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

This paper is concernedwith the target tracking problem of an autonomous surface vehicle in the presence of amaneuvering 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.

Various control methods have been developed for the target tracking of marine vehicles [2,[27][28][29][30][31][32][33][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][28][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 singlehidden-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][28][29].
Extended state observer as a key part of active disturbance rejection control method was proposed by Han [37][38][39].It is a powerful tool to deal with the nonlinear systems in 2 Complexity 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 collisionfree 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][28][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][28][29][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: Section 2 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.

Preliminaries and Problem Formulation
. .Collision Avoidance.In order to assure collision-free target tracking, the following collision avoidance potential functions are introduced [35]: where  >  > 0,  is the detection region,  is the radius of the avoidance, and  is the distance between the target and follower vehicles defined as and   ,   are positions of a target and ,  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   will affect the surge velocity when  is inside the detection region.
Taking the partial derivative of the potential function   with respect to , we can obtain [36] and the partial derivative of function   with respect to  is . .Vehicle Kinematics.The kinematics of an ASV can be expressed by using an earth-fixed frame {} and a body-fixed frame {} as shown in Figure 1.Let (  ,   ,   ) and (, , ) be the position and orientation of the target and follower, respectively.  ,   ,   denote the surge velocity, sway velocity, and angular rate of the target vehicle; , ,  represent the surge velocity, sway velocity, and angular rate of the follower vehicle, respectively.The kinematics of the target ASV is and the kinematics of follower ASV is From Figure 1, the line-of-sight range and angle between the target and the follower are defined as 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.

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.
. .Relative Dynamics.Two target tracking errors are defined as follows: where   is a desired range and  = atan2(, ) is a sideslip angle.Taking the time derivative of   and   in (8) and using ( 5) and ( 6), we have The control objective of target tracking of ASVs in the presence of unknown target kinematics is to design a surge velocity  and yaw rate  such that lim for some small constants  1 and  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: where Since   ,   , and   of the target are not available, (⋅) and (⋅) are unknown.To address it, a reduced-order extended state observer is proposed as follows [43]: where  1 ,  Since the velocities and accelerations of the target and follower ASVs are naturally bounded, Assumption 1 is reasonable.
The estimation errors are defined as follows: Taking the derivative of ( 14) along (13), we have The stability of subsystem ( 15) is stated as follows.
Lemma 2. Subsystem ( ), viewed as a system with the states being f and g and the inputs being ḟ and ġ , is input-to-state stability (ISS).
Proof.Construct the following Lyapunov function: Taking the time derivative of  1 along (15) results in where renders where 0 < where   is a class KL function,  1 ,  2 are the class K function, and . .Guidance Law Design.Based on the estimated terms f and ĝ from the RESO, an antidisturbance guidance law is proposed as follows: where  3 and  4 are positive constants; Δ 1 and Δ 2 are positive constants, which are used to limit the maximum value of control laws.Substituting ( 14) and ( 22) into (11) results in The ISS property of subsystem ( 23) is stated as follows.
Lemma 3. Subsystem ( ), viewed as a system with the states being   and   and the inputs being f and g, is ISS.
Proof.Construct the following Lyapunov function: Taking the time derivative of  2 along (23) results in where renders where 0 <  2 < 1.As a consequence, subsystem (23) where   is a class KL function,  1 and  2 are the class K functions, and with  = ||/√ 2 + Δ 2 max 1 .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.

. . 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 and The stability of the cascade of subsystem Σ 1 and subsystem Σ 2 is given by the following theorem.Theorem 4.Under Assumption , the closed-loop system cascaded by subsystem ( ) and subsystem ( ) is ISS, and the target tracking errors converge to a small neighborhood of the origin.
Proof.Subsystem Σ 1 with states (  ,   ) and exogenous inputs ( f, g) and subsystem Σ 2 with states ( f, g) and exogenous inputs ḟ and ġ are ISS.By Lemma C.4 in [47] where implying (10) with are bounded by  * and  * .Then, the errors   ,   , f and g are all bounded.Note that only uniform ultimate boundedness of closed-loop system can be achieved due to the existence of ġ and ḟ .If ġ = 0 and ḟ = 0, the closed-loop system is asymptotical stable.

Collision-Free Target Tracking
. .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: and the angle tracking error is redefined as The guidance law for collision-free target tracking based on RESO is designed as follows: where  5 and  6 are positive constants; Δ 3 and Δ 4 are positive constants, which are employed to limit the maximum value of control laws.Substituting ( 14) and ( 36) into (11) results in The ISS property of subsystem ( 37) is stated as follows.
Lemma 5. Subsystem ( ), viewed as a system with the states being   +   /  and   and the inputs being f and g, is ISS.
. .Stability Analysis.Finally, we analyze the cascade stability of subsystem (15) and subsystem (37) in a compact form: and 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 ( ), the target ASV with kinematics ( ), and the guidance law ( ) ( ).If Assumption is satisfied, the proposed guidance law can achieve the control objective described in Section . .e closed-loop system cascaded by subsystem ( ) and subsystem ( ) is ISS.Besides, ( ) outside the detection range, the tracking errors (  ,   ) converge to a small neighborhood of the origin, ( ) inside the detection range, collision avoidance is guaranteed, i.e., for all  ≥ 0, ‖[, ]  − [ t ,   ]  ‖ −  ≥  4 for some constant  4 .
Proof.Part A. The cascade system consisting of subsystem Σ 3 and subsystem Σ 4 with the relative distance  ( > ) is ISS.The proof is the same as the proof of Theorem 4.
Part B. The cascade system consisting of subsystem Σ 3 and subsystem Σ 4 with the relative distance  ( < ) is ISS.It has been proved that subsystem Σ 3 with states (  +   /  ,   ) and exogenous inputs ( f, g) and subsystem Σ 4 with states ( f, g) and exogenous inputs ḟ and ġ are ISS.By Lemma C.4 [47] A parameter selection guideline is provided as follows.The parameters  1 and  2 determine the speed of observer.The parameters  3 ,  4 ,  5 , and  6 can be chosen according to the desired response of the closed-loop system.The parameters Δ 1 , Δ 2 , Δ 3 , and Δ 4 can be used to regulate the transient performance.

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: . .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   = 0.5/ and   = 0/.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   = 6 when the distance satisfies   >  > .After 40, the distance   is changed from 6 to 2, and the tracking distance becomes  >   > .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   to track a target with time-varying velocity   = (2 + 0.1sin(2))/ and   = 0.5sin(2)/.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   when the distance is designed as   >  > .After 40, the distance   is changed from 6 to 2.The tracking distance becomes  >   >  rather than the given distance 2.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.

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.

Figure 1 :
Figure 1: A geometrical illustration for tracking a target by an ASV.

Figure 2 :
Figure 2: Trajectory of target and follower ASV.