robust trajectory tracking for a scale model autonomous ... · robust trajectory tracking for a...

Robust Trajectory Tracking for a Scale Model

Autonomous Helicopter.

Robert Mahony† and Tarek Hamel‡† Dep. of Eng., Australian Nat. Univ., ACT, 0200 Australia.

‡ I3S, UMR-CNRS 6070, 2000 route des Lucioles - Les algorithmes -

Bat Euclide B - 06903 Sophia Antipolis, France.

email:[email protected],

Phone : 61-(0)2-61.25.86.13, Fax : 61-(0)2-61.25.05.06

Revised version of paper JNRC727

Approved for publication in the International Journal of

Non-linear and Robust Control

Abstract

This paper considers the question of obtaining an a priori bound on the tracking

performance, for an arbitrary trajectory, of closed-loop control of an idealized model

of a scale model autonomous helicopter. The problem is difficult due to the presence

of small body forces that can not be directly incorporated into the control design. A

control Lyapunov function is derived for an approximate model (in which the small body

forces are neglected) using backstepping techniques. The Lyapunov function derived is

used to analyze the closed-loop performance of the full system. A theorem is proved

that provides a priori bounds on initial error and the trajectory parameters (linear

acceleration and its derivatives) that guarantees acceptable tracking performance of

the system. The analysis is expected to be of use in verification of trajectory planning

procedures.

1 Introduction

There is a growing worldwide interest in the development of scale model autonomous he-licopter systems (Schaufelberger & Geering, 1998; Koo et al. , 1998). It appears that theclassical modelling and control approaches (cf. for example (Prouty, 1995)) are not directlyapplicable due to the high actuation to inertia ratios and highly non-linear nature of the rota-tion dynamics exploited in desired flight conditions for scale model autonomous helicopters.Recently a number of authors from the control community have begun to investigate an in-tegrated non-linear dynamic model of a scale model autonomous helicopters (cf. conference

1

R. Mahony and T. Hamel, Accepted for pub., Sub. IJNRC, March 2002.

papers (Mahony et al. , 1999; Sira-Ramirez et al. , 1999; Shim et al. , 1998; E. Frazzoli &Feron, 2000; van Nieuwstadt & Murray, 1997a) and more recently the journal papers (Shak-ernia et al. , 1999; Sira-Ramirez et al. , 2000; van Nieuwstadt & Murray, 1997b)). Althoughthe model considered does not contain a sophisticated aerodynamic analysis there is hopethat by resolving the basic trajectory planning and control issues for the model considered,it will be possible to extend these developments to provide robust practical controllers forscale model autonomous helicopters. One of the key problems encountered is the presenceof significant small body forces leading to weakly non-minimum phase zero dynamics for thefull dynamic model. This is a theoretical problem that was also encountered in the investi-gation of the control of a Vertical Takeoff and Landing Jet (VTOL) (Hauser et al. , 1992;Ghanadan, 1994; Teel, 1996; Martin et al. , 1996; Murray et al. , 1996) and (Sepulchre et al., 1997, pg. 246.). Unfortunately, the differential flatness technics applicable in the case of aVTOL do not apply in general to a helicopter (Thomson & Bradley, 1987; Liceaga-Castroet al. , 1989; Mallhaupt et al. , 1997; Mahony & Lozano, 2000). Recent work (Sira-Ramirezet al. , 2000) uses trajectory planning to exploit the partial differential flatness propertiesthat do exist for the helicopter model, however, the final stabilising control design still relieson an approximation of the model. Most other authors have applied a robust control designto the model obtained by ignoring the small body forces and later analyze the performance ofthe system to ensure that for desired trajectories the un-modelled dynamics do not destroythe stability of the closed-loop system (Teel, 1996; Shakernia et al. , 1999; E. Frazzoli &Feron, 2000). Such results either take the form of monitoring the behaviour of the system inorder to ascertain when the stability guarantees of the control design are broken (cf. Lemma4.1) or provide some a priori guarantees for a restricted class of trajectories (E. Frazzoli& Feron, 2000; Shakernia et al. , 1999) (cf. Corrolary 4.5). To the authors knowledge noprevious work has proposed a robust non-linear control for a scale model helicopter for whichthere are a priori guaranteed performance bounds given for arbitrary (bounded) trajectories.

In this paper a control design based on backstepping techniques (Freeman & Kokotovic,1996) is proposed for a dynamic model of a scale model autonomous helicopter. A controlLyapunov function is derived based on the block pure feedback form (Krstic et al. , 1995) ofthe approximate dynamic model obtained by ignoring the small body forces. The trajectorytracking control design is developed independent of a local coordinate parameterisation ofthe helicopter orientation, however, Euler angles are used to parameterise the final ‘yaw’ pa-rameter that does not explicitly contribute to the flight trajectory dynamics in the idealisedmodel considered. The Lyapunov function obtained for the closed-loop system is used toanalyze the performance of the proposed control in tracking an arbitrary trajectory. Firstly,a lemma is given (Lemma 4.1) that monitors the performance of the control relative to thedecrease in the Lyapunov function on-line. A much more difficult problem is to provideguaranteed a priori bounds of tracking performance for a given trajectory that satisfiescertain constraints. Theorem 4.3 provides a result of this nature for arbitrary bounded tra-jectories. Finally, the particular case of stabilisation to a point is considered (the stationarytrajectory) and Corollary 4.5 derives a bound on the basin of attraction for the proposedcontrol law applied to this trajectory. The trajectory planning problem is a separate issueof at least equal importance to the control problem considered in this paper. Although the

2


trajectory planning problem is beyond the scope of the present paper the explicit a-prioribounds on the parameters of a trajectory that may be tracked to within a specified errorthat are given by Theorem 4.3 are expected to contribute to the design of future trajectoryplanning algorithms.

The paper is arranged into six sections. Section 2 presents the idealised model of thedynamics of a scale model autonomous helicopter. Section 3 derives a Lyapunov control foran approximate system based on that presented in Section 2 via a backstepping approach.Section 4 contains the main results of the paper and presents the analysis of the closed-loopperformance of the proposed control. Section 5 presents a series of experiments which showthe behaviour of the closed-loop system for both the approximate and complete dynamicmodels of the system.

2 Dynamic Model of Autonomous Helicopter

In this section, an idealised model for the dynamics of an autonomous scale model helicopteris presented. The model is based on a consideration of the helicopter as a combination ofrotor dynamics and rigid body dynamics of the helicopter airframe. Following the leadof recent work (Sira-Ramirez et al. , 1999; E. Frazzoli & Feron, 2000; van Nieuwstadt &Murray, 1997b; Shakernia et al. , 1999; Sira-Ramirez et al. , 2000) a simple component forcemodel of the rotor dynamics is used. A controller designed for a real world system wouldneed to consider the aerodynamic effects of the rotor dynamics in addition to the difficultnon-linear control problem associated with the airframe dynamics considered in this paperand earlier work. Some work along these lines has been published since the work in thispaper was undertaken (cf. (La Civita, 2002; Mettler, 2003) and references therein).

Let ξ ∈ R3 denote position of the centre of mass of the helicopter airframe with respect toan inertial frame of reference I := Ex, Ey, Ez where Ex etc. denote the coordinate axes inthe inertial frame. Denote the velocity of the centre of mass as v = ξ. Let A := Ea1 , Ea2 , Ea3be a body fixed frame attached to the helicopter, where Ea1 is the longitudinal axis of thehelicopter frame, Ea2 is the lateral axis and Ea3 is the vertical direction in hover conditions(cf. Fig. 1). Let R : A → I be the rotation matrix representing the orientation of A withrespect to I. The angular velocity of A is represented by a vector Ω ∈ A in the body fixedframe.

In addition to gravitational effect a helicopter is acted on by a number of forces. Theseforces seen in Equations 2-5 are explained in the following list:

Ea3u. The principal rotor provides a strong lift force termed ‘heave’, oriented along the axisEa3 in the body fixed frame, which is the principal force responsible to sustaining thehelicopter in flight.

|QM |, |QT |. Air resistance on the main and tail rotors result in torques applied directly tothe helicopter airframe |QM |, |QT | oriented around the axis of the respective rotor.

3


Ex

Ez

Ey

QM

QT

E a3

E a3 E a

2

E a1

ξ

R K w

u

P w

Figure 1: Frames of reference of scale model helicopter and principal forces acting on theairframe.

Pw. Torque control for the airframe is obtained via a combination of the rigidity of therotor blades, coriolis forces associated with the effective flapping hinge due to the rotorrigidity, and the induced torques obtained from the deformation of the main rotor diskand consequent tilting of the force vector representing the combined lift generated bythe main rotor blades. The vector w ∈ R3 represents the contribution to the torquefrom the deformation of the main rotor disk, the principal mechanism used to obtaincontrol, while the positive definite matrix P > I3 provides the modification for otherforces related to rotor rigidity.

R(η)Kw. Due to the mechanism used to obtain torque control and the consequent tiltingof the force vector associated with the main rotor lift, the torque control inputs w arecoupled directly to small body forces which effect the translational dynamics of thehelicopter airframe. Here K is a constant matrix depending on the physical parametersof the helicopter. It is defined as follows

K =1lM

0 1 0−1 0 1

lT

0 0 0

(1)

where lM and lT denote the offset of the main and tail rotor masts from the centre ofmass.

Newton’s equations of motion for a helicopter subject to the forces outlined above are:

ξ = v (2)

mv = −Ea3u+mge3 +R(η)Kw, (3)

R = Rsk(Ω), (4)

IΩ = −Ω× IΩ + |QM |e3 − |QT |e2 + Pw. (5)

In practice, the model given by Equations 2-5 is difficult to control due to the presenceof the small body forces which couple torque inputs to translational dynamics (Eq. 3). An

4


approximate model, in which the small body forces are set to zero, is used in the controldesign

ξ = v (6)

mv = −Ea3u+mge3 (7)

R = Rsk(Ω), (8)

IΩ = −Ω× IΩ + |QM |e3 − |QT |e2 + Pw. (9)

The approximate model Eqn’s 2-5 is feedback linearisable (and consequently differentiallyflat). It is a relatively straight forward exercise to obtain a path tracking control algorithmfor the approximate model (Mahony et al. , 1999; Sira-Ramirez et al. , 1999; Shim et al. ,1998; E. Frazzoli & Feron, 2000; van Nieuwstadt & Murray, 1997b). This model can alsobe used for trajectory planning problems (van Nieuwstadt & Murray, 1997b; Sira-Ramirezet al. , 2000) that exploit the differential flatness properties. However, when a controllerdesigned for the approximate system is applied to the full system dynamics Eqn’s 2-5 thereis a performance loss. It is the goal of this paper to provide an analysis of the robustness ofa controller designed in this manner. In order to provide the best analysis we have chosen touse a backstepping control design (in preference to a feedback linearisation type design) inorder to exploit the natural robustness of such methodology (Freeman & Kokotovic, 1996)in the robustness analysis undertaken in Section 4.

3 Lyapunov Based Tracking Control Design

In this section a backstepping control design is provided for the approximate model 2-5proposed in the previous section.

Let η = (φ, θ, ψ) denote the classical ‘yaw’, ‘pitch’ and ‘roll’ Euler angles commonly usedin aerodynamic applications (Goldstein, 1980, pg. 608) (cf. Fig. 2). The rotation matrixR := R(φ, θ, ψ) ∈ SO(3) representing the orientation of the airframe A relative to a fixedinertial frame may be written1 in terms of the Euler angles η = (φ, θ, ψ)

R =

cθcφ sψsθcφ − cψsφ cψsθcφ + sψsφ

cθsφ sψsθsφ + cψcφ cψsθsφ − sψcφ−sθ sψcθ cψcθ

. (10)

Remark 3.1 The Euler angles η = (φ, θ, ψ) are not a global coordinate patch on SO(3). Ifθ ≤ π

2 then the correspondence between the Euler coordinates and the rotation matrices inSO(3) is one-to-one.

1The following shorthand notation for trigonometric function is used:

cβ := cos(β), sβ := sin(β).

5


E a2

Ez

Ex

E a3

E a1φ

φ

θ

θψ

ψ

Ey

Figure 2: Classical ‘yaw-pitch-roll’ Euler angles commonly used in aerodynamic applications.Firstly, a rotation of angle φ around the axes Ez is applied, corresponding to ‘yaw’. Secondly,a rotation of angle θ around the rotated version of the Ey axis is applied, corresponding to‘pitch’ of the airframe. Lastly, a rotation of angle ψ around the axes Ea1 is applied. Thiscorresponds to ‘roll’ of the airframe around the natural axis Ea1 .

Let

ξ : R→ R3,

φ : R→ R

be smooth trajectories ξ(t) := (x(t), y(t), z(t)) and φ(t). The control objective considered isformally described in the following problem.

Problem: Find a feedback control action (u,w1, w2, w3) depending only on the measurablestates (ξ, ξ, η, η) and arbitrarily many derivatives of the smooth trajectory (ξ(t), φ(t)) suchthat the tracking error

E := (ξ(t)− ξ(t), φ(t)− φ(t)) ∈ R4 (11)

converges towards zero for all initial conditions.

Define a partial errorδ1 := ξ(t)− ξ(t). (12)

comprising that part of the tracking error associated with the position coordinates. Definea first storage function

S1 =12δT1 δ1 =

12|δ1|2. (13)

Taking the time derivative of S1 and substituting for Eq. 6 yields

d

dtS1 = δT1 (ξ − ˙

ξ)

= δT1 (v − v),

where v := ˙ξ. Let vd denote a desired value for the velocity v and choose

vd := v − 1mδ1.

With this choice one has

S1 =1mδT1 (mvd −mv) +

1mδT1 (mv −mvd)

= − 1m|δ1|2 +

1mδT1 (mv −mvd).

6


The process of backstepping continues by considering a new error

δ2 := mv −mvd.

The second storage function considered is

S2 =12|δ2|2 =

12|mv −mvd|2

Deriving S2 and recalling Eq. 7 yields

S2 = δT2 (mv −mvd),= δT2 (−Ea3u+mge3 −mvd),= δT2 (−uR(η)e3 +mge3 −mvd),

The backstepping process is continued with respect to the new variables (η, u). The controlu is explicit in the dynamics of S2 and could be used directly to partially control the errordynamics. Following this approach leads to a time scale separation of the system dynam-ics (Teel, 1996; Sepulchre et al. , 1997). Such an approach is advantageous in situationssuch as hover control where the property may be used to ensure that part of the system(eg. regulating height) is more tightly controlled than other system states. In the case ofa VTOL hovering close to the ground this can be a highly desirable characteristic of thecontrol methodology. However, in more general trajectories an artificial time-scale separa-tion imposed in the dynamics at this stage can lead to significant robustness problems. Inparticular, derivatives of the control u enter into the remaining dynamics of the system andaggressive control may lead to extreme ill-conditioning of the remaining control problem.An alternative strategy is to take a dynamic extension of u

u = u. (14)

The actual control u and its first derivative u become internal variables of a dynamic con-troller. This approach more naturally allows tradeoff between the various control objectivesof the generic trajectory tracking problem.

Let (ηd, ud) denote the desired values of η and the control u and set

udR(ηd)e3 := mge3 −mvd + δ2 +1mδ1.

The vectorial value of udR(ηd)e3 may be arbitrarily assigned for suitable values of ηd and ud.Note that ηd is not uniquely specified by this relation. The additional freedom in choosingηd is used later to control the yaw φ. It is advantageous to introduce a notation to representthe desired vector input

Xd := udR(ηd)e3 = mge3 −mvd + δ2 +1mδ1. (15)

With the above notation one has

S2 = −|δ2|2 − 1mδT2 δ1 + δT2 (Xd − uR(η)e3).

7


The process of backstepping continues by considering a third error

δ3 := udR(ηd)e3 − uR(η)e3 = Xd − uR(η)e3, (16)

the vectorial difference between the desired and true values of translation thrust uR(η)e3

and a further error that penalises the yaw

ε3 = φ− φ.The yaw component of the error term is introduced at this stage of the backstepping pro-cedure (rather than along with the initial error term δ1) in order that the relative degree ofδ3 and ε3 with respect to the controls u and w match.

Consider the storage function

S3 =12|δ3|2 +

12|ε3|2.

Deriving S3 and recalling Eq. 8 yields

S3 = δT3

(Xd − (uR(η)e3 +R(η)sk(Ω)e3)

)+ ε3(φ− ˙

φ). (17)

Remark 3.2 The derivative of Xd is computed analytically by differentiating the expressionon the right hand side of Eq. 15. The derivatives of ud and ηd are never explicitly computed.

Consider the term associated with δ3 firstly. Let (Ωd, ud) denote the desired values of Ωand the control derivative u. Analogously to the case for uR(η)e3, the full vectorial term

udR(η)e3 +R(η)sk(Ωd)e3 := Xd + δ3 + δ2 (18)

is assigned. Recalling that sk(Ωd)e3 = Ωd × e3 = −sk(e3)Ωd Eq. 18 may be written

0 1 0−1 0 00 0 1

Ω1d

Ω2d

ud

:= R(η)T

(Xd + δ3 + δ2

)(19)

It is clear that any vector specified in the right hand side of Eq. 18 may be assigned usingonly Ω1

d, Ω2d and ud. This leaves Ω3

d free to control the yaw φ. Set

Yd := udR(η)e3 +R(η)sk(Ωd)e3 = Xd + δ3 + δ2. (20)

Now consider the term associated with ε3 in Eq. 17. The desired input for this term isthe yaw velocity of the helicopter φ. Let φd denote the desired yaw velocity and choose

φd := ˙φ− ε3.

To proceed it is necessary to recap the kinematic relationship between the Euler angles andthe angular velocity of a rigid body. The generalised velocities η = (φ, θ, ψ) are related tothe angular velocity Ω via standard kinematic relationship (Goldstein, 1980, pg. 609)

η =1

cos(θ)

0 sψ cψ

0 cθcψ −cθsψcθ sθsψ sθcψ

Ω = W−1

η Ω. (21)

8


where

Wη :=

−sθ 0 1cθsψ cψ 0cθcψ −sψ 0

. (22)

Replacing η by ηd and Ω by Ωd one obtains

ηd =1

cos(θ)

0 sψ cψ

0 cθcψ −cθsψcθ sθsψ sθcψ

Ωd.

Multiplying by eT1 one obtains φd = sψcθ

Ω2d + cψ

cθΩ3d. Assume θ, ψ ∈ (−π2 , π2

)and set

Ω3d :=

cθcψ

(˙φ− ε3 − sψ

cθΩ2d

).

Thus, the non-linear coupling of the attitude dynamics leads to a dependence of Ω3d on Ω2

d

(assigned in Eq. 19).

With the choices made above one may rewrite Eq. 17 as

S3 = −|δ3|2 − δT3 δ2 − ε23 + ε3(φ− φd) + δT3 (Yd − (uR(η)e3 + uR(η)sk(Ω)e3)) . (23)

For the last stage of the backstepping algorithm new error terms are introduced

δ4 = Yd − (uR(η)e3 + uR(η)sk(Ω)e3). (24)

ε4 = φ− φd.With this choice the derivative of S3 may be written

S3 = −|δ3|2 − δT3 δ2 − ε23 + δT3 δ4 + ε3ε4.

The storage function associated with this stage of the backstepping is

S4 =12|δ4|2 +

12|ε4|2.

Thus, taking the derivative of S4 yields

S4 = δT4

(Yd −

[uR(η)e3 + 2uR(η)sk(Ω)e3 + uR(η)(Ω× e3)

])+ ε4(φ− ¨

φ). (25)

At this stage the control inputs enter into the equations through u = u, Ω via Eq. 9 and φ

as seen below.

To simplify the following analysis consider a linearising control input transformation ofequation Eq. 9. Define

w := −I−1Ω× IΩ + |QM |I−1e3 − |QT |I−1e2 + I−1Pw. (26)

With this choice Eq. 9 becomes Ω = w. Taking a second derivative of Eq. 21 and regardingonly the first component yields

φ = −eT1 W−1η WηW

−1η Ω + eT1 W

−1η w

= −eT1 W−1η WηW

−1η Ω +

sψcθw2 +

cψcθw3. (27)

9


Eq. 25 may be rewritten

S4 = δT4

(Yd − 2uR(η)sk(Ω)e3 − [uR(η)e3 − uR(η)sk(e3)w]

)

+ ε4(φ− ¨φ).

To achieve the desired control choose

uR(η)e3 − uR(η)sk(e3)w = Yd − 2uR(η)sk(Ω)e3 + δ3 + δ4 (28)

φ = ¨φ− ε4 − ε3 (29)

With these choices the derivative of S4 becomes

S4 = −|δ4|2 − δT4 δ3 − |ε4|2 − ε4ε3

It remains to show that Eqn’s 28 and 29 can be satisfied simultaneously. RewritingEq. 28 analogously to Eq. 19

0 u 0−u 0 00 0 1

w1

w2

u

= R(η)T

(Yd − 2uR(η)sk(Ω)e3 + δ3 + δ4

). (30)

As long as u 6= 0 the control signals w1, w2 and u are uniquely determined. To obtain w3

one solves for Eq. 27 yielding

cψcθw3 = ¨

φ− ε4 − ε3 + eT1 W−1η WηW

−1η Ω− sψ

cθw2. (31)

The above process fully specifies the control inputs w1, w2, w3 and u. Using Eq. 14 andEq. 26 one recovers the original control inputs u and w, that are themselves functions ofmore primitive variables of the systems such as flapping angles and thrust components. Thebackstepping procedure achieves the monotonic decrease of the Lyapunov function

L := S1 + S2 + S3 + S4.

This is easily verified by computing

L = S1 + S2 + S3 + S4

= − 1m|δ1|2 − |δ2|2 − |δ3|2 − |ε3|2 − |δ4|2 − |ε4|2 (32)

Recall that δ1 and ε3 together form the original tracking error. The error δ2 regulatesthe linear velocity of the helicopter. The additional error coordinates δ3 and δ4 incorporateinformation on the roll, pitch and their derivatives. This is natural for an under-actuatedsystem since the desired motion can only be obtained by exploiting the attitude dynamicsto control the orientation of the thrust. The yaw orientation and its velocity are stabilisedindependently using the errors ε1 and ε2. Applying Lyapunov’s second method ensures thatL → 0 along trajectories of the approximate system Eqn’s 6-9.

10


4 Analysis

In this section two results are proved that provide bounds on the evolution of the closed-loopsystem given by Eqn’s 2-5 along with the control action given by Eqn’s 30, 31, 14 and 26.

Letα = (δ1, δ2, δ3, δ4, ε3, ε4) ∈ R14

denote the vector of backstepping errors introduced in Section 3. The error dynamics forthe full idealised model Eqn’s 2-5 are easily verified to be

α =

− 1mI3

1mI3 0 0 0 0

− 1mI3 −I3 I3 0 0 00 −I3 −I3 I3 0 00 0 −I3 −I3 0 00 0 0 0 −1 10 0 0 0 −1 −1

α+

0R(η)Kw

m+1m R(η)Kw

2m2+2m+1m2 R(η)Kw

00

. (33)

The linear block is associated with the feedback linearised structure of the approximatemodel Eqn’s 6-9. The additional perturbations are due to the small body forces in the fullidealised model Eqn’s 2-5.

In the following analysis, bounds made up of norms of the errors |δi|, i = 1, . . . , 4 and|ε3| and |ε4| are used regularly. To simplify notation define

γ = (|δ1|, |δ2|, |δ3|, |δ4|, |ε3|, |ε4|) ∈ R6,

Λ = diag(1m, 1, 1, 1, 1, 1) ∈ R6×6

χ = (| ˙v|, |v(2)|, |v(3)|, | ˙φ|, |φ(2)|) ∈ R5.

Thus, a linear bound in the absolute norms of the error terms may be written πT γ + τTχ

for π ∈ R6, τ ∈ R5, constant vectors. The Lyapunov function L may be written

L =12|γ|2 (34)

and its time derivative may be written L = −γTΛγ (cf. Eq. 32).

4.1 Online Stability Analysis

Due to the presence of the small body forces in the error equations (Eq. 33) the Lyapunovfunction L may not be monotonically decreasing along solutions of the full model Eqn’s 2-5.If the perturbation is small relative to the error α then the linear dynamics in Eq. 33 willdominate the perturbations and the Lyapunov function will be decreasing. The followingtheorem provides an analogous result to that obtained in related works (Hauser et al. , 1992;Koo & Sastry, 1998).

11


Lemma 4.1 Consider the dynamics defined by Eqn’s 2-5. Let the controls w and u be givenby Eqn’s 30 and 31 and recover the control inputs w and u from Eqn’s 14 and 26. Then, theLyapunov function Eq. 34 is strictly decreasing as long as

γTΛγ ≥ σ|w|〈π0, γ〉 (35)

where

π0 =(

0, 1,m+ 1m

,2m2 +m+ 1

m2, 0, 0

)and σ =

1lM lT

√2l2T − 2lT + 1 = ||K||2.

Proof Taking the derivative of L, substituting from Eq. 33 and using Holders inequalityto bound the effect of the small body forces yields

L ≤ −γTΛγ + |δ2||R(η)Kw|+ m+ 1m|δ3||R(η)Kw|

+2m2 +m+ 1

m|δ4||R(η)Kw|

= −γTΛγ + σ|w|πT0 γ

where π0 is given in the lemma statement. The result follows directly from this inequality.4

The constant σ = ||K||2 ≈√

2/lM (for lT >> lM ) approximately measures the inverseof the offset between the centre of mass of the helicopter and the centre of the rotor disk(at which point the force u is applied). Thus, σ large corresponds to a small offset andcorrespondingly large ‘small body forces’. The larger the value of σ, the less valid is theapproximate model Eqn’s 6-9 and the more difficult it is to control the full dynamic modelEqn’s 2-5. For a scale model helicopter, a value of σ ≈ 2− 5 is typical, corresponding to anoffset of around 50cm-20cm.

4.2 Local Uniform Practical Stability Analysis

Lemma 4.1 provides a bound governing the decrease of L in terms of the size of the smallbody forces. Such a result is of significant practical importance in proving the asymptoticpractical stability of the closed-loop system to a neighbourhood of the desired trajectory.Lemma 4.1, however, does not provide an estimate of the error margin obtained in steady-state tracking. In this section it is shown that the closed-loop system with the full idealisedmodel Eqn’s 2-5 is locally uniformly practically stable and consequently the error margin forsteady-state tracking of a suitable trajectory is bounded.

Definition 4.2 Consider a system with state x. The system is termed locally uniformlypractically stable around a desired trajectory x if there exists constants k1, k2 > 0 such thatfor all |x(t)| < k1 and initial conditions |x(0)− x(0)| < k2 then there exists k3 > 0 such that

|x(t)− x(t)| < k3

for all time.

12


Rewrite Eq. 26 as follows

Pw = Iw − Ω× IΩ− |QM |e3 + |QT |e2.

Since P (u,w) > I3 it follows that ||P (u,w)||−12 ≤ 1. Taking the bound on the above equation

one obtains|w| ≤ c0|Ω|2 + c0|w|+ |w0| (36)

where|w0| = |QM |+ |QT |, c0 = ||I||2.

One may think of |w0| as a bound on the part of the control action w which is necessary tocancel the effect of the anti-torques QM and QT .

Due to the nature of the control design it is necessary to consider the torque input w3

separately from the remaining two torque inputs (w1, w2). Consider, Eq. 31 and note thatthe singularity associated with the Euler coordinates makes it impossible in general to obtainan absolute bound for the value of |w3|. This is a purely mathematical constraint due tothe imposition of a particular set of Euler coordinates and does not represent a singularityin the physical orientation of the system. By choosing different orientations for the inertialframe I with respect to which the Euler angles are measured the mathematical singularitymay be avoided. Changes of this nature, made at separate finite times, will not alter theremaining control design as long as the trajectory in the new and old representations arerelated to a smooth trajectory in the general orientation of the helicopter. Assume that theEuler coordinates are recomputed relative to a new inertial frame any time one of the anglesθ or ψ exceeds π/4. Thus, cos(ψ), cos(θ) > 1/

√2. From Eqn’s 22 and 21 the following

bounds may be computed

||W−1η ||2 ≤

1cos θ

≤√

2, ||Wη||2 ≤√

2|η|.

A bound for Eq. 31, using these estimates along with the assumed bound on the values ofcos(ψ) and cos(θ) is

|w3| ≤ 〈τ1, χ〉+ 〈π1, γ〉+ 4|Ω|2 +√

2|w2|.where

τ1 = (0, 0, 0, 0,√

2)T , π1 = (0, 0, 0, 0,√

2,√

2)T .

Let w(1,2) = (w1, w2) denote the first two torque inputs for w. Thus, multiplying Eq. 30by sk(e3) one obtains

uw(1,2) = sk(e3)R(η)T Yd + 2udiag(1, 1, 0)Ω + sk(e3)δ3 + sk(e3)δ4.

taking norms and bounding this equation one obtains

|u||w(1,2)| ≤ |Yd|+ 2|u| |Ω|+ 〈π2, γ〉, (37)

13


where π2 = (0, 0, 1, 1, 0, 0)T . Now

|w| ≤ |w(1,2)|+ |w3|≤ |w(1,2)|+ 〈τ1, χ〉+ 〈π1, γ〉+ 4|Ω|2 +

√2|w2|

≤ (1 +√

2)|u| |u| |w(1,2)|+ 〈τ1, χ〉+ 〈π1, γ〉+ 4|Ω|2 (38)

Thus, bounding |u| |w(1,2)| provides a bound for |w|, at least as long as |u| 6= 0.

A bound for |Yd| may be obtained via a rather lengthy and uninteresting calculationwhich involves computing bounds for the various desired signals |Yd|, |Xd|, |vd| and |vd|used in the backstepping procedure. One obtains

|Yd| ≤ 〈τ2, χ〉+ 〈π3, γ〉+ σc2|w| (39)

where

τ2 = (0, 0,m, 0, 0)T , c2 =2m2 + 2m+ 1

m,

π3 =(

2m+ 1m3

,2m3 +m+ 1

m3,

2(m+ 1)m

,2m+ 1m

, 0, 0)T

A bound for |u| and for |Ω| can be obtained from Eq. 24 along with the bounds obtainedfor |Yd|. In fact multiplying Eq. 24 by sk(e3) one obtains a bound for Ω(1,2) only and not forΩ3. The bound for Ω3 is obtained by bounding the first line of Eqn’s 21 and incorporatingthe expression for φd along with the bounds cos(ψ), cos(θ) > 1/

√2 discussed earlier

|Ω3| ≤ |Ω2|+ 〈τ3, χ〉+ 〈π4, γ〉.

whereτ3 = (0, 0, 0,

√2, 0)T , π4 = π1.

Thus, one obtains the following bound for |Ω|

|Ω| ≤ 2|Ω(1,2)|+ 〈τ3, χ〉+ 〈π4, γ〉

Multiplying Eq. 24 by eT3 and taking norms one obtains the following bound for |u|

|u| ≤ 〈τ4, χ〉+ 〈π5, γ〉σc3|w|,

where

τ4 = (0,m, 0, 0, 0), c3 =m+ 1m2

π5 =(m+ 1m2

,1m,

2m+ 1m

, 1, 0, 0).

Multiplying Eq. 24 by sk(e3), taking norms it is seen that |u| |Ω| may be bounded by thesame bound used for |u|. Combining the above results one obtains

|Ω| ≤ 1|u| (〈q1(u), χ〉+ 〈p1(u), γ〉+ σc4|w|) .

14


whereq1(u) = 2τ4 + |u|τ3, c4 = 2c3, p1(u) = 2π5 + |u|π4.

By combining Eqn’s 36, 38 and 39 one obtains

|w| ≤ 1|u| (〈q2(u), χ〉+ 〈p2(u), γ〉+ σc5|w|+ 2|u| |Ω|

+d1|u| |Ω|2 + |u| |w0|), (40)

where q2(u) = (1 +√

2)τ2 + |u|τ1, c5 = c2, d1 = 4 + c0 and p2(u) = (1 +√

2)π3 + |u|π1 + π2.

Finally, from Eq. 16 one obtains the two sided bound on the control u

mg − 〈τ5, χ〉 − 〈π6, γ〉 ≤ |u| ≤ mg + 〈τ5, χ〉+ 〈π6, γ〉, (41)

where

τ5 = (m, 0, 0, 0, 0)T , π6 =(

1m, 1, 1, 0, 0, 0

)T.

The following theorem uses the above development to derive an explicit relationship be-tween a guaranteed bound on the evolution of the Lyapunov function (and hence guaranteedtracking performance) in terms of quantitative bounds on initial conditions, desired trajec-tories and the physical geometry of the system (in particular the constant σ relating the sizeof the ‘small body forces’ to the torque control).

Theorem 4.3 Consider the dynamics defined by Eqn’s 2-5. Let the controls w and u begiven by Eqn’s 30 and 31 and recover the control inputs w and u using Eqn’s 14 and 26.

Let k0, k1 and k2 be the following initial bounds

|L(0)| ≤ k20, |Ω(0)| ≤ k1, |w0| ≤ (|QM |+ |QT |) ≤ k2.

Define the auxiliary constants

p03 :=

∣∣∣mgπ1 + π2 + (1 +√

2)π3 +mgd1k1π4 + 2k1(1 + d1)π5

∣∣∣ ,p1

3 := |π1 + d1k1π4| ,q03 :=

∣∣∣mgτ1 + (1 +√

2)τ2 +mgd1k1τ3 + 2k1(1 + d1)τ4∣∣∣ ,

q13 := |τ1 + d1k1τ3| ,c6 := c5 + 2k1c3 + d1k1c4.

Define two bounding functions for the possible trajectories χ in terms of the availablecontrol margin (mg −∆) ≤ u ≤ (mg + ∆)

B1(∆) :=k0

[(mg −∆)

(1− σk2

k0|π0|)

+ ∆σp13|π0| − σ

(c6 + p0

3|π0|)]

σ(p03 −∆p1

3)(q03 −∆q0

3)(42)

B2(∆) :=∆− |π6|k0

|τ5| (43)

15


Let

∆∗ := arg sup∆≥mg− k0(c4+2|π0| |π5|)

k1|π0|

min B1(∆), B2(∆)

B∗ := sup∆≥mg− k0(c4+2|π0| |π5|)

k1|π0|

min B1(∆), B2(∆)

Then; for any trajectory χ such that

|χ| ≤ B∗the closed-loop system is uniformly practically stable (cf. Def. 4.2). More precisely;

|L(t)| < k20, Ω(t) < k1 + k0|π0||π4|(mg + ∆∗) + |π0| |q1(u)|B∗, and u > mg −∆∗, (44)

for all time t ≥ 0.

Remark 4.4 i) Without any further conditions on k1 and σ there is no a priori guaran-tee that the bounds B1 and B2 are positive and the theorem may be null. In practice,the physical constraints of helicopter construction ensure that the bounds are reason-able and lead to practical bounds on possible trajectory choice. Figure 3 shows a plotof B1(∆) and B2(∆) for the example helicopter considered in Section 5.

ii) Maintaining a lower bound for |u| corresponds to maintaining a positive lift on themain rotor disk at all times.

iii) A separate bound on the angular velocity Ω is necessary due to the linearisation pro-cedure Eq. 26. This linearisation procedure enters into the analysis via the small bodyforces and is one of the main sources of complexity in the derivation. The bound KΩ

(Condition i) is slightly larger than the initial condition bound k1 since an aggressivetrajectory choice may result in larger angular velocities than the initial conditions, inpractice the angular velocity does not become unbounded for valid trajectories.

Proof In order to derive the result the conditions provided by the bounds B1 and B2 aretransformed into a set of a conditions that are less directly connected to the trajectoryplanning problem but more closely related to the manner in which the bounds necessary forthe proof are derived. Define the following two functions of u

q3(u) = q2(u) + 2k1τ4 + d1k1q1(u), p3(u) = p2(u) + 2k1π5 + d1k1p1(u),

Due to the bound B2(∆) along with Eq. 41 one has

mg −∆∗ ≤ |u| ≤ mg + ∆∗.

Using this relationship and substituting for previous definitions one obtains the structuralform of the constants p0

3, p13, q0

3 and q13

p03 −∆∗p1

3 ≤ p3(u) ≤ p03 + ∆∗p1

3, q03 −∆∗q1

3 ≤ q3(u) ≤ q03 + ∆∗q1

3 .

16


The following three conditions follow from the bounds B1 and B2. Condition i) followsfrom the limiting condition ∆∗ ≥ mg − k0(c4+2|π0| |π5|)

k1|π0| in the bounds B1 and B2 whileConditions ii) and iii) are direct consequences of bounds B1 and B2 along with the definitionsof p3(u) and q3(u):

i) |Ω(0)| < KΩ := 1(mg−∆∗)|π0| (k0(c4 + 2|π0| |π5|) + k0|π4|(mg + ∆∗) + |π0| |τ5|B∗)

ii) σ < mg−∆∗c6+|π0| |p3(u)|+ 1

k0(|p3(u)| |q3(u)|B∗+(mg−∆∗)k2|π0|)

iii) |π6|k0 + |τ5|B∗ < ∆∗.

It remains to show that these identities lead in turn to the theorem statement.

At time zero, the first bound in Eq. 44 is satisfied directly by the theorem statement.The second bound follows from Condition i). Recall Equation 41 and (at time t = 0) applythe bounds |L(0)| < k2

0 and Ω(0) < KΩ. Combining this with Condition iii) it follows thatu(0) > ∆∗. This shows that the bounds are valid at time zero. Moreover, the evolution ofthe state of the helicopter is smooth and we may proceed using a proof by contradiction.

Assume that the bounds Eq. 44 are not valid for all time. Then there exist a first timet0 such that for all 0 ≤ t < t0 the bounds Eq. 44 are valid. The contradiction is shown caseby case:

Assume that |u(t0)| = ∆∗ and that |L(t)| ≤ k20 and Ω(t) ≤ KΩ on the interval t ∈ [0, t0].

But now an argument identical to that given in the paragraph above shows that for allt ∈ [0, t0] then |u(t0)| > mg −∆∗. This ensures that the bound on |u| cannot be the firstbound which is broken.

Assume that |L(t0)| = k20 and that |u(t)| > mg − ∆∗ and Ω(t) ≤ KΩ on the interval

t ∈ [0, t0]. Using the bound on Ω(t) < KΩ one obtains

|u| |Ω| ≤ KΩ|u|, |Ω|2 ≤ KΩ|Ω|

Combining Equations 40 with the parameters given in the theorem statement one obtainsthe following relation bound on w:

|w| ≤ 1mg −∆∗ − σc6 (〈q3(u), χ〉+ 〈p3(u), γ〉+ (mg −∆∗)|w0|)

Observe that Condition ii) ensures that

σ <mg −∆∗

c6 + |π0| |p3(u)|+ a0

where a0 ≥ 0. Thus, mg − ∆∗ − σc6 > 0. Substituting into Equation 35 it follows thatL(t) < 0 if

γTΛγ ≥ σ

mg −∆∗ − σc6(γT p3(u)πT0 γ + γTπ0q3(u)Tχ+ (mg −∆∗)|w0|πT0 γ

)

But on the interval t ∈ [0, t0] one has L(t) ≤ k20, |χ| ≤ B∗ and |w0| ≤ k2. Applying these

bounds in the above equation it may be shown that this condition is equivalent to Condition

17


ii). Since L(0) < k20 is valid at time zero then L(t) < k2

0 remains strictly bounded on theinterval t ∈ [0, t0]. This ensures that the bound L(t) < k2

0 cannot be the first bound whichis broken.

Assume that Ω(t0) = KΩ and that |u(t)| > mg − ∆∗ and |L(t)| < k20 on the interval

t ∈ [0, t0]. Recall that

|Ω| ≤ 1mg −∆∗

(〈q1(u), χ〉+ 〈p1(u), γ〉+ σc4|w|) .

then substituting for the bound on |w| obtained above one obtains

|Ω| ≤ 1mg −∆∗

(〈q1(u), χ〉+ 〈p1(u), γ〉

+σc4

mg −∆∗ − σc6 (〈q3(u), χ〉+ 〈p3(u), γ〉+ (mg −∆∗)|w0|)).

Substituting for the bounds on χ and γ (from the bound on L) it may be shown2 that theright hand side is exactly the right hand side of Condition i). But then from the conditionsof the theorem |Ω(t0)| < KΩ which contradicts the assumption.

It follows by contradiction that all three bounds are valid for all time. 4

An important observation that follows from Theorem 4.3 is the explicit trade off betweenaccuracy of initial condition and the aggressiveness of the goal trajectory. This is best seenby regarding Eq. 43 where it is clear that increasing k0 directly results in a decrease in thebound B2(∆) and is expected to translate into a reduction of the margin B∗ available forthe trajectory choice. An advantage of the form in which Theorem 4.3 is stated is that thetradeoff between initial condition and tracking performance is naturally incorporated intothe analysis and no special cases need to be considered separately.

4.3 Local Asymptotic Stability for Constant Trajectories

It is interest to consider the particular case of stabilisation to a constant hover position ortracking a desired trajectory with a constant velocity. In this case no margin for the boundB∗ need be preserved since |χ| ≡ 0. Assume that the constant rotor drag required for hoveris known and pre-compensated such that the remaining air resistance terms QM and QT arenegligible.

Corollary 4.5 Consider the dynamics defined by Eqn’s 2-5. Let the controls w and u begiven by Eqn’s 30 and 31 and recover the control inputs w and u from Eqn’s 14 and 26.

Let k0 be a bound on the initial condition of the Lyapunov function

|L(0)| ≤ k20.

2The argument used here is too complicated to include here and adds little insight to the overall proof.

Interested readers may obtain the details by contacting the authors.

18


Assume that the anti-torques and the accelerations of the desired trajectories χ are zero

|χ| = 0 and |QM |+ |QT | = 0.

If there exists ∆ > 0 such that

i) |Ω(0)| < KΩ := 1(mg−∆)|π0|k0 (c4 + 2|π0| |π5|+ |π4|(mg + ∆))

ii) σ < mg−∆c6+|π0| |p3(u)|

iii) ∆ > |π6|k0

then the control w and u, given by Eqn’s 30 and 31 locally asymptotically stabilises theposture (δ1 → 0, ε3 → 0) and thus solves stabilisation and constant velocity trajectorytracking problems for the complete system Eqn’s 2-5.

Proof Note that the conditions in the Corollary statement are simplified versions of Con-ditions i), ii) and iii) used in the Proof of Theorem 4.3.

The first part of the proof is given by the proof of Theorem 4.3. That is L is alwaysbounded by k2

0. Considering that |u(t)| > ∆ and Ω(t) < KΩ for all time and using thebound on Ω(t) < KΩ one obtains

|u| |Ω| < KΩ|u|, |Ω|2 < KΩ|Ω|

Combining Equations 40 with the parameters given in the theorem statement yields thefollowing relation bound on w

|w| ≤ 1∆− σc6 〈p3(u), γ〉.

Substituting into Equation 35 it follows that L(t) < 0 if

γT (Λ− σ

∆− σc6 p3(u)πT0 )γ ≥ 0

Condition ii) ensures that

σ <∆

c6 + |π0| |p3(u)| ,

and consequently the Lyapunov function is strictly decreasing. Classical Lyapunov theorycompletes the proof. 4

5 Simulations

In this section some simulations indicating the performance of the algorithm proposed inSection 3 are presented.

19


Theorem 4.3 provides a means of trading off guarantees on tracking performance versesstabilisation robustness to initial condition error. In practice, this information is used toprovide parameter bounds for trajectory planning algorithms. Note that the bound k0 on theLyapunov function measures all the derivatives of the trajectory and not just the position.This is particularly interesting when one considers a trajectory constructed from piecewisesmooth template trajectories, such as trimming trajectories (Kaminer et al. , 1998; Hallberget al. , 1999; E. Frazzoli & Feron, 2000). The tradeoff in Theorem 4.3 provides an explicit apriori bound on the tracking performance when impulse changes in higher order derivativeterms contributing to the Lyapunov function are introduced.

The following simulations and plots are completed for an example scale model au-tonomous helicopter with parameters given in Table 1. These values represent typical valuesfor scale model autonomous helicopter capable of lifting a payload of around 3-4 Kg. Theparameters used for the dynamic model are based on preliminary measurements for a VARIO23cc scale model helicopter owned by HeuDiaSyC (CNRS Laboratory, Universite de Tech-nologie de Compiegne).

Parameter ValueMass 9.6 kgIa1 0.40 kg m2

Ia2 0.56 kg m2

Ia3 0.29 kg m2

|QM | 0.002|QT | 0.0002lT 1.2 mlM 0.27 mg 9.80 m s−2

σ 3.345

Table 1: Parameters of typical scale model autonomous helicopter.

To provide a perspective on the nature of the bounds and guarantees provided by The-orem 4.3 two plots of the bounding procedure and relationship between k0 and B∗ areprovided. A typical example of min sup bound proposed in Theorem 4.3 is shown in Figure3 for a value of k0 = 2. The values of k1 = 0.01 and k2 = 0.002 were used in the estimates.Note that the min sup of B1 and B2 is clearly defined at the intersection of the two graphs.The quadratic nature of the denominator of B1 leads to a singular asymptote in the plot ofB1 and the intersection of the upper branch with the linear bound B2 will always lead to theactual value of B∗. As a consequence of this asymptote the bound on ∆∗ never decreasesto zero. There is some physical insight in this structure since for the helicopter to achieveany trajectory it is necessary that there is some control margin mg + ∆∗ > u > mg −∆∗.Since there are always small body forces and initial trajectory errors incorporated in thestatement of Theorem 4.3 then a lower bound on ∆∗ is expected.

20


0 1 2 3 4 5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

∆

B1

(B2

)

B1

B1

B2

2.7 2.8 2.9 3 3.1−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

∆

B1

(B2

)

B1

B2

(∆*,B*)

Enlargement of dashed box shown in diagram on the right.

Figure 3: Graph of the bounds B1 and B2 with respect to ∆ for a fixed value of k0. Theright hand graph is a close up of the intersection point of the left hand figure.

Plotting a figure like Figure 3 for multiple choices of initial condition k0 it is possible todetermine a curve indicating the relationship between k0 and B∗. This relationship is shownin Figure 4 for the same choices of k1 and k2 (k1 = 0.01, k2 = 0.002).

Consider the case where k0 → 0 and consequently L(0) → 0 corresponding to initialconditions that exactly match the desired trajectory motion. In this case the bound B∗converges to a finite limit. This is a physically sensible property since it is clear that forarbitrarily aggressive trajectories the small body forces generated will destroy the track-ing properties of the algorithm no matter how small the initial estimate of the Lyapunovfunction. Effectively, the maximal value of B∗ (Bmax

∗ ≈ 0.104) provides a bound on theguaranteed asymptotic tracking performance of the closed-loop system in the presence ofsmall body forces. In the converse case, as B∗ is decreased toward zero, it appears that theinitial bound on the Lyapunov function increases to infinity. In practice, there is a physicallimit on the control action 0 < mg − ∆ < u that limits the range of ∆∗. Applying thisconstraint in Eq. 43 prevents one from using the theoretical min sup measured from Figure3 and leads to a hard constraint on the maximum initial value of k0. This physical constraint

21


prevents Corollary 4.5 from being extended to a semi-global asymptotic stability result.

0 2 4 6 8 10 12 14 16 18 200.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

k0

B*

Figure 4: Graph of the relationship between k0 and B∗ for fixed values of k1 and k2.

Two simulations of the performance of the closed-loop system have been undertaken.The first experiment considers the case of stabilisation of the helicopter dynamics to astationary configuration given an initial offset. The second considers the case of trajectorytracking. The trajectory chosen for the tracking simulations is a helix ascending in thevertical direction. For all simulations the helicopter is initially in hover with heave inputu0 = gm ≈ 96 sustaining its flight. The initial position is

ξ0 = ξ0 =

000

, φ0 = φ0 = 0.

In the first simulation the stabilisation of position of the helicopter is considered. Thedesired position ξ0 and η0 are chosen to be:

ξ0 =

12−4

, φ0 =

π

2rad.

The evolution of the dynamic state of a the scale model autonomous helicopter is shown inFigure 5 .

The second experiment concerns the problem of tracking a trajectory. In this case, thedesired trajectory was chosen as a helix ascending from a point near the centre of mass ofthe helicopter. The initial condition is characterised by the fact that L(0) ≈ 7.

22


00.5

1

01

20

2

4

xy

z

0 0.5 10

1

2

3

4

x

z

0 0.5 10

0.5

1

1.5

2

x

y

0 20 400

1

2

3

t

φ

0 20 40−15

−10

−5

0

5x 10

−3

t

θ

0 20 40−10

−5

0

5x 10

−3

t

ψ

0 20 400

0.5

1

t

x

0 20 400

0.5

1

1.5

2

t

y

0 20 400

1

2

3

4

t

z

Figure 5: Evolution of the states for the case of stabilisation of scale model autonomoushelicopter.

The velocity of the desired trajectory is

˙ξ =

0.1 cos φ0.1 sin φ−0.05

,

˙φ0 = 0.1

Thus, |χ| ≤ B∗ = 0.0142. Note that only the derivative (and higher derivatives) of thevelocity of the desired trajectory enters into the calculation of χ. Thus, the bound B∗ isusually much smaller than a bound directly on derivatives of the velocity of the trajectory.This is important since it allows one to analyze high speed trajectories as long as they arerelatively smooth. The evolution of the dynamic state for the second simulation is shown inFigure 6.

6 Conclusions

In this paper, the question of obtaining an a priori bound on the quality of tracking forarbitrary trajectories was considered for a dynamic model of a scale model autonomoushelicopter. A Lyapunov control function was derived for an approximate model (in which

23


Figure 6: Trajectory tracking for a helicopter in the presence of small body forces. Solidlines show the evolution of the the actual trajectory while the dashed lines represent theevolution of the desired one.

the small body forces are neglected) using backstepping techniques. Using the robustnessof Lyapunov control design approach the dynamics of the closed system was analyzed and atheorem was given that provides a priori bounds on initial error and the trajectory parame-ters that guarantee acceptable tracking performance of the system. The analysis is expectedto be of use in practical trajectory planning procedures. To the authors knowledge this isthe first time that quantitative a priori robustness bounds for arbitrary bounded trajectorieshave been derived for a rigid-body system with ‘small body force’ dynamic perturbations.

References

E. Frazzoli, M. Dahlen, & Feron, E. 2000. Trajectory tracking control design for autonomoushelicopters using a backstepping algorithm. Proceedings of the American Control Con-ference ACC, 4102–4107.

Freeman, R. A., & Kokotovic, P. V. 1996. Robust Nonlinear Control Design: State-space

24


0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

t

L

Figure 7: Evolution of the Lyapunov function.

and Lyapunov techniques. Systems and Control: Foundations and Applications. Boston,U.S.A.: Birkhauser.

Ghanadan, R. 1994. Nonlinear control system design via dynamic order reduction. Pages3752–3757 of: Proceedings of the Conference on Decision and Control.

Goldstein, H. 1980. Classical Mechanics. Second edn. Addison-Wesley Series in Physics.U.S.A.: Addison-Wesley.

Hallberg, E., Kaminer, I., & Pascoal, A. 1999. Flight test System for Unmanned Air Vehicles.IEEE Control Systems Magazine, 19(1), 55–65.

Hauser, J., Sastry, S., & Meyer, G. 1992. Nonlinear control design for slightly non-minimumphase systems: Applications to V/STOL aircraft. Automatica, 28(4), 665–679.

Kaminer, I., Pascoal, A., Hallberg, E., & Silvestre, C. 1998. Trajectory Tracking for au-tonomous vehicles: An integrated appraoch to guidance and control. Journal of Guid-ance and Control, 21(1), 29–38.

Koo, T. John, & Sastry, S. 1998. Output tracking control design of a helicopter model basedon approximate linearization. In: Proceedings of the IEEE Conference on Decision andControl CDC’98.

Koo, T. John, Hoffmann, Frank, Shim, Hyunchul, & Sastry, Shankar. 1998. Control Designand Implementation of Autonomous Helicopter. Invited session in the Conference onDecision and Control (CDC’98), Florida, 1998.

Krstic, M, Kanellakopoulos, I, & Kokotovic, P V. 1995. Nonlinear and Adaptive ControlDesign. Providence, Rhode Island, U.S.A.: American Mathematical Society.

25


La Civita, M. 2002 (December). Integrated Modeling and Robust Control ofor full-envelopeflight of robotic helicopters. PhD, Department of Mechanical Engineering, CarnegieMellon University, Pittsburgh, Pennsylvania.

Liceaga-Castro, E., , Bradley, R., & Castro-Linares, R. 1989. Helicopter control design usingfeedback linearization techniques. Pages 533–534 of: Proceedings of the 28th Conferenceon Decision and Control, CDC’89.

Mahony, R., & Lozano, R. 2000. (Almost) Exact Path Tracking Control for an AutonomousHelicopter in Hover Manoeuvres. In: Proceedings of the International Conference onRobotics and Automation, ICRA2000.

Mahony, R., Hamel, T., & Dzul, A. 1999. Hover Control via Approximate Lyapunov Controlfor a Model Helicopter. In: Proceedings of the 1999 Conference on Decision and Control.

Mallhaupt, P., Srinivasan, B., Levine, J., & Bouvin, D. 1997 (July). A toy more difficult tocontrol than the real thing. Pages CD–ROM Version of: Proceedings of the EuropeanControl Conference.

Martin, P., Devasia, S., & Paden, B. 1996. A different look at output tracking: Control ofa VTOL aircraft. Automatica, 32(1), 101–107.

Mettler, B. 2003. Identification modeling and characteristics of miniature rotorcraft. KluwerAcademic Publishers group, P.O. box 322, 3300 AH Dordrecht, The Netherlands.:Kluwer Academic Publishers.

Murray, R., Rathinam, M., & van Nieuwstadt, M. 1996. An Introduction to differentialflatness of mechanical systems. In: Proceedings of Ecole d’Ete, Theorie et practique dessystemes plats.

Prouty, Raymond W. 1995. Helicopter Performance, Stability and Control. Krieger Pub-lishing Company. reprint with additions, original edition 1986.

Schaufelberger, W., & Geering, H. 1998. Case Study on Helicopter Control. Invited sessionin Control of Complex Systems (COSY) Symposium, Macedonia, October, 1998.

Sepulchre, R, Jankovic, M, & Kokotovic, P. 1997. Constructive Nonlinear Control. London:Springer-Verlag.

Shakernia, O., Ma, Y., Koo, T., & Sastry, S. 1999. Landing an Unmanned Air Vehicle:Vision based motion estimation and nonlinear control. Asian Journal of Control, 1(3),128–145.

Shim, H., Koo, T., Hoffmann, F., & Sastry, S. 1998. A Comprehensive Study of ControlDesign for an Autonomous Helicopter. In: Proceedings of the 37th Conference onDecision and Control CDC’98.

Sira-Ramirez, A., Castro-Linares, R., & Liceaga-Castro, E. 1999. Regulation of the longetu-dinal dynamics of an helicopter system: A Liovillian systems approach. In: Proceedingsof the American Control Conference ACC’99.

26


Sira-Ramirez, H., Castro-Linares, R., & Liceaga-Castro, E. 2000. A Liouvillian systemsapproach for the trajectory planning-based control of helicopter models. InternationalJournal of Robust and Nonlinear Control, 10, 301–320.

Teel, A. R. 1996. A nonlinear small gain theorem for the analysis of control systems withsaturation. IEEE Transactions on Automatic Control, 41(9), 1256–1270.

Thomson, D., & Bradley, R. 1987. Recent develoments in the calculation of inverse dynamicsolutions of the helicotper equations of motion. In: Proceedings of the U.K. SimulationCouncil Triennial Conference.

van Nieuwstadt, M., & Murray, R. 1997a. Outer Flatness: Trajectory Generation for aModel Helicopter. In: Proceedings of the European Control Conference (CD-ROM).

van Nieuwstadt, M.J., & Murray, R.M. 1997b. Real time trajectory generation for differen-tially flat systems. International journal of robust and nonlinear control, may, 1–27.

27

robust trajectory tracking for a scale model autonomous ... · robust trajectory tracking for a...

Documents