Forwarding control of scale model autonomous helicopter - CiteSeerX

Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003

ThM11-5

Forwarding control of scale model autonomous helicopter: A Lyapunov control design Fr´ed´eric Mazenc INRIA Lorraine, Projet CONGE, ISGMP Bat A Ile du Saulcy 57 045 Metz Cedex 01, France

Robert E. Mahony Rogelio Lozano Department of Engineering, Heudiasyc- UTC UMR 6599 Australian National University, Centre de Recherche de Royallieu, ACT, 0200, BP 20529, 60205 Compiegne Cedex, Australia. France.

Abstract— A simple model of the dynamics of a scale model autonomous helicopter is considered. A stabilising control law is designed which guarantees the control inputs remain bounded by pre-specified control actuator limits. The approach taken exploits a combination of three non-linear control design techniques: Firstly, a passivity design technique is used to achieve practical stabilisation of the non-linear attitude dynamics of the system. Secondly, the remaining parasitic attitude dynamics are input/output linearised. Finally, forwarding techniques are used to generate a control Lyapunov function and associated control input that robustly globally asymptotically and locally exponentially stabilises the full system.

I. I NTRODUCTION Reduced scale autonomous helicopters pose a number of interesting and unanswered problems in robotics and control theory. Over the last twenty years there has been considerable work in the area of autonomous helicopter control [9]. However, much of the work done has used classical linear control theory. More recently a number of authors have considered applying non-linear control design techniques to the control of scale model autonomous helicopters (cf. conference papers [4], [12], [1] and more recently the journal papers [11], [13]). The high actuation to inertia ratios and highly non-linear nature of the attitude dynamics exploited in desired flight conditions for scale model autonomous helicopters make integrated non-linear control design highly desirable. The work presented here draws heavily on early investigation into the control of a Vertical Takeoff and Landing Jet (VTOL) [3], [15], [5], [8] and [10, pg. 246.]. In particular, we utilize the technique of forwarding [15], [7], [10], [6]. Teel [15] used a simplified model of the VTOL as an early example of applying forwarding techniques to stabilization of a system in the presence of small feedforward terms. The forwarding approach is useful when saturation on the actuators is present and provides a natural time-scale separation between the control of height and position of the closed-loop model of a helicopter. In this paper, we consider an idealised model of the helicopter based on the standard model considered in the literature [12], [11], [1]. Initially, a highly robust non-linear controller based on passivity techniques and incorporating saturation limits is designed to achieve practical stabilisation of the attitude and altitude dynamics of the helicopter. Stabilizing the altitude (or height) separately from the the

0-7803-7924-1/03/$17.00 ©2003 IEEE

lateral position of the helicopter is an approach that was first proposed by Hauser et al. [3] and has been used in several key papers on helicopter and VTOL control [3], [15]). This approach has the advantage that high gain precise control of the height of the helicopter is obtained independently of the slower coupled lateral position/attitude dynamics of the closed-loop, a property that is extremely important for manoeuvres undertaken close to the ground. In the second stage of the control design the remaining position coordinates stabilized without disturbing the stability properties of the initial two stages of control design. This is achieved by input/output linearising the parasitic attitude dynamics of the helicopter and applying a forwarding Lyapunov control design based on the approach developed in Mazenc et al. [7]. Since the first stage passivity based control ensures practical stability of the key attitude dynamics the input/output linearisation transformation is always well defined in the region of interest. The forwarding procedure makes extensive use of coordinate changes and derives explicit formulas for the Lyapunov function associated with the stability of the closed system. This provides a guaranteed robustness property that is not always present where cross terms (see [10, Section 5.2]) or exact change of coordinates (see [7, Section IV]) are utilised. The full control design proposed is discontinuous due to switching condition between the global passivity based control and the local linearisation/forwarding design. Intuitively, it is simple matter to extend the proposed design to be globally smooth, however, this would considerably complicate the technical details of the paper and does not improve the performance or the theoretical interest of the proposed design. The final control design proposed is a robust Lyapunov control that globally exponentially stabilises the full system. II. H ELICOPTER M ODEL In this section, a Lagrangian model is derived for an autonomous model helicopter in generalized coordinates. The model proposed is valid for the case where the non-linear aerodynamic effects of the main rotor are compensated by a non-linear transformation of the cyclic pitch control. Consider Figure 1. Let I = {Ex , Ey , Ez } denote a stationary right-hand inertial frame. The position of the centre of the mass of the helicopter A is denoted ξ = (x, y, z). Let

3960

A = {E1a , E2a , E3a } denote (right-hand) body fixed frame for the helicopter. The orientation of the helicopter is given by a rotation R : A → I. E3

E

Ez

b

b

E3

a

E

b

Γ = rw1 E1a + rw2 E2a + lw3 E3a

d

B

b

d

C

d

E2

h

A

c

It is convenient to work with a generalised coordinate representation of the helicopter dynamics. We use the classical ‘yaw’, ‘pitch’ and ‘roll’ Euler angles η = (φ, θ, ψ) commonly used in aerodynamic applications [2, pg. 608]. The rotation matrix R representing the orientation of the helicopter is   cθ cφ sψ sθ cφ − c ψ sφ cψ sθ cφ + s ψ sφ R :=  cθ sφ sψ sθ sφ + cψ cφ cψ sθ sφ − sψ cφ  . −sθ sψ cθ cψ cθ

1

2

lb

a

la

d

a

E

a

1

l E

c

y

0 Ex

Fig. 1. Diagram showing some of the notation used in the model of the Helicopter.

Control authority for a helicopter is obtained via lift generated by the main and tail rotors. The vector lift generated by a rotor disk lies along its axis of rotation. The tail rotor orientation is fixed in direction E2a , however, the main rotor blades are flexible and the main rotor disk tilts with the application of cyclic pitch and is subject to perturbations due to mechanical and aerodynamic effects. Following the lead of recent work [11], [13], [1] we propose an idealised dynamic model of a scale helicopter, that nevertheless incorporates the important non-linearities of the system. The model proposed uses the inclination of the main rotor disk, along with the magnitude of the thrust generated in the main and tail rotors, as the control actuation. The inclination of the rotor disk is fully actuated by the cyclic pitch inputs and its dynamics are an order of magnitude faster than the dynamics of the airframe [9]. In principal, a feedforward control with high gain feedback from cyclic pitch to rotor inclination and with reference input given by the desired inclination (specified by the control design in §-III IV) will eliminate the aerodynamic flapping effects of the main rotor from the modelled system dynamics. In practice, the helicopter must be provided with a sufficiently accurate sensor system in order that this input transformation can be undertaken. To formalise the control inputs used in the design procedure the main rotor lift is decomposed into three components (−w2 , w1 , −u) ∈ A in the body-fixed frame. The tail rotor force is written (0, −w 3 , 0) ∈ A. The choices of notation are made to identify one control, u, with the principal lift force (orientated in the negative E3a direction) of the main rotor and three torque controls w = (w 1 , w2 , w3 ) (the first two components coming from lateral components of the main rotor lift due to inclination of the main rotor disk) acting with positive sign around the three axes {E1a , E2a , E3a } on the rigid body dynamics of the helicopter airframe. Thus, the linear force applied to the helicopter is F =

−uE3a

−w

2

E1a

1

+ (w − w

centre of mass of the helicopter and l > 0 denote the tail rotor hub offset. It is easily verified that the torque applied to the helicopter is

3

)E2a .

The kinetic energy of the helicopter is m ˙ ˙ 1 T := hξ, ξi + hΩ, IΩi 2 2 where m denotes the mass of the airframe, I denotes the inertia of the airframe and Ω its angular velocity. Transforming the angular velocity into the Euler coordinates one has [2, pg. 609]    φ˙ −sθ 0 1 Ω =  cθ sψ cψ 0   θ˙  = W η˙ cθ cψ −sψ 0 ψ˙

Set I := I(η) = W T IW . One has m ˙ ˙ 1 T = hξ, ξi + hη, ˙ Iηi. ˙ 2 2 The only potential energy present is the gravitational potential given by U = mgz. The full Lagrangian L in terms of coordinates (ξ, η) is 1 1 L = mξ˙T ξ˙ + η˙ T Iη˙ − mgz. (1) 2 2 Torque applied to the helicopter transforms into generalised forces on the Euler coordinates (φ, θ, ψ) via     rw1 τφ (2) τ :=  τθ  = W −1  rw2  . lw3 τψ

The full helicopter dynamics in the generalized coordinates (ξ, η) are derived from the Euler-Lagrange equations [2]     −w2 0 mξ¨ = R  w1 − w3  +  0  (3) −u mg I¨ η = −C(η, η) ˙ η˙ + τ (4) where C(η, η) ˙ η˙ is the Coriolis matrix. Assume that the the principal axis of the inertia are aligned with the body fixed frame [9, pg. 557]. The inertia matrix I may be written as a diagonal matrix I = diag(I1 , I2 , I3 ).

I(η) := W T I h W =  I (1 + c2 ) + I c2 (c2 1

Let r > 0 denote the offset of the main rotor hub from the

3961

θ

2 2 2 θ ψ − 1) − I3 cθ cψ (I2 − I3 )cθ sψ cψ −I1 sθ

(I2 − I3 )cθ sψ cψ I2 (1 + s2ψ ) − I3 s2ψ 0

−I1 sθ 0 I1



Computing the Coriolis matrix [2] yields1 (I1 − I2 ) (I2 − I3 ) I 1 cθ C1 + C2 + C3 2 2 2 In common with all Lagrangian equations [2] the matrices I and C(η, η) ˙ satisfy  T   d d I − 2C(η, η) ˙ I − 2C(η, η) ˙ . =− dt dt C(η, η) ˙ =

Since the orthogonal components of the main rotor lift are of a much smaller magnitude than the fundamental lift u we write     −cψ sθ cφ − sψ sφ 0 mξ¨ = u  −cψ sθ sφ + sψ cφ  +  0  (5) −cψ cθ mg     cθ cφ (sψ sθ cφ − cψ sφ ) −w2   cθ sφ (sψ sθ sφ + cψ cφ ) + w1 − w 3 −sθ sψ cθ I¨ η = −C(η, η) ˙ η˙ + τ (6) where in Eq. 5, the final term represents the small body force perturbations. Writing  = r −1 and assuming l >> r then one may write   cθ cφ sψ sθ cφ − c ψ sφ   −w 2 ≤ |τ |.  c θ s φ s ψ s θ s φ + c ψ c φ  w − w 1 3 −sθ sψ cθ III. G LOBAL PASSIVITY BASED C ONTROL

The initial control task is to stabilise the attitude dynamics Eq. 6 using the control input τ ∈ mg otherwise the helicopter would be physically unable to hover. Set   mg u = satumax − z˙ (7) cψ cθ 1

Where the terms in the Coriolis matrix are given by 

C1 := 

0 ˙ ψ −θ˙

˙ −ψ 0 ˙ −φ

C := 3 2 ˙ ˙ −θs 2ψ cψ − ψ   −c2 θ s2ψ   φs 2 ˙  ˙ 2θ cψ + ψc θ c2ψ ˙ 2s ˙ φc θ 2ψ − θcθ c2ψ

−θ˙ ˙ φ 0



,



C2 := 

˙ −θs 2ψ ˙ φs 2θ 0

2 ˙ ˙ −φs 2θ cψ − θs2ψ sθ ˙ 2s −ψc θ 2ψ ˙ ˙ φs 2ψ sθ + ψs2ψ ˙ −φc θ c2ψ

 0 0 , 0

˙ −φs 2θ 0 0

˙ ˙ 2s −φc θ 2ψ + θcθ c2ψ ˙ φc θ c2ψ 0

     

where ‘sat’ denotes the saturation function  r for |r| ≤ λ satλ (r) = sgn(r)λ for |r| > λ The purpose of the control u is to assign the approximate dynamics z¨ = −z˙ to the z-dynamics of the Eq. 5 (and hence to damp the vertical velocity of the helicopter) without exceeding the actuator limits of the system. In the case considered we may assume that the initial orientation of the helicopter is not upside down, that is |ψ|, |θ| < π/2, and the control u is always positive! Similar to u there are maximum limits on the torque which can be provided by τ determined by maximum allowable deformation of the rotor blades and maximum thrust generated by the main and tail rotors. To simplify the analysis a single bound is used τmax . Consider the following Lyapunov function for the attitude dynamics of the helicopter Vη =

ηT η k0 1 T p η˙ Iη˙ + 2 2 ηT η + 1

(8)

where k0 > 0 is a positive constant. By exploiting the passivity properties of the system it is straightforward to verify that " #! T η 1 (η η)η T V˙ η = η˙ τ + k0 p . (9) − 3 η T η + 1 2 (η T η + 1) 2

Due to the specific form chosen for the Lyapunov function the term in the square brackets in Eq. 9 is bounded by the it follows that one constant 3/2. Thus, choosing k0 = τmax 3 may choose the control input τ to be # " η η˙ τmax 1 (η T η)η p τ = −sat( τmax ) (|η|) − ˙ . − 3 2 |η| ˙ 3 η T η + 1 2 (η T η + 1) 2 (10) It may be directly verified that |τ | ≤ τmax . Substituting back into the expression for V˙ η yields ˙ V˙ η = −|η|sat ˙ ( τmax ) (|η|). 2

(11)

Lemma 3.1: Consider the closed loop system given by Eqn’s 5 and 6 along with control inputs (u, τ ) given by Eq. 7 and Eq. 10. Then |u| ≤ umax ,

|τ | ≤ τmax

and η → 0,

z˙ → 0,

x ¨, y¨ → 0,

as t → ∞. Proof: Boundedness of the control inputs follows from the above discussion. Since Vη is radially unbounded in η and η˙ and decreasing (Eq. 11) bounded solutions to Eq. 6 exist for all time. Eq. 5 is a second order linear system driven by bounded inputs and consequently the solutions of ξ(t) exist on any finite time interval. Applying Lyapunov’s direct stability theorem ensures that η˙ converges to zero. Applying LaSalle theorem to the closed loop system it follows directly

3962

that η → 0 and that z˙ → 0. Now considering Eq. 5 again it is clear that x ¨, y¨ → 0.  In practice, any residual velocity x, ˙ y˙ in the translational dynamics of the helicopter will be damped by air resistance. Lemma 3.1 ensures that in the initial phase of the control algorithm the closed-loop system converges to a stable configuration. Let δ > 0 be a small positive constant which is used as a switching parameter. The first phase of the control algorithm is considered complete when |η| + |η| ˙ + |z| ˙ ≤ δ.

τ bθ = −ω − 8tθ + +

8x2 − 17ω 8x2 − 9θ − 17ω + q q 64 + (8x2 − 17ω)2 6 1 + (8x2 − 9θ − 17ω)2

x1 + 32x2 − 49 θ − 4ω 8 r q  2 , 3 1 + (8x2 − 9θ − 17ω)2 1 + x1 + 32x2 − 49 θ − 4ω 8

τ bψ = −λ − 8tψ − −

4

4

8y2 − 8x2 + 9ψ + 17λ 8y2 − 8x2 + 17λ − q q 64 + (8y2 − 8x2 + 17λ)2 6 1 + (8y2 − 8x2 + 9ψ + 17λ)2

y1 − x1 + 32y2 − 32x2 + 49 ψ + 4λ 8 r q 2 ,  3 1 + (8y2 − 8x2 + 9ψ + 17λ)2 1 + y1 − x1 + 32y2 − 32x2 + 49 ψ + 4λ 8

φ γ τ bφ = − q − q , 1 + φ2 1 + γ2

(15)

is asymptotically stable. The set

At this point the second and third phases of the control design are undertaken.

˙ |θ|, ˙ |ψ| ˙ < 1} ˙ η, η) {(ξ, ξ, ˙ |φ|, |θ|, |ψ| < π/3, |φ|,

IV. L OCAL L INEARISING AND F ORWARDING C ONTROL In this section a forwarding control design for global asymptotic stabilisation of the system is derived. In the following analysis we assume that the attitude dynamics lie in the domain specified in Section III. Following the practical stabilisation of the attitude dynamics proposed in Section III a switching control is proposed. Assume that the three components of η are norm less than π3 and that their corresponding velocities are norm smaller than one. Within this domain we switch to a control that explicitly linearises the attitude and z-dynamics of the system. The resulting linearised dynamics are taken as input dynamics to the remaining non-linear part of the system. A forwarding design ensures that the attitude dynamics remain in a prespecified neighbourhood of the origin and consequently that the linearising transformation remains well defined. Moreover, the forwarding procedure is chosen to ensure that the control derived satisfies the saturation constraints of the system. Due to space constraints the control is derived for the approximate model of the helicopter dynamics where the small body forces are not modelled. It is a straightforward, although technically cumbersome, process to extend the following development to the full non-linear model Eqn’s 5 & 6. Consider the explicit form of the system Eqn’s 5 & 6 in the absence of small body forces  x˙ 1 = x2       x˙ = −u(cψ sθ cφ + sψ sφ )   2      y˙ 1 = y2 (12) y˙ 2 = u(−cψ sθ sφ + sψ cφ )     z˙1 = z2      z˙2 = −ucψ cθ + 1     I¨ η = −C(η, η) ˙ η˙ + τ

is contained in its basin of attraction. Moreover the closedloop system is locally exponentially stable. Unfortunately, due to space constraints it is impossible to present the full proof of Lemma 4.1 in the present paper. The following discussion is a highly condensed sketch of the steps taken to construct the control Lyapunov function and their motivation. STEP 1: Linearising control transformation Consider the following control transformation

The principal result of the paper is the following Lemma 4.1: The system (12) in closed loop with inputs ! z2 z1 1 +p , (13) 1+ p u= cψ cθ 1 + z12 1 + z22 τ =C(η, η) ˙ η˙ + I−1 τb,

τb = (b τθ , τbψ , τbφ )> ,

(14)

u =

1+µ , cψ cθ

τ = C(η, η) ˙ η˙ + I−1 τb,

(16) (17)

where the new control inputs to the system are µ and τˆ. Note that the cancellation of the non-linearities in the Coriolis matrix C(η, η) ˙ η˙ does not pose any robustness problems due to the stabilisation of the attitude dynamics of the system undertaken in Section III. The partially linearised system after the control input transformation is   x˙ 1 = x2    −cψ sθ cφ −sψ sφ c s c +s s   − ψ θcψφ cθ ψ φ µ  x˙ 2 = cψ cθ      y˙ 1 = y2     y˙ = −cψ sθ +sψ cθ + −cψ sθ +sψ cθ µ 2 cψ cθ cψ cθ (18)  z˙1 = z2 , z˙2 = −µ      θ˙ = ω , ω˙ = τb  θ     ˙ ˙  ψ = λ , λ = τbψ     ˙ φ = γ , γ˙ = τbφ which has a feedforward structure [7]. STEP 2: Study the lateral dynamics independently of the yaw and altitude dynamics. The dominant restriction on control action lies in the lateral dynamics. Large accelerations in the x or y directions require large inclinations of the helicopter airframe that would cause the system to leave the practical stability region established in Section III. For this reason we study the lateral dynamics first with the goal of developing a highly robust stabilizing control that preserves the practical stability established in Section III.

3963

Once this is achieved the remaining system dynamics may be treated as perturbations and global stability can be proved. Consider the case where the yaw φ = 0 and altitude z1 are stabilized. In this case the transformed heave control µ = 0 is zero and Eq. 18 transforms to   x˙ 1 = x2       x˙ 2 = −tθ     y˙ = y 1 2 (19)  y˙ 2 = −tθ + tψ       θ˙ = ω , ω˙ = τbθ     ψ˙ = λ , λ˙ = τb ψ STEP 3: Change coordinates to separate roll and pitch dynamics. For bounded initial conditions |θ(0)| ≤ π3 , |ω(0)| ≤ 1. Consider the coordinate transformation Y1 = y 1 − x 1 ,

Y2 = y2 − x 2 ,

Then Eq. 19 decouples into   X˙ 1 = X2     X˙ = tθ 2  θ˙ = ω,     ω˙ = τbθ

X1 = −x1 ,   Y˙ 1     Y˙ 2

 ψ˙     ˙ λ

X2 = −x2 . (20)

=

Y2

=

tψ

=

λ,

(21)

= τbψ

The non-linear integrator cascades for lateral-x and lateral-y motion can now be analysed separately. STEP 4: Derive a control Lyapunov function for the decoupled systems Eq. 21 using forwarding techniques. It is a straightforward if tedious process to derive a control Lyapunov function for the decoupled systems in Eq. 21. By exploiting forwarding techniques we are able to derive a Lyapunov function that guarantees that the closed-loop trajectories remain in the practical stability region specified in Section III. The control Lyapunov functions (clf) derived can then be combined into a single clf for Eq. 19. The method used for the construction of the clf’s was based on the development in Mazenc et al. [7] and provides a guaranteed robustness property that is not always present where cross terms (see [10, Section 5.2]) or exact change of coordinates (see [7, Section IV]) are utilised. Unfortunately, there is no space in the present paper to provide the details of the construction. The following lemma has been proved. Lemma 4.2: Assume the initial conditions |θ(0)|, |ψ(0)| ≤ π 3 and |ω(0)|, |λ(0)| ≤ 1. Define the Lyapunov function    θ+ω 2 1 2 θ − 16 ln(| cos(θ)|) + 4 −x2 + 2 8 2  1 49 + 1 + −x1 − 32x2 + θ+ ω −1 8 8     ψ+λ 2 1 2 2 + 16 λ + λψ + ψ − 16 ln(| cos(ψ)|) + 4 y2 − x2 + 2 8 s  2 49 1 + 1 + y1 − x1 + 32y2 − 32x2 + ψ+ λ − 1. 8 8

U1 = 16 s



2

ω + ωθ +

Then the derivative of U1 along closed-loop trajectories of the system Eq. 19 with control τbθ , τbψ satisfies U˙ 1 ≤ −W1 , where W1 is given by 2

W1 = 16ω +  +


2 −x2 − 78 θ + 17 136 8 ω θtθ + q
2 5 1 + −x2 − 87 θ + 17 8 ω

−8x2 + 17ω + 9θ + 3

p

  2 −x1 −32x2 + 49 θ+4ω 8  2 1+ −x1 −32x2 + 49 θ+4ω 8

r

1 + (−8x2 + 9θ + 17ω)2

2 

2 (−x2 − 78 ψ + 17 136 2 8 λ) + 16λ + ψtψ + q 5 2 1 + (−x2 − 87 ψ + 17 8 λ)  

+

8y2 − 8x2 + 17λ + 9ψ + 3

p

 2 y1 −x1 +32y2 −32x2 + 49 ψ+4λ 8  2 1+ y1 −x1 +32y2 −32x2 +4λ+ 49 ψ 8

r

1 + (8y2 − 8x2 + 9ψ + 17λ)2

2 

9 π, |ω(t)| ≤ 41 Moreover, for all t ≥ 0, |θ(t)| ≤ 25 10 , |ψ(t)| ≤ 41 9 π and |λ(t)| ≤ . 25 10 By treating the influence of the altitude and yaw dynamics as perturbations to the robust stability of the lateral dynamics subsystem proved above, then, it is possible to draw on the properties of ISS systems [14] to infer global asymptotic stability of the full model. The explicit construction of a Lyapunov function, however, has the advantage of yielding robustness guarantees. STEP 5: The control Lyapunov function is extended to deal with non-trivial yaw dynamics: Consider the system obtained from Eq. 19 by setting µ = 0 and removing the constraint on the yaw dynamics in φ.  x˙ 1 = x2     t s   x˙ 2 = −tθ cφ − ψcθ φ       y˙ 1 = y2  (22) y˙ 2 = −tθ + tψ     θ˙ = ω , ω˙ = τbθ      ψ˙ = λ , λ˙ = τbψ     ˙ φ = γ , γ˙ = τbφ

Define a Lyapunov function for the yaw dyanmics  2 1 2 p V(γ, φ) = γ + 1 + φ2 − 1 2   1 2 p φ 2 γ + 1+φ −1 + p γ +2 2 1 + φ2

It is straight forward to verify that the derivative of V along the closed-loop trajectories, τbφ given by 15, satisfies 1 φ2 1 2p V˙ ≤ − 1 + γ2. − γ (23) 2 1 + φ2 2 From this construction it is possible to propose a combined Lyapunov function U2 = ln(1 + U1 ) + 11713 V(γ, φ)

(24)

The constant 11713 is chosen to ensure that the cross terms that are present in the derivative of U1 due to the non-zero

3964

.

dynamics in yaw dynamics are dominated by the stability of V. Taking the derivative of U2 along trajectories of the closed-loop system and taking some care with the bounding of cross terms one can show W1 1 φ2 1 2p U˙ 2 ≤ − 1 + γ2 − − γ 1 + U1 2 1 + φ2 2 STEP 6: The control Lyapunov function is extended to deal with non-trivial altitude dynamics: The final stage of the control design is to add the altitude dynamics. The approach taken is entirely analogous to the approach taken in STEP 5 for the yaw dynamics. Define a Lyapunov function U = U2 + 294913 V(z1 , z2 )

(25)

Taking the derivative of U along trajectories of the closedloop system and taking some care with the bounding of cross terms one can show 1 φ2 1 p W1 − − γ2 1 + γ2 U˙ ≤ − 2 2(1 + U1 ) 4 1 + φ 4 q 1 z12 1 2 − − z2 1 + z22 2 1 + z12 2

The proof of Lemma 4.1 follows from the above result along with Lyapunov’s direct method. The local exponential stability property of the closed-loop system follows from the fact that both U and W are locally at the origin lower bounded by positive definite quadratic functions. V. C ONCLUSIONS

In this paper an idealised dynamic model with six degrees of freedom for a scale model autonomous helicopter has been considered. The dynamic model was obtained using the Euler-Lagrange equation formalism. The main objective of the work was to obtain saturated control laws that achieve robust global asymptotic stability and local exponential stability of the system. To achieve this goal a non-linear control design based on passivity, linearisation and forwarding techniques was proposed. VI. REFERENCES

[5] P. Martin, S. Devasia, and B. Paden. A different look at output tracking: Control of a VTOL aircraft. Automatica, 32(1):101–107, 1996. [6] F. Mazenc. Stabilization of feedforward systems approximated by a nonlinear chain of integrators. Systems & Control Letters, 32(4):223–229, 1997. [7] F. Mazenc and L. Praly. Adding an integration and global asymptotic stabilization of feedforward systems. IEEE Transactions on Automatic Control, 41(11):1559– 1578, 1996. [8] R. Murray, M. Rathinam, and M. van Nieuwstadt. An introduction to differential flatness of mechanical systems. In Proceedings of Ecole d’Et´e, Th´eorie et practique des syst´emes plats, Grenoble, France, 1996. [9] Raymond W. Prouty. Helicopter Performance, Stability and Control. Krieger Publishing Company, 1995. reprint with additions, original edition 1986. [10] R Sepulchre, M Jankovi´ c, and P Kokotovi´ c. Constructive Nonlinear Control. Springer-Verlag, London, 1997. [11] O. Shakernia, Y. Ma, T. Koo, and S. Sastry. Landing an unmanned air vehicle: Vision based motion estimation and nonlinear control. Asian Journal of Control, 1(3):128–145, 1999. [12] A. Sira-Ramirez, R. Castro-Linares, and E. Lic´ eagaCastro. Regulation of the longetudinal dynamics of an helicopter system: A Liovillian systems approach. In Proceedings of the American Control Conference ACC’99, San Diego, California, U.S.A., 1999. [13] H. Sira-Ramirez, R. Castro-Linares, and E. LiceagaCastro. A Liouvillian systems approach for the trajectory planning-based control of helicopter models. International Journal of Robust and Nonlinear Control, 10:301–320, 2000. [14] E. D. Sontag and Y. Wang. On characterization of the input-to-state stability property. Systems and Control Letters, 24, 1995. [15] A. R. Teel. A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Transactions on Automatic Control, 41(9):1256–1270, 1996.

[1] E. Frazzoli, M. Dahleh, and E. Feron. Trajectory tracking control design for autonomous helicopters using a backstepping algorithm. In American Control Conference, Chicago, IL, U.S.A., 2000. [2] H. Goldstein. Classical Mechanics. Addison-Wesley Series in Physics. Addison-Wesley, U.S.A., second edition, 1980. [3] J. Hauser, S. Sastry, and G. Meyer. Nonlinear control design for slightly non-minimum phse systems: Applications to v/stol aircraft. Automatica, 28(4):665–679, 1992. [4] R. Mahony, T. Hamel, and A. Dzul. Hover control via approximate lyapunov control for a model helicopter. In Proceedings of the 1999 Conference on Decision and Control, Pheonix, Arizona, U.S.A., 1999.

3965