# Extended-State-Observer-Based Collision-Free Guidance Law for Target Tracking of Autonomous Surface Vehicles with Unknown Target Dynamics

Authors: Shengnan Gao; Zhouhua Peng; Dan Wang; Lu Liu
Journal: Complexity
 (2018)

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## Abstract

This paper is concerned with the target tracking problem of an autonomous surface vehicle in the presence of a maneuvering target. The velocity information of target is totally unknown to the follower vehicle, and only the relative distance and angle between the target and follower are obtained. First, a reduced-order extended state observer is used to estimate the unknown relative dynamics due to the unavailable velocity of the target. Based on the reduced-order extended state observer, an antidisturbance guidance law for target tracking is designed. The input-to-state stability of the closed-loop target tracking guidance system is analyzed via cascade theory. Furthermore, the above result is extended to the case that collisions between the target and leader are avoided during tracking, and a collision-free target tracking guidance law is developed. The main feature of the proposed guidance law is twofold. First, the target tracking can be achieved without using the velocity information of the target. Second, collision avoidance can be achieved during target tracking. Simulation results show the effectiveness of the proposed antidisturbance guidance law for tracking a maneuvering target with the arbitrary bounded velocity.


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## Body

## 1. Introduction

Recently, advanced motion control of marine vehicles has received significant attention due to its wide applications in hydrological monitoring, channel exclusion, search and rescue, biological detection, and so on [1–7]. Numerous motion control scenarios of marine vehicles have been considered including path following [8–21], path tracking [22, 23], trajectory tracking [4, 24–26], and target tracking [2, 27–34]. Target tracking is to track a maneuvering target where no information about the target behavior is known in advance except its instantaneous motion.Various control methods have been developed for the target tracking of marine vehicles [2, 27–34]. In [2], a position tracking controller is developed for an underactuated autonomous underwater vehicle based on a backstepping technique and neural networks. In [27], a straight-line target tracking controller is developed for an underactuated unmanned surface vehicle. In [28], a target tracking controller is developed for underactuated autonomous surface vehicles (ASVs) with limited control torques. In [29], a fault tolerant target tracking controller is developed for underactuated ASVs. In [2, 27–29], the velocities of the targets are known as a priori. In practice, however, the velocity information of the target may not be available by the follower. In order to track the leader in the absence of velocity information of target, a variety of methods are available [30, 31]. In [30], a robust controller is designed for target tracking of marine vessels where unknown velocity of the leader is handled by using a sliding model control. In [31], an adaptive leader-follower formation controller is designed for ASVs based on a dynamic surface control and single-hidden-layer neural networks, and an adaptive term is used to estimate the unknown velocity of target. In recent years, collision avoidance has been considered in motion control [35, 36]. However, the collision avoidance problem during target tracking is not considered in [2, 27–29].Extended state observer as a key part of active disturbance rejection control method was proposed by Han [37–39]. It is a powerful tool to deal with the nonlinear systems in the presence of large uncertainty including internal model uncertainties and external disturbances [40]. It has been widely used in numerous engineering applications [11, 41, 42]. A reduced-order extended state observer (RESO) is able to decrease the phase lag and simplify the observer structure to reduce computation load. It has the advantages of fast observation and no overshooting. Because of its advantages, it is desirable to employ the extended state observer to address the uncertainty during target tracking.In this paper, an antidisturbance guidance law for target tracking is designed based on the reduced-order extended state observer (RESO), where an ASV is requested to track a maneuvering target. Only relative line-of-sight range and angle between the follower and target are available for feedback design. At first, a RESO is used to estimate the unknown relative dynamics due to the unavailable velocity of the target. Then, an antidisturbance guidance law is proposed based on the RESO and the stability of closed-loop guidance system is analyzed via cascade analysis. The above result is extended to target tracking with collision avoidance of ASVs, and a collision-free RESO-based guidance law is developed. Simulation results are used to show the proposed collision-free guidance law for tracking a maneuvering target.The contributions of this paper is twofold. First, an antidisturbance guidance law for target tracking is designed based on the reduced-order extended state observer where target dynamics is not required to be known. Second, a collision-free guidance law for target tracking is developed such that the collision between the target and follower vehicle can be avoided. The main features of the proposed guidance law are presented in this paper as follows. First, compared with the target tracking controllers proposed in [2, 27–29] where the velocity of the target should be known in advance, the velocity of the target is not required to be known and only the relative line-of-sight distance and angle between the target and the follower are needed. Second, compared with the target tracking controllers proposed in [2, 10, 27–30, 34] where the collision avoidance problem is not considered, a collision-free RESO-based guidance law is proposed for target tracking of underactuated ASVs where the collision between the target and follower can be avoided.This paper is organized as follows: Section2 states the preliminaries and problem formulation. Section 3 gives the target tracking guidance law design and analysis. Section 4 introduces the collision-free target tracking guidance law design and analysis. Section 5 presents the simulation results. Section 6 concludes this paper.


## 2. Preliminaries and Problem Formulation

### 2.1. Collision Avoidance

In order to assure collision-free target tracking, the following collision avoidance potential functions are introduced [35]:(1)Vaρ=min⁡0,l2-R2l2-r22,where R>r>0, R is the detection region, r is the radius of the avoidance, and l is the distance between the target and follower vehicles defined as(2)l=yt-y2+xt-x2,and xt,yt are positions of a target and x,y are positions of a follower.Function (1) will be infinity when the distance between the vehicle and obstacle approaches avoidance region and is zero outside the detection region. In other words, the function Va will affect the surge velocity when l is inside the detection region.Taking the partial derivative of the potential functionVa with respect to x, we can obtain [36](3)∂Va∂x=4R2-r2l2-R2l2-r23x-xt,ifr<l<R,0,otherwise,and the partial derivative of function Va with respect to y is(4)∂Va∂y=4R2-r2l2-R2l2-r23y-yt,ifr<l<R,0,otherwise.


### 2.2. Vehicle Kinematics

The kinematics of an ASV can be expressed by using an earth-fixed frame{E} and a body-fixed frame {B} as shown in Figure 1. Let (xt,yt,ψt) and (x,y,ψ) be the position and orientation of the target and follower, respectively. ut,υt,rt denote the surge velocity, sway velocity, and angular rate of the target vehicle; u,υ,r represent the surge velocity, sway velocity, and angular rate of the follower vehicle, respectively. The kinematics of the target ASV is(5)x˙t=utcos⁡ψt-υtsin⁡ψt,y˙t=utsin⁡ψt+υtcos⁡ψt,ψ˙t=rt,and the kinematics of follower ASV is(6)x˙=ucos⁡ψ-υsin⁡ψ,y˙=usin⁡ψ+υcos⁡ψ,ψ˙=r.Figure 1
A geometrical illustration for tracking a target by an ASV.From Figure1, the line-of-sight range and angle between the target and the follower are defined as(7)ρ=yt-y2+xt-x2,β=tan-1⁡yt-yxt-x.In the following sections, we first consider the target tracking guidance law design being lack of velocity information of the target. Next, the result is extended to collision-free RESO-based guidance law design.


## 2.1. Collision Avoidance

In order to assure collision-free target tracking, the following collision avoidance potential functions are introduced [35]:(1)Vaρ=min⁡0,l2-R2l2-r22,where R>r>0, R is the detection region, r is the radius of the avoidance, and l is the distance between the target and follower vehicles defined as(2)l=yt-y2+xt-x2,and xt,yt are positions of a target and x,y are positions of a follower.Function (1) will be infinity when the distance between the vehicle and obstacle approaches avoidance region and is zero outside the detection region. In other words, the function Va will affect the surge velocity when l is inside the detection region.Taking the partial derivative of the potential functionVa with respect to x, we can obtain [36](3)∂Va∂x=4R2-r2l2-R2l2-r23x-xt,ifr<l<R,0,otherwise,and the partial derivative of function Va with respect to y is(4)∂Va∂y=4R2-r2l2-R2l2-r23y-yt,ifr<l<R,0,otherwise.


## 2.2. Vehicle Kinematics

The kinematics of an ASV can be expressed by using an earth-fixed frame{E} and a body-fixed frame {B} as shown in Figure 1. Let (xt,yt,ψt) and (x,y,ψ) be the position and orientation of the target and follower, respectively. ut,υt,rt denote the surge velocity, sway velocity, and angular rate of the target vehicle; u,υ,r represent the surge velocity, sway velocity, and angular rate of the follower vehicle, respectively. The kinematics of the target ASV is(5)x˙t=utcos⁡ψt-υtsin⁡ψt,y˙t=utsin⁡ψt+υtcos⁡ψt,ψ˙t=rt,and the kinematics of follower ASV is(6)x˙=ucos⁡ψ-υsin⁡ψ,y˙=usin⁡ψ+υcos⁡ψ,ψ˙=r.Figure 1
A geometrical illustration for tracking a target by an ASV.From Figure1, the line-of-sight range and angle between the target and the follower are defined as(7)ρ=yt-y2+xt-x2,β=tan-1⁡yt-yxt-x.In the following sections, we first consider the target tracking guidance law design being lack of velocity information of the target. Next, the result is extended to collision-free RESO-based guidance law design.


## 3. Target Tracking

At first, the relative dynamics between the target and follower is derived. Then, a RESO is used to estimate the unknown relative dynamics due to the unavailable velocity of the target. Next, an antidisturbance guidance law is designed based on the RESO. Finally, the stability of closed-loop guidance system is analyzed via cascade analysis.


### 3.1. Relative Dynamics

Two target tracking errors are defined as follows:(8)eρ=ρ-ρd,eβ=β-ψ-δ,where ρd is a desired range and δ=atan2(υ,u) is a sideslip angle. Taking the time derivative of eρ and eβ in (8) and using (5) and (6), we have(9)e˙ρ=utcos⁡β-ψt+υtsin⁡β-ψt-υsin⁡β-ψ-ucos⁡β-ψ-ρ˙d,e˙β=utsin⁡ψt-β+υtcos⁡ψt-βρ-υcos⁡β-ψ+usin⁡β-ψρ-δ˙-r.The control objective of target tracking of ASVs in the presence of unknown target kinematics is to design a surge velocityu and yaw rate r such that(10)limt→∞⁡eρ≤ϵ1,limt→∞⁡eβ≤ϵ2,for some small constants ϵ1 and ϵ2.


### 3.2. RESO Design

We first use a RESO to estimate the unknown relative dynamics due to the unavailable velocity of the target. To facilitate controller design, the relative dynamics in (9) is rewritten in the following form:(11)e˙ρ=f·-u,e˙β=g·-r,where(12)f·=utcos⁡β-ψt+υtsin⁡β-ψt-υsin⁡β-ψ+2usin2⁡β-ψ2-ρ˙d,g·=utsin⁡ψt-β+υtcos⁡ψt-βρ-υcos⁡β-ψ+usin⁡β-ψρ-δ˙.Sinceut, υt, and ψt of the target are not available, f(·) and g(·) are unknown. To address it, a reduced-order extended state observer is proposed as follows [43]:(13)p˙1=-k1p1-k12eρ+k1u,f^=p1+k1eρ,p˙2=-k2p2-k22eβ+k2r,g^=p2+k2eβ,where p1,p2 are the auxiliary states of the observer; k1,k2 are the observer gains; f^ and g^ denote the estimation of f and g. The initial values of p1 and p2 are set to p1(t0)=-k1eρ(t0) and p2(t0)=-k2eβ(t0).Assumption 1.
The time derivatives of f and g are bounded by |f˙|≤f∗ and |g˙|≤g∗, where f∗ and g∗ are positive constants.
Since the velocities and accelerations of the target and follower ASVs are naturally bounded, Assumption1 is reasonable.
The estimation errors are defined as follows:(14)f~=f^-f,g~=g^-g.
Taking the derivative of (14) along (13), we have(15)f~˙=-k1p1-k12eρ+k1u+k1f-u-f˙=-k1f~-f˙,g~˙=-k2p2-k22eβ+k2r+k3g-r-g˙=-k2g~-g˙.
The stability of subsystem (15) is stated as follows.Lemma 2.
Subsystem (15), viewed as a system with the states being f~ and g~ and the inputs being f˙ and g˙, is input-to-state stability (ISS).Proof.
Construct the following Lyapunov function:(16)V1=12f~2+12g~2.
Taking the time derivative ofV1 along (15) results in(17)V1˙=-k1f~2-f~f˙-k2g~2-g~g˙≤-λminK1E12+hE1,where E1=[f~, g~]T, K1=diag{k1,k2}, and h=[|f˙|, |g˙|]T. Noting that(18)E1≥f˙+g˙θ1λminK1≥hθ1λminK1renders(19)V1˙≤-λminK11-θ1E12,where 0<θ1<1. As a consequence, subsystem (15) is ISS, and(20)E1t≤max⁡σoE10,t,γo1f˙+γo2g˙,where σo is a class KL function, γo1,γo2 are the class K function, and(21)γo1s=γo2s=sθ1λminK1.


### 3.3. Guidance Law Design

Based on the estimated termsf^ and g^ from the RESO, an antidisturbance guidance law is proposed as follows:(22)u=k3eρeρ2+Δ12+f^,r=k4eβeβ2+Δ22+g^,where k3 and k4 are positive constants; Δ1 and Δ2 are positive constants, which are used to limit the maximum value of control laws.Substituting (14) and (22) into (11) results in(23)e˙ρ=-k3eρeρ2+Δ12-f~,e˙β=-k4eβeβ2+Δ22-g~.The ISS property of subsystem (23) is stated as follows.Lemma 3.
Subsystem (23), viewed as a system with the states being eρ and eβ and the inputs being f~ and g~, is ISS.Proof.
Construct the following Lyapunov function:(24)V2=12eρ2+12eβ2.Taking the time derivative of V2 along (23) results in(25)V2˙=-k3eρ2eρ2+Δ12-f~eρ-k4eβ2eβ2+Δ22-g~eβ≤-λminK2E22E22+Δmax1⁡2+wE2,where E2=[eρ, eβ]T, K2=diag{k3,k4}, w=[|f~|, |g~|]T, and Δmax1=max{Δ1,Δ2}.
Noting that(26)E2E2+Δmax2≥f~+g~θ2λminK2≥wθ2λminK2renders(27)V2˙≤-λminK21-θ2E22E22+Δmax1⁡2,where 0<θ2<1. As a consequence, subsystem (23) is ISS, and(28)E2t≤max⁡σcE20,t,γc1f~+γc2g~,where σc is a class KL function, γc1 and γc2 are the class K functions, and(29)γc1s=γc2s=μ-1sθ2λminK2,with μ=|s|/s2+Δmax12.
The proposed guidance law can be augmented with other methods such as PID control, adaptive control [44, 45], sliding mode control [46], and robust control at the kinetic level for achieving the desired target tracking control performance.


### 3.4. Stability Analysis

To analyze the stability of the entire closed-loop guidance system, rewrite the disturbance estimation subsystem (15) and target tracking error subsystem (23) in a compact form as(30)Σ1:e˙ρ=-k3eρeρ2+Δ12-f~,e˙β=k4eβeβ2+Δ22-g~,and(31)Σ2:f~˙=-k1f~-f˙,g~˙=-k2g~-g˙.The stability of the cascade of subsystemΣ1 and subsystem Σ2 is given by the following theorem.Theorem 4.
Under Assumption1, the closed-loop system cascaded by subsystem (15) and subsystem (23) is ISS, and the target tracking errors converge to a small neighborhood of the origin.Proof.
SubsystemΣ1 with states (eρ,eβ) and exogenous inputs (f~,g~) and subsystem Σ2 with states (f~, g~) and exogenous inputs f˙ and g˙ are ISS. By Lemma C.4 in [47], it follows that the cascade systems Σ1 and Σ2 with states (eρ,eβ,f~,g~) and exogenous inputs (f˙,g˙) are ISS, i.e., there exist class KL function σ1 and K function γc1, γo1, γc2, and γo2, such that(32)Ea≤max⁡σ1Ea0,t,γc1∘γo1f˙+γc2∘γo2g˙,where Ea=[eρ,eβ,f~,g~]. As t→∞, σ1→∞, it follows that(33)limt→∞⁡Ea≤γc1∘γo1f˙+γc2∘γo2g˙≤μ-1f∗θ1θ2λminK1λminK2+μ-1g∗θ1θ2λminK1λminK2,implying (10) with μ=|s|/s2+Δmax12. Note that |g˙| and |f˙| are bounded by g∗ and f∗. Then, the errors eρ, eβ, f~ and g~ are all bounded. Note that only uniform ultimate boundedness of closed-loop system can be achieved due to the existence of g˙ and f˙. If g˙=0 and f˙=0, the closed-loop system is asymptotical stable.


## 3.1. Relative Dynamics

Two target tracking errors are defined as follows:(8)eρ=ρ-ρd,eβ=β-ψ-δ,where ρd is a desired range and δ=atan2(υ,u) is a sideslip angle. Taking the time derivative of eρ and eβ in (8) and using (5) and (6), we have(9)e˙ρ=utcos⁡β-ψt+υtsin⁡β-ψt-υsin⁡β-ψ-ucos⁡β-ψ-ρ˙d,e˙β=utsin⁡ψt-β+υtcos⁡ψt-βρ-υcos⁡β-ψ+usin⁡β-ψρ-δ˙-r.The control objective of target tracking of ASVs in the presence of unknown target kinematics is to design a surge velocityu and yaw rate r such that(10)limt→∞⁡eρ≤ϵ1,limt→∞⁡eβ≤ϵ2,for some small constants ϵ1 and ϵ2.


## 3.2. RESO Design

We first use a RESO to estimate the unknown relative dynamics due to the unavailable velocity of the target. To facilitate controller design, the relative dynamics in (9) is rewritten in the following form:(11)e˙ρ=f·-u,e˙β=g·-r,where(12)f·=utcos⁡β-ψt+υtsin⁡β-ψt-υsin⁡β-ψ+2usin2⁡β-ψ2-ρ˙d,g·=utsin⁡ψt-β+υtcos⁡ψt-βρ-υcos⁡β-ψ+usin⁡β-ψρ-δ˙.Sinceut, υt, and ψt of the target are not available, f(·) and g(·) are unknown. To address it, a reduced-order extended state observer is proposed as follows [43]:(13)p˙1=-k1p1-k12eρ+k1u,f^=p1+k1eρ,p˙2=-k2p2-k22eβ+k2r,g^=p2+k2eβ,where p1,p2 are the auxiliary states of the observer; k1,k2 are the observer gains; f^ and g^ denote the estimation of f and g. The initial values of p1 and p2 are set to p1(t0)=-k1eρ(t0) and p2(t0)=-k2eβ(t0).Assumption 1.
The time derivatives of f and g are bounded by |f˙|≤f∗ and |g˙|≤g∗, where f∗ and g∗ are positive constants.
Since the velocities and accelerations of the target and follower ASVs are naturally bounded, Assumption1 is reasonable.
The estimation errors are defined as follows:(14)f~=f^-f,g~=g^-g.
Taking the derivative of (14) along (13), we have(15)f~˙=-k1p1-k12eρ+k1u+k1f-u-f˙=-k1f~-f˙,g~˙=-k2p2-k22eβ+k2r+k3g-r-g˙=-k2g~-g˙.
The stability of subsystem (15) is stated as follows.Lemma 2.
Subsystem (15), viewed as a system with the states being f~ and g~ and the inputs being f˙ and g˙, is input-to-state stability (ISS).Proof.
Construct the following Lyapunov function:(16)V1=12f~2+12g~2.
Taking the time derivative ofV1 along (15) results in(17)V1˙=-k1f~2-f~f˙-k2g~2-g~g˙≤-λminK1E12+hE1,where E1=[f~, g~]T, K1=diag{k1,k2}, and h=[|f˙|, |g˙|]T. Noting that(18)E1≥f˙+g˙θ1λminK1≥hθ1λminK1renders(19)V1˙≤-λminK11-θ1E12,where 0<θ1<1. As a consequence, subsystem (15) is ISS, and(20)E1t≤max⁡σoE10,t,γo1f˙+γo2g˙,where σo is a class KL function, γo1,γo2 are the class K function, and(21)γo1s=γo2s=sθ1λminK1.


## 3.3. Guidance Law Design

Based on the estimated termsf^ and g^ from the RESO, an antidisturbance guidance law is proposed as follows:(22)u=k3eρeρ2+Δ12+f^,r=k4eβeβ2+Δ22+g^,where k3 and k4 are positive constants; Δ1 and Δ2 are positive constants, which are used to limit the maximum value of control laws.Substituting (14) and (22) into (11) results in(23)e˙ρ=-k3eρeρ2+Δ12-f~,e˙β=-k4eβeβ2+Δ22-g~.The ISS property of subsystem (23) is stated as follows.Lemma 3.
Subsystem (23), viewed as a system with the states being eρ and eβ and the inputs being f~ and g~, is ISS.Proof.
Construct the following Lyapunov function:(24)V2=12eρ2+12eβ2.Taking the time derivative of V2 along (23) results in(25)V2˙=-k3eρ2eρ2+Δ12-f~eρ-k4eβ2eβ2+Δ22-g~eβ≤-λminK2E22E22+Δmax1⁡2+wE2,where E2=[eρ, eβ]T, K2=diag{k3,k4}, w=[|f~|, |g~|]T, and Δmax1=max{Δ1,Δ2}.
Noting that(26)E2E2+Δmax2≥f~+g~θ2λminK2≥wθ2λminK2renders(27)V2˙≤-λminK21-θ2E22E22+Δmax1⁡2,where 0<θ2<1. As a consequence, subsystem (23) is ISS, and(28)E2t≤max⁡σcE20,t,γc1f~+γc2g~,where σc is a class KL function, γc1 and γc2 are the class K functions, and(29)γc1s=γc2s=μ-1sθ2λminK2,with μ=|s|/s2+Δmax12.
The proposed guidance law can be augmented with other methods such as PID control, adaptive control [44, 45], sliding mode control [46], and robust control at the kinetic level for achieving the desired target tracking control performance.


## 3.4. Stability Analysis

To analyze the stability of the entire closed-loop guidance system, rewrite the disturbance estimation subsystem (15) and target tracking error subsystem (23) in a compact form as(30)Σ1:e˙ρ=-k3eρeρ2+Δ12-f~,e˙β=k4eβeβ2+Δ22-g~,and(31)Σ2:f~˙=-k1f~-f˙,g~˙=-k2g~-g˙.The stability of the cascade of subsystemΣ1 and subsystem Σ2 is given by the following theorem.Theorem 4.
Under Assumption1, the closed-loop system cascaded by subsystem (15) and subsystem (23) is ISS, and the target tracking errors converge to a small neighborhood of the origin.Proof.
SubsystemΣ1 with states (eρ,eβ) and exogenous inputs (f~,g~) and subsystem Σ2 with states (f~, g~) and exogenous inputs f˙ and g˙ are ISS. By Lemma C.4 in [47], it follows that the cascade systems Σ1 and Σ2 with states (eρ,eβ,f~,g~) and exogenous inputs (f˙,g˙) are ISS, i.e., there exist class KL function σ1 and K function γc1, γo1, γc2, and γo2, such that(32)Ea≤max⁡σ1Ea0,t,γc1∘γo1f˙+γc2∘γo2g˙,where Ea=[eρ,eβ,f~,g~]. As t→∞, σ1→∞, it follows that(33)limt→∞⁡Ea≤γc1∘γo1f˙+γc2∘γo2g˙≤μ-1f∗θ1θ2λminK1λminK2+μ-1g∗θ1θ2λminK1λminK2,implying (10) with μ=|s|/s2+Δmax12. Note that |g˙| and |f˙| are bounded by g∗ and f∗. Then, the errors eρ, eβ, f~ and g~ are all bounded. Note that only uniform ultimate boundedness of closed-loop system can be achieved due to the existence of g˙ and f˙. If g˙=0 and f˙=0, the closed-loop system is asymptotical stable.


## 4. Collision-Free Target Tracking

### 4.1. Guidance Law Design

In previous section, a target tracking controller is developed for ASVs without using the velocity of the target. In this section, a collision-free target tracking controller is developed based on a RESO and an artificial potential function. To achieve a collision-free target tracking, a desired orientation is defined as follows:(34)θr=atan2yt-y-∂Va∂y,xt-x-∂Va∂x,and the angle tracking error is redefined as(35)eβ=θr-ψ-δ.The guidance law for collision-free target tracking based on RESO is designed as follows:(36)u=k5eρ+∂Va/∂eρeρ+∂Va/∂eρ2+Δ32+f^,r=k6eβeβ2+Δ42+g^,where k5 and k6 are positive constants; Δ3 and Δ4 are positive constants, which are employed to limit the maximum value of control laws.Substituting (14) and (36) into (11) results in(37)e˙ρ=-k5eρ+∂Va/∂eρeρ+∂Va/∂eρ2+Δ32-f~,e˙β=-k6eβeβ2+Δ42-g~.The ISS property of subsystem (37) is stated as follows.Lemma 5.
Subsystem (37), viewed as a system with the states being eρ+∂Va/∂eρ and eβ and the inputs being f~ and g~, is ISS.Proof.
Construct a Lyapunov function for system (37) as(38)V3=12eρ2+12eβ2+Va.
Taking the time derivative ofV3 along (37), (3), and (4), we have(39)V3˙=-k5eρ+∂Va/∂eρ2eρ+∂Va/∂eρ2+Δ32+f~eρ+∂Va∂eρ-k6eβ2eβ2+Δ42+g~eβ≤-λminK3E32E32+Δmax2⁡2+wE3,where E3=[eρ+∂Va/∂eρ, eβ]T, K3=diag{k5,k6}, and Δmax2=max{Δ3,Δ4}.
Noting that(40)E3E32+Δmax2⁡2≥f~+g~θ3λminK3≥wθ3λminK3renders(41)V3˙≤-λminK31-θ3E32E32+Δmax2⁡2,where 0<θ3<1. As a consequence, subsystem (37) is ISS. Since V˙3 is negative definite, then V3 is not increasing inside the detection region. Since(42)limz-za→r+⁡Va=∞,where z=[x,y]T and za=[xt,yt]T, then collision avoidance is guaranteed.


### 4.2. Stability Analysis

Finally, we analyze the cascade stability of subsystem (15) and subsystem (37) in a compact form:(43)Σ3:e˙ρ=-k5eρ+∂Va/∂eρeρ+∂Va/∂eρ2+Δ32-f~,e˙β=-k8eβeβ2+Δ42-g~,and(44)Σ4:f~˙=-k1f~-f˙,g~˙=-k2g~-g˙.The following theorem presents the stability of the cascade system consisting of subsystem (15) and subsystem (37).Theorem 6.
Consider the closed-loop guidance system consisting of follower ASV with kinematics (6), the target ASV with kinematics (5), and the guidance law (36) (13). If Assumption 1 is satisfied, the proposed guidance law can achieve the control objective described in Section 3.1. The closed-loop system cascaded by subsystem (15) and subsystem (37) is ISS. Besides,
(1) outside the detection range, the tracking errors(eρ,eβ) converge to a small neighborhood of the origin,
(2) inside the detection range, collision avoidance is guaranteed, i.e., for allt≥0,[x,y]T-[xt,yt]T-r≥ϵ4 for some constant ϵ4.Proof.
Part A. The cascade system consisting of subsystem Σ3 and subsystem Σ4 with the relative distance ρ (ρ>R) is ISS. The proof is the same as the proof of Theorem 4.
Part B. The cascade system consisting of subsystem Σ3 and subsystem Σ4 with the relative distance ρ (ρ<r) is ISS. It has been proved that subsystem Σ3 with states (eρ+∂Va/∂eρ,eβ) and exogenous inputs (f~,g~) and subsystem Σ4 with states (f~, g~) and exogenous inputs f˙ and g˙ are ISS. By Lemma C.4 [47], it follows that the cascade systems Σ3 and Σ4 with states (eρ+∂Vaρ/∂eρ,eβ,f~,g~) and exogenous inputs (f˙,g˙) are ISS. Note that the errors eρ+∂Vaρ/∂eρ, eβ, f~, and g~ are all bounded. Then, collision avoidance is guaranteed [x,y]T-[xt,yt]T≥rd for all t≥0, where rd=r+ϵ4 with ϵ4 being a constant.A parameter selection guideline is provided as follows. The parametersk1 and k2 determine the speed of observer. The parameters k3, k4, k5, and k6 can be chosen according to the desired response of the closed-loop system. The parameters Δ1, Δ2, Δ3, and Δ4 can be used to regulate the transient performance.


## 4.1. Guidance Law Design

In previous section, a target tracking controller is developed for ASVs without using the velocity of the target. In this section, a collision-free target tracking controller is developed based on a RESO and an artificial potential function. To achieve a collision-free target tracking, a desired orientation is defined as follows:(34)θr=atan2yt-y-∂Va∂y,xt-x-∂Va∂x,and the angle tracking error is redefined as(35)eβ=θr-ψ-δ.The guidance law for collision-free target tracking based on RESO is designed as follows:(36)u=k5eρ+∂Va/∂eρeρ+∂Va/∂eρ2+Δ32+f^,r=k6eβeβ2+Δ42+g^,where k5 and k6 are positive constants; Δ3 and Δ4 are positive constants, which are employed to limit the maximum value of control laws.Substituting (14) and (36) into (11) results in(37)e˙ρ=-k5eρ+∂Va/∂eρeρ+∂Va/∂eρ2+Δ32-f~,e˙β=-k6eβeβ2+Δ42-g~.The ISS property of subsystem (37) is stated as follows.Lemma 5.
Subsystem (37), viewed as a system with the states being eρ+∂Va/∂eρ and eβ and the inputs being f~ and g~, is ISS.Proof.
Construct a Lyapunov function for system (37) as(38)V3=12eρ2+12eβ2+Va.
Taking the time derivative ofV3 along (37), (3), and (4), we have(39)V3˙=-k5eρ+∂Va/∂eρ2eρ+∂Va/∂eρ2+Δ32+f~eρ+∂Va∂eρ-k6eβ2eβ2+Δ42+g~eβ≤-λminK3E32E32+Δmax2⁡2+wE3,where E3=[eρ+∂Va/∂eρ, eβ]T, K3=diag{k5,k6}, and Δmax2=max{Δ3,Δ4}.
Noting that(40)E3E32+Δmax2⁡2≥f~+g~θ3λminK3≥wθ3λminK3renders(41)V3˙≤-λminK31-θ3E32E32+Δmax2⁡2,where 0<θ3<1. As a consequence, subsystem (37) is ISS. Since V˙3 is negative definite, then V3 is not increasing inside the detection region. Since(42)limz-za→r+⁡Va=∞,where z=[x,y]T and za=[xt,yt]T, then collision avoidance is guaranteed.


## 4.2. Stability Analysis

Finally, we analyze the cascade stability of subsystem (15) and subsystem (37) in a compact form:(43)Σ3:e˙ρ=-k5eρ+∂Va/∂eρeρ+∂Va/∂eρ2+Δ32-f~,e˙β=-k8eβeβ2+Δ42-g~,and(44)Σ4:f~˙=-k1f~-f˙,g~˙=-k2g~-g˙.The following theorem presents the stability of the cascade system consisting of subsystem (15) and subsystem (37).Theorem 6.
Consider the closed-loop guidance system consisting of follower ASV with kinematics (6), the target ASV with kinematics (5), and the guidance law (36) (13). If Assumption 1 is satisfied, the proposed guidance law can achieve the control objective described in Section 3.1. The closed-loop system cascaded by subsystem (15) and subsystem (37) is ISS. Besides,
(1) outside the detection range, the tracking errors(eρ,eβ) converge to a small neighborhood of the origin,
(2) inside the detection range, collision avoidance is guaranteed, i.e., for allt≥0,[x,y]T-[xt,yt]T-r≥ϵ4 for some constant ϵ4.Proof.
Part A. The cascade system consisting of subsystem Σ3 and subsystem Σ4 with the relative distance ρ (ρ>R) is ISS. The proof is the same as the proof of Theorem 4.
Part B. The cascade system consisting of subsystem Σ3 and subsystem Σ4 with the relative distance ρ (ρ<r) is ISS. It has been proved that subsystem Σ3 with states (eρ+∂Va/∂eρ,eβ) and exogenous inputs (f~,g~) and subsystem Σ4 with states (f~, g~) and exogenous inputs f˙ and g˙ are ISS. By Lemma C.4 [47], it follows that the cascade systems Σ3 and Σ4 with states (eρ+∂Vaρ/∂eρ,eβ,f~,g~) and exogenous inputs (f˙,g˙) are ISS. Note that the errors eρ+∂Vaρ/∂eρ, eβ, f~, and g~ are all bounded. Then, collision avoidance is guaranteed [x,y]T-[xt,yt]T≥rd for all t≥0, where rd=r+ϵ4 with ϵ4 being a constant.A parameter selection guideline is provided as follows. The parametersk1 and k2 determine the speed of observer. The parameters k3, k4, k5, and k6 can be chosen according to the desired response of the closed-loop system. The parameters Δ1, Δ2, Δ3, and Δ4 can be used to regulate the transient performance.


## 5. Simulation Results

In this section, simulation results are given to show the performance of the proposed guidance law for collision-free target tracking. Two cases are presented:(1) the velocity of target is constant; (2) the velocity of target is time-varying. The vehicle tracks the target with relative distance ρd (ρd>r>Rt<40s, ρd<r<Rt>40s). The initial configurations of the follower and target vehicle are (x,y,ψ)(0)=(0,0,0) and (xt,yt,ψt)(0)=(5,5,π/4), respectively. The control parameters are chosen as follows: k1=5, k2=5, k3=0.1, k4=0.2, k5=0.1, k6=0.2, Δ1=0.5, Δ2=0.5, Δ3=0.5, Δ4=0.5, r=3m, R=5m, ρd=6m(t<40s), and ρd=2m(t>40s).


### 5.1. Target Tracking and Collision Avoidance with a Target of Constant Velocity

Consider the ASV model (6) controlled by the guidance law (22) with constant velocity ut=0.5m/s and rt=0rad/s. Simulation results are shown in Figures 2–5, and Figure 2 shows the trajectories of target and follower ASV. It reveals that the follower vehicle heads for the target in a short time while holding a desired distance although the target dynamics is uncertain. Figure 3 shows that the follower vehicle tracks the target with distance ρd=6m when the distance satisfies ρd>r>R. After 40s, the distance ρd is changed from 6m to 2m, and the tracking distance becomes R>ρd>r. It reveals that the proposed guidance law can achieve collision avoidance well. The output of RESO and the relative dynamics of target are compared in Figure 4. It shows that the RESO can compensate the uncertain target kinematics efficiently. Figure 5 shows the guidance signals of the proposed guidance law. It reveals that the guidance signals are all bounded.Figure 2
Trajectory of target and follower ASV.Figure 3
The distance between vehicle and target.Figure 4
Estimation performance of the RESOs.Figure 5
Guidance law.


### 5.2. Target Tracking and Collision Avoidance with a Target of Time-Varying Velocity

The guidance law (36) is employed to track a target with time-varying velocity ut=(2+0.1sin(2t))m/s and rt=0.5sin(2t)rad/s. Figures 6–9 show the simulation results. Figure 6 shows that the follower vehicle can track the target with given relative distance. The relative distance is shown in Figure 7. It implies that the follower vehicle tracks the target with given distance ρd when the distance is designed as ρd>r>R. After 40s, the distance ρd is changed from 6m to 2m. The tracking distance becomes R>ρd>r rather than the given distance 2m. It indicates that the proposed guidance law for target tracking extended to collision avoidance is effective with the time-varying velocity of target. The output of the RESO and relative dynamics of target are compared in Figure 8. It shows that the RESO can compensate the uncertain target dynamics efficiently. Figure 9 shows the guidance signals of the proposed guidance law. It observed that the guidance signals are all bounded.Figure 6
Trajectory of target and follower ASV.Figure 7
Target tracking errors.Figure 8
Estimation performance of the RESOs.Figure 9
Guidance law.


## 5.1. Target Tracking and Collision Avoidance with a Target of Constant Velocity

Consider the ASV model (6) controlled by the guidance law (22) with constant velocity ut=0.5m/s and rt=0rad/s. Simulation results are shown in Figures 2–5, and Figure 2 shows the trajectories of target and follower ASV. It reveals that the follower vehicle heads for the target in a short time while holding a desired distance although the target dynamics is uncertain. Figure 3 shows that the follower vehicle tracks the target with distance ρd=6m when the distance satisfies ρd>r>R. After 40s, the distance ρd is changed from 6m to 2m, and the tracking distance becomes R>ρd>r. It reveals that the proposed guidance law can achieve collision avoidance well. The output of RESO and the relative dynamics of target are compared in Figure 4. It shows that the RESO can compensate the uncertain target kinematics efficiently. Figure 5 shows the guidance signals of the proposed guidance law. It reveals that the guidance signals are all bounded.Figure 2
Trajectory of target and follower ASV.Figure 3
The distance between vehicle and target.Figure 4
Estimation performance of the RESOs.Figure 5
Guidance law.


## 5.2. Target Tracking and Collision Avoidance with a Target of Time-Varying Velocity

The guidance law (36) is employed to track a target with time-varying velocity ut=(2+0.1sin(2t))m/s and rt=0.5sin(2t)rad/s. Figures 6–9 show the simulation results. Figure 6 shows that the follower vehicle can track the target with given relative distance. The relative distance is shown in Figure 7. It implies that the follower vehicle tracks the target with given distance ρd when the distance is designed as ρd>r>R. After 40s, the distance ρd is changed from 6m to 2m. The tracking distance becomes R>ρd>r rather than the given distance 2m. It indicates that the proposed guidance law for target tracking extended to collision avoidance is effective with the time-varying velocity of target. The output of the RESO and relative dynamics of target are compared in Figure 8. It shows that the RESO can compensate the uncertain target dynamics efficiently. Figure 9 shows the guidance signals of the proposed guidance law. It observed that the guidance signals are all bounded.Figure 6
Trajectory of target and follower ASV.Figure 7
Target tracking errors.Figure 8
Estimation performance of the RESOs.Figure 9
Guidance law.


## 6. Conclusions

This paper considers the target tracking problem of an ASV in the presence of a maneuvering target where the velocity information of the target is totally unknown to the follower ASV. A reduced-order extended state observer is used to estimate the unknown relative dynamics. Then, an antidisturbance guidance law is developed based on the reduced-order extended state observer for target tracking. The stability of closed-loop target tracking guidance system is analyzed via cascade analysis. Finally, the result is extended to the case that collisions between the target and leader are avoided during tracking, and a collision-free target tracking guidance law is developed. Simulation results verify the effectiveness of the proposed guidance law for tracking a maneuvering target with arbitrary velocity and having the ability to avoid collision in the presence of a maneuvering target.



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*Source: 4154670-2018-12-13.xml*