# Nonholonomic Wheeled Mobile Robot Trajectory Tracking Control Based on Improved Sliding Mode Variable Structure

Authors: Hua Cen; Bhupesh Kumar Singh
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

Several research studies are conducted based on the control of wheeled mobile robots. Nonholonomy constraints associated with wheeled mobile robots have encouraged the development of highly nonlinear control techniques. Nonholonomic wheeled mobile robot systems might be exposed to numerous payloads as per the application requirements. This can affect statically or dynamically the complete system mass, inertia, the location of the center of mass, and additional hardware constraints. Due to the nonholonomic and motion limited properties of wheeled mobile robots, the precision of trajectory tracking control is poor. The nonholonomic wheeled mobile robot tracking system is therefore being explored. The kinematic model and sliding mode control model are analyzed, and the trajectory tracking control of the robot is carried out using an enhanced variable structure based on sliding mode. The shear and sliding mode controls are designed, and the control stability is reviewed to control the trajectory of a nonholonomic wheeled mobile robot. The simulation outcomes show that the projected trajectory track control technique is able to improve the mobile robot’s control, the error of a pose is small, and the linear velocity and angular speed can be controlled. Take the linear and angular velocity as the predicted trajectory.


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## Body

## 1. Introduction

A mobile robot is a dynamic nonlinear device, with the benefits of unmanned driving and different controlling practices [1]. A wide range of applications covering survey, patrouille, emergency saving, recognition, and petrochemical features in automation industry, building structure, transport, health care, and other applications are being used by WMR from the last two decades. With the continuous improvement of robot performance, the application range of mobile robots is greatly expanded, and it is widely used in industry, agriculture, service industry, and other industries [2]. Therefore, it has become the hot spot of many scholars in recent years. Among them, wheeled mobile robot (WMR) has a wide range of practical application background [3]. This kind of system is a typical nonholonomic dynamic system because the rolling contact between the wheel and the ground must satisfy the nonholonomic constraint of pure rolling and no slipping. With the extensive feature of wheeled mobile robots (WMR) in various fields, the precision requirements for the motion trajectory control of wheeled mobile robots are also higher and higher [4, 5]. Each application has specific constraints to the robot design process, which is minimized if all compromises are interpreted by choosing the most convenient drive configuration.Pappalardo and Guida [6] analyzed the dynamics and control method of nonholonomic robots, studied the Udwadia kalaba equation and underactuated equivalent principle, and carried out corresponding motion control for the robot. Taking the dynamic model of a mobile robot as an example, numerical tests conducted to verify the effectiveness of the projected control method. In addition to preventing distinctiveness nonetheless also keeping the benefits of the mode control sliding system, Zhao et al. [7] have suggested an uncompulsive wild nonsingular fatal sliding mode control for mobile robots. The experimental results show that the stability of the control method is high, and the tracking error of the robot trajectory is small.Ernesto et al. [2] suggested that the response be improved through kinematic model when the robot is backward-oriented and with a short distance to reach the location. Spandan et al. [8] did a novel stability study for the time-related control and gave an empirical relation between the output of the control and the approximation error. In the case where the wheel velocity is not existing deprived in regard to parametrical uncertainty, Shi et al. [9] stated the controller for the case. Bessas et al. [10] have developed an integrated mode controller based on the system’s kinematic model. In consideration of several kinematic models, Onat and Ozkan [11] designed an adaptive parameter recognition controller. Based on the tracking error, the controller is moved to different kinematic models. In a Cartesian Coordinate, Lee et al. [12] developed the SMC approach angle. The risk of a switching effect due to high switching gains is a significant drawback of the switching control strategy, which can in turn lead to high frequency unmodelled dynamics affecting the robotic system. In addition, determining the uncertainty limits requires a nominal modelling of the system which, for a complex system such as WMR, is another tedious process [13].The parametric uncertainties in the actuator were approximated by the Arab and Fateh [14] fuszy logic, while the core controller used the system’s linearization approach to ignore system uncertainty. Application of response linearization-based control or converse dynamic control in real-life circumstances is not continually achievable because it needs a precise understanding of the hardware constraints. In view of the actuator limitation, Shojaei [15] recorded a strong controller with known uncertainty connection. Time-delayed control (TDC) [16] approximates all incertitude’s using immediate past-instant control input and state information. TDC output however is influenced by the TDE that occurs when the time is finite.A large number of application domains extending from surveillance, planetary exploration, patrolling, emergency rescue operations, reconnaissance, petrochemical applications, industrial automation, construction, entertainment, museum guides, personal services, intervention in extreme environments, transportation, medical care, and other industrial and nonindustrial applications are making use of wheeled mobile robots since the past two decades. Each application faces unique limitations on the robot design procedure which is reduced by selecting the most convenient drive configuration interpreting all the compromises.The shear feature and sliding mode controller were established in accordance with the sliding mode control theory in accordance with the study of the mobile robot kinematic model. The control stability is studied to diminish the posing fault, and the trajectory of nonholonomic mobile robot control is carried out.


## 2. Trajectory Tracking Control in accordance with Improved Sliding Mode Variable Structure

The trajectory tracking problem of nonholonomic WMR is to propose a virtual mobile robot model which can perfectly track the ideal space trajectory and then let the actual robot track. Depending on the spatial coordinates, the desired movement of the nonholonomic robot can be expressed [6]. Different robot states, such as angular velocity, linear velocity, and spatial location, differ in time. The tracking trajectory not only included details about the spatial location but also the kinematic model and tracking trajectory feature of the nonholonomic mobile robot with wheels, as well as information on speeds and mobile angles [17]. As shown in Figure 1, the two driving wheels are autonomously determined through 2 DC motors. The follow-up wheel only plays a supporting role in the moving process, and its influence in the kinematic model can be ignored.Figure 1
Representation of nonholonomic wheeled mobile robot.Figure1 shows nonholonomic wheeled moving robot (WMR) powered by two engines attached to a radius r at distance b of the two wheels. The vehicle length is regarded as L. The x-axis of the axle of the robot in the center of mass is located by a moving body-fixed coordinate system, and the distance offset is supposed to apply. Let P also stands for the WMR mass deprived of the driving wheels, rotor, and gearbox rotors; w be each wheel’s mass and the appropriate motor-rotor and gearbox rotor; Iw stands for the wheel MOI equivalent to the wheel axis; and Im is the MOIs of each wheel with a rotor engine and gearbox that passes through the z-axis with a wheel diameter. Three generalized coordinates, xc, yc, and θ, will specify the location of the WMR. The COM location here xc,yc is the WMR heading angle, and θ is the WMR heading angle. Again, two more variables θr and θl, indicating the rotation of the right-wing and left-wing wheels, are specified to identify some point on the wheels.


### 2.1. Nonholonomic Constraints for Wheeled Mobile Robots

As a characteristic nonholonomic control scheme, the motion of WMR is not only limited by the basic requirements of starting point and ending point but also meets the special requirements of velocity direction or trajectory shape. In addition, WMR is an essentially nonlinear under actuated drift-free dynamic system, which brings greater challenges to its control [18, 19]. However, nonintegrity makes the structure of the robot more flexible and reliable, such as allowing some joints of the robot to be driveless, which greatly reduces the manufacturing cost, quality, and energy consumption of the robot. The nonholonomic control system represented by nonholonomic wheeled mobile robot is a new type of subject, which has important practical value and theoretical significance.The motion behavior of the mobile robot is constrained by kinematics, which is determined by the structural characteristics of the robot and its motion model. In practical applications, the system is always subject to various constraints, such as geometric constraints and motion constraints, steady and unsteady constraints, double-sided constraints and single-sided constraints, integrable constraints, and nonintegrable constraints [20–23]. Under different constraints, the motion of the control system shows different characteristics. From the essence of system motion, constraints are divided into complete constraints and nonholonomic constraints. Nonholonomic constraints restrict the driving direction of robots. Holonomic constraints are dimensions relative to the state description of the system. Suppose a constraint form:(1)fq,q̇,t=0,whereq=q1,q2,⋯,qn is the generated space vector.If this form of constraint can be transformed into a form:(2)Fq,t=0,then the constraint form is integrable. Even iff in Equation (1) contains the differential quantity of generating space vector in time, it can be expressed by Equation (2), which is usually called complete constraint.A nonholonomic constraint is defined as a time derivative including variables (coordinates) in the generating system and cannot be integrated. Nonholonomic constraint is a characteristic of the mechanical system. When the dimension of the control variable is lower than the state variable, the control variable can control the state variable [24, 25]. If the system has nonholonomic constraints, the system is nonholonomic, so it cannot move in any path (for example, the car will drive forward or backward, not side-by-side) [26]. The movement of this system is not arbitrary for nonholonomic systems; nonholonomic restrictions specify a subspace in the current time of the potential speed space. The return of internal variables to the initial value in a nonholonomic system does little to ensure that the system returns to the initial state. The condition of a nonholonomic system therefore depends on the route (a sequence set of internal variables that vary according to time). The coordinate dimension for the generated space is not reduced by nonholonomic restrictions excluding the generated speed space.If the constraint of Equation (1) is integrable, its kinematic constraint is complete; that is, the velocity q1,q2,⋯,qn can be eliminated from the equation. If the constraint of Equation (1) is not integrable, then the corresponding kinematic constraint is incomplete.


### 2.2. Kinematic Model Analysis

Mobile robots through nonholonomic components are called nonholonomic mobile robots. It belongs to the category of nonholonomic constraints, and its nonholonomic constraints mainly affect the motion characteristics of nonholonomic systems. The nonholonomic constraints received by wheeled mobile robots are generally expressed as first-order differential equations.In the plane, the position and attitude of the robot can be represented by the following state vectors:(3)xmt=xmtymtθmt.x
m
,
y
m represents the position of robot global coordinate system, and θm represents the steering angle of robot global coordinate system. xm represents the pose state of the robot at time t.To keep the balance in the plane, this type of robot usually has one or more castors as additional support only. The driving control is completed by motors which are separately distributed in two wheels. In the universal organize scheme, the motion equation of the robot can be formulated as:(4)ẋmtẏmtθ̇mt=cosθmt0sinθmt001vtwt.Its nonholonomic motion constraints are as follows:(5)‐ẋmsinθm+ẏmcosθm=0.Equation (5) can be written in terms of the direction of motion that can be realized by the motion matrix(6)S=cosθmt0cosθmt001.v
t and wt are usually regarded as control variables in robot motion research. After applying the Euler integral to Equation (4), it can be written as follows:(7)xmk+1=xmk+vkΔtcosθmt,ymk+1=ymk+vkΔtsinθmt,θmk+1=θmk+wkΔt,wherek is the discrete time constant and Δt is the time interval.There are many ways to make the robot move to the ideal position by using the plane changing trajectory. The inverse kinematic challenge is easier for the robot to plane the target trajectoryxt,yt, so that the direction of the robot is always tangent to the trajectory. If the initial pose of the reference robot is on a certain trajectory, then there is a kinematic model corresponding to the physical quantity of the reference robot. It can be concluded that the control linear velocity is vt:(8)vt=±ẋ2t+ẏ2t.The positive and negative signs represent the ideal driving direction (+ is the forward direction, and - is the reverse direction). Every point on the track has a tangent angle defined as follows:(9)θt=arctan2ẏt,ẋt+kπ,wherek∈0,1 defines the ideal driving direction (0 is the forward direction, and 1 is the reverse direction). By calculating the differential of time in Equation (9), the angular velocity of the robot can be obtained:(10)wt=ẋty¨t−ẏtx¨tẋ2t+ẏ2t.By using the Equations (8) and (9), the ideal trajectory control is obtained.


### 2.3. Sliding Mode Control Principle

When the system does not satisfy the linear parameterization condition or when the system model has both structural and parametric uncertainties, a sliding mode control algorithm can be used. If the upper bound function can be found for the unknown dynamic characteristics, the sliding mode control algorithm can achieve the desired control effect by strengthening the control input, that is, using very strong control variables to deal with the unknown dynamic characteristics. Sliding mode control has strong control performance for nonlinear systems affected by uncertain dynamics, which makes the system move according to the designed sliding mode trajectory. The control method has the advantages of a simple calculation method and strong anti-interference ability.Sliding mode control for nonlinear systems influenced by unknown factors is an easy and efficient way of control. The problem can be reduced to a complex one by simple sliding mode control. The nonlinear method for the single input output can be expressed as follows:(11)yn=fy¯+u.u
∈
R and y∈R are control input variables and system output variables, respectively, and y¯=y,ẏ,⋯,yn−1T∈Rn is system state vector.In order to make the trajectory tracking errore¯=y¯−y¯d=e,ė,⋯en−11 of the controlled system converge to zero, that is, the state variables of the closed-loop system can gradually track the reference trajectory y¯d=yd,ẏd,⋯ydn−1T, a state feedback control law needs to be designed by using sliding mode controller. The PI sliding mode variable structure controller is defined as follows:(12)sy,t=et+γ∫0teτdτ.Among them,γ>0. When sy,t=0 is set, e=0 is the only solution to Equation (12). Then, the problem of tracking the desired trajectory yd is equivalent to the problem of function sy,t=0. At the same time, the selection of control law u should satisfy the following conditions:(13)12ddts2≤−ρs,whereρ>0 is a constant. Equation (13) is called sliding condition, which guarantees that the error state can converge from the initial position far away from sy,t=0 surface to sy,t=0 surface.The strategy of sliding mode controller mainly comprises 2 parts: the equivalent control variableueq is designed for the ideal sliding mode control, and the variable structure control variable usw is also designed for the nonlinear system affected by various uncertain factors. The design of sliding mode variable structure controller consists of two steps:Step 1.
Designs switching function to make sliding mode stable and dynamic.Step 2.
The sliding mode control lawu is designed to form a sliding mode region on the switching surface sy,t=0 under the condition that the condition Equation (13) is satisfied [27].For uncertain systems, using the sliding mode variable structure control method can often get the desired results. Compared with the traditional control algorithm, the sliding mode production variable structure control method is discrete; that means, the high-frequency switch control is used to switch between several control laws, which can make the system oscillate with small amplitude and high frequency near the set space surface. Such motion is defined as sliding mode motion. Since various control laws in the variable control method for sliding mode are set in advance and are not disturbed and do not rely on device parameters, this method is highly robust. Then, the concept of sliding mode control combines a trajectory tracking controller with global asymptotic stability.


### 2.4. Design Shear Function

Lemma 1.
Anyx∈R and x<∞, then φx=xsinarctanx≥0, if and only if x=0, the equal sign holds. The switching function of sliding mode variable structure control is designed based on backstepping method; firstly, the Lyapunov function is defined as Vy=1/2ye2; suppose xe=cfkωye, where fkω=1−exp−2kω/1+exp−2kω is a hyperbolic tangent function and cfkωye is a virtual feedback variable; c and k were normal. According to the characteristics of hyperbolic tangent function, we can know ωfkω≥0. Assuming ϕe=−arctanurye, then(14)ϕ̇=∂ϕe∂ur+∂ϕe∂ye=ur+ye1+urye2<0.Based on Equation (14) and Lemma 1, the following equation can be given(15)V̇y=yeẏe=ye−xeω−aω+ursinφe+aωrcosφe=‐yexeω−yeaω+uryesin−arctanurye+yeaωrcos−arctanurye≤‐yexeω−yeaω+yeaωr=‐ye2cfkωω+yeaφ̇e<0.It is concluded that as long asxe converges to cfkωye and φe converges to −arctanurye, the system state variable ye will converge to 0. According to this conclusion, the switching function is designed as follows:(16)s=s1s2=xe−cfkωyeφe+arctanurye.Through the design of sliding mode controller,s1 and s2 tend to 0, which means xe converges to cfkωye, φe converges to 0, so the system state variable ye also converges to 0.


### 2.5. Design of Sliding Mode Controller

To weaken the jitter, the exponential reaching lawṡ=−εsgns−hs is adopted, in which ε and h are greater than 0, and ε is a small positive number. The following equation can be obtained by using S type continuous function 1−e−s/1+e−s instead of symbol function:(17)ṡi=−εi1−e−si1+e−si−hisi,i=1,2.ε
i is a small positive number, and hi is a normal number.Supposeα=arctanurye, from Equation (16) and Equation (17), we can get the following results:(18)ṡi=ṡ1ṡ2=−ε11−e−s11+e−s1−h1s1−ε21−e−s21+e−s2−h2s2=ẋ−ckf′kωω̇ye−cfkωẏeϕ̇e+∂α∂uru̇r+∂α∂yeẏe=yeω−u+urcosφe−aωrsinφe−ckf′kωω̇ye−cfkωẏeωr−ω+∂α∂uru̇r+∂α∂yeẏe.From Equation (18), the kinematic trajectory tracking control law of the nonholonomic wheeled mobile robot with discrepancy drive can be obtained(19)uω=ye+urcosφe−aωrsinφe−ckf′kωω̇ye−cfkωẏe+ε11−e−s11+e−s1−h1s1ωr+∂α/∂uru̇r+∂α/∂yeursinφe+aωrcosφe+ε21−e−s2/1+e−s2−h2s21+∂a/∂yexe+a.Whena=0 is chosen, the trajectory tracking control law of discrepancy drive nonholonomic wheeled mobile robot is as follows(20)u0ω0=yeω+urcosφe−ckf′kωω̇ye−cfkωẏe+ε11−e−s11+e−s1−h1s1ωr+∂α/∂uru̇r+∂α/∂yeursinφe+ε21−e−s2/1+e−s2−h2s21+∂a/∂yexe,∂α∂ur=ye1+urye2,∂α∂ye=ur1+urye2.


### 2.6. Stability Analysis

Because the sliding mode variable structure control requires that the state quantity of the motion on the sliding surface must be the end point, when the moving point reaches the sliding surface, it must meet the following requirements:lims⟶0sṡ≤0. The following Lyapunov function is defined as [28, 29](21)V=12s12+12s22.By deriving Equation (21) and substituting it into Equation (17), we can get the following results:(22)V̇=−h1s12−h2s22−ε1s11−e−s11+e−s1−ε2s21−e−s21+e−s2.BecauseV̇≤0, so V is a monotone nonincreasing function; Vt≤V0, Vt is bounded on t∈0,∞; therefore, from Equations (16) and (21), xe, ye, φe, and ω are bounded values; it is also known from Equation (19) that ω̇ expected linear velocity ur and angular velocity ωr are bounded quantities; from error Equation (13), ẋe, ẏe, and φ̇e are bounded; that is, xe, ye, and φe are consistent continuous variables; according to Barbarat theorem, xe, ye, and φe converge to zero when t⟶∞.Theorem 2.
The Lyapunov function candidate beVe=eTMe, where M is the result of the Lyapunov equivalence. The stability aspects of the nonholonomic WMR are considered for diverse cases represented as follows:Case 1.
Whens≥€.
The time derivative ofV gives,(23)V′e=′eTMe+eTMe′=eTATM+MAe+2eTMBΔua+δ=−eTNe+2sT−kss+δ.
Additional universalization ofV′e produces:(24)V′e≤−eTNe–2k′S+2Sk≤−λminNe2+2k–k′S≤−eλminNe−2k–k′BTM
So,′Ve<0 may be conventional if e>2k–k′BTM/λminN≔ώ2. Consequently, the structure may be ultimate bound through subsequent:(25)e≤λmixMώ2λminM.
The time to rangeώ2 can initiate as tr2–t0≤Vet0–Vώ2/c0.Case 2.
WhenS<€.
The term2STδ may show as follows:(26)2sTδ≤2Sδ≤2kS=2kSTSS.
By Equations (23) and (26), time directive Ve produces(27)Ve≤−eTNe+2sTΔua+kSS≤−λminNe2+2sT−kS€+kSS.
Equation (27) computes more value €k2/2k′ when S=€k/2k′, so(28)Ve≤−λminNe2+€k22k.
Yet over,Ve<0 may conventional if e>€k2/2k′λminQ≔ώ3. Consequently, the structure is ultimate bound with the subsequent:(29)e≤λmaxMώ3λmaxM.
Correspondingly, the time to influenceώ3 can initiate as tr3–t0≤Vet0–Vώ3/c0.
Equations (23), (25), and (29) can easily be understood as ώ3<ώ2<ώ1. In addition, € is chosen very small in practice, and thus, ώ3 can be minimized by correct tuning of € and N additionally.


## 2.1. Nonholonomic Constraints for Wheeled Mobile Robots

As a characteristic nonholonomic control scheme, the motion of WMR is not only limited by the basic requirements of starting point and ending point but also meets the special requirements of velocity direction or trajectory shape. In addition, WMR is an essentially nonlinear under actuated drift-free dynamic system, which brings greater challenges to its control [18, 19]. However, nonintegrity makes the structure of the robot more flexible and reliable, such as allowing some joints of the robot to be driveless, which greatly reduces the manufacturing cost, quality, and energy consumption of the robot. The nonholonomic control system represented by nonholonomic wheeled mobile robot is a new type of subject, which has important practical value and theoretical significance.The motion behavior of the mobile robot is constrained by kinematics, which is determined by the structural characteristics of the robot and its motion model. In practical applications, the system is always subject to various constraints, such as geometric constraints and motion constraints, steady and unsteady constraints, double-sided constraints and single-sided constraints, integrable constraints, and nonintegrable constraints [20–23]. Under different constraints, the motion of the control system shows different characteristics. From the essence of system motion, constraints are divided into complete constraints and nonholonomic constraints. Nonholonomic constraints restrict the driving direction of robots. Holonomic constraints are dimensions relative to the state description of the system. Suppose a constraint form:(1)fq,q̇,t=0,whereq=q1,q2,⋯,qn is the generated space vector.If this form of constraint can be transformed into a form:(2)Fq,t=0,then the constraint form is integrable. Even iff in Equation (1) contains the differential quantity of generating space vector in time, it can be expressed by Equation (2), which is usually called complete constraint.A nonholonomic constraint is defined as a time derivative including variables (coordinates) in the generating system and cannot be integrated. Nonholonomic constraint is a characteristic of the mechanical system. When the dimension of the control variable is lower than the state variable, the control variable can control the state variable [24, 25]. If the system has nonholonomic constraints, the system is nonholonomic, so it cannot move in any path (for example, the car will drive forward or backward, not side-by-side) [26]. The movement of this system is not arbitrary for nonholonomic systems; nonholonomic restrictions specify a subspace in the current time of the potential speed space. The return of internal variables to the initial value in a nonholonomic system does little to ensure that the system returns to the initial state. The condition of a nonholonomic system therefore depends on the route (a sequence set of internal variables that vary according to time). The coordinate dimension for the generated space is not reduced by nonholonomic restrictions excluding the generated speed space.If the constraint of Equation (1) is integrable, its kinematic constraint is complete; that is, the velocity q1,q2,⋯,qn can be eliminated from the equation. If the constraint of Equation (1) is not integrable, then the corresponding kinematic constraint is incomplete.


## 2.2. Kinematic Model Analysis

Mobile robots through nonholonomic components are called nonholonomic mobile robots. It belongs to the category of nonholonomic constraints, and its nonholonomic constraints mainly affect the motion characteristics of nonholonomic systems. The nonholonomic constraints received by wheeled mobile robots are generally expressed as first-order differential equations.In the plane, the position and attitude of the robot can be represented by the following state vectors:(3)xmt=xmtymtθmt.x
m
,
y
m represents the position of robot global coordinate system, and θm represents the steering angle of robot global coordinate system. xm represents the pose state of the robot at time t.To keep the balance in the plane, this type of robot usually has one or more castors as additional support only. The driving control is completed by motors which are separately distributed in two wheels. In the universal organize scheme, the motion equation of the robot can be formulated as:(4)ẋmtẏmtθ̇mt=cosθmt0sinθmt001vtwt.Its nonholonomic motion constraints are as follows:(5)‐ẋmsinθm+ẏmcosθm=0.Equation (5) can be written in terms of the direction of motion that can be realized by the motion matrix(6)S=cosθmt0cosθmt001.v
t and wt are usually regarded as control variables in robot motion research. After applying the Euler integral to Equation (4), it can be written as follows:(7)xmk+1=xmk+vkΔtcosθmt,ymk+1=ymk+vkΔtsinθmt,θmk+1=θmk+wkΔt,wherek is the discrete time constant and Δt is the time interval.There are many ways to make the robot move to the ideal position by using the plane changing trajectory. The inverse kinematic challenge is easier for the robot to plane the target trajectoryxt,yt, so that the direction of the robot is always tangent to the trajectory. If the initial pose of the reference robot is on a certain trajectory, then there is a kinematic model corresponding to the physical quantity of the reference robot. It can be concluded that the control linear velocity is vt:(8)vt=±ẋ2t+ẏ2t.The positive and negative signs represent the ideal driving direction (+ is the forward direction, and - is the reverse direction). Every point on the track has a tangent angle defined as follows:(9)θt=arctan2ẏt,ẋt+kπ,wherek∈0,1 defines the ideal driving direction (0 is the forward direction, and 1 is the reverse direction). By calculating the differential of time in Equation (9), the angular velocity of the robot can be obtained:(10)wt=ẋty¨t−ẏtx¨tẋ2t+ẏ2t.By using the Equations (8) and (9), the ideal trajectory control is obtained.


## 2.3. Sliding Mode Control Principle

When the system does not satisfy the linear parameterization condition or when the system model has both structural and parametric uncertainties, a sliding mode control algorithm can be used. If the upper bound function can be found for the unknown dynamic characteristics, the sliding mode control algorithm can achieve the desired control effect by strengthening the control input, that is, using very strong control variables to deal with the unknown dynamic characteristics. Sliding mode control has strong control performance for nonlinear systems affected by uncertain dynamics, which makes the system move according to the designed sliding mode trajectory. The control method has the advantages of a simple calculation method and strong anti-interference ability.Sliding mode control for nonlinear systems influenced by unknown factors is an easy and efficient way of control. The problem can be reduced to a complex one by simple sliding mode control. The nonlinear method for the single input output can be expressed as follows:(11)yn=fy¯+u.u
∈
R and y∈R are control input variables and system output variables, respectively, and y¯=y,ẏ,⋯,yn−1T∈Rn is system state vector.In order to make the trajectory tracking errore¯=y¯−y¯d=e,ė,⋯en−11 of the controlled system converge to zero, that is, the state variables of the closed-loop system can gradually track the reference trajectory y¯d=yd,ẏd,⋯ydn−1T, a state feedback control law needs to be designed by using sliding mode controller. The PI sliding mode variable structure controller is defined as follows:(12)sy,t=et+γ∫0teτdτ.Among them,γ>0. When sy,t=0 is set, e=0 is the only solution to Equation (12). Then, the problem of tracking the desired trajectory yd is equivalent to the problem of function sy,t=0. At the same time, the selection of control law u should satisfy the following conditions:(13)12ddts2≤−ρs,whereρ>0 is a constant. Equation (13) is called sliding condition, which guarantees that the error state can converge from the initial position far away from sy,t=0 surface to sy,t=0 surface.The strategy of sliding mode controller mainly comprises 2 parts: the equivalent control variableueq is designed for the ideal sliding mode control, and the variable structure control variable usw is also designed for the nonlinear system affected by various uncertain factors. The design of sliding mode variable structure controller consists of two steps:Step 1.
Designs switching function to make sliding mode stable and dynamic.Step 2.
The sliding mode control lawu is designed to form a sliding mode region on the switching surface sy,t=0 under the condition that the condition Equation (13) is satisfied [27].For uncertain systems, using the sliding mode variable structure control method can often get the desired results. Compared with the traditional control algorithm, the sliding mode production variable structure control method is discrete; that means, the high-frequency switch control is used to switch between several control laws, which can make the system oscillate with small amplitude and high frequency near the set space surface. Such motion is defined as sliding mode motion. Since various control laws in the variable control method for sliding mode are set in advance and are not disturbed and do not rely on device parameters, this method is highly robust. Then, the concept of sliding mode control combines a trajectory tracking controller with global asymptotic stability.


## 2.4. Design Shear Function

Lemma 1.
Anyx∈R and x<∞, then φx=xsinarctanx≥0, if and only if x=0, the equal sign holds. The switching function of sliding mode variable structure control is designed based on backstepping method; firstly, the Lyapunov function is defined as Vy=1/2ye2; suppose xe=cfkωye, where fkω=1−exp−2kω/1+exp−2kω is a hyperbolic tangent function and cfkωye is a virtual feedback variable; c and k were normal. According to the characteristics of hyperbolic tangent function, we can know ωfkω≥0. Assuming ϕe=−arctanurye, then(14)ϕ̇=∂ϕe∂ur+∂ϕe∂ye=ur+ye1+urye2<0.Based on Equation (14) and Lemma 1, the following equation can be given(15)V̇y=yeẏe=ye−xeω−aω+ursinφe+aωrcosφe=‐yexeω−yeaω+uryesin−arctanurye+yeaωrcos−arctanurye≤‐yexeω−yeaω+yeaωr=‐ye2cfkωω+yeaφ̇e<0.It is concluded that as long asxe converges to cfkωye and φe converges to −arctanurye, the system state variable ye will converge to 0. According to this conclusion, the switching function is designed as follows:(16)s=s1s2=xe−cfkωyeφe+arctanurye.Through the design of sliding mode controller,s1 and s2 tend to 0, which means xe converges to cfkωye, φe converges to 0, so the system state variable ye also converges to 0.


## 2.5. Design of Sliding Mode Controller

To weaken the jitter, the exponential reaching lawṡ=−εsgns−hs is adopted, in which ε and h are greater than 0, and ε is a small positive number. The following equation can be obtained by using S type continuous function 1−e−s/1+e−s instead of symbol function:(17)ṡi=−εi1−e−si1+e−si−hisi,i=1,2.ε
i is a small positive number, and hi is a normal number.Supposeα=arctanurye, from Equation (16) and Equation (17), we can get the following results:(18)ṡi=ṡ1ṡ2=−ε11−e−s11+e−s1−h1s1−ε21−e−s21+e−s2−h2s2=ẋ−ckf′kωω̇ye−cfkωẏeϕ̇e+∂α∂uru̇r+∂α∂yeẏe=yeω−u+urcosφe−aωrsinφe−ckf′kωω̇ye−cfkωẏeωr−ω+∂α∂uru̇r+∂α∂yeẏe.From Equation (18), the kinematic trajectory tracking control law of the nonholonomic wheeled mobile robot with discrepancy drive can be obtained(19)uω=ye+urcosφe−aωrsinφe−ckf′kωω̇ye−cfkωẏe+ε11−e−s11+e−s1−h1s1ωr+∂α/∂uru̇r+∂α/∂yeursinφe+aωrcosφe+ε21−e−s2/1+e−s2−h2s21+∂a/∂yexe+a.Whena=0 is chosen, the trajectory tracking control law of discrepancy drive nonholonomic wheeled mobile robot is as follows(20)u0ω0=yeω+urcosφe−ckf′kωω̇ye−cfkωẏe+ε11−e−s11+e−s1−h1s1ωr+∂α/∂uru̇r+∂α/∂yeursinφe+ε21−e−s2/1+e−s2−h2s21+∂a/∂yexe,∂α∂ur=ye1+urye2,∂α∂ye=ur1+urye2.


## 2.6. Stability Analysis

Because the sliding mode variable structure control requires that the state quantity of the motion on the sliding surface must be the end point, when the moving point reaches the sliding surface, it must meet the following requirements:lims⟶0sṡ≤0. The following Lyapunov function is defined as [28, 29](21)V=12s12+12s22.By deriving Equation (21) and substituting it into Equation (17), we can get the following results:(22)V̇=−h1s12−h2s22−ε1s11−e−s11+e−s1−ε2s21−e−s21+e−s2.BecauseV̇≤0, so V is a monotone nonincreasing function; Vt≤V0, Vt is bounded on t∈0,∞; therefore, from Equations (16) and (21), xe, ye, φe, and ω are bounded values; it is also known from Equation (19) that ω̇ expected linear velocity ur and angular velocity ωr are bounded quantities; from error Equation (13), ẋe, ẏe, and φ̇e are bounded; that is, xe, ye, and φe are consistent continuous variables; according to Barbarat theorem, xe, ye, and φe converge to zero when t⟶∞.Theorem 2.
The Lyapunov function candidate beVe=eTMe, where M is the result of the Lyapunov equivalence. The stability aspects of the nonholonomic WMR are considered for diverse cases represented as follows:Case 1.
Whens≥€.
The time derivative ofV gives,(23)V′e=′eTMe+eTMe′=eTATM+MAe+2eTMBΔua+δ=−eTNe+2sT−kss+δ.
Additional universalization ofV′e produces:(24)V′e≤−eTNe–2k′S+2Sk≤−λminNe2+2k–k′S≤−eλminNe−2k–k′BTM
So,′Ve<0 may be conventional if e>2k–k′BTM/λminN≔ώ2. Consequently, the structure may be ultimate bound through subsequent:(25)e≤λmixMώ2λminM.
The time to rangeώ2 can initiate as tr2–t0≤Vet0–Vώ2/c0.Case 2.
WhenS<€.
The term2STδ may show as follows:(26)2sTδ≤2Sδ≤2kS=2kSTSS.
By Equations (23) and (26), time directive Ve produces(27)Ve≤−eTNe+2sTΔua+kSS≤−λminNe2+2sT−kS€+kSS.
Equation (27) computes more value €k2/2k′ when S=€k/2k′, so(28)Ve≤−λminNe2+€k22k.
Yet over,Ve<0 may conventional if e>€k2/2k′λminQ≔ώ3. Consequently, the structure is ultimate bound with the subsequent:(29)e≤λmaxMώ3λmaxM.
Correspondingly, the time to influenceώ3 can initiate as tr3–t0≤Vet0–Vώ3/c0.
Equations (23), (25), and (29) can easily be understood as ώ3<ώ2<ώ1. In addition, € is chosen very small in practice, and thus, ώ3 can be minimized by correct tuning of € and N additionally.


## 3. Simulation Experiment Verification

Considering that line and circle are common expected trajectory forms in practical application, this paper selects linear and circular trajectories to simulate and verify the above tracking control law and carries out the corresponding trajectory tracking controller simulation for nonholonomic WMR. The excessive acceleration caused by excessive speed will destroy the nonholonomic constraints of the robot, which is easy to cause the lateral movement or longitudinal sliding of the mobile robot. To avoid the above problems, the speed of the mobile robot must be limited. The maximum speed is 1.5 m/s, and the extreme angular velocity is 1.5 rad/s. The experiments of WMR are considered as follows: the hardware components for the robotic stages are nominated asm=34 kg, d=0.14 m, r=0.085 m, b=0.504 m, I=4.89 kg m2, Iw=0.08 kg m2, k=0.36 kg m2, I=6.13 kg m2, Iw=0.08 kg m2, and K=0.33 kg m. Due to the communiqué interruption among the robot and control station and the related sensor suite and hardware, the sampling time h is considered judiciously 100 ms. In experimentation with wheel location and velocity, the wheel encoders (incremental encoder of square design with 500 ppr) are used. Moreover, when the wheel velocities θr and θl are achieved, the following kinematic relationships integrate the remaining state variables.


### 3.1. Straight Track

When the trajectory is a straight line, the control input of the mobile robot is selected as follows: the primary linear velocity is 0.4 m/s, the angular velocity is 0.3 rad/s, and the controller parameters are set to 1.5. The simulation results are shown in Figures2–5.Figure 2
Track tracking effect.Figure 3
Change curve of pose error.Figure 4
Linear velocity variation curve.Figure 5
Variation curve of angular velocity.It can be seen from Figure2 that under the action of the control law, the nonholonomic WMR system can realize the straight-line trajectory tracking at the abscissa of about 5 m and finally converge to the given linear reference trajectory. The pose error can eventually converge to 0 and can be seen in Figure 3. The nonholonomic wheeled mobile robot device will, as shown in Figures 4 and 5, converge the anticipated linear velocity and the expected angular velocity under the control rule, which checks the effectiveness of the trajectory tracking process.


### 3.2. Circular Trajectory

When the trajectory is circular, the control input of the mobile robot is selected as follows: the primary linear velocity is 1 m/s, the angular velocity is 1 rad/s, and the controller parameters are set to 1. Figures6–9 show the results of the simulation.Figure 6
Track tracking effect.Figure 7
Change curve of pose error.Figure 8
Linear velocity variation curve.Figure 9
Variation curve of angular velocity.From Figures6–9 that under the action of the designed control law, the circular motion trajectory always has a good fit with the expected trajectory, and the pose error can eventually converge to 0, and the linear velocity and angular velocity can also converge to the expected situation after different times. Therefore, the efficiency of the control law in tracking the circular trajectory of the nonholonomic WMR is verified.In summary, it can be seen from the results of the above simulation of two tracking types on a nonholonomic mobile robot that the nonholonomic mobile robot system can progressively pursue the reference trajectory under the effect of a designed trajectory tracking law, the position error can converge to zero quickly, and the input of the speed control can converge also to the referee quickly. The efficacy of the proposed law on control is verified.


## 3.1. Straight Track

When the trajectory is a straight line, the control input of the mobile robot is selected as follows: the primary linear velocity is 0.4 m/s, the angular velocity is 0.3 rad/s, and the controller parameters are set to 1.5. The simulation results are shown in Figures2–5.Figure 2
Track tracking effect.Figure 3
Change curve of pose error.Figure 4
Linear velocity variation curve.Figure 5
Variation curve of angular velocity.It can be seen from Figure2 that under the action of the control law, the nonholonomic WMR system can realize the straight-line trajectory tracking at the abscissa of about 5 m and finally converge to the given linear reference trajectory. The pose error can eventually converge to 0 and can be seen in Figure 3. The nonholonomic wheeled mobile robot device will, as shown in Figures 4 and 5, converge the anticipated linear velocity and the expected angular velocity under the control rule, which checks the effectiveness of the trajectory tracking process.


## 3.2. Circular Trajectory

When the trajectory is circular, the control input of the mobile robot is selected as follows: the primary linear velocity is 1 m/s, the angular velocity is 1 rad/s, and the controller parameters are set to 1. Figures6–9 show the results of the simulation.Figure 6
Track tracking effect.Figure 7
Change curve of pose error.Figure 8
Linear velocity variation curve.Figure 9
Variation curve of angular velocity.From Figures6–9 that under the action of the designed control law, the circular motion trajectory always has a good fit with the expected trajectory, and the pose error can eventually converge to 0, and the linear velocity and angular velocity can also converge to the expected situation after different times. Therefore, the efficiency of the control law in tracking the circular trajectory of the nonholonomic WMR is verified.In summary, it can be seen from the results of the above simulation of two tracking types on a nonholonomic mobile robot that the nonholonomic mobile robot system can progressively pursue the reference trajectory under the effect of a designed trajectory tracking law, the position error can converge to zero quickly, and the input of the speed control can converge also to the referee quickly. The efficacy of the proposed law on control is verified.


## 4. Conclusion

In this paper, the nonholonomic WMR trajectory control system based on improved variable structure of sliding mode is projected. Based on analyzing the principle of sliding mode control, the improved method is designed by adding stability analysis to reduce the pose error of trajectory tracking control. The experimental results verify the high control precision effect of the designed method and provide a basic theoretical reference for practical application.



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*Source: 2974839-2021-06-18.xml*