A fast finite-time convergent guidance law with nonlinear disturbance observer for unmanned aerial vehicles collision avoidance

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A fast finite-time convergent guidance law with nonlinear disturbance observer for unmanned aerial vehicles collision avoidance

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Shandong University of Science and Technology, Qingdao 266590, China

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Article history: Received 29 May 2018 Received in revised form 8 December 2018 Accepted 7 January 2019 Available online xxxx

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Ning Zhang, Wendong Gai ∗ , Maiying Zhong, Jing Zhang

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Keywords: Collision avoidance Guidance law Collision cone Finite time convergent Nonlinear disturbance observer

In this paper, a fast finite-time convergent guidance law with nonlinear disturbance observer is proposed for the problem of unmanned aerial vehicle collision avoidance. First, a linear time-varying collision avoidance model based on a collision cone is established. Then a fast finite-time convergent guidance law and a nonlinear disturbance observer are designed to ensure the safety of UAV collision avoidance. In addition, the stability of a nonlinear collision avoidance system is proved by the finite-time convergence stability theory, and the stability conditions are used to design the guidance coefficients. The simulation results conclusively demonstrate that this method can achieve collision avoidance in the presence of an unknown acceleration of obstacle. Moreover, the performance of collision avoidance using this method is greater than the collision avoidance method based on normal finite time convergent guidance. © 2019 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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Unmanned aerial vehicles (UAVs) safety concerns have restricted their development and application. UAV can’t share the same airspace with aircraft due to the safety of UAV collision avoidance which has become a formidable challenge. In recent years, many researchers have done a lot of research on collision avoidance [1,2]. Collision detection is a key step before collision avoidance. Collision cone approach is an effective way of collision detection [3], and Closest Point of Approach (CPA) was used for multiple UAVs collision avoidance in [4]. A revised collision cone method was presented in [5] and the validity of the detection conditions was proved. An appropriate collision avoidance algorithm is selected to drive the UAV to fly away from the hazardous areas if collision is detected. Proportional navigation (PN) guidance law and its variants, known as biased-PN guidance law, have been widely used in UAV collision avoidance and missile interception due to their simplicity and availability [6–9]. But these guidance laws are applicable for the non-manoeuvring obstacle. Their performance may degrade when the obstacle’s acceleration is unknown. The optimal PN guidance law for UAV collision avoidance was investigated in [10] and the key parameter of this guidance law was the collision avoidance time. Accurate collision avoidance time estimation plays important

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*

Corresponding author. E-mail address: [email protected] (W. Gai).

https://doi.org/10.1016/j.ast.2019.01.021 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

roles in ensuring good performance of the optimal guidance laws while the poor collision avoidance estimation severely degrades the guidance performance [11–13]. A nonlinear dynamic inversion (NDI) guidance law is another guidance law widely used in UAV collision avoidance [14]. In [14], it was proved that NDI guidance law was same as differential geometry guidance law [15], but there was no exact solution for guidance coefficients in [14]. However, the disturbance such as the unknown acceleration of obstacle is few considered in the existing NDI guidance laws for UAV avoidance. In addition, all the above guidance laws used in UAV collision avoidance are asymptotically convergent but not finite time convergent. The finite time convergent method is critical for UAV collision avoidance. Collision avoidance time is a controllable parameter which is related to upper bound of settling time. In comparison, sliding mode control (SMC) plays an important role in missile and UAV flight control because it is not sensitive to disturbances and uncertainties [16,17]. The sliding mode guidance can capture a manoeuvring target without the need for precise target acceleration information [18]. However, the drawback of conventional sliding mode controller is asymptotically convergent [19]. Thus, the finite time convergent (FTC) guidance laws are designed such as non-singular terminal sliding-mode (NTSM) guidance law and fast terminal sliding-mode (FTSM) guidance law [20–22]. The upper bound of time-to-go can be derived by FTC stability theory. Otherwise, FTC guidance laws have the problem of high-frequency chattering. Thus, two methods are taken to alleviate the chattering phenomenon and maintained the disturbance rejection performance.

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One way is adaptive sliding mode guidance law [23–27]. Reference [28] proposed a new adaptive FTC guidance law for missiles to intercept manoeuvring targets. This guidance law generated smooth acceleration commands and stabilized the relative lateral velocity in finite time. However, this way would largely suppress the chattering phenomenon but not completely eliminate it. The other way is higher-order sliding mode guidance law [29]. A lateral guidance law for UAV based on high-order sliding mode was presented in [30]. This guidance provided very good performance in the presence of wind, generated the smooth and graceful manoeuvers and reduced the chattering effect and retained robustness and accuracy, but high-order derivatives of the switching function are required. Then a composite guidance law based on higher-order sliding mode controller and finite time nonlinear disturbance observer (FTDOB) was proposed in [31], and a higher-order sliding mode (HOSM) differentiator was designed such that the line of sight angular rate can be estimated but it is rather complex. It is to be noted that these guidance laws are only used for missile interception but not be used directly for UAV collision avoidance. In addition, to design and apply these FTC guidance laws in UAV collision avoidance, a saturation function is used instead of the sign function for the purpose of removing the chattering. However, this method brings a finite steady state error and degrades the system robustness. So it is necessary to estimate the disturbances and uncertainties and compensate them in collision avoidance guidance laws. Disturbance/uncertainty estimation and attenuation techniques are effective methods to deal with disturbances and uncertainties [32]. Some observers are widely applied, such as nonlinear disturbance observer (NDOB) [33,34], unknown input observer (UIO) [35], extended state observer (ESO) [36,37]. A fast nonsingular terminal sliding-mode (FNTSM) guidance law was established to dealing with the relationship between line of sight (LOS) angle and the LOS angle rate, the NDOB was designed to compensate for uncertainties in [38]. Then a non-smooth disturbance observer (NSDOB) was presented to solve the chattering problem in [39], but NSDOB may not be used in practice because the upper bound of derivative of target acceleration is difficult to measure. ESO required less information about the target acceleration, especially in the absence of the upper bound information of timevarying target acceleration. Ref. [40] reported an adaptive NTSM guidance law to intercept manoeuvring target, and an ESO was designed to estimate the unknown disturbance. In [41], an active disturbance rejection control (ADRC) guidance law was proposed for UAV collision avoidance. This guidance law includes the states and disturbances estimated by ESO. Compared with ESO, NDOB is simpler to be designed and applied if the guidance laws have no states to be reconstructed. In view of the above-mentioned results, a collision avoidance method based on fast finite time convergent (FFTC) guidance law with NDOB is proposed in this paper. The main contributions of this paper are summarized as follows: 1) Both finite time convergent (FTC) guidance law and fast finite time convergent (FFTC) guidance law are presented for UAV collision avoidance. The upper bounds of finite convergent time are given. 2) The NDOB is designed for FFTC guidance law to deal with disturbance such as the unknown acceleration of obstacle, and a collision avoidance time estimation method is presented. 3) The stability of the collision avoidance system with NDOB is proved by the finite-time stability theory, and the stability conditions are used to design the guidance coefficients. The organization of this paper is as follows. Section 2 is devoted to the collision detection approach, and collision avoidance model is presented in it. In Section 3, a FTC and a FFTC guidance law with NDOB are presented, and the stability of a collision avoidance

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Fig. 1. Geometric configuration of the collision cone.

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system is proved. Section 4 discusses the results of the numerical simulations performed with MATLAB. Finally, some conclusions and remarks are given in section 5.

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2. Model of collision avoidance

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For collision avoidance, it is important to use a collision detection approach which can predict any possible collision with an obstacle and compute an alternate aiming direction for the UAV to avoid the obstacle. The geometric configuration of the collision cone [3] is illustrated in Fig. 1. As shown in Fig. 1, the collision cone is defined by four points denoted as A, B, C , and D. A and C are the locations of the UAV and obstacle, respectively, B and D are the aiming points of collision avoidance. The collision cone boundary distance R b is defined by the length of A B or A D. A B is the collision cone upper bound and A D is the lower bound. The UAV has two collision avoidance strategies when the collision may occur: one is to avoid the collision along the cone boundary A D; another is to avoid the collision along the cone boundary A B. The UAV should avoid collision in the opposite direction of obstacle’s flight if it is less than π /2, otherwise, the UAV should avoid collision along the direction of the obstacle’s flight if it is greater than π /2. Therefore, the direction of the guidance command can be determined by

⎧ ⎪ ⎨ a(t ) > 0

π 2

Relative velocity V rel can be obtained by

102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

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= V cos(ψrel − ψ) + V obs cos(π + ψobs − ψrel )

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V rel = V − V obs

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In reality, both two collision cone boundaries are acceptable when the velocity of UAV is greater than that of obstacle. As shown in Fig. 1, V is the velocity of the UAV, V ob is the velocity of the obstacle, ψ and ψob are the heading angles of the UAV and obstacle, λ LOS angle, γ is the angle between the boundary of the collision cone and LOS, and ε is the angle between the relative velocity vector and LOS. 

99

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(1)

2



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≤ |ψ0 − ψOB | < π

⎪ ⎩ a(t ) < 0 0 < |ψ0 − ψOB | < π

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(2)

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3



where V is the velocity vector of UAV and V obs is the velocity vector of obstacle. Safety distance R s is the distance that guarantees the safety of the UAV. It is a constant determined by the designer. The UAV will collide with the obstacle if Eq. (3) is satisfied. Then it will switch to the path of collision avoidance.

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|λ − ψrel | = ε < γ

(3)

The model of collision avoidance can now be described as follows. According to Fig. 1, the relative motion equations of the UAV and obstacle are expressed as



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R˙ b (t ) = V r = V cos(μ − ψ) − V ob cos(μ − ψob )

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˙ = V μ = − V sin(μ − ψ) + V ob sin(μ − ψob ) R b (t )μ

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where V r is the velocity along the collision cone boundary, and ˙ is V μ is the velocity orthogonal to the collision cone boundary. μ the derivative of the collision cone boundary angle. R˙ b (t ) is the derivative of collision cone boundary distance. Differentiating V μ with respect to time, it is derived

V˙ μ = −

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(4)

Vr Vμ R b (t )

+ au − aob

(5)

au = V ψ˙ cos(μ − ψ) + V˙ sin(μ − ψ) aob = V˙ ob cos(μ − ψob ) − V ob ψ˙ ob sin(μ − ψob )

(6)

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Here, au and aob are the accelerations of the UAV and obstacle perpendicular to the collision cone boundary, respectively. ¨ can be obtained according to Eqs. (4), (5) and (6). Then μ

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μ¨ = −2

Vr Vμ R b2 (t )

−

au R b (t )

+

aob R b (t )

(7)

˙ is chosen as the state x1 . Then, the timeBased on Eq. (7), μ varying model of collision avoidance is

⎧ ⎪ ⎨ x˙ 1 = −2 V r x1 − u + d(t ) R b (t ) R b (t ) R b (t ) ⎪ ⎩ y=x

(8)

1

where u is the control input which is equal to guidance command au . d(t ) is the disturbance such as the unknown acceleration of the obstacle. ˙ conThe condition of collision avoidance achievement is that μ ˙ = 0 in finite time before UAV reaches verges to its equilibrium μ the collision point B or D.

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3.1. The fundamentals of finite time stability theory To analyse the stability of collision avoidance system, some conception of finite-time convergence is introduced here and it is hereafter referred as the FTC guidance law. The definition of finite time stability for a nonlinear time-varying system is as follows.

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 lim v (t ; t 0 , x0 ) = 0, t ∈ t 0 , T (x0 )



t → T (x0 )

v (t ; t 0 , x0 ) = 0,

t ≥ T (x0 )

Definition 1. [42] Consider a nonlinear time-varying system

x˙ = f (x, t ),

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(10)

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The system equilibrium x = 0 is called local finite time stable if it is Lyapunov stable with finite time convergence in a neighborhood of the origin U ⊂ U 0 . If U = Rn , then the system is global finite time stable. Lemma 1. [42] Consider the nonlinear time-varying system (Eq. (9)), supposed that there exist any real numbers c > 0 and 0 < k < 1, and a C1 (continuously differentiable) function V (x, t ) defined in a neighborhood Uˆ ⊂ Rn , such that V (x, t ) is positive definite on Uˆ and satisfies

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x ∈ Rn

(11)

where f : U 0 × R → Rn is continuous on U 0 × R, U 0 is an open neighborhood of the origin x = 0 and f (0, t ) = 0. The equilibrium point x = 0 is said to be finite time convergence if for any given

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Then the origin of nonlinear system (Eq. (9)) is finite time stable. The upper bound of settling time which is dependent on x0 can be obtained by

T x (x0 ) ≤

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V 1−k (x0 )

(12)

c (1 − k)

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In this section, a FTC guidance law is first proposed for UAV collision avoidance. Then, an improved guidance law which is called fast finite time convergent (FFTC) guidance law is presented. And the convergence rates of the two guidance laws are compared. The control system of UAV collision avoidance is shown in Fig. 2.

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Theorem 1. Consider the collision avoidance system (Eq. (8)), if there exists a control u such that the system state x satisfies

(13)

Proof. In the neighborhood of origin Uˆ ⊂ Rn , a C1 positive-definite function V 1 is chosen as

1 2

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where β is a positive constant and 0 < σ < 1. Then the derivative of ˙ =0 collision cone boundary angle can convergent to its equilibrium μ in finite time.

V1 = (9)

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V˙ (x, t ) + cV k (x, t ) ≤ 0

β|x|σ sgn(x) ≤0 x x˙ + R b (t )

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initial time t 0 and initial state x(t 0 ) = x0 ∈ U , a neighborhood of the origin U ⊂ U 0 , there exists a non-negative settling time T , which is dependent on x0 , such that every solution of the nonlinear system (Eq. (9)), x = v (t ; t 0 , x0 ) ∈ U /{0}, satisfies



3. Fast finite time convergent guidance law

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3.2. FFTC guidance law design

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Fig. 2. Control system of collision avoidance.

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where



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x2

(14)

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Then, the time-derivative of V 1 along the solution of Eq. (13) satisfies

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β V˙ 1 = xx˙ ≤ − |x|σ +1 , R b (t )

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∀t > 0

(15)

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During the time horizon of collision avoidance process, we have

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R˙ b (t ) < 0,

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V˙ 1 < −

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T t1 <

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V1 2 ,

∀t > 0

(17)

|x(0)| R b (0) β(1 − σ )

(18)

β|x|σ sgn(x) ≤0 x −2 x− + + R b (t ) R b (t ) R b (t ) R b (t )

R˙ b (t )

u

d(t )

(19)

Then, the control u (Eq. (20)) can be derived by Eq. (19).

xx˙ ≤ −

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β|x|σ +1 R b (0) − ct

(26)

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The upper bound of settling time (Eq. (21)) can be obtained by solving Eq. (26). Proof is complete.

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Most of the FTC guidance laws are used in missile interception, such as Ref. [20–22]. However, the missile interception is different from the collision avoidance. Therefore, according to the new model of collision avoidance, a collision avoidance system with FTC guidance law is established and it is finite time stable in Theorems 1 and 2. The upper bound of settling time can be adjusted by tuning parameters β and σ . To increase convergence rate of the derivative of collision cone boundary angle, a FFTC guidance law is designed according to the adaptive backstepping fast terminal sliding mode (FTSM) controller in [43].

u = N R˙ b (t )x + d(t ) + α x + β|x|σ sgn(x)

(27)

where N is navigation constant, and N > 2. Proof is complete.

where N is navigation constant, and N > 2. d(t ) is the disturbance of collision avoidance system and it is difficult to be obtained in practical system. β is a positive constant and the convergence 0 < σ < 1. α is a constant and α > 0. And α is a parameter which can increase convergence rate in FFTC guidance law.

The collision avoidance system is finite time stable which means the derivative of collision cone boundary angle converges

Theorem 3. Consider the collision avoidance system (Eq. (8)), if there exists a control u such that the system state x satisfies

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R b (0)

σ +1

According to Eq. (18), it is inferred that the convergence rate increases as the parameter β is increased. Moreover, the convergence rate increases as the parameter σ is decreased when |x(0)| < 1 rad/s. Combining Eqs. (8) and (13), we have

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2β

1 −σ

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(16)

According to Lemma 1, the derivative of collision cone boundary angle converges to zero in finite time, and the upper bound of settling time is derived

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∀t > 0

Substituting Eq. (16) into Eq. (15), we obtain

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0 ≤ R b (t ) < R b (0),

and

Substituting Eq. (25) into Eq. (23) yields

u = − N R˙ b (t )x + d(t ) + β|x|σ sgn(x)

(20)

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to zero in finite time. That is to say, the relative velocity V rel overlaps with the collision cone boundary A D or A B in Fig. 1 in finite time. Then, the UAV flights along with the collision cone boundary. And the collision avoidance is achieved until UAV reaches the collision point B or D.

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β|x|σ sgn(x) ≤0 x x˙ + + R b (t ) R b (t )

αx

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(28)

where β > 0, and α > 0. Then the derivative of collision cone boundary ˙ = 0 in fast finite time. angle can convergent to its equilibrium μ

V2 =

(21)

x˙ =

( N − 2) R˙ b (t ) x− R b (t )

R b (t )

(22)

where N > 0, R b (t ) > 0 and R˙ b (t ) < 0. Then, combining Eqs. (13) and (22), we obtain

( N − 2) R˙ b (t ) 2 β|x|σ sgn(x) = x x˙ + x ≤0 R b (t ) R b (t )

(23)

(24)

Then, the collision cone boundary distance R b (t ) can be simplified as

R b (t ) = R b (0) − ct

(29)

β|x|σ +1 R b (t ) R b (t ) α |x|σ +1 β|x|σ +1 ≤− − R b (t ) R b (t ) (α + β)|x|σ +1 ≤− , ∀t > 0 R b (t )

V˙ 2 ≤ −

α x2

(25)

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σ +1

∀t > 0

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(30)

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(31)

According to Lemma 1, the derivative of collision cone boundary angle converges to zero in finite time, and the upper bound of settling time is derived 1 −σ

|x(0)| R b (0) (α + β)(1 − σ )

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−

2(α + β) V˙ 2 < − V2 2 , R b (0)

T t3 <

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108

Substituting Eq. (16) into Eq. (30), we obtain

According to Theorem 1, the collision avoidance system (Eq. (8)) is finite time stable. Suppose that the derivative of collision boundary distance is approximately a constant, we have

R˙ b (t ) = −c , c > 0

2

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x2

Due to 0 < σ < 1, the time-derivative of V 2 along the solution of Eq. (28) satisfies

Proof. Substituting Eq. (20) into collision avoidance system (Eq. (8)) givens

β|x|σ sgn(x)

1

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−c R b (0)  |x(0)|1−σ  T t2 ≤ 1 − e β(1−σ ) c where −c is an approximation of R˙ b (t ) and c > 0.

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Proof. In the neighborhood of origin Uˆ ⊂ Rn , a C1 positive-definite function V 2 is chosen as

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Theorem 2. The collision avoidance system (Eq. (8)) is finite time stable if the guidance law in Eq. (20) is adopted. The upper bound of settling time can be obtained

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(32)

In Eq. (32), it is inferred that the convergence rate increases as the parameter α is increased. Proof is complete.

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Theorem 4. The collision avoidance system (Eq. (8)) is finite time stable if the guidance law in Eq. (27) is adopted. The upper bound of settling time for FFTC guidance law (Eq. (27)) is less than the FTC guidance law (Eq. (20)).

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Proof. Substituting Eq. (27) into Eq. (8) produces

( N − 2) R˙ b (t ) β|x|σ sgn(x) αx x˙ = x− − R b (t ) R b (t ) R b (t )

(33)

where N > 0, R b (t ) > 0 and R˙ b (t ) < 0. Combining Eqs. (28) and (33), we obtain









( N − 2) R˙ b (t ) 2 β|x|σ sgn(x) = x x˙ + x ≤0 + R b (t ) R b (t ) R b (t )

αx

(34)

According to Theorem 3, the collision avoidance system (Eq. (8)) with FFTC guidance law (Eq. (27)) is finite time stable. The upper bound of settling time can be directly obtained by solving Eqs. (30) and (34).

T t4 ≤

23

R b (0)  c

1−

−c |x(0)|1−σ e (α+β)(1−σ )



(35)

Comparing Eqs. (21) and (35), we obtain that

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T t2 < T t4

(36)

The upper bound of settling time for FFTC guidance law (Eq. (27)) is less than the FTC guidance law (Eq. (20)). Proof is complete. In Theorems 3 and 4, a different guidance law Eq. (27) is used to improve the convergence rate. According to the Eq. (36), the FTTC guidance law has the faster convergence rate the normal FTC guidance law proposed in Theorems 1 and 2. 3.3. Nonlinear disturbance observer design The guidance laws (Eqs. (20) and (27)) are impractical because the real disturbance d(t ) in collision avoidance process is unknown and can’t be measured accurately. Thus, the NDOB is designed to obtain the estimation of the disturbance and the real guidance laws should be as follows.

u = − N R˙ b (t )x(t ) + dˆ (t ) + β|x|σ sgn(x)

(37)

and

49 50

where dˆ (t ) is the estimation of the disturbance by NDOB. The col-

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(38)

lision avoidance system (Eq. (8)) can be descripted as a general single input/single output (SISO) system like



x˙ = f (x) + g 1 (x)u + g 2 (x)d y = h ( x)

Then NDOB is designed as



dˆ = z + p (x)

69

(41)

p ( x)

(39)



= l1 x1 + l2 x31



(42)

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where l1 and l2 are constants to be designed and the function l (x) can be derived by p (x).

78

∂ p ( x) l ( x) = = l1 + 3l2 x21 ∂x

81

(43)

Then the stability of NDOB is analysed. The estimation error is defined as

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e = d − dˆ

(44)

Suppose that the disturbances are bounded and slowly time varying. The estimate of disturbance dˆ can track the real disturbance d in system (Eq. (37)) asymptotically if nonlinear function p (x) is chosen such that l(x) g 2 (x) > 0 holds [44], which implies that

∂ p ( x) e˙ (t ) + g (x)e (t ) = e˙ (t ) + l(x) g 2 (x)e (t ) = 0 ∂x 2

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(45)

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is globally asymptotically stable. The NDOB is designed to estimate the dˆ (t ) in guidance law. Thus the disturbance can be compensated during collision process.

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Theorem 5. Consider the collision avoidance system (Eq. (8)) with the nonlinear disturbance observer in Eq. (41), state x1 converges to a neighborhood of equilibrium point x1 = 0 in finite time then evolve in it if FFTC guidance law (Eq. (38)) is adopted.

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Proof. In the neighborhood of origin Uˆ ⊂ Rn , a C1 positive-definite function is chosen as

1

106 107 108

(46)

109

According to Theorem 3 and Eq. (38), the time-derivative of V 3 satisfies

111

V3 =

2

x21

V˙ 3 = x1 −2

≤−

V˙ 3 ≤ − where

R˙ b (t ) R b (t )

x1 −

u R b (t )

+

d



R b (t )

σ +1

V3 2

2β

(47)

R b (t )

σ +1

V˙ 3 ≤ −

R b (t )

σ +1

119 120 121

(48)

V3 2 −

R b (t )

122 123 124 125

|x1 e | R b (t )

2β(1 − ξ )

116 118

(49)

126 127

Suppose that there exists a scalar 0 < ξ < 1 such that inequality (Eq. (49)) can be expressed as

2βξ

113

117

|x1 e | 2α + V3 − R b (t ) R b (t )

V3 2 +

112

115

R b (t )

α x21 β|x1 |σ +1 |x1 e | + − R b (t ) R b (t ) R b (t ) 2β

110

114

α > 0 and 0 < R b (t ) < R b (0). According to Eq. (46), we have

V˙ 3 ≤ − (40)

70 71

ˆ Substituting Eq. (46) into Eq. (47), we obtain where e = d − d.

⎧ x = x1 ⎪ ⎪ ⎪ ⎪ ⎪ Vr ⎪ ⎪ f (x) = −2 x1 ⎪ ⎪ R b (t ) ⎨ ⎪ g 1 ( x) = − ⎪ ⎪ ⎪ R b (t ) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ g ( x) = 2 R b (t )

68

where z is the internal state of the NDOB and dˆ is the estimate of unknown disturbance and also the output of nonlinear disturbance observer. p (x) is a nonlinear function that is designed as

where

1

67

z˙ = −l(x) g 2 (x) z − l(x)( g 2 (x) p (x) + f (x) + g 1 (x)u )

u = − N R˙ b (t )x(t ) + dˆ (t ) + α x + β|x|σ sgn(x)

48

5

σ +1

V3 2

|x1 e | + R b (t )

128 129 130 131

(50)

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If

2

σ +1

3

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4 5

10

13 14 15 16 17 18 19 20 21 22 23

R b (t )

σ +1

V3

2

26 27 28 29 30 31 32

(52)

|x1 e | ≤ 2β(1 − ξ )

(53)

which means that the trajectories of the closed-loop system is bounded in finite time as





lim x1 ∈ x1 |xσ1 ≤

ξ →ξ0

|e | . β(1 − ξ )

(54)

where 0 < ξ < 1, β > 0. Thus, the state x1 will converge into the neighborhood of equilibrium point x1 = 0 in finite time if the NDOB is globally asymptotically stable. The upper bound of settling time is

24 25

V3 2

lim T t5 ≤

ξ →ξ0

x(0)1−σ R b (0)

ξβ(1 − σ )

(55)

Then the state x1 will converge to equilibrium point x1 = 0 if the β is large enough such that V˙ 3 < 0 when V 3 (x1 ) is out of a certain bounded region which contains the equilibrium point x1 = 0. Proof is complete.

35 36 37 38 39 40 41

It is similar to prove that FTC guidance law (Eq. (37)) with NDOB is finite time stable according to Theorem 5. The FTTC collision avoidance system with nonlinear disturbance observer is also finite time stable in Theorem 5 and the upper bound of settling time can be obtained by Eq. (55). All of the above theorems give only the upper bound of settling time, which is generally large and not accurate, and a collision avoidance time estimation method is given below.

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Collision avoidance time is a key parameter during UAV collision avoidance. Accurate estimation of collision avoidance time is helpful to improve the quality of UAV collision avoidance and reduce the maneuvering range of UAV [45]. Collision avoidance time can be estimated by the following method once FFTC guidance law is designed. 1) In the case of disturbance d(t ) = 0: The FFTC guidance law (Eq. (38)) can be expressed as

˙ + β|μ ˙ |σ sgn(μ ˙ ) + αμ ˙ ud(t )=0 = − N R˙ b (t )μ

0

V sin ψ + V ob sin(π + ψob )

72



V cos ψ + V ob cos(π + ψob )

⎧ y T 1 + T 2 − y OB = tan ψOB (x T 1 + T 2 − xOB ) ⎪ ⎪ ⎪ ⎪ ⎪ y T 1 + T 2 − y T 1 = tan ψ( T 1 )(x T 1 + T 2 − x T 1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T 1 ⎪ ⎪ ⎨ x T 1 = x0 + V cos ψ(t )dt ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ T 1 ⎪ ⎪ ⎪ ⎪ y = y0 + V sin ψ(t )dt ⎪ ⎪ ⎩ T1

(59)

74 76 77 78 79 80 81 82 83 84 85 86 87 88

(60)

89 90 91 92 93 94 95

where (x T 1 + T 2 , y T 1 + T 2 ) is the collision avoidance point A or D in Fig. 1. (x T 1 , y T 1 ) is the UAV position at T 1 . T 2 can be derived by

(x T 1 + T 2 − x T 1

73 75

0

T2 =

70

)2

+ ( y T 1 +T 2 − y T 1

)2

V

97 98 99

(61)

100 101

The total collision avoidance time T can be derived by

T = T1 + T2

96

102 103

(62)

104 105

2) In the case of lumped disturbance d(t ) = 0: The FFTC guidance law (Eq. (38)) can be written as

106 107

(63)

ˆ dˆ is the estimation of the disturbance, and u is where u = d, used to compensate the disturbance d(t ). The disturbance could be well estimated by the NDOB which is globally asymptotically stable. The estimation error of the disturbance u − d(t ) is bounded and small. Therefore, the collision avoidance time estimated in the case of d(t ) = 0 is similar to the situation in the case of d(t ) = 0.

108 109 110 111 112 113 114 115 116 117

4. Simulation results

118 119

(56)

˙ can be obtained by Eq. (4). The collision avoidance time T where μ includes two parts: the first part is the time T 1 of relative velocity tracking collision cone boundary; the second part is the time T 2 of the UAV tracking the close collision point A or D in Fig. 1. Thus the dynamic model of UAV in lateral plane can be simplified as

⎧ u ˙ t ) == d(t )=0 ⎪ ψ( ⎪ ⎪ V ⎪ ⎨ t ⎪ ⎪ ⎪ ψ(t ) = ψ0 + ψ˙ dt ⎪ ⎩

ud(t ) =0 = ud(t )=0 + u

3.4. Collision avoidance time estimation

68

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Substituting Eqs. (2), (56), (57) and (59) into Eq. (58), the function whose root is T 1 is obtained. T 1 can be solved by general iteration method. The UAV path of second part is easy to be obtained if disturbance d(t ) = 0. UAV should keep the UAV heading angle at T 1 . Thus ψ˙ = 0 should be satisfied during the second part of collision time. The collision avoidance point (x T 1 + T 2 , y T 1 + T 2 ) can be derived by solving Eq. (60).



42 43

(58)

ψrel = π + tan−1

33 34

67 69

ψrel − μ = 0

σ +1

2βξ

According to Lemma 1, the decrease of V 3 (x1 ) in finite time drives the trajectories of the closed-loop system into

11 12

(51)

where

V˙ 3 < −

8 9

|x1 e | 2β(1 − ξ )

then

6 7

where ψ˙ is heading angle rate of UAV. Eq. (58) should be satisfied at T 1 if ψrel tracks the collision cone boundary A D in Fig. 1.

(57)

4.1. Simulation setup

120 121

To demonstrate the performance of the proposed FFTC guidance law, a UAV model from [46] is used in the simulation. The UAV collision avoidance is achieved by changing the UAV heading angle in lateral plane. The UAV begins collision avoidance if Eq. (3) is satisfied. The UAV will track the target point when the collision avoidance is completed. Safety distance is R s = 2 km and the other initial simulation conditions are listed in Tables 1 and 2. In Case I, the FTC guidance law with sign function is compared to the FTC guidance law with saturation function. The FFTC guidance law is compared with the FTC guidance law in Case II.

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Table 1 Initial conditions of the UAV.

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UAV

Case I

Case II

Case III

Case IV

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Initial position (km) Target position (km) Initial speed (m/s) Heading angle (rad)

(0, −17) (0, 17)

(0, −21) (0, 20)

(0, −19) (0, 19)

(0, −19) (0, 40)

70

170 π /2

210 π /2

190 π /2

190 π /2

72

71 73 74

8 9 10

75

Table 2 Initial conditions of obstacle.

76 77

11 12 13 14 15 16 17

Obstacle

Case I

Case II

Case III

Case IV

Initial position (km) Initial speed (m/s) Heading angle (rad)

(−9, 16)

(−13, −13)

(19, 0)

(19, 0)

(−29, 15)

190

190

190

190

190

−π /3

π /4

0

π

0

78 79 80 81 82 83 84

18 19

85

Table 3 Parameters of the FFTC guidance law and NDOB.

20

86 87

21

FFTC guidance law

NDOB

23

Parameters

Value

Parameters

Value

89

24

N

3

l1

10

90

25

β

10

l2

10

91

26

σ

0.3

p (x)

10x1 + 100x31

92

27

α

100

l(x)

10x1 + 300x21

93

22

Fig. 3. Trajectory of the UAV and obstacle in Case I.

94

28

95

29 30 31

Table 4 Collision avoidance time estimation.

96 97

32

UAV

Case I

Case II

Case III

33

Collision avoidance time T 1 (s) Collision avoidance time T 2 (s) Total collision avoidance time T (s)

20.3

40.6

60.5

56.3

30.5

80.5

36.3

43

37.1

74.8

100.8

76.9

103.5

93.4

105.3

34 35 36 37

Case IV

98 99 100 101 102 103 104

38

105

39 40 41 42 43 44 45

In Case III, the unknown acceleration of the obstacle is added to demonstrate the performance of the proposed FFTC guidance law. UAV avoids two obstacles with the different unknown acceleration in Case IV. To remove the chattering, the sign function is usually replaced by a saturation function satε (x) expressed as



46 47 48

satε (x) =

x/ε

|x| ≤ ε

sign(x)

|x| > ε

(64)

49 50 51 52

where ε = 0.5 is a small constant. Based on Section 3, the parameters of FFTC guidance law and NDOB are listed in Table 3. The collision avoidance time estimations are listed in Table 4.

53 54

4.2. Case I

55 56 57 58 59 60 61 62 63 64 65 66

88

In Case I, the difference between the initial heading angles of the UAV and obstacle is 5π /6. The velocity of the UAV is less than that of the obstacle. The process of the collision avoidance is as follows. In Fig. 3, the UAV and obstacle are closest when they are at the fifth sample circle, and the sample circles of the UAV do not intersect the sample circles of the obstacle during collision avoidance. The lateral offset is the x-axis direction distance between the UAV collision avoidance trajectory and normal flight path. Clearly, the lateral offset of FTC guidance law with the saturation function is less than that with the sign function.

106 107 108 109 110

Fig. 4. Distance between the UAV and obstacle in Case I. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

111 112 113

In Fig. 4, at 105 s, the closest distance between the UAV and obstacle is 2 km, which is equal to safety distance R s . Thus, the collision avoidance is successful. The collision avoidance time is very close to the estimation value of 100.8 s in Table 4. The error is 4.2 s. There is a time error between the estimated collision avoidance time T and the simulated collision avoidance time because the integral operations in Eqs. (57) and (60) are solved by numerical method. In Fig. 5, the acceleration command of FTC guidance law with sign function is chattering after the state x1 converges to x1 = 0, but another acceleration command is smoother because saturation function is adopted. Saturation function is an effective way to reduce the chatting.

114 115 116 117 118 119 120 121 122 123 124 125 126 127

4.3. Case II

128 129

In Case II, the difference between the initial heading angles of the UAV and obstacle is π /4. The velocity of the UAV is greater than that of the obstacle. The proposed collision avoidance method

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81 82

16 17

Fig. 5. Acceleration commands in Case I.

83 84

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85

19 20

Fig. 7. Distance between the UAV and obstacle in Case II.

86

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87

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38 39

105

Fig. 6. Trajectories of the UAV and obstacle in Case II.

106

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

based on the FFTC guidance law is compared with a collision avoidance method based on the FTC guidance law. The flight paths of the collision avoidance are as follows. From Fig. 6, the UAV and obstacle are closest when they are at the fifth sample circle. The manoeuvring range of collision avoidance based on the FTC guidance is narrower than that based on the FFTC guidance. In Fig. 7, the distance between the UAV and obstacle decreases from the beginning, and becomes the minimum at 80 s. The closest distance between the UAV and obstacle is 2 km, which is equal to the safety distance. Therefore, we can infer that the collision avoidance is successful. During the entire collision process, the distance between the UAV and obstacle based on FFTC guidance is shorter than that based on the FTC guidance. The collision avoidance time is very close to the estimation value of 76.9 s in Table 4. The error is 3.1 s. According to Fig. 8, the initial acceleration command with the FFTC guidance law is larger than with the FTC guidance law. The acceleration command in FTTC guidance law converges to zero at 65 s. But the acceleration command in FTC guidance law can’t converge to zero before 70 s. The convergent time with the FFTC guidance law is less than with the FTC guidance law. According to Fig. 9, the derivative of collision cone boundary angle in FFTC guidance law with saturation function can converge to zero in finite time. But in FTC guidance law it converges to a neighborhood of zero due to the saturation function is adopted.

Fig. 8. Acceleration commands in Case II.

107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131

Fig. 9. Derivative of collision cone boundary angle in Case II.

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82

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83 84

18

Fig. 12. Acceleration commands in Case III.

19

Fig. 10. Trajectory of the UAV and obstacle in Case III.

20

85 86

21

87

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88

23

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25

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35

101

36

102

37

Fig. 13. Real disturbance and lumped disturbance.

105

39 40 41 42 43 44 45

Fig. 11. Distance between UAV and obstacle in Case III.

46 47 48

The derivative of collision cone boundary angle starts to increase at 76.9 s once the collision avoidance is achieved.

According to Fig. 12, the acceleration commands with FFTC law are influenced by the unknown acceleration of the obstacle, and the amplitude of the acceleration command varies with the disturbance to drive state x1 converges to zero in finite time. In Fig. 13, the real disturbance is d(t ) = V˙ ob cos(μ − ψob ) because the heading angle of the obstacle ψob is a constant. The unknown obstacle acceleration can be estimated well by the NDOB during the collision avoidance process. The UAV collision avoidance is successful.

4.4. Case III

4.5. Case IV

In Case III, the difference between the initial heading angles of the UAV and obstacle is π /2. The velocity of the UAV is equal to the obstacle. The unknown acceleration of the obstacle is V˙ ob = 10 cos(t ). The flight paths of the collision avoidance are as follows. In Fig. 10, the UAV and obstacle are closest when they are at the fourth sample circle, and the sample circles of the UAV do not intersect the sample circles of the obstacle simultaneously. It is revealed that the distance between the UAV and obstacle is greater than or equal to safety distance. The UAV collision avoidance is successful. From Fig. 11, at 107 s, the closest distance between the UAV and obstacle is 2 km which is equal to safety distance. Thus, the collision avoidance is successful. The collision avoidance time is very close to the estimation value of 103.5 s in Table 4. The error is 3.5 s.

In Case IV, there are two obstacles during UAV collision avoidance process. The unknown accelerations of the obstacles are V˙ ob1 = 10 cos(t ) and V˙ ob2 = 5 cos(t ), respectively. The flight paths of the collision avoidance are as follows. In Fig. 14, the UAV and the obstacle1 are closest when they are at the third sample circle. The UAV and the obstacle2 are closest when they are at the seventh sample. The circle sample circles of the UAV do not intersect the sample circles of the obstacles simultaneously. From Fig. 15, at 93 s, the closest distance between the UAV and obstacle1 is 2 km which is equal to safety distance. At 199 s, the closest distance between the UAV and obstacle2 is also equal to safety distance. Thus, the collision avoidance is successful. The collision avoidance time is very close to the estimation value in Table 4. The errors are 0.4 s and 0.3 s, respectively.

53 54 55 56 57 58 59 60 61 62 63 64 65 66

107 108 109 110 111 112 113 114 116 117

51 52

106

115

49 50

103 104

38

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83 84

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Fig. 16. Acceleration command in Case IV.

19

87

21 22

85 86

20

Fig. 14. Trajectory of the UAV and obstacles in Case IV.

88

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34

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36

102

37

103

38

104

39

105

40

Fig. 17. Real disturbance and lumped disturbance.

108

42 43 44 45 46

Fig. 15. Distances between UAV and obstacles in Case IV.

47 48 49 50 51 52 53 54 55 56 57 58 59 60

According to Fig. 16, the first collision avoidance is achieved at 93 s. Then the UAV started to avoid the obstacle2 and the new acceleration command is calculated at the same time. The amplitude of the acceleration command varies with the disturbance to drive state x1 converges to zero in finite time. According to Fig. 17, the real disturbance of obstacle1 can be estimated well by the NDOB before the first collision avoidance is achieved. At 93 s, the disturbance of obstacle2 starts being estimated by NDOB and the disturbance can be estimated well during the second collision avoidance process. The estimated disturbances are compensated by FFTC guidance law during collision avoidance. This FFTC guidance law is suitable for collision avoidance of multiple UAVs.

61 62

5. Conclusions

63 64 65 66

106 107

41

This paper presents an automatic collision avoidance method based on the finite time convergent guidance law, and it could complete the simulation. The main conclusions are as follows:

1) A FFTC guidance law with NDOB is proposed and its upper bound of settling time is derived. A method is designed to estimate the collision avoidance time and the time error between estimated and simulated collision avoidance time is acceptable. 2) The stability of guidance laws and nonlinear collision avoidance system are proved by the finite-time convergence stability theory. Parameters of the guidance laws and NDOB can be designed by the stability domain. 3) The numerical results indicate that this FFTC law can deal with disturbance such as the unknown obstacle acceleration. In addition, the convergent time of the collision avoidance with the proposed FFTC guidance law is shorter than with the FTC guidance law. The collision avoidance system with FTTC guidance law can offer a greater finite-time convergence rate than FTC guidance law. This system can achieve collision avoidance in the presence of an unknown acceleration of obstacle and it is also suitable for collision avoidance of multiple UAVs. In the future, this FFTC guidance law will be applied to the vertical plane to realize UAV collision avoidance in three-dimensions. In addition, the guidance law should be designed to reduce the chatting for the performance improvement. It is necessary for guidance law to estimate the unknown disturbances as accurately as possible for further study.

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Conflict of interest statement

2 3

None declared.

4 5

Acknowledgements

6 7 8 9 10 11 12

This work is supported by the National Natural Science Foundation of China (No. 61603220, No. 61333005, No. 61733009), Shandong Provincial Natural Science Foundation (No. ZR2014FQ008), Research Fund for the Taishan Scholar Project of Shandong Province of China, the SDUST Young Teachers Teaching Talent Training Plan (No. BJRC20180503).

13 14

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