A Hybrid Feedback Control Strategy for Autonomous Waypoint Transitioning and Loitering of Unmanned Aerial Vehicles

Transcription

1 A Hybrid Feedback Control Strategy for Autonoous Waypoint Transitioning and Loitering of Unanned Aerial Vehicles Dean W. Sith III a, Ricardo G. Sanfelice b a Guidance Navigation and Control Center. Raytheon Missile Systes, Tucson, AZ b Coputer Engineering Departent. University of California, Santa Cruz, CA Abstract We consider the proble of autonoously controlling a fixed-wing aerial vehicle to visit a neighborhood of a pre-defined waypoint, and when nearby it, loiter around it. To solve this proble, we propose a hybrid feedback control strategy that unites two state-feedback controllers: a global controller capable of steering or transitioning the vehicle to nearby the waypoint and a local controller capable of steering the vehicle about a loitering radius. The aerial vehicle is odeled on a level flight plane with syste perforance characterized in ters of the aerodynaic, propulsion, and ass properties. Thrust and bank angle are the control inputs. Asyptotic stability properties of the individual control algoriths, which are designed using backstepping, as well as of the closed-loop syste, which includes a hybrid algorith uniting the two controllers, are established. In particular, for this application of hybrid feedback control, Lyapunov functions and hybrid systes theory are eployed to establish stability properties of the set of points defining loitering. The analytical results are confired nuerically by siulations. 1. Introduction 1.1. Background Autonoous navigation of unanned aerial vehicles (UAVs) requires algoriths that are capable of accurately controlling the otion of the vehicle with only liited control authority. Such algoriths should be capable of precisely controlling the position, orientation, and velocity of the vehicle by properly generating forces and torques through thrusters, propellers, oving surfaces, etc. A wide range of tasks for such vehicles can be recast as the proble of steering the vehicle to a path given by closed curve, e.g., a straight line or a loitering pattern. To accoplish such tasks, control algoriths capable of aintaining the vehicle aloft along the given path are required. Recent results in the literature deonstrate that feedback control algoriths can be designed to steer UAVs along different paths by reshaping the vector fields that describe the otion of the vehicle under the effect of a guidance law defined by a particular Preprint subitted to Elsevier August 25, 2017

2 set of differential equations [1, 2]. The authors of [2] argue that their reshaping approach is explicit (in the sense that the task to accoplish is pre-defined) and that avoids potential coplications of trajectory planning and tracking. While perforance is not assessed in [2], the perforance of tracking control algoriths given in ters of linear, static state-feedback laws deteriorates when curved paths are to be tracked. A study of perforance of tracking algoriths was reported in [3], where a nonlinear guidance logic for trajectory tracking of UAVs that overcoes the inherent liitation of linear feedback control in following curved paths is proposed. An alternative to the vector field shaping technique entioned above is to rely on ultiple controllers, each of the designed to accoplish a different task, and appropriately switch aong the. In such approach, a supervisory control algorith onitors the state of the vehicle and, based on the given ission, deterines which controller should be applied at each instant. The switching logic should be capable of piecing together the individual controllers to achieve the desired vehicle otion while precluding chattering around the set of points defining the switching condition. While the approach allows independent design of the individual controllers, the eergence of discrete (or discontinuous) dynaics is unavoidable, which ay ake the analysis ore involved (e.g., see the exaple in [4], which shows that two controllers a local and a global continuous-tie controller cannot be united using a continuous-tie supervisor). Furtherore, the condition triggering the switches between the controllers has to involve eory so as to avoid chattering [5]. Fortunately, recent advances in the literature of hybrid systes have ade systeatic design of control algoriths piecing together individual controllers possible [6]. Interestingly, the design of such systes can be perfored to yield a closed-loop syste that not only is chattering-free but also is robust with respect to easureent noise, actuator errors, and external disturbances; see, e.g., [7]. Due to these unique capabilities, the said hybrid systes approach to the cobination of ultiple controllers has been successfully eployed in different applications, such as the stabilization of an inverted pendulu [6] and of the position and orientation of a obile robot [8]. Furtherore, the technique has been extended in [9] to allow for the cobination of ulti-objective controllers, including state-feedback laws as well as open-loop control laws. In the context of perforance, a trajectory-based approach was also eployed in [10] to generate dwell-tie and hysteresis-based control strategies that guarantee an input-output stability property characterizing closed-loop syste perforance. More related to the application studied in this paper, algoriths for vehicles that use ultiple controllers coordinated by a supervisory algorith also lead to a hybrid syste and have been proposed in the literature. A review of such works and an algorith for the control of single and ultiple UAVs appeared in [11], where a hybrid autoaton with odes corresponding to each control task is proposed and an exaple of an altitude hybrid controller for a fixed wing UAV is presented. A general forulation of the otion planning proble for dynaical systes with syetries, which, in particular, includes odels of vehicles, appeared in [12], where a general language for trajectory genera- 2

3 tion using otion priitives; see also the robustification via supervisory hybrid feedback control [13] Contributions In this paper, we eploy the hybrid approach outlined above to provide a solution to the proble of autonoously controlling a fixed wing aerial vehicle to visit a neighborhood of a pre-defined waypoint, and when nearby it, loiter around it without chattering. More precisely, we propose a hybrid feedback control strategy that unites two state-feedback controllers: a global controller capable of steering or transitioning the vehicle to nearby the waypoint and a local controller capable of steering the vehicle about a loitering radius. Following [14], the aerial vehicle is odeled on a level flight plane with syste perforance characterized in ters of the aerodynaic, propulsion, and ass properties. The resulting odel is nonlinear and with thrust and bank angle being its control inputs. This nonlinear UAV bank-to-turn odel partially resebles a ship course controller odel [15], where heading is controlled indirectly through the heading rate. For this vehicle odel and eploying Lyapunov stability theory, we establish key asyptotic stability properties of the individual control algoriths designed. Both the local and global controllers are designed using the backstepping control design technique [16, 17]. With the region of attraction induced by each controller being characterized, the closed-loop syste incorporating a hybrid algorith uniting the individual controllers is shown to be asyptotically stable using stability tools for hybrid dynaical systes. We are not aware of a siilar solution for this UAV proble, for which the application of hybrid systes theory leads to a hybrid feedback control algorith that is chattering-free and with rigorously established properties of the region of attraction Organization of the paper The reainder of the paper is organized as follows. Section 2 forulates the proble to solve, introduces the odel of the vehicle, and proposes the structure of the hybrid controller to be designed. The ain results follow in Section 3. This section starts by recasting the proble of interest as a set stabilization proble. Then, it provides the design of the local and global controllers in two steps: first, a controller when actuation is through thrust and heading angle (Section 3.1) and, second, using backstepping, a controller when the actuation is through thrust and bank angle (Section 3.2). In Section 3.3, a hybrid controller cobining the two previously designed controllers and the properties it confers to the closed-loop syste are presented. In Section 4, the proposed control law and the results are validated in several siulations Notation and Noenclature 3

4 x Euclidean nor of x A Set to be stabilized c level set, constant C Hybrid syste flow set C D Drag coefficient C D0 Zero lift drag coefficient C L Lift coefficient D Hybrid syste jup set D Drag, N e Error function f Hybrid syste flow ap G Hybrid syste jup ap g Gravity, /s 2 h H k K K L p q r Altitude, Hybrid syste Controller gain Drag Polar Coefficient Faily of continuous functions that are zero at zero, strictly increasing, and unbounded Lift, N Local switching variable Controller index Turn radius, R Magnitude of position vector, R Real nubers s duy variable S ref Wing reference area 2 T Thrust, N U Doain of hybrid syste v Airspeed, /s V Lyapunov function W Weight, N x x position, y y position, z State vector of hybrid syste α class κ function κ Controller ρ Air density, kg/ 3 µ Local controller rotation hysteresis Ψ Heading, deg Φ Bank angle Subscripts 0 Initial Condition b Body Frae C Coanded 2. Proble Forulation We consider the proble of stabilizing an aerial vehicle to loiter around a given waypoint with specified velocity. Figure 1 depicts the scenario of proble to be solved, where v is the airspeed of the vehicle, (x, y) R 2 describe its position, and Ψ its heading angle. In this way, the vehicle s velocity and heading angle are related by where v = ([ ẋ ẋ 2 + ẏ 2, Ψ = ẏ ]), (1) : R 2 \ 0} [ π, π] (2) defines the angle, positive in a clockwise direction, between the vector input and the positive vertical axis (x). Without loss of generality, the waypoint is assued to be at the origin of the position coordinates (x, y) and the desired radius for loitering is R C. Note that is undefined when x and y are 0. The forces on the UAV are described in the body frae (x b, y b ) and can be transfored into the plane of the absolute frae (x, y) using the heading angle Ψ, via [ ] cos Ψ sin Ψ (3) sin Ψ cos Ψ 4

5 Several assuptions are ade in order to focus the analysis for the scope of this paper. These assuptions are listed below: UAV odeled as a point ass; Level flight in the (x, y) plane; Gravitational acceleration is constant; Aerodynaic properties odeled as a siple drag polar with a axiu lift coefficient; Propulsion perforance is liited by a axiu thrust; Mass is constant (no fuel burn); Vehicle banks to turn and the bank angle Φ satisfies Φ ( π 2, π 2 ). These assuptions are typical in the conceptual design phase of an aircraft where syste level aerodynaic and propulsion perforance are balanced to achieve ission objectives. Figure 1: Waypoint Geoetry Free Body Diagra The equations of otion for the UAV are deterined by using first principles. The free body diagra for the aircraft is shown in Fig. 2. Suing the forces in the body axes shown in Fig. 2 gives the equations F xb = T D, (4) F yb = L sin Φ, (5) F h = W L cos Φ, (6) 5

6 Figure 2: Level Flight Free Body Diagra. where T is thrust, D is drag, L is lift, W is weight, and Φ is bank angle. Since the proble is siplified to only consider otion in the plane, the su of forces in the vertical direction in Eq. (6) are assued to be equal to zero, which defines the lift required for the UAV to aintain altitude and is given by F h = 0 L = W cos Φ. (7) 2.2. Aerodynaic Model Aerodynaic forces such as lift and drag are given by L(v, Φ) = C L (v, Φ)S ref ρv 2 2, D(v, Φ) = C D(v, Φ)S ref ρv 2 2, (8) where lift and drag are expressed as coefficients (as defined in the noenclature) and depend explicitly on the values of v and Φ. Substituting Eq. (8) into Eq. (7) allows one to solve for the required lift coefficient to balance the weight of the UAV, which is given by C L (v, Φ) = 2W S ref ρv 2 cos Φ. (9) The drag coefficient (C D ) can be expanded into a zero lift drag coponent (C D0 ) and an induced drag coponent as follows: Eq. (10) is known as a drag polar [18]. C D (v, Φ) = C D0 + KC L (v, Φ) 2. (10) 2.3. State and Control Boundaries The iniu speed and axiu bank angle of the UAV are liited by the axiu lift coefficient C L ax. Explicit expressions for the iniu speed and axiu bank angle of a UAV obtained fro Eq. (10) are given as follows: v in (Φ) = 2W S ref ρc L ax cos Φ [ 2W S ref ρc L ax, ) (11) 6

7 Φ ax = cos 1 ( 2W S ref ρv 2 C L ax ) (12) The axiu speed of the UAV is defined by the speed at which drag equals the axiu thrust (T ax ) of the UAV. Starting by setting the ax thrust T ax equal to drag and substituting in the expressions for drag given in Eq. (10) and the expression for required lift given in Eq. (9) gives an algebraic expression involving velocity. Solving for velocity gives the axiu velocity T ax + T 2 ax 16C D0 W v ax = 2 K, (13) ρc D0 which occurs at a bank angle of zero Hybrid Syste Model As outlined in Section 1.2, our solution to the nonlinear control proble stated above consists of a hybrid feedback control strategy that unites two statefeedback controllers: a global controller capable of steering or transitioning the vehicle to nearby the waypoint and a local controller capable of steering the vehicle about a loitering radius. In this way, the closed-loop syste resulting when applying our proposed hybrid control algorith is a hybrid dynaical syste. The state, which we denote by z, of the proposed aerial vehicle controlled by a hybrid control algorith is given by z = [x y v Ψ q] U := R R [v in, v ax ] [ π, π] 1, 2} (14) where q is a switching variable that describes which controller is being utilized. A value q = 2 corresponds to the global (transit) controller and q = 1 corresponds to the local (loiter) controller being in the loop. The global controller will be designed to guide the vehicle at speed v C2 to nearby the waypoint while the local controller will be designed to aneuver at a desired speed v C1 about the loiter circle of radius R C centered around the waypoint; see Figure 1. The paraeter R C defines the desired loiter circle radius used in the feedback law. The velocity is liited by the iniu and axiu steady level speed of the UAV (v in and v ax ). For each q 1, 2}, the UAV s thrust (T q ) and bank angle (Φ q ) define the control inputs of the UAV and are constrained as follows: (T q, Φ q ) [T in, T ax ] [ Φ ax, Φ ax ]. (15) The switching between two controllers allows the dynaics of the UAV to be odeled as a hybrid syste. The switch of controllers is a discrete event, while the otion of the UAV is a continuous flow of real tie. The hybrid fraework we eploy in this work is presented in [6], where a generic hybrid syste, H, is given by four objects (C,f,D,G) defining its data: Flow ap: a single-valued ap f defining the flows (or continuous evolution) of H; 7

8 Flow set: a set C specifying the points where flows are possible; Jup ap: a set valued ap G defining the jups (or discrete evolution) of H; Jup set: a set D specifying the points where jups are possible. Then, for the control proble in this paper, a hybrid syste H = (C, f, D, G) representing the closed-loop syste fro controlling the UAV using the transit and loiter controllers has state space U and can be written in the copact for H : z U ż = f(z, κ(z)) z C z + = G(z) z D, (16) where κ is the feedback law applied to the control input of the UAV, which is given by κ1 (x, y, v, Ψ) if q = 1 (local/loiter) κ(z) = (17) κ 2 (x, y, v, Ψ) if q = 2 (global/transit). As it is ade precise below, each feedback law has corresponding flow set (C q ) and jup set (D q ) that specify the doain where the control law is applied and where to switch to another control law respectively. The flow and jup ap for the UAV hybrid syste are given by f(z, (T q, Φ q )) = G(z) = v cos Ψ v sin Ψ T q D(v,Φ q) g v tan Φ q 0 x y v Ψ 3 q (18) (19) The flow ap defines a differential equation that corresponds to the equations of otion for the UAV in the plane under the effect of the feedback law, as discussed previously, and the jup ap toggles which controller is coanding thrust and bank angle. The events triggering jups are deterined by the following choice of the flow and jup sets: C := C 1 C 2, D := D 1 D 2 8

9 where C 1 := z U : q = 1, R(x, y) R C d, R(x, y) 0} C 2 := z U : q = 2, 1 ( (R(x, y) RC ) 2 + (v v C2 ) 2 + (Ψ Ψ C2 (x, y)) 2) } c, R(x, y) 0 2 D 1 := z U : q = 1, R(x, y) R C d, R(x, y) 0} D 2 := z U : q = 2, 1 ( (R(x, y) RC ) 2 + (v v C2 ) 2 + (Ψ Ψ C2 (x, y)) 2) } c, R(x, y) 0 2 (20) where R(x, y) = x 2 + y 2. The flow and jup sets were selected to ensure stability of the hybrid controller. This was done by ensuring that the boundary for the global jup set (D 2 ) is contained in a sub-level set of the basin of attraction induced by the local controller. The details of this selection are discussed in the uniting global and local controller section, which is Section 3.3. For the special case of R = 0, the hybrid syste is undefined. However, solutions cannot reain at R = 0 because of the lower bound on velocity given in Eq. (23). 3. Design of Hybrid Controller for Transit and Loitering The transit and loiter control laws for the UAV have different objectives. The transit control law will coand a thrust to approach the loiter circle at transit velocity, while coanding a heading to point the UAV s velocity vector towards the loiter circle. In contrast, the loiter control law will coand thrust (less than the axiu) to achieve the desired loiter velocity, while gradually coanding a heading to ake the velocity vector of the UAV tangent to the desired loiter circle. Fig. 3 shows a block diagra of the closed-loop syste where a hybrid controller perfors the selection of transit and loiter ode. Figure 3: Control Syste Block Diagra. The supervisory logic within the hybrid controller switches between the controllers to steer the UAV to the loiter circle and then loiter about it. The objective of the hybrid controller in loiter ode is to stabilize the UAV into 9

10 a loiter pattern defined by a constant radius R C and a constant velocity v C, which iplicitly defines a desired heading Ψ C. The target set of points for the state coponents of the UAV (x, y, v, Ψ) and of the controller (q) is given by A = (x, y, v, Ψ, q) U : R(x, y) = R C, v = v Cq, Ψ = Ψ Cq (x, y), q = 1}. (21) Naely, the goal is to steer R to R C at speed v Cq with q = 1. For both the local and global controller, an error quantity for position, velocity, and heading angle will be defined. For each q 1, 2} the error is given by e 1,q R(x, y) R C e q = e 2,q e 3,q := v v Cq Ψ Ψ Cq. (22) Asyptotic stability of the set A will be proven using the backstepping ethod[17] by first showing that the syste can be stabilized by controlling Ψ directly. Section 3.1 covers the proof of stability for the loiter and transit controllers individually through Ψ actuation. Section 3.2 applies the backstepping ethod to the results fro Section 3.1 to show that both the loiter and transit controllers can be stabilized individually with actuation through Φ. Section 3.3 cobines the transit and loiter controllers fro Section 3.2 to create a united hybrid controller that aintains the stability properties of the individual controllers Control design for the case of actuation through (T, Ψ) In this case, the state of interest becoes z = [x y v q] Ũ := R R [v in, v ax ] 1, 2} (23) and the error syste is defined as ẽ q = [ ẽ1,q ] := ẽ 2,q [ ] R(x, y) RC. (24) v v Cq The state error equation has continuous dynaics given by [ x R(x,y) ẽ q = v cos Ψ q + y R(x,y) v sin Ψ ] q T q D(v) (25) where the control input is defined as (T q, Ψ q ). Note that in this case, drag is only a function of velocity (i.e., D(v)) instead of velocity and bank angle. A candidate Lyapunov function is given by 1 Ṽ q (ẽ q ) = 1 2ẽ q ẽ q. (26) 1 Note that the Lyapunov function given in Eq. (26) is suitable for global asyptotic stability of the origin of the error syste as it is positive definite and radially unbounded in fact, it is lower and upper bounded by (quadratic) K functions. 10

11 Note that the target set of points fro ẽ q to stabilize is given by à = (x, y, v) : R(x, y) = R C, v = v Cq, q = 1}, (27) and the flow and jup sets are defined as C 1 := z Ũ : q = 1, R(x, y) R C d, } R 0 C 2 := z Ũ : q = 2, 1 ( (R(x, y) RC ) 2 + (v v C1 ) 2) } c, R 0 2 D 1 := z Ũ : q = 1, R(x, y) R C d, } R 0 D 2 := z Ũ : q = 2, 1 ( (R(x, y) RC ) 2 + (v v C1 ) 2) } c, R 0, (28) 2 where c > (v ax v in ) 2 (29) d > 2 c. (30) Using Eq. (25), the variation of Ṽq with tie can be expressed as [ x R(x,y) Ṽq(ẽ q ), v cos Ψ q + y R(x,y) v sin Ψ ] q T q D(v) = Ṙ(R(x, y) R C)+ T q D(v) (v v Cq ). (31) where Ṙ is the first entry of Eq. (25). The thrust control input affects the second ter in Eq. (31). To ensure that this ter is negative, a siple control law is one that guarantees that the thrust is larger than the drag if the current speed is saller than the coanded speed, that is, v v Cq < 0, or saller than the drag if the current speed is larger than the current coanded speed, that is, v v Cq > 0. When the desired speed is achieved, that is, v = v Cq, the thrust can be set to be equal to the drag to aintain this speed. These properties are satisfied by proportional control law, that is, T q = D(v) k Tq (v v Cq ), (32) where k Tq > 0. It should be noted that though Eq. (32) applies to both the global and local regions, each region ay have a different target velocity (v Cq ), effectively creating two different controllers. Plugging the expression for the coanded thrust in Eq. (32) into Eq. (31) leads to [ x R(x,y) Ṽq(ẽ q ), v cos Ψ q + y R(x,y) v sin Ψ ] q T q D(v) = Ṙ(R(x, y) R C) k T q (v v Cq ) 2. (33) 11

12 Figure 4: Quiver Plot of Global Controller Coanded Heading Global/Transit Controller (q=2) The objective of the transit (global) controller (q = 2) is to steer the UAV fro every point in C 2 to a point in ( D 2 ) in finite tie, so as to switch to the loiter (local) controller. In ters of the state variables, this eans pointing the velocity vector of the UAV towards the loiter radius. The value of the coanded heading in the global region ( C 2 ) is given by the control law defined as ([ ]) x Ψ C2 (x, y) = (x, y) R 2 such that z ȳ C 2 (34) where 2 x = sign(r C R(x, y))x ȳ = sign(r C R(x, y))y Note that the global heading controller is not defined on the loiter circle. This is because the local controller will be used in that region. Fig. 4 shows a quiver plot of the global controller heading with respect to a generic loiter circle. The direction of the arrows in the plot indicate the heading. The global controller coands a heading that is orthogonal and pointing towards the loiter circle. Proposition 1. For every function D, constant > 0, velocity set-point v C2 [v in, v ax ], loiter radius R C > 0, there exists γ 2 K such that the closed-loop syste resulting fro controlling the error syste [ x R(x,y) ẽ 2 = v cos Ψ 2 + y R(x,y) v sin Ψ ] 2, T 2 D(v) 2 The sign function returns +1 if the input is positive, -1 if the input is negative, and 0 if the input is 0. 12

13 with the controller [ ] [ T2 D(v) kt2 (v v = κ Ψ 2 (x, y, v) := C2 ) 2 Ψ C2 (x, y) ], (35) where k T2 > 0 and Ψ C2 is given in Eq. (34), is such that Ṽ2(ẽ 2 ), ẽ 2 γ 2 ( ẽ 2 ) for all ẽ 2 R 2 such that [x y v 2] C 2. In particular, an appropriate choice for γ 2 is kt2 s 2 γ 2 (s) = in 2, v } ins 2 s 0. Proof Following Eq. (33), we have that Ṽ2(ẽ 2 ), ẽ 2 = Ṙ(R(x, y) R C) k T 2 (v v C2 ) 2, where the thrust controller in Eq. (35) is already included in the second ter. Substituting the expression for Ṙ given in the first entry of Eq. (25) results in Ṽ2(ẽ 2 ), ẽ 2 = ( x R(x, y) v cos Ψ + ) y R(x, y) v sin Ψ (R(x, y) R C ) k T 2 (v v C2 ) 2. Fro Eq. (34) a triangle can be drawn as shown in Fig. 5 to derive Figure 5: Global Controller Triangle x cos Ψ C2 = R(x, y), (36) ȳ sin Ψ C2 = R(x, y). (37) The substitution of Eq. (36) and Eq. (37) fro the heading controller gives ( ) x Ṽ2(ẽ 2 ), ẽ 2 = R(x, y) v x R(x, y) + y R(x, y) v ȳ (R(x, y) R C ) k T 2 (v v C2 ) 2, R(x, y) 13

14 which can be siplified as shown below. ( x Ṽ2(ẽ 2 ), ẽ 2 + y 2 ) 2 = v sign(r C R(x, y)) R(x, y) 2 (R(x, y) R C ) k T 2 (v v C2 ) 2. = v sign(r C R(x, y))(r(x, y) R C ) k T 2 (v v C2 ) 2. = v R(x, y) R C k T 2 (v v C2 ) 2. (38) Substituting the error paraeters fro Eq. (24) into Eq. (38), naely, ẽ 2 = [ẽ 1,2 ẽ 2,2 ], results in Ṽ2(ẽ 2 ), ẽ 2 = v ẽ 1,2 k T 2 (ẽ 2,2 ) 2. (39) Since v is lower bounded by v in, γ 2 can be defined as 3 kt2 s 2 γ 2 (s) = in 2, v } ins 2 s 0. (40) The first property in Proposition 1 establishes stability and unifor attractivity of the origin of the global error coordinates (e 2 ) for the closed-loop syste using the global controller on the flow set (C 2 ) Local Controller The objective of the local controller (q = 1) is to steer the state of the UAV to the desired loiter circle at the desired velocity. For the local controller, the set point velocity (v C2 ) used in Eq. (32) to deterine the thrust input is the desired loiter speed. The expression for the local controller coanded heading shown in Eq. (42) is inspired by the control law in [1]. We incorporate a discrete dynaic paraeter p 1, 1} designating the direction of rotation about the 3 This bound follows considering the generic function V (x) = ax bx 1, where x 1 > 0, a > 0, and b > 0. Note if x 1 x 2 then b x 1 = b ax x 1, x 2 } = b x. Since x = x x2 2 2 x, we get b x 1 b x. If x 1 < x 2 then ax = a x 2 a x 2 2 = a x }. Then we have V (x) α(x) := in a x 2 2, b x 2 14

15 loiter circle. 4 Then, the control law is defined by ([ 2 y(rc R(x, y) Ψ C1 (x, y) = 2 ) + p2xr(x, y)r C x(r 2 C R(x, y) 2 ) p2yr(x, y)r C ]) (x, y) R 2 such that z C 1 (42) which is well-defined and single valued on C 1. The local controller coands a Figure 6: Quiver plot of local controller coanded heading (CW for p=1 and CCW for p=-1). heading that gradually approaches the loiter circle tangentially and settles on an equilibriu that can rotate clockwise or counter-clockwise depending on p. Selecting p = 1 gives a counter-clockwise rotation, and selecting p = 1 gives a clockwise rotation. Fig. 6 shows a quiver plot of the local controller heading with respect to a generic loiter circle. The direction of the arrows in the plot indicate the heading. Proposition 2. For every constant > 0, velocity set-point v C1 [v in, v ax ], loiter radius R C > 0, and paraeter p 1, 1}, there exists γ 1 K such that the closed-loop syste resulting fro controlling the error syste ẽ 1 = [ x R(x,y) v cos Ψ 1 + T 1 D(v) y R(x,y) v sin Ψ 1 ], 4 This paraeter could becoe a state variable with discrete dynaics [ ] [ ] 1 x cos Ψ p = 1 and µ p + R(x,y) y sin Ψ = p when [ ] [ ] 1 x cos Ψ p = 1 and µ R(x,y) y sin Ψ (41) and continuous dynaics ṗ = 0 where µ [0, 1] is a constant paraeter that defines hysteresis for switching the value of p. The local controller is hybrid because of the switching of p. Switching is based on the cross product of unit position and velocity vectors as shown in Eq. (41). The selection of rotation direction in Eq. (41) utilizes the pre-existing rotation of the UAV around the loiter point based on the cross-product of the UAV s initial position and velocity vectors. 15

16 with the controller [ ] [ T1 D(v) kt1 (v v = κ Ψ 1 (x, y, v) := C1 ) 1 Ψ C1 (x, y) ], (43) where k T1 > 0 and Ψ C1 is given in Eq. (42), satisfies Ṽ1(ẽ 1 ), ẽ 1 γ 1 ( ẽ 1 ) for all ẽ 1 R 2 such that [x y v 1] C 1. In particular, an appropriate choice for γ 1 is } v in R C γ 1 (s) = in (R C + d), k T 1 s 2 s 0, 2 + RC 2 Proof In this proof, R(x, y) will be denoted as R to siplify notation. Following Eq. (33), we have that Ṽ1(ẽ 1 ), ẽ 1 = Ṙ(R R C) k T 1 (v v C1 ) 2, where the thrust controller in Eq. (43) is already included in the second ter. Substituting the expression for Ṙ given in the first entry of Eq. (25) results in ( x Ṽ1(ẽ 1 ), ẽ 1 = R v cos Ψ + y ) R v sin Ψ (R R C ) k T 1 (v v C1 ) 2. A triangle can be drawn fro Eq. (42) as shown in Fig. 7 to derive expressions for the sin and cos of Ψ C1. The and ± have been substituted for p fro Eq. (42) to account for both directions of rotation. Figure 7: Local Controller Triangle The hypotenuse of the triangle in the figure can be derived using the law of cosines starting with the expressions for the other legs of the triangle taken 16

17 fro Eq. (42). The expressions in Eq. (44) and Eq. (45) follow naturally fro the expression for the hypotenuse of the triangle in Fig. 7. cos Ψ C1 sin Ψ C1 = x(r2 C R2 ) 2yRR C R(R 2 C + R2 ) = y(r2 C R2 ) ± 2xRR C R(R 2 C + R2 ) (44) (45) The further substitution of Eq. (44) and Eq. (45) fro the heading controller gives ( ( x x(r 2 Ṽ1(ẽ 1 ), ẽ 1 = R v C R 2 ) ) 2yRR C R(RC y ( y(r 2 R2 ) R v C R 2 )) ) ± 2xRR C R(RC 2 + R2 ) (R R C ) k(v v C 1 ) 2. This expression can be siplified as shown below Ṽ1(ẽ 1 ), ẽ 1 = v(r2 C R2 ) (R 2 + RC 2 ) (R R C) k T 1 (v v C1 ) 2. Ṽ1(ẽ 1 ), ẽ 1 = v(r C + R)(R C R) (R 2 + RC 2 ) (R R C ) k T 1 (v v C1 ) 2. Ṽ1(ẽ 1 ), ẽ 1 = v(r + R C)(R R C ) 2 (R 2 + RC 2 ) k T 1 (v v C1 ) 2. (46) Substituting the error paraeters fro Eq. (24) into Eq. (46) naely, ẽ 1 = [ẽ 1,1 ẽ 2,1 ] results in Ṽ1(ẽ 1 ), ẽ 1 = v(r + R C)(ẽ 1,1 ) 2 (R 2 + RC 2 ) k T 1 (ẽ 2,1 ) 2, (47) which allows the definition of γ 1 as v in R C γ 1 (s) = in (R C + d) 2 + RC 2, k T 1 } s 2 s 0, (48) Then, the origin of the local error coordinates (ẽ 1 ) for the closed-loop syste ipleenting the local controller on the flow set ( C 1 ) is asyptotically stable. Since the origin of ẽ 1 corresponds to the points (x, y) in the loiter circle and loiter velocity equal to v Cq, this also iplies that the heading of the UAV on the loiter circle is tangent to the loiter circle as illustrated in Fig. 6. The following result establishes that, for each q 1, 2}, the constraint [T in, T ax ] is forward invariant under the feedback laws in (35) and (43) assigning T. 17

18 Lea 1. Under the effect of (32), ẽ 2,q evolves according to ẽ 2,q = k T q ẽ2,q and, since k Tq > 0, we have that ẽ 2,q = 0 is globally asyptotically stable. Furtherore, every solution to this subsyste with T q (0) [T in, T ax ] is such that T q (t) [T in, T ax ] t 0. (49) Reark The stability property of the origin (for ẽ 2,q ) follows directly fro the resulting dynaics of ẽ 2,q, which are obtained by plugging (32) into ẽ 2,q. In fact, under such feedback, ẽ 2,q (t) = ẽ 2,q (0)e k Tq t t 0 (50) for any initial condition ẽ 2,q (0). The property in (49) is a consequence of the resulting for of T in (32), since the substitution of (50) into (32) gives T = D(v) ẽ 2,q (0)e k Tq t (51) 3.2. Control design for the case of actuation through (T, Φ) via backstepping The results in the previous section establish properties of the UAV when Ψ is a control input. However, for the hybrid syste with flow ap given in Eq. (18), Ψ is a function of the control input Φ. Using the ethod of backstepping, asyptotic stability will be proved for the hybrid syste with Φ as a control input. Recalling the error syste given in Eq. (25), and adding and subtracting the desired heading Ψ Cq fro the heading state Ψ, we obtain [ x R(x,y) v cos ( ) Ψ Ψ Cq + Ψ Cq + y R(x,y) v sin ( ) ] Ψ Ψ Cq + Ψ Cq ė q = T q D(v,Φ) Now recall e 3,q fro Eq. (22) as the difference between the current heading Ψ and the coanded heading Ψ Cq and evaluate its derivative, that is,. where, according to Eq. (18), e 3,q = Ψ Ψ Cq ė 3,q = Ψ z Ψ Cq (z), f(z, κ(z)), Ψ = g v tan Φ q. The dynaics of e q are then x R(x,y) v cos ( ) e 3,q + Ψ Cq + y R(x,y) v sin ( ) e 3,q + Ψ Cq ė q = T q D(v,Φ) g v tan Φ q z Ψ Cq (z), f(z, κ(z)). (52) 18

19 Using V q (e q ) = Ṽq(ẽ q ) e 3,q 2 (53) for a Lyapunov function candidate, the derivative can be expressed as ( x V q (e q ), ė q = R v cos ( ) y e 3,q + Ψ Cq + R v sin ( ) ) e 3,q + Ψ Cq (R R C ) k T q (v v Cq ) 2 ( g + e 3,q v tan Φ q z Ψ Cq (z), f(z, κ(z)) ) ( x V q (e q ), ė q = R(x, y) v ( ) cos e 3,q cos Ψ Cq sin e 3,q sin Ψ Cq y + R(x, y) v ( ) ) sin e 3,q cos Ψ Cq + cos e 3,q sin Ψ Cq (R(x, y) R C ) k T q (v v Cq ) 2 + e 3,q ( g v tan Φ q z Ψ Cq (z), f(z, κ(z)) ) (54) Fro here, the control input Φ q can be forulated for the specific global and local controllers to ake the origin asyptotically stable for the syste in Eq. (52) on the set U \ z : R(x, y) = 0} Global Controller Backstepping The global controller is designed to ake the origin of the error syste e q asyptotically stable. This is achieved by inverting the tangent function containing Φ 2, the control input, and copensating for positive ters that coe when differentiating the Lyapunov function. This results in the controller Φ C2 (x, y, v, Ψ, e 3,2 ) ( = tan 1 k Φ 2 (x, y, v, e 3,2 )ve 3,2 + v ) g g zψ C2 (z), f(z, κ 2 (x, y, v, Ψ)) ( = tan 1 k Φ 2 (x, y, v, e 3,2 )ve 3,2 g where the function k Φ2 is of the for yv 2 gr(x, y) 2 cos Ψ + xv 2 ) gr(x, y) 2 sin Ψ, (55) k Φ2 (x, y, v, e 3,2 ) = ν + ξ(e 3,2 )v R(x, y) R C, (56) ν > 0, and s ξ(s) such that for each s R ξ(s) > 1 2 if s π 2 and ξ(s) 0 if s < π 2. (57) With the proposed definition of the function ξ we have that there exists β (0, 1) s.t. cos s + ξs 2 β. 5 Making the gain k Φ2 a function of the syste state 5 The definition of ζ 0 for s ( π π 2 2 ) iplies that ζs2 0 and, since cos s > 0, on this region we have the existence of such β > 0. For the case when s π, we can solve 2 19

20 allows for additional copensation for ters that arise when differentiating the Lyapunov function. Different values of ν and ξ can be selected, still satisfying the constraints in Eq. (57), as e 3,2 approaches zero, to prevent chattering of the controller coands. Proposition 3. For every constant > 0, velocity set-point v C2 [v in, v ax ], and loiter radius R C > 0, there exists γ 2 K such that the closed-loop syste resulting fro controlling the error syste x R(x,y) v cos (e 3,2 + Ψ C2 ) + ė 2 = with the controller [ ] T2 = κ Φ 2 (x, y, v, Ψ, Ψ C2 ) := 2 g T 2 D(v,Φ) y R(x,y) v sin (e 3,2 + Ψ C2 ) v tan Φ 2 z Ψ C2 (z), f(z, κ 2 (z)) [ D(v, Φ) kt2 (v v C2 ) Φ C2 (x, y, v, Ψ, e 3,2 ), (58) ], (59) where k T2 > 0, Ψ C2 is given in Eq. (34), and Φ C2 is given in Eq. (55), satisfies 1. V 2 (e 2 ), ė 2 γ 2 ( e 2 ) for all e 2 R 3 such that [x y v Ψ] C 2 In particular, an appropriate choice of γ 2 is vin βs γ 2 (s) = in, k T 2 s 2 } 3 3, νs2 s 0. 3 Proof In this proof, R(x, y) will be denoted as R to siplify notation. Substituting in Eq. (54) the expressions for cos Ψ C2 and sin Ψ C2 given in Eq. (36) and Eq. (37) gives ( x ( V 2 (e 2 ), ė 2 = R v x cos e 3,2 R sin e y ) 3,2 + y ( R R v x sin e 3,2 R + cos e y )) 3,2 (R R C ) R k T 2 (v v C2 ) 2 ( g ) + e 3,2 v tan Φ 2 z Ψ C2 (z), f(z, κ 2 (z)) ( x = v (cos 2 ) y e 3,2 + v (cos 2 )) e 3,2 sign(r R C )(R R C ) R 2 k T 2 (v v C2 ) 2 R 2 ( g ) + e 3,2 v tan Φ 2 z Ψ C2 (z), f(z, κ 2 (z)) = v R R C cos e 3,2 k T 2 (v v C2 ) 2 ( g ) + e 3,2 v tan Φ 2 z Ψ C2 (z), f(z, κ 2 (z)). (60) for the inflection point by taking derivatives, which gives sin s + 2ζs = 0, or equivalently, sin s = 2ζ. Selecting ζ > 1/2 oves the inflection point to s ( π π ) due to the fact that s 2 2 sin s 1 s π. As a consequence, there exists a lower bound β (0, 1) as required.. s 2 20

21 Taking the partial derivative of Ψ C2 with respect to the hybrid syste state and coputing the dot product with the global flow ap gives [ ] [ y x z Ψ C2 (z), f(z, κ 2 (z)) = R 2 R v cos Ψ v sin Ψ T 2 D = yv xv cos Ψ + R2 R which can be substituted into Eq. (60) resulting in ] g v tan Φ sin Ψ, (61) V 2 (e 2 ), ė 2 = v R R C cos e 3,2 k T 2 (v v C2 ) 2 ( g + e 3,2 v tan Φ 2 + yv xv ) cos Ψ R2 R 2 sin Ψ. Since R 0, the further substitution of the Φ C2 controller defined in Eq. (55) siplifies the expression further to V 2 (e 2 ), ė 2 = v R R C cos e 3,2 k T 2 (v v C2 ) 2 k Φ2 (x, y, v, e 3,2 )e 2 3,2. (62) The expression in Eq. (56) for k Φ2 and the error ters fro Eq. (22) can be substituted to give V 2 (e 2 ), ė 2 = v R R C cos e 3,2 k T 2 (v v C2 ) 2 (ν + ξv R R C )e 2 3,2 = v e 1,2 ( cos (e 3,2 ) + ξe 2 ) k T2 e 2 2,2 3,2 νe2 3,2 Since cos (e 3,2 ) + ξe 2 3,2 β as defined by Eq. (57), we have, for each e 2 where γ 2 is V 2 (e 2 ), ė 2 v e 1,2 β k T 2 e 2 2,2 ν(e 3,2) 2 γ 2 ( e 2 ). (63) vin βs γ 2 (s) = in, k T 2 s 2 } 3 3, νs2 s 0. 6 (64) 3 6 This bound follows considering the generic function V (x) = νx βx2 2 + γx 1, where x 1 > 0, β > 0 and ν > 0. If x 1 x 2 and x 1 x 3 then γ x 1 = γ ax x 1, x 2, x 3 } = γ x. Since x = x x2 2 + x2 3 3 x, we get γ x 1 γ x. Also, if x 2 > x 1 and 3 x 2 > x 3 then βx 2 2 = β x 2 β x 2 = β x Siilarly, if x 3 > x 1 and x 3 > x 2 then } νx 2 3 = ν x 2 ν x 2 2 = ν x 2 2. Then we have V (x) α( x ) := in ν x , β x 2 3, γ x 3 21

22 Local Controller Backstepping Siilar to the global controller, the local controller is designed to ake the origin of the error syste asyptotically stable. This is achieved by inverting the tangent function containing Φ 1, the control input, and copensating for positive ters that coe when differentiating the Lyapunov function. In this way the proposed controller is given by Φ C1 (x, y, Ψ, Ψ C1 ) [ = tan 1 k Φ 1 (x, y, v, Ψ, Ψ C1 )ve 3,1 + v ] g g zψ C1 (z), f(z, κ 1 (z)) [ = tan 1 k Φ 1 (x, y, v, Ψ, Ψ C1 )ve 3,1 g yv 2 gr(x, y) 2 cos Ψ + xv 2 ] gr(x, y) 2 sin Ψ (65) where k Φ1 is given by k Φ1 (x, y, v, Ψ, Ψ C1 ) =χ + γ(e 3,1 )v (R C + R(x, y))(r(x, y) R C ) 2 RC 2 + R(x, y)2 + sin e 3,1 ζ 2 v2r(x, y)r C p R(x, y) R C RC 2 + R(x, (66) y)2 where χ > 0, ζ (0, π 2 ) and s γ(s) is a sooth function such that γ(s) > 1 2 if s π 2, and γ(s) 0 if s < π 2. (67) With the proposed definition of the function γ we have that there exists β (0, 1) s.t. cos s + γ(s)s 2 β s 0. Note that aking the gain k Φ1 a function of the syste s state allows for additional copensation of ters that arise when differentiating the Lyapunov function V 1. Proposition 4. For every constant > 0, velocity set-point v C1 [v in, v ax ], loiter radius R C > 0, there exists γ 1 K such that the closed-loop syste resulting fro controlling the error syste x R(x,y) v cos (e 3,1 + Ψ C1 ) + y R(x,y) v sin (e 3,1 + Ψ C1 ) ė 1 = T 1 D(v,Φ 1), g v tan Φ 1 z Ψ C1 (z), f(z, κ 1 (z)) with the controller [ ] T1 = κ Φ 1 (x, y, v, Ψ, Ψ C1 ) := 1 [ D(v, Φ1 ) k T1 (v v C1 ) Φ C1 (x, y, Ψ, Ψ C1 ) ], (68) where k T1 > 0 and Ψ C1 is given in Eq. (42) and Φ C1 is given in Eq. (65), satisfies, for each ɛ > 0, 1. V 1 (e 1 ), ė 1 γ 1 ( e 1 ) + ɛ for all e 1 R 3 such that [x y v Ψ] C 1 22

23 where ζ is chosen such that v ax d R C 2(R C + d) sin ζ < ɛ (69) In particular, an appropriate choice of γ 1 is v in βr C γ 1 (s) = in (RC 2 + (R C + d) 2 ), k } T 1, χ s 2 s 0 Proof In this proof, R(x, y) will be denoted as R to siplify notation. Substituting in Eq. (54) the expressions for cos Ψ C2 and sin Ψ C2 given in Eq. (44) and Eq. (45) gives ( ( x V 1 (e 1 ), ė 1 = R v x(rc 2 cos e R2 ) p2yrr C y(rc 2 3,1 R(RC 2 + sin e ) R2 ) + p2xrr C 3,1 R2 ) R(RC 2 + R2 ) + y ( R v x(rc 2 sin e R2 ) p2yrr C y(rc 2 3,1 R(RC cos e )) R2 ) + p2xrr C 3,1 R2 ) R(RC 2 + R2 ) (R R C ) k T 1 (v v C1 ) 2 ( g ) + e 3,1 v tan Φ 1 z Ψ C1 (z), f(z, κ 1 (z)) which can be siplified to RC V 1 (e 1 ), ė 1 = (v 2 cos e ) R2 3,1 RC 2 + vp sin e 2RR C R2 3,1 RC 2 + (R R C ) k T 1 (v v C1 ) 2 R2 ( g ) + e 3,1 v tan Φ 1 z Ψ C1 (z), f(z, κ 1 (z)) (70) = v(r R C) ( cos RC 2 + e3,1 (R 2 R2 C R 2 ) k T1 (v v C1 ) 2 ) p2rr C sin e 3,1 ( g ) + e 3,1 v tan Φ 1 z Ψ C1 (z), f(z, κ 1 (z)). (71) Taking the partial derivative of Ψ C1 with respect to the hybrid syste state 23

24 gives Ψ C1 x = tan 1 y(rc 2 R 2 )+p2xrr C x(r 2 C R 2 ) p2yrr C x 1 = 1 + ( y(r C 2 R 2 )+p2xrr C x(r 2 C R 2 ) p2yrr C ) 2 ( p2rrc (x(r 2 C R 2 ) p2yrr C ) (R 2 C R 2 )(y(r 2 C R 2 ) ) + p2xrr C ) (x(r 2 C R 2 ) p2yrr C ) 2 = 4yR2 R 2 C y(r C 2 R 2 ) 2 R 2 (R C 2 R 2 ) 2 + 4R 4 R 2 C = y R 2 Ψ C1 y = tan 1 y(r 2 C R 2 )+p2xrr C x(r 2 C R 2 ) p2yrr C y 1 = 1 + ( y(r C 2 R 2 )+p2xrr C x(r 2 C R 2 ) p2yrr C ) 2 ( 2 (x(rc R 2 ) p2yrr C ) + p2rr C (R 2 C R 2 )(y(r 2 C R 2 ) ) + p2xrr C ) (x(r 2 C R 2 ) p2yrr C ) 2 = 4xR2 R 2 C + x(r C 2 R 2 ) 2 R 2 (R C 2 R 2 ) 2 + 4R 4 R 2 C = x R 2. Then, coputing the dot product with the local flow ap gives [ ] [ y x z Ψ C1 (z), f(z, κ 1 (z)) = R 2 R = yv R 2 which can be substituted into Eq. (71) resulting in v cos Ψ v sin Ψ T 1 D(v, Φ) xv cos Ψ + sin Ψ, (72) R2 V 1 (e 1 ), ė 1 = v(r R C) ( cos RC 2 + e3,1 (R 2 R2 C R 2 ) k T1 (v v C1 ) 2 ) p2rr C sin e 3,1 ( g + e 3,1 v tan Φ 1 + yv xv ) cos Ψ R2 R 2 sin Ψ Since R 0, further substituting the local controller given in Eq. (65) and grouping ters results in V 1 (e 1 ), ė 1 = v(r R C) R 2 C + R2 ( cos e3,1 (R 2 C R 2 ) p2rr C sin e 3,1 ) k T 1 (v v C1 ) 2 k Φ1 e 2 3,1 = v(r R C) 2 R 2 C + R2 cos e 3,1 (R C + R) v(r R C) R 2 C + R2 p2rr C sin e 3,1 k T 1 (v v C1 ) 2 k Φ1 e 2 3,1 ] g v tan Φ

25 Substituting the definition of the gain function in (66) results in V 1 (e 1 ), ė 1 = v(r R C) 2 RC 2 + (cos e 3,1 + γ(e 3,1 )e 2 3,1)(R C + R) R2 v(r R C) RC 2 + p2rr R2 C sin e 3,1 (1 + sign (R R C ) sign (sin e 3,1 ) p e 3,1 2 ) ζ 2 k T 1 (v v C1 ) 2 χe 2 3,1. (73) For the case of sign (R R C ) sign (sin e 3,1 ) p 0 we have V 1 (e 1 ), ė 1 = v(r R C) 2 RC 2 + (cos e 3,1 + γ(e 3,1 )e 2 3,1)(R C + R) R2 v R R C RC 2 + 2RR C sin e 3,1 (1 + e 3,1 2 ) R2 ζ 2 k T 1 (v v C1 ) 2 χe 2 3,1. Substituting the error ters fro Eq. (22) and the bound fro Eq. (67) eans that the expression can be bounded by V 1 (e 1 ), ė 1 v(e 1,1) 2 RC 2 + β(r R2 C + R) k T 1 (e 2,1 ) 2 χ(e 3,1 ) 2 γ 1 ( e 1 ) where γ 1 is defined as v in βr C γ 1 (s) = in (RC 2 + (R C + d) 2 ), k } T 1, χ s 2 s 0 (74) For the case of sign (R R C ) sign (sin e 3,1 ) p < 0 Eq. (73) can be bounded by ( ) V 1 (e 1 ), ė 1 v(r R C) 2 RC 2 + β(r C + R) v R R C R2 RC 2 + 2RR e 3,1 2 C sin e 3,1 R2 ζ 2 1 k T 1 (v v C1 ) 2 χe 2 3,1 (75) where we have used Eq. (67). For Eq. (75), when e 3,1 ζ, the bound given in Eq. (74) holds. The case of e 3,1 < ζ defines the value of ɛ given in Proposition 4. For this case we have V 1 (e 1 ), ė 1 = v(r R C) 2 RC 2 + (cos e 3,1 + γ(e 3,1 )e 2 3,1)(R C + R) R2 + v R R C RC 2 + 2RR C sin e 3,1 (1 e 3,1 2 ) R2 ζ 2 k T 1 (v v C1 ) 2 χe 2 3,1. 25

26 Where the second ter fro the right hand side can upper bounded as v R R C RC 2 + 2RR C sin e 3,1 (1 e 3,1 2 ) R2 ζ 2 < v axd 2(R C + d) sin ζ e 3,1 < ζ. R C (76) Given ɛ > 0 and the bound in Eq. (76) pick ζ (0, π 2 ) such that Eq. (69) holds. Then V 1 (e 1 ), ė 1 γ 1 ( e 1 ) + ɛ for all e 1 R 3 such that [x y v Ψ] C 1 where γ 1 is v in βr C γ 1 (s) = in (RC 2 + (R C + d) 2 ), k } T 1, χ s 2 s Uniting Global and Local Controller The global and local controllers proposed in Section 3.1 are united using a hybrid controller with a logic variable q 1, 2} leading to the closed-loop syste in Eq. (16); see [6, Exaple 3.23] for a siilar forulation. The construction in Eqs. (17)-(20) of the closed loop in Eq. (16) is such that the sets C 1, C 2, D 1, and D 2 deterine when switches between the local and global controllers should occur. The definitions of these sets depend on the positive paraeters c and d. The flow and jup sets can either have a disk shape (c R C ) or an elliptical shape (c < R C ) depending on the values selected for c and d. Next, we provide a design ethodology for these paraeters. For starters, the paraeter c is chosen so that jups fro using the global controller (q = 2) to using the local controller (q = 1) occur at points in D 2 that are also in the c-sublevel set of V 1, where V 1 is defined in (53). More precisely, we design c such that D 2 L V1 (c) (77) where L V1 (c) := (x, y, v, Ψ) R R [v in, v ax ] [ π, π] : V 1 (e 1 ) c} The set D 2 is defined in (20) as the c-sublevel of e 2 V 2 (e 2 ), and, as established by Proposition 3, is globally attractive. In this way, since C 2 is the closed copleent of D 2 for q = 2 and c > 0, we have that solutions fro C 2 reach D 2 in finite tie. Then, fro points z in D 2, which due to our construction in Eq. (77) are points that are also in L V1 (c), we have that R(x, y) R c 2c. To guarantee that after jups fro D 2 the state z belongs to C 1, we design the paraeters c and d to satisfy d > 2c (78) This is the relationship between c and d given in Eq. (78). With these choices, we have D 2 C 1. 26

27 Additionally, c is chosen such that C 2 z U : R = R C } = so as to avoid reaching the singularity of Ψ C2 given in Eq. (34), which is used in the global controller given in Eq. (59) for points along the loiter circle. This requireent leads to the additional constraint on c given by c > 1 2 ( π 2 + (v ax v in ) 2) (79) Now, according to Proposition 4, fro points in C 1 we have that solutions are such that the error e 1 satisfies 7 V 1 (e 1 (t)) exp( ᾱ(t t 0))V 1 (e 1 (t 0)) + (1 exp( ᾱ(t t 0)))ɛ over each interval of flow (starting at soe tie t 0) with q = 1, where ᾱ = 2 in vinβr C (R 2 C +(R C+d) 2 ), k T 1, χ }. Hence, for each such solutions, t e 1 (t) is such that t V 1 (e 1 (t)) = 1 2 e 1(t) 2 converges to [0, ɛ], and we have that li e 1(t) 2ɛ t as long as the solution stays in C 1. To ensure the latter property, we pick 2ɛ < d which guarantees that e 1 (t) 2ɛ for all t in the doain of e 1, which, in particular, iplies that R(x(t), y(t)) R c < d for each such t. Hence, the solutions reain in C 1. When the paraeters c and d are chosen as above, the closed-loop syste in Eq. (16) with data as in Eqs. (17)-(20), where κ 1 is given as in (68) and κ 2 as in (59) has the following properties: For every point in C 2, solutions with q = 2 approach D 2, converge to it in finite tie, and a switch to the local controller (q = 1) occurs. For every point in C 1, solutions with q = 1 stay in C 1 and the error quantity t e 1 (t) approaches the set [0, 2ɛ]. For every point in D q, a jup that toggles q occurs. These properties are validated nuerically in the next section. Although the design of the individual controllers and the supervisory algorith leading to the closed-loop syste in Eq. (16) with data as in Eqs. (17)-(20) is done in noinal, disturbance-free conditions, the design guarantees certain 7 With soe abuse of notation, solutions are paraeterized by t only. 27

28 robustness to disturbances. Certainly, persistent disturbances such as unodeled dynaics and easureent noise will prevent trajectories fro converging to the set A in (21). When the nor of those disturbances is sall, trajectories fro copact sets are expected to converge to points that are nearby the noinal set A in (21) this property can be foralized by redefining the flow and jup sets as closed sets, which would necessarily lead to a saller basin of attraction, and applying the results in [6]. Furtherore, due to the use of hysteresis in the switching echanis, the proposed controller does not suffer fro chattering along the boundary of the jup set as long as the disturbances are sall enough, which can be quantified as a function of the constants c and d in the proposed controller. Due to the properties of the Lyapunov-like functions in Propositions 1-4, the tools in [19] for the study of input-to-state stability and in [20] for the robust design of controllers using control Lyapunov functions can be used to precisely characterize the effect of large disturbances in the closed-loop syste. 4. Nuerical Analysis The hybrid syste for the UAV transition between transit and loiter was ipleented in the Hybrid Equations (HyEQ) Toolbox [21] to further explore and verify the analytical results presented. Note that the fact that the flow set C and D overlap requires iposing a desired seantics when flows are possible fro such overlaps. For instance, a trajectory that reaches a point in the boundary of C 2 that is also in D 2, could experience a jup, in which case the value of q for that trajectory will becoe equal to 1, or potentially flow for soe tie along the coon boundary of those two sets. While either behavior is appropriate, since the choice of the constants in Section 3.3 guarantee that all trajectories approach the target set A, the siulations are perfored using forcing seantics: whenever a trajectory reaches the jup set, a jup is executed. Table 1: Siulation Paraeters UAV Value Units Scenario Value Units Gain Value W N µ ξ see Eq. (80) C Lax v C1 160 s k T C D v C2 200 s k T kg K 59 - ρ ζ S ref g 9.81 s γ see Eq. (81) 2 T [ ] N χ 10 ν 10 The inputs and gains used for the presented siulation results are listed below in Table 1. Several of the gain paraeters listed in the table vary as a function of the vehicle s flight conditions while still satisfying the constraints 28

29 established in the control law analysis. The criteria for varying the gains is listed below. ξ = 3 2π 2. (80) γ = 3 2π 1. (81) The local/global switching criteria for the siulation were set as c = 1 2 (π2 + (v ax v in ) 2 ) + 1 (82) d = 2c (83) where Eq. (82) and Eq. (83) were selected to satisfy Eq. (79) and Eq. (78) respectively. Siulation results are shown to deonstrate the perforance of the algorith Outside the Loiter Circle (Disk) For the scenario starting outside the loiter circle a loiter radius (R C ) of 350 was used. Figure 8 shows a perforance tie history for the trajectory. The vehicle is initially headed in the wrong direction, but the global controller reverses heading to fly towards the center of the loiter circle. The hysteresis between C 1 and C 2 prevents chatter between the local and global controllers. The vehicle switches fro the global controller to the local controller at a point inside the local controller flow set and follows a trajectory that approaches the loiter circle on a tangent and ultiately tracks the loiter circle. The switch Figure 8: Siulation results starting outside the loiter circle. fro the global controller to the local controller occurs at approxiately 7 seconds. In the global control region, the vehicle attepts to accelerate to the coanded global controller velocity. For the first two seconds the vehicle does not accelerate quickly because of the high drag while turning. Once the vehicle has copleted its turn it accelerates ore quickly to the global coanded velocity until the switch to the local controller. In the local control region, the 29

30 thrust is coanded to its iniu while the vehicle decelerates to the coanded loiter velocity Inside the Loiter Circle (Disk) Figure 9 shows a perforance tie history for a trajectory starting in the local controller set inside the loiter circle with radius (R C ) of As shown in the position graph, the local controller set C 1 is shaped like a disk because c > R C. The vehicle stays in C 1 indefinitely and no controller switching occurs as the position converges to the loiter circle and the velocity converges to the coanded loiter velocity. The vehicle starts at the center of the loiter circle Figure 9: Siulation results starting inside the loiter circle with disk shaped local controller set and gradually spirals outward for a tangential approach which is constrained by the vehicle s turn radius. The vehicle started out initially faster than the desired loiter velocity and coands the iniu thrust until approxiately 7 seconds at which point the thrust gradually approaches the drag as intended by the control law given in Eq.( 32) Inside the Loiter Circle (Donut) To deonstrate a scenario with a donut shaped local controller set C 1, a loiter circle with radius (R C ) of 5000 was used. Figure 10 shows a perforance tie history for a trajectory starting in the global controller set C 2 inside the loiter circle. At the beginning of the trajectory the vehicle accelerates in a straight line towards the loiter circle. At 12 seconds into the trajectory the vehicle has nearly reached the coanded transit velocity when it enters the local controller set C 1 and the coanded velocity switches. Since the vehicle needs to decelerate to reach the coanded loiter velocity the iniu thrust is coanded. Upon reaching C 1 the vehicle begins banking to aneuver onto the loiter circle trajectory. The siulation results confir that the controller design provides the desired perforance within each of the local and global odes. Also, the design of the local and global sets prevents chattering between the two odes. 30

31 Figure 10: Siulation results starting inside the loiter circle with donut shaped local controller set 5. Conclusion A hybrid control syste for UAV waypoint loitering was analyzed and deonstrated. Asyptotic stability was established, and verified by siulation, for the UAV syste odeled in a plane with thrust and bank angle as control inputs. There are several opportunities for future work. One could be to expand the control algorith to be three diensions by including altitude and account for tie varying ass. Another possibility could be to expand the algorith to use other odes such as changing altitude. Though this paper focused on a UAV as the syste being controlled, the algorith could be applied to other systes such as boats or wheeled vehicles. 6. Acknowledgent The first author would like to thank his faily for patiently giving up evenings and weekends to let work on this project. Research by R. G. Sanfelice was partially supported by NSF Grants no. ECS and CNS , and by AFOSR Grants no. FA and FA References [1] E. W. Frew, D. A. Lawrence, C. Dixon, J. Elston, and W. J. Pisano. Lyapunov guidance vector fields for unanned aircraft applications. In 2007 Aerican Control Conference, pages , July [2] D. Lawrence, E. Frew, and W. Pisano. Lyapunov vector fields for autonoous unanned flight control. Journal of Guidance, Control, and Dynaics, 31(5): , [3] S. Park, J. Deyst, and J. How. A new nonlinear guidance logic for trajectory tracking. In AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, Rhode Island,