2014 22nd Mediterranean Conference on Control and Automation (MED) University of Palermo. June 16-19, 2014. Palermo, Italy
Robust Formation and Reconfiguration Control of Multiple VTOL UAVs: Design and Flight Test Fang Liao , Xiangxu Dong, Feng Lin National University of Singapore, 117411, Singapore,
Rodney Teo DSO National Laboratories, 118230, Singapore,
Jian Liang Wang Nanyang Technological University, 639798, Singapore
Abstract— In this paper, a distributed robust feedback control strategy with inter-vehicle collision avoidance is proposed for formation and reconfiguration control of a team of VTOL UAVs. A potential-field approach is used to generate a desired velocity for each UAV which ensures that the team of UAVs can perform formation and reconfiguration, avoid inter-vehicle collision as well as track a specified virtual leader. Each UAV is controlled to track its desired velocity subject to dynamic constraints. The proposed feedback control is robust against error disturbances due to dynamic constraints and measurement noise. A formation flight test of three quadrotor UAVs demonstrates the effectiveness and robustness of the proposed formation control strategy.
I. I NTRODUCTION Formation flying is a disciplined flight of a team of aircraft. It requires the team of aircraft follow a predefined trajectory while maintaining a desired spatial pattern. Formation flight of multiple unmanned aerial vehicles (UAVs) has broad applications such as terrain and utilities inspection [10], search and rescue [5], disaster monitoring [15], aerial mapping [3], traffic monitoring [12], reconnaissance mission [16], and surveillance [17]. In recent years, motivated by the wide range of possible military and civilian applications, formation control of multiple UAVs has attracted significant attention from the UAV research community. In the existing literature, formation control approaches are classified into the following four basic strategies: multiple-input multiple-output (MIMO) strategy, leader-follower (LF) strategy, virtual structure (VS) strategy, and behavior-based strategy. MIMO strategy [7], [8], [13], [18] treats the team of UAVs as a MIMO plant. Optimality and stability are the primary advantages of MIMO strategy. However, high information requirement is its main disadvantage. In LF strategy [4], [6], [11], [19], each follower is required to have at least one leader. The advantage of this strategy is to simplify formation control problems to individual tracking problems. Its disadvantage is that the leader is a single point of failure for the formation. VS strategy [9], [14] considers the entire formation as a rigid body. Its main advantage is able to track the planned formation trajectory accurately as the virtual center is on the planned
trajectory. One of its disadvantages is that it is based on an open-loop control. Vehicles will get out of formation if the virtual structure moves too fast to track or the system is affected by disturbance. The behavioral strategy [1], [2] combines the outputs of multiple controllers designed for achieving different and possibly competing behaviors, such as collision avoidance, obstacle avoidance, goal seeking, trajectory tracking and formation keeping behaviors. All the behaviors are summed together in the behavioral strategy through suitable weight coefficients which set the relative priority between them. It is natural to derive control strategies when UAVs have multiple competing objectives. However, the behaviors may destructively interfere each other. Since different strategy has different applications and its own advantages and disadvantages, in this paper, a combined strategy is adopted for formation control and collision avoidance of a team of UAVs with dynamic constraints such as acceleration, velocity, turn rate and flight path angle constraints. An unified distributed cascade feedback control design is proposed for the team of UAVs to form and reconfigure formation flight and avoid inter-vehicle collision. Throughout this paper, the notation I𝑛 represents the identity matrix of order 𝑛 and 1𝑛 represents the vector [1 1, ⋅ ⋅ ⋅ , 1]𝑇 with 𝑛 elements. II. UAV P LATFORM The UAVs used in the formation flight are the commercial quadrotors, see Figure 1. As these quadrotor UAVs are equipped with NAZA controllers that implement the attitude and heaving control, they are quite stable. The limitation of the UAV is specified in Table 1.
Fang Liao, corresponding author, is with Temasek Laboratories, 5A Engineering Drive 1, Singapore 117411, E-mail:
[email protected]
978-1-4799-5901-3/14/$31.00 ©2014 IEEE
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TABLE I L IMITATION OF THE Q UADROTOR UAV Roll angle Pitch angle Roll rate Pitch rate Yaw rate Cruise speed
Minimum Value −0.3 rad −0.3 rad −0.9 rad/s −0.9 rad/s −1.8 rad/s 0 m/s
Maximum Value 0.3 rad 0.3 rad 0.9 rad/s 0.9 rad/s 1.8 rad/s 5 m/s
Since an unconnected communication network may not form formation flying, we assumed that Assumption 1: The communication network 𝒢 is always connected, namely, 𝑟𝑎𝑛𝑘(L𝒢 ) = 𝑛 − 1. To avoid collision among UAVs, we define the intervehicle collision avoidance set of the vehicle 𝑖 as △
𝒟𝑖 = {𝑗 ∈ 𝒱 : 𝑑𝑖𝑗 (𝑡) ≤ 𝑅𝑎 , 𝑖 ∕= 𝑗}
(6)
where 𝑅𝑎 is the collision detection distance. B. Outer-loop Control Fig. 1.
Consider the following potential function for a team of 𝑛 UAVs.
Quadrotor
𝑉 (𝑡) = 𝑉 𝑓 (𝑡) + 𝑉 𝑎 (𝑡) + 𝑉 𝑡 (𝑡) III. ROBUST F ORMATION AND R ECONFIGURATION C ONTROL D ESIGN
Here 𝑉 𝑓 (𝑡) ∈ ℜ in (7) is the potential function of formation described by
In this section, a unified distributed robust formation and reconfiguration control strategy with inter-vehicle collision avoidance is proposed for a team of UAVs with dynamic constraints. This strategy is a combination of behavior strategy, MIMO strategy and LF strategy. Potential-Field approach is used to guarantee the stability of formation and inter-vehicle collision avoidance. A cascade control structure is presented for the design of formation control. A. Communication Network △
△
𝑉 𝑓 (𝑡) =
𝑛 𝑛 1∑ ∑ 𝜔𝑖𝑗 𝑉𝑖𝑗𝑓 (𝑡) 2 𝑖=1
(8)
𝑗=1,𝑗∕=𝑖
where 𝜔𝑖𝑗 is defined as in (5) and 𝑉𝑖𝑗𝑓 (𝑡) ∈ ℜ is the potential function of the vehicles 𝑖 and 𝑗 given by 𝑉𝑖𝑗𝑓 (𝑡) =
1 𝑇 [p𝑖 (𝑡) − p𝑗 (𝑡) − (p𝑓 𝑖 − p𝑓 𝑗 )] 2 [p𝑖 (𝑡) − p𝑗 (𝑡) − (p𝑓 𝑖 − p𝑓 𝑗 )]
(9)
3
Let 𝒢 = (𝒱, ℰ) denote the undirected communication network with the vertex set 𝒱 representing the vehicles 𝒱 = {1, 2, ⋅ ⋅ ⋅ , 𝑛}
(7)
(1)
and the edge set ℰ representing the communication links between the vehicles
with p𝑖 (𝑡), p𝑗 (𝑡) ∈ ℜ the positions of UAVs 𝑖 and 𝑗 in NED frame and p𝑓 𝑖 , p𝑓 𝑗 ∈ ℜ3 the assigned relative positions of UAVs 𝑖 and 𝑗 in the desired formation configuration. 𝑉 𝑎 (𝑡) ∈ ℜ in (7) is the potential function of inter-vehicle collision avoidance 𝑛 𝑛 1∑ ∑ 𝑉 𝑎 (𝑡) = 𝑉𝑖𝑗𝑎 (𝑡) (10) 2 𝑖=1 𝑗=1,𝑗∕=𝑖
△
ℰ = {𝑒𝑖𝑗 : 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛, 𝑗 ∈ 𝒩𝑖 }
(2)
where 𝒩𝑖 is the neighbor set of the vehicle 𝑖 △
𝒩𝑖 = {𝑗 ∈ 𝒱 : information from UAV 𝑗 is received}
(3)
The graph Laplacian L𝒢 of the communication network 𝒢 is defined element-wise as ⎧ if 𝑖 ∕= 𝑗 ⎨ −𝜔𝑛𝑖𝑗 , △ ∑ (4) (L𝒢 )𝑖𝑗 = 𝜔𝑖𝑘 , if 𝑖 = 𝑗 ⎩
𝑉𝑖𝑗𝑎 (𝑡)
where ∈ ℜ the inter-vehicle collision avoidance potential function of the vehicles 𝑖 and 𝑗 given by { (𝜌𝑎 −𝑑𝑖𝑗 (𝑡))/(𝜌𝑎 −𝑅𝑎 ) 𝜖𝑎 , if 𝑗 ∈ 𝒟𝑖 𝑉𝑖𝑗𝑎 (𝑡) = (11) 0, otherwise with 𝑑𝑖𝑗 (𝑡) ∈ ℜ the distance between UAVs 𝑖 and 𝑗, 𝑅𝑎 the collision detection distance, 𝜖𝑎 ∈ ℜ a very small positive scalar and 𝜌𝑎 < 𝑅𝑎 . 𝑉 𝑡 (𝑡) ∈ ℜ in (7) is the potential function of follower tracking a pre-defined virtual leader
𝑘=1,𝑘∕=𝑖
𝑡
𝑉 (𝑡) =
where the weight 𝜔𝑖𝑗 is assigned to each communication link 𝑒𝑖𝑗 and given by { 1, if information from UAV 𝑗 is received 𝜔𝑖𝑗 = (5) 0, otherwise Obviously, the graph Laplacian L𝒢 is with the property that zero is one of its eigenvalues and all its nonzero eigenvalues are positive (Gershgorin Circle Theorem), namely, L𝒢 ≥ 0. Furthermore, L𝒢 has a right eigenvector 1𝑛 and a left eigenvector 1𝑇𝑛 corresponding to 0, namely, L𝒢 1𝑛 = 0 and 1𝑇𝑛 L𝒢 = 0.
𝑛 ∑
𝑐𝑖 𝑉𝑖𝑡 (𝑡)
(12)
𝑖=1
where the weight 𝑐𝑖 ≥ 0 satisfies { 𝑐 , if information from virtual leader is obtained 𝑐𝑖 = 𝑖 0, otherwise with
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𝑛 ∑
(13)
𝑐𝑖 > 0, and 𝑉𝑖𝑡 (𝑡) ∈ ℜ is given by
𝑖=0
𝑉𝑖𝑡 (𝑡) =
1 𝑇 [p𝑖 (𝑡) − p𝑣 − (p𝑓 𝑖 − p𝑓 𝑣 )] 2 [p𝑖 (𝑡) − p𝑣 − (p𝑓 𝑖 − p𝑓 𝑣 )]
(14)
with p𝑣 ∈ ℜ3 the waypoint of the virtual leader in NED frame and p𝑓 𝑣 ∈ ℜ3 the assigned relative position of the virtual leader in the desired formation configuration. Remark 1: If the communication network 𝒢 defined in (1)-(5) is connected, the formation configuration is determined by the relative position between neighbors. Hence, the potential function 𝑉 𝑓 (𝑡) = 0 ensures that the formation is maintained if the communication network 𝒢 is connected. From (11). it is noted that the potential function 𝑉𝑖𝑗𝑎 (𝑡) will tend to infinity if 𝑑𝑖𝑗 (𝑡) tends to zero by choosing 𝜌𝑎 that is close to 𝑅𝑎 and choosing a sufficient small 𝜖𝑎 > 0. Therefore, 𝑉 𝑎 (𝑡) = 0 can avoid the inter-vehicle collision between UAVs. If the virtual leader’s information is received by at least one UAV in the team and the communication network 𝒢 is connected, the potential function 𝑉 (𝑡) = 0 ensures that the team of UAVs maintains the formation and tracks the virtual leader with guaranteed inter-vehicle collision avoidance. Theorem 1: Consider a team of 𝑛 UAVs satisfying Assumption 1. For a pre-defined virtual leader connected with at least one UAV in the team, the team of UAVs asymptotically achieves formation flight and follows the virtual leader with guaranteed inter-vehicle collision avoidance if all the individual UAVs fly in the following corresponding speed. ⎧ ⎨∑ p˙ 𝑖 (𝑡) = K𝑓 [p𝑖 (𝑡) − p𝑗 (𝑡) − (p𝑓 𝑖 − p𝑓 𝑗 )] ⎩ 𝑗∈𝒩𝑖
+𝑐𝑖 [p𝑖 (𝑡) − p𝑣 − (p𝑓 𝑖 − p𝑓 𝑣 )] ⎫ ⎬ ∑ + 𝛼𝑖𝑗 (𝑡)[p𝑖 (𝑡) − p𝑗 (𝑡)] ⎭ 𝑗∈𝒟𝑖
𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛
(15)
where K𝑓 ∈ ℜ3×3 < 0 is a symmetric negative definite 𝑛 ∑ 𝑐𝑖 > 0, matrix, 𝑐𝑖 ∈ ℜ ≥ 0 is defined in (13) and satisfies 𝑖=0
𝒩𝑖 and 𝒟𝑖 are defined in (3) and (6), respectively, and (𝜌 −𝑑𝑖𝑗 (𝑡))/(𝜌𝑎 −𝑅𝑎 )
𝛼𝑖𝑗 (𝑡)
=
−
𝜖𝑎 𝑎
ln 𝜖𝑎 𝑑𝑖𝑗 (𝑡)(𝜌𝑎 − 𝑅𝑎 )
(16)
Proof: The proof is omitted due to space limit. ♦ Remark 2: Theorem 1 presents a distributed approach to generate the desired velocity for each UAV. If all the UAV fly in their corresponding desired velocities, it is guaranteed that the team of UAVs can perform formation flight/reconfiguration and track the given virtual leader as well as avoid inter-vehicle collision. Remark 3: The larger the control gain K𝑓 in (15) is, the more robust the system against the disturbance is. However, there is trade-off between robustness and velocity constraints. Remark 4: Uniform disturbances will not affect the formation flight and inter-vehicle collision avoidance of a team of UAVs. However, they will affect the performance of the team of UAVs tracking the virtual leader. The above discusses the outer-loop of the cascade control that generates the desired velocity for each UAV. In the
following, the inner-loop of the cascade control will be discussed to drive the velocity of each UAV to track its desired velocity subject to the dynamic constraints. C. Inner-loop Control 𝑇 Denote p˙ 𝑟 (𝑡) = [p˙ 𝑟1 , p˙ 𝑟2 , ⋅ ⋅ ⋅ , p˙ 𝑟𝑛 ] ∈ ℜ3𝑛 the desire velocity of the group of vehicles with the desired velocity of the 𝑖th UAV p˙ 𝑟𝑖 given by (15). Assumption 2: The desired velocity p˙ 𝑟 (𝑡) is a slowly ¨ 𝑟 (𝑡) = 0. changing vector and assume that p Theorem 2: Under Assumptions 1 and 2, the velocity of 𝑇 ˙ the team of UAVs p(𝑡) = [p˙ 1 , p˙ 2 , ⋅ ⋅ ⋅ , p˙ 𝑛 ] converge to 𝑇 the desired velocity p˙ 𝑟 (𝑡) = [p˙ 𝑟1 , p˙ 𝑟2 , ⋅ ⋅ ⋅ , p˙ 𝑟𝑛 ] asymptotically if all the individual UAVs fly in the following corresponding acceleration. ⎧ ⎨∑ ¨ 𝑖 (𝑡)=K𝑣 p [p˙ 𝑖 (𝑡) − p˙ 𝑟𝑖 (𝑡) − (p˙ 𝑗 (𝑡) − p˙ 𝑟𝑗 (𝑡))] ⎩ 𝑗∈𝒩𝑖
+𝑐𝑖 [p˙ 𝑖 (𝑡) − p˙ 𝑟𝑖 (𝑡)]} ,
𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛 (17)
where K𝑣 ∈ ℜ3×3 < 0 is a symmetric negative definite 𝑛 ∑ 𝑐𝑖 > 0, matrix, 𝑐𝑖 ∈ ℜ ≥ 0 is defined in (13) and satisfies 𝑖=0
and 𝒩𝑖 is defined in (3). Proof: The proof is omitted due to space limit. ♦ After consideration of the control constraints and measurement noise, the closed inner-loop system of the team of vehicles becomes { ¨ (𝑡) = K𝑉 (L+C) [p(𝑡)− ˙ p p˙ 𝑟 (𝑡)]+L𝑁 w𝑐 (𝑡) (18) G: ˙ z(𝑡) = p(𝑡) − p˙ 𝑟 (𝑡) where w𝑐 (𝑡) ∈ ℜ3𝑛 is the disturbance error due to control constraints, wind disturbance and measurement noise, and z(𝑡) is the performance output. Since uniform disturbances will not affect formation flying and inter-vehicle collision avoidance of a team of UAVs as discussed in Remark 4, here we introduce L𝑁 = L𝑛 ⊗ I3 with L𝑛 = 𝑛I𝑛 − 1𝑛 1𝑇𝑛 to remove uniform disturbances and only consider nonuniform disturbances. In the following theorem, an 𝐻∞ control design approach is proposed to determine the gain matrix K𝑉 = I𝑛 ⊗ K𝑣 for the velocity tracking system (18). Assumption 3: The disturbance error w𝑐 (𝑡) is bounded. Lemma 1: Assume that Assumptions 1-3 hold. Consider the system G as in (18). For a given scalar 𝜇 > 0, supposed that there exist a matrix Y ∈ ℜ3×3 and a symmetric positive definite matrix Q ∈ ℜ3×3 > 0 satisfying ⎡ ⎤ (L𝒢 + C𝑓 ) ⊗ (Y + Y𝑇 ) L𝑛 ⊗ Q I3𝑛 ⎣ L𝑛 ⊗ Q −𝜇I3𝑛 0 ⎦ 330𝑠 The assigned relative position p𝑓 𝑣 of the virtual leader in the desired formation shape is the same as that of UAV 2, i.e., p𝑓 𝑣 = p𝑓 𝑟2 . The waypoints p𝑣 of the virtual leader in NED frame are specified as in Table 3. The yaw angle 𝜓𝑣
where 𝛾𝑓 and 𝜓𝑓 are the flight path angle and yaw angle of the desired formation shape. Similarly, to make the yaw angles of the team of UAVs converge to the yaw angle 𝜓𝑣 of the virtual leader, the reference yaw rate is given by ⎧ ⎫ ⎨∑ ⎬ 𝜓˙ 𝑟𝑖 (𝑡) = K𝜓 [𝜓𝑖 (𝑡) − 𝜓𝑗 (𝑡)] + 𝑐𝑖 [𝜓𝑖 (𝑡) − 𝜓𝑣 ] ⎩ ⎭
TABLE II WAYPOINT OF V IRTUAL L EADER 20𝑠 ≤ 𝑡 < 90𝑠
𝑡 < 20𝑠 ⎡
⎡
⎤
−56.3810 ⎣ 20.5227 ⎦ −5
0 ⎣ 0 ⎦ −5
p𝑣
⎤
145𝑠 ≤ 𝑡 < 200𝑠
200𝑠 ≤ 𝑡 < 305𝑠
⎡
⎡
90𝑠 ≤ 𝑡 < 145𝑠 ⎡
⎤ −56.3810 ⎣ 50.5227 ⎦ −5 𝑡 ≥ 305𝑠
𝑗∈𝒩𝑖
𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛 (24)
p𝑣
where K𝜓 ∈ ℜ3×3 < 0 is a symmetric negative definite matrix and 𝜓𝑖 is the yaw angle of UAV 𝑖. IV. C ONTROLLER PARAMETERS AND S IMULATION Three quadrotor UAVs (shown in Figure 1) are used to demonstrate formation flight and reconfiguration. According to the conditions in Section II, the controller parameters are not unique. They can be calculated or chosen as follows. K𝑓 𝜖𝑎
= −0.1I3 , =
0.1,
K𝑣 = −0.35I3 ,
𝜌𝑎 = 4.5,
𝑅𝑎 = 5
K𝜓 = −0.5 (25)
⎤
−26.3810 ⎣ 50.5227 ⎦ −5
−26.3810 ⎣ 0.5227 ⎦ −5
of the virtual leader is specified by ⎧ 2.7925, 2.7925 − 0.0814(𝑡 − 75), ⎨ 1.5708, 1.5708 − 0.1047(𝑡 − 130), 𝜓𝑣 = 0, −0.1047(𝑡 − 185), ⎩ −1.5708,
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⎤
⎡
⎤ −26.3810 ⎣ −49.4773 ⎦ −5
𝑡 < 75𝑠 75𝑠 ≤ 𝑡 < 90𝑠 90𝑠 ≥ 𝑡 < 130𝑠 130𝑠 ≤ 𝑡 < 145𝑠 145𝑠 ≥ 𝑡 < 185𝑠 185𝑠 ≤ 𝑡 < 200𝑠 𝑡 ≥ 200𝑠
Figures 2 shows the simulation results of the three quadrotor UAVs controlled by the proposed formation control approach. From Figure 2, it is observed that the three UAVs are initially not in formation. They first form the triangle formation and make translational and rotational movement. Then the triangle formation is extended to a big one and reconfigured to the alongside formation, and this alongside formation rotates 270𝑜 clockwise around UAV 2 to form the tandem formation. Finally, the tandem formation make translational movement. It is also observed that the path of virtual leader (green dash line) is well followed by the team of UAVs.
3m. It need be mentioned that the accuracy of the low cost GPS is 1m for horizontal position and 3m for height.
20
Fig. 3.
10
Formation flight test of three quadrotor UAVs
triangle formation 0 −10
VI. C ONCLUSIONS
tandem formation
X(m)
−20
In this paper, a distributed robust formation and reconfiguration control strategy is proposed for VTOL UAVs with dynamic constraints. The proposed approach is based on potential field approach and consensus conception. It is robust against bounded disturbances due to dynamic constraints and measurement noises. The formation flight test demonstrates the effectiveness of the proposed approach.
−30 −40 alongside formation −50
extended triangle
−60 −70
path of virtual leader (green dash line)
−80 −80
−60
−40
−20 UAV1
0 Y(m) UAV2
20
40
60
80
UAV3
R EFERENCES Fig. 2.
Simulation result of X-Y view of formation trajectory
V. F LIGHT T EST OF F ORMATION C ONTROL Flight test is the most important step for controller validating and potential problems identifying. This section provides the outcomes of the autonomous formation flight experiments. The procedure for the formation test of the three quadrotor UAVs is as follows. First, the ground station gives a takeoff command to the three UAVs. The three UAVs take off automatically and hover at the specified positions. Then the ground station gives a formation command to the three UAVs. The three UAVs start to complete the whole formation test process. In this flight test, the communication rate is 10𝐻𝑧. The formation control loop including outer-loop and inner-loop control is running in 10𝐻𝑧 while the command convertor runs in 50𝐻𝑧. Figure 3 shows a photo of the three UAV formation test. The results from the formation flight test are shown in Figures 4-5. Figure 4 shows the horizontal trajectory of three UAVs flying in formation. It is noted that the trajectory drifts from the path of the virtual leader due to the strong northwest wind. However, the formation shape can still be maintained. Figure 5 shows the height of UAVs and distances between UAVs, where the maximum steady state horizontal distance error is 2.5m and the maximum steady state height error is
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20 10 0 −10
North (m)
−20 −30 −40 −50 −60 path of virtual leader
−70 −80 −80
−60
−40
−20 UAV1
Fig. 4.
0 East (m) UAV2
20
wind
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400
UAV3
Flight test result of X-Y view of formation trajectory
distance in X−Y plane(m)
25 20 15 10 5
0
50
100
150 d
12
200 time(s) d
250
13
d
23
12
Height(m)
10 8 6 4 2
0
50
100
150 UAV
1
Fig. 5.
200 time(s) UAV
2
Flight test result of distance and height of UAVs
1445
250 UAV
3
