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Airship control

Leonardo Solaque1 and Simon Lacroix2

1 : De los Andes UniversityCarrera 1ra No. 18A-10Bogota, [email protected]

2 : LAAS-CNRS7, Avenue du Colonel RocheToulouse, [email protected]

Summary. This chapter presents an approach to control small sized airships toreach a given goal position, following a planned trajectory. Two sections are de-voted to modeling and identification: a simplification of the airship dynamics whenthe airship speed is stabilized yields to a decoupling of the lateral and longitudi-nal dynamics. Various controllers to stabilize the speed, altitude and heading areproposed and analyzed. An investigation on the generation of feasible trajectoriesis proposed, and a controller dedicated to follow the generated trajectories is in-troduced. Simulations results illustrate the developments, as well as experimentalresults obtained with two di!erent airships.

1 Introduction

The ever on-going developments in a wide spectrum of technologies, rangingfrom actuator, sensors and computing devices to energy and materials willensure lighter than air machines a promising future. There is undoubtedly aregain of interest in this domain, as shown by recent industrial developmentson heavy loads transportation projects 1, High Altitude Long Enduranceplatforms and surveillance applications. As for small size unmanned radio-controlled models, which size is of the order of a few tens of cubic meters,their domain of operation is mainly advertising or aerial photography. Theyare easy to operate, they can safely fly at very low altitudes (down to a fewmeters), and they can hover a long time over a particular area, while beingable to fly at a few tens of kilometers per hour, still consuming little energy.1 Such as the ATG Skycats – up-to-date information on this project is available at

www.worldskycat.com

2 Solaque and Lacroix

Their main enemy is the wind (see [EBB+98] for a detailed and convincing re-view of the pros and cons of small size airships with regards to helicopters andplanes). Let’s also note that some specific applications of unmanned blimpsare more and more seriously considered throughout the world, from planetaryexploration to military applications, as shown by numerous contributions inthe AIAA Lighter Than Air conferences and European Airship Conventionsfor instance [AIA05, AIR04].

The first mentions of the development of unmanned autonomous blimpscan be found in the literature of the early 90’s [Bos93], but it is only recentlythat various projects have reached e!ective achievements. One of the mostadvanced is Aurora, a project held at the Information Technology Institute ofCampinas, Brazil, mainly devoted to flight control [dPBG+99a, AdPR+01].Other projects are held at the university of Virginia [Tur00], at the univer-sity of Stuttgart [KSK01], the university of Wales [BSS00], the university ofEvry [HB03b]. An interesting characteristics of such projects is that they mixvarious innovative technological developments and fundamental research.

Chapter overview and outline

Flight control is of course the first problem to tackle in order to endow anairship with the ability to autonomously achieve any mission. Di"culties arisemainly because of the non-linearities of the airship model and the wind pertur-bations that strongly a!ects the flight. We focus here on this problem, aimingat defining means to control small sized airships to reach a given goal position,following a planned trajectory. Section 2 presents a complete dynamic airshipmodel, and simplifications made to ease the synthesis of control laws. Underthe hypotheses that the speed is regulated (“cruise flight hypotheses”), thesesimplifications allow to decouple the lateral and longitudinal dynamics, yield-ing two sub-models. Section 3 then presents means to estimate the parametersof the established models, so that proper control laws can be defined.

The overall approach to reach a goal while following a trajectory is sketchedin figure 1. It relies on three independent controllers that regulate the speed,heading and altitude, the heading reference being set by the path followingcontrol loop. Controls laws to stabilize speed, altitude and heading duringcruise flight are presented in section 4, and section 5 presents how feasibletrajectories can be planned and executed. References to previous work in theliterature are presented throughout the sections, as well as simulation resultsand experimental results obtained either with the airships Karma and UrAn(figure 2).

2 Airship modeling

The considered airships have a classic “cigar shaped” structure. The availablecontrol parameters are presented in figure 3: the main thrusters are mounted

Airship control 3

Fig. 1. Overall control system scheme. Three independent controllers are in chargeof regulating the speed, heading and altitude of the airship, on the basis of theestimated state. When a trajectory to follow is defined, a control loop sets thereference for the heading.

Fig. 2. The two airships UrAn (Andes University, 28 m3) and Karma (LAAS, 18m3).

on a vectorized axis, thus enabling vertical take o! and providing additionallift at slow speeds, where no aerodynamic lift is possible. Rudders allowscontrol in the longitudinal and lateral planes, and an additional tail rotorenables lateral control at slow speed, where the rudders control surfaces arenot e"cient2.2 Only UrAn is actually equipped with such a thruster.


Fig. 3. The available control parameters. Right: detail of the vectorized thrusters ofUrAn.

2.1 Frames and kinematic model

Three frames are defined to describe the blimp motion (Fig.4): the globalframe R0 fixed to an arbitrary point on the earth, oriented along the NEDconvention. The airship body frame Rd has its origin at the center of volume(CV) of the hull [GR98]. CV is chosen as the origin of this frame becauseit is assumed to coincide with the center of buoyancy, where the aerostaticlift force is applied. Finally, the aerostatic frame aligned with the direction ofmotion is Ra.

Zo

Xd

CV

!"

Zd

Yd#

Ro

Yo

Xo

RaRd

CG

Fig. 4. Considered frames

The xd axis of Rd is coincident with the symmetry axis of the envelope,the (xd, zd) plane coincides with the longitudinal plane and the orientationof Rd with respect to R0 is given by the Euler angles roll !, pitch " and

Airship control 5

yaw #. The xa axis of Ra is coincident with the airship aerodynamic velocityvA|Ra

= (Mad )!1(vdl ! vw), where vdl and vw respectively denote the airship’s

velocity and the winds velocity with respect to Rd, and Mad is expressed as

in eq.2. $ is the angle of attack within the (xd, yd) plane and % is the skidangle within the (xd, yd) plane. The orientation matrix Md

0 between the globalframe R0 and local frame Rd is given by:

Mdo =

!

"c#c" !s#c!+ s!c#s" s!s# + s"c#c!c"s# c#c!+ s"s#s! !c#s!+ c!s"s#!s" c"s! c"c!

#

$ (1)

and the transformation matrix Mad between the local frame Rd and the

aeronautic frame Ra is:

Mad =

!

"c$c% !c$s% !s$s% c% 0

s$c% !s$s% c$

#

$ (2)

where sx (resp. cx) denotes the function sin(x) (resp. cos(x)).

2.2 Dynamic model

The dynamic model of the airship is established on the basis of the followingassumptions:

• The hull is considered as a solid: aero-elastic phenomena are ignored, andso are the motion of helium inside the hull (no phenomenon of inertialadded fluid due to such motion is considered);

• the mass of the blimp and its volume are considered as constant;• the blimp displaces volume: the phenomenon of added fluid induces a

variation of inertia and mass and is significant (proportional to the volumeof air displaced by the hull);

• the center of buoyancy is assumed to coincide with the CV;• as the speed for a small blimp is generally low (low Match number), the

couplings between dynamics and thermal phenomena are neglected andthe density of air is not locally modified by the system’s motion;

• the Earth is considered as flat over the flight area.

These assumptions are reasonable for the considered airships, and allowto apply the rigid body mechanics theory. Through the use of Newton’s lawsof mechanics, aero-dynamical theory [BL03] with Kirsho!’s law and Bryson’stheory, the dynamic model with respect to the Rd frame can be written as:

Mdv = !Td(vdlr) + Ta(vA) + ga + Tp (3)

Where:


• v is the airship’s speeds states vector within Rd, which is composed of thelinear velocities vdl = [u, v, w]T and angular velocities vdr = [p, q, r]T . Forcontrol purpose, it is expressed in the frame R0.

• the 6"6 matrix Md is composed of masses, inertias and the correspondingcoupling terms:

Md =%

mI3 !mAmA IN

&=

!

''''''"

m 0 0 0 mzg 00 m 0 !mzg 0 mxg

0 0 m 0 !mxg 00 !mzg 0 Ix 0 !Ixz

mzg 0 !mxg 0 Iy 00 mxg 0 !Ixz 0 Iz

#

(((((($(4)

where (xg, zg) is the gravity center position.• Td is the dynamic force vector and contains centrifugal and Coriolis terms:

Td(vdlr) =

!

''''''"

mwq ! mvr ! mxgq2 + mzgpr ! mxgr2

mur ! mwp + mzgqr + mxgpqmvp ! muq ! mzgp2 + mxgpr ! mzgq2

!mzgur + mzgwp ! Ixzpq ! (Iy ! Iz)qr!mzgvr ! mxgvp + mzgwq + mxguq + Ixz(p2 ! r2) ! (Iz ! Ix)pr

!mxgwp + mxgur ! (Ix ! Iy)pq + Ixzqr

#

(((((($

(5)• ga gathers the gravity and buoyancy forces (FG is the airship’s weight, FB

is the buoyancy lift):

ga =

!

''''''"

!(FG ! FB)s"(FG ! FB)c"s!(FG ! FB)c"c!!zgFGc"s!

!zgFGs" ! (xgFG ! xcFB)c"c!(xgFG ! xcFB)c"s!

#

(((((($(6)

• Tp represents the controls applied on the airship: it contains the torquesof the vectored thrust FM at (Ox, Oz) position and of the tail rotor thrustFrc at (xrc, zrc) position. The norm and the direction µ of the propellersare adjustable within the longitudinal plane and the direction of the tailrotor thrust within the lateral plane.

Tp =

!

''''''"

FMcµFrc

!FMsµFrczrc

FMOzcµ + FMOxsµFrcxrc

#

(((((($(7)

Airship control 7

• Ta = A ˙vA !D(vdr)vA + Tsta(v2a) is the aerodynamic and moments vector,

where:– A is the 6 " 6 symmetric matrix of added masses, inertia at center of

gravity (CG), and coupling terms of the fluid (virtual mass and inertiaterms – [Tuc26] presents how these coe"cients can be estimated):

A ˙vA =

!

''''''"

a11 0 0 0 a15 00 a22 0 a24 0 a26

0 0 a33 0 a35 00 a42 0 a44 0 a46

a51 0 a53 0 a55 00 a62 0 a64 0 a66

#

(((((($

!

''''''"

VXa|Rd

VYa|Rd

VZa|Rd

pqr

#

(((((($(8)

– vA = [va, vdr], where va = vdl ! vw is the vector of aerodynamictranslational velocity, vw = [uw, vw, ww]T being the wind velocity inRd. A study on the influence of wind over airship can be found in[Azi02].

– D(vdr)vA is a vector that contains the centrifugal and Coriolis terms:

D1(vdr)vA =

!

''''''"

0 a22r !a33qD1z1 0 a33p

(a11 ! xm22)q !a22p 0D1z2 !(a62 + a35)q D1z3

(a35 + x2m22)q !a42ra62p !a15qD1z4 (a15 + a42)q !a53p

a24r !a35q a26ra35q !a15q 0

!a24p ! a26r a15q 0!a64q (a55 ! a66)r 0D1z5 0 !a64r

(a44 ! a55)q a46r 0

#

(((((($

!

''''''"

VXa|Rd

VYa|Rd

VZa|Rd

pqr

#

(((((($

(9)

where D1z1 = pm13 + r(xm11 ! a11), D1z2 = pm33 + r(a15 + xm13),D1z3 = (a62+a35)r+a24p, D1z4 = !(a51+a24!xm13)p!(a26!x2m11)ry D1z5 = a64p + (a66 ! a44)r.

– Tsta(v2a) represents the aerodynamic forces and moments at CG, which

are proportional to the form of the hull and the square of the aerody-namic velocity.

Tsta(va2) =

!

''''''"

12&va

2SrefCT12&va

2SrefCL12&va

2SrefCN

! 12&va

2SrefLrefCl

! 12&va

2SrefLrefCm

! 12&va

2SrefLrefCn

#

(((((($=

!

''''''"

Fx

Fy

Fz

L0

M0

N0

#

(((((($(10)


where Sref and Lref depend of the airship geometry. CT , CL, CN , Cl,Cm and Cn are, respectively, tangential, normal, lateral, roll, pitch and yawstationary coe"cients.

2.3 Simplified model

In order to obtain a tractable dynamic model, additional simplifications canbe made, by restricting the domain of operation and decoupling lateral andlongitudinal controls. In the case of a cruise flight in the absence of wind, theequation that links the blimp velocity and thrust control comes down to:

u =1

(m ! a11)(Fmcosµ +

12&V 2

a SrefCT ) (11)

Provided that the airship speed is regulated (u = u0) and that it is flyingalong a straight line ('g = 0, v = 0), the evolution model of the altitude canbe simplified. The involved variables in the model are z, w, u, ", q and 'e:considering that w # u and $ $ 0, we have:

z = !u sin "" = q

q = ((mxg!a35!x2m22)uq!zgFG sin !!(xgFG!xcFB) cos !+FmOz cos µ+FmOx sin µ! 12"u2Sref Lref CmN )

Iy!a55

(12)By linearizing around the operating point,!(xgFG!xcFB) cos "+FmOz cosµ $

0, " $ 0 % sin " $ " and cos " $ 1, we have:

z = !u sin "" = q

q = k2|#eq + k1|#e

" + k3|#e'e

(13)

where k2|#e= (mxg!a35!x2m22)u

Iy!a55, k1|#e

$ !zgFG

Iy!a55and k3|#e

$ ! "u2Sref Lref CmN

2(Iy!a55) .Equation (13) corresponds to a second order control system in ", and a thirdorder system in z. Considering the dynamic of " similar as a first order, themodel of the altitude z can be seen as a first order system plus an integralterm.

Considering now a constant speed motion in the plane (x, y) with 'e =0 and w = 0, the dynamic equations of the blimp and its position in thehorizontal plane can be written as:

x = u cos# ! v sin#y = u sin# ! v cos#

# = r

r = ur(mxg+a26!x2m11)+Frcxrc! 12"u2Sref Lref CnN

(Iz!a66)

(14)

The involved variables in this model are x, y, u, v, #, r and 'g. Consideringv # u, % $ 0 and Frc = 0, the equations (14) can be simplified in:

Airship control 9

x = u cos#y = u sin## = r

r = k2|#gr + k1|#g

'g

(15)

Note that the coe"cients k1|#gand k2|#g

can be found similarly to theabove altitude model reduction procedure (a similar application of simplifieddynamics for an airplane model in cruise flight can be found in [FNC93]).

3 Model identification

One way to estimate the aerodynamic model parameters is the use of wind-tunnels experiments with Munk’s hull equations [Mun36]. Jones and DeLau-rier [JD83] estimate the aerodynamic coe"cients of an airship from some the-oretical results and wind-tunnel data. Gomes and Ramos [GR98] propose thatthe virtual masses and inertia can be basically estimated with the Munk’s hullequations. Furthermore, they show that the aero-dynamic coe"cients of equa-tion (10) can be obtained either from direct measurement in a wind-tunnel,from the geometrical characteristics of the blimp, or from aerodynamic stabil-ity derivatives. The coe"cients proposed by Hygounenc and Soueres [HS02b]have been determined by means of wind-tunnel tests made with a scaled modelof Karma (see photo 5).

Fig. 5. Scale model of Karma

Few publications address the estimation of airship aerodynamical param-eters from real world flight data. The dynamic identification of an airship


is presented in [JBMZ04], where the authors develop a general aerodynamicmodel at the reference operation condition of the airship, based on Munk’stheory and Jones and DeLaurier’s contributions. Yamasaky and Goto [YG03a]with the objective to construct models for the development of control laws,present two experiments in order to identify the airship flight dynamics: theseexperiments consist in constrained flight tests and indoors free-flying tests.

In the following section (section 3.1), we present an approach to estimatethe aerodynamic parameters from cruise flight data. Section 3.2 analyzes theparametric variations and the validity of the models simplifications presentedin section 2.

3.1 Estimation of the aerodynamic parameters

To avoid experimental aerodynamic measurements in wind tunnels, we adopteda numerical technique that determines the system coe"cients (() by fittingtime series data obtained from state measurements of the system. The di"-culty arises from the fact that the acquired data are corrupted by noise: wepresent here an approach to parameters identification based on the reduc-tion of Ta(vA) when the airship is in cruise flight, using the Kalman filter asa parameters identifier. The unknown parameters are in the state vector ofthe filter, and are constant in the particular case of a steady state regime:(k = (k!1. Therefore, the state space is:

xk+1 = f(xk,(k, uk) + )k(k = (k!1 + )i

(16)

where x, y, u and ( are, respectively, the state system, the output system,the input system and the unknown coe"cients, and )k,i are gaussian noises.The observation function is:

yk = h(xk) + *k (17)

The expression of Ta(vA) is cumbersome. With the assumptions presentedsection 2 (cruise flight), we can simplify Ta(vA) and still obtain a generalrepresentation which includes the aerodynamic phenomena:

Ta(vA) =

!

''''''"

k1(cT1 + cT2$+ cT3'e)k1(cL1 + cL2% + cL3'g)k1(cN1 + cN2$+ cN3%)k2(cl1 + cl2$+ cl3'e)

k2(cm1 + cm2% + cm3'g)k2(cn1 + cn2$+ cn3%)

#

(((((($(18)

where k1 = 12&Sref and k2 = 1

2&SrefLref are constants depending on theairship geometry. This simpler model of the aerodynamics of the airship incruise flight also simplifies the formulation of the control laws.

Airship control 11

Estimation in simulation

To determine the performance and reliability of our method, we have carriedout simulation tests, comparing the results of the Extended Kalman Filter(EKF) and the Unscented Kalman Filter (UKF, in [CM03] the authors presentan application of UKF on spacecraft attitude estimation) as parametric iden-tifiers, with known system parameters. These parameters, that we are goingto take as real, were identified by wind-tunnel test, as Hygounenc and Souerespresent in [HS02b].

We use N = 1500 samples and the Euler method to integrate the di!er-ential equations with a step size +T = 0.1. Notice that the integration stepsize may di!er from the sampling time. The state measurements are corruptedwith an additive Gaussian noise (similar to the on-board real sensors). Re-sults are shown in the left side of Table 2. Figure 6 shows the evolution ofthe estimates of the parameters CT2, Cm1 and Cm3, and table 1 shows themean and variance of the error e = |x!x|

|x| for the blimp state (speed vector[u, v, w, p, q, r]T in Rd and position vector [x, y, z,!, ",#]T in R0). For bothmodel and parameters, the UKF approach converges more rapidly to a betterestimate than the EKF.

Fig. 6. Evolution of the parameters CT2, Cm1 and Cm3

Estimation with real world data for Karma.

Once made the study of the algorithms with simulation data, we use real dataacquired by the GPS and the compass on-board Karma, that respectivelyprovides estimates of (x, y, z, x, y, z) and (!, ",#) expressed in R0, at a GPSrate of 1Hz and a compass rate of 10Hz. Data acquired during a 923 second


Table 1. Table of error

State Mean UKF Variance UKF Mean EKF Variance EKFu 0.0177 0.0004 0.0190 0.0006v 0.0034 0.0006 0.0058 0.0007w 0.0156 0.0003 0.0287 0.0007p 0.0016 0.0001 0.0017 0.0002q 0.0019 0.0001 0.0023 0.0002r 0.0057 0.0001 0.0020 0.0002x 0.0439 0.0015 0.0639 0.0016y 0.0670 0.0025 0.0209 0.0027z 0.0217 0.0038 0.0086 0.0040! 0.0626 0.0009 0.0651 0.0009" 0.0179 0.0001 0.0209 0.0003# 0.0153 0.0171 0.0603 0.0018

long trajectory are exploited to estimate the airship model parameters usingan UKF. Figure 7 shows the convergence for some model parameters, and theright side of table 2 shows all the estimated values of the model parametersof the Karma with their covariance.

Fig. 7. Convergence of some model parameters with time.

Airship control 13

Table 2. Left side: Parameter estimation from simulated data and right side: Pa-rameter estimation from real world data

Parameter Real EKF UKFa11 1.5276 1.2633 1.9660a22 21.0933 20.8660 21.1896a33 20.4220 20.2018 21.4239a44 16.3905 15.9151 16.8548a55 382.1290 380.0030 380.1463a66 388.0972 379.9988 384.0155a15 = a51 0 0.1309 0.0001a24 = a42 0 !0.0958 0.1628a26 = a62 !69.6939 !59.9030 !69.8978a35 = a53 67.7045 70.0684 70.3361a46 0 0.0801 0.0101m13 1.2801 2.0621 1.5235m33 !49.7019 !48.0249 !48.5013xm11 25.6919 23.0748 24.5183xm22 23.6878 20.0129 21.0075xm13 !4.5582 !9.1165 !5.4699x2m11 !173.4906 !150.0044 !170.8227x2m22 !166.3538 !149.9994 !158.8524CT1 ! !1.8974 !0.6579CT2 ! 0.1071 0.1069CT3 ! !2.8752 !0.6877CL1 ! 50.8756 15.4789CL2 ! !15.9874 !11.5582CL3 ! !1.2234 !7.2243CN1 ! !0.0417 !0.0664CN2 ! 0.5487 0.5789CN3 ! 1.0258 0.389Cl1 ! 0.0205 0.0304Cl2 ! !0.1919 !0.1271Cl3 ! 0.0170 0.0266Cm1 ! !0.0405 !0.0415Cm2 ! 0.7975 0.9982Cm3 ! 0.2930 0.2173Cn1 ! 1.0833 0.6207Cn2 ! !0.8574 !0.7589Cn3 ! 0.0450 !0.0010

Parameter UKF Covariance(Pii)a11 11.2428 0.1580a22 90.4922 0.0925a33 70.5841 0.0922a44 42.3387 0.0901a55 383.7979 0.0837a66 419.9314 0.0872a15 = a51 6.9103 0.1309a24 = a42 1.2382 0.1240a26 = a62 !195.3407 0.1269a35 = a53 59.4323 0.1053a46 !28.5030 0.1053m13 33.7772 0.0621m33 !93.7707 0.0982xm11 76.4093 0.0905xm22 54.7163 0.0192xm13 75.3240 0.0962x2m11 !201.9972 0.0335x2m22 !224.8353 0.0896CT1 !2.9074 0.1290CT2 !0.2250 0.0446CT3 !0.7970 0.0767CL1 15.0799 0.0744CL2 !7.6177 0.0644CL3 !3.2706 0.0249CN1 !2.1196 0.0676CN2 !13.9818 0.0949CN3 0.6837 0.0508Cl1 5.1576 0.0538Cl2 2.9208 0.1509Cl3 1.0168 0.0582Cm1 !0.0725 0.1442Cm2 !1.8937 0.0814Cm3 1.1017 0.0762Cn1 !0.1082 0.0942Cn2 !0.5101 0.0415Cn3 0.0115 0.0227

3.2 Analysis of the validity of the reduced models

The approach to airship parameters estimation in cruise flight, presented insection 3.1, enables to knows the parameters of simplified models. To makesure the control laws derived from the reduced models are sound, an analysisof the reduced models validity is required. The classical techniques used forsystem identification such step response and extended least squares enable


to say that the airship global dynamics can be represented by the simplifiedmodels. This section presents such an analysis, using an Extended KalmanFilter to estimate the state of each simplified model and the evolution of theirparameters.

Speed vs thrust propeller va/T

Let us consider the model (11) with the state estimation , = [x,-, . ]T . Thecorresponding Euler model is given by:

x(k + 1) = x(k) + Tm$(k) (-(k)u(k) ! x(k))

.(k + 1) = .(k)-(k + 1) = -(k)

(19)

It is possible to demonstrate that the matrix F (the linearized form of fof the state equation) can be written as:

F =

!

"(1 ! Tm

$ ) Tmu$

Tm$2 (!-u + x)

0 1 00 0 1

#

$ (20)

Figure 8 shows the evolution of state estimation and figure 9 present astability and statistic analysis obtained in simulation.

Altitude vs symmetrical deflection z/!e

Considering the model (13) with ,1 = [z, ", q, kz, k1, k2, k3]T as states for theestimation, the Euler model can be written as:

z(k + 1) = z(k) + Tmkz(k)"(k)"(k + 1) = "(k) + Tmq(k)q(k + 1) = q(k)(1 + Tmk1(k)) + Tm(k2(k)"(k) + k3(k)'e(k))

(21)

The F matrix is given by:

F =

!

''''''''"

1 Tmkz 0 Tm" 0 0 00 1 Tm 0 0 0 00 Tmk2 (1 + Tmk1) 0 Tmq Tm" Tm'e0 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

#

(((((((($

(22)

Figures 10 and 11 present the results obtained. They show the parametricconvergence of the model.

Airship control 15

0 50 100 150 2000

2

4

6

8

10

12

14Estimated of the speed state, x in blue and x* in red

Spee

d

Time [s]0 50 100 150 200

−0.5

0

0.5

1

1.5

2Variance

$Sp

eed

Time [s]

0 50 100 150 2000

2

4

6

8

% in blue and & in red

% an

d &

Time [s]0 50 100 150 200

−0.5

0

0.5

1

1.5

2Variance of $

% in blue and $

& in red

$% a

nd $&

Time [s]

Fig. 8. Evolution of speed, $ and % states

Heading vs dis-symmetrical deflection "/!g

For the model (15), the states for the estimation ,1 = [#, r, k1, k2]T , and theEuler model is given by:

#(k + 1) = #(k) + Tmr(k)r(k + 1) = r(k)(1 + Tmk2(k)) + Tmk1(k)'g(k) (23)

The F matrix can be written as:

F =

!

''"

1 Tm 0 00 (1 + Tmk2) Tm'g Tmr0 0 1 00 0 0 1

#

(($ (24)

The system states (# and r) are presented in figure 12, and an analysis ofestimation errors and pole movements can be seen in figure 13.

In summary, figures 8, 10 and 12 respectively show the validity of the sim-plified speed, altitude and heading models: the estimated parameters converge


−1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2−0.02

−0.01

0

0.01

0.02Pole movement

Imag

inar

y ax

is

Real axis

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1Gaussian distribution

error, x=[u − u*]

f(x)

Fig. 9. Analysis: poles movement and statistic error of speed state

to a fixed values. Figures 9, 11 and 13 show that the poles remain in the leftside of the complex plane when system evolves in cruise flight.

Once the simplified model study had been done and the test of the algo-rithms on simulation been made, we used real flight data. Figures 14 and 15show the speed state evolution and the corresponding stability analysis whenthe airship was moving in the cruise flight phase.

4 Control

Now that we have a tractable model with estimates of its coe"cients, onecan tackle the control problem. Airship flight control issues have now beenaddressed in various contributions in the literature. Elfes et al [EMH+05]present a control architecture where the yaw, pitch and altitude of the airshipare controlled on the basis of adaptive adjustments of PI and PID controllers.Paiva et al [dPBG+99b] describe a control strategy with a PID controllerfor the longitudinal velocity, and a PD controller for altitude and heading.Azinheira et al [AdPRB00] present a heading regulator based on H" controltechnique, exploited in a path following strategy. Moutinho and Azinheria

Airship control 17

0 50 100 1500

50

100Estimation: Altitude state, z in blue & z* in red

z &

'g*1

00 in b

lack

Time [s]0 50 100 150

−1012

Variance

$z

Time [s]

0 50 100 150−0.2

0

0.2Estimation: Pitch state, x in blue & x* in red

! [

deg]

Time [s]0 50 100 150

−1012

Variance

$!

Time [s]

0 50 100 150−0.4−0.2

00.2

Estimation: q state, x in blue & x* in red

q [

deg]

Time [s]0 50 100 150

−1012

Variance$

q

Time [s]

0 50 100 150−10

0

10kz in blue, k1 in red, k2 in black & k3 in magenta

% &

&

Time [s]0 50 100 150

024

Variances

$kz, $

k1, $

k2 &

$k3

Time [s]

Fig. 10. Evolution of z, ", q, kz, k1,k2 and k3 states

−1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0−5

0

5Pole movement

Ima

gin

ary

axis

Real axis

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.5


error, x=[z − z*]

f(x)

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250

0.5


error, x=[! − !*]

f(x)

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250

0.5


error, x=[q − q*]f(

x)

Fig. 11. Analysis: poles movementand statistic error of z, ", q states

0 50 100 150−0.2

0

0.2

0.4

0.6Estimation: " state, " in blue and "* in red

" &

'g in

bla

ck

Time [s]0 50 100 150

−1

0

1

2Variance

$"

Time [s]

0 50 100 150−0.02

0

0.02Estimation: d"/dt state, r in blue and r* in red

r [d

eg

]

Time [s]0 50 100 150

−1

0

1

2Variance

$r

Time [s]

0 50 100 150−1

−0.5

0

k1 in blue and k2 in red

k1 a

nd

k2

Time [s]0 50 100 150

−1

0

1

2

Variances: $k1 in blue and $k2 in red

$k1

an

d $

k2

Time [s]

Fig. 12. Evolution of #, r, k1 and k2

states

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−1

−0.5

0

0.5

1Pole movement

Imag

inar

y ax

is

Real axis

−8 −6 −4 −2 0 2 4 6x 10−4

0

0.5


error, x=[" − "*]

f(x)

−8 −6 −4 −2 0 2 4 6x 10−3

0

0.5


error, x=[r − r*]

f(x)

Fig. 13. Analysis: poles movementand statistic error of # and r states


0 50 100 150 200 250 300 350 400 450 5000

5

10

15Measured State (blue) & Estimated State (red .)

Tiem [s]Sp

eed,

u=V

x

0 50 100 150 200 250 300 350 400 450 5000

0.5

1Coefficient [k]

Iterations

Stat

ic ga

in [k

]

0 50 100 150 200 250 300 350 400 450 5000

5

10

15Coefficient [&]

Iterations

time

cons

tant

[&]

Fig. 14. Evolution of speed, $ and %states obtained on real world data

−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0−0.1

−0.05

0

0.05

0.1Pole mouvement

Imag

inar

y ax

is

Real axis

−6 −4 −2 0 2 4 60

0.2

0.4

0.6

0.8


error, x=[u − u*]

f(x)

Fig. 15. Poles movement and statisticerror of speed state based on real worlddata

[MA05] present a dynamic inversion controller (feedback linearization) for thelateral and longitudinal control, and an other feedback linearization approachcombined with PID controllers has been proposed by Zhang and Ostrowski[ZO99]. Some authors opt for fuzzy control approaches, such as Rao et al[JRZGX05], who describe a heading controller for navigation task based onfuzzy control optimized by genetic algorithms, and Diaz et al [DSG06] whopropose a parallel distributed compensation for cruise flight phase based onfuzzy techniques. A robust stability augmentation system for yawing motionhas been presented by Yong-Hwan MO et al [YG03b]. A control strategy withback-stepping techniques was proposed by Hygounenc and Soueres [HS02b].

In this section, we present and analyze three di!erent types of controls forthe airship: PID control, Generalized Predictive Control (GPC) and non-linearcontrol by extended linearization. All these regulators are conceived as SingleInput-Single Output (SISO) systems. These controllers are established on adecomposition of the system into two independent lateral and longitudinalsubsystems [HS02a], and are the following:

• a control of the aerodynamic speed acting on the main thrusters,• an altitude controller acting on the tail-fins by the 'e signal,• and a heading controller acting on the tail-fins by the 'g signal.

Thus, the problem consists in designing control laws so that the airshipoutput will follow as closely as possible the reference signals ur, zr and #r,respecting the cruise flight hypotheses for the validity of simplified models,that is a similar behavior to the answer in open loop (respecting the naturalmovement of the airship).

4.1 PID control

Every simplified model presented in section 3.2 can be represented by:

Airship control 19

x = Ax + Buy = Cx

(25)

where x is a n" 1 state vector, y is a p" 1 output vector and u is a q " 1input vector (A, B, C and D are respectively, n " n, n " p, q " n and q " pmatrices). The control law u defined by a PID controller can be written as :

u = Kpum + Ki

)(uref ! um)dt + Kd

dum

dt(26)

where uref is the reference and um is the controlled variable. The propor-tional, integral and derivative gains Kp, Ki and Kd can be determined thanksto state feedback and pole placement theory.

Finally, it is recommended use an anti-windup strategy to avoid saturationof the integral action term. The scheme that represents the PID control withits saturation is presented in figure 16

Fig. 16. PID regulator scheme

Speed controller.

With the open-loop control model for the aerodynamic speed written as

H(s) =va(s)T (s)

=-

.s + 1(27)

and the control input T (thrust propeller) given by

T (s) = kpva +ki

s(vr ! va) (28)

where vr is the reference speed, the closed-loop characteristic polynomialbecomes


s2 ! 1.(1 + -kp)s +

-

.ki (29)

Now, defining the polynomial s2 + 2/wns + w2n based on performance

criteria and equaling with eq. (29), the coe"cients kp and ki can be found.

Altitude and heading controller.

With a control model for the altitude written as

H(s) =z(s)'e(s)

=kzk3

s3 ! k2s2 ! k1s(30)

and the control input

'e(s) = kpz +ki

s(zr ! z) + kd" + k2dq (31)

where zr is the reference altitude, the closed-loop characteristic polynomialbecomes

s4 ! s3(k2 + k3k2d) ! s2(k1 + k3kd) ! s(kpk3kz) + kzk3ki (32)

Based on the performance criteria a polynomial (s2 + 2/wns + w2n)(s +

1/.1)(s + 1/.2) can be formulated. Equaling it with eq. (32), the coe"cientskp, kd, ki and k2d to control the altitude can be determined.

The gains to control the heading are derived with the same formulation.

Simulation results.

Figure 17 shows simulation results of the three PID controllers. Stabilizationof each output state is presented and some couplings between regulators areshowed. There is a lost in altitude when the airship turns, because it alwaysmade flights with a little overload, for security reasons (gravity force is greaterthan buoyancy force). Figure 18 illustrates within 3D space how the controllersstabilize a cruise flight during which 4 di!erent heading references are givenby an operator.

4.2 Generalized Predictive Control

General Predictive Control was introduced by Clarke et al [DC87], and hasbecome popular in both industry and academy [Cla88, EC95, IL98]. It is aclass of predictive control based on the minimization of an objective functionto generate the control law.

A SISO system can be described by the following general form:

A(z!1)y(t) = z!dB(z!1)u(t ! 1) + C(z!1)e(t) (33)

Airship control 21

0 100 200 300−200

0

200

400Heading Control

Time [s]"r in

red

& "

s in b

lue,

[deg

]0 100 200 300

−5

0

5Angular Speed [(

"]

Time [s]r [

deg

/s]

0 100 200 300−20

0

20

40Signal Control of Heading

Time [s]

' g [deg

]

0 100 200 3000

5

10Speed Control

Time [s]u r in re

d &

u s in b

lue,

[m/s

]

0 100 200 3000

20

40

60Signal Control of Speed

Time [s]

T [N

]

0 100 200 300−2

0

2Speed on the axe Y & Z in Rd

Time [s]

v blue

& w

red, [

m/s

]

0 100 200 300−20

−10

0

10Altitude Control

Time [s]

z r in re

d &

z s in b

lue,

[m]

0 100 200 300−40

−20

0

20Angular Speed [(

!]

Time [s]

q [d

eg/s

]

0 100 200 300−50

0

50Signal Control of Altitute

Time [s]

' e [deg

]

Fig. 17. Results of the PID controllers: from top to bottom, evolution of the con-trolled parameter with respect to the reference, evolution of the control input, andevolution of the derivatives of the controlled parameters.

where u(t) and y(t) are the control and output signals of the system ande(t) is a zero mean white noise. A, B and C are polynomials with the z op-erator and d is the dead time of the system. This model is called “ControllerAuto-Regressive Moving-Average (CARMA)” model. When the disturbancesare non-stationary, the CARMA model is not appropriated and we must con-sider an integrated CARMA model (CARIMA) [DC87]. Considering the samenotation follow by Camacho [EC95] the CARIMA model becomes:

A(z!1)y(t) = z!dB(z!1)u(t ! 1) + C(z!1)e(t)+

(34)

with

+ = 1 ! z!1 (35)

The GPC algorithm consists of applying a control sequence that minimizesa cost function of the form:


−200−100

0100

200300

400

−500−400

−300−200

−1000−6

−4

−2

0

2

4

6

8

10

12

Position on axis X [m], NorthPosition on axis −Y [m], West

Posit

ion

on a

xis −

Z [m

], Up

Airship position during the flight

start

recovery of the system

cruise flight

Fig. 18. Motion resulting from the application of the three PID controllers, withthe states stabilized in figure 17.

J(N1, N2, Nu) = E{N2*

j=N1

'(j)[y(t+ j|t)!w(t+ j)]2 +Nu*

j=1

((j)[+u(t+ j!1)]2}

(36)where E{&} is the expectation operator and y(t + j|t) is an optimum j

step ahead prediction of the system output on data up to time t, N1 and N2

are the minimum and maximum costing horizons, Nu is the control horizon,'(j) and ((j) are weighting sequences and w(t + j) is the future referencetrajectory.

The objective of GPC control is then to find the future control sequenceu(t), u(t+1), ... so that the future system output y(t+j) will follow as closelyas possible w(t + j), of course by minimizing the cost function J(N1, N2, N3)(see figure 19).

Speed controller

The following discrete equivalence can be obtained when the simplified speedmodel is discretized with sample time ts = 0.1s:

H(z!1) =0.0198z!1

1 ! 0.9802z!1(37)

Note that for this system the delay d is equal to zero and the noise poly-nomial is equal to 1.

Airship control 23

Fig. 19. GPC regulator scheme

Now with the algorithm presented by Camacho [EC95] and the parametervalues N1 = 1, N2 = N = 10 and ( = 1.2 the control law becomes:

T (k) = T (k ! 1) ! 4.7404va(k) + 4.0845va(k ! 1) + 0.0151va|ref(k + 1)+0.0288va|ref(k + 2) + · · · + 0.1084va|ref(k + 10)

(38)

Altitude controller

Let us consider the following discrete altitude model:

H(z!1) =0.000603 + 0.002424z!1 + 0.000609z!2

1 ! 3.021z!1 + 3.041z!2 ! 1.02z!3(39)

If d = 0, C(z!1) = 1 and the parameter values N1 = 1, N2 = N = 10 and( = 15, the calculated control becomes:

'e(k) = 0.9484'e(k ! 1) ! 0.0516'e(k ! 2) ! 24.2363z(k) + 65.6447z(k ! 1) ! 59.8995z(k ! 2)+18.3751z(k! 3) + 0.00004zref(k + 1) + 0.00031zref(k + 2) + · · · + 0.0384zref(k + 10)

(40)

Heading controller

Considering the discrete heading model:

H(z!1) =0.001839z!1 + 0.001722z!2

1 ! 1.822z!1 + 0.8219z!2(41)

and knowing d = 0, C(z!1) = 1, the parameter values N1 = 1, N2 = N =10 and ( = 0.6, the heading control law is written as:

'g(k) = 0.9717'g(k ! 1) ! 0.0282'g(k ! 2) ! 38.5727#(k) + 64.3467#(k ! 1) ! 26.4822#(k ! 2)+0.0029#ref(k + 1) + 0.0108#ref(k + 2) + · · · + 0.1609#ref(k + 10)

(42)


Simulation results

Figures 20 and 21 show the results of the GPC controller, with the sameairship model and references as in section 4.1.

0 100 200 300−200

0

200

400Heading Control

Time [s]"r in

red

& "

s in b

lue,

[deg

]

0 100 200 300−5

0

5

10Angular Speed [(

"]

Time [s]

r [de

g /s

]

0 100 200 300−50

0


Time [s]

' g [deg

]

0 100 200 3000

5

10Speed Control

Time [s]u r in re

d &

u s in b

lue,

[m/s

]

0 100 200 3000

20

40


Time [s]

T [N

]

0 100 200 300−4

−2

0


Time [s]

v blue

& w

red, [

m/s

]

0 100 200 300−15

−10

−5

0Altitude Control

Time [s]z r in

red

& z s in

blu

e, [m

]

0 100 200 300−20

0

20Angular Speed [(

!]

Time [s]

q [d

eg/s

]

0 100 200 300−50

0


Time [s]

' e [deg

]

Fig. 20. Results of the GPC controllers

4.3 Non-linear control by extended linearization (ELC)

The linearization of non-linear system equations on equilibrium points is avery useful technique applied by control engineers. Recent control techniqueshave been formulated according to local or global validity. When a linearizedmodel is not maintained, di!erent linearized models corresponding to the op-eration points of the system are needed. These set of linearized models iscalled tangent linear model (TLM).

A non-linear system can be described by:

x = f(x, u) (43)

Airship control 25

−300 −200 −100 0 100 200 300 400

−400−300

−200−100

0100

0

2

4

6

8

10

12


Posit

ion

on a

xis −

Z [m

], Up


start

cruise flight


Fig. 21. Motion resulting from the application of the GPC controllers

where f is a vector field parametrized by the input u and the states x.Using the Taylor approximation and considering that the system works on anoperation or balance point (x0, u0), the approximated linearized function fcan be written as:

f(x, u) = f(x0, u0)+0f

0x(x0, u0)(x!x0)+

0f

0u(x0, u0)(u!u0)+1(x!x0, u!u0)

(44)If the higher power terms are negligible, the equation (44) becomes:

'x =0f

0x(x0, u0)'x +

0f

0u(x0, u0)'u (45)

this is a linear representation of equation (43) and the set of linearizedmodels can be represented by:

'x = F (x0, u0)'x + G(x0, u0)'u (46)Considering equation (46) and feedback theory with pole placement tech-

nique, a local control law can be formulated:

'u = !K(x0, u0)'x + 'v (47)integrating this control local law, it is possible to find a non-linear control

law in all the system work space [FNC93]. Finally, this law can be expressedby:


u = 2(x, v) (48)

Note that this law is valid only with small variations of the reference signal.The scheme that represents the nonlinear control by extended linearizationwith an anti-windup is presented in figure 22.

Fig. 22. Extended linearization regulator scheme

To begin the calculation procedure of the control law by extended lineariza-tion method, it is necessary to guarantee that the control model depends onlyon the desired state variable and satisfy the local governability %f

%u (x0, u0) '= 0.The following developments require both conditions to be true.

Speed controller

The speed dynamics can be represented by:

mu = Fm cosµ +12&u2SrefCTt (49)

where u is the state variable and Fm is the control input. The balancepoint (du/dt = 0) is given by:

u20 =

2Fm0 cosµ

&SrefCTt(50)

The TLM model can be expressed as:

'u = "Sref CT tuo

m 'u + cosµm 'Fm

'u = a(uo)'u + b(uo)'Fm(51)

If the desired system performance is represented by:

Airship control 27

s2 + 23wos + w2o (52)

the control law is:

'Fm = !K(u0, Fm0)'u +)

('uref ! 'u)dt (53)

where 'uref is the speed reference, and considering the feedback theorywith pole placement technique, it is possible to formulate the characteristicpolynomial of the closed-loop system and then, considering equation (52), thecoe"cient values can be determined.

k(uo) = !2&wom!"Sref CTtuo

cosµ m

ka(uo) = mw2o

cosµ(54)

According to equation (48), the control law after the integration is:

Fm(t) =!23womu(t)

cosµ! 1

2&SrefCTtu2(t)

cosµ+

mw2o

cosµ

)(uref (t) ! u(t))dt (55)

Altitude controller

Let us consider the following altitude model

z = !u sin "" = q

q = k2q + k1" + k3'e

(56)

working on the operation point

'e0 = 0"0 = 0q0 = 0

(57)

The TLM model becomes:

'z = !u cos "0'"'# = 'q

'q = k2'q + k1'" + k3''e

(58)

If the control law is expressed as:

''e = !K(z0, "0, q0, 'e0)'x +)

('zref ! 'z)dt (59)

where 'x is the variation of the state vector, K = [kz, k!, kq]T , and thedesired performance represented by


(s2 + 2/wns + w2n)(s + 1/.1)(s + 1/.2) (60)

the control law in closed-loop and after the integration becomes:

'e = w2n

!u cos !k3$1$2

+(zref ! z)dt ! $2+$1+2'wn

$1$2k3q+

! 1+2'wn($1$2)+$1$2(w2n+k1)

$1$2k3" + 2'wn+$1+$2

$1$2k3u cos ! z

(61)

Heading controller

Similarly to the altitude controller and the desired performance as (s2 +2/wns + w2

n)(.s + 1), the heading controller can be expressed as:

'g =w2

n

k1.

)(#ref ! #)dt ! 23wn. + 1

k1.r ! w2

n. + 23wn

k1.# (62)

Simulation results

Figures 23 and 24 show the results of the ELC controller, with the sameairship model and references as in section 4.1.

In general it is possible to see that the controllers follow the references(speed, altitude and heading controls), even though there are couplings be-tween the controlled states. The main di!erence in the controllers’ perfor-mance is in the control signal that they generate. The GPC controllers tryto minimize the pursuit error and the control signal energy, consequently theresponse time is smaller (see Fig. 20). The PID and Nonlinear controls havesimilar results with smaller control signals (see Fig. 17 and Fig. 23). Never-theless, it remains the an advantage of the nonlinear controllers is to have thepossibility of adapting automatically the parameters with the system operat-ing point.

4.4 Experimental results

Test with UrAn

The speed and altitude controllers have been tested at the UrAn airship, undera cruise flight condition. UrAn is a 28m3 airship built by the MiniZep com-pany, and is the support for research held at the university of De los Andes(Bogota-Colombia). It is equipped with a mini computer that communicateswith the ground via a Wifi link, an RF receiver interconnected with the ac-tuators, the computer and the ground remote control, a GPS, a barometricaltimeter, an inertial sensor that provides the airship attitude and headingand a home-made anemometer.

Figure 25 shows the aerodynamic speed va and the signal control T withthe three proposed controllers. As can be seen, the flight is quite perturbed,

Airship control 29

0 100 200 300−200

0

200

400Heading Control

Time [s]"r in

red

& "

s in b

lue,

[deg

]0 100 200 300

−5

0

5

10Angular Speed [(

"]

Time [s]r [

deg

/s]

0 100 200 300−20

0

20


Time [s]

' g [deg

]

0 100 200 3000

5

10Speed Control

Time [s]u r in re

d &

u s in b

lue,

[m/s

]

0 100 200 3000

20

40


Time [s]

T [N

]

0 100 200 300−4

−2

0


Time [s]

v blue

& w

red, [

m/s

]

0 100 200 300−15

−10

−5

0Altitude Control

Time [s]

z r in re

d &

z s in b

lue,

[m]

0 100 200 300−20

0

20

40Angular Speed [(

!]

Time [s]

q [d

eg/s

]

0 100 200 300−50

0


Time [s]

' e [deg

]

Fig. 23. Results of the ELC controllers

the system having sometimes di"culties to follow the speed reference, andthe some saturated control signals are generated. Nevertheless, the non-linearcontroller is the one that behaves the best.

A less perturbed flight is shown figure 26, the saturated signals beinggenerated when the airship is moving towards the wind direction or against.Unfortunately, UrAn and Karma airships are not equipped with a wind sen-sors.

Figure 27 shows the altitude z and the signal control 'e when a GPCregulator is applied to the blimp. Strong control inputs maintain the altitudeclose to the reference signal, because the sensor resolution (±1m) is low.

Test with Karma

The speed and heading controllers have been tested on-board with Karmaairship. The Karma hull was made by the Zodiac company, its volume is18m3 and is the support for research held at the LAAS/CNRS laboratory(Toulouse-France). Its instrumentation is similar to UrAn’s.


−200−100

0100

200300

400

−500

−400

−300

−200

−100

0

1000

2

4

6

8

10

12


Posit

ion

on a

xis −

Z [m

], Up


start


cruise flight

Fig. 24. Motion resulting from the application of the ELC controllers

Figure 28 shows the aerodynamic speed va and the signal control T withtwo proposed controllers (PID and GPC). As it can be seen, the control signalpresents saturations when the reference is high and the system cannot followit. Nevertheless, the controllers performances are good.

Figure 29 shows the #ref and output system when a PID heading regulatoris applied to Karma. Saturation on the reference signals assure smooth changeson the control signal and proximity to the operation point.

Change in reference signal and a GPC regulator are presented in figure30. As it can be seen the system follows the reference with low e!orts of thecontrol signal.

In summary, the performance of the proposed regulators is acceptable,showing the validity of the established models. Other tests with adaptivecontrols based on the gradient method and Lyapunov functions have beendeveloped [SGL04]: in general, these regulators present control signals strongerthan the GPC controllers.

5 Path planning and following

Various contribution on path following or waypoint reaching for airships canbe found in the literature. Hygounenc [HS03] shows a lateral control strat-egy with a path following controller to drive the airship within an horizontalplane. In [EMH+05] the authors propose an approach called “orienteering”:

Airship control 31

200 400 600 800 1000 1200 1400 16000

20

40

60

80

Signal Control

Time [s]

Thru

st p

rope

ller [

N]200 400 600 800 1000 1200 1400 1600

0

5

10Reference (red) & Outpu (blue) system

Time [s]

Aero

dyna

mic

spee

d [m

/s]

200 400 600 800 1000 1200 1400 1600012345

Type of Control

Time [s]

ELC=

1,G

PC=2

,PID

=4

Fig. 25. Speed control on real system (UrAn), first result

the control objective is defined in terms of a waypoint to reach, rather than interms of a trajectory to follow. Similar work is presented in [RLG+05], wherethe approach in order to obtain #ref is expressed in terms of the deviationfrom a nth waypoint. When the distance between the airship and the nth pointis less than 20m, the mission planner treats the nth waypoint as reached andthe (n + 1)th waypoint is set as the new target point. In [RA05], the authorspresent an approach in which visual signals are used to achieve a road follow-ing task – similar work can be found in [ZO99], in the context of an indoorvision-guided blimp. In [AdPRB00], the authors presents two approaches toensure the guidance of an airship: one is based on a H" control techniqueand the other based on a PI control.

Fewer work tackle the motion planning problem. [KO03] describes somerandomized motion planning algorithm, considering systems with both kine-matic and dynamic constraints. A work based on the shortest paths for the lat-eral navigation of an autonomous unmanned airship is presented in [HB03a];the idea is the characterization of the shortest paths taking into account thedynamics and actuators limitations of the airship.

We consider here the problems of determining an appropriate trajectorywithin the 2D plane to reach a given goal, and of ensuring that the trajectory isproperly followed. We consider the case where the airship is evolving in cruise


300 400 500 600 700 800 900 1000 1100 1200 13000

20

40

60

80Signal Control

Time [s]

Thru

st p

rope

ller [

%]

300 400 500 600 700 800 900 1000 1100 1200 13000

5

10

15reference (red) & Outpu (blue) system

Time [s]

Aero

dyna

mic

spee

d [m

/s]

300 400 500 600 700 800 900 1000 1100 1200 1300012345

Type of Control

Time [s]

ELC=

1,G

PC=2

,PID

=4

Fig. 26. Speed control on real system (UrAn), second result

flight, with the equilibrium conditions mentioned in section 2: the systemdynamics can be linearized.

5.1 Path planning

If the longitudinal dynamics are stabilized to the steady values uref and zref ,considering that the skid angle % remains negligible and the control directlyacts on #, the lateral airship’s dynamics can be described by the system:

x = u cos#y = u sin## = 'g/&

(63)

where 'g is the control to navigate in the 2D horizontal plane. This modelpresents two kinematic constraints: one that forces the vehicle to move tangen-tially to its main axis, and a second constraint due to the bound on the rudderangle prevents the blimp from turning with a radius of curvature lower thana given threshold &. This kinematics model was studied by Dubins [Dub57],who considered the problem of characterizing the shortest paths for a particlemoving forward with a constant velocity. In [BSBL94], Soueres presents thesame model and provides solutions with optimal control, the shortest path

Airship control 33

160 180 200 220 240 260 280 300 320−40

−20

0

20

40Signal Control with GPC

Time [s]

' e [deg

]

160 180 200 220 240 260 280 300 320−21.5

−21

−20.5

−20

−19.5

−19

−18.5Reference (red) & Outpu (blue) system

Time [s]

z [m

]

Fig. 27. Altitude control on real system (UrAn)

problem being equivalent to the minimum-time when the module of linearvelocity is kept constant.

Rewriting the system (63), x = f(x)u + g(x)'g, it is possible to show thatit is controllable. The main di"culty that arises is the under-actuation of thesystem, and an other is the coupling between position and direction, knownas the non-holonomic constraint x sin# + y cos# = 0.

The problem of steering the system (63) from an initial configuration /i(x)to a final configuration /f (x), is the trajectory generation problem, that canbe formulated as the minimization of J('g):

J('g) =) tf

t0

L(x(t, 'g), 'g(t))dt (64)

under the constraints

x = f(x)u + g(x)'gx(t0) = x0

x(tf ) = xf

'gmin ( 'g ( 'gmax

(65)

Then, the problem is to find the admissible control 'g ) Rm that minimizesthe time to reach the final state xf from the initial state x0. To solve this


250 300 350 400 450 500 550 6000

50

100

Signal Control

Time [s]

Fore

cepr

opel

ler [%

]

250 300 350 400 450 500 550 600

0

2

4

6

8Reference (red) and Output (blue −−) system

Time [s]Sp

eed

[m/s

]

250 300 350 400 450 500 550 6001

2

3

4

5FO=1,GPC=2,PID2=3,PID=4

Time [s]

Type

of C

ontro

l

Fig. 28. Speed control on real system (Karma)

problem, we apply the Pontryagin’s Maximum Principle (PMP) to obtain anoptimal trajectory x#(t) defined on [0, T ] with a '#g time-optimal referencecontrol. The Hamiltonian H is defined by:

H = (1u cos# + (2u sin# + (3'g (66)

So we have the adjoint vector, ( : [0, T ] ) R3, satisfying ( = !%H%x and

((t) '= 0 for every t ) [0, T ], and if x#, (# and '#g verify these conditions, thenwe have an extremal solution.

The extremal control can have di!erent values, 'g = 'gmax, 'g = 'gminand 'g = 0, they give circular (to the right and to the left), and straight trajec-tories, respectively. This type of controller is called “bang-zero-bang control”.The geometric approach allows to complete the total solution (see [BSBL94]),it uses the movement symmetry properties of the blimp and divides the statespace, such that only six families of curves can describe the airship’s move-ment within the 2D plane. The calculation of these trajectories is very fastand the trajectory that has the smaller distance is the optimal one.

Note that the control 'g cannot vary continuously along a given trajectory('g $ angular speed #). In order to make the angular speed r a variable ofthe system, two dynamic extensions are proposed by controlling the variation

Airship control 35

1000 1050 1100 11500

100

200

300

400

Reference (red) & Outpu (blue−−) system

Time [s]

" [d

eg]

1000 1050 1100 1150

0

10

20

30

40

Signal Control (PID)

Time [s]

' g [deg

]

Fig. 29. Heading control on real system (Karma), first result

of the angular speed r instead of the angular velocity. The first extension iscontrolling by 'g and the other one is controlling by !kpsir + 'g.

First dynamic extension

If the variation of the angular speed r is directly controlled by 'g, the newsystem is:


r = kg'g

(67)

where kg is a constant depending on the system dynamics. Assuming thatthe system (67) is controllable and considering that the cost function to beminimized is an energy criterion (similar to the shortest path problem whenthe velocity is kept constant):

J('g) =) tf

t0

< 'g(t), 'g(t) > dt (68)


1360 1370 1380 1390 1400 1410 1420 1430 1440 1450−460

−440

−420

−400

−380

−360Reference (red) & Outpu (blue−−) system

Time [s]

" [d

eg]

1360 1370 1380 1390 1400 1410 1420 1430 1440 1450−40

−30

−20

−10

0

10

20

30Signal Control (GPC)

Time [s]

' g [deg

]

Fig. 30. Heading control on real system (Karma), second result

an optimal control '#g can be found if there is an input 'g(t) for everyt ) [0, T ] which minimizes the cost function J . So the Hamiltonian for thesystem (67) becomes:

H =12(u2 + '2g) + (1u cos# + (2u sin# + (3ur + (4'g

Second dynamic extension

If the variation of the angular speed r is directly controlled by !kpsir + 'g(similar behavior to a first order system), the system can be written as:


r = !krr + 'g

(69)

where kr is a constant depending on the system dynamics. Similarly toabove, there is an optimal control u# that steers the system (69) from thestarting point /i(x) to the final point /f (x) and the Hamiltonian H can beexpressed by:

Airship control 37

H =12(u2 + '2g) + (1u cos# + (2u sin# + (3ur + (4('g + ukrr)

This class of systems gives non-convex problems and in general it is di"cultto find a solution. One possibility to solve it is using numerical methods: thetechnique which we describe here was developed by Fernandes et al. [CFL94].

The Fernandes’ method: Let us consider an input control 'g ) L2([0, T ])and denoting by {ek}"k=1 an orthonormal basis for L2([0, T ])3, the continuouscontrol law 'g can be expressed as:

'g ="*

k=1

($kei 2k!t

Tk + %ke

!i 2k!tT

k ) (70)

and with the Ritz Approximation4, the function 'g can be approximatedby truncating its expansion up to some rank N . The new control law and theobjective function become:

'g =,N

k=1 $kek

J('g) =+ tf

t0< 'g(t), 'g(t) > dt *

,Nk=1 |$k|2

(71)

where $ = $1,$2, . . . ,$N ) RN . The configuration /f (x) = /(T ) is thesolution at time T applying the control 'g. In order to steer the system to/f (x), an additional term must be added to the cost function,

J($) =N*

k=1

|$k|2 + 4||f($) ! qf ||2 (72)

where /f = f($) and qf is the goal position. Note that the new finite-dimensional problem converges to the exact solution when N % + (see[CFL94]).

Results

The results obtained by applying the Soueres method and the Fernandes’numerical optimization approach to the di!erential systems (63), (67) and(69) minimizing the respective objective functions, are show in figures 31 and32.3 L2 denotes a Hilbert space4 The Ritz Approximation Theory approximates the solution using solutions of

some finite-dimensional problems.


0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

Movement within horizontal plane (2D), Dubins in red & SmoothingM1 in blue & SmoothingM2 in black

Y po

sitio

n [m

], Ea

st

X position [m], North

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80" evolution

" [d

eg]

Time [s]

0 5 10 15 20 25 30 35 40 45 50 55

−10

0

10

d"/dt evolution

d"/d

t [de

g/se

c]

Time [s]

Smoothing (model 1)Smoothing (model 2)Dubins

start goal

direction of movement

Fig. 31. Trajectories generated by systems 63 (Dubins), 67 (smoothing model 1)and 69 (smoothing model 2), for an initial configuration &i = [0, 0, 0] and a finalconfiguration &f = [250,!250, 0]

.

5.2 path following

Once a trajectory is planned, a dedicated control to ensure the path followingis required. As usually done in mobile robotics, the consideration of a Frenetframe enables to formulate the path following problem as the regulation ofthe lateral distance and the orientation between the airship and its projectionperpendicularly to the point s on the path 3. The error dynamics can bewritten as:

d = va sin#e

#e = #d ! #r(73)

where d is the lateral distance between CV 5 and the frame Rr and #e is theangular error between the blimp’s orientation and the reference’ orientationof the point s (figure 33).5 Volume center of the hull

Airship control 39

0 50 100 150 200 250

−50

0

50

Movement within horizontal plane (2D), Dubins in red & SmoothingM1 in blue & SmoothingM2 in black

Y po

sitio

n [m

], Ea

st

X position [m], North

0 5 10 15 20 25 30 35 40 45−100

−50

0

50" evolution

" [d

eg]

Time [s]

0 5 10 15 20 25 30 35 40 45 50−10

−5

0

5

10d"/dt evolution

d"/d

t [de

g/se

c]

Time [s]

Smoothing (Model 1)Smoothing (Model 1)Dubins

start goal

direction of movement

Fig. 32. Trajectories generated by systems 63 (Dubins), 67 (smoothing model 1)and 69 (smoothing model 2), for an initial configuration &i = [0, 0, 0] and a finalconfiguration &f = [250,!250,!90].

Knowing that the dynamic of the angular speed in the cruise flight canbe represented by #d = !k1(d#d + k2(ref#ref , and inspired by non-linearcontrol (feedback linearization control), #ref can be write as:

#ref =k1(ref#d + #r + v

k2(ref

(74)

Under the hypothesis that the airship is flying near the trajectory 3 (#e

and d remains small), the system (73) becomes:

'd = a'#e

'#e = 'v(75)

where a = va cos#e0. The control is intended to make d % 0 and #e % 0.For this end, a stabilizing feedback with integral control law is applied. Thecharacteristic polynomial can be determined by:


Fig. 33. Definition of the Frenet frame and of the parameters to regulate.

------

s !a 0!kd (s ! k(e) ka

1 0 s

------= s3 ! s2(k(e) ! s(akd) + aka

Now, if the performance system in close-loop is given by (s3 + s2(23w0 +1$ ) + s(wo

2 + 2&wo

$ ) + w2o$ ) the values of k(e , kd and ka, are defined by the

Routh-Hurwitz criterion:

k(e = !23wo + 1/.

kd = !w2o+2&wo/$

a

ka = w2o

a$

(76)

where d < | 1" | and #e ) [!)2

)2 ]. The control law for the system (75) is:

v = k(e#e + kdd + ka

)(!d)dt (77)

Finally, the reference #ref for the blimp heading control system and thatassures to follow the trajectory 3 can be written by:

#ref =k1(ref#d + #r + k(e#e + kdd + ka

+(!d)dt

k2(ref

(78)

Results

The results of path following, when the path is planned by the Dubins model,are shown in figure 34 and the airship evolution within the 3D space is pre-sented in Fig. 35.

Airship control 41

0 100 200 300 400−100

0

100

200

300Refred and Sysblue(without wind) & green (with wind)

Time [s]

X posit

ion [m

]

0 100 200 300 400

−2000

200400

Refred and Sysblue(without wind) & green (with wind)

Time [s]

Y posit

ion [m

]

0 100 200 300 400−400

−200

0


Time [s]

"di

rect

ion [d

eg]

−100 0 100 200 300−400−200

0200400

2D position, Refred and Sysblue(without wind) & green (with wind)

X position [m]

Y po

sitio

n [m

]

0 100 200 300 400−40

−20

0

20

40derror [m], Sysblue(without wind) & green (with wind)

Time [s]

Erro

r [m

]

0 100 200 300 400−100

−50

0

50

100"error [deg], Sysblue(without wind) & green (with wind)

Time [s]

Erro

r [de

g]

Start

wind (1.5m/s)

Goalwithout wind

with wind

Fig. 34. Results of the control to follow a trajectory, Dubins model.

The results of our path following method and 3D airship movement, whenthe path is planned by the second dynamic extension model, are shown infigures 36 and 37.

These results compare the performance of the regulators when the airshipis disturbed or not by wind. The second dynamic model has presented atrajectory more adapted to the airship dynamics and then, the lost of altitudewhen the blimp turns is less important.

6 Conclusion

This chapter presented a complete model of an airship, which can be decoupledinto simpler sub-models under the assumption that it is evolving at a stabi-lized speed. An identification technique based on the Unscented Kalman Filterproved to give better convergence than one based on an Extended KalmanFilter, and has been successfully applied with the airship Karma. A globalcontrol strategy that integrates a path planner, a path follower and elemen-tary controllers has been proposed, and experimental results of cruise flightstabilization with the airships UrAn and Karma have been obtained.


−200

−100

0

100

200

300

−600−500

−400−300

−200−1000

100200

300400−20

0

20

40

Posit

ion

on a

xis −

Z [m

], Up

Airship position within 3D, tracking task


Wind (1.5m/s)

Start

Goal

without wind

with wind

Fig. 35. Evolution within 3D space, Dubins model.

This work proves that small size airships can successfully be automaticallycontrolled, provided they evolve in cruise flight conditions. More work is tobe performed when the airship is facing wind gusts and abrupt variations.

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Airship control 43

0 100 200 300 400−200

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X posit

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Time [s]

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