Guidance Control of a Small Unmanned Aerial Vehicle with a Delta Wing

Transcription

1 Proceeings of Australasian Conference on Robotics an Automation, -4 Dec 13, University of New South Wales, Syney Australia Guiance Control of a Small Unmanne Aerial Vehicle with a Delta Wing Shuichi TAJIMA, Takuya AKASAKA, Makoto KUMON an Kazuo OKABE : Kumamoto University, Japan s.tajima@ick.mech.kumamoto-u.ac.jp, kumon@gpo.kumamoto-u.ac.jp : Sky Remote Inc., Japan Abstract A small an light unmanne aerial vehicle (UAV) with a elta-shape main wing calle a kiteplane is consiere in this paper. The UAV is suitable for groun observation because of its slow flying spee, which can be achieve ue to its light weight an large main wing; however, the aircraft can be greatly isturbe by a control comman (set point) upate or a win isturbance. In orer to compensate for this problem, this paper proposes a bankan-turn approach for path-following control base on nonlinear attitue control. The bank angle of the kiteplane is commane so as to track the esire path, an the reference signal is esigne using stability analysis. The results of a numerical simulation are also presente to valiate the propose approach. 1 Introuction Rescue activities an surveillance missions in angerous areas, such as those affecte by natural isasters, are important in orer to save victims an obtain important ata. Those areas are not always easily accessible by orinary means since the infrastructure (incluing the stanar moes of transportation) can be severely amage. Therefore, unmanne aerial vehicles (UAVs) are expecte to be use becausethey areable to be operate remotely. UAVs can be classifie into two categories accoring to their configuration: fixe-wing UAVs an rotary-wing UAVs. Both types of UAVs have been stuie actively (e.g., [Mueller an DeLaurier, 1; Chao et al., 1]). Fixe-wing UAVs are efficient in terms of energy consumption since they can glie, but they normally fly rather fast, which is not suitable for measuring objects on the groun. On the other han, rotary-wing UAVs such as helicopters are able to hover in one location, which provies the ability to capture terrain ata but limits the enurance because of the high energy consumption. In this paper, a lightweight fixe-wing UAV with a large main wing that realizes a slow flight spee in orer to compensate for the stanar fixe-wing UAV rawbacks is consiere for surveillance missions. This UAV is name a kiteplane because its main wing is mae of cloth an has a elta triangle shape like a kite, as shown in Fig. 1. Aileron Figure 1: Kiteplane Main Wing Engine Elevator Ruer Nagata [Nagata et al., 4] propose PID control for the kiteplane because of its simple ynamics, an the author [Kumon et al., 6] propose a fuzzy-logic base nonlinear flight controller for waypoint tracking. These methos use the position of the aircraft to irectly control its flight path without consiering the attitue ue to its stable ynamics. Although this approach succeee at simplifying the controller structure, large attitue eviations coul be triggere by set-point changes after the aircraft passe waypoints or by strong win turbulence, which might lea to significant overshoot or oscillation. In orer to improve the flight-path-following perfor-

2 Proceeings of Australasian Conference on Robotics an Automation, -4 Dec 13, University of New South Wales, Syney Australia mance, this paper proposes realizing bank-an-turn for a straight line an a circular orbit using a roll angle controller with the nonlinear attitue control from the kiteplane propose by Akasaka [Akasaka an Kumon, 1]. The present paper is organize as follows: In the next section, the kiteplane an its ynamics are briefly introuce. Then, Sec. 3 first introuces Akasaka s controller, an the path-following controller is propose. In orer to valiate the propose approach, Sec. 4 shows the numerical simulation results, an the conclusions follow in (Sec. 5). Small UAV with Delta Wing: Kiteplane.1 Kiteplane A kiteplane is a UAV that has a kite-like main wing in the shape of a elta, as shown in Fig. 1. The main wing is light an flexible because it is mae of cloth, an the UAV is capable of carrying a large payloa. The wing s flexibility provies safety an robustness if it crashes into thegroun. Thecenterofmassislocateunerthemain wing, an the ailerons are attache at iheral angles. The kiteplane s length, wingspan, an height are 116 mm, 9 mm, an 4 mm, respectively, an it weighs approximately 4.5 kg. The kiteplane has three control surfaces: the ailerons, the elevator, an the ruer, an they are actuate by servomotors. The propeller that generates the thrust require to fly is riven by a brushless motor, an its rotational spee is controlle by a spee controller. A global positioning system (GPS) is installe to measure the elta wing s position, an a three-imensional accelerometer, a three-imensional rate gyro, an a three-imensional magnetometer are also installe to estimate the attitue of the aircraft. In this paper, it is assume that the three-imensional position, attitue an their time erivatives are available with respect to the inertial frame.. Dynamics of the Kiteplane The ynamics of the kiteplane are presente in this subsection. It is assume that the total aeroynamic force an torque can be moele as a sum of the forces acting on the wings. The frames consiere in this paper are shown in Fig., an the quantities with respect to the inertial frame an boy frame are enote by subscripts I an B when istinction is necessary. The aeroynamic forces that affect the main wing, the ailerons, the elevator, an the ruer (with respect to the boy frame) are enote as f m, f a, f e, an f r, respectively, an the thrust generate by the propeller is ZI XI YI XB ZB Aileron YB Engine MainWing Elevator Ruer Figure : Inertial frame an boy frame inicate by T as in [Kumon et al., 6]. The aeroynamic parameters, such as the air spee an angle of attack, have nonlinear effects on those forces, an the forces affecting the ailerons, the elevator, an the ruer are also governe by control inputs since those wings are control surfaces. The aeroynamicforceon the elevatorf e with respect to the wing frame, which is the frame aroun the wing with its X an Z axes in the rag an lift irections, respectively, is enote as f e W. In this paper, the following aeroynamic moel is introuce: f e W = 1 ρv S e C D 1 ρv S e C L, (1) where ρ, V, S e, C L, an C D represent the air ensity, flow spee, elevator area, lift coefficient, an rag coefficient, respectively. C L an C D are consiere linear functions of the control input to the elevator. f e W is transforme to the boy frame to compute f e. Other aeroynamic terms, f m, f a, an f r, are also moele in a similar manner an then summe up to obtain f B, which is shown below. Since the flow spee V can be compute from the win an the velocity of the aircraft itself, which is a part of the system state, the moel contains nonlinear terms with respect to the state of the system. It is consiere possible for the magnitue of the thrust T to be commane irectly since the rotation spee of the propeller for the real platform is well controlle ue to an inner-loop spee controller with a high-power electric motor. ConsieringtheUAVasarigiboy, themotionequation with respect to the center of mass can be given as follows: m t [ xi y I z I ] T = fi, () I B t ω B +ω B I B ω B = n B, (3)

3 Proceeings of Australasian Conference on Robotics an Automation, -4 Dec 13, University of New South Wales, Syney Australia where f I n B = q f B q = q ( fm + f a + f e + f r + T = l m f m +l a f a +l e f e +l r f r +l T T. ) q Here, x I, y I, an z I inicate the position of the boy, an ω B represents the angular velocity. The attitue is represente by a quaternion q, an represents the multiplication of quaternions. f I an f B represent the forces acting on the center of mass with respect to the boy frame an the inertial frame, respectively, an n B enotes the torque on the boy. The translation between f I an f B can be given by using a quaternion as shown above, an x represents a quaternion with the vector x for its imaginary part (i.e., x = [,x T] T ), an shows the conjugate operation of the quaternion. m an I B represent the mass of the boy an the inertial matrix, respectively. l m, l a, l e, l r, an l T represent the position of the aeroynamic forces acting on the wings. Further etails of the quaternion operations can be foun in [Murray et al., 1994]. In this paper, the attitue is represente by the quaternion, an its ynamics are given as follows: t q = 1 ω I q = 1 q ω B (4) Let the state of the system be represente by (x I,y I,z I, t x I, t y I, t z I,q,ω B ). Recalling the fact that f B an n B can be compute if the state an control inputs are given, the evolution of the system can be obtaine by the ynamics (), (3), an (4). 3 Guiance Control of Kiteplane The propose guiance controller consists of inner-loop attitue control an outer-loop path-following control. In the following subsections, the inner-loop attitue controller propose by Akasaka [Akasaka an Kumon, 1] is shown first, an then, the path-following controller for straight-line reference paths an circular orbits is propose, which is the main contribution of this paper. 3.1 Attitue control Attitue control for a kiteplane was propose by Akasaka [Akasaka an Kumon, 1], an this subsection briefly introuces the metho. Base on the ynamics shown above, the ynamics of the attitue subsystem can be represente as follows: q = 1 q ω B ω B = I B 1 (n B ω B I B ω B ) (5) The attitue an its reference are represente by quaternions, an they are enote by q an q, respectively. The reference angular velocity is efine as ω B, an the following hols as in (4): t q = 1 q ω B. The real an imaginary parts of those quaternions are enote as [ ] [ ] t t q =, q p = p an the attitue error enote by e q is efine as [ ] et e q q q =. e p Here, n B is the quantity that primarily governs the attitue motion, an it is assume that n B can be moele as n B = Gu+n, (6) where u represents the control input vector, an G an n represent a non-singular known matrix an a boune isturbance term, respectively. The ifference between ω B an ω B is efine as ν, an its imaginary part is efine as ν p as above, or ν ω B ω B = [ ν p ]. An augmente reference signal enote by ξ is efine as ξ = ω B K q ( e t p+te p +e p p), (7) where K q shows a positive symmetric matrix. Since ξ is ifferentiable, the control input u is efine as } u = G { K 1 ν ν +ω B I B ω B +I B ξ, (8) where K ν is a positive symmetric matrix. The control input (8) enables the UAV to follow the esire attitue, which can be shown as follows: Consier a positive scalar function of e q as V q = e q T e q, an a simple calculation shows that the time erivative of V q can be evaluate as V q = ( e t p+te p +e p p) T ν p. Ifν p isequaltoorverycloseto K q ( e t p+te p +e p p), then e q convergesclosetothe origin. Inorertovaliate the above, another scalar positive function is introuce, V ν = 1 e ν T I B e ν (9)

4 Proceeings of Australasian Conference on Robotics an Automation, -4 Dec 13, University of New South Wales, Syney Australia where e ν ν p ( K q ( e t p+te p +e p p)). Using the input (8), ifferentiating V ν with respect to time leas to V ν e T ν n k ν e ν, (1) where k ν shows the minimum eigenvalue of K ν. Since n is boune, e ν will converge sufficiently close to the origin by selecting a large enough k ν. 3. Guiance control In this paper, the path to follow is mainly consiere in two-imensional space, an the altitue is controlle inepenently. The altitue controller is shown first, an then, the main path-following controller is propose. Altitue control The objective for the altitue control escribe in this paper is to maintain a constant height uring flight. The current altitue, its reference, an the reference altitue change rate are enote by z, z, an v z, respectively. Taking the limit of the control input range into account, v z is efine as v z = tanh{k z (z z )}, (11) where k z represents a positive constant. Since the altitue is primarily ominate by thrust, the altitue controller is efine as follows: T = k t ( t z v z)+t, (1) where k t an T represent a positive constant feeback gain an the trim input for the throttle, respectively. Path-following control It is common to approximate a UAV as a point mass an to limit the path to follow straight lines an circles or arcs since the well-known Dubin s moel [Dubins, 1957] can achieve the shortest path between two given points by using a combination of those sections. Therefore, the path-following control of a straight line an a circular orbit is consiere in this paper. Since the inner-loop controller controls the attitue, the roll angle, or bank, is consiere as the input to control the position of the UAV, an the angle is enote by θ. The heaing angle is also controlle to converge to the esire irection, but its transient response may affect the path-following performance. In orer to take this effect into account, the current heaing angle an the esire angle, which are enote as α an α, respectively, are also consiere. Because of the attitue controller, it can be assume that α α. Point mass moel As the simplest moel, the UAV ynamics are approximate by a point mass, an the motion equation can be given as follows: m t x I = T cosα+lsinθsinα Dcosα (13) m t y I = T sinα Lsinθcosα Dsinα, (14) where L an D enote the lift an rag force affecting the UAV. Circle case Denote the reference circle raius as r an the istance from the center to the current UAV position as r. Without loosing the generality, the center of the reference circle is assume to be the origin of the coorinate frame. Defining the error e r r r, the control objective can be efine to make e r as t. α is given as the irection tangential to the reference path in this case. Let k 1 an k represent the appropriate positive feeback gains, an assume cos(α α ). The bank angle θ is efine as follows: sinθ = k 1e r +k ṙ+(t D)sin(α α )+mr t α,(15) Lcos(α α ) when the right han sie is within the range of 1 to 1; otherwise, θ will be selecte to clip to π or π. The reference roll angle (15) can be valiate as follows. The motion equations (13) an (14) can be rewritten with respect to the raial irection as m t r = (T D)sin(α α ) Lsinθcos(α α ) +mr t α. (16) Introucing a small positive constant ǫ, the scalar function V r is efine as V r = k 1 +ǫk e r + m ( t e r) +ǫme r t e r. (17) When ǫ is sufficiently small, it is obvious that V r is greater than or equal to, an there exists a positive constant B 1 that satisfies the following inequality: V r B 1 { e r +( t e r) }. Using (16) an (15), the erivative of V r with respect to time can be evaluate as follows, an it can be shown

5 Proceeings of Australasian Conference on Robotics an Automation, -4 Dec 13, University of New South Wales, Syney Australia that there exists a positive constant B : t V r = (k 1 +ǫk )e r t e r t e r { k 1 e r +k t e r +ǫm( t e r) ǫe r {k 1 e r +k t e r} B {e r +( t e r) } B B 1 V r. (18) (18) implies that the raius r converges to the esire value r exponentially as far as the input θ can be realize by (15). Straight line case Denotethe referencestraightline asax+by = c, where a, b, an c are the parametersof the line, an the control objective is such that the position of the UAV (x I an y I ) converges to this reference line. Define the error from the reference to the current position of the UAV as e l = ax I +by I c. Let k 3 an k 4 be positive constants. Assuming asinα bcosα, the input θ is efine to satisfy sinθ = k 3e l k 4 t e l (T D)(acosα+bsinα), L(asinα bcosα) (19) as far as the right-han sie is within the range of 1 to 1; otherwise, θ will be selecte to clip to π or π. The convergence of e l to can be implie as follows. Introucing a small positive constant µ, a scalar function V l is efine as V l = k 3 +µk 4 e l + m { } t e l +µme l t e l.() As shown in the circularreferencecase, if µ is sufficiently small, there exists a positive constant C 1 that hols to { } V l C 1 (e l + t e l ). (1) Using (13), (14), an (19), the erivative of V l with respect to time can be evaluate as t V l = (k 3 +µk 4 )e l t e l + t e l(a 1 +a sinθ) { } +µm t e l +µe l (a 1 +a sinθ), wherea 1 = (T D)(acosα+bsinα)ana = L(asinα bcosα). Since a sinθ = a 1 k 3 e l k 4 t e l, there exists a positive constant C that satisfies { } t V l C (e l + t e l ) C C 1 V l, () } which proves the statement. Input conversion Although the path-following commans efine above arefortherollanglesθ anyawα,itisnecessarytoconvert those quantities to the reference quaternion in orer to supply the information to the attitue controller. The reference quaternion is enote as q. Then, the following is use for the conversion: q = = cos α sin α cos θ cos α sin θ cos α sin θ sin α cos θ sin α 1 4 Numerical simulation cos θ sin θ. (3) In orer to verify the propose control scheme, the propose metho was teste through numerical simulations. Figure 3: Reference path Figure 4: Flight path

6 Proceeings of Australasian Conference on Robotics an Automation, -4 Dec 13, University of New South Wales, Syney Australia error between the current position an the esire path was also teste (Fig. 9). It was not easy to tune the PID gains for both the straight reference segment an the circular curves simultaneously by consiering the control surface saturation, an the achieve performance was poorer than the propose metho, as shown in the figure. Figure 5: Path-following error Figure 6: Altitue 4.1 Path-following guiance The oval-like path with connecting arcs an line segments shown in Fig. 3 is consiere as the reference path to follow. The UAV was locate 5 m offset from the reference path at the initial state, but its irection was aligne with the path irection. The altitue at the initial state was the same as the reference. The path followe by the UAV in the x y plane is shown in Fig. 4. The figure shows that the UAV converge to the esire path smoothly an tracke it well. Figure5showsthepath-followingerrorse l ane r with respect to time. The errors reache exponentially an remaine close to the origin, although they oscillate because the controller was erive base on the approximate moel in this paper. Figure 6 shows the response of the altitue uring the flight. It eviate at the initial transient perio an converge back to the esire value. This quantity also fluctuate aroun the esire value, but it was also kept close enough to the reference. Figure 7 an Fig. 8 show the attitue error in the quaternion an the euler angle. The real part eviate within the range of(.5,.5), an the imaginary part elements were also maintaine close to throughout the flight, which valiates that the attitue controller succeee to track the reference roll an yaw angles generate by the outer-loop path-following mechanism. As a comparison, the PID controller that utilize the Figure 9: Flight path by PID control 4. Discussion Although the propose controller is evelope base on a point mass an approximate ynamics for path following, an it is assume that the attitue can be perfectly controlle by the inner loop in orer to achieve pathfollowing control, the above simulation showe that the UAV coul follow the reference path almost perfectly, which valiates the propose approach. Oscillation close to the reference path might be able to be suppresse by tuning the feeback gain, but input saturation may reuce the performance, an the gain tuning process nees to be consiere further. The noise in the measurement is not consiere in this paper, an it may also eteriorate the performance. 5 Conclusion This paper propose a path-following control system base on nonlinear attitue control for a small UAV with a elta wing. Introucing simple point-mass ynamics, the stability of the path-following control system was consiere, an the approach was valiate through numerical simulations. References Takuya Akasaka an Makoto Kumon. Robust attitue control system for kite plane. In Proceeings of Sys-

7 Proceeings of Australasian Conference on Robotics an Automation, -4 Dec 13, University of New South Wales, Syney Australia Figure 7: Quaternion error roll[eg] pitch[eg] yaw[eg] Time[s] Time[s] Time[s] Roll Pitch Yaw Figure 8: Attitue error in Euler angle tem Integration 1, pages , 1. (in Japanese). HaiYang Chao, YongCan Cao, an YangQuan Chen. Autopilots for small unmanne aerial vehicles: a survey. International Journal of Control, Automation an Systems, 8(1):36 44, 1. Lester E. Dubins. On curves of minimal length with a constraint on average curvature, an with prescribe initial an terminal positions an tangents. American Journal of Mathematics, 79(3): , Makoto Kumon, Yuya Uo, Hajime Michihira, Masanobu Nagata, Ikuro Mizumoto, an Zenta Iwai. Autopilot system for kiteplane. IEEE/ASME Transactions on Mechatronics, 11(5):615 64, oct 6. Th Mueller an James D DeLaurier. An overview of micro air vehicle aeroynamics. Progress in Astronautics an Aeronautics, 195:1 1, 1. R. M. Murray, Z. Li, an S. S. Sastry. A Mathematical Introuction to Robotic Manipluation. CRC Press, Masanobu Nagata, Makoto Kumon, Ryuichi Kouzawa, Ikuro Mizumoto, an Zenta Iwai. Automatic flight path control of small unmanne aircraft with eltawing. In Proc. of the Int. Conf. on Control, Automat. an Syst., pages , Bangkok, Thailan, August 4.