# Adaptive Fuzzy Path Tracking Control for Mobile Robots with Unknown Control Direction

Authors: Qifei Du; Lin Sha; Wuxi Shi; Liankun Sun
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In order to synthesize controllers for wheeled mobile robots (WMRs), some design techniques are usually based on the assumption that the center of mass is at the center of the robot itself. Nevertheless, the exact position of the center of mass is not easy to measure, thus WMRs is a typical uncertain nonlinear system with unknown control direction. Based on the fast terminal sliding mode control, an adaptive fuzzy path tracking control scheme is proposed for mobile robots with unknown control direction. In this scheme, the fuzzy system is used to approximate unknown functions, and a robust controller is constructed to compensate for the approximation error. The Nussbaum-type functions are integrated into the robust controller to estimate the unknown control direction. It is proved that all the signals in the closed-loop system are bounded, and the tracking error converges to a small neighborhood of the origin in a limited time. The effectiveness of the proposed scheme is illustrated by a simulation example.


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## Body

## 1. Introduction

The mobile robot system is a typical nonholonomic system, and the research of tracking control has always remained one of the most challenging tasks in the area of mobile robot system.According to whether the tracking trajectory is a function of time, tracking control is divided into trajectory tracking and path tracking. For the trajectory tracking, based on the kinematics model, the backstepping method, neural network method, fuzzy neural network method, and input-output linearization method were developed in [1–4]. For the path tracking, when the center of mass of the robot is exactly at the geometric center of the wheel axis, the path tracking problem was studied in [5], while the center of mass of the robot is located on the central axis of the two driving wheels, the path tracking problem was developed in [6]. The assumption in [5, 6] that the center of mass is located on the geometric center of the wheel axis or the central axis of the two driving wheels is a good idea for an actual operating robot system; however, the exact position of the center of mass is not easy to measure when it is actually running; thus, the mobile robot system is a typical uncertain nonlinear system. By using the universal approximation of the fuzzy system [7] and the traditional linear sliding mode control techniques, the tracking control scheme were presented for the mobile robot system with uncertainty [8–12]. However, the traditional linear sliding mode control can only achieve infinite time asymptotic convergence and cannot achieve finite time tracking. Moreover, in order to improve convergence speed, it is necessary to increase the design parameters in the sliding mode control, which can increase the gain of the controller and result in the saturation of the control input. Terminal sliding mode (TSM) [13], as an effective finite-time convergence method, has received wide attention by many scholars [14–25]. In recent years, the fast terminal sliding mode (FTSM) [18] and nonsingular Terminal sliding mode (NTSM) [19] had been applied to robot control. For example, in [19, 21, 22], the tracking control was proposed for manipulator based on NTSM, and a simple terminal sliding mode control was carried out and was presented for mobile robot in [23–25]. However, the proposed methods in [23–25] were a known system whose center of mass is located at the midpoint of the drive axle. When the adaptive fuzzy system is used to control the robot, the approximation error of the fuzzy system is inevitable. In order to suppress the approximation error, it is usually necessary to introduce a robust controller [26–28]. However, in [26–28], it is necessary to assume that the control direction of the system is known. For a mobile robot system with an uncertain center of mass, the control direction is difficult to obtain. Since Nussbaum first proposed the Nussbaum-type function in 1983 and successfully solved the control of the first-order linear system with unknown control direction [29], the Nussbaum-type function has been widely used in the design of control systems [30–33]. For example, the adaptive fuzzy control was proposed for the multiinput multioutput nonlinear system with unknown control direction [30, 31] and for the strict feedback nonlinear system with unknown control direction [32, 33]. In this paper, the Nussbaum-type function technique is applied to the control of the mobile robot system with uncertain center of mass to solve the control problem of unknown control direction of the mobile robot caused by the uncertain center of mass.In this paper, the path tracking control was proposed for a mobile robot with uncertain center of mass and unknown control direction. The fuzzy system is used to approximate the unknown function in the robot system, and an indirect adaptive fuzzy controller is designed by using FTSM, designing a robust controller to compensate the fuzzy approximation error, and integrating the Nussbaum-type function into the robust controller to estimate the unknown control direction. Based on the Lyapunov stability analysis, an adaptive law is designed for unknown parameters, and it is proved that this method can not only ensure that all signals in the closed-loop system are bounded but also make the tracking error converge to a small neighborhood of the origin within a finite time. The simulation results verify the effectiveness of the method in this paper.


## 2. Problem Description

The structure of the wheeled robot is shown in Figure1. The robot has three wheels, and the front left and right wheels are driving wheels. To provide power for the vehicle body, the rear wheel is a follow-up universal wheel, which can not only move with the movement direction of the front wheel but also ensure the balance of the robot.Figure 1
Wheeled mobile robot model.Because wheeled robot is affected by loads, its center of mass is usually not at the center of the robot itself. In Figure1, it is assumed that the center of mass of the robot is at point C, and the midpoint of the connecting axis of the first two driving wheels of the robot is point P. The distance from point P to point C is set as L, and the angle between the robot’s forward direction axis and the line PC is set as ϕ. The linear velocity of the progress of robot is u1, and the angular velocity of the robot rotation is u2. Let the coordinate of point C as x1,x2,x3, where x1 is the abscissa and x2 is the ordinate, and let the horizontal abscissa be X1-axis, and the angle between the axis of motion direction of the vehicle body and the X1-axis is x3, and the P coordinate is x1′,x2′,x3′; then, the kinematics equation of point P is(1)x˙1′=u1cosx3,x˙2′=u1sinx3,x˙3′=u2.The location relation between pointP and Cx1,x2,x3 can be written as(2)x1=x1′−Lcosx3−ϕ,x2=x2′−Lsinx3−ϕ,x3=x3′.By using (1), (2) yields(3)x˙1=u1cosx3+Lu2sinx3−ϕ,x˙2=u1sinx3−Lu2cosx3−ϕ,x˙3=u2,where x˙1, x˙2, and x˙3 are the velocity components moving in all directions at point C, so (3) is the kinematics model of the robot with the center of mass Cx1,x2,x3 as the reference point.In order to achieve the ideal control, it is assumed that path function isfx1,x2=0, and the tracking error function is z=fx1,x2; when the parameters L and ϕ are unknown, under the effect of the amount of control ux1,x2,x3, the system is made to move along the set path, namely, for a given value φ, there is the time t1, and when t>t1, the tracking error z=fx1t,x2t<φ.In this paper, it is assumed that the robot runs at linear speedu1 and the input angular velocity u2 is controlled. The derivative of the tracking error is(4)z˙=fx1⋅x˙1+fx2⋅x˙2.Substitute (3) into (4), and we have(5)z˙=fx1cosx3+fx2sinx3u1+fx1sinx3−ϕ−fx2cosx3−ϕLu2.Letx=x1,x2,x3T and hx=fx1⋅cosx3+fx2⋅sinx3u1:(6)gx=fx1⋅sinx3−ϕ−fx2⋅cosx3−ϕL.Letu2=u; then, (5) can be rewritten as(7)z˙=hx+gxu.Assumption 1.
There is a constantg0 such that 0<gx<g0.
The control law is designed as(8)u=−hx−kzgx,where k>0. Substituting (8) into (7), we obtain z˙=−kz.
Obviously, the tracking errorz is going to converge to zero. Since the exact position of the center of mass C is unknown, that is, the distance L and angle ϕ are unknown, the control gain gx is actually an unknown nonlinear function, and it difficult to design and implement the control law (8). Considering the universal approximation of the fuzzy system, the fuzzy system is used to approximate the unknown function gx, and the Nussbaum-type function technique is used to estimate the unknown control direction, and the terminal sliding mode control is used to make the convergence of tracking error in a limited time.Definition 1.
(see [29]). A function Nζ with the following form is called Nussbaum-type function:lims⟶+∞sup1/s∫0sNζdζ=+∞ and lims⟶+∞inf1/s∫0sNζdζ=−∞. The common Nussbaum-type functions are ζ2cosζ, ζ2sinζ, and expζ2cosπ/2ζ.Lemma 1.
(see [34]). Vt and ζt are smooth functions defined in the interval 0,tf and Vt≥0, ∀t∈0,tf, and Nt=expζ2cosπ/2ζ is a smooth even Nussbaum-type function. If the following inequality holds, Vt≤c0+∫0tgτNζτ+1ζ˙τdτ, ∀t∈0,tf, where gt is a time-varying parameter, and gt∈I=l−,l+, 0∉I, c0 is a reasonable constant; then, Vt, ζt, and ∫0tgτNζτ+1ζ˙τdτ must be bounded in 0,tf.Lemma 2.
(see [20]). If the continuous function Vt>0 satisfies the inequality, V˙t+αVt+βVq/pt≤0, ∀t>t0, then Vt will converge to the equilibrium point in finite time ts, where ts≤t0+p/αp−qlnαV1−q/pt0+β/β, where α>0, β>0, p and q are odd, and q<p.


## 3. Design of Tracking Controller

In order to realize the tracking error convergence in a finite time, the sliding mode surface is designed as(9)s=z˙+α⋅z+β⋅zq/p=0,where α>0, β>0, p and q are odd numbers, and q<p<2q. For the sake of convenience, γ=q/p are assigned in this paper. The designed control law is designed as(10)u=1gx−hx−α⋅z−β⋅zγ.Substituting (10) into (7), we obtain z˙+α⋅z+β⋅zγ=0.According to Lemma1, z will converge in a limited time.We know that the nonlinear functiongx is unknown in (10), so controller (10) cannot be implemented. In this paper, the fuzzy logic system is adopted to approximate the nonlinear function gx. The form of the fuzzy rule base is as follows: Rl:ifx1isF1land,……,andxnisFnl,thenyisGl,l=1,2,3,…,M, where Fili=1,2,…,n and Gl are fuzzy sets, respectively, belonging to functions μFilxi and μGly, both of which are Gaussian, where M is fuzzy rule number. x=x1,x2,…,xnT∈Rn is the input vector of the fuzzy system, and y∈R is the output variable. Using single-valued fuzzy generator, product inference rule, and central average fuzzy eliminator, the output form of the fuzzy system can be expressed as follows:(11)g^x|θ=∑l=1My˜l∏i=1nμFilxi∑l=1M∏i=1nμFilxi=θTψx,where θ=θ1,θ2,…,θMT is the adaptive variable vector and θl=y˜l is the point corresponding to the maximum value of μGlψx=ψ1x,…,ψMxT is the fuzzy basis function vector, where ψlx=∏i=1nμFilxi/∑l=1M∏i=1nμFilxi, the control law (10) may have singularity problems, so the fuzzy system (11) g^x|θ=θTψx is adopted to approximate the unknown function gx, and the equivalent control law is designed as follows:(12)ueq=1g^x|θ+ε⋅signg^x|θ−hx−α⋅z−β⋅zγ,where ε>0, signg^x|θ=1g^x|θ≥0−1g^x|θ<0.To compensate the approximation error of the fuzzy system, the control law is designed as follows:(13)u=ueq+ur,where robust control ur will be designed later. Substituting (12) into (6), we obtain(14)z˙=−αz−βzγ−g^x|θueq+gxueq+gxur−ε⋅signg^x|θueq.The optimal parameter of the adaptive vector is defined asθ∗=argminθ∈Ωgsupx∈Dxg^x|θ−gx, where Dx is the definition domain of input variables of the fuzzy system and Ωg is the allowable set of adaptive parameter θ.Define the minimum approximation error asω=g^x|θ∗−gx, and define Φ=θ^−θ∗, then (14) can be written as(15)z˙=−αz−βzγ−ΦTψxueq−ωueq−ε⋅signg^x|θueq+gxur.Based on universal approximation theorem,ω is bounded but not easy to measure. In this paper, we assume that there is a constant ρ∗>0 so that ω≤ρ∗, due to the unknown ρ∗, define δ^g is the estimated value of ρ∗, and let δ¯g=δ^g−ρ∗.In (15), the sign of robust control gain gx is unknown, which makes it difficult to design a robust controller ur. However, Nussbaum-type function technique is a feasible method to solve such unknown problems; thus, Nussbaum-type function Nζ=expζ2cosπ/2ζ is introduced into the design of a robust controller ur. The robust controller ur is designed as follows:(16)ur=Nζδ^g+ε2ueq2zδ^g+εueqz+σ2,ζ˙=δ^g+ε2ueq2z2δ^g+εueqz+σ2,where σ is the time-varying parameter. The following parameter adaptive law is designed as(17)θ^˙=ηψxueqz,(18)δ^˙g=μueqz,(19)σ˙=−λσ,where η>0, μ>0, λ>0, and σ0≠0.Theorem 1.
The adaptive fuzzy controller (13) and the adaptive law of unknown parameters (17)–(19) are adopted to the robot system (7); then,(1)
All signals in a closed-loop system are bounded(2)
The tracking errorz will converge to a small neighborhood zz|≤w/α−α∗ of the origin in finite time ts, where ts≤2/α01−γ0lnα0V′1−γ0t0+β0/β0Proof.
(1)
Consider the following Lyapunov function candidate:(20)V=12z2+12ηΦTΦ+12μδ¯g2+12λσ2.The derivative of (20) is(21)V˙=zz˙+1ηΦTθ^˙+1μδ¯gδ^˙g+1λσσ˙.From (15), we have(22)zz˙=−αz2−βzγ+1−ΦTψxueqz−ωueqz−ε⋅signg^x|θueqz+gxurz.Substituting (22) into (21), we obtain(23)V˙=−αz2−βzγ+1−ΦTψxueqz−ωueqz+gxurz−ε⋅signg^x|θueqz+1ηΦTθ^˙+1μδ¯gδ^˙g+1λσσ˙.Substituting (17) into (23) and because ω≤ρ∗, we obtain(24)V˙≤−αz2−βzγ+1+ρ∗+εueqz+gxurz+1μδ¯gδ^˙g+1λσσ˙.Sinceδ¯g=δ^g−ρ∗, we obtain(25)V˙≤−αz2−βzγ+1+δ^g+εueqz−δ¯gueqz+gxurz+1μδ¯gδ^˙g+1λσσ˙.Substituting (18) into (25), we obtain(26)V˙≤−αz2−βzγ+1+δ^g+εueqz+gxurz+1λσσ˙.Substituting the robust control law (16) into (26) yields(27)V˙≤−αz2−βzγ+1+δ^g+εueqz+1λσσ˙+gxNζζ˙.By using (16), (27) can be rewritten as(28)V˙≤−αz2−βzγ+1+gxtNζ+1ζ˙+1λσσ˙+δ^g+εueqz−δ^g+ε2ueq2z2δ^g+εueqz+σ2.Since(29)δ^g+εueqz−δ^g+ε2ueq2z2δ^g+εueqz+σ2≤σ2,using (28) and (29), we can obtain(30)V˙≤−αz2−βzγ+1+gxtNζ+1ζ˙+σ2+1λσσ˙.Substituting (19) into (29), we have(31)V˙≤−αz2−βzγ+1+gxtNζ+1ζ˙.Integrating (31), we obtain(32)Vt−V0≤−∫0tαz2−∫0tβzγ+1+∫0tgxτNζ+1ζ˙dτ.The following resulted from (32):(33)Vt≤V0−∫0tαz2−∫0tαz2−∫0tβzγ+1/22+∫0tgτNζ+1ζ˙dτ≤V0+∫0tgτNζ+1ζ˙dτ.According to Assumption1, gt∈−g0,g0 and 0∉−g0,g0; therefore, by using Lemma 1, Vt, ∫0tgτNζ+1ζ˙dτ and ζt are bounded in the interval 0,tf; these conclusions are also feasible at tf=+∞. Therefore, z, θ^, δ^g, and σ are also bounded. From (12) and (16), we can know that ueq and ur are also bounded. Therefore, according to (13), the control law u is bounded.(2)
Sinceω≤ρ∗, we get the following equation from (22):(34)zz˙≤−αz2−βzγ+1+ΦTψxueqz+ρ∗+εueqz+gxurz.(34) can be rewritten as(35)zz˙≤−z2α−ΦTψxueq+ρ∗+εueq+gxurz−1−βzγ+1.Letw=supt>0ΦTψx+ρ∗+εueq+gxur; then, (35) can be rewritten as(36)zz˙≤−12z2⋅2α−wz−1−2γ+1/2β12z2γ+1/2.IfV′=1/2z2, from (36), we obtain(37)V′≤−2α¯tV′−2γ+1/2βV′γ+1/2,whereα¯t=α−wz−1.For any small constantα∗>0, an appropriate choice of α is made to α¯t=α−wz−1≥α∗>0; thus, it can be obtained from equation (37) that(38)V′≤−2α∗V′−2γ+1/2βV′γ+1/2.Equation (38) can be written as(39)V′≤−α0V′−β0V′γ0,whereα0=2α∗, β0=2γ+1/2β, and γ0=γ+1/2. According to Lemma 2 and α¯t=α−wz−1≥α∗>0, a reasonable choice of α is made. Then, the tracking error z will converge to a small neighborhood of the origin zz|≤w/α−α∗ in a finite time ts, where ts≤2/α01−γ0lnα0V′1−γ0t0+β0/β0.


## 4. Simulation Experiment

The following simulation experiments will be used to verify the effectiveness of the control method designed in this paper. Let the desired path of the robot befx1,x2=x12+x22−1=0, that is, a circle with radius 1, and set the tracking error as z=fx1,x2=x12+x22−1; then, z˙=2u1x1cosx3+x2sinx3+2Lu2x1sinx3−ϕ−x2cosx3−ϕ. According to the above description, we have hx=2u2x1cosx3+x2sinx3 and gx=2Lu2x1sinx3−ϕ−x2cox3−ϕ.The position of the center of massC cannot be determined from the previous description, so the control gain gx is also unknown. By using the fuzzy system g^x|θg=θgTψx to approximate gx, the membership functions inputted are as follows: μFi1xi=exp−xi−1.5/22, μFi2xi=exp−xi/22, and μFi3xi=exp−xi+1.5/22, all of which are Gaussian, and i=1,2,3, so the fuzzy system designed in this paper has 27 rules.In the kinematics model (3), it is assumed that L=0.3, ϕ=π/6, the initial attitude coordinate of the robot is 0.4,0.2,−π/8, and its linear velocity u1=1. Set the parameter in the controller ε=0.1, and each component of the initial value in the parameter θ^ is randomly selected within the interval −1,1, ζ0=1.2, δ^g0=0.1, and σ0=0.1 The parameters in the adaptive law are as follows: η=3.8, μ=0.6, and λ=0.5.Under the action of FTSM adaptive fuzzy controller, the robot’s tracking effect on the desired pathfx1,x2=x12+x22−1=0, namely, the circular trajectory, which is shown in Figure 2.Figure 2
Expected path and tracking path.Figure3 shows the error curve of robot path tracking. Figure 4 is the curve of the controller signal u.Figure 3
Error curve.Figure 4
Controller signal.From the above simulation results, it can be seen that all the signals in the closed-loop system are bounded, the error convergence of robot path tracking can be achieved, and the tracking errorz converges to a small neighborhood of the origin z≤1.98×10−3 after time ts=4.58s.


## 5. Conclusion

In this paper, an adaptive fuzzy path tracking control method has been developed for the mobile robot system with unknown mass center position. In this method, the Nussbaum-type function is used to estimate the unknown control direction, and the terminal sliding mode is used to ensure the convergence of trajectory tracking in a limited time. It is proved that the scheme can not only guarantee that all signals in the closed-loop system are bounded but also make the path tracking error of the mobile robot converge to a small neighborhood near the origin in finite time. The effectiveness of the proposed scheme is illustrated by a simulation. The research of adaptive fuzzy path tracking control method is still in the stage of theoretical research. The next step will be how to introduce engineering practice to solve practical engineering problems.



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