Trajectory Tracking Strategy for Sliding Mode Control with Double Closed-Loop for Lawn Mowing Robot Based on ESO

The mowing robots work with a multivariable strong coupling underactuated system that is mostly troubled by difficulty controlling and unsatisfactory accuracy. Especially, the frequent external disturbances and parameter changes are likely to get missed and heavy cutting. In this paper, a new trajectory tracking control method based on extended state observer (ESO) is introduced with a particular focus on dual closed-loop sliding mode. Firstly, from the perspective of kinematics, a speed assistant controller was designed to generate the speed control quantity, and secondly, a sliding mode control algorithm based on the improved Fractional Power Rate Reaching Law (IFPRRL) was programmed to control the drive motor that tracked the speed control quantity. By means of comparison, our improved algorithm presented faster arrival time and better robustness along with similar jittering. At the same time, the robustness of the system was further enhanced with the help of an optimized ESO to tackle unmodeled disturbances and uncertain disturbances during the operation. Finally, the experimental analysis of the motor drive circuit and the trajectory tracking control system of the lawn mowing robot were both carried out respectively. The analysis shows that the performance of the proposed reaching law sliding mode control algorithm had some new pleasing changes, such as adjustment time and robustness. The circular trajectory and the detour mowing trajectory were respectively tracked in the double closed-loop sliding mode designed in this paper. The experimental goal was to ensure that the error vector Pe = (x Axis position error xe, y Axis position error ye, Angle error θe) all remaining at (0.01m, 0.01m, 0.01rad) were 5.34s and 5.36s, respectively, and both could be finally converged to 0. The results show that the newly developed controller based on ESO presented smaller arrival time and stronger robustness. The dual-closed-loop control of sliding-mode trajectory tracking method was capable to meet the real-time and precision requirements of the lawnmower robot for quick trajectory tracking.


I. INTRODUCTION
As a basic crop for urban greening and garden decoration, lawns are widely planted in parks, golf courses, private houses and other places. Beautiful and tidy lawns give people a feeling of refreshment. High-quality lawns therefore require a lot of manpower and material resources to maintain, especially the pruning job involving a complicated process and high cost. In order to achieve the purpose of reducing manpower and cost, more and more countries began to use lawn robots to take over the mowing job. Thanks to decades of development, the relevant technical level has made progress that is quite impressive [1]. Nonetheless, the working process of the lawn mower robot may have its mowing trajectory deviated due to the changeable terrain and intricate obstacles, leading to such misoperation as missing and re-cutting in some areas. This gave birth to a higher requirement for real-time planning of the mowing path, for accurate path tracking is the key to achieve full coverage of the mowing area with no deviation. Targeting the trajectory tracking of the lawn mower robot, industry professionals have achieved initial results from their in-depth research. In this work, the path tracking mechanism was investigated as to the lawn mower robot with a three-wheel differential drive.
At present, the trajectory tracking control methods of threewheel differential drive robots are commonly involved with PID control [2][3][4], back-stepping [5][6][7], and intelligent control [8][9][10][11] as well as adaptive control [12][13][14][15][16] and sliding mode variable structure control [17][18][19][20][21], etc. As a simple and reliable working mode, PID control is widely applied in tracking the trajectory of mobile robots. Victor Julio E. Normey-Rico et al. created a PID path tracking controller based on the kinematics model, and verified the feasibility of the PID controller on differentially driven mobile robots [2]. Literature [3] determined the PID controller parameters according to the minimum root mean square error (RMSE) of the rotational deviation of each wheel, proving that PID may have certain tracking advantages. Note that when it encounters the complex system of non-linear, underactuated and strong coupling such as lawn mowing robot, it may not be able to perform well in real-time and robustness [22]. Inversion control is a design method of forward-backward recursion, suitable for online control, real-time performance. Literature [5][6][7] studied the trajectory tracking problem of mobile robots under different conditions, introducing a back-stepping control concept. The simulation results showed that it performed well in a static environment, but held back by each simulation being allowed to only solve for a fixed time, not that much satisfactory as to the performance of the nonlinear system. The neural networks features amazing learning ability and has somehow been witnessed in the field of tracking control of wheeled mobile robots. Literature [8] worked with artificial neural network (ANN) to provide a strict trajectory tracking control method. Literature [9] proposed a kind of special tracking control strategy based on a three-layer neural network, which cleared up tracking problem by way of the Learning by Demonstration (LFD). In addition to using data learning and demonstration learning, Literature [10] also suggested a vision-based tracking model, which is a typical vision method for robot tracking associated with wavelet decomposition and artificial neural network (WD-ANN), which helped implement model building and trajectory tracking. Although the intelligent control function represented by neural network is powerful, the engineering application is far from being reliable, scalable, and maintainable due to the complex calculation and high hardware requirements. The wheel-style mobile robot is required to adapt to the complex and changeable working environment, so its mathematical model is difficult to establish accurately, which may bring many uncertain factors to the accurate control of the system. The study of adaptive control is aimed at systems with a certain degree of uncertainty, so some scholars tend to use of adaptive control algorithms in trajectory tracking control. Martins Felipe N. et al. developed an adaptive controller, which enabled the parameters of the robot dynamics to be updated online, so that the system worked with smaller errors and better performance. [12] Literature [13] explored uncertain nonholonomic tracking of mobile robot paths. Combined with the actuator dynamics, a simple adaptive control method was established by the use of adaptive control techniques to handle all uncertainties with good results. Literature [14] successfully constructed an adaptive robust tracking controller, which could offset the parameter uncertainty in a wheeled mobile robot asymptotically and accurately. The research of the Literature [15] was connected with a new scheme of adaptive virtual speed controller and torque control law. To obtain the feedforward compensation, disturbance observers were placed for estimating lumped disturbances. The adaptive control method can well solve the uncertain disturbance, but through a hard process due to the cumbersome design, high cost and complex realization.
The variable structure control is nonlinear and was proposed by the former Soviet Union scholars Emelyanov and Utkin [23] since 1950s, and has developed into a relatively independent research branch. This is a non-fixed system structure that may change regularly with the current state of the system, forming a unique "sliding mode" that moves along the state trajectory. Therefore, it is also called Sliding Mode Control (SMC). Because the "sliding mode" can be created by itself and need no more job from system parameters and disturbances, it is quite impressive for simple design, fast response and strong robustness. It is worth noting that after the state reaches the sliding surface, it would be difficult to ensure that it could stay on the sliding surface rather than travel back and forth, resulting in jittering. This is unbearable for general controller actuators. That's why how to reduce the jittering is vital in sliding mode control, hence coming into the vision of peers.
Commonly used methods for suppressing jittering may cover boundary layer [24][25][26], filtering mode [27][28], highorder sliding mode control [29][30][31][32][33][34][35] or reaching law [36][37][38][39][40], etc. The boundary layer method involves the use of a saturation function instead of a switching function, which may effectively suppress jittering, yet at the expense of system robustness. The filtering mode is proven to be effective on smoothly filtering the control signal and eliminating jittering. Unfortunately, the credits are reduced because of difficulties in maintaining the post-filtering stability of the system, making the process of stability analysis more complicated. The high-order sliding mode can not only effectively suppress output jittering, but also ensure good robustness, especially in some high-order nonlinear systems. This feature comes along at the cost of the complexity of the closed-loop system and should be used with caution due to design complexity [32]. The reaching lawology was proposed by Gao Weibing [36], who worked out three commonly used ways: index reaching law, isokinetic reaching law, and sub-order reaching law. This method can subtly adjust the reaching law parameters to reduce high-frequency buffeting under the premise of ensuring the dynamic quality of the system, thus drawing a lot of eyeballs from peers. For example, Literature [37] discussed an Exponential Reaching Law (ERL) method, used an exponential function that dynamically adapted to changes in the controlled system to design a nonlinear approach law. Control input jittering was reduced while maintaining the high tracking performance of the controller in steady state. Literature [38] improved ERL and recommended an Enhanced Exponential Reaching Law (EERL), which had been successfully applied to wind power generation systems. Reference [39] introduced a Power Rate Exponential Reaching Law (PRERL) reaching law to improve controller arrival time and reduce jittering. Reference [40] set its focus on a Fractional Power Rate Reaching Law (FPRRL), which was an improved version in terms of arrival time, robustness, and reduced jitter when compared to the conventional reaching law. The reaching law is regarded as a good control scheme, characterized by a simple design and control dynamics.
To sum up, accurately tracking the rotational speed of the driving wheels of the mowing robot requires both strong robustness and real-time performance within the allowable range of jittering. For this purpose, inspired by the literature [40], this work developed an optimized FPRRL driving motor sliding mode control algorithm, which obtained better control performance. At the same time, a double-closed loop was included to establish a new sliding mode control strategy, which aimed toaccurately track the predetermined trajectory of the mowing robot. This work made the following main contributions: 1) On the basis of the original FPEEL, a new reaching term was added without changing the original parameters, which reduced the arrival time and improved the robustness.
2) Considering the unmodeled disturbance and external disturbance of the system, an extended state observer was designed. An adaptive compensation term for external disturbance was therefore introduced into the control law to improve the robustness of the system.
3) A double closed-loop sliding mode controller was set up to ensure finite-time convergence and realize the tracking of the predetermined trajectory of the lawnmower robot.
The main structure of this paper is as follows: Section 2 describes the system as a whole; Section 3 elaborates the modeling of the lawn mower, the design of the extended state observer, the derivation process of the trajectory tracking controller and the stability proof; Section 4 discusses the simulation results; Section 5 gives the conclusion.

II. SYSTEM OVERVIEW
The mowing robot mobile platform studied in this paper worked with a three-wheel working structure, in which the front wheel was the guide, and the rear two wheels were the driving wheels, with the steering to be realized by differential driving. The driving wheel was driven by two DC motors, and the speed was fed back through the encoder; the position of the body was given by the positioning system.
To realize the tracking of the desired trajectory, the control system was expected to give the output voltage of the driving motor of the mowing robot. To achieve this goal, the desired trajectory was asked to be given in the first place. The angular velocity and linear velocity control laws were estimated by the kinematic controller. Subsequently, the voltage output of the driving wheel was estimated by the speed tracking controller in order to form a closed loop. In this way, controllability of the voltage, and thus tracking of the desired trajectory, was fixed. The process is shown in Figure 1

III. DESIGN THE CONTROL ALGORITHM
The algorithm part of this paper includes the kinematic model, the derivation of the motor drive model, the design of the extended state observer, the design of the kinematics controller, the deduction of the speed control algorithm of the driving wheel motor based on the IFPRRL sliding mode control and its stability proof.

A. KINEMATICS MODELING
Kinematics model is the basis for studying the trajectory tracking of sports robots [41]. By establishing a Cartesian coordinate system (as shown in Figure For the kinematics model, the system inputs are linear velocity v and angular velocity  , while for the mowing robot, there are only two drive motors that can be directly controlled. So, it is necessary to design a kinematics controller to transform the system control v and  The relationship between the linear velocity v and the angular velocity  into controlling the rotation speed of the two driving wheels of the robot.  In the figure 3, the center of the steering circle of the robot is N, the geometric center is M, the angle of rotation of the robot is θ, and the distance between the two wheel axes is 2L, OM is the turning radius r. The relationship between the linear velocity v and the linear velocity of the left and right driving wheel vl, vr is = 2 The angular velocity of the left and right wheels are the same, so Thus the relation ship of the angular velocity ω and v, vr is obtained Combining (3) and (5), we can get the relationship between q and vl, vr Substitute (3) and (5) into equation (2) to obtain the relationship between the state of the lawn mower robot and the rotation speed of the left and right wheels.

B. MODEL OF MOTOR DRIVE
The mowing robot used in this work was equipped with a permanent magnet DC motor as the drive motor, which is a device that converts electrical energy into mechanical energy through magnetic coupling. There were two essential parts: the rotor (or armature) and the stator. The armature rotated in the stator frame of the motor [42]. Its equivalent circuit diagram is shown as in Fig. 4. Where Ra is the total resistance of the armature circuit (including the internal resistance of the power electronic device and the resistance connected in the main circuit), which is a constant, Vc is the back electromotive force generated by the internal coil when the motor is working, Va is the voltage across the motor, La is the total inductance of the main circuit and other inductors connected to it, Te is the electromagnetic torque, a  angular speed of rotor rotation, a i armature circuit current.
Evidently, when the load is added, the speed of the motor will decrease as the load increases. To make sure that the motor runs stably and safely at the given speed, the motor operation process was analyzed in depth so as to work out the motor modeling and dynamic structure design. According to the motor model, the voltage balance equation of the main circuit is established as follows Where v k is the speed constant determined by the magnetic flux density of the permanent magnet, the number of rotor windings, and the physical properties of the iron core.
Then operate the motor torque balance equation, we get ++ Where t k is the torque constant determined by the magnetic flux density of the permanent magnet, the number of rotor windings and the physical properties of the iron core, J is the moment of inertia of the rotor and motor load, and B is the damping constant of the entire mechanical rotating system. Substituting (12) into (11) Combine (11) and (13) to get a complete description of the DC motor: Laplace transformation is performed on equation (14) to If the disturbance around the steady state is considered, and the initial condition of the motor is assumed to be zero, all variables refer to the offset from the reference state, and the above equation is to become From the above, the transfer function of the entire motor is: According to formula (17), the state space equation of the DC motor drive model is written as: () ut is the control voltage input, and () dt is the uncertainty disturbance of the system.

C. EXTENDED STATE OBSERVER
Since the total uncertainty disturbance () dt in the system (18) was unknown, it was necessary to estimate it. ESO was a novel observation algorithm that was responsible for online estimation of internal and external disturbances [43].
Define new state variables * and get a new state space equation (19): To obtain an estimate of the unknown disturbance d(t), a thirdorder ESO extended state observer (20) was designed: 3  is an adjustable state observer parameter. Define estimation error That is, the characteristic polynomial becomes ( ) The three eigenvalues were all

D. DOUBLE CLOSED-LOOP SLIDING MODE CONTROLLER
The lawnmower robot is a multi-variable and strongly coupled nonlinear underactuated control system, which has such disadvantages as the changing terrain of the working environment, the intricate obstacles, and the difficulty in establishing the mathematical model accurately. challenge. To deal with the above problems, this work designed a tracking control method in a double closed-loop sliding mode based on ESO. The control block diagram is shown in Figure 6   . Therefore, the design control law is: Stability proof: 1 Stability proof:  For this,  could be obtained as follows In the equation, the first four cases correspond to the calculation formulas of  when the four quadrants 1  , 2  , 3  and 4  respectively from top to bottom, and the last four cases are respectively the  calculation formula corresponding to the four-part coordinate axis. It is worth noting here that the initial value of n is 0. When  changed from the quadrant I to quadrant IV, the value of n was incremented by 1. Conversely, when  changed from the quadrant IV to quadrant I, the value of n was decremented by 1.
θ obtained by formula (30) is the angle required by the position control law formulas (28) and (29). If θ was equal to θd, then the ideal trajectory tracking control law could be realized. In reality, θ and θd cannot be exactly the same, especially the initial stage of control being very different. This may cause the entire closed-loop control system to fall in an unstable state.
For this reason, the angle  obtained by formula (29) may be regarded as an ideal value, that is, take In reality, the difference between θ and θd may lead to an inaccurate position control law, resulting in the instability of the entire closed-loop system. Therefore, it is necessary to design a faster algorithm in attitude control law than the position control law to make θ track θd in a quicker way.
The previous task guarantees (x, y) tracking. Secondly, the accurate tracking of  makes it necessary to create the attitude control law w, so as to see the angular velocity tracking. Take

V s s k s s k s
, so, the system was asymptotically stable, which means that in a limited time, 3 0, 0 e s  →→ . The designed closed-loop system is composed of inner and outer loops. The inner loop is the attitude subsystem, including an attitude subsystem sliding mode controller; the outer loop is the position subsystem to generate the intermediate command signal θd, which is passed to the inner loop system. The inner loop system follows the sliding mode control law to track this intermediate command signal. The closed loop system is structured as shown in the red box in Figure 6. Since d  needs to be implemented when designing the inner loop controller, a continuous value is required for vd, so that the control law u1 and u2 are continuous values. Therefore, u1 and u2 are not allowed to include switching functions.
In the attitude control law equation (34), the derivation of d  is needed. This is a complicated process. In this work, a linear second-order differentiator was merited to implement d  .
In the equation, the input signal to be differentiated is n(t), 1  is to track the signal, and 2  is the estimation of the first derivative of the signal, A is the differentiator parameter, the initial value of the differentiator is 1 (0) 0 To achieve stable inner-loop control with sliding mode, this work was performed by way of making the inner-loop convergence speed greater than the outer-loop convergence speed to achieve θ fast tracking θd, thus ensuring the stability of the closed-loop system. In this system, the control gain coefficients of the inner and outer loops were by adjusted. This is a guarantee that the convergence speed of the inner loop is greater than that of the outer loop.

2) SLIDING MODE CONTROLLER BASED ON IFPRRL
As aforementioned, we have got the control laws v and  , and the left and right wheel speeds could be obtained by formula (7) nl, nr. The issue on how to make the left and right wheel drive motors of the lawn mower robot track nl, nr stood up to be discussed below. The driving motors were the same to the left wheel and the right wheel, and won't be described separately in this paper.
According to the state space equation (18), select the error x is the set value of the rotational angular velocity. The first step is to determine the sliding mode surface function 4 s , as shown in formula (36) 41 ee s c n n =+ (36) To ensure the existence of the designed sliding surface, 1 c needs to satisfy the Hurwitz condition, that is, 1 c >0. Taking the derivative of 4 s , we get Although this reaching law suppressed jittering well, it had a better arrival time for systems with small step changes. Nonetheless, when applied to a system with large step changes, a larger k value had to be selected to ensure the arrival time, except for an excessively large k that may lead to increased jittering. Therefore, it is crucial to reduce the arrival time within a reasonable range. In this regard, this work managed to get an improved FPRRL, which is shown in formula (39) after the improvement ( ) are strictly positive, so it has no effect on the stability of the system. In the improved reaching law, a new exponential approach term

IV. SIMULATION RESULTS AND ANALYSIS A. SIMULATION AND ANALYSIS OF DRIVE MOTOR SPEED CONTROL
Firstly, taking the robot-driven DC motor as the controlled object, the correctness and effectiveness of the improved FPRRL sliding mode control algorithm in this paper were verified. The simulation environment was MATLAB2018b, the drive motor parameters were Ra=2, La=0.02 H, J=10.1 kg.m 2 , kv =1.8V/(rad/s), kt=17.2Nm/A, ESO bandwidth 60 o  = . Set the initial speed to 0rad/s and the target speed to 2rad/s. Its speed was regulated, respectively, by means of PID controller, two mature reaching law control methods EERL [38], FPRRL [40] and improved FPRRL. The key formulas and parameter settings used in simulation are shown in Table 1.
Figure9 shows the tracking effect. As evidenced in the figure, the PID control shows a faster rising speed, but the overshoot of 9.45% and the adjustment time of 1.3s produced an unsatisfactory control effect. Except for the PI controller, all three reaching law sliding mode control methods presented similar arrival times, where FPRRL was slightly better than EERL, and the improved FPRRL was slightly better than FPRRL. Note that the improvement here is not an improvement by adjusting the size of the parameters. Rather, a new structure did the job to improve the control performance under similar parameters.
Figure10 shows the output of the controller. The improved FPRRL was compared with the control outputs of the PI, EERL, FPRRL algorithms. It is plain to see that, except for PI control, jittering exists in all three sliding mode control methods. Among them, the jittering of EERL is the largest, and the amplitude is within ±0.005V. The FPRRL and the improved FPRRL have almost the same jitter amplitude, within ±0.003V. The improved FPRRL reaching law proposed in this paper made good performance in jittering control. That is, the advantage of low jittering of FPRRL may be maintained on the premise of an optimized arrival time.
Figure11 shows the achievement of three sliding mode surfaces of EERL, FPRRL and improved FPRRL. It can be seen that the improved FPRRL gave the shortest arrival time, followed by FPRRL and finally EERL. As expected, there were sliding modes in all three sliding modes, but the buffeting of FPRRL and improved FPRRL was significantly smaller than that of EERL. The results showed that the improved FPRRL expressed a faster arrival time than the previous one under the premise of similar system robustness and jittering.

B. ROBUSTNESS ANALYSIS OF CONTROLLER UNDER STEP DISTURBANCE
To verify the robustness of the improved FPRRL algorithm, this work made a suddenly increase in the interference at t=2.5s (formula 42). If the controller was robust enough, it must be able to overcome the disturbance and maintain a good tracking effect. Figure12 is the response curve after applying disturbance, and Figure 13 is the controller output. Obviously, both EERL and PI controller performance were greatly affected. Among them, the EERL error was the largest, up to 0.17rad, followed by PI control, which also hit 0.085rad. The maximum error between FPRRL and the improved FPRRL was not more than 0.033rad, and the maximum error of the improved FPRRL was slightly lower than that of the FPRRL, and the maximum error was 0.03rad. Additionally, the recovery time from disturbance also varied. The PI controller had the longest adjustment time of 0.826s, followed by EERL with 0.564s, FPRRL with 0.241s, and the improved FPRRL with the shortest adjustment time of 0.235s. The results showed that the algorithm proposed in this paper had improved robustness.

C. SIMULATION AND ANALYSIS
Taking the lawn mower robot as the controlled object, the path tracking simulation experiment was then carried out. Improved FPRRL sliding mode controller used parameter selection as the same as above. The parameters in the motion controller were a1=a2=3, b1=b2=10, a3=3.0, 3 b =0.5. In the attitude control law, the differentiator parameter was selected as 100. The actual parameters of the robot are as follows: the radius of the driving wheel is 0.1m, the distance between the two wheels is 0.3m, and the distance between the center line of the wheel and the front end is 0.12m.
First select a circular trajectory for tracking. The center of the reference trajectory (desired trajectory) was (0m, 0m), with the radius of 1 m. As shown in formula (43): Robot The initial pose of the global coordinate is (x0, y0, θ0) = (0.6m, 1 m, 0rad), the initial linear velocity and steering angle are (v0, w0) = (0 m/s, 0 rad/s). The simulation result of circular trajectory tracking is shown in the figure 14 shown.
As indicated in Figure 14, θe is stable within 0.01rad after 0.94s, and xe is within 3.5s After stabilizing within 0.01m, ye stabilized within 0.01m after 5.34s. The simulation error of the circular arc trajectory of the platform continued to converge with the increase of time, and finally tends to 0. And Fig.  14(h)(i) shows the output jittering of the controller of the left wheel and the right wheel are both kept within the range of ±0.02V. Under the condition that the initial errors are xe=0.4m and ye=1m respectively, the proposed controller, as expected, made the lawnmower robot converge to the desired trajectory smoothly and achieved satisfactory tracking effect. Judging from the tracking situation of θ in Fig. 12(b), θ was continuous during the movement of the robot, which successfully solved the problem of control system jitter caused by the angle flip during the control process. To simulate the real mowing path, this work picked a 6m × 10m rectangular lawn as the lawn mowing robot's work site, in which the path planning was detoured. The simulation environment and controller parameters were the same as above, and the robot's initial pose (x 0,y0,θ0)= (1m , 0 m, 0rad), the initial linear velocity and steering angle were (v0,w0)= (0 m/s, 0 rad/s). Set the starting position of the path as (0m, 1m), the straight line length of the path as 10m, the set speed of the straight line as 0.1m/s, the arc radius as 0.5m, and the turning velocity as app. 0.01m/s, and the whole tracking process was introduced with disturbance (44).
Where () l dt is the left wheel interference, () r dt is the right wheel interference, () nt is Gaussian white noise, its amplitude is [-1,1], and the signal-to-noise ratio is 3dB. The tracking effect is shown in Figure 15.   Figure 15 displays that the simulation error of the tracking trajectory continues to converge with time, xe, ye, θe are kept within the range of 0.01m, 0.01m, 0.01rad, and the time is 3.55s, 5.36s, 0.99s respectively. And look in Figures (h) and (i), the control voltage of the left and right wheels varies little, and the jittering is not greater than ±0.05V. Furthermore, the designed ESO did a good job on observing the interference of the superposition of sinusoidal and Gaussian white noise, and then compensated the controller to ensure the robustness of the system.
The simulation results showed that the sliding mode control with double closed-loop designed in this work came along with several typical advantages, such as real-time performance, robustness and tracking accuracy. When used to track the trajectory of the lawn mowing robot, this proposed methodology was able to satisfy the tracking requirements of the predetermined trajectory, and served as valuable reference for achieving full coverage of the mowing area and reducing missed cuts and re-cuts.

V. CONCLUSION
In this work, the lawnmower robot was regarded as a cascaded system composed of motor drive equations and vehicle body kinematics equations. The whole design was aimed at a sliding mode controller for driving motor and a sliding mode trajectory tracking controller with dual closed-loop for lawn mowing robot. The following conclusions are drawn: 1) In this paper, a new reaching law sliding mode controller is introduced to control the speed of the drive motor through simulation experiments. The results showed that under similar buffeting, the new reaching law got faster arrival time and better robustness than the traditional sliding mode algorithm.
2) For unmodeled disturbances and external disturbances, an extended state observer was developed to achieve effective observation of external disturbances and compensate them in control, further improving the robustness of the system.
3) The sliding mode controller with double closed-loop designed was proven to control the lawn mowing robot to effectively track the circular trajectory. It guaranteed that the error vectors Pe= (xe, ye, θe) remaining in the range of (0.01m, 0.01m, 0.01rad) are 3.5s, 5.34s and 0.94s at the time frame, respectively, and finally converged to 0. Accordingly, the control voltage jittering was controlled within ±0.02V. 4) In the scenario where both the left and right wheels were accompanied by the superimposed interference of sine and white noise, the circuitous mowing trajectory was effectively tracked while the tracking error continued to converge with the increase of time. Specifically, xe, ye, θe kept in the range of 0.01m, 0.01m, 0.01rad at the time frame of 3.55s, 5.36s, 0.99s respectively. The voltage jittering was controlled within ±0.05V, prevailing at anti-interference ability.