1
Dynamic Modeling, Trajectory Optimization, and Control of a Flexible Kiteplane Ryan James Caverly, Student Member, IEEE, and James Richard Forbes, Member, IEEE
Abstract—This paper investigates dynamic modeling, trajectory optimization, and control of a flexible kiteplane used for wind-energy harvesting. The individual components of the kiteplane, including flexible wings and a rigid fuselage, are modeled separately and then constrained together using the nullspace method. The flexible wings of the kiteplane are modeled as flexible plates, and the Rayleigh-Ritz method is used to discretize the partial differential equation that describes the strain energy stored in the wing. The attitude of the kiteplane is described by the direction cosine matrix (DCM) directly and a proportionalintegral-derivative (PID) control law that makes use of the DCM is implemented for attitude control. An unsteady aerodynamic model based on Theodorsen’s lift model is used in simulation to allow for an accurate model under transient conditions. An optimal trajectory is found using a simplified dynamic model and solving a finite-dimensional constrained optimization problem. Numerical simulations of the optimal trajectories are performed to demonstrate the kiteplane’s energy-harvesting capability. Index Terms—Kiteplane, wind-energy harvesting, dynamic modeling, trajectory optimization, attitude control, flexible structure, direction cosine matrix.
I. I NTRODUCTION HE harvesting of wind energy at high altitudes has been proposed as a potential source of renewable energy. Motivation to do so stems from the fact that high-altitude winds are much stronger than low-altitude winds. Traditional wind turbines are only able to access low-altitude winds whereas airborne wind energy (AWE) technology is able to extract energy from energy-dense high-altitude winds [1]–[3]. In this paper, an AWE system is considered, where the flying structure is tethered to a ground-based winch linked to an electric generator [1], [2], [4]–[6]. Wind energy is harvested by performing cyclical trajectories that typically involve an energy harvesting phase followed by a energy consuming phase, where the energy consumed is less than the energy harvested for the trajectory to be generative. Kiteplanes tethered to a ground-based winch are an attractive way to harvest the energy present in high-altitude winds [6]. Kiteplanes are kites with control surfaces similar to a conventional aircraft, giving them the ability to fly without consistent tether tension [6]. The control surfaces of the kiteplane simplify the task of stabilizing the kiteplane and allow for implementations with a single tether that generates electricity when reeled out. Wind-energy harvesting kites are typically designed to be lightweight, which can introduce
T
R. J. Caverly is with the Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA (e-mail:
[email protected]). J. R. Forbes is with the Department of Mechanical Engineering, McGill University, Montreal, QC, Canada (email:
[email protected]).
significant flexibility to the structure of the kite [6]. This flexibility can significantly affect the dynamics of the kite and complicate controller design. Although the modeling of flexible aircraft wings has been considered in the context of lightweight long-endurance aircraft [7], [8] and a nonlinear aeroelastic model of a kite is presented in [9], to our knowledge [10] is the only paper that considers dynamic modeling of a kiteplane with flexible wings. In order for energy-harvesting kites to be viable source of renewable energy, they must produce significantly more power than they consume. Efficient control techniques are critical to ensuring that energy-harvesting kites meet this requirement [11]. In particular, trajectory optimization is often used to maximize the energy harvested by kites [11]–[13]. Unfortunately, there exists little literature on the trajectory optimization of kiteplanes [14]. In this paper, the dynamic model of a kiteplane with flexible wings is presented, where the fuselage of the kiteplane is modeled as a rigid cylinder and the wings are modeled as flexible plates. An unsteady aerodynamic model based on Theodorsen’s lift model [15] is included in the dynamic model to more accurately represent the transient behavior of the aeroelastic structure. An optimal trajectory is presented that is found by solving a finite-dimensional constrained optimization problem using a simplified dynamic model of the kiteplane. An attitude control law based on the direction cosine matrix (DCM) is used to track the desired trajectory. The novel contribution of this paper is the trajectory optimization of a flexible kiteplane to maximize its power generation and a more detailed presentation of the dynamic model from [10]. The remainder of this paper proceeds as follows. In Section II a preliminary discussion on the reference frames, the attitude parameterization, and notation used are discussed. The dynamic model of the kiteplane is developed in Section III, including the high-fidelity model and the simplified model. In Section IV the attitude control law is presented, along with an optimal trajectory that maximizes the energy harvested by the kiteplane. Numerical results of the kiteplane are provided in Section V, and concluding remarks are found in Section VI. II. P RELIMINARIES A. Reference Frames Multiple reference frames are used to describe the attitude of the various bodies included in the dynamic model of the kiteplane, as shown in Fig. 1. The frame Fa is a groundfixed inertial reference frame, Ff is a reference frame affixed to the center of the kiteplane’s fuselage, and Fr and Fl are
2
Rn → Rn×n , is defined as T
Fr f
Fa
r
a× = −a×
Ff
Fl l
a
Fig. 1. Visualization of the reference frames used to describe the attitude of the components of the flexible kiteplane.
0 = a3 −a2
−a3 0 a1
a2 −a1 , 0
where a = [a1 a2 a3 ] . The uncross operator, (·)v : Rn×n → Rn , is defined as 0 −a3 a2 0 −a1 , A = −AT = a3 −a2 a1 0 T
T where Av = [a1 a2 a3 ] .
III. DYNAMIC M ODELING reference frames affixed to the intersection of the root and the trailing edge of the right and the leading edge of the left wing, respectively. Notice that the origin of Fi is denoted as point i, where i = a, r, l, f . B. Attitude Parameterization The DCM is used directly to describe the attitude of the reference frames relative to each other, and is not parameterized further using Euler angle, quaternions, or any other parameterization. The DCM, written as Cba , is a matrix representation of the attitude of Fb relative to Fa satisfying Cba ∈ R3×3 , CT ba Cba = 1, and det(Cba ) = +1. Additionally, the DCM can be used to change the frame of reference a physical vector is represented in. For example, consider va and vb , which are representations of → v in Fa and Fb , respectively. − They are related through the DCM by vb = Cba va . The DCM is chosen to represent attitude due to its ability to represent attitude globally and uniquely [16]. The kiteplane modeled in this paper may fly with a large pitch angle or even with its wings perpendicular to the ground, which motivates the use of the DCM. This avoids the kinematic singularities associated with Euler angles and the two-to-one mapping associated with quaternions [16]. C. Notation In this paper a physical vector is denoted by a lowercase letter with an arrow underneath, for example, → v . The rep− resentation of a physical vector in a given reference frame is identified by the same lowercase letter in boldface with a subscript that indicates the reference frame that the vector is resolved in. For example, va is the representation of → v − in Fa . A physical vector that represents a relative position between two points has a superscript to identify the points. For example, → v pq is a physical vector representing the position of − point p relative to point q. The time-derivative of a physical vector with respect to a given reference frame is denoted by a superscript dot followed by a letter identifying the frame. For example, → v a is the time-derivative of → v with respect to Fa . − − The cross and uncross operators are used throughout this paper and are defined as follows. The cross operator, (·)× :
·
The flexible kiteplane is modeled as individual components and the components are then constrained together using the null-space method [17]. The kinetic and potential energies of each individual component are first derived, followed by the derivation of the generalized forces and moments acting on the kiteplane, the constraints between components, the unsteady aerodynamic model used, and the development of the equations of motion of the kiteplane using a Lagrangian approach. A simplified dynamic model is also presented, which is used in Section IV-B to generate an optimal energyharvesting trajectory for the kiteplane. Due to the complicated nature of the dynamic model, a summary of the model is presented in Table I. Table I provides information on the degrees of freedom of the model as well as the number of aerodynamic forces acting on the aircraft. A. Kinetic and Potential Energies 1) Kiteplane Wing: Each wing of the kiteplane is modeled as a flexible plate that can bend and twist. The plate’s strain energy is given by [18] " 2 Z ∂2w ∂2w Eh3 1 + Vplate = 2 2 ∂x2 ∂y 2 A 12(1 − ν ) 2 2 ! # ∂2w ∂2w ∂ w − 2(1 − ν) − dA, (1) ∂x2 ∂y 2 ∂x∂y where E is the elastic modulus of the plate, h is the thickness of the plate, ν is the Poisson ratio of the plate and w is the of the displacement of the plate perpendicular to the undeformed plane of the plate. A Rayleigh-Ritz discretization is used with multiple basis functions to capture the first few vibrational modes of the plates. Specifically, w(x, y, t) = x(x)Y(y)qe (t) where x(x) = x1 (x) x2 (x) · · · xn (x) , y(y) = y1 (y) y2 (y) · · · ym (y) , Y(y)
=
qe (t)
=
diag{y(y), y(y), . . . , y(y)} ∈ Rn×nm , e T e q1 (t) q2e (t) · · · qnm (t) .
The basis functions xi , i = 1, . . . , n and yj , j = 1, . . . , m are chosen to satisfy the geometric boundary conditions of the plate. In this paper three basis functions are chosen
3
in both the x and y directions to satisfy the clampedfree-free-free geometric boundary conditions of the wing: x(x) = x2 x3 x4 , y(y) = 1 y y 2 . Therefore, qe (t) = T [q1e (t) · · · q9e (t)] . Note that alternatively a finite element method could be used to model the flexible wings, which is in fact a specific form of the Rayleigh-Ritz method [19]. The fidelity of the model can be adjusted by increasing or decreasing the number of basis functions. Substituting w(·, ·, ·) into (1) and performing the chain T rule of differentiation on w(x, y, t) gives Vplate = 21 qe Ke qe where Z 2 T d2 YT d2 xT d2 x Td x e Y + 2νY x K = D YT dx2 dx2 dx2 dy 2 A ! dYT dxT dx dY d2 YT T d2 Y x x 2 + 2(1 − ν) + dA. (2) dy 2 dy dy dx dx dy Notice that Ke is a constant matrix with respect to time. The strain energies associated with the right and left wings of the T T kiteplane are VBr = 21 qBr KBr qBr and VBl = 12 qBl KBl qBl , Br Bl e respectively, where K = K = diag{0, K }, qBr = i h T T T T pra qer , rra rra a is the position vector of point r a relative to point a expressed in Fa , pra is the column matrix containing the nine entries of the DCM Cra , and qer is the column matrix containing the elastic coordinates associated with the right wing. The column matrix qBl is defined similarly with the coordinates of the left wing. The kinetic energy of the right wing is found to be Z a a r dm1 a · → r dm1 a dm, TBr a/a = 12 → − − =
·
1 2ν
Br Br T
·
·
MBr ν Br ,
a
where → r dm1 a is the time rate of change with respect to Fa − of the physical vector that represents the position of a mass element dm1 located on the right wing relative to point a, r Br r × mBr CT mBr 1 −CT ra 13 H ra cr r r MBr = ? IB mBr Gr , r ? ? mBr Fr Cra is the DCM associated withRthe attitude of Fb relative to R × 1r Fa , Hr = Br xr Yr dV , Gr = Br rdm 13 xr Yr dV , Fr = r R R T T × × 1r Yr xr xr Yr dV , IrBr r = − Br rrdm1 r rdm σdV is the r Br rr second moment of mass of the wing relative to point r, cB = r R dm1 r r σdV is the first moment of mass of the wing relative Br r to point r, σ is the density of the right wing, mBr is the mass T of the wing, 1 is the identity matrix, 13 = 0 0 1 , and h T i T T T ν Br = r˙ ra ωrra q˙ er . Similarly, the kinetic energy of a T
the left wing is given by TBl a/a = 21 ν Bl MBl ν Bl , where the augmented velocity and mass matrix are identical to those of the right wing, with the subscript r replaced by l. The gravitational potential energy associated with the right wing can be calculated as Z g UBr a = − g ·− r dm1 a dm, − → Br → =
ra mBr g1T 3 (ra
+
cmr CT ), ra rr
where g = −g → a 3 , g is the gravitational constant, → a 3 is the − − → − basis vector of Fa that points upwards, the position vector of the center of mass of the wing relative to point r expressed T in Fr is rcmr = [c/2 l/2 0] , c is the chord length of the r wing, and l is the span of the wing. Similarly, the gravitational potential of the left wing is la T cml UBgl a = mBl g1T ). 3 (ra + Cla rl
The total potential energy of each wing can be written as UBr a = UBgr a + VBr and UBl a = UBgl a + VBl . 2) Kiteplane Fuselage: The fuselage of the kiteplane is modeled as a rigid cylindrical body. The kinetic energy of the fuselage is fa h i m 1 r˙ a 0 T T Bf TBf a/a = 21 r˙ fa a ωff a B f ωff a 0 Ir f =
1 Bf T Bf Bf M ν . 2ν
The gravitational potential of the fuselage is fa UBgf a = mBf g1T 3 ra .
B. Generalized Forces and Moments The aerodynamic forces considered in this paper include lift and drag acting on the wings, the horizontal stabilizers, the vertical stabilizer, and the fuselage. The drag induced by sideslip on the fuselage is also considered. The aerodynamic forces acting on each wing are discretized into three point loads equally spaced along the aerodynamic centerline of the wing. The aerodynamic forces acting on the horizontal and vertical stabilizers are discretized into single point loads applied on the aerodynamic centerline at the mid-span point of the horizontal and vertical stabilizers. The aerodynamic forces acting on the fuselage are discretized into a single point load acting on the aerodynamic center of the fuselage. The tether force acting on the fuselage is applied at the kiteplane’s center of mass. The principle of virtual work is employed to derive the generalized forces and moments acting on the system. The virtual work associated with one of the aerodynamic loads on the right wing is δW
f Lr1 · δ → r r1 a (x1 , y1 ), − → − Lr T = fr 1 Cra δrra1 a (x1 , y1 ),
=
(3)
where f Lr1 is the vector representing the lift force applied at → − the first aerodynamic load application point of the right wing 1 and is expressed as fLr when resolved in Fr , → r r1 a (x1 , y1 ) is r − the vector associated with the position of the first aerodynamic load application point (x1 , y1 ) relative to point a and is expressed as rra1 a (x1 , y1 ) when resolved in Fa . The virtual displacement of rra1 a is h i T r1 r rr ) δrra1 a = 1 ∂ (Crara δqBr , CT 1 x(x )Y(y ) 1 1 ra 3 ∂p
T
1 where rrr1 r = [x1 y1 wr (x1 , y1 , t)] . The contents of fLr r are discussed in detail in Section III-E. The virtual work of (3)
4
h
can be rewritten as δW
= δqBr
T
ωff a
T
q˙ er
T
q˙ el
T
1
T
r˙ ra f
r1 r CT ra rr ∂pra
∂(
)
T
T Lr Cra fr 1 ,
YT (y1 )xT (x1 )1T 3 Cra T
= δqBr f Lr1 .
(4)
Equation (4) can be used to find the generalized forces and moments associated with the remaining aerodynamic loads on both the right and left wings. A very similar expression to (4) is used to find the generalized forces and moments acting on the fuselage without the terms relating to the Rayleigh-Ritz discretization of the wings. The complete column matrix of generalized forces moments is compiled in the column hP and P BT P BT iT P Bi BrT matrix f = f f l f f , where f is the sum of the generalized forces and moments acting on body Bi , i = r, f, l. C. Constraints Each segment of the kiteplane is constrained together to form the complete flexible kiteplane. This is done by kinematically constraining the velocities of the root of each wing to be equal to the velocity of the point on the fuselage where they should be connected. For example, ×
1 0 0 Υ = 1 0 0 1 0
iT
ˆ where , by ν = Υν, ×
rf −CT f a rf Crf 0 × T lf −Cf a rf Clf 0 0 1
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
is constructed based upon the kinematic constraints between the augmented velocities. Note that the columns of Υ lie in the null space of Ξ, that is ΞΥ = 0. D. Equations of Motion Lagrange’s equation is used to derive the equations of motion of the flexible kiteplane: T ∂LSa/a T d ∂LSa/a − = f + ΞT λ, (5) dt ∂ q˙ ∂q where LSa/a = TSa/a − USa , TSa/a = TBr a/a + TBl a/a + TBf a/a , USa = UBr a +UBl a +UBf a , and λ is a column matrix that contains Lagrange multipliers. The total kinetic energy of the system can also be written as
rf ˙ fa a + CT r˙ ra ωff a = 0, a −r f a rf
˙ TSa/a = 21 ν T Mν = 21 q˙ T ST MSq,
where rrf f is the position vector of point r relative to point f expressed in Ff , and rlf f is the position vector of point l relative to point f expressed in Ff . A similar constraint can be written for the left wing. The angular velocity of each wing is constrained to be identical to the angular velocity of the fuselage, that is ωrra = Crf ωff a and ωlla = Clf ωff a . In addition, there is a kinematic constraint on the DCM ia ˙ ia parameters: Ξia = 0 [20], where i (p )p T ia pia 0 0 1 p1 iaT 0 p 0 ia ia 2 . pia Ξia (p ) = , p = T T 2 i pia pia 0 ia 2 1 p3 × × −pia −pia −1 2 1
˙ and S = where M = diag{MBr , MBl , MBf }, ν = Sq, fa la diag{Sra , S , S }. Using (6), the first term of Lagrange’s r l f equation is found to be T d ∂LSa/a ˙ = ST Mν˙ + S˙ T Mν + ST Mν. (7) dt ∂ q˙
All of the constraints are written together as Ξq˙ = 0, where rf × f a 1 0 0 0 0 0 −1 CT r S fa f f 0 Sra 0 0 0 0 0 −Crf Sff a r × 0 lf fa 0 0 1 0 0 −1 CT r S fa f f fa , Ξ= la 0 0 0 0 Sl 0 0 −Clf Sf 0 Ξra 0 0 0 0 0 0 r la 0 0 0 0 Ξl 0 0 0 0 0 0 0 0 0 0 Ξra f h T i T T T ˙ ia . Additionally, q˙ = q˙ Br q˙ Bl q˙ Bf and ωiia = Sia i p h i T T T T the augmented velocities, ν = ν Br ν Bl ν Bf , can be related to a set of reduced augmented velocities, νˆ =
(6)
Premultiplying (5) by ΥT ΓT and knowing SΓ = 1 and ΞΓΥ = 0 gives ˙ ΥT Mν˙ + ΥT ΓT S˙ T Mν + ΥT Mν ∂LSa/a T − ΥT ΓT = ΥT ΓT f . (8) ∂q Notice the Lagrange multipliers have been removed. Applying the identities found in [20], (8) can be rewritten as ΥT Mν˙ + ΥT ΓT Kq + ΥT fnon = ΥT ΓT f , ˙ where fnon = ΩMν + × Mν + anon , Ω fa ra× la× diag{0, ωr , 0, 0, ωl , 0, 0, ωf }, mBr g13 mB vBr × Cra 13 + rcgr× Cra r˙ ra r a r 0 mBl g13 × × anon = , cgl mBl vBl Cla 13 + rl Cla r˙ la a 0 mBf g13 cgf × Bf × fa ˙ mBf v Cf a 13 + rf Cf a ra
(9) =
5 ×
ii vBi = σBi HBi q˙ ei − cB ωiia , i = r, l, and vBf i Bf f × f a ˙ νˆ into (8) gives ˆ˙ + Υ −cf ωf . Substituting ν˙ = Υν
¯ν ¯ + ¯fnon = f¯, ˆ˙ + Kq M
(10)
Υ Γ K, ¯fnon
¯ ¯ where = ΥT MΥ, K = M T T ¯ ˙ Υ fnon + MΥνˆ , f = Υ ΓT f .
=
T
T
=
TABLE I S UMMARY OF KITEPLANE DYNAMIC MODEL PRESENTED IN S ECTIONS III-A TO III-E. Property Rigid degrees of freedom Flexible degrees of freedom Aerodynamic model states Aerodynamic force locations
Value 6 (3 translational, 3 rotational) 18 (9 each wing) 30 10
E. Unsteady Aerodynamic Model The unsteady aerodynamic model presented in this paper is based on an approximation of Theodorsen’s aerodynamic model in state-space form [15], [21]. Theodorsen’s model extends quasi-steady thin airfoil theory by accounting for addedmass forces and the effect of wake vorticity. Theodorsen’s model can be represented as a transfer function or state-space model using one of the approximations found in [21]. The lift coefficient at a given point along the aerodynamic centerline of the wing can be found as the output of the state-space system [21] x˙ L CL (t) = y¯L
¯ Lu ¯L = AL (U∞ )xL + B ¯ L (U∞ )xL + D ¯ Lu ¯L , = C
C1 ,
where CD (t) = a + bCL2 (t), and the constants a and b are characteristics of the chosen airfoil. Note that when calculating the lift and drag forces applied to the wings, only a third of the wing’s surface area is used to find each of the three lift forces applied to each wing.
(11)
¯L = u
2
(12)
where U∞ is the magnitude of the free stream velocity of the airflow relative to the airfoil, ¯ L (U∞ ) = C2 CL (U∞ ), ¯ L = BL 0 , C B ¯ L = C2 DL D
T applied relative to point a, and 11 = 1 0 0 . Similarly, the drag force applied at a given point is
ia/a ia/a 2 1 fDi = − ρ S(a + bC (t))
vr ,
v r r L 2 air
αqc,e , α˙ mc,e
C1 is the added-mass coefficient of the airfoil, C2 is the quasisteady lift coefficient of the airfoil or the slope of the lift curve of the airfoil, αqc,e is the effective angle of attack at the quarter-chord position, αmc,e is the effective angle of attack at the mid-chord position, and (AL (U∞ ), BL , CL (U∞ ), DL ) is based on the chosen approximation to Theodorsen’s model. The chosen approximation is discussed in the Appendix. The effective angle of attack at the quarter-chord and midchord positions is found by calculating the angle between the velocity vector of the chosen point relative to the velocity of the surrounding air and the physical basis vector along the chord of the airfoil pointing towards its leading edge. Note that it is unlikely that a measurement of α˙ mc,e is available in practice, so a measurement of αmc,e is filtered to obtain α˙ mc,e . In order to implement this model, separate state-space systems are required for each point where the aerodynamic forces are to be computed. In this paper, ten state-space systems are required to account for the three application points on each wing, and single application points on the two horizontal stabilizers, the vertical stabilizer, and the fuselage. The lift force generated by the calculated lift coefficient is
ia/a × ia/a 1 fLi
11 vr , r = 2 ρair SCL (t) vr 2
where ρair is the density of the air, S is the surface area of the wing, CL (t) is the output of the unsteady aerodynamic ia/a model, vi is the velocity of the point where the lift force is
F. Simplified Dynamic Model A simplified planar dynamic model of the kiteplane is used in Section IV-B when designing an optimal trajectory for the kiteplane. This simplified model approximates the kiteplane as a particle with aerodynamic lift and drag forces, a gravitational force, and a tether force acting on it. A similar point-mass model is often used to solve optimal control problems involving kite systems [13], [22], [23]. The equations of motion of the model are D T g m¨rfa a = fL a + fa + fa + fa ,
(13)
T where m is the mass of the kiteplane, rfa a = 0 y z is the position of the kiteplane relative to point a expressed in Fa and constrained to planar motion,
f w/a T × f w/a 1 fL
Cf a 1 1 v f a = 2 ρair SCL,α α vf 2
is the lift force expressed in Fa ,
f w/a 2 2 f w/a 1 fD = − ρ S(a + bC α )
v
CT a L,α f a vf f 2 air 2
is the drag force expressed in Fa , fTa = −T rfa a / rfa a 2 is the tether force exerted on the kiteplane expressed in Fa , fga = T 0 0 −mg is the gravitational force expressed in Fa , ρair is the density of air, which is assumed to be constant, S is the surface area of the kiteplane, CL,α is the slope of the kiteplane’s lift curve, α is the kiteplane’s angle of attack, Cf a is the direction cosine matrix representing the attitude ofFf f w/a f w/a aw/a relative to Fa , vf = Cf a va = Cf a r˙ fa a + va is the velocity of the kiteplane relative to wind with respect to Fa aw/a expressed in Ff , va is the velocity of point a relative to the wind with respect to Fa expressed in Fa , and 0 ≤ T ≤ Tmax is the tension in the tether.
6
IV. C ONTROL F ORMULATION
z pos. (m)
500
400 -810
-790
-780
-750
0
0
where Kp = diag{kp,1 , kp,2 , kp,3 }, Kd = diag{kd,1 , kd,2 , kd,3 }, 0 < kp,i < ∞, i = 1, 2, 3, 0 < kd,i < ∞, i = 1, 2, 3, 0 < k < ∞, Cf d is the DCM representing the relative attitude between Ff and a desired reference frame, Fd , and (·)v is the uncross operator. Notice that the control law of (15) uses the DCM directly, rather than using an Euler angle or quaternion parametrization. This avoids the kinematic singularities associated with Euler angles and the unwinding of quaternions, leading to an attitude control law that is well-defined globally. The control law of (15) determines the control torques to be applied to the kiteplane, which then have to be applied by the kiteplane’s control surfaces. Although omitted in this paper, the pitch, roll, and yaw moments can be mapped to the appropriate control surface deflections by solving a control allocation problem [29]–[31].
500
20
40
0
20
Time (s) 600
T (N)
θ (deg)
40
Time (s)
50
(14)
Consider the PID control law of the form [28] v τ c = K p Cf d − CT − Kd ωff a fd Z t v fa Cf d − CT − kω dτ, (15) + Ki fd f
-750
400
-850
A. Attitude Control
-760
600
-800
It is assumed that the tether has significant longitudinal flexibility. This flexibility is modeled by a single lumped mass system, whose equations of motion are described by
0
400 200
-50
0 0
20
40
0
20
Time (s)
40
Time (s)
4000
P (W)
where qe is the flexible coordinate that describes the longitudinal extension of the tether, mt is the mass of the tether, ct is the longitudinal damping coefficient of the tether, and kt is the longitudinal stiffness coefficient of the tether. Although (14) is a relatively simple cable dynamic model, lumped mass models are widely used to approximate the continuous dynamics of cables [24]–[27].
The tether force exerted on the kiteplane is given by fTa c f 2 = ct q˙e + kt qe , where ct q˙e is the damping force in the tether and kt qe is the stiffness force in the tether. In this paper, it is assumed the control input of the tether is simply a force applied to it, T , which is equivalent to a torque applied to a massless winch. The control forces applied to the tether can be converted to control torques by dividing by the chosen winch radius.
-770
y pos. (m)
2
mt q¨e + ct q˙e + kt qe = T,
-800
z pos. (m)
rfa a
fTa c f = − fTa c f 2
f a .
ra
600
y pos. (m)
The control of the kiteplane is divided into two separate problems: attitude control and trajectory generation. The attitude control of the kiteplane ensures that the kiteplane tracks the attitude given by the desired trajectory. The desired trajectory dictates the desired pitch angle and the tension to be applied to the tether. It is assumed that the tether can only pull on the kite, since cables only operate in tension, and the tether is attached to a ground-based winch at the origin of Fa . The tether force acting on the fuselage is
2000 0 -2000 0
5
10
15
20
25
30
35
40
Time (s)
Fig. 2. Optimized trajectory plots of the kiteplane’s horizontal and vertical positions versus time, pitch angle versus time, tether tension versus time, instantaneous power generated versus time, and average instantaneous power generated (dashed line) versus time. The net energy harvested throughout the trajectory is 19.6 kJ (average of 490 W net instantaneous power over 40 seconds).
B. Optimal Trajectory An optimal trajectory is designed to maximize the net energy harvested by the kiteplane. Optimally-designed energy harvesting trajectories for kites are often three-dimensional in nature [11]–[13]. In order to simplify the optimization process, only planar trajectories are considered in this paper. Although this restriction naturally leads to less energy harvested, the approach presented in this section is relatively simple and does yield significant energy harvested. The optimization problem employs the simplified dynamic model of the kiteplane from Section III-F in first-order form, given by x3 , x4 x˙ = f(x, u) = 1 0 1 0 L D T g fa + fa + fa + fa m 0 0 1 T T where x = x1 x2 x3 x4 = y z y˙ z˙ . The objective is to minimize Z t1 (yy ˙ + zz)T ˙ p J1 (u) = − dt, (16) 2 y + z2 t0
7
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γz,j ln (−1) (z0 − z(t1 )) + zoff
. (18)
The objective function Jc (u) is composed of logarithmic barrier functions that guarantee |y(t1 ) − y0 | ≤ yoff and |z(t1 ) − z0 | ≤ zoff are satisfied at every iteration of the numerical optimization solver [32]. The optimization problem is posed as the minimization of J2 (u) subject to −θmin ≤ u1 ≤ θmax , 0 N ≤ u2 ≤ Tmax , −θoff ≤ θ(t1 ) − θ(t0 ) ≤ θoff , −Toff ≤ T (t1 ) − T (t0 ) ≤ Toff , and −αmax ≤ α ≤ αmax . For this numerical problem, Tmax = 500 N, θmin = −40◦ , θmax = 40◦ , αmax = 20◦ , y0 = −800 m, z0 = 400 m, y˙ 0 = z˙0 = 0 m/s, yoff = 5 m, zoff = 10 m, θoff = 5.7◦ , Toff = 10 N, γy,1 = γy,2 = 1 J, γz,1 = γz,2 = 1 J, and t1 = 40 s. Wind in the negative y-axis is modeled as 6.15(z)1/7 m/s [33]. The finite-dimensional constrained optimization problem is solved in matlab using fmincon with the control input parameterized as a piecewise linear function divided into forty equally spaced segments. The initial guesses used with fmincon are u1,k = θeq and u2,k = Teq , k = 1, . . . , 40, where θeq and Teq are respectively the pitch angle and tether tension that maintain the position of the kiteplane at y = y0 and z = z0 . In this numerical example, θeq = 0.65◦ and Teq = 61 N. This choice of initial control inputs ensures that the initial evaluation of J2 (u) is finite, since the terminal constraints embedded in the barrier functions are satisfied. Since an interior-point method is used, at each subsequent iteration of the numerical optimizer the terminal constraints remain satisfied, while the net energy harvested throughout the trajectory is increased. The optimal
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where
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Fig. 3. Kiteplane in simulation with exaggerated wing deflections.
φ (deg)
J2 (u) = J1 (u) + Jc (u),
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T T where u = u1 u2 = θ T is the control input, and −J1 (u) is the net energy harvested throughout the trajectory. The control input is subject to the constraints 0 N ≤ T ≤ Tmax and θmin ≤ θ ≤ θmax . The angle of attack of the kiteplane is also constrained as |α| ≤ αmax . The initial conditions T are given as x(0) = y0 z0 y˙ 0 z˙0 and the terminal constraints are |y(t1 ) − y0 | ≤ yoff and |z(t1 ) − z0 | ≤ zoff , where yoff and zoff are the maximum allowable terminal position offsets in the y and z axis, respectively. The terminal constraint forces the kiteplane to complete an approximately closed trajectory that begins and ends approximately at the same position. Additional constraints on the control inputs is made to ensure that the initial and final inputs are approximately the same, which allows for truly cyclical trajectories. These constraints are given by |θ(t1 ) − θ(t0 )| ≤ θoff and |T (t1 ) − T (t0 )| ≤ Toff , where θoff and Toff are the maximum allowable offsets in initial and final inputs. The terminal time t1 is prescribed as well. In order to use an interior-point method, the objective function of (16) is augmented as
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Fig. 4. Results with the optimal trajectory from Section V-B: Plots of fa fa the kiteplane’s horizontal and vertical positions (ra2 and ra3 ) versus time, leading-edge wingtip deflection, w(c, l), versus time, Euler angles extracted from Cf a versus time, wind speed in each axis versus time, and angle of attack versus time.
control inputs, the corresponding kiteplane trajectory and the instantaneous power generated are shown in Fig. 2. Notice that the control input constraints are satisfied and the constraint on the terminal position is satisfied to within the desired limits, while 19.6 kJ net energy is harvested (average of 490 W net instantaneous power over 40 seconds). V. N UMERICAL R ESULTS Numerical simulations of the kiteplane model developed in Section III with the attitude controller proposed in Section IV-A and the optimal trajectory from IV-B is performed to demonstrate the kiteplane’s ability to harvest energy. A. Simulation Parameters The main properties of the kiteplane include a wingspan of 6.5 m, an overall length of 4 m, and a mass of 1 kg. Each wing has a chord length of 0.75 m, surface area of 2.25 m2 ,
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Fig. 5. Results with the optimal trajectory from Section V-B: Plots of the net instantaneous power, tether tension, and reel-in rate versus time. An average net instantaneous power of 447 W is generated throughout the trajectory.
an aspect ratio of 8.33, thickness of 0.02 m, a modulus of elasticity of 100 MPa, a lift curve slope of 6, an added-mass coefficient of 0.1, and drag coefficients of a = 0.1 and b = 0.00406. The chosen modulus of elasticity corresponds to an extremely flexible wing, which is comparable to the inflatable wings of the kiteplane used in [6]. The fuselage has a length of 4 m, a radius of 0.25 m, a lift curve slope of 2π, an addedmass coefficient of 0.1, drag coefficients of a = 0.1 and b = 0.00406, and a sideslip drag coefficient of 1.17. The horizontal stabilizer has a surface area of 0.45 m2 , a lift curve slope of 6, an added-mass coefficient of 0.1, and drag coefficients of a = 0.1 and b = 0.00406. The vertical stabilizer has a surface area of 0.225 m2 , a lift curve slope of 6, an added-mass coefficient of 0.1, and drag coefficients of a = 0.1 and b = 0.00406. The initial conditions are (θ(0), φ(0), ψ(0)) = (0◦ , 0◦ , 0◦ ), T qer (0) = qel (0) = 0, rfa a (0) = [0 − 800 400] m. The instantaneous power generated is found by calculating the net power generated by the tether and subtracting the power used by the attitude control system: Pnet = Ptether + Pattitude =
˙ τ T ω f a , where rθ˙ is the reel-in rate multiplied − fTa c f 2 rθ− c f by the radius of the winch, or equivalently the linear velocity of the tether. An animation of the simulated kiteplane is shown in Fig. 3. pa/a In simulation the wind is modeled as va = wpa a + 1/7 pa/a T pa 6.15 13 ra 12 m/s [33], where va is the wind velocity relative to a at a given point p resolved in Fa , rpa a is the posiT tion vector of p relative to a resolved in Fa , 12 = [0 1 0] , pa pa T pa pa 2 pa wa = wa,1 wa,2 wa,3 , and wa,i ∼ N (0, σ ), i = 1, 2, 3, are Gaussian white noise wind gusts with standard deviation σ = 1 m/s and characteristic length l = 1 s [34]. B. Optimal Trajectory Results The optimal trajectory from Section IV-B is implemented in simulation. The control gains used for the attitude control law include k = 0.05 s, kp,j = 40, 000 N·m, kd,j = 200 N·m·s, and ki,j = 4, 000 N·m/s, j = 1, 2, 3. Figure 4 includes plots
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Fig. 6. Results with the optimal trajectory from Section V-B: Plot of the kiteplane’s vertical position versus its horizontal position.
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Fig. 7. Results with three cycles of the optimal trajectory from Section V-B: fa fa Plots of the kiteplane’s horizontal and vertical positions (ra2 and ra3 ) versus time, leading-edge wingtip deflection, w(c, l), versus time, Euler angles extracted from Cf a versus time, wind speed in each axis versus time, and angle of attack versus time.
of the kiteplane’s roll, pitch, and yaw angles versus time as extracted from the DCM Cf a , the kiteplane’s horizontal and vertical positions versus time, the deflection of the right wing’s leading-edge wingtip versus time, and the kiteplane’s angle of attack. Figure 5 includes plots of the instantaneous power generated, tether tension, and reel-in rate versus time. A plot of the kiteplane’s trajectory in the horizontal-vertical plane is provided in Fig. 6. A simulation is also performed with three cycles of the
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Fig. 8. Results with three cycles of the optimal trajectory from Section V-B: Plots of the net instantaneous power, tether tension, and reel-in rate versus time. An average net instantaneous power of 490 W is generated throughout the trajectory.
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Fig. 9. Results with three cycles of the optimal trajectory from Section V-B: Plot of the kiteplane’s vertical position versus its horizontal position, where each cycle is denoted by a different color and the start and end of each cycle is denoted by a marker.
optimal trajectory, in order to demonstrate the cyclical nature of the trajectory. Figure 7 includes plots of the kiteplane’s roll, pitch, and yaw angles versus time as extracted from the DCM Cf a , the kiteplane’s horizontal and vertical positions versus time, the deflection of the right wing’s leading-edge wingtip versus time, and the kiteplane’s angle of attack. Figure 8 includes plots of the instantaneous power generated, tether tension, and reel-in rate versus time. A plot of the kiteplane’s trajectory in the horizontal-vertical plane is provided in Fig. 9 and a plot of the kiteplane’s three-dimensional trajectory is shown in Fig. 10. C. Discussion As shown in Fig. 5, the optimized trajectory yields an average net instantaneous power of 447 W over the 40 second
fa ra1
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Fig. 10. Results with three cycles of the optimal trajectory from Section V-B: Plot of the kiteplane’s three-dimensional position, where each cycle is denoted by a different color and the start and end of each cycle is denoted by a marker.
cycle, or 17.88 kJ of net energy harvested. If the kiteplane repeatedly flew this trajectory for 24 hours, the energy harvested would be approximately 10.7 kW·hr or 38.6 MJ. This is enough energy to cover one third the daily electricity consumption of an average American residential dwelling (30 kW·hr) [35]. It is quite reasonable to assume that a larger kiteplane with less restrictive bounds on pitch angle and tether tension would lead to increased energy-harvesting capabilities. These modifications, along with a setup that features many wind energy-harvesting kiteplanes, could lead to a scenario where the energy harvested by kiteplanes powers an entire community. Simulation results with three cycles of the optimal trajectory, shown in Fig. 8, yield slightly more energy harvested, with an average net instantaneous power of 490 W over the 120 seconds, or 19.6 kJ of net energy harvested per cycle. The results of the three cycle simulation show that the kiteplane’s trajectory does approximately return to its starting location, even in the presence of significant wind gusts in three dimensions, while harvesting roughly the same amount of energy per cycle as the single cycle simulation results. It is important to note that due to wind gusts the kiteplane is not expected to return to the exact same starting position, as illustrated in simulation, which is why a hard constraint that would enforce this is not included in the trajectory optimization of Section IV-B. VI. C ONCLUSION A dynamic model of a kiteplane with flexible wings has been developed that uses the DCM directly to describe the attitude of the kiteplane and includes an unsteady aerodynamic model based on Theodorsen’s lift model. An optimal trajectory was generated to maximize the net energy harvested by the kiteplane using a simplified dynamic model of the kiteplane. The optimal trajectory simulation results with the complete three-dimensional, unsteady aerodynamic model demonstrated
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the substantial energy-harvesting capabilities of kiteplanes as a source of renewable energy. The optimal trajectory was also shown to maintain its energy harvesting and cyclical properties over multiple trajectory cycles, which demonstrates the feasibility of deploying a kiteplane for energy harvesting over long periods of time. Future work will include the development of threedimensional optimal trajectories, which could potentially allow for a substantial increase in the energy harvested by the kiteplane. A PPENDIX In this appendix, the linear system used to approximate Theodorsen’s lift model [15] is presented. Note that v = U∞ /c and b = c/2, where U∞ is the free-stream velocity and c is the chord length of the airfoil. The chosen approximation is [21], [36] −0.3455v −0.01365v 2 1 AL (v) = , BL = , 1 0 0 CL (v) = 0.1081b2 v 0.006825b2 v 2 , DL = 0.5. Higher-order approximations to Theodorsen’s lift model can be found in [21]. R EFERENCES [1] L. Fagiano and M. Milanese, “Airborne wind energy: an overview,” in American Control Conference, Montreal, Canada, June 2012, pp. 3132– 3143. [2] M. Diehl, “Airborne wind energy: Basic concepts and physical foundations,” in Airborne Wind Energy. Berlin Heidelberg: Springer-Verlag, 2014, pp. 3–22. [3] C. Vermillion, T. Grunnagle, R. Lim, and I. Kolmanovsky, “Model-based plant design and hierarchical control of a prototype lighter-than-air wind energy system, with experimental flight test results,” IEEE Transactions on Control Systems Technology, vol. 22, no. 2, pp. 531–542, March 2014. [4] M. Canale, L. Fagiano, and M. Milanese, “High altitude wind energy generation using controlled power kites,” IEEE Transactions on Control Systems Technology, vol. 18, no. 2, pp. 279–293, March 2010. [5] L. Fagiano, M. Milanese, and D. Piga, “High-altitude wind power generation,” IEEE Transactions on Energy Conversion, vol. 25, no. 1, pp. 168–180, March 2010. [6] E. J. Terink, J. Breukels, R. Chmehl, and W. J. Ockels, “Flight dynamics and stability of a tethered inflatable kiteplane,” Journal of Aircraft, vol. 48, no. 2, pp. 503–513, March–April 2011. [7] I. Tuzcu, P. Marzocca, E. Cestino, G. Romeo, and G. Frulla, “Stability and control of a high-altitude, long endurance UAV,” Journal of Guidance, Control, and Dynamics, vol. 30, no. 3, pp. 713–721, May–June 2007. [8] M. J. Patil, D. H. Hodges, and C. E. S. Cesnik, “Nonlinear aeroelasticity and flight dynamics of high-altitude long-endurance aircraft,” in Structures, Structural Dyanmics, and Materials Conference, April 1999. [9] A. Bosch, R. Schmehl, P. Tiso, and D. Rixen, “Dynamic nonlinear aeroelastic model of a kite for power generation,” Journal of Guidance, Control, and Dynamics, vol. 37, no. 5, pp. 1426–1436, September– October 2014. [10] R. J. Caverly and J. R. Forbes, “Flexible kiteplane modeling and control with an unsteady aerodynamic model,” in American Control Conference, Boston, MA, July 2016, pp. 4972–4977. [11] M. Milanese, L. Fagiano, and D. Piga, “Control as a key technology for a radical innovation in wind energy generation,” in American Control Conference, Baltimore, MD, 2010, pp. 2361–2377. [12] A. Ilzhofer, B. Houska, and M. Diehl, “Nonlinear MPC of kites under varying wind conditions for a new class of large-scale wind power generators,” International Journal of Robust and Nonlinear Control, vol. 17, no. 17, pp. 1590–1599, 2007.
[13] L. Fagiano, M. Milanese, and D. Piga, “Optimization of airborne wind energy generators,” International Journal of Robust and Nonlinear Control, vol. 22, no. 18, pp. 2055–2083, 2012. [14] J. Gillis, J. Goos, K. Geebelen, J. Swevers, and M. Diehl, “Optimal periodic control of power harvesting tethered airplanes: How to fly fast without wind and without propellor?” in American Control Conference, Montreal, Canada, June 2012, pp. 2527–2532. [15] T. Theodorsen, “General theory of aerodynamic instability and the mechanism of flutter,” NACA, Tech. Rep. 496, 1935. [16] N. A. Chaturvedi, A. K. Sanyal, and N. H. McClamroch, “Rigid-body attitude control,” IEEE Control Systems Magazine, vol. 31, no. 3, pp. 30–51, 2011. [17] J. Garcia de Jalón and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems. New York: Springer-Verlag, 1994. [18] A. W. Leissa and M. S. Qatu, Vibrations of Continuous Systems. New York, New York: McGraw-Hill, Inc., 2011. [19] L. Meirovitch and M. K. Kwak, “Convergence of the classical RayleighRitz method and the finite element method,” AIAA Journal, vol. 28, no. 8, pp. 1509–1516, 1990. [20] A. H. J. de Ruiter and J. R. Forbes, “General identities for parameterizations of SO(3) with applications,” Journal of Applied Mechanics, vol. 81, no. 7, p. 071007, 2014. [21] S. L. Brunton and C. W. Rowley, “Empirical state-space representations for Theodorsen’s lift model,” Journal of Fluids and Structures, vol. 38, pp. 174–186, 2013. [22] P. Williams, B. Lansdorp, and W. J. Ockels, “Optimal crosswind towing and power generation with tethered kites,” Journal of Guidance, Control, and Dynamics, vol. 31, no. 1, pp. 81–93, January–February 2008. [23] B. Houska and M. Diehl, “Optimal control of towing kites,” in IEEE Conference on Decision and Control, San Diego, CA, December 2006, pp. 2693–2697. [24] F. R. Driscoll, R. G. Lueck, and M. Nahon, “Development and validation of a lumped-mass dynamics model of a deep-sea ROV system,” Applied Ocean Research, vol. 22, pp. 169–182, 2000. [25] M. Nahon, G. Gilardi, and C. Lambert, “Dynamics/control of a radio telescope receiver supported by a tethered aerostat,” Journal of Guidance, Control, and Dynamics, vol. 25, no. 6, pp. 1107–1115, Nov.–Dec. 2002. [26] T. Walton and H. Polacheck, “Calculation of transient motion of submerged cables,” Mathematics of Computation, vol. 14, pp. 27–46, 1960. [27] O. Nagatomi, M. Nakamura, and W. Koterayama, “Dynamic simulation and field experiment of submarine cable during laying and recovery,” in Proceedings of the International Offshore and Polar Engineering Conference, Kitakyushu, Japan, May 26–31 2002, pp. 255–262. [28] F. Goodarzi, D. Lee, and T. Lee, “Geometric nonlinear PID control of a quadrotor UAV on SE(3),” in European Control Conference, 2013, pp. 3845–3850. [29] W. C. Durham, “Constrained control allocation: Three-moment problem,” Journal of Guidance, Control, and Dynamics, vol. 17, no. 2, pp. 330–336, March–April 1994. [30] O. Härkegård, “Dynamic control allocation using constrained quadratic programming,” Journal of Guidance, Control, and Dynamics, vol. 27, no. 6, pp. 1028–1034, November–December 2004. [31] M. Bodson, “Evaluation of optimization methods for control allocation,” Journal of Guidance, Control, and Dynamics, vol. 25, no. 4, pp. 703– 711, July–August 2002. [32] F. A. Potra and S. J. Wright, “Interior-point methods,” Journal of Computational and Applied Mathematics, vol. 124, no. 1–2, pp. 281– 302, 2000. [33] M. W. Isaacs, J. B. Hoagg, I. I. Hussein, and D. Olinger, “Retrospective cost adaptive control for a ground tethered energy system,” in European Control Conference, 2011, pp. 824–829. [34] C. E. Rasmussen and C. K. Williams, Gaussian Processes for Machine Learning. MIT Press, 2006. [35] U.S. Energy Information Administration. Residential energy consumption survey (RECS). [Online]. Available: http://www.eia.gov/consumption/residential/data/2009/ [36] R. T. Jones, “Operational treatment of the non-uniform lift theory in airplane dynamics,” NACA, Tech. Rep. 629, 1938.
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Ryan James Caverly (S’13) received the B.Eng. degree in mechanical engineering (Hons.) from McGill University, Montreal, QC, Canada, in 2013, and the M.Sc.Eng. degree in aerospace engineering from the University of Michigan, Ann Arbor, MI, USA, in 2015. He is currently pursuing the Ph.D. degree in aerospace engineering at the University of Michigan, Ann Arbor, MI, USA. His research interests include dynamics and control systems, with a focus on robotic and aerospace applications, as well as robust control techniques.
James Richard Forbes (M’11) received the B.A.Sc. degree in mechanical engineering (Hons., Co-op) from the University of Waterloo, Waterloo, ON, Canada in 2006, and the M.A.Sc. and Ph.D. degrees in aerospace science and engineering from the University of Toronto Institute for Aerospace Studies (UTIAS), Toronto, ON, Canada, in 2008 and 2011, respectively. James is currently Assistant Professor of Mechanical Engineering at McGill University, Montreal, QC, Canada. James is also a member of the Centre for Intelligent Machines. The focus of James’ research is dynamics, estimation, and control of robotic and aerospace systems. He is the co-author of Spacecraft Dynamics and Control: An Introduction published by Wiley in 2013.
