DESIGN OF CONTROL LAWS FOR MANEUVERED FORMATION FLIGHT

Transcription

1 DESIGN OF CONTROL LAWS FOR MANEUVERED FORMATION FLIGHT Giampiero Campa^, Marcello R. Brad Seanor^, Mario G. Perhinschi^ Department o Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 26506/6106, Abstract This paper presents identiication, control synthesis and simulation results or an YF-22 aircrat model designed, built, and instrumented at West Virginia University. The goal o the project is the experimental demonstration o ormation light or a set o 3 o the above models. In the planned light coniguration, a pilot on the ground maintains controls o the leader aircrat while a wingman aircrat is required to maintain a pre-deined position and orientation with respect to the leader. The identiication o both a linear model and a nonlinear model o the aircrat rom light data is discussed irst. Then, the design o the control scheme is presented and discussed. Using the developed nonlinear model, the control laws or a maneuvered light o the ormation are then simulated with Simulink and displayed with the Virtual Reality Toolbox. Simulation studies have been perormed to evaluate the eects o speciic parameters and the system robustness to atmospheric turbulence. The results o this analysis have allowed the ormulation o speciic guidelines or the design o the electronic payload or ormation light. 1 Introduction Autonomous ormation light is an important research area in the aerospace community. The aerodynamic beneits o ormation light, and in particular close ormation light, have been well documented [1]. The investigation o control issues related to a leader-wingman ormation has lead to the introduction o dierent types o compensationtype controllers [2]. In Re. [3] a ormation light control scheme was proposed based on the concept o Formation Geometry Center, also known as the Formation Virtual Leader. More complex control laws based upon Linear Quadratic Regulator and Dynamic Inversion (DI) approaches have also been proposed [4]. This paper presents design results o the ormation control laws to be implemented on a set o 3 YF-22 aircrat models that are designed, built, and instrumented at West Virginia University (WVU). One o the 3 WVU YF-22 models is shown in Figure 1. The model eatures an 8 t. uselage length with a 6.5 t. wingspan or an approximate take-o weight o 48 lbs, including a 12 lbs electronic payload consisting on a PC-104 light computer, a complete set o sensors, a GPS receiver and a set o RF modems used or data transmission. Figure 1 - WVU YF-22 aircrat model The aircrat models are currently undergoing individual light-tests with light-testing o the ormation control laws to be perormed in the early Due to the limitations on the light range, the WVU YF-22 models will be expected to perorm airly tight maneuvers at high Euler angles and moderately high angular rates. Thereore, a speciic issue is the design o a control scheme allowing or ormation control under these light conditions. Another objective is to design a ormation control scheme with a limited amount o inormation exchange (between leader and wingman) needed to maintain the predeined ormation geometry. The paper is organized as ollows. The second section describes the identiication o a linear and nonlinear single aircrat model rom collected light data. The third section outlines the geometric characteristics o the ormation. The ourth section outlines the design o the ormation control laws. The inal sections will present the simulation and visualization environments, together with the main results. The symbols used throughout the paper are very standard, but readers less amiliar with light mechanics could consult [7] or download the FDC manual [10] as a reerence. 2 System Identiication o the WVU YF-22 Flight data or several maneuvers were collected or parameter identiication purposes using the ollowing onboard instrumentation: Absolute and dierential pressure sensors: (SenSym ASCX15AN and SenSym ASCX01DN) to measure H and V (altitude and Proessor, ^ Research Assistant Proessor

2 Inertial Measurement Unit (Crossbow DMU-VGX) to measure Ax, Ay, Az, p, q, r, ϕ, θ, (accelerations, roll pitch and yaw rates, roll and pitch angles). Custom designed nose probe to measure α and β (attack and sideslip angles). Potentiometers on the control suraces to measure δe, δa, δr (elevators, ailerons and rudders delections). During the light, the PC-104 based on-board computer collects in real time (at a rate o 100Hz) all o the above signals using the integrated data acquisition card (Diamond MM 32), and stores them on a lash-card or post-light downloading. A set o light data was used or the actual parameter estimation process, while a second set o data was used or validation purposes. Turn maneuvers, plus doublets on each control suraces, (typical or collecting data or parameter identiication purposes), were perormed. 2.1 Linear Model Identiication The linear model identiication was perormed with a 3-step process. First, the light data time histories were inputted to a Simulink scheme providing smoothing and rearrangement o the signals. Next, a batch Matlab ile perormed the actual identiication algorithm. The last step o the model identiication process was the validation o the linear model using time histories o the control surace delections rom the validation light data Following the identiication study, the estimated linear lateral-directional aerodynamic model is given by: β β p p = r r φ φ (1) δ A δ R 0 0 The estimated longitudinal model is given by: v v α α = q q θ θ δ E This model represent the aircrat in a steady and level light at 42 m/s, 336m o altitude, with alpha and theta o 3 deg. This linear model was mainly used or control synthesis. (2) For simulation purposes, a ull nonlinear aircrat model was considered highly desirable i not necessary. 2.2 Linear Model Identiication The identiication o the mathematical model o a nonlinear system is a more challenging issue [5,6]. Most o the nonlinear identiication eorts rely on both good physical insight [5] and some orm o optimization algorithm like Steepest descent or Newton-Raphson [6]. The general nonlinear model o an aircrat system can be expressed (see or example [7]) as: x = ( x, δ, G, FA( x, δ), M A( x, δ)); (3) y = g( x, δ, G, FA( x, δ), M A( x, δ)); where x is the state vector (linear and angular positions and velocity), y is the output vector (linear and angular accelerations), δ is the input vector (surace delections), G is a vector o geometric parameters and inertia coeicients, F A and M A are aerodynamic orces and moments acting on the aircrat; inally, and g are the known analytic unctions that express the dynamics and kinematics o a rigid body. The aerodynamic orces and moments are expressed using the aerodynamic coeicients C D, C Y, C L, C l, C m, C n : CD( x, δ) bcl( x, δ) FA = qs CY( x, δ), MA = qs ccm( x, δ) (4) CL( x, δ) bcn( x, δ) where S is the wing platorm area, q the dynamic pressure, b the wingspan, and c the mean aerodynamic chord. The aerodynamic coeicients are oten approximated by aine unctions in x and δ ; or example, or the lit coeicient: CL( x, δ) = cl0 + cl αα + clqq+ cl δ δ (5) e e where, c L0 and the other three coeicients are usually called the derivatives o C L. When the above approximations are considered satisactory, the nonlinear aircrat model is completely determined by its aerodynamic derivatives as well as by its inertial and geometric coeicients (which can typically be evaluated experimentally). In this eort, the inertial and geometric characteristics o the WVU YF-22 model were determined with an experimental set-up; thus, the remaining critical issue was the determination o the aerodynamic derivatives. Formulas to calculate the entries o the matrices o the linear model in (1) and (2) rom the values o the aerodynamic derivatives and geometric-inertial parameters are well known [7]. By inverting such ormulas, an initial value or all the main aircrat aerodynamic derivatives was then calculated rom the matrices in (1) and (2) together with the measured geometric and inertial parameters. Next, a parameter optimization routine based on routines available with the Matlab Optimization Toolbox was set up. Speciically, a Matlab routine was written such that it could take as an input the aerodynamic derivatives to be estimated, perorm a simulation with the nonlinear model resulting rom those derivatives, compare the outputs with the real data, and return the dierence (to be minimized) to

3 the caller unction. The mincon unction which eatures the constrained optimization o a multivariable unction using a Sequential Quadratic Programming (SQP) technique [8] - was then used to ind the set o aerodynamic derivatives providing the best it with the light data, starting rom the initial set o aerodynamics derivatives calculated rom the linear models. The importance o starting the minimization rom a set o already accurate derivatives should be emphasized; in act, this last optimization can be considered a reinement o the parameters. A lesson learned was that in order or such an optimization to avoid local minima and converge successully, care must be taken in the selection o the cost unction. Speciically, the selected cost unction contained 3 terms, a term expressing the RMS o the deviation between real and predicted output, a requency based term expressing the lowest spectral components o the deviation, and a term expressing the dierence between the current linearized model (obtained by perorming a numerical linearization algorithm on the current nonlinear model) and the base linear model in equations (1) and (2). A inal validation o the nonlinear model was then conducted using the validation light data set, similarly to what was done or the linear model. As shown in Figure 2, the agreement between simulated and real data is substantial. C L0 = , C Lα = , C Lq = , C LδE = , C m0 = , C mα = , C mq = , C mδe = Lateral-Directional Aerodynamic derivatives: C Y0 = , C Yb = , C Yp = , C Yr = , C YδA = , C YδR = , C l0 = , C lb = , C lp = , C lr = , C lδa = , C lδr = , C n0 = 0, C nb = , C np = , C nr = , C nδa = , C nδr = Formation Control Problem : Geometry From a geometric point o view the ormation light control problem can be naturally decomposed into two independent problems: a level plane tracking problem and a vertical plane tracking problem. Figure 3 Formation Geometry Figure 2 Linear and nonlinear model prediction versus real light data The resulting aircrat nonlinear model is given by: Geometric and Inertial Data (60% uel load): c = m, b = m, S= m 2, I xx = Kg m 2, I yy = Kg m 2, I zz = Kg m 2, Ixz = Kg m 2, mass = Kg, T (engine thrust orce) = N Longitudinal Aerodynamic derivatives: C D0 = , C Dα = ,C Dq = 0, C DδE = , Figure 3 shows the level plane ormation geometry. All trajectory measurements, i.e., leader/wingman position and velocity, are deined with respect to a pre-deined Earth- Fixed Reerence x o y plane and are measured by the on-board GPSs. The pre-deined ormation geometric parameters are the orward clearance c and lateral clearance l. The orward distance error c and lateral distance error l can be calculated rom the trajectory measurements and ormation geometric parameters using the relationships: VLy ( xl xw ) VLx ( yl yw ) l = lc (6) V Lxy ( ) + ( ) VLy yl yw VLx xl xw = c (7) V Lxy

4 2 2 where VLxy = VLx + VLy is the projection o the leader s velocity onto the x y plane. Accordingly, the relative orward and lateral speed o the wingman are deined as the time derivatives o the orward and lateral distance respectively and are needed or ormation control purposes which can be calculated as: VLxVWy VLyVWx l = + ( + c ) Ω L (8) VLxy VLxVWx VLyVWy + = VLxy ( l+ lc ) Ω L (9) VLxy The trajectory-induced angular velocity in the x y plane (around the vertical axis) Ω L is computed as: Ω L ψ L = ( qlsinφl + rlcos φl) cosθ (10) L At nominal conditions, the leader and wingman aircrat are separated by a vertical clearance h c. The vertical distance error h, can then be calculated by: h = zl zw hc (11) while its time derivative is given by: h = VLz V (12) Wz The vertical control problem is reduced to the issue o maintaining the vertical clearance h c. Based on the above subdivision o the ormation geometry, the design o the control laws are separated into 3 channels to control respectively the lateral, orward, and vertical distance. The resulting design is outlined in the ollowing sections. 4 Design o the Formation Control Laws Ideally, to achieve the best trajectory tracking perormance, the ormation light control laws, - that is, the wingman light control laws - should be based on a ull inormation tracking strategy. This concept can be expressed as: Wingman s control inputs = Leader s control inputs + State error eedback where the control inputs include delections or the throttle, elevator, aileron and rudder, while state error eedback consists o internal state variable errors (such as errors in angular rate and Euler angles) and trajectory state variable errors (such as orward distance, lateral distance, and vertical distance) between the leader and wingman. The rationale behind this approach is the act that i both aircrat were lying in the same position then the leader acts as a reerence system or the wingman, so the state eedback control measures the dierences between the leader (reerence) state and the wingman state, and provides corrections to the wingman control inputs in order to correct these dierences. In reality, the desired wingman position is shited with respect to the leader s position; thereore, extra compensation might be needed to account or the dierence in the trajectory variables between the leader and the (ideal) wingman. It should be noted that in this ull inormation approach all the leader s states and control inputs are needed to calculate the wingman control inputs; thereore, a high communication bandwidth between the leader and wingman is required. An additional goal was to ormulate a criteria or limiting the amount o data to be exchanged rom leader and wingman aircrat while maintaining the geometry o the ormation. This issue has direct implications on the required perormance and, thereore, the cost o commercially available RF modems. Thereore only a ew o the leader outputs and states are actually used to calculate the wingman s control inputs. 4.1 Lateral Distance Control The objective o the lateral distance control is to minimize the lateral distance error l. Since bank angle rate changes are substantially higher than rate changes in the lateral position, the lateral dynamics exhibit a typical two-timescale eature. Thereore, the design o the control system can be decomposed into two successive phases, that is, the design o an inner loop controlling the bank angle and augmenting the lateral-directional stability, and the design o an outer loop which tries to maintain a desired lateral clearance with respect to the leader. The resulting linear control law is given below and shown in Figure 4, where the subscripts L and W indicate respectively the wingman and leader aircrat. Inner loop control law: δaw = δal + Kp pw + K φ ( φw φd ) (13) δrw = δrl + Kr r W (14) Outer loop control law: φ = φ + Kl + Kl (15) d L l l At this point, given the basic lateral-directional linear model o the aircrat (1), and the actuators dynamics: 1 Ga () s = s (16) the inner loop controller can be designed based on the aircrat lateral-directional linear model. The outer loop design requires a suitable kinematic reerence model. Such a model can be obtained by considering the aircrat perorming a steady state coordinated turn where the lit orce balance and centriugal orce balance equations apply. This leads to the ollowing expression: g Ω W = tanφw V (17) Wxy Additionally, by assuming a straight and level light condition o the leader and identical speed or the wingman and the leader, it results that Ω = Ω W while (8) takes on the ollowing simple orm: l = V Wxy sin ( Ω) (18) where Ω = ΩW Ω L. The linearization o the above two equations (around the standard level-straight light

5 condition) o the wingman provides the ollowing model o the trajectory dynamics: g Ω = φw VWxy (19) l = V Ω Wxy Thus, the ull linear model or lateral distance controller design is the combination o equations (1), (16), and (19). Classic root-locus based compensation design tools can then be applied to the model or evaluating the controller gains 26. The basic design speciication is to assign the damping ratio o the dominant poles a value around 0.7. The resulting values or the parameters o the dierent control laws are given by: Kp = 0.05, Kφ = 0.312, Kr = 0.3 K = 1.76, K = l l (20) 4.2 Forward Distance Control The objective o the orward distance control is to minimize the orward distance error. The orward distance control law, shown in Figure 5, is given by: δtw = δtl + K K + (21) The cascade o two irst order linear models can approximate the orward dynamics. The 1 st model represents the engine response in terms o throttle to thrust; this model has been obtained through an experimental analysis o the perormance o the jet engines installed on the WVU YF-22 aircrat; a 2 nd model represents the airspeed response in terms o thrust to airspeed, which is approximately derived rom the knowledge o nominal thrust, airspeed, mass, and the assumption that the change o aerodynamic drag is in proportional to the change o airspeed. The ollowing equation represents the resulting complete transer unction o throttle to airspeed: Gvt () s = GVT () s GTT () s = (22) s 1+ s It should be noted that this model also represents the transer unction rom throttle (o wingman) to orward velocity o (9) under the assumed level-straight constant speed light condition. The parameters o the orward distance controller outlined in (21) were then determined through a root locus-based compensator design: K = 4.76, K = 1.19 (23) 4.3 Vertical Distance Control The objective o the vertical distance control is to minimize the vertical distance error h. As with the lateral distance case, the problem exhibits a two time scale structure; thus, the control scheme can be designed using an inner loop controller - which is basically a pitch angle controller - and an outer loop controller providing altitude control. A linear control law that accomplishes the above scheme is given by the ollowing ormulas. Inner loop control law: δew = δel + Kq q W + K θ ( θw θd ) (24) Outer loop control law: θd = θl + K z δz+ Kzδz (25) The design is based on the linear short period model in (2) in addition to a linearized kinematic model: θ = q h = VWzθ (26) With a root locus-based design 20 the parameters o the vertical controller are ound to be: K = 0.56, K = 1.66, K = 1.59, K = 5.22 (27) q θ z z 5 Simulation Results A Simulink scheme eaturing the models o the WVU YF- 22 aircrat and the ormation controller was developed and implemented. Given the multi-object nature o the problem, the design o a visualization environment ully integrated with the simulation was considered to be critical. The Virtual Reality Toolbox (VRT) was selected as the visualization environment since it allows or objects and events o a virtual world (coded in VRML 2.0 or higher [9]) to be driven by signals rom Matlab/Simulink. The resulting scenery rom a view behind the wingman is shown in Figure 4. Figure 4 VRT visualization (behind wingman) A preliminary analysis was conducted to assess i delections o the control suraces rom the leader in (13), (14), and (24) were actually necessary to have desirable tracking perormance. This analysis showed that the elimination o these signals rom the leader aircrat did not signiicantly decrease the perormance o the control scheme. Consequently, these signals were no longer used (constant trim values were used instead o them) thereore saving communication bandwidth or light testing. Next, a simulation study was conducted to assess the need or the bank angle rom the leader in the wingman control laws in (15). The main results or this study is shown in Figure 5 where the leader s lateral maneuvering bank angle is about 30.

6 in terms o maximum orward, lateral and vertical errors with lower leader s bank angles. Figure 5 Trajectories with/without the bank angle rom leader aircrat It can be seen rom the simulation that the deormation o the ormation in terms o the lateral distance o the wingman rom the leader is unacceptably large (as much as 450 eet), when the leader s bank angle is not available or ormation control purposes. Based on the above considerations, it was decided that the electronic instrumentation o the wingman models will be required to include a three-axis angular rate gyro measuring roll rate p w, pitch rate q w and yaw rate q w, a vertical gyro measuring pitch angle θ w and bank angle ϕ w and a GPS receiver measuring the position o the wingman aircrat x w, y w, z w and its velocity vector, V wx, V wy, V wz. In addition, it was concluded that the pitch angle θ L and bank angle ϕ L o the leader aircrat, along with its positions and velocity vector (x L, y L, z L and its velocity vector, V Lx, V Ly, V Lz ) are required by the wingman s control system. Figure 6 -Trajectories in level plane Another simulation study was conducted to assess the perormance o the ormation controller ollowing o an S shape light trajectory in level plane (Figure 6) with a maximum leader bank angle o approx. 50 deg (as recorded in typical light tests o the WVU YF-22 models); The ormation geometry was: c = 100 t; l c = 100 t, h c = 0 t, with initial position error = 300 t, l = 300 t, h = 0 t. The simulations were conducted with and without the modeling o wind gusts acting in speciic light segments. A more detailed analysis o the results shows that, as expected, the control scheme provides better perormance Conclusions This paper presents an approach or the design o linear control laws to maintain speciied geometry or a ormation o research aircrat models. The design is based on compensation-type controllers or minimizing tracking errors along the orward, lateral, and vertical axes. The analysis shows that the availability o the Euler angles rom the leader aircrat is critical or the wingman to maintain the assigned ormation geometry throughout the maneuvered light. An additional goal was to evaluate the criteria to limit the necessary data communication between the leader and the wingman. The design has been veriied through a set o simulation studies interacing the aircrat models and the control schemes in Simulink with a VRT environment. The results o the simulation show a desirable perormance o the ormation control schemes. Reerences 1 - Pachter M., D Azzo J.J., Proud A. W., Tight Formation Flight Control, Journal o Guidance, Control, and Dynamics, Vol. 24, No. 2, March-April 2001, pp Proud A.W. Close Formation Flight Control, MS Thesis, AFIT/GE/ENG/99M-24, School o Engineering, Air Force Institute o technology (AU), Wright-Patterson AFB, OH, March Giulietti F., Pollini L., Innocenti M., Formation Flight control: A Behavioral Approach, Proceedings o the 2001 AIAA GNC Conerence, AIAA Paper , Montreal, Canada, August Singh S.N., Pachter M., Chandler P., Banda S., Rasmussen S., Schumacher, C.J. Input-Output Invertibility and Sliding Mode Control or Close Formation Flying o Multiple UAVs, Proceedings o the 2000 AIAA GNC Conerence, Denver, CO, August Ljung, L.,: System Identiication: Theory or the User, 2 nd Ed., PTR Prentice Hall, Upper Saddle River, Englewood Clis, NJ, Maine, R.E., Ili, K.W., Identiication o Dynamic Systems: Theory and Formulation, NASA RF 1168, June Stevens, B. and Lewis, F. Aircrat Control and Simulation, John Wiley & Sons, NY, Hock, W. and K. Schittowski, A Comparative Perormance Evaluation o 27 Nonlinear Programming Codes, Computing, Vol. 30, p. 335, The VRML Web Repository (Dec. 2002): Rauw, M.O.: "FDC A Simulink Toolbox or Flight Dynamics and Control Analysis". Zeist, The Netherlands, ISBN: ,