Landing a VTOL Unmanned Aerial Vehicle on a moving platform using optical flow

Transcription

1 Laning a VTOL Unmanne Aerial Vehicle on a moving platform using optical flow B. HÉRISSÉ, T. HAMEL, R. MAHONY, F-X. RUSSOTTO 4 Abstract This paper presents a nonlinear controller for a Vertical Take-off an Laning (VTOL) Unmanne Aerial Vehicle (UAV) that exploits a measurement of the average optical flow to enable hover an laning control on a moving platform, such as, for example, the eck of a sea going vessel. The VTOL vehicle is assume to be equippe with a minimum sensor suite (a camera an an IMU), manoeuvring over a texture flat target plane. Two ifferent tasks are consiere in this paper: the first concerns stabilizing the vehicle relative to the moving platform, that is maintaining a constant offset from a moving reference. The secon concerns regulation of automatic vertical laning onto a moving platform. Rigorous analysis of system stability is provie an simulations are presente. Experimental results are provie for a quarotor UAV to emonstrate the performance of the propose control strategy. Keywors: Optical Flow, Automatic laning, Unmanne Aerial Vehicle, Nonlinear control I. INTRODUCTION Recent avances in technology an a long list of potential applications have le to a growing interest in aerial robotic vehicles []. Unmanne aerial vehicles are an ieal solution for many inoor an outoor applications that presently jeoparize human or material safety, such as for example; monitoring traffic congestion, regular inspection of infrastructure such as briges, am walls an power cables, investigation of hazarous environments, etc. An important capability for a subset of potential applications, particularly those associate with maritime scenarios, is the ability to autonomously lan the vehicle on a moving platform such as the eck of a sea going vessel, or inee any laning pa attache to a moving vehicle. The associate capability of stabilizing the motion of the UAV with respect to a ynamic moving environment is itself of significance in a wie range of more general applications. Autonomous laning of UAV on moving platforms has been investigate using a moel of the vertical motion of laning platform [], [], a tether-guie [4] or known target motion [5], [6]. The main iea of prior work is base on obtaining a preiction of the motion of the moving laning pa to provie a fee-forwar compensation uring ONERA - The French Aerospace Lab, BP 8, F-9 Palaiseau ceex, France ( bruno.herisse@onera.fr) IS UNSA-CNRS, route es Lucioles - Les Algorithmes - Bt. Euclie B, BP, 69 Sophia Antipolis - ceex France (phone: + () ; fax: + () ; thamel@is.unice.fr) School of Engineering, Australian National University, Canberra ACT,, Australia ( Robert.Mahony@anu.eu.au) 4 CEA, List, Interactive Robotics Laboratory, 8 route u Panorama, BP6, Fontenay-aux-Roses, 965, France ( francois-xavier.russotto@cea.fr) the laning manoeuvre. This approach has the avantage that, shoul a reliable preictive moel of the motion of the laning pa be etermine, the resulting performance of the laning manoeuvre is of high quality. The approach suffers from the isavantage that in many applications of interest it is ifficult to etermine a reliable preictive moel of the motion of the laning pa either because the motion of the laning pa is primarily stochastic an no preictive moel is vali, or ue to the limite amount of ata available to the UAV uring the laning manoeuvre. In such situations the vehicle control algorithm must fall back on feeback control strategies. An approach that stems from the insight into the behaviour of flying insects an animals uses visual flow [7] as feeback for aerial vehicles in the control of motion in ynamic environments. Since optical flow provies relative velocity an proximity information with respect to the local environment [8], it is an ieal cue that can be use to perform laning control strategies [9], [7], as well as obstacle avoiance [], [], [], terrain following [], [4], [5], visual servo control [6], or even in both localization an control [7]. It is rare that moving obstacles are consiere in prior literature using optical flow, however it is well known that insects show great capabilities in achieving laning tasks on moving objects such as, for example, a bee laning on a flower. Moreover, the full vehicle ynamics analysis is rarely iscusse in the majority of work on the analysis of insect flight behaviour, since the flight regime of insects is highly ampe ue to their high rag to mass ratios. The control strategies that have been observe in the various biological stuies o not necessarily generalise to high-inertia, low-rag aerial vehicles. In this paper, an optical flow base control law for hovering flight an laning manoeuvre on a moving platform is propose using only IMU an optical flow measurements. The image information consiere is the average optical flow obtaine from a texture target plane, using aitional information provie by an embee IMU for erotation of the flow. A non-linear PI-type controller is esigne for hovering flight while another nonlinear controller, exploiting the vertical optical flow (similar to the inverse of the wellknown time-to-contact), is propose for vertical laning. It is necessary to assume boune ynamics of the moving platform, however, no preictive moel of the platform is require to obtain the esire close-loop performance. To prove global stability an convergence of the close-loop system, Lyapunov analysis is use both for the stabilisation of the hovering flight relative to a static plane an for the vertical laning relative to a horizontal plane moving with unknown (boune) ynamics in the vertical irection. In

2 practice, the stabilisation an vertical laning also works with lateral motion. Experimental results are obtaine on a quarotor UAV capable of quasi-stationary flight evelope at CEA (French Atomic Energy Commission). A high gain controller is use to stabilise the orientation ynamics of the vehicle, an approach classically known in aeronautics as guiance an control (or hierarchical control) [8], an the stabilisation an laning control is evelope for the resulting reuce translational ynamics of the vehicle. The propose closeloop control schemes emonstrate efficiency an performance for the hovering flight an vertical laning manoeuvre. The material presente in this present paper is an extension of the prior work [9]. It incorporates the groun effect, consiers the situation of target is moving, contains etaile proof of the system stability an incorporates aitional simulations an experiments. The boy of the paper consists of six sections followe by a conclusion. Section II presents the funamental equations of motion for an X4-flyer UAV. In Section III, funamental equations of optical flow are presente. Sections IV an V present the propose control strategies for hovering an vertical laning manoeuvre respectively. Section VI escribes simulations results an Section VII escribes the experimental results obtaine on the qua-rotor vehicle. Fig. : The quarotor UAV evelope in Centre Energie Atomique, an use for the experimental results in the paper. II. UAV DYNAMIC MODEL AND TIME SCALE SEPARATION The VTOL UAV is represente by a rigi boy of mass m an of tensor of inertia I along with external forces ue to gravity an forces an torques provie by rotors. To escribe the motion of the UAV, two reference frames are introuce: an inertial reference frame I associate with the vector basis [e, e, e ] an a boy-fixe frame B attache to the UAV at the center of mass an associate with the vector basis [e b, e b, e b ]. The position an the linear velocity of the UAV in I are respectively enote ξ = (x, y, z) T an v = (ẋ, ẏ, ż) T. The orientation of the UAV is given by the orientation matrix R SO() from B to I. Finally, let Ω = (Ω, Ω, Ω ) T be the angular velocity of the UAV efine in B. A translational force F an a control torque Γ are applie to the UAV. The translational force F combines thrust, lift, rag an gravity components. For a miniature VTOL UAV e b e b B e b I e e Fig. : Definition of the boy-fixe frame B an the inertial frame I in quasi-stationary flight one can reasonably assume that the aeroynamic forces are always in irection e b, since the thrust force preominates over other components []. The gravitational force can be separate from other forces an the ynamics of the VTOL UAV can be written as: ξ = v () m v = T Re + mge + () ϵṙ = R Ω, Ω = ϵω () ϵi Ω = Ω I Ω + Γ, Γ = ϵ Γ (4) In the above notation, g is the acceleration ue to gravity, an T a scalar input terme the thrust or heave, applie in irection e b = Re, the thir-axis unit vector. The term represents constant (or slowly time varying unmoele) forces. The matrix Ω enotes the skew-symmetric matrix associate to the vector prouct Ω x := Ω x for any x. The positive parameter < ϵ < is introuce for timescale separation between the translation an orientation ynamics. It means that the orientation ynamics of the VTOL UAV are compensate with separate high gain control loop (Γ = Γ/ϵ ). For this hierarchical control, the time-scale separation between the translational ynamics (slow time-scale) an the orientation ynamics (fast time-scale) can be use to esign position an orientation controllers uner simplifying assumptions. Although reuce-orer subsystems can hence be consiere for control esign, the stability must be analyze by consiering the complete close-loop system [8]. In this paper, however, we will focus on the control esign for the translational ynamics. Thus, the full vectorial term T Re will be consiere as control input for these ynamics. We will assign its esire value u (T Re ) = T R e. Assuming that actuator ynamics can be neglecte, the value T is consiere to be instantaneously reache by T. For the orientation ynamics of ()-(4), a high gain controller is use to ensure that the orientation R of the UAV converges to the esire orientation R. The resulting control problem is simplifie to ξ = v, m v = u + mge + (5) Thus, we consier only control of the translational ynamics (5) with a irect control input u. This common approach is use in practice an may be justifie theoretically using singular perturbation theory []. e

3 III. OPTICAL FLOW EQUATIONS In this section image plane kinematics an spherical optical flow are erive. The camera is assume to be attache to the center of mass of the vehicle so that the camera frame coincies with the boy-fixe frame. A. Kinematics of an image point uner spherical projection We compute optical flow in spherical coorinates in orer to exploit the passivity-like property iscusse in []. The main avantage is that, in spherical coorinates, the optical flow is expresse in a simple form. Moreover, it is shown in [] that optical flow equations can be numerically compute from an image plane to a spherical retina. A Jacobian matrix relates temporal erivatives (velocities) in the spherical coorinate system to those in the image frame. Motivate by this iscussion, we make the assumption that the image surface of the camera is spherical with unit image raius. S W Ω η V ṗ = Ω p P π p(v V P ) Fig. : Image kinematics for spherical camera image geometry Define P = (X, Y, Z) R as a visible target point, possibly moving, expresse in the camera frame. The image point observe by the spherical camera is enote p an is the projection of P onto the image surface S of the camera. Thus, p = P (6) P The time erivative ṗ is the kinematics of the image point, also calle optical flow equations, on the spherical surface. The kinematics of an image point for a spherical camera of image surface raius unity are [], [8] V P ṗ = Ω p π p P (V V P ), (7) where π p = (I pp ) is the projection π p : R T p S, the tangent space of the sphere S at the point p S. The vectors V = R v an V P are expresse in the boy-fixe frame an represent respectively the translational velocity of P the center of mass of the vehicle an the translational velocity of the target point P. Let η I enote the unit normal of a target plane [6] (Fig. ). Define := (t), to be the orthogonal istance from the target surface to the origin of frame B, measure as a positive scalar. Thus, for any point P on the target surface (t) = P, R η where P is expresse in the boy-fixe frame an η is expresse in the inertial frame. For a target point, one has P = (t) p, R η = (t) cos (θ) where θ is the angle between the inertial irection η an the observe target point p. Substituting this relationship into (7) yiels cos (θ) ṗ = Ω p (t) π p(v V P ) (8) B. Average optical flow Measuring the optical flow is a key aspect of the practical implementation of the control algorithms propose in the sequel. The optical flow ṗ can be compute using a range of algorithms (correlation-base techniques, features-base approaches, ifferential techniques, etc) [4]. Note that ue to the rotational ego-motion of the camera, (8) involves the angular velocity as well as the linear velocity [8]. For the control problem we efine an inertial average optical flow from the integral of all observe optical flow correcte for rotational angular velocity. By integrating optical flow over an aperture, in this case a soli angle on the sphere, we obtain information on the scale velocity of the vehicle. We assume that the target is a texture plane moving with a pure translational velocity V t B (no rotational velocity). Thus, for any target points P on the plane, V P = V t. We also assume that the normal irection η is known an the available ata are ṗ, R an Ω where R an Ω are estimate from the IMU ata [5]. The average optical flow is obtaine by integrating the observe optical flow over a soli angle W of the sphere aroun the pole normal to the target plane (Fig. ). The average of the optical flow on the soli angle W is given by (see the appenix for more etails): ϕ = ṗ p = π (sin θ ) Ω R η Q(V V t) W where the parameter θ an the matrix Q epen on the size of the soli angle W. It can be verifie that Q = R (R t ΛRt )R is a symmetric positive efinite matrix. The matrix Λ is a constant iagonal matrix epening on parameters of the soli angle W an R t represents the orientation matrix of the target plane with respect to the inertial frame. For instance, if W is the hemisphere centere at η, corresponing to the visual image of the infinite target plane, it can be shown that [6] Λ = π () 4 (9)

4 From (9) it is straightforwar to obtain a measurement of the inertial average optical flow correcte for rotational angular velocity w = (R t Λ R t )R(ϕ + π (sin θ ) Ω R η) () By expressing it with respect to the rigi boy motion, it yiels: w = v v t + noise () where v t = RV t is the translational velocity of the target plane expresse in the inertial frame. Note that, for theoretical analysis provie in Sections IV an V, the noise of equation () is ignore. Its effects on the convergence is consiere in Section VI. Remark.: In the particular situation where the target plane is stationary (v t = ), () becomes w = v + noise () IV. STABILIZATION OF THE HOVERING FLIGHT OVER A TEXTURED TARGET In this section a control esign ensuring hovering flight over a texture flat plane is propose. The control problem consiere is the stabilization of the linear velocity about zero espite unmoele constant (or slowly time varying) ynamics. In particular, the velocity of the target will be assume to be constant ( v t ). A PI-type non-linear controller epening only on the measurable variable w = (v v t )/ = ṽ/ () is propose for the translational ynamics (5). The result is state in the following theorem. Theorem 4.: Assume that η is known an invariant an is a constant. Consier the ynamics (5) an assume that the control input u is chosen as u = k P w + k I wτ + mge, k P, k I > (4) Then, for any initial conitions = () >, the linear velocity error ṽ converges asymptotically to zero. More precisely: ) = ṽ, η converges to an (t) >, t. ) the horizontal velocity ṽ = π η ṽ converges to zero. Proof: Proof of item (): Recall the ynamics of the vehicle (5) an consier the component ṽ = ṽ, η in irection η. One obtains: Note that ṽ follows: m ṽ = k P ṽ k I ṽ τ +, η (5) =. Equation (5) can also be written as k I τ, η (6) ( ) = k P k I ln (7) m = k P where = e,η /k I. Note that the control law is well efine an smooth for >. For any initial conitions such that = () >, efine the Lyapunov function caniate L η by L η = m [ ( ( ) ) ] + k I ln + (8) Since function u (u(ln u ) + ), u >, it is straightforwar to verify that L η. Differentiating L η an recalling equation (7), one obtains L η = k P (9) This implies that L η < L η () as long as (t) >. Two ifferent cases may occur epening on the initial value of L η : L η () < k I an L η () k I. From the expression of the Lyapunov function (8), the first case (L η () < k I ) implies that there exists ε > such that (t) > ε >, t. Consequently, remains strictly positive an equation (7) is well efine for all time. Application of LaSalle s principle shows that the invariant set is containe in the set efine by L η =. This implies that in the invariant set. Recalling (7), it is straightforwar to show that converges asymptotically to. For the secon situation (L η () k I ), we have to show that for all time. Assume that there exists a first time t such that (t ) < an < (t ) <. If we show that there exists a secon time t > t such that (t ) = an (t ) > then, L η (t ) < k I an conitions of the first case are verifie, an the result follows. We procee using a proof by contraiction. Assume that for all time t > t, (t) <. This implies (t) < (t ) <, t > t. Thus, recalling equation (7), it follows that there exists ε > such that (t) > ε >, t > t. As a consequence, there exists a time T > t such that converges to ( ) when t tens to T. Recalling equation (7), it yiels: > k P m >, t > t () Integrating this equation, it follows: (t ) > k ( ) P m ln, t > t () (t ) Since converges to, converges to +. This contraicts the fact that <, t > t. It follows that (t) >, t an consequently (t) converges to. Proof of item (): Let ṽ be the planar velocity π η ṽ I. Consier the component perpenicular to η of the control law (4), u ṽ = π η u = k P + k ṽ I τ + mgπ ηe () Recall the ynamics of the vehicle (5) an consier the component perpenicular to η. Substituting the control law () into (5), one obtains m ṽ = k P ṽ k I ṽ τ + () where = π η. Let δ be the following variable: δ = ṽ τ k I (4) 4

5 Differentiating δ, it yiels δ = ṽ Consier the following Lyapunov function caniate: δ L πη = k I + m δ (5) (6) where δ = ṽ /. Differentiating L πη an recalling equation (), one obtains ( k P + m ) / L πη = δ (7) Using the fact that (, ) converges to (, ), one can insure that there exists a time T an ε > such that (k P + m/) > ε >, t > T Therefore, L πη < ε δ, t > T. Moreover, it is straightforwar to verify that L πη remains boune in [, T ] by noticing that L πη < max L πη, t [, T ] min This implies that L πη (t) < L πη (T ), t > T. To show that δ converges to, we nee to show that L πη is uniformly continuous. Then, application of Barbalat s Lemma (see []) will conclue the proof. To this purpose it is sufficient to show that Lπη is boune. Since an are boune, it remains to show that δ an δ are boune to satisfy the conition. δ an δ are boune since L πη is boune. Moreover, it is straightforwar to show that δ is boune using its expression: δ = k P m δ k I m δ δ Thus, L πη is uniformly continuous, hence δ converges to. Finally, using the fact that ṽ = ṽ η + ṽ, it follows that ṽ converges to zero. V. LANDING CONTROL ON A MOVING TEXTURED TARGET In this section we consier the laning manoeuvre of the aerial robot on a horizontal plane moving vertically. The primary goal is to aress the question of the vertical laning on a moving platform (target) with unknown ynamics. The most important application concerns laning on a eck of a ship in high seas an tough weather [], [4], [5], [6]. A common moel of the vertical motion z G of the platform as the motion of the ship involve by the sea waves is []: n z G = a i cos (ω i t + ϕ i ) (8) i= where a i, ω i, ϕ i are unknown constants. The classical approach estimates the parameters of motion an uses these to a a fee-forwar compensation term in the control input. In this paper, we consier a more general vertical motion z G of the platform with respect to the inertial frame I. We assume that z G is a smooth function of class C (z G an ż G are continuous functions of time t) such that z G is boune by a known value. We assume that the target plane belongs to the plane x-y of the inertial frame so that h is the height of the vehicle with respect to the moving platform. The vertical velocity of the target plane is ż G e. Consequently, from () with v t = ż G e, it is straightforwar to verify that hence, Define w = w, e = v ż G e h w z = w = ḣ h w = (,, ω ) T, ω >, (9) as the esire average optical flow. Note that the vertical component of the inertial average optical flow acts analogously to optical flow ivergence. It is straightforwar to show that if w w one has (v x, v y ) = (, ) an v z = h ω exp( ω t). Consequently, h(t) = h exp( ω t) insuring a smooth vertical laning. In practice, it is impossible to exactly track ω an it is necessary to implement a feeback system. We propose to use the previous control law (4) for the x-y ynamics to stabilise the vehicle over the laning pa. We still nee to provie a control scheme for the remaining egree of freeom (h z z G ). In particular, we fix a esire set point, ω, for the flow ivergence (the flow in the normal irection to the target plane, equal to the inverse of the time-to-contact) an esign a control law that regulates (ḣ/h + ω ) aroun. The controller is a irect application of the controller propose in [6], along with a complete an more rigorous proof of the exponential convergence an stability of (h, ḣ) to (, ) espite unknown ynamics an unknown terms. Consier the ynamics m v = b(t)u + mge + (t) () where b(t) is a slowly time varying parameter that moels the groun effect (b ). An approximate moel for b(t) can be foun in [7], [] b(t) = ( D h(t) + l ) () where l an D can be ientifie on a physical system. Note that l > D so that b > when h = (see Figure 4). Note also that max(b(t)) = b max is obtaine when h =. Theorem 5.: Consier the ynamics of the vertical component of () an assume that the vertical component u z of the thrust vector u is the control input. Choose u z as u z = mk(w z ω ) + mg () Assume that z G is at least C, assume that z G, (t) are boune an uniformly continuous, an assume that b(t). Choose the control gain k such that k > z max + m z G max + mg b max mω () 5

6 ..8 groun effect b Differentiating ζ an recalling equations (6) yiels ζ = αζ (9).6 Since ζ = h, it follows that on [, T max ) b h (m) Fig. 4: Groun effect b with l =.5m an D =.5m Then, for all initial conitions such that h > (h z() z G () ): ) the thir component of the ifferential equation () along with () is smooth an non-singular. This implies that the solution (h(t), ḣ(t)) is well efine for all time t. ) h(t) > remains positive an (h, ḣ) converge exponentially to zero. ) the control law () is boune for all time an ḧ. Proof: In the first step, we prove that the the thir component of the ifferential equation () along with () is smooth an non-singular while h(t) >. This implies that there exists a time T max > such that the solution (h(t), ḣ(t)) exists an is well efine on t [, T max ). In a secon step we prove item () while showing that T max = an finally we will prove item (). Proof of item () for t [, T max ): Firstly, recall that the ynamics of the consiere system are ecouple an recall the ynamics of the thir component of () m v z = b(t)u z + mg + z (t) (4) It follows that the height ynamics can be written: mḧ = mkb(t)(w z ω ) z + (b(t) )mg + m z G (5) ) (ḣ = mkb(t) h + α(t) (6) where, ( α(t) = ω + z g (b(t) ) mkb(t) kb(t) z ) G kb(t) (7) Recalling conition (), it is straightforwar to show that α(t) is a positive an boune function (α(t) >, t ). The ynamics (6) are well efine as long as h(t) >, hence there exists a first time T max, possibly infinite, such that (h, ḣ) is well efine on [, T max ). Proof of item () for t [, T max ): Define the following virtual state on [, T max ) ζ(t) = h(t) exp ( ḧ(τ) kb(τ) τ ) (8) h exp ( α max t) < ζ(t) < h exp ( α min t) It remains to show that t ḧ(τ) kb(τ) τ is boune on [, Tmax ) to insure, using (8), that there exist ϵ, ϵ > such that ϵ h exp ( α max t) < h(t) < ϵ h exp ( α min t) for all time t [, T max ). We will show that T max = using continuity. This will ensure that (h, ḣ) is well efine on [, ), an h(t) converges exponentially to. To o this we must prove that t ḧ(τ) kb(τ) τ is boune by stuying the evolution of (ḣ, ḧ). Proof that the sign of ḣ(t) oes not change more than once an ḣ(t) is boune. Two situations may occur: ḣ() : to show that there exists a time T on [, T max) such that ḣ(t ) <, assume the converse; that is, ḣ for all time t. Thus, from (6) where α(t) >, ḧ < an, by exploiting (8) where b, ) ( ) (ḣ(t) ḣ ζ(t) h exp h exp k ḣ k Since ζ < α min ζ, it follows that ζ is exponentially ecreasing. Therefore, there exists a time T such that ζ(t ) < h exp ( ḣ/k). This contraicts the assumption. ḣ() < : to show that ḣ <, t [, T max), assume the converse; that is, there exists T such that ḣ(t ) = an ḣ(t) <, t < T. Since ḣ is continuous an recalling (6), it follows that there exists δ > an ϵ > such that ḧ(t) < ϵ, t [T δ, T ]. Recalling (8) one has that ζ(t) = h(t) exp ( T δ ) ḧ(τ) t kb(τ) τ + ḧ(τ) T δ kb(τ) τ Using the fact that ḧ(t) < ϵ, t [T δ, T ], it follows that ( ) T δ ḧ(τ) ζ(t) < h(t) exp kb(τ) τ, t [T δ, T ] Moreover, since b(t), t [, T max ), ( T δ ) ḧ(τ) t ζ(t) h(t) exp kb(τ) τ + ḧ(τ) T δ k τ ( ) ) T δ ḧ(τ) (ḣ(t) h(t) exp kb(τ) τ exp k Using the fact that ḣ(t ) =, one obtains ( ) T δ ḧ(τ) ζ(t ) h(t ) exp kb(τ) τ This proves the contraiction. 6

7 To show that ḣ is lower boune, let J be the following storage function: J = ḣ (4) Differentiating J an recalling equations (6) yiels ) J = kb(t)ḣ (ḣ + αh h (4) It follows that J is negative as long as ḣ > αh. Since there exists a time T such that ḣ <, t > T, it follows that h > is upper boune. Consequently, ḣ is boune. Due to the variation of b(t), bouneness of ḣ is not sufficient to conclue that t ḧ(τ) kb(τ) τ is boune. Therefore, it is necessary to stuy the evolution of ḧ. Proof that the sign of ḧ oes not change more than once. In the following we assume without loss of generality that ḣ <, therefore ḣ(t) < for all time. ḧ() : to show that there exists a time T such that ḧ(t ) >, assume the converse; that is, ḧ for all time t. Since ḣ is negative an ecreasing, it follows that h is strictly monotonically ecreasing an cannot have a positive limit. Consequently, from (6), there exists a time T such that ḣ(t )/h(t ) < α. Hence ḧ(t ) > an the contraiction follows. ḧ() > : to show that ḧ(t) for all time, assume the converse; that is, there exists T an δ > such that ḧ(t δ) = an ḧ <, t (T δ, T ]. This implies that (ḣ/h)(t δ) = α an ḣ/h > α, t (T δ, T ]. Using the fact that ḣ/h = α at time (T δ) an ḣ/h < α, t (T δ, T ] while ḣ is negative an ecreasing an h is positive an ecreasing, the contraiction follows. Using the fact that there exists a time T [, T max ) from which ḣ < is boune an ḧ, t [T, T max), it is straightforwar to verify that t ḧ(τ) kb(τ) τ remains boune on [T, T max ). Therefore, since ζ is exponentially ecreasing, one can ensure that h remains positive an exponentially ecreasing on [T, T max ). Now, we prove that T max = an thus that ζ is well efine on [, ). Assume that T max, it means that there exists a positive number δ such that h(t) > (by continuity) an such that t ḧ(τ) kb(τ) τ is unboune on [Tmax, T max +δ). This contraicts the above iscussion. It follows that h converges exponentially to. Moreover, using (4) an (4) with irect application of the Input-to-State-Stable (ISS) argument, it follows that ḣ is exponentially stable. Proof of item () for t [, ): Now, we prove that the controller () is boune by proving that ḧ. Analogously to the proof of Barbalat s Lemma, we procee by contraiction. Assume that ḧ oes not converge to. Since (ḧ/kb)(τ) τ an b(t) are boune an there exists a time T such that ḧ, t > T, it follows that there exists ϵ > an two sequences (T n ) n R + an (δ n ) n R + such that (i) T n + as n +, ḧ (ii) kb (T n δ n ) = ϵ/ an ḧ kb (T n) = ϵ, (iii) ϵ/ ḧ kb (t) ϵ, t [T n δ n, T n ]. We nee to show that (δ n ) n is lower boune by a strictly positive number. Using the fact that ḧ/kb ϵ, t [T n δ n, T n ] an recalling (6), one has ) (ḣ h + α ϵ Integrating this inequality within [T n δ n, T n ] one obtains ( ) h(t) t ln (α + ϵ) τ h(t n δ n ) T n δ n Given that one has T n δ n (α + ϵ) τ ( ) h(t) ln h(t n δ n ) T n δ n (α max + ϵ) τ, (α max + ϵ) δ n Hence, h(t) h(t n δ n ) exp ( (α max + ϵ)δ n ), t [T n δ n, T n ] an therefore, using the fact that ḣ < an increasing (ḧ ), ḣ h ḣ h(t n δ n ) exp ((α max + ϵ) δ n ) ḣ h ḣ(t n δ n ) h(t n δ n ) exp ((α max + ϵ) δ n ) (4) Using (ḧ/kb)(t n δ n ) = ϵ/ an (ḧ/kb)(t n) = ϵ, one also has ḣ h (T n) = ϵ α(t n ) ḣ h (T n δ n ) = ϵ α(t n δ n ) Recalling inequality (4), it follows that exp ((α max + ϵ) δ n ) ϵ + α(t n ) ϵ/ + α(t n δ n ) Now, we nee the uniform continuity of α(t). To show this, we first nee to show that /b(t) is itself uniformly continuous (see (7)). The result is straightforwar to show using () an the fact that (h, ḣ) converge to (ḃ/b is boune, then /b is uniformly continuous). Using assumptions of the theorem an the fact that /b is uniformly continuous, it follows that α(t) is uniformly continuous. Thus, there exists γ > such that T n t γ α(t n ) α(t) ϵ/4. Consiering the case δ n < γ, one obtains α(t n ) α(t n δ n ) ϵ/4 an, therefore exp ((α max + ϵ) δ n ) ϵ/4 + α(t n δ n ) ϵ/ + α(t n δ n ) Using the fact that α(t) is boune, it is straightforwar to verify that ϵ/4 + α(t) ϵ/ + α(t) ϵ/4 + α max ϵ/ + α max, t > 7

8 This implies that exp ((α max + ϵ) δ n ) ϵ/4 + α max ϵ/ + α max >, n Consequently, there exists δ > such that δ n δ for all n. The next step of the proof is a irect application of the proof of Barbalat s Lemma. By efinitions (ii)-(iii), for all t [T n δ n, T n ] an for all n we have ( ) ḧ kb (t) = ḧ kb (t) = ḧ ḧ kb (T n) kb (T n) ḧ kb (t) ( ) ḧ kb (T n) ḧ kb (T n) ḧ kb (t) ϵ ϵ ϵ As a consequence, (ḧ/kb)(τ) τ converges to + which contraicts the hypothesis. Using the fact that ḧ(t) converges to, it is straightforwar to verify that: ḣ/h α(t) when t tens to +. This means that the escent spee epens on α(t). ḧ is boune an therefore the control law () is boune. Remark 5.: Note that the stability of the control law (4) use for the lateral ynamics uring the laning manoeuvre can also be prove in the case where an b are constant; the proof is similar to the secon part of the proof of Theorem 4. using the fact that ḣ is boune an converges to. The authors o not have a formal proof of stability in the case where both b(t) an (t) vary over time or in case where the lateral ynamics of the target plane is not zero. Nevertheless, if these variables vary sufficiently slowly (ḃ(t), (t) an v t ), then the robustness of Theorem 4. will ensure stability. Characterising the stability conitions is a ifficult problem that remains open. that the vertical optical flow w z remains positive for all time even if it oes not reach ω an the height h = z + z G converges exponentially to. We also notice that the height remains positive uring the manoeuvre, implying that the vehicle oes not collie with the platform. Figure 6 shows the result with a moving platform. We keep the same parameters as before. The vertical motion of the platform is chosen as z G = a G sin (πf G t) with a G =.m an f G =.s It is straightforwar to verify that conition () hols. Note that, uring the simulation, z G is assume to be unknown. This means that no fee-forwar compensations is performe. Figure 6 shows the close-loop trajectory of the vertical motion of the vehicle. Observe that the vertical optical flow remains positive for all time even if it oes not reach ω an the height h = z + z G converges exponentially to espite the fact that the vertical motion of the platform is unknown. Figure 7 shows the same result with an aitional noise on the measure optical flow. One observes that the convergence is not affecte, even close to the touchown. OpticalFlow wz (/s) height h (m) vertical optical flow height h= z+z G position ( z) (m) position z Fig. 5: Simulation of vertical laning on a static platform using controller () VI. SIMULATIONS In orer to evaluate the efficiency of the propose servo control technique, Matlab simulations of the vertical laning of an iealise quarotor (4) on static or moving platform are presente. The simulations presente consier only the vertical laning problem of the vehicle on a static an a moving platform. The mass of the vehicle is chosen m =.85kg. It correspons to the physical mass of the quarotor use for experiments. The control gain is set to k =, the error z is chosen z =.. For the parameter b efine in (), incorporating the groun effect, we have chosen l =.5m an D =.5m. The esire set point ω is set to.5s. Using the above values of the ifferent parameters involve in the vertical motion (6), it is straightforwar to show that conition () is verifie. Figure 5 shows the close-loop trajectory of the vertical motion of the vehicle. One can verify Figure 8 shows the result with a stochastically moving platform. The vertical motion of the platform is now the sum of n = 7 sinusoial signals (see equation (8)), where parameters a i [,.], ω i [, 6] an ϕ i [, π] are chosen stochastically. The bouns of the parameters are chosen to ensure that the conition () is verifie. During the simulation, z G is still assume to be unknown. Once again, one observes the expecte behaviour, the height h = z + z G converges exponentially to espite the unknown motion of the platform. In Figure 9, a phase iagram for ifferent trajectories is presente. For each trajectory, parameters are chosen stochastically an analogously to the previous simulation. z [, ] is also chosen stochastically. As for the set point ω an initial conitions, they are chosen such that the figure is unerstanable: h [, 4], ḣ [ 4, 6] an ω [.5, 4.5]. 8

9 OpticalFlow wz (/s) vertical optical flow OpticalFlow wz (/s) vertical optical flow height h (m) height h= z+z G position ( z) (m) position z Fig. 6: Simulation of vertical laning on an oscillating platform using controller () height h (m) height h= z+z G position ( z) (m) position z Fig. 8: Simulation of vertical laning on a stochastically moving platform using controller () OpticalFlow wz (/s) height h (m) vertical optical flow height h= z+z G position ( z) (m) position z height h (m) Fig. 7: Simulation of vertical laning on an oscillating platform with noisy optical flow height spee (m/s) Fig. 9: Phase iagram of the ynamical system for 9 inepenent simulations The figure shows robustness of the approach since the expecte behaviour is observe for all trajectories. One can observe that the trajectories satisfy the result of theorem 5., that is h(t) > remains positive for all time an (h, ḣ) converge to zero. One also verify that there exists a time T such that ḣ <, t T (see the proof of the theorem). VII. EXPERIMENTAL RESULTS In this section, we present two experiments that emonstrate the performance of the propose control scheme on a physical vehicle. The UAV use for the experimentation is the quarotor, constructe by the CEA (Fig. ), a vertical take off an laning vehicle ieally suite for stationary an quasi stationary flight [8]. A. Prototype escription The X4-flyer is equippe with a set of four electronic boars esigne by the CEA. Each electronic boar inclues a micro-controller an has a particular function. The first boar integrates motor controllers which regulate the rotation spee of the four propellers. The secon boar integrates an Inertial Measurement Unit (IMU) consisting of low cost MEMS accelerometers, that give the gravity components in the boy frame, angular rate sensors an magnetometers. On the thir boar, a Digital Signal Processing (DSP), running at 5 MIPS, is embee an performs the control algorithm of the orientation ynamics an filtering computations. The final boar provies a serial wireless communication between the operator s joystick an the vehicle. An embee camera with a view angle of 7 egrees pointing irectly own, transmits vieo to a groun station (PC) via a wireless.4 GHz analogue link. A Lithium-Polymer battery provies nearly minutes of flight time. The loae weight of the prototype is about 85g. In parallel the vieo signal, the X4-flyer sens inertial ata to the groun station at a frequency of 5Hz. The ata is processe by the groun station PC an incorporate into the control algorithm. Desire orientation an esire thrust are generate on the groun station PC an sent to the rone. 9

10 A key challenge for the implementation lies in the relatively large time latency between the inertial ata an visual features. For orientation ynamics, an embee high gain controller in the DSP running at 66Hz, inepenently ensures the exponential stability of the orientation towars the esire set point. B. Experiments The target plane use is a large boar painte with ranom contrast textures (Fig. ). It is hel an move manually. A Pyramial implementation of the Lucas-Kanae [9] algorithm is use to compute the optical flow. The efficiency of the algorithm is increase by efocusing the camera to low-pass filter images. The fiel of view of the aperture is of aroun the irection of observation η. Optical flow is compute on points on this aperture an a least-square estimation of motion parameters is use to obtain robust measurements of the average optical flow w []. Fig. : Hovering flight above the laning pa Given that the ivergent flow magnitue is relatively small compare to the lateral flow in the forwar an backwars irections [] an since only the ivergent flow is use for laning manoeuvre, the control approach is split into two sequential phases. In the first phase the vehicle is stabilize over the laning plane. Once the velocity has stablise to zero the laning phase is initiate. During the experiments, the yaw velocity is also separately regulate to zero. This has no effect on the propose control scheme. The laning pa has been move both vertically an laterally to show performances of the control algorithms. For the vertical laning, the esire set point w is set to (,,.) T. This ensures a relatively rapi escent (approximatively in s). Note that no measurements of the relative position ξ = ξ ξ G of the UAV with respect to the platform are available. Nevertheless, an estimation of the UAV s relative position can be compute from the average optical flow using ξ ξ h = wγ(τ) τ = ( τ ) w exp w z δ τ where ξ enotes the relative position of the UAV with respect to the platform: ξ = ξ ξ G. Note that ( τ ) ( ) τ ḣ γ(τ) = exp w z δ = exp h δ = h(τ) h In Figures an, the components of the relative position ( ξ ξ )/h are presente. Figure shows the result using controller (4) for the stabilisation of the X4-flyer with respect to the platform (from s to 4s) an controller () for the vertical laning manoeuvre (from 4s). For the stabilisation phase, the platform is moving laterally (from s to s) an vertically (from s to 4s). During the laning manoeuvre (t 4s) the platform is moving only vertically. Note that, uring the laning phase, controller (4) is still use for the x-y ynamics. This ensures that the vehicle remains stable over the laning pa. Figure shows the exponential escent of the height while the lateral position remains stable. Note that the relative position (y y G )/h converges aroun, this is ue to an initial bias of the inertial measurements in y-irection that has been compensate by the integral term in the controller (4). Note also that, contrary to what was expecte, the height h is slowly oscillating uring the laning phase. This implies that conition () is not verifie for all time t an therefore, the positivity of α(t) (see Section V) is not always guarantee. This problem is mainly ue to the fact that experimental constraints (large time latency, outer loop s sampling time which is of 5Hz) prevent us from choosing a higher gain k which strictly respect the conition. The vehicle lans at time 8s. We notice that, ue to the laning gear, the final position is not h. Figure shows the result when the laning pa is moving with high oscillations. The controller for the stabilisation of the X4-flyer with respect to the platform is use from s to 55s an the controller for laning is use from t = 55s while oscillations start from s. During laning manoeuvre, the height h is highly oscillating, which means that the gain k is not high enough to compensate the oscillations. Nevertheless, the istance with the groun remains positive which insures the non-collision with the moving target an the UAV is even able to lan with a satisfactory behaviour. These results can be watche on the vieo accompanying the paper or at the following url: VIII. CONCLUDING REMARKS This paper presente a nonlinear controller for vertical laning of a VTOL UAV using the measurement of average optical flow on a spherical camera along with the IMU ata. The originality of our approach lies in the fact that neither linear velocity nor istance with the target is reconstructe. Inertial ata is use only for erotation of the flow an the propose approach is an image base visual control algorithm. Both stabilisation an vertical laning with respect to a moving platform were consiere an a rigorous analysis of the stability of the close-loop systems was provie. Simulations provie a clear picture of the preicte response of the propose algorithm. The experimental results inicate

11 (x x G )/h (y y G )/h h/h (x x G )/h (y y G )/h h/h STABILISATION STABILISATION LANDING lateral motion vertical motion vertical motion Fig. : Vertical laning on a moving platform STABILISATION STABILISATION LANDING no motion vertical motion vertical motion Fig. : Vertical laning with high oscillations some of the ifficulties with obtaining high gain feeback control an show that the propose scheme is effective even if the assumptions in the theorems on t necessarily hol. There are several irections in which further work is of interest. The practical limitations of real worl systems limit magnitue of the feeback gain that can be applie an lea to limitations in the applicability of the approach in the presence of aggressive motion of the environment. Improving the time response of the optical sensors woul alreay provie a major improvement in the close-loop response an alleviate much of this ifficulty. The consieration of a feefowar compensation coul alleviate the epenence on high gain feeback when the environmental motion can be moelle. How to accomplish this within the image base paraigm is a challenge. Finally, although the robustness of the propose approach inicates that small variation of orientation an error in estimation of the normal irection will not estroy the stability analysis obtaine, it is of interest to consier the situation where the platform orientation is time varying an the normal of the platform is not assume to be known. Acknowlegments: This work was partially fune by Naviflow grant an by ANR project SCUAV (ANR-6-ROBO- 7) an by the Australian Research Council through the ARC Discovery Project DP8859, Image-base teleoperation of semi-autonomous robotic vehicles. APPENDIX In this appenix, we provie erivation of optical flow integration escribe in Eq. (9) an use in Paragraph III-B. Using the notations of Section III, consier the average of the optical flow in the irection η over a soli angle W of S. Define (α e, α a ) to be the spherical coorinates of η where α e is the elevation angle an α a is the azimuth angle. With these parameters, efine R t to be the orientation matrix from a frame of reference with η in the z-axis assuming no yaw rotation to the inertial frame I c (α e ) c (α a ) s (α a ) s (α e ) c (α a ) R t = c (α e ) s (α a ) c (α a ) s (α e ) s (α a ) s (α e ) c (α e ) Define θ as the angle associate with the apex angle θ of the soli angle W. Then: ϕ = ṗ p = π (sin θ ) Ω R η Q(V V t) W where, Q = R (R t ΛRt )R is a symmetric positive efinite matrix. The matrix Λ is a positive iagonal matrix epening on the soli angle W. It can be written as Λ = π q p, R η q W = θ θ= π ϕ= (I qq ) q, R t η sin θ θ ϕ where q = (s (θ) c (ϕ), s (θ) s (ϕ), c (θ)). Eventually, straightforwar but teious calculations verify that: Λ = π (sin θ ) 4 λ 4 λ where λ = (sin θ) 4 (sin θ ). REFERENCES [] K. P. Valavanis. Avances in Unmanne Aerial Vehicles. Springer, 7. [] L. Marconi, A. Isiori, an A. Serrani. Autonomous vertical laning on an oscillating platform: an internal-moel base approach. Automatica, 8:,. [] Xilin Yang, Hemanshu Pota, Matt Garratt, an Valery Ugrinovskii. Preiction of vertical motions for laning operations of uavs. In Proceeings of the 47th IEEE Conference on Decision an Control, Cancun, Mexico, December 8. [4] So-Ryeok Oh, Kaustubh Pathak, Sunil K. Agrawal, Hemanshu Roy Pota, an Matt Garratt. Approaches for a tether-guie laning of an autonomous helicopter. IEEE Transactions on Robotics, ():56 544, 6. [5] Srikanth Saripalli, James F. Montgomery, an Gaurav S. Sukhatme. Visually-guie laning of an unmanne aerial vehicle. IEEE transactions on robotics an automation, 9():7 8,. [6] Cory S. Sharp, Omi Shakernia, an S. Shankar Sastry. A vision system for laning an unmanne aerial vehicle. In IEEE International Conference on Robotics an Automation,. [7] M.V. Srinivasan, S.W. Zhang, J. S. Chahl, E. Barth, an S. Venkatesh. How honeybees make grazing lanings on flat surfaces. Biological Cybernetics, 8:7 8,. for all x R, s (x) = sin (x), c (x) = cos (x), t(x) = tan (x)

12 receive the Grauate egree Bruno HERISS E from the Ecole Superieure Electricit e (SUPELEC), a french Grane Ecole of Engineering in Energy an information Science, an a Research Master egree in signal, telecommunications an image processing from SUPELEC in joint authorization with the University of Rennes in 7. After years as a Ph.D. stuent at the Interactive Robotics Laboratory at CEA List, he obtaine his octorate egree in Robotics from the University of NiceSophia Antipolis in. Since, he has been a research engineer at ONERA, the French Aerospace Lab. His current research interests inclue localization an navigation of unmanne aerial vehicles. [8] J. Koenerink an A. van Doorn. Facts on optic flow. Biol. Cybern., 56:47 54, 987. [9] F. Ruffier an N. Franceschini. Visually guie micro-aerial vehicle: automatic take off, terrain following, laning an win reaction. In Proceeings of international conference on robotics an automation, LA, New Orleans, April 4. [] Geoffrey L. 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Optimisation et implementation e lois e commane embarquees pour la teleopration intuitive e micro rones aeriens X4flyer. PhD thesis, universite Nice Sophia Antipolis, 7. [8] N. Guenar, T. Hamel, an R. Mahony. A practical visual servo control for an unmanne aerial vehicle. IEEE Transactions on Robotics, 4(): 4, 8. [9] B. Lucas an T. Kanae. An iterative image registration technique with an application to stereo vision. In Proceeings of the Seventh International Joint Conference on Artificial Intelligence, pages , Vancouver, 98. [] S. Umeyama. Least-squares estimation of transformation parameters between two point patterns. IEEE Trans. PAMI, (4):76 8, 99. [] J. S. Chahl, M. V. Srinivasan, an S. W. Zhang. Laning strategies in honeybees an applications to uninhabite airborne vehicles. The International Journal of Robotics Research, ():, 4. Tarek HAMEL receive his Bachelor of Engineering from the Institut Electronique et Automatique Annaba, Algeria, in 99. He conucte his Ph.D. research at the University of Technologie of Compi`egne (UTC), France, an receive his octorate egree in Robotics from the UTC in 995. After two years as a research assistant at the University of Technology of Compi`egne, he joine the Centre Etues e M`ecanique Iles e France in 997 as an associate professor. In / he spent one year as CNRS researcher at the Heuiasyc Laboratory. Since, he has been full Professor at the IS UNSA-CNRS laboratory of the University of Nice-Sophia Antipolis, France. His research interests inclue control theory an robotics with particular focus on nonlinear control, vision-base control an estimation an filtering on Lie groups. He is involve in applications of these techniques to the control of Unmanne Aerial Vehicles an Mobile Robots. Robert MAHONY obtaine a science egree majoring in applie mathematics an geology from the Australian National University (ANU) in 989. After working for a year as a geophysicist processing marine seismic ata he returne to stuy at ANU an obtaine a Ph.D. in systems engineering in 994. Between 994 an 997 he worke as a Research Fellow in the Cooperative Research Centre for Robust an Aaptive Systems base in the Research School of Information Sciences an Engineering, ANU, Australia. From 997 to 999 he hel a post as a post-octoral fellow in the CNRS laboratory for Heuristics Diagnostics an complex systems (Heuiasyc), Compiegne University of Technology, FRANCE. Between 999 an he hel a Logan Fellowship in the Department of Engineering an Computer Science at Monash University, Melbourne, Australia. Since July he has hel the post of senior lecturer in mechatronics at the Department of Engineering, ANU, Canberra, Australia. His research interests are in non-linear control theory with applications in mechanical systems an motion systems, mathematical systems theory an geometric optimisation techniques with applications in linear algebra an igital signal processing. Franc ois-xavier RUSSOTTO receive in 997 the Grauate egree from the Ecole Superieure Electricit e (SUPELEC), a french Grane Ecole of Engineering in Energy an Information Science, Paris, France. He worke next as a evelopment engineer in automation at Thales Optronics SA (TOSA), on esign of servomechanism for aircraft embee systems, an as a project manager at Peugeot Citroen Automobiles SA (PSA), on esign of innovative Man-Machine Interface for automobile. He now works as a project manager in Robotics R&D, mainly focuse on Supervisory Control for robotic systems, at CEA.