46] For the following system If all the solutions of the system (14) are asymptotically stable at the origin, which satisfy the condition and The solution of system (15) satisfies it has positive constant 2 v l which satisfies the following inequality

Chebyshev neural network
The Chebyshev polynomial can be obtained from the following second order recursive equation [27]         where * W denotes the desired weight vector,  is the approximation error, and there exists known constants 0  ,which where  is an appropriate constant, g is a time-varying parameter restricted in the unknown closed interval , S s s

Controller design
Step 1: Define the position tracking error as: Then, the derivative of 1 e can be obtained as follows Define the Lyapunov function with PP-TBLF to satisfy the objective condition 1 1 , 0 e r t    as follows With (11) and (12)  For facilitating controller design, assuming that 1 can be written by letting pass i  through the IHSFTD as: Then, the virtual control input 2  is defined by the following equality: Then, combining (32) and (33) Step 2: Choose the Lyapunov function candidate 2 V as where 1 0  .
The variables 1  is defined as where 1  denote the estimate of 1  and 1  is given later.
However, the MAX, AVG and SD of the tracking errors in the proposed ANNBC are smaller than that of PD, which are indicated in Table 2. Furthermore, it can be seen from Fig.2 Fig.3(c). It can also be clearly seen in Table 3 Fig.4(a) and Fig.4(c). Furthermore, in Fig.4(b) and Fig.4(d), the tracking error of ANNBC still keeps within the prescribed performance bound and the MAX, AVG and SD of the tracking error 1 e and 5 e of ANNBC are still better than that of PD, which is shown in Table 4 and Table 5. The stiffness adjustability can be confirmed by Fig. 4(e) and Fig. 4(f) in the same way.
Compared with PD tracking, the ANNBC still has good highprecision tracking performance and robustness under disturbance.
This also illustrates that the proposed neural network can effectively compensate for the error caused by the disturbance.
Additionally, it can be seen from Figs.5(a)-(b) that the control input torque of ANNBC is still better than PD, and ANNBC does not oscillate violently with the addition of disturbance. Thus, we can draw the conclusion that when existing disturbance, the proposed ANNBC can still track the desired signal with higher precision tracking performance and stronger robustness than PD.   Above all the analysis of the sinusoidal trajectory tracking experiments, the simulation results confirmed that the proposed ANNBC can track the desired signal with disturbance and without disturbance. Besides, the comparison between ANNBC and PD presents ANNBC control approach can obtain highprecision tracking performance than PD for VSA system with output constrain, input saturation and disturbance.

Conclusion
In this paper, we focus on dynamical analysis and adaptive control of the VSA wherein the system has strong coupling nonlinearity. Based on the principle of backstepping, the presented ANNBC scheme combined with the IHSFTD, CNN, HTF, NF, and PP-TBLF compels system state to approximate the given signal with a small error. The stability analysis of the VSA system is demonstrated by Lyapunov criterion. Furthermore, the main simulation results verify the validity of the proposed ANNBC. Since it is the first time to control this nonlinear model without using FL, we did not get the most satisfactory tracking accuracy. The research of control strategies for HVSAs are still the challenging work. In the future work, we will be focused on extending the robust tracking for other types of HVSAs and try to improve the control precision by optimizing the controller and its parameters. In addition, more control strategies and the effect comparisons between them will also be used in the follow-up studies.

Figure 1
The mechanism transmission schematic diagram of VSA based on the VLAM