Adaptive Optimal Path Following for High Wind Flights

Transcription

1 Milano (Italy) August - September, 11 Aaptive Optimal Path Following for High Win Flights Ashwini Ratnoo P.B. Sujit Mangal Kothari Postoctoral Fellow, Department of Aerospace Engineering, Technion-Israel Institute of Technology, Haifa, 3, Israel. ( ashwini@aeroyne.technion.ac.il) Research Scientist, Department of Electrical an Computer Engineering, University of Porto, Portugal ( sujit@fe.up.pt) Grauate stuent, Department of Engineering, University of Leicester, Unite Kingom-LE1 7RH. ( mk1@le.ac.uk) Abstract: Unmanne aerial vehicle path following is aresse as an infinite horizon regulator problem. Using the linear quaratic regulator technique an optimal guiance law is erive. The state weighting matrix is chosen as a function of the position error an this aaptive nature of the cost function controls the UAV errors tightly. Numerical simulations are carrie out for straight line an circular paths uner various win conitions. Results show consierably lower position errors as compare to an existing guiance law. Path following with wins up to half the vehicle airspee is achieve. Keywors: Guiance systems, Optimal control, Path planning 1. INTRODUCTION Unmanne aerial vehicles (UAVs) have been successfully eploye in various applications that inclue search an rescue, surveillance, patrol an monitoring missions. They have the capability to provie information remotely an act like an extene sensor. With avances in flight control an miniaturization of UAVs, they are foun to play a key role in urban environments. Of particular interest is path following in high wins. Urban environments have unstructure win patterns an with high magnitues causing very high errors in UAV path following. Most of the mission paths can be efine using a set of waypoint an loiters maneuvers, where way-points are straight line segments while loiters are circular orbits. The path following controller has to accurately track the esire path in presence of win isturbances. The path following control law must have low computational complexity to be use in real-time. There is wie boy of literature on eveloping guiance laws for UAVs to follow straight line, curve trajectories an loiter maneuvers in the presence of wins. A piecewiseaffine control law with Lyapunov base controller for the UAV to follow a path is presente by Shehab an Rorigues [5]. The authors i not consier win isturbances in their work. Kaminer et al. [] evelope an optimal control law for multiple UAV trajectory following with coorination. The trajectory is parameterize as a function of time to follow which becomes an issue when UAV is initially locate away from the path. A L 1 aaptive controller for UAV path following taking win isturbances into account is evelope by Cao et al. [7]. Nelson et al. [] evelope a simple an effective path following controller using vector fiels. Vector fiels are evelope for both straight line an circular paths in the presence of wins. Rysyk [] evelope guiance laws for following straight line an circular paths base on helmsman behavior. The guiance law changes the course of the vehicle base on the win magnitue an irection which is estimate online. Lawrence et al. [] evelope a generalize vector fiel metho with lyapunov stability for ifferent maneuvers. Ceccarelli et al. [7] esigne a guiance law for a MAV taking win into consieration for continuous monitoring a target in the camera fiel of view. Breivik an Fossen [7] presente guiance laws for path following in D an 3D using control lyapunov functions. The approach oes not take win into account. Park et al. [7] presente a non-linear guiance law () that generates a reference target point on the path an uses this reference point as a target to maneuver the aircraft along the path. The authors present theoretical results an show results through experiments uner win isturbances. A typical path following comman as a constant gain linear function of position an velocity errors is incapable of placing a har boun on the position error. As the main contribution of the present work, we present an optimal UAV path following guiance law as an infinite horizon linear quaratic regulator (LQR) solution with an aaptive running cost where the state weighting gains are functions of the position error itself. The state weighting matrix is chosen as a function of the current an the maximum allowable position error an this aaptive nature of the cost function controls the UAV errors tightly in presence of wins. The guiance law oes not require path curvature information an is base on position an heaing errors of the UAV. Copyright by the International Feeration of Automatic Control (IFAC) 195

2 Milano (Italy) August - September, 11 The organization of the rest of the paper is as follows: The UAV path following problem is efine in Section followe by the aaptive optimal guiance law erivation in Section 3. Guiance law implementation an UAV kinematics are escribe in Section. Section 5 presents the simulations stuies on the guiance laws with comparative results. Section contains concluing remarks.. PROBLEM FORMULATION 3.1 Infinite horizon LQR Consier the general infinite-horizon nonlinear regulator problem of the form: Minimize J = 1 [x T Qx + Ru (t)]t () t with respect to the state x an control u subject to the linear system ynamics ẋ = Ax + Bu (5) an erive the feeback control u = R 1 B T Px () where P is obtaine from the algebraic Riccati equation A T P + PA PBR 1 B T P + Q = (7) an Q an R > are esign parameters. 3. Trajectory Tracking Problem an LQR Solution Fig. 1. Formulation Geometry Consier a UAV flying with a constant spee v in a plane as shown Fig. 1. The UAV mission is to follow a esire path as shown with the ashe line in Fig. 1. The UAV has a position error of with respect to the esire path an also a velocity error calculate as = v sin(ψ ψ p ) (1) wherev, ψ an ψ p are the UAV spee, heaing angle, an esire path angle, respectively. The path following problem is to formulate a control law u for the UAV to keep the errors, an small in presence of changing esire path geometries an wins. We consier the path following problem with a boun on the allowable error as b () where b is the esire position error ban with respect to the path. Also, u is the lateral acceleration which can be relate to the UAV heaing angle rate ψ, as u = v ψ (3) 3. ADAPTIVE OPTIMAL GUIDANCE LAW The problem of UAV path following is inherently an infinite horizon regulator problem where the guiance objective is to annul the UAV position an velocity errors with respect to a esire path. Contrary to finite time missile guiance applications, the UAV regulator solutions shoul keep the errors close to zero for all times an not just at the termination of the flight. We use aaptive running cost on the error states in an infinite horizon regulator formulation for the UAV path following problem. Linearizing UAV ynamics with respect to the esire heaing as v = = v sin(ψ ψ p ) () v = v( ψ ψ p ) cos(ψ ψ p ) (9) In the absence of path curvature information, we assume ψ p =. Using (3) an substituting for ψ p in (9) with small angle (ψ ψ p ), we have v = v ψ = u (1) Representing () an (1) in the stanar form we have x = [ v ], A = ẋ = Ax + Bu (11) 1, B = 1 (1) Consier the general infinite-horizon linear quaratic regulator problem where we minimize J = 1 [x T Qx + Ru (t)]t (13) t with respect to x an u subject to (11). Here, Q an R are the state weighting matrix an the control weighting matrix, respectively. Also, Q an R are the two esign parameters available to us for obtaining the esire control. We assume R = 1 (1) an Q to be an aaptive matrix with q Q = 1 q (15) We check the conitions for the LQR control solution to be locally asymptotically stable as, 19

3 Milano (Italy) August - September, 11 (1) {A, B} shoul be controllable in the omain of interest. Using (1) we have, {B,A B} = 1 (1) () f(x) = Ax C 1. From (1), we euce Ax = C 1 (17) Y W1 u R θ UAV ψ v ψp W esire straight line path (3) f() =. Using (17), we have f() = (1) O X (a) Straight-line () B. From (1) we have B as a constant non-zero matrix. (5) Q an R >. From (15) an (1) these conitions are met. Solving algebraic Riccati equation Optimal solution for the optimal control problem (13) is given as, u UAV ψ v ψp where P > is efine as, p11 p P = 1 p 1 p u = R 1 B T Px (19) is obtaine by solving the algebraic Riccati equation, () R' Rc esire circular path A T P + PA PBR 1 B T P + Q = (1) Solving for P using (1), (1), (15) an () in (1), we have p11 p 1 p11 p p11 p 1 1 p 1 p p 1 p p 1 p 1 p11 p [ 1 ] 1 q + 1 p 1 p q = () Simplifying an equating iniviual elements, we have, p 1 = q 1 (3) p = q 1 + q () p 11 = q 1 q 1 + q (5) Aaptive feeback control Using (1), (), (3), () an (5) in (19), we have the optimal control solution as, u p11 p = [ 1 ] 1 p 1 p v [ ] = q 1 + +q 1 + q v () We propose an aaptive gain q1 for a running penalty on as, q1 = b b (7) where b is the maximum allowable position error as consiere in (). The aaptive state weight q 1 increases Fig.. Path geometries (b) Circular the penalty as the UAV position error increases an gets closer to the maximum allowable limit. Very close to the maximum allowable limit the UAV uses maximum control to keep itself in the esire error ban. Note that, the weight remains close to unity for UAV flights with negligible error. This way the error is controlle tightly in case there are suen changes in the esire path or win isturbances. Since there are no practical concerns on the gain q associate with v, we chose q = 1 () Using (7) an () in (), we have the aaptive optimal guiance law () as u b = b + b b + 1 v (9). GUIDANCE LAW IMPLEMENTATION UAV trajectory planning involves two stanar geometries, namely, the straight line/way-point guiance an circular/loiter guiance. The guiance law of (9) can be implemente on the two stanar geometries an the relate kinematics are escribe next. 197

4 Milano (Italy) August - September, 11.1 Straight-line following Consier a straight line path forme by the way-points W1-W as shown in Fig. (a). Let R be the isplacement of UAV with respect to point W1 making an angle θ with the reference. The UAV position an velocity errors can be calculate as, Y position (m) Path an = R sin(θ ψ p ) (3) 1 5 Win Direction respectively. = v sin(ψ ψ p ), (31) X position (m) (a) Trajectories. Circular path following Consier a circular path of raius R c as shown in Fig. (b). Let R be the isplacement of the UAV with respect to the center of the circular path. The UAV position an velocity errors can be calculate as, an = R R c (3) 1.3 Effect of win = v sin(ψ ψ p ), (33) Motion of a UAV with respect to groun or stationary esire paths is affecte by win isturbances. In presence of win the groun velocity components, say v gx an v gy, of the UAV are given as v gx = v cos ψ + x (3) v gy = v sin ψ + y (35) where x an y are the components of the win spee along X an Y irection. Note that though is inepenent of win, v in presence win can be expresse as v = v sin(ψ ψ p ) + sin(ψ w ψ p ) (3) where ψ w is the irection of win with reference. Note that kinematics escribe in this section is use to simulate trajectories in the next section. However, in real life scenarios a GPS/INS aie UAV can compute onboar its groun position an groun velocity an then an v can irectly be compute with respect to any path. 5. SIMULATION RESULTS Using the guiance law erive in (9) we simulate various path following scenarios. Simulations are carrie out for straight line an circular paths consiering the two as the basic geometries for a general path planning problem. For comparative stuies we use well known non-linear guiance law ()Park et al. [7] an present the results here. The UAV spee, v, is consiere 5 m/sec with a minimum turn raius r min = 3v = 75 m. Lateral acceleration: u (m/sec ) (b) Position error (c) Lateral accelerations Fig. 3. Straight line following in % win 5.1 Straight line following Consier a straight line path passing through the origin of the X-Y plane with a unity slope as shown by a ashe line in Fig. 3(a) with a ashe black line. We consier win with =.v blowing perpenicular to the esire path with ψ w = 135. The parameters use for are: b = an for - L1 = 15. Trajectories for an are plotte in Fig.3 (a) with blue an re colors, respectively. The trajectory is affecte highly by the cross win. The corresponing position errors are shown Fig. 3(b) where has a maximum UAV position error of m an the transient of the position error settles quickly to the steay state. Whilst has a maximum position error is as high as 9.5 m an the guiance law has much higher settling time. Lateral acceleration profiles for 19

5 Milano (Italy) August - September, 11 1 =.3v =.v =.5v 3 Path Y (m) (a) Position error for X (m) (a) Trajectories 5 =.3v =.v =.5v (b) Position error for Fig.. Position error comparison in 3, an 5 % win following a straight line path the case are plotte in Fig. 3(c). For, as shown in blue color, the aaptive gains of (9) quickly increase as the win inuces the position error. This higher control input quickly reuces the position error an both the errors an the control input go to zero at a fast rate. The lateral acceleration is lower initially resulting in a higher position error cause ue to win. Further, we vary the win spee magnitue an compare the performance of the two guiance laws. The position error for is plotte in Fig. (a) with an approximate maximum of 3, an 9.5 m for a win spee of 3,, 5% of the UAV airspee. The performance, as shown in Fig. (b), eteriorates consierably in presence of high wins an corresponing maximum position errors are 1.5, 19 an 5 m, respectively. 5. Circular path following Next, we consier a circular path of raius 5 m centere at the origin an a UAV initially following the circle with no errors. The win is consiere blowing at a spee 3% of the UAV air spee in a irection perpenicular to the UAV initial heaing. The trajectories are plotte in Fig. 5(a) where the blue an re color paths show an trajectories, respectively. The esire circular path is shown by a ashe curve. Again, the win is less affective against the law where a maximum position error, as plotte in Fig. 5 (b), of. m in observe. The corresponing maximum error for is aroun 1 m. Note that the position errors o not converge to zero as in the straight line following case Lateral acceleration:u (m/sec ) (b) Position error (c) Lateral accelerations Fig. 5. Circular path following in 3% win since the UAV woul require a centripetal acceleration to follow the circular path an nees position an velocity errors to generate it. The corresponing lateral acceleration profiles, as plotte in Fig. 5(c), shows a higher initial control response by the resulting in superior error control. Next, we stuy the effect of win magnitue on circular path following with win spees 5, 35 an 5 % UAV air spee. All other parameters are the same as previously. The position errors for are plotte in Fig. (a) with a maximum of 3.3,. an 1. m for =.5v,.35v, 5v, respectively. The corresponing maximum error values for, as shown in Fig. (b), are 1.9, 1.1 an 5. m, respectively. A consierable position error improvement is gaine using the propose guiance law. 199

6 Milano (Italy) August - September, 11 Error in istance (m) 1 1 =.5v =.35v =.5v (a) Position error for =.5v =.35v =.5v (b) Position error for ference, New York, NY, USA, 7, pp , oi:1.119/acc.7.3 D.R. Nelson, D.B. Barber, T.W. McLain an R.W. Bear Vector Fiel Path Following for Small Unmanne Air Vehicles Proceeings of the American Control Conference, Minneapolis, Minnesota, USA,, pp , oi:1.119/acc..157 R. Rysyk UAV Path Following for Constant Line-of-Sight AIAA Unmanne Unlimite Systems, Technologies an Operations Aerospace, Lan an Sea Conference, San Diego, CA, USA, 3, AIAA-3-. D.A. Lawrence, E.W. Frew an W.J. Pisano Lyapunov Vector Fiels for Autonomous UAV Flight Control AIAA Guiance, Navigation an Control Conference an Exhibit, Hilton Hea, South Carolina, USA, 7, AIAA N. Ceccarelli, J.J. Enright, E.Frazzoli, S.J. Rasmussen an C.J. Schumacher Micro UAV Path Planning for Reconnaissance in Win Proceeings of the American Control Conference, New York City, NY, USA, 7, pp , oi:1.119/acc M. Breivik an T.I. Fossen Principles of Guiance-Base Path Following in D an 3D Proceeings of the IEEE Conference on Decision an Control, Seville, Spain, 5, pp. 7-3, oi:1.119/cdc S. Park, J. Deystt an J.P. How Performance an Lyapunov Stability of a Nonlinear Path-Following Guiance Metho Journal of Guiance, Control, an Dynamics, vol. 3, no., pp , oi:1.51/ Fig.. Position error comparison in 5, 35 an 5 % win following a circular path. CONCLUSIONS An aaptive optimal UAV guiance law is presente for high win flights. Infinite horizon LQR formulation is use to solve the guiance problem after linearizing the error ynamics. An aaptive state weighting matrix is use for tighter control of UAV errors in high isturbances. The guiance law results is consierably low errors as compare to an existing law in the literature. Simulations are carrie out for win spees up to half the UAV airspee for straight line an circular path following. REFERENCES S. Shehab an L. Rorigues Preliminary Results on UAV Path Following Using Piecewise-Affine Control Proceeings of the IEEE Conference on Control Applications, Toronto, Canaa, 5, pp oi:1.119/cca I. Kaminer, O. Yakimenko, A. Pascoal an R. Ghabcheloo Path Generation, Path Following an Coorinate Control for Time Critical Missions of Multiple UAVs Proceeings of the American Control Conference, Minneapolis, Minnesota, USA,, pp , oi:1.119/acc C. Cao an N. Hovakimyan, I. Kaminer, V.V. Patel an V. Dobrokhoov Stabilization of Cascae Systems via L1 Aaptive Controller with Application to a UAV Path Following Problem an Flight Test Results Proceeings of the American Control Con- 199