Safety-critical Obstacle Avoidance Control of Autonomous Surface Vehicles with Uncertainties and Disturbances

Authors: gege  dong
ying  hao
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-14 13:14:38

Abstract: This paper proposes a safety-critical obstacle avoidance control approach for autonomous surface vehicles (ASVs) with disturbances and uncertainties. The existing exponential control barrier functions (ECBF) are extended to handle unknown disturbances, leading to the development of input-to-state safe exponential control barrier functions (ISSf-ECBFs). An extended state observer is used to estimate unknown external marine disturbances and internal model uncertainties, based on which an anti-disturbance controller is designed. Based on the proposed ISSf-ECBFs,  a quadratic programming problem is formulated to determine the optimal control input. It is proven that the closed-loop system is input-to-state safe and the errors of the closed-loop system are uniformly ultimately  bounded. Simulations validate the effectiveness of the proposed control strategy.

Main paper: 
 i. introduction: 
 Autonomous Surface Vehicles (ASVs) are gaining attention for their ability to enhance maritime operations [1] - [3] . With advanced sensors and navigation systems, ASVs can navigate complex environments and perform diverse tasks [4] . They are increasingly utilized in search and rescue, fisheries management, hydrographic surveying, and offshore energy, making them a focal point for researchers in ASV control [5] - [7] .ASVs navigating in dynamic marine environments face numerous challenges, primarily internal model uncertainties and external disturbances [8] . Internal uncertainties arise from modeling inaccuracies, parameter variations, and sensor noise. Additionally, ASVs must navigate unpredictable ocean conditions, such as waves, currents, and winds. These factors can adversely affect the performance of control strategy. To address this challenge, researchers have proposed various methods to enhance the robustness of the system, such as sliding mode control [9] , adaptive control, and neural network control [10] .The Extended State Observer (ESO) can estimate disturbances in real-time and dynamically adjust the control strategy. By treating internal model uncertainties and external disturbances as lumped disturbances for estimation, the reliance on the model can be reduced, thereby enhancing the robustness of the system.In complex maritime environments, ASVs face significant threats from various obstacles, including vessels, islands, and reefs [11] . To mitigate these risks, researchers have proposed several obstacle avoidance strategies, such as the artificial potential method [12] , the velocity obstacle method [13] , and the dynamic window approach [14] . Control barrier functions (CBFs), introduced in [15] , have proven effective in ensuring real-time safety. In [16] , the nominal controller was modified to formally adhere to safety constraints for successful obstacle avoidance. However, the control strategy in [16] did not account for model uncertainties or disturbances. To address this, [17] introduced input-to-state safe control barrier functions (ISSf-CBFs). Furthermore, [18] proposed a framework to ensure safety for uncertain nonlinear systems with structured parametric uncertainty. In [19] , a collision avoidance strategy for ASVs was proposed using ISSf-CBFs. However, these functions have a relative degree of one, limiting their use in higher-order systems. To address this, [20] introduced exponential control barrier functions (ECBFs). [21] further explored ISSf-ECBFs under known perturbation bounds, but measuring such disturbances is challenging. Therefore, a safety-critical controller based on ISSf-ECBFs is crucial for ASVs dealing with unknown model uncertainties and external disturbances.This paper presents a safety-critical control strategy for Autonomous Surface Vehicles (ASVs) that accounts for external marine disturbances and internal model uncertainties. The key contributions are as follows:1) While the existing method [21] constructs safety constraints only under known disturbances or their upper bounds, this paper extends the results of input-to-state safe control barrier functions (ISSf-ECBFs) to develop safety constraints for unknown disturbances.2) Unlike previous work [12] , [22] - [24] , this paper formulates a safety-critical controller based on ISSf-ECBFs by constructing a quadratic programming problem to facilitate collision avoidance with obstacles.The structure of the paper includes the following sections: The preliminaries and problem statement are covered in Section II. Section III gives the safety-critical controller design and Section IV is stability and safety analysis. Simulations are carried out in Section V. Section VI summarizes this article.In this paper, the notation ∥•∥ denotes the 2-norm of a vector, and R represents the set of real numbers. The symbols λ min (•) and λ max (•) indicate the smallest and largest eigenvalues of a symmetric matrix, respectively.Let β(r) be a scalar continuous function defined for r ∈[-b, a). If a = ∞, b = 0, and β(r) → ∞ as r → ∞, then β(r) belongs to class K ∞ . If a, b = ∞, β(r) → ∞ as r → ∞, and β(r) → -∞ as r → -∞, it represents an extended class K ∞ , denoted as K ∞,e .Consider the following systemEQUATIONwhere x(t) ∈ R n denotes state and u ∈ R m denotes control input. The term d w denotes bounded disturbances. The function f (x) ∈ R n and g(x) ∈ R n×m are locally Lipschitz continuous.Definition 1. [25] The set C ∈ R n is described asEQUATIONwhere h(•) ∈ R n → R represents a continuously differentiable function, and C is referred to as the safe set. If for all x 0 ∈ C it holds that x(t) ∈ C for every t ∈ I(x 0 ), then the set C is considered forward invariant. Consequently, the system described by (1) with d w (t) = 0 can be deemed safe on C.Definition 2. The relative degree of S(x) : R n → R with respect to the system (1) refers to the number of derivatives required along the dynamics of (1) before the control input u explicitly appears.[17] For system (1), an extended set C d ⊃ C is expressed as followsC d ≜ {x ∈ R n | S(x) + β d (||d w (t)|| ∞ ) ≥ 0} ∂C d ≜ {x ∈ R n | S(x) + β d (||d w (t)|| ∞ ) = 0} Int(C d ) ≜ {x ∈ R n | S(x) + β d (||d w (t)|| ∞ ) > 0} (3)where ||d w || ∞ ≤ dw , a positive constant, and S(x) is a continuous function, with (ISSf-ECBF [17] ) Considering the sets C d defined by (3), S(x), which has a relative degree ρ > 1, qualifies as an ISSf-ECBF for the system described in (1). This holds true if, for all x ∈ R n , there exist a bound ∥d w ∥ ∞ ≤ τw and a function γ(β d (•) ∈ K ∞,e .•) ∈ K ∞,e that satisfies sup u∈U [L ρ f S(x) + L g L ρ-1 f S(x)u + ( ∂(L ρ-1 f S(x)) ∂x ) T d w +T T s H s ] ≥ -γ(∥d w ∥ ∞ ) (4) The terms L ρ f and L g L ρ-1 f represent the Lie derivatives of the function S(x). T s = p 0 p 1 . . . p ι T where p i is positive constant. H s = [S(x) L f S(x) . . . L ρ-1 f S(x)] T .Lemma 1. If S(x) functions as an ISSf-ECBF for the system (1) in the set C, then any controller u ∈ U that is Lipschitz continuous and valid for all x ∈ R n must satisfyEQUATION)This implies that the set C d is forward invariant. In other words, the system (1) is ISSf on the set C.The kinematics and kinetics of ASV can be described as:EQUATIONwhere η(t) = p(t) ψ(t) T ∈ R 3 represents the position andEQUATIONThe vector ν(t) = u(t) v(t) r(t) T ∈ R 3 represents the surge velocity, sway velocity, and yaw velocity, respectively. The matrix M denotes the inertial matrix. f (ν) represents the Coriolis and centripetal matrix, damping matrix, and unmodeled hydrodynamics. The vector τ (t) signifies the forces produced by the actuators. The external disturbances, caused by wind, waves, and ocean currents, are represented byd w (t) = d w1 (t) d w2 (t) d w3 (t) T ∈ R 3 .Letting q(t) = R(ψ)ν(t), (6) can be rewritten asEQUATIONwhereξ = RM -1 (d w + f (ν)) + Ṙν.The desired parameterized path is set as p 0 (θ) = [x 0 (θ), y 0 (θ), ψ 0 (θ)]T , ψ 0 (θ) = arctan(y θ 0 (θ)/x θ 0 (θ)) where θ represents path variable. y θ 0 (θ) and x θ 0 (θ) is the partial derivative of y 0 (θ) and x 0 (θ), respectively. In addition, it is assumed that p θ 0 (θ) is bounded.The safety-critical obstacle avoidance controller of ASV is required to achieve the following tasks:(1) Geometric task: Ensure that the ASV follows the desired path, meaning thatlim t→∞ ∥p(t) -p 0 (θ)∥ < l 1 (9)where l 1 ∈ R denotes a small positive constant.(2) Dynamic task: The derivative of the path variable θ converge to the desired speedEQUATION)where u d (t) represents desired speed and l 2 is a small positive constant.(3) Obstacle avoidance task: To prevent collisions between the ASV and obstacles, the following condition must be metEQUATION)where pk (t), r k , and d k represent the position, the radius, and the minimum obstacle avoidance distance of the kth obstacle.While the previous studies have made substantial progress, they were mainly directed at scenarios with known disturbances or predefined upper bounds. To alleviate this limitation, the following theorem is presented to account for unknown disturbances.Theorem 1. Given the ISSf-ECBF S(x) as defined in Definition 5 for the system (1) on the set C, if there exists a bound ∥d w ∥ ∞ ≤ dw such that for every x ∈ R n , the following inequality holdssup u∈U [L ρ f S(x) + L g L ρ-1 f S(x)u + T T s H s -( ∂(L ρ-1 f S(x)) ∂x ) T ( ∂(L ρ-1 f S(x)) ∂x )] ≥ 0 (12)and the admissible control set satisfies asEQUATION)Then, we can obtain that the system (1) is ISSf on C.Proof. For u ∈ U(x), one hasEQUATIONAdding and subtractingEQUATIONwhich is of the form (4).Remark 1. Compared with [21] , the proposed ISSf-ECBF can deal with unknown perturbations. Although asymptotically stable ESO is used in reference 21, in practice, the disturbance estimation error is difficult to be 0. Thus, it is essential to develop ISSf-ECBFs that ensure safety in the presence of unknown disturbances.In this section, we will focus on designing a safety-critical controller. The control architecture for the proposed strategy is illustrated in Figure 1 . Firstly, we utilize the ESO to obtain the estimations of the model uncertainties, external disturbances in this part. In addition, the ESO relies on the following assumption. where ξ * is a positive constant.Then, the ESO is devised to estimate model uncertainties, external disturbances.EQUATIONwhere q(t) and ξ(t) represent the estimates of q(t) and ξ(t).The observer matrices K 1 K 2 T = 2wI 3 w 2 I 3 T where w is the observer bandwidth. Defining q(t) = q(t) -q(t), and ξ(t) = ξ(t) -ξ(t) are the estimates. The dynamics of q(t) and ξ(t) can be written asEQUATIONNext, ( 18) can be rewritten asEQUATIONwhere E o (t) = qT (t) ξT (t)T ∈ ℜ 6 andT = -K 1 I 3 -K 2 0 3 , D = 0 3 I 3 .Then, the following tracking error is defined as e 1 = pp 0 (θ). By taking the derivative of e 1 and using (6), we can get ė1 = q -p θ 0 (θ) θ.Let u d -ϑ(t) = θ(t), one can obtainEQUATIONThe kinematic guidance law q d is designed as follows to stabilize e 1 :EQUATIONand θ = -ℓ(ϑEQUATIONwhere k 1 = diag{k 11 , k 12 , k 13 }, ℓ and µ are positive constants.To proceed, defining e 2 = q-qd , where qd is the estimate of q d . qd can be obtained by using the following filtering scheme:EQUATION)where t d is a positive constant. LetEQUATIONAnd qd ≜ a = a 1 a 2 a 3 T . a j is bounded by |a j | ≤ a * j , j = 1, 2, 3, where a * j is a positive constant. For details, please refer to [26] .Then, the time derivative of e 2 yieldsEQUATION)To stabilize e 2 , the anti-disturbance control law is developed as follows:EQUATION)wherek 2 = diag{k 21 , k 22 , k 23 }. Denote τ = τ c + τ e .In this part, considering the collision with obstacles and ASV to design the optimal surge and sway force of safety conditions. From (8), we can getEQUATIONwhere p denotes [x(t), y(t)] T , q denotes R 2 (ψ)[u, v] T . ξ2 , ξ2 and τ 2 is the first two dimensions of ξ, ξ and τ , respectively. pk = [x k , y k ] T is position of kth obstacle. We choose the following candidate ISSf-ECBFEQUATIONwhere pek = p -pk , s = [p T , qT ] T . To achieve the objective of obstacle avoidance, the set C can be obtainedC = p ∈ R 2 : S k (s) = ∥p ek ∥ 2 -(r k + d k ) 2 ≥ 0 (30)For ease of notation, it is denoted by S k in the sequel. The safety constraint with S k (s) is described asU = τ 2 : L 2 f S k + L g L f S k τ 2 -( ∂(L f S k ) ∂x ) T ( ∂(L f S k ) ∂x ) + T T s H s ≥ 0 (31)whereL 2 f S k = 2q T q + 2p T ek ξ2 , L g L f S k = 2p T ek , T s = [β 2 , 2β] T .For the ASV, ensuring safety takes precedence over geometric objectives. Based on the safety constraint (31), the following quadratic programming problem is constructed.EQUATIONwhere ϕ = 2q T q --(∂(L f S k ) ∂x ) T ( ∂(L f S k ) ∂x) + 2p T ek ξ2 + T T s H s . The τ * is obtained by solving the above quadratic programming problem.Remark 2. The proposed safety-critical controller can avoid obstacles while ensuring minimal impact on the given tracking task.In this section, we will conduct stability and safety analysis of the closed-loop system.Lemma 2. The observer error subsystem in (19) is ISS, and the error signals being q and f are bounded byEQUATIONwhereEQUATIONwhere ς 1 ∈ R is a positive constant.Proof. Choose the following Lyapunov functionEQUATIONTaking (34) into account, one hasEQUATION)It follows that the observer error subsystem described by (19) is ISS. It is important to note that V 1 is bounded and satisfies the inequality([λ min (N )]/2)||E o (t)|| 2 ≤ V 1 ≤ ([λ max (N )]/2)∥E o (t)∥ 2 .From this, we can derive (33).Next, we will outline the stability analysis of the closedloop system. Lemma 3. Taking into account the error dynamics represented by (21) and (26) , the error signals e 1 , e 2 , e r , ϑ, and γ are uniformly ultimately bounded byEQUATION)whereQ = diag {1, 1/ℓµ}, γ 2 = 2ς 2 (1 -ϵ)/λ max (Q).Proof. The constructed Lyapunov function isV 2 = 1 2 (e T 1 e 1 + e T 2 e 2 + e T d e d ) + ϑ 2 2ℓµ .According to (20) , ( 23)-( 27) , the time derivative of V 2 is a )V2 = e T 1 (EQUATIONFinally, we can obtainEQUATIONChoose λ min (k 1 ) -1 2 > 0, 1 t d -1 2 > 0. Then, de- fine ς 2 = min(λ min (k 1 ) -1 2 , λ min (k 2 ), 1 t d -1 2 , 1 µ ), ϖ = ∥M τ e ∥, E = e TEQUATION)It can be established that the error subsystem related to obstacle avoidance control is ISS. Additionally, the errors of the closed-loop system satisfies (37).The subsequent lemma presents the safety analysis of the ASV.Lemma 4. Given the dynamics of the ASV as outlined in (6), if p(t 0 ) ∈ U and τ * ∈ U are satisfied for all t > t 0 , the closed-loop system will be ISSf.Proof. According to Lemma 1, if τ * ∈ U holds, then C is forward invariant, meaning that the set C is ISSf. In other words, as long as p(t 0 ) is within C, the position p(t) will remain in C indefinitely. Therefore, the closed-loop system is ISSf.Theorem 2. The closed-loop system is shown to achieve ISSf, indicating that collision avoidance is feasible. Furthermore, all error signals in the closed-loop system are uniformly ultimately bounded.Proof. According to Lemma 4, the ASV will meet the safety constraint, meaning that the safety objective (11) is fulfilled. We employ Lemmas 2, 3, and [27, Lemma 1] , which enable us to deduce that the closed-loop system is ISSf. The norm ||E|| is uniformly ultimately bounded byEQUATIONGiven that E is bounded, we can deduce that e 1 and ϑ are also bounded. As a result, it follows that ∥p(t) -p 0 (θ)∥ = ∥e 1 ∥ and || θ(t) -u d || remain bounded, that is, ( 9) and ( 10) hold.Remark 3.To validate the effectiveness of the proposed control strategy, this paper conducts simulations using Cybership II in [28] . The simulation parameters are set as follows: w = 40, Fig. 2 illustrates the effectiveness of the safety-critical controller proposed for the autonomous vessel. The upper section demonstrates that the vessel prioritizes obstacle avoidance to ensure safety. Once the obstacle avoidance operation is complete, it can proceed with the tracking task. Fig. 3 illustrates the observation effect of the extended state observer. The velocity and aggregated disturbances of the ASV can be EQUATIONThis paper introduces a safety-critical control strategy for ASVs that considers external marine disturbances and internal model uncertainties. Initially, an anti-disturbance controller is devised based on the estimation of lumped disturbances using an ESO. Following this, a quadratic optimization problem is established by incorporating ISSf-ECBFs to enforce safety constraints on the control inputs. By solving this problem, a safety-critical controller is derived, significantly improving the safety and robustness of the system. The closed-loop system is demonstrated to be ISSf, with error signals shown to be uniformly ultimately bounded. The effectiveness of the proposed control strategy is verified through simulation results. 
 
 Appendix:  



Reviews: review: title: title: This paper proposes a safety-critical obstacle avoidance control approach for autonomous surface vehicles (ASVs) with disturbances and uncertainties. The feasibility of the designed control approach is proven via the simulation example. However, the following suggestions need to be considered in the revised manuscript to further improve the quality of this paper.
main_review: main_review: 1. How does the extension of ISSf-ECBFs to handle unknown disturbances improve safety constraints compared to the method in [21]? 
2. How does the safety-critical controller formulated in this paper using ISSf-ECBFs differ from the approaches in [12], [22]-[24]? 
3. What are the main benefits of using a quadratic programming problem for collision avoidance with obstacles?
title: title: This paper is well structured, logical and of practical value and is recommended for publication.
main_review: main_review: The paper introduces a significant advancement in the field of autonomous surface vehicles (ASVs) by addressing safety-critical obstacle avoidance in the presence of disturbances and uncertainties. The extension of exponential control barrier functions (ECBFs) to account for unknown disturbances through the development of input-to-state safe exponential control barrier functions (ISSf-ECBFs) represents a noteworthy contribution. This enhancement ensures that ASVs can maintain safety even when faced with unpredictable external conditions, a critical requirement for real-world applications. The reviewer has the following questions to discuss with the authors:

1. How well does the proposed ISSf-ECBFs approach perform in real-world scenarios where disturbances might be more complex or varied than those simulated? Are there plans for testing the system in real maritime environments?

2. The paper relies on an extended state observer to estimate unknown external disturbances and internal uncertainties. How sensitive is the overall control strategy to inaccuracies in these estimations? What happens if the observer’s estimations are significantly off?

3. While the paper addresses unknown disturbances, how does the control strategy perform under extreme or highly irregular disturbances that might exceed typical operational conditions? Is there a threshold beyond which the safety guarantees no longer hold?

4. Over extended periods of operation, how does the control strategy account for potential changes in the performance of ASVs or wear-and-tear that could affect the accuracy of the state observer and overall system safety?
title: title: Accept
main_review: main_review: This paper presents a safety-critical obstacle avoidance control strategy for autonomous surface vehicles, addressing external marine disturbances and internal model uncertainties. It extends the Input-State Safety Index Control Barrier Function to handle unknown disturbances and designs an extended state observer for estimating lumped disturbances. Based on the observer's results, a disturbance rejection controller is designed. The closed-loop system's input-state safety and the boundedness of the error signals are proven, and the effectiveness of the control strategy is validated through simulations.

1.How accurately can the unknown lumped disturbances be estimated in practice?

2.What is the computational efficiency of the quadratic programming problem, and is it suitable for real-time control applications?

3.How robust is the control strategy against model uncertainties and disturbances in various marine environments?

4.Are there more concise or efficient control strategies that can meet the requirements for safety-critical obstacle avoidance?