Quadratic Integral Sliding Mode Control for Nonlinear Harmonic Gear Drive Systems with Mismatched Uncertainties

For a class of nonlinear harmonic gear drive systems with mismatched uncertainties, a novel robust control method is presented on the basis of quadratic integral sliding mode surface, and the closed-loop system has satisfying performance and strong robustness against mismatched uncertainties and nonlinear disturbances. Considering time-varying nonlinear torques and parameters variations which are caused by nonlinear frictions and backlash, a nonlinear harmonic gear drive system mathematic model is established and the effect of nonlinear parts is compensated during control system design. It is proven that the quadratic integral sliding mode surface can be reached in finite time and the closed-loop system is asymptotic stable robustly. The simulation studies are carried out in comparison with traditional linear sliding mode control and integral sliding mode control, verifying the effectiveness of the proposed method.


Introduction
Harmonic gear is a new type of drive which is realized by elastic deformation movement.Instead of using rigid component, it uses a flexible component to realize the drive.Therefore, it has many special properties which cannot be obtained in other drives.The harmonic gear is composed of three main components, including a circular spline, a flexspline, and a wave generator.Through the action of the wave generator, a movable deformation wave is generated in the flexspline and the flexspline engages with the circular spline to achieve the transmission.The harmonic gear has advantages of large range of transmission ratio, multiteeth gearing, large carrying capacity, high precision, and running smoothly without impact.Because of its own characteristics, harmonic gear is widely applied in aviation control systems, instrumentation, robotics, and other fields [1][2][3][4], where drive systems are required to be capable of small volume, light weight, high transmission precision, and small hysteresis.
Despite those advantages, many issues will occur during runtime, such as assembly errors, gear wears, and deterioration of working environment.Due to kinetic errors and nonlinear elastic deformations, these issues will cause motion delay, transmission errors, and inaccurate tracking, which leads to the fact that it is very important to focus on angular transmission errors in harmonic gear drive systems.
In order to reduce the errors, many researchers have done wide investigations on force analysis and movement analysis.The stress of flexspline in harmonic gear drive is analyzed in [5].At same time, a harmonic gear drive system dynamic system is established by dynamic equation analysis and least square method in [6].In [7], the research of theoretical modeling of nonlinear torsional behavior in harmonic gear is carried out.These researches abovementioned focus on harmonic gear drive systems' dynamic model, analysis of errors in harmonic gear drive systems, or nonlinear frictions model, which illustrates the fact that angular transmission errors of harmonic gear drive system have been widely analyzed and have been mathematically modeled in various conventional studies and make it practical to compensate the error by control methods.
However, there are fewer researches on this part and one primary reason for this is harmonic gear drive system, a class of nonlinear systems, which are affected by mismatched uncertainties such as nonlinear torques and working condition deterioration.It is hard to control such nonlinear systems with mismatched uncertainties and nonlinear disturbances.Consequently, researches in this area are developed slowly.Previous research [8] designs a PID controller for a class of harmonic drive system with frictions to reduce transmission errors, and as a result, harmonic gear drive system's performance is enhanced by error compensation.The results show that using control method to reduce error is feasible but [8] failed to consider mismatched uncertainties and nonlinear disturbances, which are inevitable in practice, while [9] designing a sliding mode controller based on linear sliding mode surface, which has a better performance than PID controller, and taking disturbances into consideration.
Though it has its improvements, there are also some limitations in [9] such as ignoring some critical torques in modeling and designing, using linear sliding mode surface and no attention to uncertainties.
Sliding mode control (SMC) is a new variable structure robust control method.By choosing the sliding surface and the reaching law, it can make the system response quickly with strong robustness and have insensibility to external interferences and parameters variations.As an important method of nonlinear control, it has been widely researched in recent years.
In [10], Xie Jian et al. design a sliding mode controller for hydraulic pump-control-motor system and use simulation results to illustrate that the sliding mode controller has strong anti-interference ability and good tracking performance when compared with PID controller, improving the control accuracy and stability of the system.In [11], by using boundary layer method to restrain the chatting, NE Sadr et al. design a sliding mode controller for missile autopilot and the system has strong robustness to uncertainties and disturbances.Besides, Murat Furat et al. design a secondorder integral sliding mode controller for a SISO system in [12], making the system overcome the parameter fluctuation and uncertainties caused by the external load.Furthermore, Azar A T et al. design an adaptive sliding mode controller for the Furuta Pendulum in [13].In comparison with other controllers, the simulation result shows that the sliding mode control has advantages of fast response and small errors.In addition, Ginoya D et al. design a sliding mode controller using extended disturbance observer for a system with uncertainties in [14] and the closed-loop system has robustness to the uncertainties.Hess RA et al. design a slide mode controller for a nonlinear unmanned aerial vehicle in [15], and the controller has a good performance on nonlinear system.
Investigations in [10][11][12][13][14][15] illustrate that the sliding mode control method finds widel applications on nonlinear system control and has a good performance.Besides, the closedloop system has insensitivity to disturbance and uncertainties, which means that the system has strong robustness against uncertainties including parameters variations and disturbances.However, the insensitivity to uncertainties is realized only in sliding phase.If system is not on the sliding mode surface, the system will be vulnerable to uncertainties.Additionally, the robustness is only limited to matched uncertainties.The so-called integral sliding mode control (ISMC) is proposed to improve such situation but, unfortunately, the performance of ISMC is conservative.
A class of nonlinear harmonic gear drive systems with mismatched uncertainties is studied and the primary contributions of this paper are as follows.(1) A novel quadratic integral sliding mode controller design method is presented on the basis of quadratic integral sliding mode surface.(2) Considering frictions, time-varying nonlinear torques, and parameters variations which are caused by backlash, the influence of nonlinear parts on control system has been studied.The mathematical models of nonlinear torques and frictions are presented and the effect of nonlinear parts is compensated during control system design.(3) In system modeling, take frictions and nonlinear elastic deformation into consideration and the mathematical model of harmonic gear drive system is established.(4) By using Lyapunov stability theory, it is proven that the quadratic integral sliding mode surface can be reached in finite time and the closed-loop system is robustly asymptotic stable.(5) In comparison with the traditional linear sliding mode control (which can be abbreviated as SMC) and the integral sliding mode control (which can be abbreviated as ISMC), the simulation results demonstrated that quadratic integral sliding mode control (which can be abbreviated as QISMC) stabilizes the system rapidly with fast response, short rise time, small overshoot, small error, and strong robustness against the mismatched uncertainties and nonlinear disturbances.
The remainder of this paper is organized as follows.Section 2 gives details on harmonic gear drive systems modeling and illustrations for backlash and frictions.In Section 3, the quadratic integral sliding surface is presented; furthermore, its corresponding quadratic integral sliding mode controller is designed in Section 4. Section 5 proves that quadratic integral sliding mode surface can be reached in finite time and the closed-loop system is robustly asymptotic stable.The effectiveness of the quadratic integral sliding mode control is verified in comparison with SMC and ISMC in Section 6, and Section 7 draws the conclusions of this paper.

Harmonic Gear Drive System
Harmonic gear drive system (Figure 1) is mainly divided into the DC motor system and the harmonic gear system.The DC motor is driven by controller and outputs torques to the wave generator in harmonic gear system; as a result, torques are transmitted by engagement to a flexspline.A mathematical description of harmonic gear drive system will be given in this part.

DC Motor System
. Using Kirchhoff 's law of electricity, DC motor system can be presented by the following: where   is the motor armature current, () is the motor armature voltage,  is the equivalent resistance,   is the back-EMF coefficient, and   is the angel speed of motor rotor. is the motor armature inductance, which is generally ignored due to its tiny value.2.2.Harmonic Gear System.As Figure 2 [16] shows, harmonic gear system can be simplified as a typical two-mass system including a circular spline, a flexspline, and a wave generator.Harmonic gear transmission system satisfies following dynamic equations: where   is the angular displacement of the wave generator,   is the angular displacement of the flexspline,   is the torques on the wave generator, and   is the torques on the flexspline.  is the equivalent frictions on the wave generator and   is the equivalent frictions on the flexspline.  is the moment of inertia of the wave generator and   is the moment of inertia of the flexspline.The kinetic model of harmonic gear drive system satisfies the following: where  is the reduction ratio,   is the torsional stiffness of the harmonic gear, and   is the motor torque coefficient.
(1) gives Substituting   into (4), it would be followed that Substituting ( 3) and ( 6) into (2), one can obtain harmonic gear drive systems kinetic model: Considering the frictions model [16], because the angle range in accurate control is small, it is acceptable to choose the average Coulomb's frictions   as whole frictions in harmonic gear drive systems.
The static frictions   is about 3.88% larger than   ; then one can obtain And frictions on wave generator   () is where In the same way, frictions on the flexspline   () would be as follows: where The gears will heat because of frictions in gear drive system.Correspondingly, a gap which is called backlash [17] is designed as the expansion space for gears (shown in Figure 3).These gaps will continue to increase due to wears and frictions caused by repeated stop, variable speed, the working environment, and other reasons, finally leading to motion delay and inaccurate tracking.
A mathematical expression of backlash is introduced in [18], considering the dead-zone model, and defines the elastic deformation torsion angle Δ() caused by backlash as where  is the width of backlash and Δ is the transmission error caused by nonelastic deformation torsion angle, which can be approximated by random number model.The Δ() dead-zone model can be presented by Figure 4.Because of Δ(), a nonlinear torque   () =   Δ() is attached to harmonic gear drive systems.In addition, it is inevitable that system parameters and characteristics will drift slowly due to factors such as gear wear and environment.As a result, choose Δ as the state parameters variations and Δ as the control signals variations.
Choose state vectors as and harmonic gear drive systems can be presented as follows: It is shown in (13) and ( 14) that the system of harmonic gear drive system is a class of nonlinear system with mismatched uncertainties including parameters perturbation and external disturbance.

Quadratic Integral Sliding Mode Design
In this part, a quadratic integral sliding mode surface is designed to compensate the influence of mismatched uncertainties and a detailed analysis of system in sliding mode is given to verify the effectiveness of proposed method.
The quadratic integral sliding mode control law will be designed as where  0 stands for the abovementioned control law of nominal system, which is responsible for the performance of the nominal system and satisfies Assumption 3.
While  1 stands for discontinuous control action that compensates the mismatched uncertainties, based on quadratic integral sliding mode surface, the definition of  1 will be introduced in (32).
In order to compensate the effect of mismatched uncertainties, the quadratic integral sliding mode surface is designed as Here ( 0 ) is the initial values of the state vector and () ∈  is demonstrated in ( 19) with the arbitrary scalar  > 0.
The sliding mode surface ( 18) is in a quadratic form; therefore, ( 18) is a quadratic integral sliding mode surface.Assuming that the quadratic integral sliding mode surface  = 0 can be reached, it leads to One can obtain the equivalent control of  1 : Substituting  1 into (15), then the system in sliding mode will be It is evident that One can obtain the conclusion It is obvious that the effect of uncertainties is compensated.Equation (31) indicates that if the quadratic integral sliding mode surface is reached and remains there, the effect of uncertainties Δ(, ) + Δ(, ) 0 can be compensated, achieving robustness to mismatched uncertainties.Besides, the system will be on sliding mode surface from any initial state; consequently, the system is robust to mismatched uncertainties from any initial state.

Quadratic Integral Sliding Mode Controller Design
The design of the abovementioned quadratic integral sliding mode controller will be illustrated in this part.The structure of the quadratic integral sliding mode controller is shown in Figure 5.
As was mentioned earlier, the quadratic integral sliding mode control law will be designed as  =  0 +  1 .Then  1 is the discontinuous control action that compensates the mismatched uncertainties, with definition as where  0 ,  1 ,  0 , and  1 ∈  are scalars defined as follows: The two approaching parameters  1 and  2 in (33) and (35) are arbitrary positive scalars, and  = ( +  0 ‖‖ +  1 ‖‖ 2 ) −1 .The sliding mode control will inevitably cause chattering due to switching time lag, spatial lag, and inertia effect.Chattering will affect the control precision and reduce systems control quality, in more serious cases, and even destroy control elements.In most investigations, the sign function is replaced with the saturation function to reduce the chattering.To some degree, this method can weaken the chattering, but the saturation function is a discontinuous function, making the result conservative.In this paper, the sign function is replaced by the hyperbolic tangent function, which is continuous and smooth.
Hyperbolic tangent function is shown as follows: tanh (   ) = e / − e −/ e / + e −/ (37) where  stands for the quadratic integral sliding mode and  > 0 is a positive scalar, which decides the change rate of hyperbolic tangent inflection point.

Harmonic Gear Drive Systems Stability
The reachability of the sliding mode surface and the stability of closed-loop system under quadratic sliding mode control will be proved in this part.
This completes the proof.r In conclusion, it is proved that the quadratic sliding mode surface (18) is reachable and the closed-loop nonlinear system with mismatched uncertainties in sliding mode is asymptotically stable with robustness to mismatched uncertainties.The system's dynamic performance is depended on  0 , and various control theories (such as LQR, H-infinity control) could be applied in  0 designing according to different requirements.

Simulation Studies
What deserved our attention is that the quadratic integral sliding mode control could be applied to the uncertain systems with linear time-invariant nominal plants, where (, ) = , (, ) = ,  is the system matrix, and  is the input matrix.
Simulation results in comparison with SMC and ISMC, shown in Table 2 and Figures 6-12.
As the simulation results indicated, when choosing SMC, the system has acceptable overshoot but has a larger steady state error and much longer stable time.When choosing ISMC, the system rises rapidly with a fewer steady state error but has larger control values and maximum overshoot, which will inevitably cause impact to the system.
When choosing QISMC, the system will steady rapidly with an acceptable steady state error.In comparison with SMC, control values rise a little and the relative increment is 0.2% ∼ 4%.The system has minimum overshoot, minimum stable time, and minimum error values, which satisfy the requirements of high precision, fast response, and smooth operation.
Considering that the system is affected by nonlinear mismatched uncertainties, simulation results are presented in Tables 3 and 4 and Figures 13-19.
The simulation results without uncertainties (in Table 2) are compared with simulation results with uncertainties (in Table 3), and then system deterioration caused by uncertainties is demonstrated in Table 4, which will indicate system robustness against mismatched uncertainties.
As shown in results, when system is affected by uncertainties, ISMC has the most serious deterioration and simulation curves cannot keep smooth.Despite the fact that SMC can still stabilize the system, the control system quality will not be accepted after deterioration.In comparison with SMC and ISMC, QISMC can stabilized the system with a series advantages such as fast response, fewer rise time, fewer overshoot, and fewer small error.The relative control value    increment lies in 0.2% ∼ 5.3%.Simulation results illustrate that QISMC has strong robustness to nonlinear mismatched uncertainties and disturbance.

Conclusions
In order to compensate harmonic gear transmission errors by control method, this paper studies quadratic integral sliding mode control for a class of nonlinear harmonic gear drive systems with mismatched uncertainties, where a new quadratic integral sliding mode controller design method based on quadratic integral sliding mode surface is presented.Considering the nonlinear torques which are caused by backlash and frictions, the models of nonlinear frictions and torques are presented and the influence of nonlinear parts is compensated during control system design.In system modeling, frictions and nonlinear elastic deformations are taken into consideration and then the mathematical model of harmonic gear drive system is established.By using Lyapunov stability theory, it is proven that the quadratic integral sliding mode can be reached in finite time and closed-loop systems are asymptotic stable robustly.Simulation studies are carried out and the results verify the effectiveness of the proposed method.Compared with the traditional linear sliding mode control (SMC) and the integral sliding mode control (ISMC), the quadratic integral sliding mode control stabilizes the system rapidly with shorter rise time, lower overshoot, smaller errors, and strong robustness against mismatched uncertainties and nonlinear disturbances.The results indicate that QISMC has a good performance with strong robustness on a class of nonlinear system with mismatched uncertainties.

Figure 1 :
Figure 1: Structure of Harmonic gear drive system.

Figure 5 :
Figure 5: Structure of control system for harmonic gear drive systems.

Table 2 :
Simulation data without uncertainties.

Table 3 :
Simulation data with uncertainties.