Formation Control Laws for Autonomous Flight Vehicles

Transcription

1 Formation Control Laws for Autonomous Flight Vehicles Martina Chiaramonti, Fabrizio Giulietti and Giovanni Mengali Abstract- The problem of aircraft formation dynamics and control is investigated from the viewpoint of formation architecture. Three different formation structures, Leader-Wingman, Virtual Leader and Behavioral approaches are introduced. A [9]. In the Virtual Leader structure, instead, each aircraft receives the same information, namely the trajectory of the Virtual '. comparative study is made using an unified approach through a suitable control law based on the dynamic inversion approach. Leader. Ths latter may be a real aircraft or more commonly I. INTRODUCTION Aircraft formation control has become an active research area in recent years. In a manner similar to migratory birds, aircraft in a formation may experience a substantial drag reduction using the vortex upwash created by the leading aircraft[l]. This in turn implies potential fuel reduction with significant benefits both for military and commercial employments. However, there are other important aspects that motivate the growing attention towards this subject. For instance, it is believed that aircraft formation control will play a fundamental role in future aerospace scenarios, where unmanned air vehicles will be required to maneuver in swarms for purposes of surveillance, reconnaissance and rescue in hostile environments[2]. Finally, autonomous formation flight systems are widely investigated due to their close connection to the aerial refuelling problem[3], [4]. Although aircraft formation benefits have been theoretically known for many years, systematic flight tests have been conducted only very recently, especially after NASA launched the Autonomous Formation Flight Program in Most of the existing literature is concerned with the problem of aircraft formation control[5], [6], [7], [8], [9] and/or aerodynamic interference modelling[10], [11], [12], while a fixed formation structure is assumed. In particular, the Leader-Wingman, Virtual Leader and Behavioral structures, have been studied thoroughly [5], [6], [7], [8], [9], [13], [14]. In the Leader-Wingman structure the first aircraft in the formation is designated as the Leader, with the rest of aircraft (Wingmen) treated as followers. While the Leader aircraft maintains a prescribed trajectory, the followers refer their position to the other aircraft in the formation. More precisely, Wingmen track a fixed relative distance from the neighboring aircraft and, accordingly, form a chain. In this structure a rear aircraft usually exhibits a poorer response than its reference due to error propagations. Nevertheless, the Leader-Wingman structure, for its simplicity, is widely employed in control and management of multiple vehicle formations [5], [6], [7], [8], absence of error propagations and, 2) the formation behavior is prescribed by simply specifying the behavior of the Virtual Leader. The disadvantage is that there is not explicit feedback to the formation. Actually each aircraft has no information about its distance from the followers and therefore may not be able to avoid collisions. A quite different strategy is represented by a Behavioral approach. In this formation structure, the control action of each aircraft is a weighted average of the control for each behavior. In the context of aircraft formation control this idea has been first introduced by Anderson and Robbins[ 13]. Then, it has been further exploited by Giulietti et al.[15] with the introduction of an imaginary point in the formation called the Formation Geometry Center (FGC). Each aircraft in the formation is required to maintain a prescribed distance from the FGC. The position of this point depends on the relative distances between all aircraft in the formation. This allows each aircraft to have the capability of sensing other vehicles' movements from the nominal position and to maneuver in order to reconstitute the formation geometry. This characteristic is very similar to the natural behavior of migratory birds[ 16]. The strength of this approach is that there is explicit feedback to the formation so that problems of collision avoidance are easily taken into account. Another important advantage is that aircraft in a formation tend to cooperate each other, hence there is not a single point of failure in the structure. On the other hand, the behavioral approach is more complex than the others because the information needed, in this case, are more plentiful. The aim of this work is to build a formation flight control system based on the dynamic inversion theory. II. FORMATION MODELING A three degrees-of-freedom point mass model is chosen to represent each aircraft in the formation. This simplified model, is well suited for dealing with trajectory tracking problems. A. Reference Frames Martina Chiaramonti and Giovanni Mengali are with the Department of Aerospace Engineering, University of Pisa, Pisa, Italy Aircraft dynamics are described with the aid of two righthand reference frames: A local vertical frame Tv and a wind an ideal point in the formation that aircraft must track. The advantage of a Virtual Leader structure is twofold: 1) all aircraft in the formation exert the same transient due to the martina.chiaramonti@ing.unipi.it, g.mengali@ing.unipi.itaxsfmet.bthvehirognsnteisatnou axsfmew.bthvehirognsnteisatnou Fabrizio Giulietti is with the Department of Aerospace Engineering, University of Bologna, Forli, Italy fabrizio.giulietti@mail.ingfo.unibo.it position of the aircraft. It is assumed that Tv has axes parallel

2 to an inertial frame TI. As usual, the axis xw of 'Tw is i A directed along the velocity vector V and the axis Zw lies in the plane of symmetry of the aircraft. D di The transformation between the vertical frame Tv and the DdO wind-axes frame Tw is defined through the usual rotation matrix Twv = Twv((b,'Y,c) [17] where b,'y and f / " di > correspond to aircraft heading, flight-path and roll angles. For convenience, the flight-path angle 'y is assumed to have the sign reversed with respect to the usual notations. This means, for instance, that the aircraft rate of climb is given Til by -V sin'y, where V is the aircraft velocity. B. Aircraft Equations of Motion Fig. 1. Definition of geometric quantities with actual (A) and desired (D) It is assumed that the equations of motion of the generic positions of ith aircraft aircraft in the formation are * T-D V T-D +gi (1) m glgn sin (2) V cos a g relative to the reference point in the local wind-axes frame 'Y = nco o-co 3 w are (see fig. I ) where A L (4) mg is the load factor, L and D are lift and drag forces, T is the engine thrust, m is the aircraft mass and g is the gravitational acceleration. Using the aircraft drag polar, the d V - RIV - Q(d - d) + d (7) drag coefficient CD is given by CD = CDO + kc% (5) where CL is the lift coefficient, CDO is the zero-lift drag coefficient and k is the induced drag factor. Accordingly, the where Q is the cross-product matrix and R is the transformaaerodynamic drag force is evaluated as: tion matrix between the local and the G-centered vertical D qsd + k (nmmg)2 (6) frames[17]. Note the the footer i has been dropped in the SC0+k6qs D previous equation for the sake of simplicity. Let x, y and u be the state, the output and the input vector, respectively. where q = p is the dynamic pressure, p is the atmo- Let [d -d]d, = [di di, dj dj, - - dk dkl, - and define: spheric density and S is the wing surface. C. Distance Dynamics One of the main issues in a formation flight control system is to provide each aircraft with the ability to maintain a prescribed distance from a certain reference point G. T [ X This may coincide with the formation leader (either real or T virtual), a neighboring aircraft, or a certain point within the Y [Yi, Y2, Y31 = X (9) aircraft formation. u [Ul, U2, U3 (10) Assume that ri and r, are the positions of i-th aircraft and reference point with respect to the origin 0 of the inertial frame. Let di be the relative actual position between G and the i-th aircraft, while the corresponding desired position is represented by an overbarred symbol. Also assume that an orthogonal wind-axes frame Tw is "attached" to the The chosen inputs are useful for the dynamic inversion reference point G whose orientation is defined through Euler procedure. The usual commands [17] T, q5 and n can be angles ~, t, &b (aircraft heading, flight-path and roll angles) recovered by taking into account the limitations on the above and let V be the velocity of G. Following Giulietti and derivatives [18]. Mengali[ 17] the position dynamics of the generic aircraft The following differential equations describe the dynamics

3 of the generic aircraft with respect to the formation center * - - ~~~~~~~~~~~~~+ ±1 = V + [cos 'ycos /cos cos4 + cos' sin fcos sin + ( + ia g +sin'ysin ] - QQT~~ v n +-13cos 5 = Pi+ +b sn 0 -/ 3P1 +P2(A+B 29 V X3 +B 13) (P3 U3 + P4 + P5 U2) Xl [- sinq5cos +P6([Q1COSq5+Q2sin 5] V - Q3 sl V x -[V (n-cos z5cos Y)] X3 + (+Q4 V X3) + (u1 + P7 + P8 u3 + P U2) 12 (1 1) (16) 2 -[(cos 0 sin &b - sin q sin 'sin &b). -c For a definition of the quantities appearing in the previous + (cos cos -sin 0 sin -y sin 0) cosy sin equations the reader is referred to the Appendix. The time + _ ~-g sin cos(12 derivatives can be rewritten in a more compact form that [nc si V highlights the input-output relationships. They are as follows g n sin sin / 13 (recall that y x) V cos'y F [ (7 [(sin5sinb -cosq5sin'ysin b) cos cos+ I2 IF+IE U2 (17) (-sinq5cos - cosq5sin'ysin b) cos -sin+ L Y3 J L U3 I - g cos 6 (n cos -cos (I) cos bcos'ysiny] V - V + (13) where + (. + gnsinsin'y X2 [F1 1 Tgh sin 2 - v l1 ~ v Cos )F12 F21 F31] III. DYNAMIC INVERSION (18) ( 0 K6X3 +K9X3 K5X3 Because the input variables do not appear explicitly in E = 13 A4 1X + Alo 13 A8 13 Eqs. (11), (12) and (13), we proceed deriving these expres- 12 P5 X1 + P9 12 P3 X1 + P8 12 sions w.r.t. the time [19]: and sil = V+A+K1X2 +K2X2 +K3u2X2+ Fl =A+KiX2 +K2X2 +K4([QQcos+Q2sinq5] V+ K4([QQcosq5+Q2sinq5] V-Q3 [Vj] x1+ Q3 v ], [+Q4ThX3)+ n sin + + '+ Q4 (K7 +Kll) ([R1 sin q+r2 cos] V+ V X3) +KsU3 z3+ K6 U2 X3+ V J ~~~~~~~~~~~~2gn sin22_ -gn + R3 cos0 +(K7 + Kll) ([R1 sinq + R2 cosq5 V+ + V- x+ + [2gn sin 25gTn+R3 COs] x+ +[ nsijr4] V X2)+K83+Klol3 (. nhsiflc/5r40n(19) +V ) 12)+ Kgx3+ F21 =A1 + (A2 + A3) Xl + As (A+ + Klo X3 + K9 U2 X3 (+ +BB SinX h ccos 12-9 X3 +B 13)+ (14) +A6( [Rl sin q + R2 COS V+ ( 1 2 1l i (A2 -[A3) xi + A4 U2 X1+ [+ 22g (A g n + R3BCOS 14- sin5 T Cos1 + A5 ~A+B V 2-9VX3+B V 13) + +K-nTsiflqR4 + A6 ([Rl sin q5+ 2 COSq5 V-P+ ( -+sin Th Cos _ F 9Thosin 5,-9ThR3 osn ~129 q A+B g x- V1+B V~13) + +(,+ Tsinq5R4)\ +P6([Q1cosq+Q2sinq] V-Q3 [ V]] + {Ul+ A7 + Agt3 + Ag+ Aio 2}1x3 + (5+ Q4 ThV ) )+P x+p x (15) (21)

4 The dynamic system can now be linearized. The new A = cos y; inputs are: B s'= Coy; u2l -E 17F + E Fv21 (22) LU13_ L[V3 D = sin ; from which, we get F [ 1 [V1 (23) F = siny; 03J V3 G = sin ; The following control laws are assumed: H = cosb; Vl = yld + kll (Yld -Y1) + kl2 (Yld- Yl) (24) Qi =-(ADH-GAE); = P2d + - k21 (P2d P2) + - k22 (Y2d Y2) V3 = Y3d + k31 (Y3d - 3) + k32 (Y3d - Y3) (26) (25) Q2 (CF-BGAD-ABGE); where the subscript "d" means desired value. Q3 gc; Because we aim to keeping the correct formation geometry, =g B; the actual distance is set to the desired value, that is Pid = C Yid = Yid = 0, i = 1, 2, 3. Therefore: Al = [-Q,4 sin b + Ql cos b + Q2 b cos + cos q + Q2 sin q] V; vi =-kll l-kl2yl (27) + Q2 sin (] V + [Ql =-k2l2-yk22y2 (28) [sinq V3= -k31y3 - k32y3 (29) A2 = [v] The kij values have been found by solving, for each channel, a pole assignment problem. A3-Q= [ VCos IV. CONCLUSIONS V The problem of aircraft formation dynamics and control A4 = Q3 [sinj] has been investigated from the viewpoint of formation archi- Vj tecture. The use of the Formation Geometry Center concept As --Q3 [n 1 is attractive. In fact, collisions between aircraft may be L V I easily avoided because each aircraft in the formation has sin b the capability of sensing the other aircraft movements. The A6 = + Q 4 employment of a control law based on the dynamic inversion A7 Q in concept has been investigated. V V. APPENDIX A8- Q sin Equations (14)-(16) contain a number of variables that are t cos defined in the following. Ag = Q4n V A [cos ' cos& cos cos + cos 'ysin &cos 'sin+ Alo Q4n + sin 'y sin] V; R = GAE - ADH; A = V A ~~~~~~~~~~~ ~ R=CF - BGAE (cos -y cos 0 cos cos FJ+ cos -y sin 0/ cos sin 0I+ R2=CFB A - ADBG; D G + sin y siny) + V (-' sin, cos 0 cos cosb+ R=g cos'y; - < sin cos 0Ccos cos - sin 0<cos05 Cos 0 R g sin _ ~~~~~~~~~~~~~~~~cosa +C0sin 5 0cos cos0y cos +ycosin ysin y+cos ysin 'si; v B= g cosy; K2B V ~-sinq5 B =-g'tsinqi; K3 =B V

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