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Nonlinear speed estimation of a GPS-free UAVGiovanni L. Santosuosso a , Khadidja Benzemrane b & Gilney Damm ba Dipartimento di Ingegneria Elettronica, Universitá di Roma Tor Vergata, Via del Politecnico1, 00133 Rome, Italyb Laboratoire IBISC – CNRS/Université d'Evry Val d'Essonne, 40 rue du Pelvoux, 91020 EvryCedex, France

Version of record first published: 02 Nov 2011.

To cite this article: Giovanni L. Santosuosso, Khadidja Benzemrane & Gilney Damm (2011): Nonlinear speed estimation of aGPS-free UAV, International Journal of Control, 84:11, 1873-1885

To link to this article: http://dx.doi.org/10.1080/00207179.2011.627685

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International Journal of ControlVol. 84, No. 11, November 2011, 1873–1885

Nonlinear speed estimation of a GPS-free UAV

Giovanni L. Santosuossoa, Khadidja Benzemraneb and Gilney Dammb*

aDipartimento di Ingegneria Elettronica, Universita di Roma Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy;bLaboratoire IBISC – CNRS/Universite d’Evry Val d’Essonne, 40 rue du Pelvoux, 91020 Evry Cedex, France

(Received 22 November 2010; final version received 22 September 2011)

In this article, the problem of robust state observer design for a class of unmanned aerial vehicles (UAVs) isaddressed. A prototype four-rotors helicopter robot for indoors and outdoors applications is considered: thedrone is not equipped with GPS related devices, so we describe a strategy to estimate its translational velocityvector based on acceleration, angles and angular speeds measurements only. Since the linearised system is non-observable at the equilibrium point, a nonlinear observability verification is performed for persistently excitingtrajectories. A global exponential solution to this open problem is provided in the framework of adaptiveobservation theory when exact measurements are available. A modified observer is presented to enhance velocityestimation robustness in the realistic case of noisy measurements. The results are compared with a classicalestimation strategy based on the extended Kalman filter to test the algorithm’s performance.

Keywords: nonlinear observability; nonlinear observers; adaptive observers

1. Introduction

The speed estimation problem has an important rolein the context of vehicle control. For land movingrobots, the odometry time derivative has a satisfactoryperformance while for large aerial vehicles (manned orunmanned) velocity estimations can be obtained viaapproximate derivation of the successive measure-ments from GPS-sensors, motivated by the smallresulting errors compared to the measured variables.For fast aircrafts the standard procedure is integratingthe acceleration and coupling this result to thederivative of GPS measurements. In fact, signals(translational acceleration, angles and angular velo-city) delivered by the embedded inertial measurementunit (IMU) can be used to obtain position informationthrough a double integration process. However,because of sensor drift, such procedure requires highprecision IMU. This sensor cannot be used for smallsize unmanned aerial vehicles (UAVs) where cost,weight and volume are the most important constraints.

For such applications, two critical issues arise inan ‘open loop’ strategy, such as direct accelerationintegration: an unknown constant estimation error isproduced even when exact acceleration measurementsare available while a random drift is induced by noisyacceleration estimations. In practice, numerical inte-gration along with measurement noise induces a veryfast growing velocity measurement error.

Reasonable approaches to overcome these problemsare represented by sensor fusion techniques, in order to

compensate the lack of precision of low-cost IMUs. In

that context the information obtained from the GPS is

used to bound the integration error and initial condi-

tions for acceleration integration are provided by GPS

devices (Vik and Fossen 2001). This technique providesbounded errors that are related to the GPS order of

magnitude precision and sample rate. Currently used

GPS may assure a precision of 5–10m with a sample

rate of 1–2Hz. The resulting errors are usually small

compared to the size of aerial vehicles and the distance

to obstacles. In the same way, the control systems

usually applied to these vehicles are robust enough toaccept the residual disturbance on the speed estimation

provided by these methods.Unfortunately, the above estimation approach

cannot be implemented on small drones less than 1m

wide and flying at low speed that we consider in this

note. This issue is even more relevant in indoors orsimply urban applications as inspecting bridges and

industrial facilities which cannot rely on GPS, since

satellite’s signals are shaded by the inspected structure.A technical solution when no obstacles are

present would involve the use of a D-GPS system.

These systems are known as centimetric GPS and

have a precision of some centimetres, but they arevery expensive also; heavyweight and low-range

*Corresponding author. Email: gilney.damm@ibisc.fr

ISSN 0020–7179 print/ISSN 1366–5820 online

� 2011 Taylor & Francis

http://dx.doi.org/10.1080/00207179.2011.627685

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operating equipment. In the same way, Dopplermeasurements coupled with GPS can actually increasevelocity estimation accuracy, but they can only be usedin special conditions of smoothness and continuity ofobstacles, as well as a good knowledge of their positionand shape.

In Metni and Hamel (2007), a control law based oncomputer vision is proposed for quasi-stationaryflights above a planar structure such as bridges toperform inspection tasks. Likewise, estimates of thepose and velocity of a rigid body can be provided by analgorithm based on IMU data and images provided bya camera (Baldwin, Mahony, Trumph, Hamel, andCheviron 2007; Hamel and Mahony et al. 2007;LeBras et al. 2007; Herisse et al. 2008). The maindrawback of this method is the large amount ofcomputer resources needed. It is still not possible toapply vision algorithms in the small micro-controllersused in the considered UAVs. Displacing the imageprocessing algorithms to computers at a land base ishardly implemented for two reasons: first, the timedelay induced by remote computing and communica-tion, along with the danger of losing the contact withthe drone could inevitably imply disruptive deviceinstability; a second important drawback is thesensibility of cameras and vision algorithms to lightvariations, associated with outdoor drone displace-ment. Vision can be used in an hierarchical displaywhere a slow controller gives path planning referencesto a fast stabilising one, as in Langelaan and Rock(2005), where an unscented Kalman filter is applied tothe navigation problem and a solution for simulta-neous localisation and mapping when no GPS signal isavailable is proposed. The result described inLangelaan and Rock (2005) relies on an IMU alongwith a monocular camera only and possible losses ofvision or radio contact do not affect the UAVsstability. In the field of fusion techniques, an interest-ing solution to the problem of estimation of theposition and velocity of UAVs is proposed in Vissiere,Martin, and Petit (2007a, b; Vissiere et al. 2008). Thealgorithm based on Kalman filtering techniques usesthe magnetic field disturbances to improve the estima-tion error due to the use of low-cost IMU; it may beuseful for indoor applications.

In this article, we consider the problem of transla-tional velocity estimation from measurements oftranslational accelerations in the local referenceframe, Euler angles in the global reference frame andangular speeds in the local reference frame. Sincetranslational velocity knowledge is necessary for anefficient drone control design, most literature aboutsmall flying robots control assumes that at least thetranslational speeds are available (Pounds, Mahony,Hynes, and Roberts 2002; Castillo, Dzul, and Lozano

2004; Lozano, Dzul, and Castillo 2004; Guenardt,

Hamel, and Moreaut 2005). This is motivated by the

fact that it is hardly possible for a human pilot

(inboard or in tele-operation) to regulate a drone by

giving references and trajectories as inputs based on

the sole acceleration measurement. Note that in the

general case, the reliable estimation of the speed vector

is still an open problem. In fact, the acceleration in the

local (body fixed) frame is not a standard measurement

from IMU, but even if this information is available the

classical open-loop integration using this information

would roughly lead to the same results obtained with

the standard acceleration measurement in the global

reference frame.Recently, Bonnabel, Martin, and Rouchon (2006),

Vik and Fossen (2001), Zhao and Slotine (2005),

Benallegue et al. (2008), Besnard et al. (2007),

Boutayeb et al. (2008), Benzemrane et al. (2007),

Benzemrane (2009), Madani and Benallegue (2007)

and Pflimlin et al. (2007) have proposed observers to

be implemented on UAVs. The critical issue in this

domain comes from the fact that the translation

velocity is classically known as non-observable. A

key contribution of this study is to show that although

the system linearised approximation is non-observable

in the origin, the full nonlinear system may become

observable under suitable sufficient conditions, thus

allowing its estimation.Motivated by the above issues, in this note we

describe a novel observation strategy that solves the

problem of the speed estimation of an UAV, when the

translational acceleration (in the local reference frame),

the angles and the angular speeds (in the global

reference frame) are available for measurement. We

focus our analysis on a prototype drone – a four-rotors

helicopter robot shown in Figure 1 – produced to

Figure 1. Prototype drone.

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operate in an urban environment at the LaboratoireIBISC – CNRS, Universite d’Evry, which is notequipped with GPS-related devices (Pradel,Benzemrane, Damm, and Azouz 2007). We provide aglobal exponential solution to this open problem in thelight of classical adaptive estimation theory when exactmeasurements are available. A modified estimator ispresented to enhance robustness in velocity estimationin the case of noisy acceleration measurements. Thesimulation results obtained on the model of the realdrone – even in the presence of noise – provide anadditional algorithm validation test. The proposedscheme is relatively simple and this is a key issue sincethe algorithm has to be implemented in smallembedded micro-controllers without heavy computa-tion burden.

The results in this article are inspired by two designprinciples. The first one is that by increasing the use ofmodel based information, it is possible to better exploitsensors estimations, which is obviously obtained via amuch larger modelling effort and theoretical complex-ity. Second, by exploiting the reference frame changesof coordinates, there are new information that mayarise when some rather mild conditions of persistenceof excitation (PE) are fulfilled. This property in turn issatisfied when the considered aerial vehicle hasdecoupled rotational and translational dynamics,since we aim at constructing an observer whichestimates translational states based on rotationalinformation. These design principles can be implemen-ted on any vehicle such that rotational dynamics can bestabilised independently on the estimation of transla-tional velocities, like helicopters and vertical take offand landing (VTOL) airplanes.

The nonlinear observability study introduced in thisarticle finds a natural complement in the design of anextended Kalman filter (EKF) for the considered UAV,which is used in Section 4 to illustrate and compare thebehaviour of the proposed adaptive observer. The testsperformed show that the adaptive observer has a betterperformance in the case of internal and externaldisturbances, such as wind, unmodelled dynamics andmeasurement noise. Furthermore, the proposed obser-vers were designed based on adaptive theory and asconsequence, present a self-tuning behaviour mucheasier to set than the EKF.

This article is organised as follows: in Section 2 wedescribe the drone model and formulate the problemaddressed in this article, analysing the observabilityproperty of the system. This is guaranteed introducinga suitable PE assumption which is used in Section 3 toconstruct the observation algorithms and to prove thatthe estimation error converges exponentially to zero.In Section 4 we report some simulations that illustratethe observer performance and stress the influence of

some parameters on the observer behaviour. Finally, inSection 5 we conclude the note with few remarks alongwith an outline of our future research on this topic.

2. Problem formulation

In this section, we describe (Azouz, Benzemrane,Damm, and Pradel 2007) the UAV dynamic modelequations and state the problem that we solve in thefollowing sections. It is important to remark that theconsidered drone is a modified quadrotor that presentsthe interesting feature of swinging two of its motors,similar to a VTOL airplane, that is represented as twoadditional inputs than the standard quadrotors. Thisfeature is not used in this study, since it does notaddress the UAVs control, but it could be aninteresting feature in a different setting.

Let �1 ¼4[x, y, z]T be the UAV position vector

represented in the global reference frame, let �2 ¼4

[�, �, ]T be the Euler angles vector represented in theglobal reference frame (roll pitch and yaw, respec-tively); we also assume that �1 ¼

4[u, v,w]T is the speed

vector represented in the local reference frame (surge,sway and heave, respectively) and �2 ¼

4[ p, q, r]T is the

angular speed vector represented in the local referenceframe. The UAV model in state space form collects theset of the first-order differential equations expressing_�1, _�2, _�1, _�2, as a function of �1, �2, �1, �2. To be morespecific, the linear velocity in the fixed reference frame_�1 ¼ ð _x, _y, _zÞT can be expressed as a function of �2, �1,via the set of differential equalities

_x ¼ cosð�Þcosð Þu

þ sinð�Þsinð�Þcosð Þ � cosð�Þsinð Þð Þv

þ cosð�Þsinð�Þcosð Þ þ sinð�Þsinð Þð Þw,

_y ¼ cosð�Þsinð Þu

þ sinð�Þsinð�Þsinð Þ þ cosð�Þcosð Þð Þv

þ cosð�Þsinð�Þsinð Þ � sinð�Þcosð Þð Þw,

_z ¼ �sinð�Þuþ sinð�Þcosð�Þvþ cosð�Þcosð�Þw:

ð1Þ

The time derivatives of the Euler angles _�2 ¼( _�, _�, _ )satisfy the following equations:

_� ¼ pþ sinð�Þqþ cosð�Þrð Þtanð�Þ,

_� ¼ cosð�Þq� sinð�Þr,

_ ¼ sinð�Þqþ cosð�Þrð Þcosð�Þ�1:

ð2Þ

We assume that the UAV four rotors turn, respec-tively, at speed !1, !2, !3 and !4. Support of rotors 1and 3 can also swivel around their pitch axes (angles &1and &3 represent the orientation of each support).These quantities represent the system’s control inputfunctions and are related to the time derivatives of the

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speed vectors, respectively, _�1 ¼ ½ _u _v _w�T and _�2 ¼½ _p _q _r�T which are accelerations in the local reference

frame and comply with the mechanical equations of

the UAV. In particular, we have

_u ¼ ð�qwþ rv� g sinð�ÞÞ �kTmð!2

1 sinð&1Þ þ !23 sinð&3ÞÞ,

_v ¼ ð�ruþ pwþ g sinð�Þcosð�ÞÞ,

_w ¼ ð�pvþ quþ g cosð�Þcosð�ÞÞ

�kTmð!2

1 cosð&1Þ þ !22 þ !

23 cosð&3Þ þ !

24Þ, ð3Þ

where g is the gravity acceleration constant, m is the

drone mass, kT is a constant relating the UAV rotor

speeds and resulting thrust. In the angular setting

we obtain

Ixx _p¼�ðIzz� IyyÞrqþ lbkTð!21 cosð&1Þ �!

23 cosð&3ÞÞ

� qIrð!1 cosð&1Þ þ!2þ!3 cosð&3Þ þ!4Þ

� kMð!21 sinð&1Þ þ!

23 sinð&3ÞÞ,

Iyy _q¼�ðIxx� IzzÞpr� rIrð!1 sinð&1Þ þ!3 sinð&3ÞÞ

þ pIrð!1 cosð&1Þ þ!2þ!3 cosð&3Þ þ!4Þ

þ lbkTð!22�!

24Þ þ uGkTð�!

21 sinð&1Þ þ!

23 sinð&3ÞÞ,

Izz _r¼�ðIyy� IxxÞpq� lbkTð!21 sinð&1Þ �!

23 sinð&3ÞÞ

þ kMð!21 cosð&1Þ �!

22þ!

23 cosð&3Þ �!

24Þ

þ qIrð!1 sinð&1Þ þ!3 sinð&3ÞÞ,

ð4Þ

where Ixx, Iyy, Izz are the inertia moments, kM is the

constant relating rotor speeds and torque, lb is the

length of each drone’s arm and Ir is the rotor’s inertia

moment constant.In this article we assume that the following

quantities are available for measurement: orientation

(Euler angles in the global reference frame), angular

velocities (in the local reference frame) and transla-

tional acceleration (in the local reference frame) of the

vehicle. The coefficients kT and kM as well as different

parameters of the model are experimentally identified.

In this context,1 defining the vectors

X ¼ ½�, �, , p, q, r, u, v,w�T,

ym ¼ ½�, �, , p, q, r, _u, _v, _w�T,

� ¼ ½!1,!2,!3,!4, &1, &3�T,

8><>: ð5Þ

Equations (2)–(4) can be written in a compact

notation as

_X ¼ f ðX,�Þ, X 2 R9,

ym ¼ hðX,�Þ, y 2 R6,

(ð6Þ

where f: R9�R

6!R

9 is a smooth function depending

on the state and the control input �, and h(X,�) is

obtained by combining the second equality in (5) along

with (3), expressing _u, _v, _w as functions of X and �.

The article objective may then be stated as: givensystem (6), design an observation strategy yieldingexponentially converging estimates of the state X fromthe measurable output ym, to be eventually included inany control strategy of system (6).

In practical terms, the observation algorithmfocuses on the estimation of the unmeasurable statevariables �1, based on the measurable variables �2, _�1and �2 given by the standard sensors embedded in thedrone. We remark that a necessary condition for thesynthesis of any observer is the local observabilityproperty to be verified. To this purpose, we recall thefollowing definition.

Definition 1 (Isidori 1989; Nijmeijer and van der Shaft1990): Let G denote the set of all finite linearcombinations of the Lie derivatives of h1, . . . , hp withrespect to f for various values of u¼ constant. Let dGdenote the set of all their gradients. If we can find nlinearly independent vectors within dG, then the systemis locally observable.

The observability matrix O¼4dG is given by

O ¼

L0f h

. . .

Lp�1f h

0B@

1CA:

The system is locally observable if O has full rank.Thus in the case we consider in this article, theestimation strategy has to be coupled with a trackingcontrol algorithm driving the angular speeds [ p, q, r]T

to bounded and non-vanishing reference trajectories[ pr(t), qr(t), rr(t)]

T that guarantee the rank of matrix Oto be different from zero. To be more specific, let �(t)be the matrix defined as

�ðtÞ ¼

0 r �q

�r 0 p

q �p 0

264

375: ð7Þ

In this article we assume �(t) is persistently exciting(PE), as formally stated below.

Hypothesis 1: The matrix �(t) is bounded and PE, i.e.there are positive integers T �, K�1 and K�2 such that

K�1I �

Z tþT �

t

�ð�ÞT�ð�Þd� � K�2I for all t 2 R: ð8Þ

Claim 1: Hypothesis (1) is satisfied for any speedvector represented in the local reference frame�2¼ [ p, q, r]T, which is bounded and PE, i.e. such that

c�1I �

Z tþT �

t

�2ð�ÞT�2ð�Þd� � c�2I for all t 2 R ð9Þ

for suitable positive integers T �, c�1 and c�2.

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Proof: To prove the claim we show that any �2(t)which does not comply with (8) does not complywith (9) also. In fact, if �(t) is not PE, then thereexists a non-zero vector s¼ [s1, s2, s3]

T such thatlimt!1k�(t)sk2¼ 0. By expanding the term k�(t)sk2,recalling (7), we deduce that the limit identityabove implies

s3 limt!1

qðtÞ ¼ s2 limt!1

rðtÞ; s3 limt!1

pðtÞ ¼ s1 limt!1

rðtÞ;

s2 limt!1

pðtÞ ¼ s1 limt!1

qðtÞ:

ð10Þ

Property (10) can be expressed concisely as�2(t)¼Tr(t)Vþ �(t) where V2R

3, V 6¼ 0 is any nonzerovector, Tr(t) is any scalar time function and �(t) is anyvector function such that limt!1�(t)¼ 0. It is straight-forward to show that any �2(t) in the form above doesnot satisfy (9), and this proves the claim. œ

Remark 1: By virtue of Claim 1, Hypothesis 1 isguaranteed for any angular speed vector �2(t) thatspans periodically a three-dimensional surface, forinstance a non-vanishing periodic orbit. Besides, thenature of the vehicle provides vibrations in realapplications that may already give enoughpersistence of excitation which relies on angularspeeds (oscillation frequency) rather than angles(oscillation amplitudes).

Remark 2: The necessity of Hypothesis 1 clarifiesthat our estimation strategy can be implemented onthat set of aerial vehicles satisfying PE condition. Allhelicopters and quadrotors fulfil this requirementbecause their rotational and translational dynamicsare decoupled and the rotational states are availablefor measurement with standard sensors (IMU, infraredsensors, magnetometers). For this reason, it ispossible to track a PE reference independently ofestimation results. These could then be used in an outerloop that may be used to stabilise the translationaldynamics.

3. Observer design

In this section we describe the estimation strategyfollowing the techniques presented in Marino andTomei (1995) and Marino and Santossuosso (2001), aswell as an outline of its stability proof. Two differentalgorithms that estimate the linear velocity of thesystem described by Equations (2)–(4) are presented.The first algorithm considers the ideal case with nonoise on the measurement outputs while the secondtakes into account additive measurement noise on thetranslational accelerations.

The measured output can be expressed as

ym¼4½ y1, y2�

T¼ �, �, , p, q, r|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

y1

, _u, _v, _w|fflffl{zfflffl}y2

24

35T

,

where the vector y2 ¼ _�1 ¼ ½ _u, _v, _w�T is available for

measurement, so that Equation (3) is re-written as

_�1ðtÞ ¼ �ðtÞ�1ðtÞ þ�ðtÞ,

y2 ¼ _�1,ð11Þ

with the matrix � as in (7) and the vector � is

defined as

�ðtÞ ¼

�g sinð�Þ �kTmð!2

1 sinð&1Þ þ !23 sinð&3ÞÞ

g sinð�Þcosð�Þ

g cosð�Þcosð�Þ �kTmð!2

1 cosð&1Þ

þ!22 þ !

23 cosð&3Þ þ !

24Þ

8<:

9=;

26666664

37777775:

In the following we design a reduced-order observer

which estimates the translational velocity of the UAV

recalling that, in order to guarantee that p(t), q(t) and

r(t) are different from zero, the control inputs have to

be chosen such that the system oscillates around the

equilibrium point.

3.1 Basic algorithm

In this section, we construct an algorithm to estimate

the time function �1(t) assuming, by virtue of (11), that

the quantity Y1¼4½ _�1ðtÞ ��ðtÞ� ¼ �ðtÞ�1ðtÞ is measur-

able, recalling that both the time derivative of the

function to be estimated y2 ¼ _�1ðtÞ and the regression

matrix �(t) are available for measurement. We remark

that many adaptive estimation techniques (Marino and

Tomei 1995) are shown to guarantee exponential

function estimation when the time derivative of the

regression matrix is also available for measurement,

thus the first step in the estimation strategy is to

replace the regression identity Y1¼�(t)�1(t) with

another identity Y2 ¼4M(t)�1(t), where M(t) is a

matrix (to be defined) whose time derivative _M is

available for measurement. To this purpose, let

M(t)2R3�R

3 be the state of the filter

_M ¼ ��Mþ�ðtÞ ð12Þ

with arbitrary initial condition M(0)2R3�R

3, where

�2Rþ is a positive tuning parameter chosen by the

designer. The matrix M(t) is also PE, as stated below.

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Claim 2: The matrix M is PE, i.e. there exist positiveintegers K�M1 and K�M2 such that

K�M1I �

Z tþT �

t

Mð�ÞTMð�Þd� � K�M2I for all t 2 R,

ð13Þ

where T � is the positive real defined in Hypothesis 1.

Proof: By contradiction, assume that there exists avector s2R

3 with ksk¼ 1 such that limt!1M(t)s¼ 0.Then by setting DM(t)¼M(tþT �)s�M(t)s wehave that limt!1DM(t)¼ 0. Recalling that by virtueof (12) DMðtÞ ¼ ��

R tþT �t Mð�Þs d� þ

R tþT �t �ð�Þs d�,

we deduce that limt!1

R tþT �t �ð�Þs d� ¼ 0, contradict-

ing Hypothesis (1). œ

Define the system

_ ¼ ��þ _�1 ��ðtÞ þM _�1

¼ ��þ y2 ��ðtÞ þMy2 ð14Þ

with state 2R3, arbitrary initial condition (0)2R

3.The task of (12) and (14) is to obtain a measurableestimate of the vector Y2¼M�1 with M measurablealong with its time derivative. In fact, by setting ¼4�M�1 we have

_ ¼ �� �M�1ð Þ ¼ ��, ð15Þ

which implies that exponentially converges to zero.Consider the filter

_Q ¼ ��QþMTM ð16Þ

with state Q2R3�R

3 and any initial condition suchthat Q(0)> 0, where �2R

þ is a positive tuningparameter chosen by the designer. From (16) byusing the arguments in Marino and Tomei (1995) wedemonstrate the following result.

Claim 3: Assume that Q(0)> 0; then the matrix Q(t)is bounded and positive definite, i.e. there exists apositive constant c2 such that Q(t)> c2I for all t� 0.

Proof: Each entry of the matrix Q(t) is the state of alinear asymptotically stable filter driven by a boundedinput, so that Q(t) is bounded. In order to prove thatQ(t) is positive definite, from (16) we have that

QðtÞ ¼ e��tQð0Þ þ

Z t

0

e�� t��ð ÞMð�ÞTMð�Þd�:

Recall that e��tQ(0)> 0 andR t0 e�� t��ð ÞMð�ÞT�

Mð�Þd� � 0 for all t> 0, thus QðtÞ � e��T�

Qð0Þ forall 0� t�T �; however from (16) we also inferthat Q(T �)�

R T �0 e�� T ���ð ÞMð�ÞTMð�Þd� � e��T

�

�R T �0 Mð�ÞT

�

Mð�Þd�, that by virtue of (13) in Claim 2yields QðT �Þ � e��T

�

K�M2I: By repeating iterativelythese arguments we have that QðtÞ � e�2�T

�

K�M2I in

each interval iT � � t� (iþ 1)T � with i¼ 1, 2, . . . ,1,

concluding that Q(t)� c2I where c2 ¼ minfe��T�

Qð0Þ,

e�2�T�

K�M2g. œ

Thus the idea behind the observation strategy in

this note is to seek an estimate

�1 ¼

u

v

w

264

375

of �1 such that by setting the estimation error variable~�1¼4�1 � �1, its time derivative can be written in

the form

_~�1 ¼ �ð�Q ~�1 þ Þ, ð17Þ

where � 2Rþ is a positive tuning parameter to be

chosen by the designer and is a quantity to be defined

within the synthesis process. Note that must not be

chosen necessarily as a measurable quantity, rather it

has to converge exponentially to zero. If this is the

case, then the positive definiteness of Q(t) implies that

limt!1 ~�1ðtÞ ¼ 0 exponentially. Since �1 ¼ �1 � ~�1, the

computation of _�1 along with (17) yields

_�1 ¼ �ð�Q�1 þQ�1 � Þ þ y2: ð18Þ

From the last expression we infer that has to include

the non-measurable term Q�1 for the dynamics _�1 to be

implementable. Notice that d(Q�1)/dt¼��(Q�1)þMTM�1þQy2; this circumstance suggests that has

to be set equal to Q�1 plus the states of some stable

linear filters with eigenvalue �� fed by terms

compensating MTM�1 and Qy2. In particular, defining

these stable filters as

_� ¼ ���þ ¼4� ��þ þM�1, ð19Þ

_� ¼ ��� þQy2 þ �M��ðtÞ½ �T�¼4� �� þQy2 � _MT�

ð20Þ

with vector states �2R3, �2R

3, arbitrary initial

conditions �(0)2R3 and �(0)2R

3 and setting

4¼Q�1 �MT�� �, ð21Þ

by taking into account (19), (20) we obtain

_ ¼ �� �MT: ð22Þ

From (22) we infer that is the state of a stable filter

driven by the exponentially vanishing input , so that

limt!1 (t)¼ 0 exponentially, which implies that

limt!1 ~�1ðtÞ ¼ 0 exponentially. Equality (21) along

with (18) yields the dynamics for �1, the estimate of

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�1, given by

_�1 ¼ �ð�Q�1 þMT�þ �Þ þ y2: ð23Þ

Previous arguments can be summarised as follows.

Proposition 1: Consider the UAV model described by

(2)–(4). Assume that the variables �2, �2, _�1 are availablefor measurement and that Hypothesis 1 holds. Then for

all �2Rþ, �2R

þ, � 2Rþ, the dynamic observer

described by (12), (14), (16), (19), (20) and (23) with

state M(t)2R3�R

3, Q(t)2R3�R

3, (t)2R3,

�(t)2R3, �(t)2R

3 and �1ðtÞ 2 R3 is such that the

vector �1 � �1 converges to zero globally exponentially

for any initial conditions M(0)2R3�R

3, Q(0)2

R3�R

3, (0)2R3, �(0)2R

3, �(0)2R3 and �1ð0Þ 2 R

3

with Q(0)> 0.

Proof: The result has been already outlined in the

previous arguments; a rigorous demonstration follows

from standard Lyapunov theory. In particular, con-

sider the class of Lyapunov functions

V ¼1

2�1 ~�1k k

2þ1

2�2 k k

2þ1

2�3 k k

2, ð24Þ

where �1, �2, �3, are positive real numbers to be defined

later. By computing the time derivative _V and recalling

(15), (17), (22), we obtain

_V ¼ ���3 k k2���2 k k

2��2 TMT ~z

� ��1 ~�T1Q ~�1 þ ��1 ~�T1 : ð25Þ

Note that Hypothesis 1 implies that matrix M(t) is also

bounded and PE. From (16) by using the arguments in

Marino and Tomei (1995) it is shown that if Q(0)> 0,

then matrix Q(t) is positive definite for all t� 0. Setting

the positive constant c1¼ supt2[0,1)kM(t)k2, we deduce

�2 TMT �

�2�

4 k k2þ

�2�c21 k k

2,

� T ~�1 ��c22

~�1k k2þ�

2c2 k k2,

where c2 is the bound defined in Claim 3. The

substitution of the previous inequalities in (25) yields

_V � �

��3��

�2�c21

�k k2�

��2��

�2�

4þ�1�

2c2

� k k2

� �1�

�c2 �

c22

�~�1k k

2:

By setting �1¼ 1, �2 ¼2��c2

, �3 ¼2�2c

21

�� ¼4�c2

1

�2�c2, we finally

obtain

_V � �

��c22

�~�1k k

2�

��

2c2

� k k2�

�2�c21�2c2

�k k2:

ð26Þ

Let �1 ¼ min �c2,�2 ,�

� �: Previous inequality implies

that _V � ��1V, which guarantees the convergence of

all errors exponentially to zero. œ

3.2 Extension of the basic algorithm

In this section, the former algorithm is modified in

order to quantify the observer performance in the case

of measurement noise applied on the velocity and

acceleration signals. We assume now that the noisy

acceleration a� and the noisy velocity matrix ��(t) to

be available for measurement, given by

a� ¼

_uþ�11ðtÞ

_vþ�12ðtÞ

_wþ�13ðtÞ

264

375,

��ðtÞ ¼

0 rþ�21ðtÞ �q��22ðtÞ

�r��21ðtÞ 0 pþ�23ðtÞ

qþ�22ðtÞ �p��23ðtÞ 0

264

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where �ij(t), i¼ 1, 2, j¼ 1, 2, 3 are bounded measure-

ment noises, i.e. such that

�ijðtÞ�� �� � �M, for all t4 0, i ¼ 1, 2, j ¼ 1, 2, 3,

ð27Þ

with �M2Rþ a suitable positive bound. In this

framework, the measurement noises play the role of

‘small’ non-vanishing exogenous inputs affecting the

estimation error dynamics. The observation strategy in

this setting focuses on the synthesis of a ‘robust’

observer, i.e. such that noise influence on estimation

error is ‘small’. To comply with this requirement, extra

tuning parameters will be introduced in the observer

design, which also aims at input-to-state stability (ISS)

property (see Khalil (1996) for ISS definition): in

practice, the observer estimation error is attracted

inside an open ball whose radius depends on the

measurement noise. To adapt ISS property to the

design we consider in this note, we recall the following

result, that is demonstrated via standard Lyapunov

theory.

Lemma 1: Consider the system

_x ¼ AðtÞxþ BðtÞu

with state x(t)2Rn, arbitrary initial condition x(0)2R

n

and input u(t)2Rm. Assume that:

(i) there is a positive scalar parameter �m such

that vTA(t)v�� �mkvk2 for all t� 0 and for

any v2Rn;

(ii) there is a positive scalar parameter �M such that

kB(t)k2<�M for all t� 0.

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If there are suitable positive constants ku, uM, �u such

that uðtÞ�� ��2� ku expð��utÞ þ u2M i.e. u(t) is exponen-

tially attracted to a ball with radius uM, then there are

suitable positive constants k1, k2, �x such that

xðtÞ�� ��2� k1 expð��xtÞ þ k2u

2M

ð28Þ

i.e. x(t) is also exponentially attracted to a ball whose

radius is proportional by a suitable factor k2 to the bound

uM of the input u(t).

Proof: Consider the Lyapunov function VðtÞ ¼12 xðtÞ�� ��2: By computing its time derivative, we have

_VðtÞ ¼ xTAðtÞxþ xTBðtÞu: ð29Þ

By completing the squares, recalling (i), (ii), we

obtain xTA(t)x�� �mkxk2 and xTBðtÞu � �m

2 kxðtÞk2þ

�M2�mkuðtÞk2, which substituted into (29) and recalling

the definition of V yields

_VðtÞ � ��mVðtÞ þ�M2�m

ku expð��utÞ þ u2M�

:

Set �� ¼ minf�m2 , �ug. Previous inequality implies that

V(t)�W(t), where W(t) complies with the differential

equality

_WðtÞ ¼ ��mWðtÞ þ�M2�m

ku expð���tÞ þ u2M

� ð30Þ

with initial condition Wð0Þ ¼ Vð0Þ ¼ 12 xð0Þ�� ��2. The

solution of (30) is

WðtÞ ¼Wð0Þexpð��mtÞ

þ�M2�m

ku�m� ��

e���t� e��mt

�þu2M�m

1� e��mt ��

:

Neglecting the negative terms in previous equality,

we have

WðtÞ �Wð0Þ expð��mtÞ þ�M2�m

ku�m � ��

e���t þ

u2M�m

� ;

recalling that 12 kxðtÞk

2 ¼ VðtÞ �WðtÞ and Wð0Þ �

expð��mtÞ �12 kxð0Þk

2 exp(���t), we deduce that (28)

is verified by setting �x¼ ��, k1 ¼

�Mku�m �m���ð Þ

þ kxð0Þk2

and k2 ¼�M�2m: œ

The first step in the estimation strategy is the

definition of the filters with states M�2R3�R

3,

�2R3, whose role is the same of filters (12), (14) in

previous section:

_M� ¼ � �þk

4

� �M� þ��ðtÞ , ð31Þ

_� ¼ � �þk

4

� �� þ a� ��ðtÞ þM�a� ð32Þ

with arbitrary initial conditions, where k2Rþ, �2R

þ

are tuning parameters chosen by the designer.

By setting �¼ ��M��1 and �1(t)¼ [�11(t),�12(t),

�13(t)]2R3 �2¼ [�21(t), �22(t), �23(t)]

T2R

3 and

N2ðtÞ ¼

�vðtÞ wðtÞ 0

uðtÞ 0 �wðtÞ

0 �uðtÞ vðtÞ

264

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recalling that �1 ¼4[u, v,w]T is the speed vector

represented in the local reference frame, it is straight-

forward to verify that

_� ¼ � �þk

4

� �� þ ðM� þ IÞ�1 þN2�2: ð33Þ

It is shown in the following that the dynamics of

�(t) in (33) comply with the hypotheses of Lemma 1,

i.e. �(t) is exponentially attracted inside a ball whose

radius depends on the bounds of �1(t) and �2(t).

Consider the filter with matrix state Q�2R3�R

3, with

initial condition such that jQ�(0)j> 0, defined as

_Q� ¼ ��Q� �k

4Q�Q

T�Q� þMT

�M�, ð34Þ

where �2Rþ is a tuning parameter chosen by the

designer. The filter (34) is a generalisation of system

(16); by following arguments similar to the ones of

previous section we show a result that generalises

Claim 3.

Claim 4: Assume that Q�(0)> 0; then the matrix

Q�(t) is bounded and positive definite, i.e. Q�(t)> 0

for all t� 0.

Proof: First, recall that Q� is bounded. In fact, set

x�¼Q�s where s2R3 is any vector such that ksk¼ 1.

The computation of the time derivative of x�¼Q�s, by

virtue of (34) yields

_x� ¼ � �Iþk

4Q�Q

T�

� �x� þMT

�M�s: ð35Þ

From (35) we infer that x� complies with Lemma 1

with��Iþ k

4Q�ðtÞQT�ðtÞ

in place of A(t) and � in place

of �m, with MT�ðtÞM�ðtÞ in place of B(t) and s2R

3 in

place of u(t). Thus x�¼Q�s is exponentially attracted

inside a ball whose radius depends on supt2 0,1½ Þ ���MT�ðtÞM�ðtÞ

��, so that (being s2R3 arbitrary), we

conclude that Q� is bounded; in particular, there exists

cQ2Rþ such that

supt2 0,1½ Þ

Q�ðtÞ�� �� � cQ: ð36Þ

In order to prove that Q� is positive definite,

set ��¼ sTQ�s, where s2R3 is any vector such that

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ksk¼ 1. The computation of the time derivative of ��,by virtue of (34) along with (36) yields

d��=dt � ����� þ M�s

�� ��2, ð37Þ

where �� ¼ �þ k4 c

2Q: From the inequality above we

infer that

��ðtÞ � e���t��ð0Þ þ

Z t

0

e��� t��ð Þ M�ð�Þs

�� ��2d�:Recall that e��

�t��ð0Þ4 0 andR t0 e��� t��ð Þ �

kM�ð�Þsk2d� � 0 for all t> 0, thus ��ðtÞ � e��

�t��ð0Þfor all 0� t�T �, where T � is the positive real

defined in Hypothesis 1. From (16) we have

that ��(T�)�

R T �0 e��

� T ���ð ÞkM�ð�Þsk2d� � e��

�T � �R T �0 kM�ð�Þsk

2d�. Remark that by following the same

arguments in Claim 2 we deduce that M� is PE, in

particular there exists a suitable positive bound K��2such that

R T �0 kM�ð�Þsk

2d� � K��2, which implies

��ðT�Þ � e��

�T �K��2: By repeating iteratively these

arguments we have that ��ðtÞ � e�2��T �K��2 in each

interval iT � � t� (iþ 1) T � with i¼ 1, 2, . . . ,1, so that

��(t)� q�, where q� ¼ minfe�T�

��ð0Þ, e�2T �K��2g and

we conclude that kQ�(t)k� q�. œ

Consider the two filters with states ��2R3 and

��2R3, whose task is analogous to the one of (19),

(20), respectively, in the previous section,

_�� ¼ ���� þ �, ð38Þ

_�� ¼ ���� �k

4Q�Q

T��� þQ�a� � _M�

T��

�k

4Q�Q

T�M

T���, ð39Þ

with arbitrary initial conditions.We introduce an estimate of the linear velocity

�1 ¼

u

v

w

264

375

that satisfy the differential equations:

_�1 ¼ �ð�Q��1 þMT��� þ ��Þ þ a�, ð40Þ

with � 2Rþ tuning parameter to be chosen by the

designer.Under Hypothesis 1, if the variables �2, �2, a� are

available for measurement, the dynamic observer

described by (31), (32), (34), (38), (39) and (40) with

state M�(t)2R3�R

3, �(t)2R3, Q�(t)2R

3�R

3,

��(t)2R3, ��(t)2R

3 and �1ðtÞ 2 R3 is a generalisation

of the observer presented in Section 3.1. In fact, by

setting the estimation error variables ~�1¼4�1 � �1 and

�¼4Q�1 �MT

��� � ��, we obtain:

_ � ¼ �

��þ

k

4Q�Q

T�

� � �MT

�� �Q��1, ð41Þ

_~�1 ¼ �ð�Q� ~�1 þ �Þ � �1: ð42Þ

The iterative application of Lemma 1 to the systems(33), (41) and (42) leads to the demonstration of theconvergence properties of the estimation schemeintroduced above, where the parameter k adds adegree of freedom to the system and the observerguarantees the exponential convergence of the estima-tion error to a region around zero as well as therobustness with respect to the noises �1 and �2.

Proposition 2: Consider the UAV model described by(2)–(4). Assume that the variables �2, �2, a� are availablefor measurement and that Hypothesis 1 holds. Then forall k2R

þ, �2Rþ, �2R

þ, � 2Rþ, the dynamic observer

described by (31), (32), (34), (38), (39), and (40) withstate M�(t)2R

3�R

3, �(t)2R3, Q�(t)2R

3�R

3,��(t)2R

3, ��(t)2R3 and �1ðtÞ 2 R

3 is such that

�1 � �1k k2� g1 expð��2tÞ þ g2�

2M ð43Þ

for suitable positive reals �2, g1, g2, where �M is givenin (27), for any initial conditions M�(0)2R

3�R

3,Q�(0)2R

3�R

3, �(0)2R3, ��(0)2R

3, �1ð0Þ 2 R3

with Q�(0)> 0.

Proof: System (33) complies with the hypotheses ofLemma 1 with � in place of x and [�1(t), �2(t)] inplace of u(t), noting that hypothesis (i) is triviallyverified and point (ii) is a consequence of theboundedness of the matrices M�(t) and N2(t). Thus,by (27), � is exponentially attracted in a ball whoseradius depends on �M. This property implies thatsystem (41) also complies with Lemma 1 with � inplace of x(t) and [�(t), �1(t)] in place of u(t). Infact, (41) complies with hypothesis (i) with�½�þ k

4Q�ðtÞQT�ðtÞ� in place of A(t) and (ii) follows

noting that MT�ðtÞ and Q�(t) are bounded, so that the

vector �(t) is exponentially attracted in a ball withradius depending on �M. Note that from Claim 4 wehave that if Q�(t)> 0, then matrix Q�(t) is boundedand positive definite for all t� 0. Hence system (42)verifies the hypotheses of Lemma 1 with ~�1ðtÞ in placeof x(t) and [ �(t), �1(t)] in place of u(t), leading finallyto inequality (43). œ

4. Simulation results

4.1 Illustration of the adaptive observer performances

In this section, we illustrate the observer designedabove to estimate the linear velocity of an UAV based

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on the measurable angles, angular velocities and linear

accelerations. These simulations also illustrate the

effect of the tuning parameters �, � and � on the

observer performances. In all simulations the observer

estimations have initial conditions set to zero, while the

desired states are time varying (and different from zero

at t¼ 0). In all simulations both observers track the

desired time varying states, with different perfor-

mances and noise rejections. We have used the

following parameters values:

Consider the first version of the observer that doesnot take into account the measurement noise. Figure 2

shows the angular speeds, the angles and theaccelerations.

We choose in our simulation an angular speedvector �2(t) that spans periodically a three-dimensionalsurface, in particular a non-vanishing periodic orbit.By virtue of Claim 1, this choice guarantees that PE

Hypothesis 1 is satisfied. The time history of the threespeeds to be observed is shown in Figure 3(a), andobserver estimates in Figure 3(b), while estimationerrors going exponentially to zero are shown in

Figure 3(c).

We have then considered the first observer in thecase of additive measurement noise. The measuredacceleration is presented in Figure 4 where the noise �is 10% of the measured acceleration.

The simulation is undertaken using the secondversion of the observer, where the disturbance isattenuated following the design parameter k. A correctchoice of parameter � (which guarantees thatc2¼ �min(Q�) is sufficiently big) also improves theconvergence speed of the observer. One may see inFigure 5(a) the velocity estimated by the observer,while Figure 5(b) describes the estimation time historyand Figure 5(c) illustrates the observation error goingexponentially to a residual set given by the noiseamplitude.

m¼ 2.500 kg kT¼ 10�5N s2

lb¼ 23 cm IR¼ 100.10�7 kgm2

Ixx¼ 22,493� 10�7 kgm2 Iyy¼ 222611.10�7 kgm2

Izz¼ 32,5130� 10�7 kgm2 kM¼ 9.10�5N s2muG¼ 0.032m

Figure 2. (a) Angular speed, (b) angles and (c) acceleration.

Figure 3. Undisturbed measurements: (a) Linear speed,(b) speed estimation and (c) estimation error.

Figure 4. Measured acceleration.

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4.2 Comparison with an EKF

The aim now is to compare the observer proposed in

Section 3.2 with an EKF, which is an establishedbenchmark for estimation techniques. The EKF

described here is based on the dynamic model of theUAV given by the continuous-time Equations (2)–(4).

We considered the corresponding discrete-time modelthat can be written as

Xkþ1 ¼ f ðXk,Uk,WkÞ,

Yk ¼ hðXk,ZkÞ:

�ð44Þ

The discrete-time state vector is Xk¼ [�k, �k, k, pk,qk, rk, uk, vk, wk]. Uk is the control input. Wk and Zk,

respectively, represent process noise and measurementnoise at time step k; they are zero mean white Gaussian

sequences of random variables.The behaviour of the EKF designed to estimate the

linear velocity of an UAV based on the measurable

angles, angular velocities and linear accelerations isillustrated by numerical simulation and compared to

the estimate provided by the observer described inSection 3.2. In all simulations, the observer estimationshave initial conditions set to zero, while the desired

states are time varying (and different from zero att¼ 0). Model parameters are the same as those

presented in Section 4.1 and we have consideredadditive measurement noise on the acceleration

(see Figure 5). First, we compare the performance ofour estimation strategy with EKF by setting k¼ 50,

reporting the results in Figure 6.They show that the algorithm in Section 3.2

significantly outperforms the EKF, since itguarantees in steady-state condition a much smaller

estimation error. Notice in this simulation (with

k¼ 50), the estimation error convergence of the

algorithm in this note is slower than EKF; this is

reasonable, since our strategy involves a cascade of

filters, the transient performance of the entire algo-

rithm being the sum of transient performances of the

filters in the cascade. However, the total number of

tuning parameters in our estimator is smaller than the

number of parameters to be set in the EKF. As a

consequence, the tuning procedure of our algorithms,

verified in our simulations, is much more simple than

that of the EKF. This is easily explained first by the

small number of parameters, but as well as their clear

effect on the convergence (through the analysis of the

Lyapunov function). Secondly, several dynamical

variables behave as an automatic gain, this character-

istic coming from adaptive theory make our observers

rather self-tuning compared to the EKF.By decreasing the value of k, in particular by setting

k¼ 10 and performing the simulation for this setting,

we have a faster convergence rate, as displayed in

Figure 7. Finally, Figure 8 shows that by choosing

k¼ 0, we obtain a steady state estimation error

increase.This phenomenon can be explained as follows:

From expression (34) we infer that the bigger is k, the

smaller is the norm of Q�; the latter quantity in turn is

related, by virtue of (42), to the inverse of the

estimation error convergence rate. Thus by decreasing

k we may obtain a faster convergence rate but may

have a greater sensibility to disturbances also; this

property is inherently associated with a high gain

estimation strategy.

Figure 6. Comparison of the performances of the EKF andthe adaptive observer for k¼ 50.

Figure 5. Noisy measurements: (a) Linear speed, (b) speedestimation and (c) estimation error.

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In conclusion, the simulation tests show that thebest performance on the algorithm in Section 3.2 areobtained via a fine tuning of the parameter k in order toguarantee a small steady-state estimation error out-performing EKF and simultaneously a convergencerate typical of a classic EKF approach.

5. Conclusion

In this study, the UAV drone speed estimationproblem has been addressed when no GPS-related

device is available. This is a realistic feature in manypractical applications and has motivated recent workson the field. The results established for the specificprototype in this study hold also for the entire class ofUAV such that rotational and translational dynamicsmay be decoupled, as helicopters and VTOL aircrafts.In this context, the first theoretical issue arises by thefact that the linearised system is non-observable at theequilibrium point. Thus a nonlinear observabilityverification is performed for PE trajectories so thatunder some mild sufficient conditions an estimationstrategy is shown to guarantee exponential estimationof the translational speed, by using the tools ofadaptive observation theory.

A modified scheme is also proposed in order toobtain enhanced robustness property with respect tomeasurement noise. The theoretical results are illu-strated via computer simulations, stressing the noiseattenuation features. They also provide useful insightabout the parameter tuning effect to achieve asatisfactory observer performance. Furthermore, com-puter simulations are used to compare the proposedestimation strategy with the classical EKF. They showthat the adaptive observer guarantees good perfor-mances even in the presence of measurement noise,with greater disturbances attenuation with respect tothe one provided by the EKF.

Future research will be dedicated to the design of asimplified algorithm to possibly trade off noiseimmunity against faster convergence along withcoupling the proposed observer to a control systemdesigned to stabilise the four-rotors helicopter.Additional work will be to implement the consideredscheme in the UAV available at IBISC laboratory andto extend current results to the standard accelerationmeasurements provided by low-cost IMUs.

Note

1. We will not consider position estimation. For thisreason, the state vector does not include it.

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