# Trajectory Tracking of Autonomous Ground Vehicles with Actuator Dead Zones

Authors: Pengfei Zhang; Qiyuan Chen; Tingting Yang
Journal: International Journal of Computer Games Technology
 (2021)

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## Abstract

This paper investigates the trajectory tracking problem of autonomous ground vehicles (AGVs). The dynamics considered feature external disturbances, model uncertainties, and actuator dead zones. First, a novel time-varying yaw guidance law is proposed based on the line of sight method. By a state transformation, the AGV is proved to realize trajectory tracking control under the premise of eliminating guidance deviation. Second, a fixed time dead zone compensation control method is introduced to ensure the yaw angle tracking of the presented guidance. Furthermore, an improved fixed-time disturbance observer is proposed to compensate for the influence of the actuator dead zone on disturbance observation. Finally, the trajectory tracking control strategy is designed, and simulation comparison shows the effectiveness of the compensate method. The CarSim–MATLAB cosimulation shows that the proposed control strategy effectively makes the AGV follow the reference trajectory.


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## Body

## 1. Introduction

Trajectory tracking control of autonomous ground vehicles (AGVs) plays an essential role in domestic and industrial use for increasing safety, accuracy, and efficiency demands, such as self-guiding uncrewed driving vehicles, platform supplying, and minesweeping [1]. Critical to the trajectory tracking problem of an AGV is its capability for accurate and reliable control to follow the desired trajectory by the planner [2]. The high dynamic nonlinearity, uncertainties and disturbances, and the mechanical limitations of the vehicle itself make the AGV trajectory tracking control challenging.Various research results exist in the literature. Some existing works investigate the trajectory tracking problem of AGVs utilizing the kinematic model [3, 4]. However, the driving condition may be limited without reckoning the dynamics model. For this reason, the trajectory tracking control problem of a class of autonomous vehicles is investigated with parametric uncertainties, external disturbances, and overactuated features by introducing a novel adaptive hierarchical control framework with the linear matrix inequality technique [5]. Then, an augmented state variable and a nonlinear observer are trained to design a torque overlay-base robust steering wheel angle control with the backstepping method [6]. A robust H∞ dynamic output-feedback controller is designed to control the vehicle motion without using the lateral velocity information [7]. Besides, an accurate and efficient clothoid approximation approach is presented using Bezier curves [8]. A model predictive control law incorporating neural-dynamic optimization is introduced [9].Compared with the above approaches, the finite-time control method has faster convergence speed, better disturbance rejection property, and better robustness against uncertainties [10]. Further, a fixed-time control method can guarantee global exponential convergence within a designated time independent of the initial conditions [11]. Thus, a fixed-time control scheme of AGVs might be of great significance. Apart from that, the actuator dead-zones with uncertainty issues have seldom been investigated in the trajectory tracking control design. Due to the mechanical limitations of the vehicle itself, dead zones in yaw actuators should be considered. Dead zones are commonly observed in servo valves or DC servo motors of AGVs but are not discussed in the above papers [12, 13]. When the control input falls in the dead zone, the actuator will have no output signal, resulting in steady-state errors and degraded performance in the system. How to address the actuator dead-zone issue in the fixed-time trajectory tracking control problem of AGVs still requires extensive research.The trajectory tracking control law should achieve disturbance rejection to enhance accuracy and reliability. The LQR technique investigates a hierarchical adaptive trajectory tracking control for autonomous vehicles [14]. A compensation method combining NTSM and ARDC is presented [4]. A kinematic MPC handling the disruptions on road curvature and a PID feedback control of yaw rate rejecting uncertainties and modeling errors are designed to follow the reference trajectory [15]. The intelligent control method of interactive control paradigm-based robust lateral stability controller design for autonomous automobile trajectory tracking with uncertain disturbance is proposed [16]. Above all, the EKF, LQR, and MPC are usually simplified with limited available model parameters. However, time-varying and nonlinear terms always exist, which cannot be regarded as white Gaussian noise. The intelligent control method generally depends on experience and enough off-line training. Although the ARDC method may effectively solve the uncertain systems and be easy to implement, observer error convergence’s settling time is inaccurate. The DOBC method is a good option for rejecting disturbances in such a case. Mainly, a fixed-time observer can estimate the perturbance within a given settling time [17]. However, since the existence of dead zones, it is difficult to obtain the complete control signal. Thus, existing fixed/finite-time observers cannot directly be applied in the tracking control problem.Motivated by the above observations, this paper is aimed at giving a fixed-time observer-based control method, rendering the AGVs with yaw actuator dead zones to follow the reference trajectory. To this end, time-varying yaw guidance and corresponding state transformation are introduced to convert trajectory tracking control into a simple yaw angle tracking problem. A fixed-time observer-based control scheme solving the actuator dead-zones is constructed for the transformed system, whose dwell times are independent of the initial conditions. The main contributions of this paper are threefold:(1)
A novel time-varying yaw guidance law is proposed based on the line of sight method, which can overcome the undesired terms involved in the error dynamics(2)
Uncertainties, external disturbances, and dead zones are simultaneously considered in the tracking control problem, making the designed controller robust to more extensive and challenging. Note that there is seldom any previous work addressing the control issues in one controller(3)
Based on fixed-time control theory, a feedback compensation control scheme based on the fixed-time observer is proposed for the tracking problem of AGVs. Compared with the method suggested in the reference [17], the dwell time can be chosen independently of initial conditionsThis paper is organized as follows. Section System Modeling and the Objective explains the modeling of autonomous ground vehicles and the objective. Section Main Results contains the time-varying yaw guidance law, state transformations, trajectory tracking control framework, and convergence analysis. Numerical simulations and discussions are presented in Section Simulation.


## 2. System Modeling and the Objective

### 2.1. System Modeling

The configuration of the AGV (i.e., the position and orientation) investigated in this paper can be described by two independent coordinates or two degrees of freedom, i.e., surge and yaw. For such AGVs, an inertial frame is used, as depicted in Figure1. The states x, y, and φ are the longitudinal, lateral positions, and the vehicle’s yaw angle in inertial coordinate, respectively. According to the existing literature [4], the kinematic and kinetic model of the AGV can be described as follows:
(1)ẋ=vxcosφ−vysinφ,ẏ=vxsinφ+vycosφ,φ̇=wr,mv̇y+vxwr=Fyf+Fyr,Izẇr=LfFyf−LrFyr.Figure 1
AGV model.The statesvx, vy, and wr denote the vehicle’s longitudinal, lateral, and yaw rate. The parameter m is the mass of the vehicle; Fyf and Fyr indicate the equivalent lateral forces of the front and rear axles, respectively. Parameters Lf and Lr are the distances from the C.G. to the front and rear axle, and Iz is the yaw moment of inertia. On the basis that the steering angle is sufficiently slight, the front and rear wheel lateral forces can be linearized below [18]:
(2)Fyf=−Cfaf=−Cfβ+wrLfvx−δf,Fyr=−Crar=−Crβ−wrLrvx,whereCf and Cr denote the equivalent lateral stiffness of the front and rear axle, δf is the steering angle of the front wheel, af and ar represent the sideslip angle of the front and rear wheels, and β is the sideslip angle.In addition, due to the mechanical limitations of the vehicle itself, dead zones are commonly observed in servo valves or DC servo motors of AGVs. Hence, by combining (1)-(2) with the actuator dead-zone characteristic, the nonlinear model of the AGV considered in this paper is described as follows:
(3)ẋ=vxcosφ−vysinφ,ẏ=vxsinφ+vycosφ,φ̇=wr,v̇y=Δvy,wr,t,ẇr=fvx,wr,t+LfCfIz−1τδf,whereΔvy,wr,t=−vxwr+m−1Cfτδf−β−vx−1wxLf+Crvx−1wrLr−β and fvx,wr,t=LrCrIz−1β−vx−1wrLr−LfCfIz−1vx−1wrLf+β are unknown unmodeled dynamics and disturbances and the input τδf is the force with the dead-zone nonlinearities, which can be described by
(4)τδf=artδf−brt,δf≥brt0,−blt<δf<brtaltδf+blt,δf≤−blt,where art, alt, brt, and blt are positive constants satisfying the condition that art≥a¯r, alt≥a¯l, brt≤b¯r, and blt≤b‾l. In addition, a¯r, a¯l, b‾r, and b‾l are known positive constants.


### 2.2. The Objective

Define the reference trajectory as follows:(5)ẋref=vx,refcosφref−vy,refcosφref,ẏref=vx,refsinφref+vy,refsinφref,φ̇ref=wref,where xref and yref are the desired longitudinal and lateral positions and vx,ref and vy,ref are the desired longitudinal and lateral velocities, respectively. φref and wref are the desired yaw angle and yaw angle rate of the vehicle, respectively. By analyzing system (3), the trajectory tracking problem can be transformed into the yaw angle tracking control problem by constructing the desired yaw angle φref [4], which eventually satisfies φ⟶φref.Define the error statesxe=x−xref, ye=y−yref, and φe=φ−φref. Subsequently, the derivative of the position errors can be calculated as follows:
(6)ẋe=vxcosφ−vx,refcosφref−vysinφ+vy,refsinφref,ẏe=vxsinφ−vx,refsinφref+vycosφ−vy,refcosφref,φ̇e=wr−wref.This paper is aimed at establishing a control methodology that can stabilize the trajectory tracking error system in (6) while ensuring that all closed-loop signals are bounded.Remark 1.
The trajectory tracking problem is transformed into the yaw angle tracking control problem, and it is efficient to reduce the complexity of controller design by dimension reduction. Due to the dead-zone nonlinearities of the inputτδf, the existing fixed-time observers for disturbance estimation cannot be applied directly, which increases the challenge of the tracking problem.


## 2.1. System Modeling

The configuration of the AGV (i.e., the position and orientation) investigated in this paper can be described by two independent coordinates or two degrees of freedom, i.e., surge and yaw. For such AGVs, an inertial frame is used, as depicted in Figure1. The states x, y, and φ are the longitudinal, lateral positions, and the vehicle’s yaw angle in inertial coordinate, respectively. According to the existing literature [4], the kinematic and kinetic model of the AGV can be described as follows:
(1)ẋ=vxcosφ−vysinφ,ẏ=vxsinφ+vycosφ,φ̇=wr,mv̇y+vxwr=Fyf+Fyr,Izẇr=LfFyf−LrFyr.Figure 1
AGV model.The statesvx, vy, and wr denote the vehicle’s longitudinal, lateral, and yaw rate. The parameter m is the mass of the vehicle; Fyf and Fyr indicate the equivalent lateral forces of the front and rear axles, respectively. Parameters Lf and Lr are the distances from the C.G. to the front and rear axle, and Iz is the yaw moment of inertia. On the basis that the steering angle is sufficiently slight, the front and rear wheel lateral forces can be linearized below [18]:
(2)Fyf=−Cfaf=−Cfβ+wrLfvx−δf,Fyr=−Crar=−Crβ−wrLrvx,whereCf and Cr denote the equivalent lateral stiffness of the front and rear axle, δf is the steering angle of the front wheel, af and ar represent the sideslip angle of the front and rear wheels, and β is the sideslip angle.In addition, due to the mechanical limitations of the vehicle itself, dead zones are commonly observed in servo valves or DC servo motors of AGVs. Hence, by combining (1)-(2) with the actuator dead-zone characteristic, the nonlinear model of the AGV considered in this paper is described as follows:
(3)ẋ=vxcosφ−vysinφ,ẏ=vxsinφ+vycosφ,φ̇=wr,v̇y=Δvy,wr,t,ẇr=fvx,wr,t+LfCfIz−1τδf,whereΔvy,wr,t=−vxwr+m−1Cfτδf−β−vx−1wxLf+Crvx−1wrLr−β and fvx,wr,t=LrCrIz−1β−vx−1wrLr−LfCfIz−1vx−1wrLf+β are unknown unmodeled dynamics and disturbances and the input τδf is the force with the dead-zone nonlinearities, which can be described by
(4)τδf=artδf−brt,δf≥brt0,−blt<δf<brtaltδf+blt,δf≤−blt,where art, alt, brt, and blt are positive constants satisfying the condition that art≥a¯r, alt≥a¯l, brt≤b¯r, and blt≤b‾l. In addition, a¯r, a¯l, b‾r, and b‾l are known positive constants.


## 2.2. The Objective

Define the reference trajectory as follows:(5)ẋref=vx,refcosφref−vy,refcosφref,ẏref=vx,refsinφref+vy,refsinφref,φ̇ref=wref,where xref and yref are the desired longitudinal and lateral positions and vx,ref and vy,ref are the desired longitudinal and lateral velocities, respectively. φref and wref are the desired yaw angle and yaw angle rate of the vehicle, respectively. By analyzing system (3), the trajectory tracking problem can be transformed into the yaw angle tracking control problem by constructing the desired yaw angle φref [4], which eventually satisfies φ⟶φref.Define the error statesxe=x−xref, ye=y−yref, and φe=φ−φref. Subsequently, the derivative of the position errors can be calculated as follows:
(6)ẋe=vxcosφ−vx,refcosφref−vysinφ+vy,refsinφref,ẏe=vxsinφ−vx,refsinφref+vycosφ−vy,refcosφref,φ̇e=wr−wref.This paper is aimed at establishing a control methodology that can stabilize the trajectory tracking error system in (6) while ensuring that all closed-loop signals are bounded.Remark 1.
The trajectory tracking problem is transformed into the yaw angle tracking control problem, and it is efficient to reduce the complexity of controller design by dimension reduction. Due to the dead-zone nonlinearities of the inputτδf, the existing fixed-time observers for disturbance estimation cannot be applied directly, which increases the challenge of the tracking problem.


## 3. Main Results

### 3.1. Guidance Law Design

Define the coordinate transformation as follows:(7)z1=xecosφ+yesinφ,z2=−xesinφ+yecosφ.The dynamics ofz1,z2 can be stated as
(8)ż1=vx−vx,refcosφ−φref−vy,refsinφ−φref+z2wr,ż2=vx,refsinφ−φref−vy,refcosφ−φref−z1wr+vy.Lemma 2.
Stabilization of the systemz1,z2 equals stabilizing the system xe,ye in (6).Proof.
The Jacobian matrix betweenz1,z2 and xe,ye is cosφsinφ−sinφcosφ, whose rank is 2. This issue implies that the model transformation is differential homeomorphic. Additionally, the convergence of states z1 and z2 means the system xe,ye converges to zero. In summary, stabilization of the system z1,z2 equals that stabilization of the system xe,ye. The proof is completed.☐The above lemma indicates that the trajectory tracking problem is to design desired yaw angleφref to stabilize the system z1,z2.Remark 3.
In literature [4], the tanh· function is introduced to stabilize the error system, which is a standard method. However, in the system (8), due to the existence of undesired terms −z1wr+vy involved in the error dynamic of z2, the traditional method cannot be applied directly to stabilize the system (8). Hence, the desired yaw angle φref should be redesigned.In this paper, a novel idea of designingφref stemming from the line of sight method is proposed, and a time-varying parameter ϵ is introduced to deal with the undesired terms involved in the error dynamic of z2.Lemma 4.
The statez2 in (8) is stabilized if the following condition holds,
(9)Δ2≤1χ·z2·1+χ=1+χχ·z2,where ∆ is the line of sight range, the variables χ=Γ2−1, Γ2=vx,ref2/fz22, and fz2=−z1wr+vy.Proof.
To stabilize statez2 while overcoming the term −z1wr+vy, we introduce the time-varying parameter ϵ based on the line of sight method, which satisfies φ−φref=−arctanz2+ϵ/Δ. Substituting φ−φref into equation (8), it gives
(10)ż2=−vx,refz2+ϵΔ2+z2+ϵ2+fz2=−vx,refz2Δ2+z2+ϵ2−vx,refϵΔ2+z2+ϵ2+fz2,where −vy,refcosφ−φref+vy is replaced by fz2 to simplify. In order to stabilize state z2, the following condition eliminates the last second terms in (10):
(11)fz2=vx,refϵΔ2+z2+ϵ2,then the value of the variable ϵ can be solved as
(12)ϵ=2z2−4z22+4Δ2+z22Γ2−12Γ2−1.
The solutionϵ=2z2+4z22+4Δ2+z22Γ2−1/2Γ2−1 is deleted because ϵ goes to infinity as Γ2−1⟶0. Simultaneously, to ensure the existence of the value ϵ, the formula 4z22+4Δ2+z22Γ2−1 under the root sign should be greater than or equal to 0, that is to say, Γ2−1Δ2≥−z22+Γ2−1z22. Define χ=Γ2−1 for simplification, it gives
(13)χΔ2≥−z22+χz22,
then the following conditions are discussed:(1)
Ifχ≥0, that is to say, ∣vx,ref∣≥∣fz2∣, equation (13) is satisfied for any ∆(2)
If−1<χ<0, that is to say, ∣vx,ref∣<∣fz2∣, then Δ2≤−z21+χ/χ should be satisfied to guarantee equation (13)(3)
Ifχ<−1, due to χ=Γ2−1, there is no solution ∆ satisfying the condition (13)
In summary, the parameter∆ is chosen based on the condition (9). The proof is completed.☐The longitudinal velocityvx is always sufficiently large in practice, which results that the absolute value of speed vx is always more significant than that of term fz2; then, ∆ can be chosen without limitation. According to equations (8), (10), and (11), the derivative of z1,z2 yields:
(14)ż1=vx−vx,refcosφ−φref−vy,refsinφ−φref+z2wr,ż2=−vx,refz2Δ2+z2+ϵ2.Thus, to make the vehicle perfectly follows the reference trajectory, we only need to design the velocityvx to stabilize z1.Lemma 5.
If the longitudinal velocityvx satisfies that
(15)vx=vx,refcosφ−φref−κutanhz1−z2wr−vy,refsinφ−φref,where the parameter ku is a positive value, then the system z1,z2 in (8) converges to zero.Proof.
Substituting (15) into (8), the dynamics of z1 and z2 can be given as
(16)ż1=−κutanhz1,ż2=−vx,refz2Δ2+z2+ϵ2.
Statesz1 and z2 converge to zero with t⟶∞.☐In the after parts, we assume that longitudinal velocity has been closed-loop controlled and focuses on maintaining the yaw angleφ of the AGV on the desired value.


### 3.2. Improved Fixed-Time Disturbance Observer

In this subsection, a fixed-time disturbance observer considering inputs dead-zones is proposed. In [4], the ARDC method is utilized to estimate the uncertainties and disturbances; however, the estimation error converges to zero asymptotically. The fixed-time control method has a faster convergence speed, and the settling time is dependent on the initial conditions. This issue prohibits its applicability in practical vehicle systems.Define the desired yaw angle of AGV asφd=−arctanz2+ϵ/Δ+φref, and the error of yaw angle and its derivative are ϵ1=φ−φd and ϵ2=φ̇−φ̇d. Then, one has
(17)ϵ̇1=φ̇−φ̇d=ϵ2,ϵ̇2=φ¨−φ¨d=fvx,wr,t−φ¨d+gτδf,where fvr,wr,t refers to the unknown unmodeled dynamics and disturbances, the parameter g=LfCfIz−1 is known, and the input τδf owns the dead-zone characteristic.Assumption 6.
The unknown termḟvy,wr,t is bounded and satisfies ḟvy,wr,t=Δf<σ with σ being a positive constant.
Then, the yaw angle tracking problem can be transformed to stabilize the system (17). To facilitate the subsequent analysis process, we introduce the definition of fixed-time control from [19–21].Definition 7.
Consider a system as(18)ẋ=gt,x,x0=x0,where x∈Rn is the state vector and g:R+×Rn is the nonlinear function. The system (18) is said to be fixed-time stable, if it is globally asymptotically stable and any solution xt,x0 reaches the equilibria at some fixed-time moment, i.e., xt,x0=0, ∀t≥Tx0, where T:Rn⟶R+∪0 is the settling-time function.
Following the above definition, we defineϵ^2 and f^vy,wr,t as the estimations of states ϵ2 and fvx,wr,t in (17), and the symbol ·q as sign··q. Then, the fixed-time disturbance observer is developed based on system (17) as follows:
(19)ϵ^̇2=−φ¨d+f^vy,wr,t+κϵ~1ϵ~21/2+κϵ~2ϵ~2pϵ+a¯δft+b¯Signϵ~2+τ¯,f^̇vx,wr,t=κϵ~3Signϵ~2,
wherepϵ>1, a¯>maxa‾l−a¯l,a‾r−a¯r, b¯>maxb‾l−b¯l,b‾r−b¯r, and ϵ~2=ϵ2−ϵ^2, τ¯ is
(20)τ¯=Signδft2Signδftτ¯a+τ¯b−τ¯a+τ¯b,with τ¯a=a¯rδft−b¯r, and τ¯b=a¯lδft+bl. The constants κϵ~1,κϵ~2, and κϵ~3satisfy
(21)κϵ~1>2κϵ~3,κϵ~2>0,κϵ~3>4σ.
Due to the existence of dead zones, the complete control signal cannot be obtained. So the traditional fixed or finite time observer cannot be directly applied. This section introduces the compensation terma¯δft+b¯Signϵ~2 more significant than τδf−τ¯ to overcome the influence of the dead-zone characteristic on the observer design in (19).Lemma 8.
Consider system (17) under Assumption 6, the observer (19) can estimate the term fvx,wr,t within the fixed time.Proof.
Define the estimation errorsf~vx,wr,t=fvx,wr,t−f^vx,wr,t and ϵ~2=ϵ2−ϵ^2. According to the system (19), we can have the derivatives of f~vx,wr,t and ϵ~2 as follows:
(22)ϵ~̇2=f~vx,wr,t−κϵ~1ϵ~21/2−κϵ~2ϵ~2p1+gτδft−τ¯−ga¯δft+b¯Signf~vx,wr,t,f~̇vx,wr,t=−κϵ~3Signϵ~2+Δf.
SinceΔf<σ and κϵ~3>4σ, according to the Corollary 1 of [22], the estimation error f~vx,wr,t converges to zero in the fixed settling time:
(23)Tϵ~≤1κϵ~2pϵ−121/4κϵ~1/κϵ~2pϵ−1/pϵ+1/2+223/4κϵ~1/κϵ~21/2κϵ~11+κϵ~3+σκϵ~3−σ1−2κϵ~3/κϵ~1.
The proof is completed.☐


### 3.3. Controller Design

The trajectory tracking control strategy is designed based on the fixed-time observer in this subsection. Consider the above disturbance observer (19), the system (17) with dead-zone characteristics can be described as following second-order nonlinear form:
(24)ϵ̇1=ϵ2,ϵ̇2=f^vx,wr,t+gτδf+f~φ,wr,t−φ¨d,where f^vx,wr,t is the estimation of fvx,wr,t.For the second-order nonlinear system (24), the control law in this subsection is designed as follows:
(25)δf=−1algf^vx,wr,t+ϵ3∗−b¯l,ξ2>0,0,ξ2=0,1argf^vx,wr,t+ϵ3∗+b¯r,ξ2<0,where variables ξ1 and ξ2 are
(26)ξ1=β1ϵ1q1+ϵ21/λ+β2ϵ1q2,ξ2=ξ1λ+ϵ2,and variable ϵ3∗ is
(27)ϵ3∗=β3ξ2q3+β4ξ2q4+β1λq1ξ1λ−1ϵ1q1−1ϵ2+β2λq2ξ1λ−1×ϵ1q2−1ϵ21+ξ1λ−1ϵ21/λ−1−1,where λ=0.5, 1<q1<2,q2>2,0<q3<1,q4>1, β1, β2,β3,β4 are designed positive constants.Lemma 9.
The control law (25) can ensure that the system (24) converges to zero in fixed time with the settling time Tc, where:
(28)Tc=21−q3/2β31−q3+21−q4/2β4q4−1+2λβ1λ1−λq1+2λβ2λλq2−1.Proof.
Sincef~vx,wr,t converges to zero in fixed time with Lemma 8, for any t>Tϵ~, the dynamics of states ϵ1 and ϵ2 in (24) can be represented as:
(29)ϵ̇1=ϵ2,ϵ̇2=f^vr,wr,t+gτδf−φ¨d.
According to results in [22], the system (29) converges to zero before the fixed time Ts=Tc+Tϵ~. The proof is completed.☐In the above theorem, the state transformation is introduced:(30)ξ2=ξ1λ+ϵ2,which guarantees the nonsingularity of the control law. This characteristic can be directly derived from the nonsingularity of ϵ3∗. By the definition of ϵ3∗, we can know:
(31)ϵ3∗=β3ξ2q3+β4ξ2q4+β1λq1ξ1λ−1ϵ1q1−1ϵ2+β2λq2ξ1λ−1ϵ1q2−1ϵ2×1+ξ1λ−1ϵ21/λ−1−1.Moreover, we have(32)ϵ3∗=ξ11−λβ3ξ2q3+β4ξ2q4ξ11−λ+ϵ21/λ−1−1+β1λq1ϵ1q1−1ϵ2×ξ11−λ+ϵ21/λ−1−1+β2λq2ϵ1q2−1ϵ2ξ11−λ+ϵ21/λ−1−1,with parameter λ=0.5, hence, one has ξ11−λξ11−λ+ϵ21/λ−1−1=ξ10.5ξ10.5+ϵ21/λ−1−1≤1,ϵ2ξ11−λ+ϵ21/λ−1−1=ϵ2ξ11−λ+ϵ2−1≤1. Then, ϵ3∗ must be a nonsingular variable; hence, the control law (25) is also nonsingular.According to Lemma9, the proposed control law can stabilize the second-order nonlinear system (17) with the actuators’ dead zones in the fixed-time. Finally, the trajectory tracking control theorem for AGVs with dead zones is given.Theorem 10.
Consider the AGV dynamics in (1)-(3) with the dead zones (4), combining with the fixed-time state observer (19) and the control law (25). Under Assumption 6, the error signals f~vx,wr,t, ϵ1, and ϵ2 in the closed-loop system for the trajectory tracking control of the AGV have bounded in the fixed time Ts. Besides, the trajectory tracking control can be achieved.Proof.
According to Lemmas8 and 9, the fixed-time state observer (19) and the control law (25) can stabilize the transformed system in the fixed time. With Lemmas 2–5, the trajectory tracking control can be proved.☐


## 3.1. Guidance Law Design

Define the coordinate transformation as follows:(7)z1=xecosφ+yesinφ,z2=−xesinφ+yecosφ.The dynamics ofz1,z2 can be stated as
(8)ż1=vx−vx,refcosφ−φref−vy,refsinφ−φref+z2wr,ż2=vx,refsinφ−φref−vy,refcosφ−φref−z1wr+vy.Lemma 2.
Stabilization of the systemz1,z2 equals stabilizing the system xe,ye in (6).Proof.
The Jacobian matrix betweenz1,z2 and xe,ye is cosφsinφ−sinφcosφ, whose rank is 2. This issue implies that the model transformation is differential homeomorphic. Additionally, the convergence of states z1 and z2 means the system xe,ye converges to zero. In summary, stabilization of the system z1,z2 equals that stabilization of the system xe,ye. The proof is completed.☐The above lemma indicates that the trajectory tracking problem is to design desired yaw angleφref to stabilize the system z1,z2.Remark 3.
In literature [4], the tanh· function is introduced to stabilize the error system, which is a standard method. However, in the system (8), due to the existence of undesired terms −z1wr+vy involved in the error dynamic of z2, the traditional method cannot be applied directly to stabilize the system (8). Hence, the desired yaw angle φref should be redesigned.In this paper, a novel idea of designingφref stemming from the line of sight method is proposed, and a time-varying parameter ϵ is introduced to deal with the undesired terms involved in the error dynamic of z2.Lemma 4.
The statez2 in (8) is stabilized if the following condition holds,
(9)Δ2≤1χ·z2·1+χ=1+χχ·z2,where ∆ is the line of sight range, the variables χ=Γ2−1, Γ2=vx,ref2/fz22, and fz2=−z1wr+vy.Proof.
To stabilize statez2 while overcoming the term −z1wr+vy, we introduce the time-varying parameter ϵ based on the line of sight method, which satisfies φ−φref=−arctanz2+ϵ/Δ. Substituting φ−φref into equation (8), it gives
(10)ż2=−vx,refz2+ϵΔ2+z2+ϵ2+fz2=−vx,refz2Δ2+z2+ϵ2−vx,refϵΔ2+z2+ϵ2+fz2,where −vy,refcosφ−φref+vy is replaced by fz2 to simplify. In order to stabilize state z2, the following condition eliminates the last second terms in (10):
(11)fz2=vx,refϵΔ2+z2+ϵ2,then the value of the variable ϵ can be solved as
(12)ϵ=2z2−4z22+4Δ2+z22Γ2−12Γ2−1.
The solutionϵ=2z2+4z22+4Δ2+z22Γ2−1/2Γ2−1 is deleted because ϵ goes to infinity as Γ2−1⟶0. Simultaneously, to ensure the existence of the value ϵ, the formula 4z22+4Δ2+z22Γ2−1 under the root sign should be greater than or equal to 0, that is to say, Γ2−1Δ2≥−z22+Γ2−1z22. Define χ=Γ2−1 for simplification, it gives
(13)χΔ2≥−z22+χz22,
then the following conditions are discussed:(1)
Ifχ≥0, that is to say, ∣vx,ref∣≥∣fz2∣, equation (13) is satisfied for any ∆(2)
If−1<χ<0, that is to say, ∣vx,ref∣<∣fz2∣, then Δ2≤−z21+χ/χ should be satisfied to guarantee equation (13)(3)
Ifχ<−1, due to χ=Γ2−1, there is no solution ∆ satisfying the condition (13)
In summary, the parameter∆ is chosen based on the condition (9). The proof is completed.☐The longitudinal velocityvx is always sufficiently large in practice, which results that the absolute value of speed vx is always more significant than that of term fz2; then, ∆ can be chosen without limitation. According to equations (8), (10), and (11), the derivative of z1,z2 yields:
(14)ż1=vx−vx,refcosφ−φref−vy,refsinφ−φref+z2wr,ż2=−vx,refz2Δ2+z2+ϵ2.Thus, to make the vehicle perfectly follows the reference trajectory, we only need to design the velocityvx to stabilize z1.Lemma 5.
If the longitudinal velocityvx satisfies that
(15)vx=vx,refcosφ−φref−κutanhz1−z2wr−vy,refsinφ−φref,where the parameter ku is a positive value, then the system z1,z2 in (8) converges to zero.Proof.
Substituting (15) into (8), the dynamics of z1 and z2 can be given as
(16)ż1=−κutanhz1,ż2=−vx,refz2Δ2+z2+ϵ2.
Statesz1 and z2 converge to zero with t⟶∞.☐In the after parts, we assume that longitudinal velocity has been closed-loop controlled and focuses on maintaining the yaw angleφ of the AGV on the desired value.


## 3.2. Improved Fixed-Time Disturbance Observer

In this subsection, a fixed-time disturbance observer considering inputs dead-zones is proposed. In [4], the ARDC method is utilized to estimate the uncertainties and disturbances; however, the estimation error converges to zero asymptotically. The fixed-time control method has a faster convergence speed, and the settling time is dependent on the initial conditions. This issue prohibits its applicability in practical vehicle systems.Define the desired yaw angle of AGV asφd=−arctanz2+ϵ/Δ+φref, and the error of yaw angle and its derivative are ϵ1=φ−φd and ϵ2=φ̇−φ̇d. Then, one has
(17)ϵ̇1=φ̇−φ̇d=ϵ2,ϵ̇2=φ¨−φ¨d=fvx,wr,t−φ¨d+gτδf,where fvr,wr,t refers to the unknown unmodeled dynamics and disturbances, the parameter g=LfCfIz−1 is known, and the input τδf owns the dead-zone characteristic.Assumption 6.
The unknown termḟvy,wr,t is bounded and satisfies ḟvy,wr,t=Δf<σ with σ being a positive constant.
Then, the yaw angle tracking problem can be transformed to stabilize the system (17). To facilitate the subsequent analysis process, we introduce the definition of fixed-time control from [19–21].Definition 7.
Consider a system as(18)ẋ=gt,x,x0=x0,where x∈Rn is the state vector and g:R+×Rn is the nonlinear function. The system (18) is said to be fixed-time stable, if it is globally asymptotically stable and any solution xt,x0 reaches the equilibria at some fixed-time moment, i.e., xt,x0=0, ∀t≥Tx0, where T:Rn⟶R+∪0 is the settling-time function.
Following the above definition, we defineϵ^2 and f^vy,wr,t as the estimations of states ϵ2 and fvx,wr,t in (17), and the symbol ·q as sign··q. Then, the fixed-time disturbance observer is developed based on system (17) as follows:
(19)ϵ^̇2=−φ¨d+f^vy,wr,t+κϵ~1ϵ~21/2+κϵ~2ϵ~2pϵ+a¯δft+b¯Signϵ~2+τ¯,f^̇vx,wr,t=κϵ~3Signϵ~2,
wherepϵ>1, a¯>maxa‾l−a¯l,a‾r−a¯r, b¯>maxb‾l−b¯l,b‾r−b¯r, and ϵ~2=ϵ2−ϵ^2, τ¯ is
(20)τ¯=Signδft2Signδftτ¯a+τ¯b−τ¯a+τ¯b,with τ¯a=a¯rδft−b¯r, and τ¯b=a¯lδft+bl. The constants κϵ~1,κϵ~2, and κϵ~3satisfy
(21)κϵ~1>2κϵ~3,κϵ~2>0,κϵ~3>4σ.
Due to the existence of dead zones, the complete control signal cannot be obtained. So the traditional fixed or finite time observer cannot be directly applied. This section introduces the compensation terma¯δft+b¯Signϵ~2 more significant than τδf−τ¯ to overcome the influence of the dead-zone characteristic on the observer design in (19).Lemma 8.
Consider system (17) under Assumption 6, the observer (19) can estimate the term fvx,wr,t within the fixed time.Proof.
Define the estimation errorsf~vx,wr,t=fvx,wr,t−f^vx,wr,t and ϵ~2=ϵ2−ϵ^2. According to the system (19), we can have the derivatives of f~vx,wr,t and ϵ~2 as follows:
(22)ϵ~̇2=f~vx,wr,t−κϵ~1ϵ~21/2−κϵ~2ϵ~2p1+gτδft−τ¯−ga¯δft+b¯Signf~vx,wr,t,f~̇vx,wr,t=−κϵ~3Signϵ~2+Δf.
SinceΔf<σ and κϵ~3>4σ, according to the Corollary 1 of [22], the estimation error f~vx,wr,t converges to zero in the fixed settling time:
(23)Tϵ~≤1κϵ~2pϵ−121/4κϵ~1/κϵ~2pϵ−1/pϵ+1/2+223/4κϵ~1/κϵ~21/2κϵ~11+κϵ~3+σκϵ~3−σ1−2κϵ~3/κϵ~1.
The proof is completed.☐


## 3.3. Controller Design

The trajectory tracking control strategy is designed based on the fixed-time observer in this subsection. Consider the above disturbance observer (19), the system (17) with dead-zone characteristics can be described as following second-order nonlinear form:
(24)ϵ̇1=ϵ2,ϵ̇2=f^vx,wr,t+gτδf+f~φ,wr,t−φ¨d,where f^vx,wr,t is the estimation of fvx,wr,t.For the second-order nonlinear system (24), the control law in this subsection is designed as follows:
(25)δf=−1algf^vx,wr,t+ϵ3∗−b¯l,ξ2>0,0,ξ2=0,1argf^vx,wr,t+ϵ3∗+b¯r,ξ2<0,where variables ξ1 and ξ2 are
(26)ξ1=β1ϵ1q1+ϵ21/λ+β2ϵ1q2,ξ2=ξ1λ+ϵ2,and variable ϵ3∗ is
(27)ϵ3∗=β3ξ2q3+β4ξ2q4+β1λq1ξ1λ−1ϵ1q1−1ϵ2+β2λq2ξ1λ−1×ϵ1q2−1ϵ21+ξ1λ−1ϵ21/λ−1−1,where λ=0.5, 1<q1<2,q2>2,0<q3<1,q4>1, β1, β2,β3,β4 are designed positive constants.Lemma 9.
The control law (25) can ensure that the system (24) converges to zero in fixed time with the settling time Tc, where:
(28)Tc=21−q3/2β31−q3+21−q4/2β4q4−1+2λβ1λ1−λq1+2λβ2λλq2−1.Proof.
Sincef~vx,wr,t converges to zero in fixed time with Lemma 8, for any t>Tϵ~, the dynamics of states ϵ1 and ϵ2 in (24) can be represented as:
(29)ϵ̇1=ϵ2,ϵ̇2=f^vr,wr,t+gτδf−φ¨d.
According to results in [22], the system (29) converges to zero before the fixed time Ts=Tc+Tϵ~. The proof is completed.☐In the above theorem, the state transformation is introduced:(30)ξ2=ξ1λ+ϵ2,which guarantees the nonsingularity of the control law. This characteristic can be directly derived from the nonsingularity of ϵ3∗. By the definition of ϵ3∗, we can know:
(31)ϵ3∗=β3ξ2q3+β4ξ2q4+β1λq1ξ1λ−1ϵ1q1−1ϵ2+β2λq2ξ1λ−1ϵ1q2−1ϵ2×1+ξ1λ−1ϵ21/λ−1−1.Moreover, we have(32)ϵ3∗=ξ11−λβ3ξ2q3+β4ξ2q4ξ11−λ+ϵ21/λ−1−1+β1λq1ϵ1q1−1ϵ2×ξ11−λ+ϵ21/λ−1−1+β2λq2ϵ1q2−1ϵ2ξ11−λ+ϵ21/λ−1−1,with parameter λ=0.5, hence, one has ξ11−λξ11−λ+ϵ21/λ−1−1=ξ10.5ξ10.5+ϵ21/λ−1−1≤1,ϵ2ξ11−λ+ϵ21/λ−1−1=ϵ2ξ11−λ+ϵ2−1≤1. Then, ϵ3∗ must be a nonsingular variable; hence, the control law (25) is also nonsingular.According to Lemma9, the proposed control law can stabilize the second-order nonlinear system (17) with the actuators’ dead zones in the fixed-time. Finally, the trajectory tracking control theorem for AGVs with dead zones is given.Theorem 10.
Consider the AGV dynamics in (1)-(3) with the dead zones (4), combining with the fixed-time state observer (19) and the control law (25). Under Assumption 6, the error signals f~vx,wr,t, ϵ1, and ϵ2 in the closed-loop system for the trajectory tracking control of the AGV have bounded in the fixed time Ts. Besides, the trajectory tracking control can be achieved.Proof.
According to Lemmas8 and 9, the fixed-time state observer (19) and the control law (25) can stabilize the transformed system in the fixed time. With Lemmas 2–5, the trajectory tracking control can be proved.☐


## 4. Simulation

To verify the effectiveness of the presented method, we use CarSim and MATLAB platforms to carry out the trajectory tracking problem. We used a general AGV model derived by [4]. In this simulation, the proposed guidance laws (9), fixed-time state observer (19), and the control law (25) are applied. Let the AGV have the following initial conditions: x0,y0,φ0,vx0,vy0,wr0T=0,0,0,0,0,0T, representing the initial values of positions x and y, yaw angle φ, velocities vx, vy, and yaw angle rate wr, respectively. The reference trajectory is chosen as xd0,yd0=10m,10m, φd=0.15sin0.1×t.Table1 shows all the variables involved in the AGV model. The parameters of actuator dead-zones and control parameters are chosen as follows:
(33)ar=0.7+0.3sint,al=0.7+0.3sint,br=5+5cost,bl=5+5cost,κϵ~1=2,κϵ~2=0.3,κϵ~3=2,p1=2,q1=1.9,q2=2.1,q3=0.9,q4=1.1,β1=0.2,β2=2,β3=0.2,β4=2,a¯r=1,a¯l=1,a¯l=0.7,a¯l=0.7,b¯l=10,b¯l=10,b¯l=5,b¯l=5.Table 1
The model parameters of the AGV.
SymbolQuantityValuemMass of the vehicle1723 kgIzVehicle yaw moment of inertial4175 kg·mLfThe distances from vehicle C.G. to the front axis center1.232 mLrThe distances from vehicle C.G. to the rear axis center1.468 mCfVehicle nominal front axle cornering stiffnessN/radCrVehicle nominal rear axle cornering stiffnessN/rad


### 4.1. Disturbance Observer Compensation

To illustrate the advantage of the proposed disturbance observer, the trajectories of our improved disturbance observer and conventional fixed-time disturbance observer are shown in Figures2 and 3.Figure 2
Comparisons of estimation states.Figure 3
Comparisons of cumulative estimation errors.In this comparison, we aim at the compensation result of the dead zone of the actuator in the improved fixed time disturbance observer (Improved-FTDO) and compare it with the conventional fixed time disturbance observer (FTDO) [17]. The results show that after considering the dead zone of the actuator, the improved FTDO method proposed is better than the conventional FTDO method because the compensation problem is considered in this paper.


### 4.2. Actuator Compensation

The simulation results are shown in Figures4–7. To evaluate the effectiveness of our method, we consider the convergence comparisons between the control methods of ADCC and NTSM-ADRC in [4, 22], respectively. The same driving conditions are carried out, whose road adhesion coefficient is 0.8. We now look at the differences in the performance of yaw angle changing δf shown in Figures 4. It reveals that the maximum value of δf in our method is the minimum one in the three approaches, which means that the input of yaw angle changing δf is gentler than the other two controllers, and the ride comfort is guaranteed.Figure 4
Comparisons of inputs.Figure 5
Comparisons of observer errors.Figure 6
Comparisons of trajectories of positions.Figure 7
Comparisons of tracking errors.In Table1, only the nominal values of the parameters are given, and there are still many parametric uncertainties. For example, (1) the tire deformation; (2) the equivalent lateral stiffness Cf and Cr of the front and rear axle change caused by wheel load transfer; (3) the shift in road attachment, and (4) the distance from the front and rear axles to the center of the vehicle Lf and Lr change caused by the vehicle’s centroid position changes. Here, we consider the influence of the lateral stiffness values on the performance of the proposed control algorithm. We analyze the impact of the lateral stiffness values with unknown values of Cf and Cr in simulations. The IFDO-ADCC does not need to set different control parameters for various vehicle speeds than some controllers. It has strong robustness to the change of the external environment.Moreover, the IFDO-ADCC can dynamically estimate and compensate the system for total disturbances in fixed time. The simulation results are shown in Figure5. We can see that the IFDO-ADCC has faster estimation error convergence and good disturbance estimation performance, which reflects the excellent robustness to the uncertainties. The trajectory tracking performances of AGVs by the controllers of IFDO-ADCC, ADCC, and NTSM-ARDC are given in Figure 6. We can see that three controllers can track the reference trajectory well. Figure 7 shows the corresponding tracking error of the vehicle under different controllers. It can be seen that the trajectory tracking performance of the proposed IFDO-ADCC method is superior to ADCC and NTSM-ARDC.


### 4.3. Overall Situation

In Figure8, the CarSim and MATLAB simulation results of the trajectory tracking based on the proposed guidance laws (9), fixed-time state observer (19), and the control law (25) are given. It shows that the proposed algorithms can be applied in actual scenarios.Figure 8
Trajectories of positions in MATLAB and CarSim platforms.Compared with the ADCC method and NTSM-ARDC proposed in [4], the case of input dead zones is considered in the trajectory tracking performance in our paper. Although the control input falls in the dead-zone at a specific time when the actuator has no output signal, the performance in trajectory tracking errors is guaranteed, proving the effectiveness of the method proposed in our paper.


## 4.1. Disturbance Observer Compensation

To illustrate the advantage of the proposed disturbance observer, the trajectories of our improved disturbance observer and conventional fixed-time disturbance observer are shown in Figures2 and 3.Figure 2
Comparisons of estimation states.Figure 3
Comparisons of cumulative estimation errors.In this comparison, we aim at the compensation result of the dead zone of the actuator in the improved fixed time disturbance observer (Improved-FTDO) and compare it with the conventional fixed time disturbance observer (FTDO) [17]. The results show that after considering the dead zone of the actuator, the improved FTDO method proposed is better than the conventional FTDO method because the compensation problem is considered in this paper.


## 4.2. Actuator Compensation

The simulation results are shown in Figures4–7. To evaluate the effectiveness of our method, we consider the convergence comparisons between the control methods of ADCC and NTSM-ADRC in [4, 22], respectively. The same driving conditions are carried out, whose road adhesion coefficient is 0.8. We now look at the differences in the performance of yaw angle changing δf shown in Figures 4. It reveals that the maximum value of δf in our method is the minimum one in the three approaches, which means that the input of yaw angle changing δf is gentler than the other two controllers, and the ride comfort is guaranteed.Figure 4
Comparisons of inputs.Figure 5
Comparisons of observer errors.Figure 6
Comparisons of trajectories of positions.Figure 7
Comparisons of tracking errors.In Table1, only the nominal values of the parameters are given, and there are still many parametric uncertainties. For example, (1) the tire deformation; (2) the equivalent lateral stiffness Cf and Cr of the front and rear axle change caused by wheel load transfer; (3) the shift in road attachment, and (4) the distance from the front and rear axles to the center of the vehicle Lf and Lr change caused by the vehicle’s centroid position changes. Here, we consider the influence of the lateral stiffness values on the performance of the proposed control algorithm. We analyze the impact of the lateral stiffness values with unknown values of Cf and Cr in simulations. The IFDO-ADCC does not need to set different control parameters for various vehicle speeds than some controllers. It has strong robustness to the change of the external environment.Moreover, the IFDO-ADCC can dynamically estimate and compensate the system for total disturbances in fixed time. The simulation results are shown in Figure5. We can see that the IFDO-ADCC has faster estimation error convergence and good disturbance estimation performance, which reflects the excellent robustness to the uncertainties. The trajectory tracking performances of AGVs by the controllers of IFDO-ADCC, ADCC, and NTSM-ARDC are given in Figure 6. We can see that three controllers can track the reference trajectory well. Figure 7 shows the corresponding tracking error of the vehicle under different controllers. It can be seen that the trajectory tracking performance of the proposed IFDO-ADCC method is superior to ADCC and NTSM-ARDC.


## 4.3. Overall Situation

In Figure8, the CarSim and MATLAB simulation results of the trajectory tracking based on the proposed guidance laws (9), fixed-time state observer (19), and the control law (25) are given. It shows that the proposed algorithms can be applied in actual scenarios.Figure 8
Trajectories of positions in MATLAB and CarSim platforms.Compared with the ADCC method and NTSM-ARDC proposed in [4], the case of input dead zones is considered in the trajectory tracking performance in our paper. Although the control input falls in the dead-zone at a specific time when the actuator has no output signal, the performance in trajectory tracking errors is guaranteed, proving the effectiveness of the method proposed in our paper.


## 5. Conclusions

In this paper, based on a global diffeomorphism transformation, a fixed-time observer control method is proposed for the trajectory tracking problem of the AGV system. Trajectory tracking of AGVs subject to actuator saturation is still an open problem. Energy consumption optimal control of AGVs is another interesting problem to be studied in the future. Additionally, some nonlinear system control methods can also be applied for AGVs in the future. For example, an adaptive super-twisting sliding mode control technique is developed for forn-link rigid robotic manipulators [23]. An adaptive nonsingular integral-type second-order terminal sliding mode tracking control for nonlinear systems with uncertainties is proposed [24].



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*Source: 2914190-2021-12-24.xml*