Adaptive Synchronized Attitude Angular Velocity Tracking Control of Multi-UAVs

Transcription

1 25 American Control Conference June 8-, 25 Portland, OR, USA WeA45 Adaptive Synchronized Attitude Angular Velocity racking Control of Multi-UAVs Hugh H Liu and Jinjun Shan Abstract An adaptive synchronization strategy is proposed and applied to the attitude angular velocity tracking control of multiple Unmanned Aerial Vehicles (UAVs) It achieves global asymptotic convergence of both the attitude angular velocity tracking and the angular velocity synchronization, even in the presence of system parameter uncertainties Simulation results of multiple UAVs in coordination verify the effectiveness of the proposed approach I INRODUCION In order to achieve high efficiency and productivity, there is tremendous demand for the use of multicomposed systems in manufacturing, aerospace operations and other applications he multicomposed systems work either under cooperative or under coordinated schemes According to [], synchronization may be defined as the mutual time conformity of two or more processes Cooperation, coordination and synchronization are intimately linked subjects and usually they are used as synonyms to describe the same kind of behavior [2] For mechanical systems, motion synchronization is of great importance as soon as two or more systems have to cooperate he cooperative behavior gives flexibility and maneuverability that cannot be achieved by an individual system [3] In recent years, motion synchronization has received a lot of attention A cross-coupling concept is first proposed in [4] to address the motion synchronization problem It is combined with adaptive feedforward controller to achieve speed synchronization of two motion axes [5], as well as position and speed synchronization of two gyro motions [6] he position synchronization of multiple robot systems with only position measurements is studied in [2], where coupling errors are introduced to create interconnections that render mutual synchronization of the robots his method is also applied to synchronization control of multiple mobile robots [3] In this paper, we address the angular velocity synchronization of multiple Unmanned Aerial Vehicles (UAVs) during their coordinated flight or formation flight Motivated by the aforementioned cross-coupling concept, we propose an adaptive synchronized attitude angular velocity tracking strategy for multiple UAVs It achieves global asymptotic convergences of both the attitude angular velocity tracking and the angular velocity synchronization, even in the presence of system parameter uncertainties Hugh H Liu and Jinjun Shan are with he Institute for Aerospace Studies, University of oronto, 4925 Dufferin Street, oronto, Canada M3H 56 liu,shan@utiasutorontoca /5/$25 25 AACC 28 he remainder of this paper is organized as follows Section II introduces the equations of attitude motion of UAV in a body-fixed coordinate frame Section III presents the generalized synchronization error concept and the development of the adaptive synchronization controller Moreover, the stability analysis is given to demonstrate the global asymptotic convergence Numerical simulation results of attitude angular velocity tracking control of 4 UAVs are given in Section IV Finally, Section V offers some conclusions and recommendation on future work II AIUDE DYNAMICS OF UAV From the traditional nonlinear aircraft model, we have the following attitude dynamics for UAVs J xx P J xz Ṙ +(J zz J yy )QR J xz PQ= L () J yy Q +(J xx J zz )PR+ J xz (P 2 R 2 )=M (2) J zz Ṙ J xz Ṗ +(J yy J xx )PQ+ J xz QR = N (3) where J R 3 3 is a positive-definite moment of inertia matrix, J xx, J yy, J zz and J xz are the corresponding elements, the applied moment M R 3 [L M N], Ω R 3 [P Q R] is the attitude rate vector and P, Q, R are the roll, pitch, yaw rate, respectively Our aim is to design a controller for attitude angular velocity tracking and angular velocity tracking synchronization between all UAVs First, we introduce a pseudo attitude angle vector Θ R 3 [θ P θ Q θ R ] and θ P = P, θ Q = Q, θ R = R he pseudo attitude angle vector Θ is different from the classical Euler angle [φ θ ψ] and is introduced only for controller design purpose hus, we obtain the following augmented equations of attitude dynamics J xx θp J xz θr +(J zz J yy )QR J xz PQ= L (4) J yy θq +(J xx J zz )PR+ J xz (P 2 R 2 )=M (5) J zz θr J xz θp +(J yy J xx )PQ+ J xz QR = N (6) or in matrix format J Θ+N( Θ, J) =M (7) where the nonlinear term N( ) R 3 is (J zz J yy )QR J xz PQ N( Θ, J) = (J xx J zz )PR+ J xz (P 2 R 2 ) (J yy J xx )PQ+ J xz QR (8)

2 If n similar UAVs are considered, we have the following n attitude dynamics equations J Θ + N (Ω, J ) = M J 2 Θ 2 + N 2 (Ω 2, J 2 ) = M 2 J i Θi + N i (Ω i, J i ) = M i J n Θ n + N n (Ω n, J n ) = M n where subscript i denotes the i-th UAV Writing these n attitude dynamic equations in a matrix format, we have I Θ + N(Ω, I) =M (9) where Θ R 3n, N R 3n, M R 3n are vectors, I R 3n 3n is a diagonal matrix, and they are of the following expressions I = diag[j J 2 J i J n ] Θ = [Θ Θ 2 Θ i Θ n ] N = [N N 2 N i N n ] M = [M M 2 M i M n ] = [L M N L 2 M 2 N 2 L i M i N i L n M n N n ] III CONROLLER DESIGN A Control objective First, we define E(t) R 3n as the attitude angular velocity tracking error vector of n UAVs, Ξ(t) R 3n as the angular velocity synchronization error vector hey have the following expressions E(t) [e (t) e 2 (t) e i (t) e n (t)] () Ξ(t) [ε (t) ε 2 (t) ε i (t) ε n (t)] () e i (t) Ω d (t) Ω i (t) (2) e i (t) Θ d (t) Θ i (t) (3) where Ω d R 3 is the desired attitude angular velocity trajectory he details on synchronization error will be discussed in the subsequent section For synchronized attitude angular velocity tracking of multiple UAVs, first of all, the designed controller should guarantee the stability of the attitude angular velocity tracking errors of all involved systems Secondly, the controller should also guarantee the stability of the angular velocity synchronization errors hirdly, the controller should regulate the attitude angular velocity motions to track the desired trajectory at the same rate so that the corresponding synchronization errors go to zero simultaneously hus, the control objective becomes: E(t) and Ξ(t) as t 29 Without loss of generality, we reformat the attitude dynamic equations in Eq (9) as follows IΦ + N(Ω, I) =Y(Φ, Ω)Ψ = M (4) where Y i R 3 4 is the regression matrix and is composed of known variables, Ψ i R 4 is a constant parameter vector, Φ i R 3 is a dummy variable vector and will be defined later, Ψ R 4n, Y R 3n 4n In practice, there exist uncertainties in the system parameters, such as the moments of inertia We define Ψ(t) as the estimation of Ψ Moreover, the following expressions can be obtained Ψ = [Ψ Ψ 2 Ψ i Ψ n ] Ψ i = [J xxi J yyi J zzi J xzi ] Φ = [Φ Φ 2 Φ i Φ n ] Φ i = [Φ Pi Φ Qi Φ Ri ] (5) Y = diag[y Y 2 Y i Y n ] Y i (Φ i, Ω i ) = Φ Pi Q i R i Q i R i (Φ Ri + P i Q i ) P i R i Φ Qi P i R i P 2 i R 2 i ) P i Q i P i Q i Φ Ri Q i R i Φ Pi B Synchronization error Synchronization error is used to identify the performance of synchronization controller, ie how the trajectory of each UAV converges with respect to each other here are various ways to choose the synchronization error In general, the attitude angular velocity synchronization error Ξ(t) can be formulated as the following generalized expression Ξ(t) =E(t) (6) where Ξ(t) is a linear combination of attitude angular velocity tracking error E(t), R 3n 3n is the generalized synchronization transformation matrix If we choose the following synchronization transformation matrix 2I I I I 2I I = (7) I 2I I I I 2I we can get the corresponding attitude angular velocity synchronization error ε (t) = 2e (t) e 2 (t) e n (t) ε 2 (t) = 2e 2 (t) e 3 (t) e (t) ε 3 (t) = 2e 3 (t) e 4 (t) e 2 (t) (8) ε n (t) = 2e n (t) e n (t) e (t) In a similar way, other kinds of synchronization errors can be formed It can be seen from Eq (8) that each individual UAV s synchronization error is a linear combination of its tracking error and two adjoining UAVs tracking errors

3 C Adaptive synchronization controller According to the control objective discussed previously, the designed control moment M(t) should guarantee convergences of both the attitude angular velocity tracking error E(t) and the angular velocity synchronization error Ξ(t) simultaneously, and realize expected transient characteristic of attitude angular velocity tracking motion o realize motion synchronization between all UAVs, we may include all synchronization errors related to a certain axis into this axis s controller for each UAV However, this method can only be feasible when the UAVs number n is small If n is extreme large, including all synchronization errors into each controller will lead to intensive on-line computational work hus, in this section, we try to design controller for each axis to stabilize its attitude angular velocity tracking motion and synchronize its motion with several adjacent attitude axes First, the coupled attitude angular velocity error E (t) that contains both the attitude angular velocity tracking error E(t) and a linear combination of the angular velocity synchronization error Ξ(t) is introduced E (t) =E(t)+B Ξ(t) (9) where B diag[b B 2 B n ] is a positive coupling gain matrix, E [e e 2 e n] and its expanded expression is e (t) = e [2ε (t)+b (τ) ε 2 (τ) ε n (τ) ] e 2(t) = e 2 (t)+b 2 [2ε2 (τ) ε 3 (τ) ε (τ) ] e 3(t) = e 3 (t)+b 3 [2ε3 (τ) ε 4 (τ) ε 2 (τ) ] (2) [2εn e n(t) = e n (t)+b n (τ) ε n (τ) ε (τ) ] From Eq (2) we can see that, the synchronization error ε i (t) appears in e i(t) and e i+(t) as negative value ε i (t), while as positive value 2ε i (t) in e i (t) In this way, the coupled attitude errors are driven in opposite directions by ε i (t), which contributes to the elimination of the synchronization error ε i (t) With notation of the coupled attitude angular velocity error E (t), the coupled filtered tracking error for the ith UAV, r i (t) R 3, is further defined as r i (t) =e i (t)+λ i e i (t) (2) where Λ i R 3 3 is a constant, diagonal, positive-definite, control gain matrix For all n UAVs, the whole coupled filtered tracking error becomes r(t) =E (t)+λ E (t) (22) 3 where r(t) [r (t) r 2 (t) r n (t)], Λ R 3n 3n diag[λ Λ 2 Λ n ] o account for the uncertainties in system parameters, the controller is designed to contain an adaptation law for on-line estimation of the unknown parameters he error between the actual and estimated parameters is defined by Ψ(t) Ψ(t) Ψ(t) (23) where the parameter estimation error Ψ(t) R 4n he dummy variable Φ introduced in Eq (4) has the following expression Φ = Θ d + ΛE + B Ξ (24) Based on the dynamics equation (4), the control moment vector M(t) is designed to be M(t) =Y(Φ, Ω) Ψ(t)+Kr(t)+K s Ξ(t) (25) where K R 3n 3n and K s R 3n 3n are two constant, diagonal, positive-definite, control gain matrices he estimated parameter Ψ(t) is subject to the adaptation law Ψ = ΓY ( )r (26) where Γ R 4n 4n is a constant, diagonal, positive-definite, adaptation gain matrix By differentiating Eq (23) with respect to time, the following closed-loop dynamics for the parameter estimation error can be obtained Ψ = ΓY ( )r (27) heorem he proposed adaptive synchronization coupling controller Eqs (25) and (26) guarantees the global asymptotic convergences of both the attitude angualr velocity tracking error E(t) and the angular velocity synchronization error Ξ(t), ie, lim E(t), Ξ(t) = (28) t Proof Considering the following positive definite Lyapunov function V (r, Ψ, Ξ) [ r Ir + 2 Ψ ( Γ Ψ + Ξ ) Ks Ξ + ( Ξ ) ( BΛKs Ξ )] (29) Differentiating Eq (29) with respect to time t yields V (r, Ψ, Ξ) =r Iṙ + Ψ Γ Ψ + Ξ K s Ξ + ( Ξ ) BΛKs Ξ (3) Differentiating Eq (22) with respect to time t and consider Eqs (9, 24), we have ṙ = Ė + ΛE = Ė + B Ξ + ΛE = Θ d Θ + B Ξ + ΛE = Φ Θ (3)

4 Multiplying I at the both sides of Eq (3), substituting Eqs (9, 4, 25) into it, we obtain Iṙ = IΦ I Θ = Y(Φ, Ω)Ψ M = Y(Φ, Ω) Ψ Kr K s Ξ (32) Substituting Eqs (27, 32) into Eq (3) yields V (r, Ψ, Ξ) = r Iṙ + Ψ Γ Ψ + Ξ K s Ξ + ( Ξ ) BΛKs Ξ = r [Y(Φ, Ω) Ψ Kr K s Ξ] + Ψ Γ [ ΓY ( )r]+ Ξ K s Ξ + ( Ξ ) BΛKs Ξ = r Kr r K s Ξ + Ξ K s Ξ + ( Ξ ) BΛKs Ξ (33) Replacing r in the second term of Eq (33) with Eqs (9, 22), we have ] V ( ) = [E + B Ξ + Λ E + BΛ Ξ K s Ξ r Kr + Ξ K s Ξ + ( Ξ ) BΛKs Ξ ( = r Kr Ξ ) BKs Ξ Ξ ΛK s Ξ Because K, BK s,andλk s are all positive-definite matrices, so we can conclude that ( V ( ) = r Kr Ξ ) ( BKs Ξ ) Ξ ΛK s Ξ (34) Since V (r, Ψ, Ξ) in Eq (34), the V (r, Ψ, Ξ) given in Eq (29) is either decreasing or constant Due to the fact that V (r, Ψ, Ξ) is non-negative, we conclude that V (r, Ψ, Ξ) L ; hence r L and Ψ L With r L, we conclude from Eq (22) that E (t) L and E (t) L,and E(t), E(t), Ξ(t), Ξ(t) L based on the definition in Eq (9) Because of the boundedness of Θ d (t) and Ω d (t), we conclude that Θ(t) L and Ω(t) L from Eq (3) aking Ψ L and Ψ a constant vector, Ψ L is obtained from Eq (23) With the previous boundedness statements and the fact that Θ d (t) is also bounded, the dummy variable Φ L and Y( ) L 3 can be concluded from their definitions in Eq (24) and Eq (5) Hence, the control input M(t) L is also determined from Eq (25) he preceding information can also be used to Eq (9) and Eq (3) to get Θ(t), ṙ(t) L Until now, we have explicitly illustrated that all signals in the adaptive synchronization controller and system remain bounded during the closed-loop operation From Eq (34), we show that r(t) L 2, Ξ(t) L 2 and Ξ(t) L 2 Hence, E (t) L 2 and E (t) L 2 can be concluded from Eq (22) From Eq (9), we conclude E(t) L 2 since Ξ(t) L 2 and E (t) L 2 Moreover, we have E(t) L 2 because of Ξ(t) L 2 When Θ(t) L and Θ d (t) is bounded, Ė(t) L can be concluded Considering E(t) L 2 and Ė(t) L, we can conclude that Ξ(t) L 2 and Ξ(t) L hus, Corollary in [7] can be applied to conclude that lim t E(t) = lim Ξ(t) = t IV SIMULAION In this section, we take one example of cooperative motion of four UAVs he UAVs are chosen as Hanger 9 J-3 Piper Cub like UAV [8] he parameters, moments of inertia, given in able I are modified according to the values used in [9] hese four UAVs are required to track the following three-axis attitude angular rate profiles P d (t) = 5π cos (πt) Q d (t) = π cos (πt) (35) R d (t) = 3π cos (πt) he control and adaptation gains are also given in able I ABLE I: System parameters and control gains of multiple UAVs Symbol Value Ψ i Ψ =[6, 237, 25, 55], Ψ 2 =[3, 67, 9, 7] Ψ 3 =[2, 6, 23, 3], Ψ 4 =[474, 63, 98, ] Ψ i () Ψ, 9Ψ 2, Ψ 3, 2Ψ 4 K i K =diag[25, 25, 25], K 2 =diag[3, 3, 3] K 3 =diag[5, 5, 5], K 4 =diag[45, 65, 75] K si K s =diag[4, 4, 4], K s2 =diag[6, 6, 6] K s3 =diag[3, 3, 3], K s4 =diag[7, 75, 8] Λ i Λ =diag[8, 8, 8], Λ 2 =diag[,, ] Λ 3 =diag[3, 3, 3], Λ 4 =diag[8, 8, 8] Γ i Γ =diag[,,, ] Γ 2 =diag[25, 25, 25, 25] Γ 3 =diag[,,, ] Γ 4 =diag[3, 3, 3, 3] B i diag[,, ] Figure shows the simulation results of adaptive attitude angular velocity tracking control of four UAVs without synchronization strategy Fig 2 shows the corresponding simulation results with the proposed synchronization strategy For evaluation of control performance, we calculated the 2-norm of the attitude angular velocity tracking error E(t) and the angular velocity synchronization error Ξ(t)

5 he values in able II show that although the attitude angular velocity tracking error E(t) can be realized by using adaptive controller without synchronization straregy, there are significant differences between the attitude angular velocity tracking errors of all involved UAVs It means that the attitude angular velocity synchronization errors are large and the attitude angular velocity tracking motions are not synchronized well However, with the proposed synchronization strategy, the synchronization errors can be remarkably reduced ake the attitude angular velocity tracking motion of z-axis as an example Without synchronization strategy, the 2-norm of the attitude angular velocity tracking errors and the attitude angular velocity synchronization errors for four UAVs are 293, 3534, 5949, 483 and 35228, 9243, 78, 592 (deg/sec), respectively With the synchronization strategy, the corresponding 2- norm values are 2386, 249, 2728, 2463 and 3846, 747, 27, 49 (deg/sec), respectively he 2-norm of the attitude angular velocity tracking errors have been regulated to be very close values In this way, the synchronization errors are reduced observably ABLE II: Performance evaluation of various control strategies close formation flight, with consideration of the coupled aerodynamics REFERENCES [] I I Blekhman, P S Landa, and M G Rosenblum, Synchronization and chaotization in interacting dynamical systems, ASME Applied Mechanics Review, vol 48, no, pp , 995 [2] A Rodriguez-Angeles and H Nijmeijer, Mutual synchronization of robots via estimated state feedback: A cooperative approach, IEEE ransactions on Control Systems echnology, vol 2, no 4, pp , 24 [3] H Nijmeijer and A Rodriguez-Angeles, Synchronization of Mechanical Systems Singapore: World Scientific, 23 [4] Y Koren, Cross-coupled biaxial computer controls for manufacturing systems, ASME Journal of Dynamic Systems, Measurement, and Control, vol 2, no 2, pp , 98 [5] M omizuka, J S Hu, C Chiu, and Kamano, Synchronization of two motion control axes under adaptive feedforward control, ASME Journal of Dynamic Systems, Measurement, and Control, vol 4, no 6, pp 96 23, 992 [6] L F Yang and W H Chang, Synchronization of twin-gyro precession under cross-coupled adaptive feedforward control, Journal of Guidance, Control and Dynamics, vol 9, no 3, pp , May-Jun 996 [7] D M Dawson, J Hu, and C Burg, Nonlinear Control of Electric Machinery Marcel Dekker, Inc, 998 [8] air8htm [9] Errors I V No synchronization ė x ė y ė z ε x ε y ε z With synchronization strategy ė x ė y ė z ε x ε y ε z V CONCLUSIONS In this paper, an adaptive synchronization control approach is applied to the attitude angular velocity tracking of multiple Unmanned Aerial Vehicles (UAVs) Different from the previous work on adaptive tracking control, the global asymptotic convergences of both the attitude angular velocity tracking error and the angular velocity synchronization error can be achieved using the proposed adaptive synchronization controller, even in the presence of unknown parameters he introduction of the generalized synchronization error in controller provides significant enhancement in achieving synchronization behavior among all UAVs Future work currently under our consideration includes: ) adaptive synchronization control of multiple UAVs based on more complicated modeling (such as 6 DOF aircraft model with both translational motion and attitude motion); 2) adaptive synchronization control for multiple UAVs in 32 attitude velocity tracking error of (deg/s) attitude velocity tracking error of I (deg/s)

6 attitude velocity tracking error of (deg/s) standard velocity synchronization error of yaw axis(deg/s) I V Fig : Adaptive attitude angular velocity tracking control of multiple UAVs attitude velocity tracking error of V (deg/s) I V I V I V standard velocity synchronization error of pitch axis(deg/s) I V I V Fig 2: Adaptive attitude angular velocity tracking control of multiple UAVs with synchronization strategy 33