Bounded Neural Network Control for Target Tracking of Underactuated Autonomous Surface Vehicles in the Presence of Uncertain Target Dynamics

Neural network-based dynamic target enclosing control for uncertain nonlinear multi-agent systems over signed networks

Neural networks have significant advantages in the estimation of uncertainty dynamics, which can afford highly accurate prediction outcomes and enhance control robustness. With this in mind, this study presents a neural network-based method to investigate the uncertain target enclosing control problem for multi-agent systems over signed networks. Firstly, a nominal target enclosing controller is constructed by adding the target information component into the classical bipartite consensus error, in which the multi-agent system can be grouped to enclose the target from opposite sides. Secondly, the uncertain dynamics of the target and matched/unmatched disturbances of agents are estimated to generate the feedforward control components by adopting the neural network approximation. Therefore, high-cost sensors are unnecessary for applications that require obtaining high-order information about a target, such as velocity and acceleration, while still ensuring accurate target-enclosing control. Additionally, the proposed target enclosing controller exhibits improved robustness in the presence of both matched and unmatched disturbances. To further demonstrate its effectiveness, numerical simulations are conducted.

Concurrent Learning-Based Adaptive Control of Underactuated Robotic Systems With Guaranteed Transient Performance for Both Actuated and Unactuated Motions

With the wide applications of underactuated robotic systems, more complex tasks and higher safety demands are put forward. However, it is still an open issue to utilize "fewer" control inputs to satisfy control accuracy and transient performance with theoretical and practical guarantee, especially for unactuated variables. To this end, for underactuated robotic systems, this article designs an adaptive tracking controller to realize exponential convergence results, rather than only asymptotic stability or boundedness; meanwhile, unactuated states exponentially converge to a small enough bound, which is adjustable by control gains. The maximum motion ranges and convergence speed of all variables both exhibit satisfactory performance with higher safety and efficiency. Here, a data-driven concurrent learning (CL) method is proposed to compensate for unknown dynamics/disturbances and improve the estimate accuracy of parameters/weights, without the need for persistency of excitation or linear parametrization (LP) conditions. Then, a disturbance judgment mechanism is utilized to eliminate the detrimental impacts of external disturbances. As far as we know, for general underactuated systems with uncertainties/disturbances, it is the first time to theoretically and practically ensure transient performance and exponential convergence speed for unactuated states, and simultaneously obtain the exponential tracking result of actuated motions. Both theoretical analysis and hardware experiment results illustrate the effectiveness of the designed controller.

Neural Network Control of Underactuated Surface Vehicles With Prescribed Trajectory Tracking Performance

This article is concerned with the fast and accurate trajectory tracking control problem for a sort of underactuated surface vehicle under model uncertainties and environmental disturbances. A novel neural networks (NNs)-based prescribed performance control strategy is proposed to solve the problem. In the control design, a new type of performance function is constructed which provides a way to predefine the settling time and accuracy, straightforward. Then, a pair of barrier functions are employed to combat not only the position error but also the virtual control input. This evades the possible singularity or discontinuity of the control solution. Next, an initialization technique is exploited, removing the requirement for the initial condition of the control system. Finally, two NNs are employed to deal with the unknown ship nonlinearities. The performance analysis not only demonstrates the effectiveness of the proposed approach but also reveals its robustness against disturbances and unknown reference trajectory derivatives. There is, thus, no need to acquire such knowledge or employ specialized tools to handle disturbances. The theoretical findings are illustrated by a simulation study.

Design and Experimental Validation of Deep Reinforcement Learning-Based Fast Trajectory Planning and Control for Mobile Robot in Unknown Environment

This article is concerned with the problem of planning optimal maneuver trajectories and guiding the mobile robot toward target positions in uncertain environments for exploration purposes. A hierarchical deep learning-based control framework is proposed which consists of an upper level motion planning layer and a lower level waypoint tracking layer. In the motion planning phase, a recurrent deep neural network (RDNN)-based algorithm is adopted to predict the optimal maneuver profiles for the mobile robot. This approach is built upon a recently proposed idea of using deep neural networks (DNNs) to approximate the optimal motion trajectories, which has been validated that a fast approximation performance can be achieved. To further enhance the network prediction performance, a recurrent network model capable of fully exploiting the inherent relationship between preoptimized system state and control pairs is advocated. In the lower level, a deep reinforcement learning (DRL)-based collision-free control algorithm is established to achieve the waypoint tracking task in an uncertain environment (e.g., the existence of unexpected obstacles). Since this approach allows the control policy to directly learn from human demonstration data, the time required by the training process can be significantly reduced. Moreover, a noisy prioritized experience replay (PER) algorithm is proposed to improve the exploring rate of control policy. The effectiveness of applying the proposed deep learning-based control is validated by executing a number of simulation and experimental case studies. The simulation result shows that the proposed DRL method outperforms the vanilla PER algorithm in terms of training speed. Experimental videos are also uploaded, and the corresponding results confirm that the proposed strategy is able to fulfill the autonomous exploration mission with improved motion planning performance, enhanced collision avoidance ability, and less training time.

Collision-free automatic berthing of maritime autonomous surface ships via safety-certified active disturbance rejection control

This paper addresses the automatic berthing of a maritime autonomous surface ship operating in a confined water environment subject to static obstacles, dynamic obstacles, thruster constraints, and space constraints due to shorelines. A safety-certified active disturbance rejection control (ADRC) method is proposed for achieving the automatic berthing task of an MASS in the presence of model uncertainties and ocean disturbances. An extended state observer (ESO) based on a second-order robust exact differentiator (RED) is employed to estimate an extended state vector consisting of internal model uncertainties and external ocean disturbances. With the aid of the RED-based ESO, a nominal ADRC law is designed to achieve the position and heading stabilization. To avoid collisions with static obstacles, dynamic obstacles, and shorelines, input-to-state safe high-order control barrier functions are used to guarantee safety. Optimized control signals are obtained based on a constrained quadratic programming (QP) problem within safety constraints. In order to translate the control signals into the individual thruster command, a constrained QP problem is further used to search for optimized commands in real time. It is proven that the closed-loop automatic berthing system is input-to-state stable. By using the proposed method, the MASS is able to reach the desired position and heading with collision avoidance. Simulation results verify the effectiveness of the proposed safety-certified ADRC method for automatic berthing.

Event-Triggered Finite-Time Formation Control of Underactuated Multiple ASVs with Prescribed Performance and Collision Avoidance

In this paper, an event-triggered finite-time controller is proposed for solving the formation control problems of underactuated multiple autonomous surface vessels (ASVs), including asymmetric mass matrix, collision avoidance, maintaining communication distances and prescribed performance. First, to not only avoid collisions between the follower and leader but also maintain an effective communication distance, a desired tracking distance is designed to be maintained. Second, an improved barrier Lyapunov function (BLF) is proposed to implement the tracking error constraint. In addition, the relative threshold event-triggering strategy effectively solves the communication pressure problem and greatly saves communication resources. Finally, based on coordinate transformation, line of sight (LOS) and dynamic surface control (DSC), a comprehensive finite-time formation control method is proposed to avoid collisions and maintain communication distance. All the signals of the proposed control system can be stabilized in finite time (PFS). The numerical simulation results verify the effectiveness of the proposed control system.

IBLF-Based Adaptive Neural Control of State-Constrained Uncertain Stochastic Nonlinear Systems

In this article, the adaptive neural backstepping control approaches are designed for uncertain stochastic nonlinear systems with full-state constraints. According to the symmetry of constraint boundary, two cases of controlled systems subject to symmetric and asymmetric constraints are studied, respectively. Then, corresponding adaptive neural controllers are developed by virtue of backstepping design procedure and the learning ability of radial basis function neural network (RBFNN). It is worth mentioning that the integral Barrier Lyapunov function (IBLF), as an effective tool, is first applied to solve the above constraint problems. As a result, the state constraints are avoided from being transformed into error constraints via the proposed schemes. In addition, based on Lyapunov stability analysis, it is demonstrated that the errors can converge to a small neighborhood of zero, the full states do not exceed the given constraint bounds, and all signals in the closed-loop systems are semiglobally uniformly ultimately bounded (SGUUB) in probability. Finally, the numerical simulation results are provided to exhibit the effectiveness of the proposed control approaches.

Spherical Formation Tracking Control of Nonlinear Second-Order Agents With Adaptive Neural Flow Estimate

This article addresses the spherical formation tracking control problem of nonlinear second-order vehicles moving in flowfields under both undirected networks and directed, strongly connected networks. Different from the previous adaptive estimate of the time-invariant parameters of flowfields, the flowfields under our consideration are spatial and absolutely unknown dynamics. Adaptive neural networks (ANNs) with the novel cooperative adaptive algorithms are proposed to approximate the flowfield acting on the channel of each vehicle's velocity (i.e., the mismatched flowfield) and the flowfield pushing the acceleration (i.e., the matched flowfield), respectively. For the purpose of avoiding the complex derivation derived from backstepping, the novel first-order filters are generated by dynamic surface based on barrier functions and relative positions of neighbors. The proposed control algorithms and adaptive upgrade law are fully distributed without using any global information of the graph. The uniform boundedness is analyzed in the Lyapunov sense. Simulation results are given to verify the theoretical analysis.

Network-Based Line-of-Sight Path Tracking of Underactuated Unmanned Surface Vehicles With Experiment Results

This article deals with the problem of network-based path-tracking control of an underactuated unmanned surface vehicle subject to model uncertainties and unknown disturbances over a wireless network. A two-level network-based control architecture is proposed, including a local inner loop and a remote outer loop. In the remote outer loop, an event-triggered line-of-sight guidance law is designed to achieve path tracking while reducing the network burden for the remote control at the kinematic level. In the local inner loop, an extended state observer is employed to estimate the unknown disturbances due to the model uncertainties and environmental disturbances. Based on the estimated information from the extended state observer, an event-triggered anti-disturbance control law is developed to reduce the execution rate of actuators at the kinetic level. The stability of the closed-loop path-tracking system is proved based on the input-to-state stability and cascade stability theory. The effectiveness of the proposed network-based method for path tracking of the USV is verified via experiments.

Tip Position Control and Vibration Suppression of a Planar Two-Link Rigid-Flexible Underactuated Manipulator

When a flexible link manipulator lacks a joint motor, how to use the remaining motors to achieve the control objective is a challenge, and the research in this direction is limited. This article presents a tip position control and vibration suppression approach for a planar two-link rigid-flexible (TLRF) underactuated manipulator with a passive first joint. First, we establish a dynamic model of the system by using the assumed mode method (AMM) and the Lagrangian modeling method. Then, we obtain the dynamic coupling relationship of the two links based on the dynamic model. According to this dynamic coupling relationship, we find that the passive rigid link can be controlled indirectly by controlling the active flexible link. Thus, we calculate the target angles of the two links by using the inverse kinematic method and design a controller for the active flexible link to stabilize it at its target angle and to suppress its vibration. Next, we optimize the parameters of this controller by using the genetic algorithm (GA). GA helps us simultaneously stabilize the passive rigid link at its target angle while realizing the control objective of the active flexible link. The simulation results demonstrate the effectiveness of the proposed control approach.

Anti-Saturation-Based Adaptive Sliding-Mode Control for Active Suspension Systems With Time-Varying Vertical Displacement and Speed Constraints

In this article, an adaptive sliding-mode control scheme is developed for a class of uncertain quarter vehicle active suspension systems with time-varying vertical displacement and speed constraints, in which the input saturation is considered. The integral terminal SMC is adopted to improve convergence accuracy and avoid singular problems. In addition, neural networks are used to model unknown terms in the system and the backstepping technique is taken into account to design the actual controller. To guarantee that the time-varying state constraints are not violated, the corresponding Barrier Lyapunov functions are constructed. At the same time, a continuous differentiable asymmetric saturation model is developed to improve the stability of the system. Then, the Lyapunov stability theory is used to verify that all signals of the resulting system are semi globally uniformly ultimately bounded, time-varying state constraints are not violated, and error variables can converge to the small neighborhood of 0. Finally, results of the simulation of the designed control strategy are given to further prove the effectiveness.

Predictor-Based Neural Dynamic Surface Control for Bipartite Tracking of a Class of Nonlinear Multiagent Systems

This article is concerned with bipartite tracking for a class of nonlinear multiagent systems under a signed directed graph, where the followers are with unknown virtual control gains. In the predictor-based neural dynamic surface control (NDSC) framework, a bipartite tracking control strategy is proposed by the introduction of predictors and the minimal number of learning parameters (MNLPs) technology along with the graph theory. Different from the traditional NDSC, the predictor-based NDSC utilizes prediction errors to update the neural network for improving system transient performance. The MNLPs technology is employed to avoid the problem of "explosion of learning parameters". It is proved that all closed-loop signals steered by the proposed control strategy are bounded, and the system achieves bipartite consensus. Simulation results verify the efficiency and effectiveness of the strategy.

Fixed-time consensus tracking control with connectivity preservation for strict-feedback nonlinear multi-agent systems

This paper deliberates fixed-time consensus tracking control for strict-feedback nonlinear multi-agent systems with limited communication/sensing range constraints. First, both potential function and coordinate error transformation surface are designed to make the constraints implicit. Next, based on the synthesis of neural network and adaptive technology, the fixed-time virtual variable is proposed without the upper bounds of estimation errors and disturbances. Then, a fixed-time distributed consensus tracking protocol is designed under backstepping method with a fixed-time differentiator to avoid singularity. Lyapunov stability analysis demonstrates that the closed-loop system under the designed control strategy can accomplish the convergence within fixed time, simultaneously connectivity preservation can be guaranteed. Finally, numerical emulation corroborates the availability of the designed control strategy.

Path following Control of an Underactuated Catamaran for Recovery Maneuvers

This paper focuses on the autonomous recovery maneuvers of an unknown underactuated practical catamaran, which returns to its initial position corresponding to the man overboard (MOB) by simply adjusting the rate of turn. This paper investigates the completion of model-based path following control for not only the traditional Williamson turn, but also complex recovery routes under time-varying disturbances. The main difficulty of model-based path following control for predicting the hydrodynamic derivatives of a practical catamaran was solved by the approximated calculation of a diagonal matrix. The second key problem of differential calculation for an underactuated model in the case of complex reference trajectories under severe disturbances was investigated. Even though this paper employs a diagonal matrix with unknown nonlinear terms, the experimental test using a small craft with payloads by remote control demonstrated the sway force per yaw moment in turning cases. Adaptive backstepping mechanisms with unknown parameters were proven by the Lyapunov theory as well as the passive-boundedness of the sway dynamics, guaranteeing the stability of sway motion in the case of unavailable sway control. The effectiveness of the algorithms of the guiding concept and error dynamics is demonstrated by the numerical simulations.

TD-Based Adaptive Output Feedback Control of Ship Heading with Stochastic Noise and Unknown Actuator Dead-Zone Input

To meet the demand of ship control, a new heading control tactic is explored using switching theory. Different from the previous results, the stochastic noise and switched control are considered in the Norrbin nonlinear mathematical model concurrently to discuss ship heading control. Then, for the resulting systems, the adaptive control issue is addressed, while the dead-zone input is embedded and unknown. By establishing switched state observers for the corresponding subsystems, the conservatism of the common state observer can be reduced greatly, and the analysis can be achieved under the switching signal satisfying the average dwell time (ADT). The “explosion of terms” problem occurring in the backstepping technique is well remedied via an innovative tracking differentiator (TD) technology, which is an innovation in itself. According to the theory proof, under the developed control tactic, the resulting signals are then to be bounded in probability, with the tracking goal being achieved well. The theoretical design result was analyzed, and the corresponding validity is given through simulation experiments.

Robust Adaptive Self-Structuring Neural Network Bounded Target Tracking Control of Underactuated Surface Vessels

This paper studies the target-tracking problem of underactuated surface vessels with model uncertainties and external unknown disturbances. A composite robust adaptive self-structuring neural-network-bounded controller is proposed to improve system performance and avoid input saturation. An extended state observer is proposed to estimate the uncertain nonlinear term, including the unknown velocity of the tracking target, when only the measurement values of the line-of-sight range and angle can be obtained. An adaptive self-structuring neural network is developed to approximate model uncertainties and external unknown disturbances, which can effectively optimize the structure of the neural network to reduce the computational burden by adjusting the number of neurons online. The input-to-state stability of the total closed-loop system is analyzed by the cascade stability theorem. The simulation results verify the effectiveness of the proposed method.

Visual Navigation and Landing Control of an Unmanned Aerial Vehicle on a Moving Autonomous Surface Vehicle via Adaptive Learning

This article presents a visual navigation and landing control paradigm for an unmanned aerial vehicle (UAV) to land on a moving autonomous surface vehicle (ASV). Therein, an adaptive learning navigation rule with a multilayer nested guidance is designed to pinpoint the position of the ASV and to guide and control the UAV to fulfill horizontal tracking and vertical descending in a narrow landing region of the ASV by means of merely relative position feedback. To ensure the feasibility of the proposed control law, asymptotical stability conditions are derived based on Lyapunov stability theory. Landing experimental results are reported for a UAV-ASV system consisting of an M-100 UAV and a self-developed three-meters-long HUSTER-30 ASV on a lake to substantiate the efficacy of the proposed landing control method.

Distributed Path Following of Multiple Under-Actuated Autonomous Surface Vehicles Based on Data-Driven Neural Predictors via Integral Concurrent Learning

This article addresses the problem of distributed path following of multiple under-actuated autonomous surface vehicles (ASVs) with completely unknown kinetic models. An integrated distributed guidance and learning control architecture is proposed for achieving a time-varying formation. Specifically, a robust distributed guidance law at the kinematic level is developed based on a consensus approach, a path-following mechanism, and an extended state observer. At the kinetic level, a model-free kinetic control law based on data-driven neural predictors via integral concurrent learning is designed such that the kinetic model can be learned by using recorded data. The advantage of the proposed method is two-folds. First, the proposed formation controllers are able to achieve various time-varying formations without using the velocities of neighboring vehicles. Second, the proposed control law is model-free without any parameter information on kinetic models. Simulation results substantiate the effectiveness of the proposed robust distributed guidance and model-free control laws for multiple under-actuated ASVs with fully unknown kinetic models.

Data-Driven Adaptive Disturbance Observers for Model-Free Trajectory Tracking Control of Maritime Autonomous Surface Ships

In this article, we address the disturbance/ uncertainty estimation of maritime autonomous surface ships (MASSs) with unknown internal dynamics, unknown external disturbances, and unknown input gains. In contrast to existing disturbance observers where some prior knowledge on kinetic model parameters such as the control input gains is available in advance, reduced- and full-order data-driven adaptive disturbance observers (DADOs) are proposed for estimating unknown input gains, as well as total disturbance composed of unknown internal dynamics and external disturbances. An advantage of the proposed DADOs is that the total disturbance and input gains can be simultaneously estimated with guaranteed convergence via data-driven adaption. We apply the proposed full-order DADO for the trajectory tracking control of an MASS without kinetic modeling and present a model-free trajectory tracking control law for the ship based on the DADO and a backstepping technique. We report the simulation results to substantiate the efficacy of the proposed DADO approach to model-free trajectory tracking control of an autonomous surface ship without knowing its dynamics.

Swarm velocity guidance based distributed finite-time coordinated path-following for uncertain under-actuated autonomous surface vehicles

This article mainly researches the problem of distributed finite-time coordinated path-following for under-actuated autonomous surface vehicles (ASVs) within a network swarm. Each vehicle in swarm system suffers from velocity restrictions and multiple uncertainties including parameter perturbations and time-varying environment disturbances. Based on the constructed bionic swarm pattern and potential function, the swarm velocity guidance (SVG) with self-organization and collision avoidance is developed to guide ASV surge velocities and heading angles simultaneously. A distributed observer by adding correction terms to the vehicle model is involved to identify the lumped uncertainties, and the estimations are utilized as feed-forward compensation to weaken the uncertainty impact, thus achieving high tracking precision. By using asymmetric barrier Lyapunov function, the uncertainty observer based distributed surge and heading kinetics controllers under physical restrictions are devised to guarantee that the guided signals generated by SVG are tracked within finite time. Through simulation studies of swarm path-following, it is demonstrated that the designed control approach is feasible and efficient for multiple uncertain under-actuated ASVs.

CED: A Distance for Complex Mass Functions

Evidence theory is an effective methodology for modeling and processing uncertainty that has been widely applied in various fields. In evidence theory, a number of distance measures have been presented, which play an important role in representing the degree of difference between pieces of evidence. However, the existing evidential distances focus on traditional basic belief assignments (BBAs) modeled in terms of real numbers and are not compatible with complex BBAs (CBBAs) extended to the complex plane. Therefore, in this article, a generalized evidential distance measure called the complex evidential distance (CED) is proposed, which can measure the difference or dissimilarity between CBBAs in complex evidence theory. This is the first work to consider distance measures for CBBAs, and it provides a promising way to measure the differences between pieces of evidence in a more general framework of complex plane space. Furthermore, the CED is a strict distance metric with the properties of nonnegativity, nondegeneracy, symmetry, and triangle inequality that satisfies the axioms of a distance. In particular, when the CBBAs degenerate into classical BBAs, the CED will degenerate into Jousselme et al.'s distance. Therefore, the proposed CED is a generalization of the traditional evidential distance, but it has a greater ability to measure the difference or dissimilarity between pieces of evidence. Finally, a decision-making algorithm for pattern recognition is devised based on the CED and is applied to a medical diagnosis problem to illustrate its practicability.

Cascaded Correlation Refinement for Robust Deep Tracking

Recent deep trackers have shown superior performance in visual tracking. In this article, we propose a cascaded correlation refinement approach to facilitate the robustness of deep tracking. The core idea is to address accurate target localization and reliable model update in a collaborative way. To this end, our approach cascades multiple stages of correlation refinement to progressively refine target localization. Thus, the localized object could be used to learn an accurate on-the-fly model for improving the reliability of model update. Meanwhile, we introduce an explicit measure to identify the tracking failure and then leverage a simple yet effective look-back scheme to adaptively incorporate the initial model and on-the-fly model to update the tracking model. As a result, the tracking model can be used to localize the target more accurately. Extensive experiments on OTB2013, OTB2015, VOT2016, VOT2018, UAV123, and GOT-10k demonstrate that the proposed tracker achieves the best robustness against the state of the arts.

Bounded neural adaptive formation control of multiple underactuated AUVs under uncertain dynamics

This paper studies the leader-following formation control problem of multiple underactuated autonomous underwater vehicles (AUVs) under uncertain dynamics and limited control torques. A multi-layer neural network-based estimation model is designed to handle the unknown follower dynamics. The backstepping approach, a neural estimation model, as well as a saturation function, are employed to propose a bounded formation control law. Then, a Lyapunov-based stability analysis ensures a maximum bound for all the closed-loop system variables and guarantees that the formation errors between vehicles ultimately converge to a bounded compact set. The outstanding properties of the designed controller are highlighted as follows. First, only the leader position and given formation are required without any leader velocity information requirement. Second, update laws of the neural network weight are extracted using the estimation errors instead of tracking ones, which can effectively enhance the transient characteristics of the control system. Third, the control torques are bounded within predefined bounds. At the end, extensive simulations are given for a number of AUVs to verify the efficiency of the presented formation control scheme.

Swarm Control for Self-Organized System With Fixed and Switching Topology

In this article, we propose the swarm control for a self-organized system with fixed and switching topology, which can realize aggregation, dispersion, or switching formation when swarm moves. The self-organized system can automatically construct the communication topology for intelligent units in swarm. Swarm control can realize aggregation and dispersion of intelligent units based on its communication topology when swarm moves. The proposed swarm control, in which distances between the related intelligent units are time varying, is different from traditional swarm consensus or swarm formation maintenance. To design swarm control, we define the normalization adjacency matrix and normalization degree matrix based on communication topology. The communication topology is automatically generated based on relation-invariable persistent formation. Depending on whether the communication topology changes or not, the swarm control can be classified as fixed topology and switching topology. Then, the swarm control with fixed and switching topology is designed and analyzed, respectively. The swarm control can realize stability asymptotically when topology is fixed and realize stability in finite time when topology is switched. The simulation results show that the proposed approaches are effective.

Cooperative Path Following Ring-Networked Under-Actuated Autonomous Surface Vehicles: Algorithms and Experimental Results

This paper addresses the cooperative path following the problem of ring-networked under-actuated autonomous surface vehicles on a closed curve. A cooperative guidance law is proposed at the kinematic level such that a symmetric formation pattern is achieved. Specifically, individual guidance laws of surge speed and angular rate are developed by using a backstepping technique and a line-of-sight guidance method. Then, a coordination design is proposed to update the path variables under a ring-networked topology. The equilibrium point of the closed-loop system has been proven to be globally asymptotically stable. The result is extended to the cooperative path following the lack of sharing of a global reference velocity, and a distributed observer is designed to recover the reference velocity to each vehicle. Moreover, the cooperative path following the presence of an unknown sideslip is considered, and an extended state observer is developed to compensate for the effect of the unknown sideslip. Both simulation and experimental results are provided to illustrate the effectiveness of the proposed cooperative guidance law for the path following over a closed curve.