SCIENCE CHINA Information Sciences
. RESEARCH PAPER . Special Focus on Formation Control of Unmanned Systems
July 2017, Vol. 60 070204:1–070204:13 doi: 10.1007/s11432-016-9074-8
Fault-tolerant cooperative control for multiple UAVs based on sliding mode techniques Peng LI1 , Xiang YU2 , Xiaoyan PENG3 , Zhiqiang ZHENG1 & Youmin ZHANG2* 1College
of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha 410073, China; 2Department of Mechanical and Industrial Engineering, Concordia University, Montreal H3G 1M8, Canada; 3State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China Received December 14, 2016; accepted March 13, 2017; published online May 31, 2017
Abstract This paper proposes a fault-tolerant cooperative control (FTCC) design approach for multiple unmanned aerial vehicles (UAVs), where the outer-loop control and the inner-loop fault accommodation are explicitly considered. The reference signals for the inner-loop of the follower UAV can be directly produced by resorting to a proportional control. In the presence of actuator faults, the estimation of the fault information can be completed within finite time. Moreover, the control of the inner-loop is reconfigured based on the fault information adaptation and sliding mode techniques, such that the deleterious effects due to failed actuators can be compensated within finite time. Simulations of UAV cooperative flight are conducted to illustrate the effectiveness of this FTCC scheme. Keywords fault-tolerant cooperative control, unmanned aerial vehicle, actuator faults, finite-time fault accommodation Citation Li P, Yu X, Peng X Y, et al. Fault-tolerant cooperative control for multiple UAVs based on sliding mode techniques. Sci China Inf Sci, 2017, 60(7): 070204, doi: 10.1007/s11432-016-9074-8
1
Introduction
During the past few years, cooperative flight of unmanned aerial vehicles (UAVs) has been a central focus due to its wide prospect in both civilian and military applications [1–3]. As compared to a single UAV, UAVs arranged in a cooperative manner can complete a complicated mission effectively and safely. Hence, a group of UAVs via cooperative control have potential of being applied for forest fire surveillance [4, 5], natural resource exploration [6], research and rescue [7,8], and power line/grid inspection [9], respectively. Generally, there exist four major approaches to maintaining an organized topology of multiple UAVs. (1) In a leader-follower architecture, one UAV is responsible of leading the group direction while others are demanded to keep a specific distance relative to the leader [10]. (2) The real leading UAV is replaced by a virtual one, based on which all the group members pursue a unified goal [11]. This type of cooperative control scheme is regarded as virtual leader method. (3) The nature of bird flocking and fish schooling is mimic to achieve UAV cooperative control [12, 13]. A weighted average is employed to determine the * Corresponding author (email:
[email protected]) c Science China Press and Springer-Verlag Berlin Heidelberg 2017
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control of each UAV in accordance with the individual behaviour. (4) A differential game approach is recognized as an appropriate option in the situation that each UAV in a group has the independently selected objective [14]. As aforementioned, the last decade has witnessed the development of UAV formation. Requirements that drive the expectation for cooperative flight consist of reliability and safety. However, the closed-loop UAV system may exhibit poor performance or instability breaking the formation topology, if subjected to in-flight failures. Hence, fault-tolerant control that can ensure the safety of handicapped system has drawn significant attention [15–21]. Communication faults pose a severe challenge, since the performance of formation flight highly relies on the coordination and communication among group members. The wide majority of contributions concentrate on fault-tolerant cooperative control (FTCC) algorithms against communication defects/delays [22–28]. The principal idea behind the existing methods is to reorganize the formation topology via the healthy communication links. In addition, actuators in a UAV play a dominating role in bridging control commands and actual control efforts. The desired maneuvers and the cooperative flight can be achieved, as long as actuators onboard function well. Nonetheless, actuators in a UAV may encounter different sorts of faults, including stuck, partial loss of effectiveness, outage, and control surface impairment. Without any corrective and prompt reactions, the prescribed mission cannot be accomplished while the safety of UAV cooperative flight is indubitably exacerbated. Sliding mode control (SMC) is an efficient nonlinear control method, and it is used to design flight control system in the presence of actuator faults. In [29], a SMC is designed for a civil aircraft with actuator and sensor faults, and an adaptation mechanism is proposed to handle the actuator faults. In [30], a novel fault-tolerant attitude tracking control scheme is developed for flexible spacecraft with partial loss of actuator effectiveness fault under the consideration of actuator satuation. In [31, 32], higher-order SMC is used to design the fault-tolerant control system for hypersonic vehicles. The pivotal preliminary works are reported [33–36], in an effort to maintain the formation topology in the presence of UAV actuator faults. By resorting to a leader-follower architecture, SMC [33] and robust control [34] techniques are deployed to reconfigure the controllers, such that the impact of actuator faults can be eliminated. More recently, a leader-follower based FTCC methodology is presented for multiple UAVs in [35], where actuator faults, actuator saturation, and potential collisions are accounted for. In [36], the FTCC algorithm attempts to ensure the prescribed formation of multiple UAVs. Particularly, in response to actuator faults, the reference mechanism is gracefully degraded with consideration of UAV capabilities, while finite-time adaptation of control gains is achieved. Despite that recent successes in FTCC design have been gained, one issue related to actuator fault compensation needs to be addressed. It is emphasized in [15, 16], that the amount of fault recovery time is closely related to fault characteristics and system operating conditions. The physical restrictions imposed on UAV inputs and outputs are likely exceeded if fault accommodation cannot complete within an allowable amount of time. In the case that the safety constraints are violated, the breakdown cannot be prevented, inducing UAV collisions and mission failures. From both safety and mission perspectives, more emphasis on handling faults within finite time has to be put. Moreover, to the best of the authors’ knowledge, there are few papers addressing the finite-time FTCC design problem under actuator faults. Motivated by the preceding discussion, this paper presents a novel FTCC algorithm by means of finitetime SMC technique in the leader-follower context. Therefore, the developed FTCC scheme is capable of offering finite-time accommodation for actuator faults in cooperative flight. The main contributions of this paper are highlighted as follows. (1) The adaptation of the fault information can be completed within finite time. (2) The reconfigurable control is designed based on the fault information adaptation and sliding mode techniques, enforcing the tracking errors to the sliding surface within finite time, despite that actuator faults are present in the UAV group. (3) The UAV dynamics are explicitly incorporated into the FTCC design as compared to the existing single-integrator or double-integrator dynamics in UAV cooperative control. With the aid of the proposed design, corrective reactions can be exposed and the specified formation of multiple UAVs can be maintained. The rest of this paper is arranged as follows. The problem to be investigated in this study is formulated in Section 2. An FTCC scheme with combination of both inner-loop and outer-loop controllers is presented
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Table 1 Symbol
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Nomenclature
Interpretation
Subscript L
The leader UAV
Subscript iF
The ith follower
x, y
The positions with respect to x-axis and y-axis of earth reference frame
V
The forward velocity
φ
The heading angle between the forward velocity and x-axis
VLx , VLy
The forward velocity projected along x and y axes of the earth-fixed reference frame
fi
The actual forward separation between the leader and the ith follower UAV
fid
The desired forward separation between the leader and the ith follower UAV
∆fi
The forward error between the actual separation distance and the desired one
li
The actual lateral separation between the leader and the ith follower UAV
lid
The desired lateral separation between the leader and the ith follower UAV
∆li
The lateral error between the actual separation distance and the desired one
α, β
The angle of attack (AOA), sideslip angle
p, q, r
The angular rates of roll, pitch, and yaw
ω
The heading angular rate
θ, ψ
The pitch angle and yaw angle
δe , δT , δa , δr
The deflections of elevator, throttle, aileron, and rudder
in Section 3, for handling actuator malfunctions in follower UAVs. Simulation results are illustrated in Section 4, together with performance assessment. Finally, concluding remarks are given in Section 5.
2 2.1
Problem statement Geometry of UAVs cooperative flight
The kinematic models of the leader UAV and the ith follower in the group are described as ˙ XL = VL cos φL , Y˙ L = VL sin φL , φ˙ = ω , L L ˙ XiF = ViF cos φiF , Y˙ iF = ViF sin φiF , φ˙ = ω , iF
(1)
(2)
iF
where the symbols can be referred to Table 1. A trigonometric representation of the heading angle is VLy sin φL = q , 2 +V2 VLx Ly
(3)
where VLx and VLy stand for the forward velocity projected along x and y axes of the earth-fixed reference frame, respectively. As depicted in Figure 1, the group of UAVs is formed by a leader-follower topology. Therefore, the separation errors between the leader UAV and the ith follower can be written as ∆fi = (XL − XiF ) cos φL + (YL − YiF ) sin φL −fid , {z } | fi
∆li = − (XL − XiF ) sin φL + (YL − YiF ) cos φL −lid . | {z } li
(4)
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Y VL L
YL
li The leader UAV fi ViF
iF
YiF The ith follower UAV
0
XiF Figure 1
XL
X
(Color online) The formation geometry.
Subsequently, differentiating Eq. (4) results in " # " # " # ∆f˙i VL − li ωL − cos (φiF − φL ) = + ViF . ∆l˙i −fi ωL − sin (φiF − φL ) 2.2
(5)
UAV dynamics
Under a certain trimming condition, the linearized model of the leader UAV can be represented as ( x˙ L = AxL + BuL , xL (0) = xL0 , yL = CxL , T
(6)
T
where xL = [VL , αL , qL , θL , βL , pL , rL , ψL ] ∈ ℜ8 , yL = [VL , ψL − βL ] ∈ ℜ2 , uL = [δLe , δLT , δLa , δLr ]T ∈ ℜ4 denote the state vector, output vector, and control input vector, respectively. In (6), A, B, and C are system matrices with appropriate dimensions. Supposing that the ith follower UAV is the same type as the leader, the dynamics can be written as ( x˙ iF = AxiF + BuiF , xiF (0) = xiF0 , (7) yiF = CxiF . By introducing the integral of tracking error into the system states, the augmented system can be described as #"R # " # " # # " " I 0 εL 0 −C εL dt uL + rL , (8) + = 0 x˙ L 0 A xL B where rL stands for the vector of uniformly bounded reference signals and εL = rL − yL . In the interest R of simplicity, define xLa = εxLLdt ∈ ℜ10 , Aa = 00 −C ∈ ℜ10×10 , Ba = B0 ∈ ℜ10×4 , Ga = I0 ∈ ℜ10×2 . A Hence, Eq. (8) can be rewritten as x˙ La = Aa xLa + Ba uL + Ga rL .
(9)
The augmented representation of the ith follower can be obtained in a manner similar to (9): x˙ iFa = Aa xiFa + Ba uiF + Ga riF .
(10)
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Reference generator
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SMC
The leader UAV
Actuators
Faults Formation specifications
Figure 2
2.3
Main focus Outer-loop control
Sliding mode FTC
Actuators
The ith follower UAV
(Color online) The conceptual block diagram of the proposed FTCC.
UAV actuator fault model
Note that how to overcome actuator fault effects of follower UAVs is concerned in this work. Gain fault and bias fault are the faults commonly present in flight actuators. Gain fault can be identified as a multiplicative-type fault, inducing the degradation of actuator effectiveness. Bias fault is naturally an additive-type fault, resulting in a specific drift from the true value. As a consequence, the actuator fault mode of the ith follower UAV is formed as ufiF = (I − ρiF ) uiF + κiF ,
(11)
where ρiF = diag {ρiF,1 , ρiF,2 , ρiF,3 , ρiF,4 } denotes the loss of effectiveness of the actuators in the ith T follower and 0 6 ρiF,j < 1. κiF = [κiF,1 , κiF,2 , κiF,3 , κiF,4 ] represents the bias faults. Thus, the handicapped follower UAV can be expressed as x˙ iFa = Aa xiFa + Ba (I − ρiF ) uiF + Ba κiF + Ga riF . 2.4
(12)
Objective
The objective of this study is to develop an FTCC scheme such that: (1) The required leader-follower formation can be guaranteed despite that actuator malfunctions exist in the ith follower UAV; (2) The process of actuator fault accommodation can be completed within a finite amount of time by using sliding mode approaches.
3
Fault-tolerant cooperative control design
The conceptual structure of the proposed FTCC system is illustrated in Figure 2. As reported in the authors’ previous work [36], the FTCC scheme is composed by three units: (1) the reference generator to determine the flight velocity and attitude of the leader UAV; (2) the outer-loop control for the ith follower UAV to generate the feasible commands; and (3) the inner-loop fault-tolerant control (FTC) for the ith follower UAV to counteract the adverse effects resulting from actuator faults. In regard to the reference generator, it is profoundly important to consider the actuation systems capability of the overall UAV group. At the design stage of the leader reference generation, more emphasis must be placed on the kinematics constraint of formation flight. For the sake of space, the design procedure is omitted in this paper, while the details can be referred to [36]. 3.1
Outer-loop control design
In the context of leader-follower based FTCC, the references of the ith follower are produced in light of the distance error model. Focusing on the outer-loop of the ith follower UAV, the goal is to generate the
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commands, which can be tracked by the inner-loop FTC. Using a proportional controller with kiF1 > 0 and kiF2 > 0, Eq. (5) can be further formed as " # " # " # " # ∆f˙i VL − li ωL − cos (φiF − φL ) −kiF1 ∆fi = + ViF = . (13) ∆l˙i −fi ωL − sin (φiF − φL ) −kiF2 ∆li Using (13), one can obtain "
cos (φiF − φL ) sin (φiF − φL )
#
ViF =
"
kiF1 ∆fi + VL − li ωL kiF2 ∆li − fi ωL
#
.
For facilitating the generation of the inner-loop reference signals, let ( hiF1 = kiF1 ∆fi + VL − li ωL , hiF2 = kiF2 ∆li − fi ωL .
Subsequently, the reference commands for the ith follower UAV are generated as q r ViF = h2iF1 + h2iF2 , φriF
=
φL + arctan (hiF2 /hiF1 ) ,
φL + arctan (hiF2 /hiF1 ) + π, φ + arctan (h /h ) − π, L iF2 iF1
(14)
(15)
(16)
if hiF2 > 0, if hiF1 > 0, hiF2 < 0,
(17)
if hiF1 < 0, hiF2 < 0.
Remark 1. In comparison to the outer-loop design presented in [34,36], the proposed method is capable of ensuring the velocity command of the follower UAV always positive. Moreover, the heading angle reference can be yielded straightforwardly as indicated in (17), while this reference must be obtained via the integral of the angular rate in [34, 36]. 3.2
Inner-loop FTC design
The inner-loop FTC is to track the references delivered from the outer-loop in the case of actuator faults. In order to facilitate the design process of the sliding mode fault-tolerant controller (SMFTC), Eq. (12) can be rewritten as x˙ iFa = Aa xiFa + Ba uiF + Ba (κiF − ρiF uiF ) + Ga riF . (18) Letting ∆fault = κiF − ρiF uiF , Eq. (18) can be thereby recast as x˙ iFa = Aa xiFa + Ba uiF + Ba ∆fault + Ga riF .
(19)
Remark 2. The “uncertain” term ∆fault contains the information of both gain and bias faults. Additionally, it is assumed that there exist two unknown constants ε1 and ε2 such that the norm of ∆fault satisfying k∆fault k 6 ε1 + ε2 kxiFa k. Since ∆fault = κiF − ρiF uiF and uiF are functions of the states, this assumption is reasonable. In this study, an ideal reference model with uniformly bounded reference inputs is exploited as x˙ m = Am xm + Bm riF ,
(20)
where Am ∈ ℜ10×10 is a Hurwitz matrix and Bm ∈ ℜ10×2 . Note that Am and Bm are known matrices determined by UAV performance specifications. By comparing the faulty model (19) and the reference model (20), one can obtain e˙ = x˙ iFa − x˙ m = Aa xiFa + Ba uiF + Ba ∆fault + (Ga − Bm )riF − Am xm
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= Am e + (Aa − Am )xiFa + Ba uiF + Ba ∆fault + (Ga − Bm )riF ,
(21)
where e = xiFa − xm . When the matching conditions Aa − Am = Ba M and Ga = Bm hold, the error dynamics can be represented as e˙ = Am e + Ba M xiFa + Ba uiF + Ba ∆fault . (22) To this end, the main objective of inner-loop FTC is, without the requirement of the information of upper bound of perturbation ∆fault , to develop an adaptive SMFTC, allowing the faulty system of (18) to track the reference model of (20). In general, the design procedure of SMFTC can be divided into two phases. First, it is necessary to establish a sliding surface for the faulty system, Z t S = Ne − N (Am + Ba K) edτ , (23) 0
where N ∈ ℜ4×10 is a constant matrix with full rank. It is expected that not only the sliding motion can occur on the manifold, but also the closed-loop system can preserve the desired dynamic performance. It is worth mentioning that N is selected to guarantee that N Ba is nonsingular and the matrix K ∈ ℜ4×10 is designed to satisfy Re [λ (Am + Ba K)] < 0. (24) As long as N ∈ ℜ4×10 , K ∈ ℜ4×10 , and the developed SMFTC are properly designed, the tracking error dynamics can be driven to the sliding surface S = 0 and remain thereafter. Thus, the resulting closed-loop system can be stabilized and the tracking performance can be preserved. After presenting the sliding surface, the next step is to design an appropriate SMFTC law in the event of actuator faults such that: (1) the sliding surface will attract the trajectories of (22); and (2) these trajectories will remain on the sliding surface for all subsequent time. The upper bound of ∆fault is the key to construct the SMFTC law with the switching portion dominating the influence of actuator faults. Nevertheless, it is difficult to obtain such kind of knowledge in practice due to the complication of ∆fault . To address this issue, two adaptive gains εˆ1 and εˆ2 are applied to adapt the unknown parameters ε1 and ε2 , respectively. The adaptation errors are defined as ε˜1 (t) = εˆ1 (t) − ε1 and ε˜2 (t) = εˆ2 (t) − ε2 , respectively. The adaptive SMFTC for (22) is constructed as uiF = un + uad , un = −M xiFa + Ke − η(N Ba )−1
(25) S , kSk
(26)
BaT N T S , kS T N Ba k
(27)
where η is a positive constant. The adaptive gains can be achieved as
T
1 S N Ba , S = 6 0, kSk εˆ˙1 = γ1 0, S = 0,
(28)
uad = − (ˆ ε1 + εˆ2 kxiFa k)
T
1 S N Ba kxiFa k , S = 6 0, kSk εˆ˙2 = γ2 0, S = 0,
(29)
where γ1 and γ2 are positive design constants, affecting the adaptation rates of ε1 and ε2 , respectively. Theorem 1. Consider the tracking error dynamics (22) in the presence of UAV actuator faults. If the proposed SMFTC law formed by (25)–(27), the sliding function (23), and the adaptation laws in (28) and (29) are employed, then the dynamics of (22) are globally asymptotically stable.
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Proof.
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Define a Lyapunov function candidate as 1 1 V = kSk + γ1 ε˜21 + γ2 ε˜22 . 2 2
(30)
Differentiating V with respect to time gives S T S˙ V˙ = + γ1 ε˜1 ε˜˙1 + γ2 ε˜2 ε˜˙2 . kSk
(31)
According to (22) and (23), one can render S˙ = N e˙ − N (Am + Ba K)e = N Ba M xiFa + N Ba uiF + N Ba ∆fault − N Ba Ke.
(32)
Moreover, by noting that ε˜˙1 (t) = εˆ˙1 (t),
ε˜˙2 (t) = εˆ˙2 (t),
(33)
and applying the matching conditions, it is yielded that ST V˙ = (N Ba M xiFa + N Ba uiF + N Ba ∆fault − N Ba Ke) + γ1 ε˜1 ε˜˙1 + γ2 ε˜2 ε˜˙2 kSk ST = kSk +
−1
N Ba M xiFa + N Ba −M xiFa + Ke − η(N Ba )
ST (N Ba ∆fault − N Ba Ke) + γ1 ε˜1 ε˜˙1 + γ2 ε˜2 ε˜˙2 kSk
BTN TS S
− (ˆ ε1 + εˆ2 kxiFa k) aT
S N Ba kSk
! S N Ba BaT N T S
+ N Ba ∆fault + γ1 ε˜1 ε˜˙1 + γ2 ε˜2 ε˜˙2 −η − (ˆ ε1 + εˆ2 kxiFa k) T
S N Ba kSk
T
T
S N Ba
S N Ba 6 −η − (ˆ ε1 + εˆ2 kxiFa k) + (ε1 + ε2 kxiFa k) kSk kSk
T
T
S N Ba kxiFa k
S N Ba + ε˜2 + ε˜1 kSk kSk
!!
ST = kSk
= −η < 0.
(34)
In accordance with [37], the inequality (34) guarantees that V and S will approach to zero in finite time tf 6 t0 + V (tη 0 ) . Since the value of V is bounded, ε˜i , i = 1, 2 (and hence εˆi ) are bounded as well. In addition to the boundedness of ε˜1 and ε˜2 , the solution of εˆ2 is
Z tf
S T N Ba kxiFa k 1 εˆ2 = dτ +ˆ ε2 (0), S = 6 0. (35) γ2 t0 kSk Provided that εˆ2 is bounded and the integral is nonnegative, the state variable of xiFa must be bounded for t0 6 t 6 tf . When the handicapped system is in the sliding mode, the equivalent control can be attained by writing S˙ = 0 as ueq = −M xiFa − ∆fault + Ke.
(36)
One can obtain the closed-loop error dynamic equation after the malfunctioned system entering the sliding manifold by substituting (36) into (22). Subsequently, the resulting equation is e˙ = (Am + Ba K)e.
(37)
It is concluded that the dynamic response of e is globally asymptotic stable if the condition in (24) is satisfied. This completes the proof.
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Remark 3. The developed adaptive SMFTC can ensure the finite-time stability of the closed-loop system under UAV actuator malfunctions, while the control parameters are tuned in response to the UAV status using the adaptive laws. When comparing to the design approach in [36], the advantages of the investigated method are that: (1) the gains εˆ1 and εˆ2 can be adapted within finite time, which can be seen from (30) and (34); and (2) the fault compensation can be completed within finite time. In addition, the UAV dynamics are explicitly incorporated into the FTCC design, which is different from the normally used single-integrator or double-integrator dynamics in UAV cooperative control problems. Remark 4. In real applications, S cannot become exactly zero in finite time due to nonlinearities, noises, and switching delays. Thus the values of εˆ1 and εˆ2 may become boundless. One option of overcoming this difficulty is to modify the adaptive tuning laws (28) and (29) by exploiting the dead zone technique [38]:
T
1 S CBa , kSk > Φ, kSk (38) εˆ˙1 = γ1 0, kSk 6 Φ,
T
1 S CBa kxiFa k , kSk > Φ, kSk (39) εˆ˙2 = γ2 0, kSk 6 Φ, where Φ is a small positive constant.
4 4.1
Simulation results and evaluation UAV model and design parameters
The dynamics of the selected UAV can be referred to [34]. The system matrices are given as follows: −0.0334 −2.9770 0 −9.81 −1.0750 0.2453 −0.0016 −4.1330 0.9800 0 , BLong = 0.3470 −4.1330 , ALong = (40) 0 0.0077 −140.20 −4.435 0 −140.22 0 0 1.0000 0 0 0
ALat
−0.7320 0.0143 −0.9960 0.0706
−893.00 −9.0590 2.0440 = 101.673 0.0186 −1.2830 0
A=
"
ALong
0
0
ALat
0 #
,B =
1.0000 " BLong 0
0 0 0 0 BLat
,
#
,
0
0.2440
328.653 308.498 , BLat = 47.528 102.891 0 0 " # 1 0 00 0 0 0 0 C= . 0 0 0 0 −1 0 0 1
The reference model for the leader and follower UAVs is selected as " # " # Am,long 0 Bm,long 0 x˙ m (t) = xm (t) + r(t), 0 Am,lat 0 Bm,lat where r(t) can be either rL or riF (t), and 0 −1.000 0 0 0 −1.600 2.200 −2.100 −1.900 −18.30 Am,long = −5.100 6.100 −10.00 1.600 −9.500 , −247.1 342.8 −68.90 −250.3 −1204.2 0 0 0 1.000 0
Bm,long
1 0 = 0, 0 0
(41)
(42)
(43)
(44)
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Am,lat
0
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1.0000
0
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0
−1.0000
0.2826 −0.7876 −0.0562 −1.1188 −0.5025 = 169.7872 188.4091 −251.3218 −76.1241 −361.3527 , 92.0458 244.7857 −51.8445 −41.9064 −189.1341 0 0 0 1.0000 0
Bm,lat
1 0 = 0. 0 0
(45)
The outer-loop control gains for followers are designed as kiF1 = kiF2 = 0.3. The matrices N and K in the sliding manifold are designed as " # 0.4892 −0.6988 0.0039 0.7213 2.7907 Klong = , 0.8442 −1.0116 0.6340 0.0107 1.7533 " # 0 −0.0001 0.0054 −0.0161 0 Nlong = , 0 0.0001 −0.0025 0.0172 0 " # (46) −0.2820 0.2763 −0.2050 0.1272 0.2678 Klat = , 0.4564 −0.1817 −0.1225 −0.2069 −0.3387 " # 0 −0.0001 0 −0.0071 0 Nlat = . 0 0.0143 −0.2411 −0.0007 0 The designed constant η = 1, and the adaptive rate parameters γ1 = γ2 = 10. 4.2
Simulation scenarios
Three UAVs with the triangular topology of the cooperative flight are selected in the simulation runs. The desired distances between the leader and the follower UAV#1 corresponding to the forward and lateral directions are f1d = 100 m and l1d = −100 m, respectively. The desired distances between the leader and the follower UAV#2 with respect to the forward and lateral directions are f2d = 100 m and l2d = 100 m. The initial positions of the leader UAV, follower UAV#1, follower UAV#2 are (200, 200), (0, 400) and (0, 0), respectively. The gain and bias faults in the follower UAV#1 are involved in the simulation, which are represented as ( ( 0, 0 6 t < 10, 0, 0 6 t < 10, ρ1F,1 = κ1F,1 = 0.5, t > 10, −5, t > 10, ( ( 0, 0 6 t < 10, 0, 0 6 t < 10, κ1F,2 = ρ1F,2 = 0.3, t > 10, −8, t > 10, ( ( (47) 0, 0 6 t < 10, 0, 0 6 t < 10, κ1F,3 = ρ1F,3 = 0.3, t > 10, 7, t > 10, ( ( 0, 0 6 t < 10, 0, 0 6 t < 10, ρ1F,4 = κ1F,4 = 0.3, t > 10, −7, t > 10. 4.3
Performance evaluation
The trajectories of the UAV group are shown in Figure 3. When the errors are present due to the various initial positions of each UAV, the FTCC can enforce the group members UAV#1 and UAV#2 to keep the specific distances with respect to the leader UAV. From Figure 3, it can be observed that the formation topology of the cooperative flight can be ensured by resorting to the developed FTCC scheme, despite that the follower UAV#1 encounters both the gain and bias faults at t = 10 s. The velocity response of the faulty UAV is highlighted in Figure 4. Although the actuator faults take place in the follower UAV#1, the velocity is able to return to the reference value with a short period of adjustment. As can be observed from Figure 5, the tracking performance of the heading angle is satisfactory even in the presence of the actuator malfunctions. Thus, the outputs of the faulty UAV are capable of converging to
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Leader Follower #1 Follower #2
1400 1200 1000 800 y (m)
600 400 200 0 −200 −400 −600 −500
0
Figure 3
500
1000 x (m)
1500
2000
2500
(Color online) The trajectories of 3 UAVs.
80 Velocity of UAV#1 Reference
V (m/s)
60 40 20 Faults occur at t =10 s
0 −20
Figure 4
0
5
10
15
20
25 30 Time (s)
35
40
45
50
(Color online) The velocity of the follower UAV#1 and the reference velocity.
40
φ (°)
20 0 Faults occur at t = 10 s −20 −40
Figure 5
Heading angle of UAV#1 Reference 0
5
10
15
20
25 30 Time (s)
35
40
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(Color online) The heading angle of the follower UAV#1 and the reference heading angle.
the intended references within finite time under the designed FTCC. The deflections of the actuators of UAV#1 are depicted in Figures 6–9. The key observation lies in that the actuators of the faulty UAV can be appropriately utilized, not only to accommodate the gain and bias faults, but also to achieve the required maneuvers which are needed to keep the specific distances with respect to the leader UAV. As a result, the forward and lateral distances of the follower UAV#1 relative to the leader UAV can be preserved at the desired values under the actuator fault condition.
5
Conclusion
Actuator faults in a group of multiple UAVs dramatically compromise the UAVs safety. An FTCC architecture, with integration of both outer-loop and inner-loop suitable for cooperative control, is presented
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25
20 15
15
δT
δe (°)
20 Faults occur at t=10 s
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5 0
Faults occur at t=10 s
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Time (s) Figure 6
The deflection of δe in the follower UAV#1.
Figure 7
40
δr (°)
δa (°)
40
45
50
The deflection of δT in the follower UAV#1.
20
Faults occur at t=10 s
0 −20
Figure 8
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30
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−40 0
25 30 Time (s)
10 0
Faults occur at t=10 s
−10 5
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The deflection of δa in the follower UAV#1.
−20
0
Figure 9
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The deflection of δr in the follower UAV#1.
against actuator faults. The significant merits of the developed method lie in three aspects. (1) The adaptation of the fault information can be completed within finite time; (2) The finite-time stability of the faulty UAV can be guaranteed so that the formation topology can still be maintained; and (3) The UAV dynamics are explicitly taken into account over the design process. The simulations show that the investigated scheme can be successfully used to deal with scenarios involving actuator faults. Acknowledgements This work was supported in part by Natural Sciences and Engineering Research Council of Canada, National Natural Science Foundation of China (Grant Nos. 51575167, 61403407, 61573282, 61603130), Shaanxi Province Natural Science Foundation (Grant No. 2015JZ020), Hunan Province Natural Science Foundation (Grant No. 2017JJ3041), and Fundamental Research Funds for the Central Universities (Grant No. 531107040965). The authors would like to thank the support from the Collaborative Innovation Center of Intelligent New Energy Vehicle and the Hunan Collaborative Innovation Center for Green Car. Thanks also to the associate editor and anonymous reviewers for the constructive comments. Conflict of interest
The authors declare that they have no conflict of interest.
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